tag:blogger.com,1999:blog-78505406400450815152024-03-13T11:26:38.883-07:00Mission examguruhttp://www.blogger.com/profile/01591375209227117455noreply@blogger.comBlogger54125tag:blogger.com,1999:blog-7850540640045081515.post-72419421289783191842019-09-14T19:06:00.000-07:002019-09-14T19:06:09.250-07:00G.K<div dir="ltr" style="text-align: left;" trbidi="on">
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guruhttp://www.blogger.com/profile/01591375209227117455noreply@blogger.comtag:blogger.com,1999:blog-7850540640045081515.post-32050321359221604482019-09-09T23:23:00.000-07:002019-09-14T18:55:55.223-07:00Filmfare Awards 2019 list .<div dir="ltr" style="text-align: left;" trbidi="on">
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<img alt="Image result for filmfare awards 2019 pdf download" height="334" src="https://www.filmfare.com/awards/filmfare-awards-east-2019/images/main-image.jpg" width="640" /><br />
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<div style="background-color: #fcfcfc; color: #333333; font-family: Roboto, Arial, sans-serif; font-size: 18.667px; margin: 0px; padding: 0px;">
<table border="1" style="border-collapse: collapse; border-spacing: 0px; border: 1px solid rgb(228, 229, 229); margin: 0px 0px 20px; padding: 0px; width: 669px;"><tbody style="margin: 0px; padding: 0px;">
<tr style="background-color: #f7f7f7; margin: 0px; padding: 0px;"><th bgcolor:="" grey="" style="background: rgb(102, 102, 102); border: 1px solid rgb(0, 0, 0); color: #f2f2f2; margin: 0px; padding: 5px 8px; text-align: center;" width="40%">Category</th><th bgcolor:="" grey="" style="background: rgb(102, 102, 102); border: 1px solid rgb(0, 0, 0); color: #f2f2f2; margin: 0px; padding: 5px 8px; text-align: center;">Winner</th></tr>
<tr style="margin: 0px; padding: 0px;"><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Best Film</td><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Raazi</td></tr>
<tr style="background-color: #f7f7f7; margin: 0px; padding: 0px;"><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Critics Best Film</td><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Andhadhun</td></tr>
<tr style="margin: 0px; padding: 0px;"><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Best Actor in a Leading Role (Male)</td><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Ranbir Kapoor (Sanju)</td></tr>
<tr style="background-color: #f7f7f7; margin: 0px; padding: 0px;"><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Critics Best Actor in Leading Role (Male)</td><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Ayushmann Khurrana (Andhadhun), Ranveer Singh (Padmaavat)<br /><a href="https://www.blogger.com/null" name="more" style="-webkit-font-smoothing: antialiased !important; color: #0000cc; margin: 0px; padding: 0px; text-decoration-line: underline; text-rendering: optimizelegibility !important; transition: color 0.15s linear 0s;"></a></td></tr>
<tr style="margin: 0px; padding: 0px;"><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Best Actor in a Leading Role (Female)</td><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Alia Bhatt (Raazi)</td></tr>
<tr style="background-color: #f7f7f7; margin: 0px; padding: 0px;"><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Critics Best Actor in Leading Role (Female)</td><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Neena Gupta (Badhaai Ho)</td></tr>
<tr style="margin: 0px; padding: 0px;"><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Best Director</td><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Meghna Gulzar (Raazi)</td></tr>
<tr style="background-color: #f7f7f7; margin: 0px; padding: 0px;"><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Best Actor in Supporting Role (Male)</td><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Gajarj Rao (Badhaai Ho), Vicky Kaushal (Sanju)</td></tr>
<tr style="margin: 0px; padding: 0px;"><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Best Actor in Supporting Role (Female)</td><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Surekha Sikhri (Badhaai Ho)</td></tr>
<tr style="background-color: #f7f7f7; margin: 0px; padding: 0px;"><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Best Music Album</td><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Sanjay Leela Bhansali (Patmaavat)</td></tr>
<tr style="margin: 0px; padding: 0px;"><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Best Lyrics</td><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Gulzar (Raazi)</td></tr>
<tr style="background-color: #f7f7f7; margin: 0px; padding: 0px;"><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Best Playback Singer (Male)</td><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Arijit Singh (Raazi)</td></tr>
<tr style="margin: 0px; padding: 0px;"><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Best Playback Singer (Female)</td><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Shreya Ghoshal (Padmaavat)</td></tr>
<tr style="background-color: #f7f7f7; margin: 0px; padding: 0px;"><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Life Time Achievement Award</td><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Sridevi</td></tr>
<tr style="margin: 0px; padding: 0px;"><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Best Debut Female</td><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Sara Ali Khan (Kedar Nath)</td></tr>
<tr style="background-color: #f7f7f7; margin: 0px; padding: 0px;"><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Best Debut Male</td><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Ishaan Khattar (Beyond the Clouds)</td></tr>
<tr style="margin: 0px; padding: 0px;"><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">50 Years of Outstanding Contribution to Cinema</td><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Hema Malini</td></tr>
<tr style="background-color: #f7f7f7; margin: 0px; padding: 0px;"><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">People’s Choice Award for Best Short Film</td><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Plus Minus</td></tr>
<tr style="margin: 0px; padding: 0px;"><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Best Short Film in Fiction</td><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Rogan Josh</td></tr>
<tr style="background-color: #f7f7f7; margin: 0px; padding: 0px;"><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Best Short Film in Non-Fiction</td><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">The Soccer City</td></tr>
<tr style="margin: 0px; padding: 0px;"><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Best Actor Female in Short Film</td><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Kirti Kulhari (Maya)</td></tr>
<tr style="background-color: #f7f7f7; margin: 0px; padding: 0px;"><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Best Actor Male in Short Film</td><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Hussain Dalal (Shameless)</td></tr>
<tr style="margin: 0px; padding: 0px;"><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Best Original Story</td><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Abubhav Sinha (Mulk)</td></tr>
<tr style="background-color: #f7f7f7; margin: 0px; padding: 0px;"><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Beat Screenplay</td><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Sriram Raghavan, Arijit Biswas, Pooja Ladha Surti, Yogesh Chandekar, Hemanth Rao (Andhadhun)</td></tr>
<tr style="margin: 0px; padding: 0px;"><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Best Dialogue</td><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Akshat Ghildial (Badhaai Ho)</td></tr>
<tr style="background-color: #f7f7f7; margin: 0px; padding: 0px;"><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Best Editing</td><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Pooja Ladha Surti (Andhadhun)</td></tr>
<tr style="margin: 0px; padding: 0px;"><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Best Action</td><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Vikram Dahiya, Sunil Rodriguez (Mukkabaaz)</td></tr>
<tr style="background-color: #f7f7f7; margin: 0px; padding: 0px;"><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Best Background Score</td><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Daniel B George (Andhadhun)</td></tr>
<tr style="margin: 0px; padding: 0px;"><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Best Choreography</td><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Kurti Mahesh Midya, Jyothi D Tommar (Patmaavat)</td></tr>
<tr style="background-color: #f7f7f7; margin: 0px; padding: 0px;"><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Best Cinematography</td><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Pankaj Kumar (Tumbbad)</td></tr>
<tr style="margin: 0px; padding: 0px;"><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Best Costume</td><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Sheetal Sharma (Manto)</td></tr>
<tr style="background-color: #f7f7f7; margin: 0px; padding: 0px;"><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Best Production Design</td><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Nitin Zihani Choudhary, Rakesh Yadav (Tumbbad)</td></tr>
<tr style="margin: 0px; padding: 0px;"><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Best Sound Design</td><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Kunal Sharma (Tumbbad)</td></tr>
<tr style="background-color: #f7f7f7; margin: 0px; padding: 0px;"><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Best VFX</td><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Red Chillies VFX (Zero)</td></tr>
</tbody></table>
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<h2 style="background: linear-gradient(rgb(255, 255, 255) 0%, rgb(240, 240, 240) 100%) rgb(252, 252, 252); border-color: rgb(52, 152, 219) rgb(204, 204, 204) rgb(204, 204, 204); border-image: initial; border-style: solid; border-width: 3px 1px 1px; box-shadow: rgb(255, 255, 255) 0px -1px 0px inset; box-sizing: border-box; color: #333333; font-family: Roboto, Arial, sans-serif; font-size: 24px; line-height: 1.25; margin: 1.5em 0px 10px; padding: 7px 7px 7px 2em; position: relative;">
<img alt="Image result for filmfare awards 2019" height="360" src="https://i.ytimg.com/vi/E0Drd4gVSDc/maxresdefault.jpg" width="640" /></h2>
<h2 style="background: linear-gradient(rgb(255, 255, 255) 0%, rgb(240, 240, 240) 100%) rgb(252, 252, 252); border-color: rgb(52, 152, 219) rgb(204, 204, 204) rgb(204, 204, 204); border-image: initial; border-style: solid; border-width: 3px 1px 1px; box-shadow: rgb(255, 255, 255) 0px -1px 0px inset; box-sizing: border-box; color: #333333; font-family: Roboto, Arial, sans-serif; font-size: 24px; line-height: 1.25; margin: 1.5em 0px 10px; padding: 7px 7px 7px 2em; position: relative;">
63rd Filmfare Awards 2018</h2>
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<table border="1" style="border-collapse: collapse; border-spacing: 0px; border: 1px solid rgb(228, 229, 229); margin: 0px 0px 20px; padding: 0px; width: 669px;"><tbody style="margin: 0px; padding: 0px;">
<tr style="background-color: #f7f7f7; margin: 0px; padding: 0px;"><th bgcolor:="" grey="" style="background: rgb(102, 102, 102); border: 1px solid rgb(0, 0, 0); color: #f2f2f2; margin: 0px; padding: 5px 8px; text-align: center;" width="40%">Category</th><th bgcolor:="" grey="" style="background: rgb(102, 102, 102); border: 1px solid rgb(0, 0, 0); color: #f2f2f2; margin: 0px; padding: 5px 8px; text-align: center;">Winner</th></tr>
<tr style="margin: 0px; padding: 0px;"><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Best Film</td><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">'Hindi Medium'</td></tr>
<tr style="background-color: #f7f7f7; margin: 0px; padding: 0px;"><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Critics' Award for Best Film</td><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">'Newton'</td></tr>
<tr style="margin: 0px; padding: 0px;"><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Best Actor In A Leading Role (Female)</td><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Vidya Balan for 'Tumhari Sulu'</td></tr>
<tr style="background-color: #f7f7f7; margin: 0px; padding: 0px;"><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Best Actor In A Leading Role (Male)</td><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Irrfan Khan for 'Hindi Medium'</td></tr>
<tr style="margin: 0px; padding: 0px;"><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Critics' Award for Best Actor (Male)</td><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Rajkummar Rao for 'Trapped'</td></tr>
<tr style="background-color: #f7f7f7; margin: 0px; padding: 0px;"><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Critics' Award for Best Actor (Female)</td><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Zaira Wasim for 'Secret Superstar'</td></tr>
<tr style="margin: 0px; padding: 0px;"><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Best Director</td><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Ashwiny Iyer Tiwari for 'Bareilly Ki Barfi'</td></tr>
<tr style="background-color: #f7f7f7; margin: 0px; padding: 0px;"><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Best Debut Director</td><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Konkona Sensharma for 'A Death in the Gunj'</td></tr>
<tr style="margin: 0px; padding: 0px;"><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Best Actor In A Supporting Role (Male)</td><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Rajkummar Rao for 'Bareilly Ki Barfi'</td></tr>
<tr style="background-color: #f7f7f7; margin: 0px; padding: 0px;"><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Best Actor in a Supporting Role (Female)</td><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Meher Vij for 'Secret Superstar'</td></tr>
<tr style="margin: 0px; padding: 0px;"><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Best Dialogue</td><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Hitesh Kewalya for 'Shubh Mangal Saavdhan'</td></tr>
<tr style="background-color: #f7f7f7; margin: 0px; padding: 0px;"><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Best Screenplay</td><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Shubhashish Bhutiani for 'Mukti Bhavan'</td></tr>
<tr style="margin: 0px; padding: 0px;"><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Best Original Story</td><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Amit Masurkar for 'Newton'</td></tr>
<tr style="background-color: #f7f7f7; margin: 0px; padding: 0px;"><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Best Actor (Male) in a Short Film</td><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Jackie Shroff for 'Khujli'</td></tr>
<tr style="margin: 0px; padding: 0px;"><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Best Actor (Female) in a Short Film</td><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Shefali Shah for 'Juice'</td></tr>
<tr style="background-color: #f7f7f7; margin: 0px; padding: 0px;"><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">People's Choice Award for Best Short Film</td><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">'Anahut'</td></tr>
<tr style="margin: 0px; padding: 0px;"><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Best Short Film (Fiction)</td><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">'Juice'</td></tr>
<tr style="background-color: #f7f7f7; margin: 0px; padding: 0px;"><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Best Short Film (Non Fiction)</td><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">'Invisible Wings'</td></tr>
<tr style="margin: 0px; padding: 0px;"><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Best Music Album</td><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Pritam for 'Jagga Jasoos'</td></tr>
<tr style="background-color: #f7f7f7; margin: 0px; padding: 0px;"><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Best Playback Singer (Male)</td><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Arijit Singh for 'Roke na ruke naina' - 'Badrinath Ki Dulhania'</td></tr>
<tr style="margin: 0px; padding: 0px;"><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Best Playback Singer (Female)</td><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Meghna Mishra for Nachdi phira - Secret Superstar</td></tr>
<tr style="background-color: #f7f7f7; margin: 0px; padding: 0px;"><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Best Lyrics</td><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Amitabh Bhattacharya for Ullu ka pattha - Jagga Jasoos</td></tr>
<tr style="margin: 0px; padding: 0px;"><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Lifetime Achievement Award</td><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Mala Sinha and Bappi Lahiri</td></tr>
<tr style="background-color: #f7f7f7; margin: 0px; padding: 0px;"><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Best Choreography</td><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Vijay Ganguly and Ruel Dausan Varindani for Galti se mistake - Jagga Jasoos</td></tr>
<tr style="margin: 0px; padding: 0px;"><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Best Background Score</td><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Pritam for Jagga Jasoos</td></tr>
<tr style="background-color: #f7f7f7; margin: 0px; padding: 0px;"><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Best Action</td><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Tom Struthers for Tiger Zinda Hai</td></tr>
<tr style="margin: 0px; padding: 0px;"><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Best Cinematography</td><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Sirsha Ray for A Death In The Gunj</td></tr>
<tr style="background-color: #f7f7f7; margin: 0px; padding: 0px;"><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Best Editing</td><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Nitin Baid for Trapped</td></tr>
<tr style="margin: 0px; padding: 0px;"><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Best Production Design</td><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Parul Sondh for Daddy</td></tr>
<tr style="background-color: #f7f7f7; margin: 0px; padding: 0px;"><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Best Sound Design</td><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Anish John for Trapped</td></tr>
<tr style="margin: 0px; padding: 0px;"><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Best Costume</td><td style="border: 1px solid rgb(172, 172, 172); margin: 0px; padding: 5px 8px;">Rohit Chaturvedi for A Death In The Gunj</td></tr>
</tbody></table>
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guruhttp://www.blogger.com/profile/01591375209227117455noreply@blogger.comtag:blogger.com,1999:blog-7850540640045081515.post-59900472914479100622016-01-09T19:52:00.000-08:002019-09-14T18:55:55.332-07:00 PUNCHLINES<div dir="ltr" style="text-align: left;" trbidi="on">
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<o:p></o:p></div>
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<b><span style="color: red; font-size: 16.0pt; line-height: 115%;"> AIRLINE PUNCHLINE<o:p></o:p></span></b></div>
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<span style="font-size: 12.0pt;">Austrian Airlines<o:p></o:p></span></div>
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<span style="font-size: 12.0pt;">The most friendly airlines<o:p></o:p></span></div>
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<span style="font-size: 12.0pt;">Cathay Pacific<o:p></o:p></span></div>
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<span style="font-size: 12.0pt;">The heart of Asia<o:p></o:p></span></div>
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<span style="font-size: 12.0pt;">Indian Airlines<o:p></o:p></span></div>
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<span style="font-size: 12.0pt;">Your home in the sky<o:p></o:p></span></div>
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<span style="font-size: 12.0pt;">Singapore Airlines<o:p></o:p></span></div>
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<span style="font-size: 12.0pt;">A great way to fly<o:p></o:p></span></div>
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<span style="font-size: 12.0pt;">Lufthansa<o:p></o:p></span></div>
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<span style="font-size: 12.0pt;">There is no better way to fly<o:p></o:p></span></div>
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<span style="font-size: 12.0pt;">Jet Airways<o:p></o:p></span></div>
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<span style="font-size: 12.0pt;">The joy of flying<o:p></o:p></span></div>
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<span style="font-size: 12.0pt;">Air Sahara<o:p></o:p></span></div>
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<span style="font-size: 12.0pt;">Emotionally yours<o:p></o:p></span></div>
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<span style="font-size: 12.0pt;">Thai Airways<o:p></o:p></span></div>
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<span style="font-size: 12.0pt;">Smooth as silk<o:p></o:p></span></div>
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<span style="font-size: 12.0pt;">Sri Lankan Airlines<o:p></o:p></span></div>
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<span style="font-size: 12.0pt;">You are our world<o:p></o:p></span></div>
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<span style="font-size: 12.0pt;">British Airlines<o:p></o:p></span></div>
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<span style="font-size: 12.0pt;">It’s time to change the way you look at things</span></div>
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<b><span style="color: white; font-size: 12.0pt; mso-themecolor: background1;">FINANCIAL INSTITUTIONS PUNCHLINE<o:p></o:p></span></b></div>
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<b><span style="font-size: 12.0pt;">State Bank of India<o:p></o:p></span></b></div>
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<span style="font-size: 12.0pt;">With You – All the way<b><o:p></o:p></b></span></div>
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<b><span style="font-size: 12.0pt;">Punjab National Bank<o:p></o:p></span></b></div>
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<span style="font-size: 12.0pt;">The name you can bank upon<b><o:p></o:p></b></span></div>
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<b><span style="font-size: 12.0pt;">Citibank<o:p></o:p></span></b></div>
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<span style="font-size: 12.0pt;">Where money lives<b><o:p></o:p></b></span></div>
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<b><span style="font-size: 12.0pt;">ICICI Bank<o:p></o:p></span></b></div>
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<span style="font-size: 12.0pt;">We make banking a pleasure<b><o:p></o:p></b></span></div>
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<b><span style="font-size: 12.0pt;">ICICI Prudential Life Insurance<o:p></o:p></span></b></div>
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<span style="font-size: 12.0pt;">We cover you. At every step of life<b><o:p></o:p></b></span></div>
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<b><span style="font-size: 12.0pt;">LIC We know India better<o:p></o:p></span></b></div>
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<span style="font-size: 12.0pt; mso-bidi-font-weight: bold;">We know
India better<o:p></o:p></span></div>
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<b><span style="font-size: 12.0pt;">Royal Sundaram<o:p></o:p></span></b></div>
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<span style="font-size: 12.0pt;">Beyond Expectations<b><o:p></o:p></b></span></div>
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<b><span style="font-size: 12.0pt;">New Indian Assurance<o:p></o:p></span></b></div>
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<span style="font-size: 12.0pt;">India’s largest non-life insurance Company<b><o:p></o:p></b></span></div>
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<b><span style="font-size: 12.0pt;">Global Trust Bank<o:p></o:p></span></b></div>
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<span style="font-size: 12.0pt;">Experience a new way<b><o:p></o:p></b></span></div>
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<b><span style="font-size: 12.0pt;">IDBI<o:p></o:p></span></b></div>
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<span style="font-size: 12.0pt;">What can I do for you<b><o:p></o:p></b></span></div>
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<td style="border-right: solid windowtext 1.0pt; border: solid #4F81BD 1.0pt; height: 15.95pt; mso-border-alt: solid #4F81BD 1.0pt; mso-border-right-alt: solid windowtext .5pt; mso-border-themecolor: accent1; mso-border-themecolor: accent1; padding: 0in 5.4pt 0in 5.4pt; width: 169.25pt;" valign="top" width="226"><div class="MsoNormal" style="margin-bottom: 0.0001pt; text-indent: 4.5pt;">
<b><span style="font-size: 12.0pt;">Union Bank of India<o:p></o:p></span></b></div>
</td>
<td style="border-left: none; border: solid #4F81BD 1.0pt; height: 15.95pt; mso-border-left-alt: solid windowtext .5pt; mso-border-themecolor: accent1; padding: 0in 5.4pt 0in 5.4pt; width: 249.6pt;" valign="top" width="333"><div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span style="font-size: 12.0pt;">Good people to bank with<b><o:p></o:p></b></span></div>
</td>
</tr>
<tr style="height: 15.35pt; mso-yfti-irow: 11;">
<td style="border-bottom: none; border-left: solid #4F81BD 1.0pt; border-right: solid windowtext 1.0pt; border-top: none; height: 15.35pt; mso-border-left-alt: solid #4F81BD 1.0pt; mso-border-left-themecolor: accent1; mso-border-left-themecolor: accent1; mso-border-right-alt: solid windowtext .5pt; padding: 0in 5.4pt 0in 5.4pt; width: 169.25pt;" valign="top" width="226"><div class="MsoNormal" style="margin-bottom: 0.0001pt; text-indent: 4.5pt;">
<b><span style="font-size: 12.0pt;">HDFC Standard Life Insurance<o:p></o:p></span></b></div>
</td>
<td style="border-right: solid #4F81BD 1.0pt; border: none; height: 15.35pt; mso-border-left-alt: solid windowtext .5pt; mso-border-right-themecolor: accent1; padding: 0in 5.4pt 0in 5.4pt; width: 249.6pt;" valign="top" width="333"><div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span style="font-size: 12.0pt;">Service that makes you smile<b><o:p></o:p></b></span></div>
</td>
</tr>
<tr style="height: 15.35pt; mso-yfti-irow: 12;">
<td style="border-right: solid windowtext 1.0pt; border: solid #4F81BD 1.0pt; height: 15.35pt; mso-border-alt: solid #4F81BD 1.0pt; mso-border-right-alt: solid windowtext .5pt; mso-border-themecolor: accent1; mso-border-themecolor: accent1; padding: 0in 5.4pt 0in 5.4pt; width: 169.25pt;" valign="top" width="226"><div class="MsoNormal" style="margin-bottom: 0.0001pt; text-indent: 4.5pt;">
<b><span style="font-size: 12.0pt;">SBI Life Insurance Co.<o:p></o:p></span></b></div>
</td>
<td style="border-left: none; border: solid #4F81BD 1.0pt; height: 15.35pt; mso-border-left-alt: solid windowtext .5pt; mso-border-themecolor: accent1; padding: 0in 5.4pt 0in 5.4pt; width: 249.6pt;" valign="top" width="333"><div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span style="font-size: 12.0pt;">With us you are sure<b><o:p></o:p></b></span></div>
</td>
</tr>
<tr style="height: 15.35pt; mso-yfti-irow: 13;">
<td style="border-bottom: none; border-left: solid #4F81BD 1.0pt; border-right: solid windowtext 1.0pt; border-top: none; height: 15.35pt; mso-border-left-alt: solid #4F81BD 1.0pt; mso-border-left-themecolor: accent1; mso-border-left-themecolor: accent1; mso-border-right-alt: solid windowtext .5pt; padding: 0in 5.4pt 0in 5.4pt; width: 169.25pt;" valign="top" width="226"><div class="MsoNormal" style="margin-bottom: 0.0001pt; text-indent: 4.5pt;">
<b><span style="font-size: 12.0pt;">HSBC<o:p></o:p></span></b></div>
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<span style="font-size: 12.0pt;">The world’s local bank<b><o:p></o:p></b></span></div>
</td>
</tr>
<tr style="height: 15.35pt; mso-yfti-irow: 14;">
<td style="border-right: solid windowtext 1.0pt; border: solid #4F81BD 1.0pt; height: 15.35pt; mso-border-alt: solid #4F81BD 1.0pt; mso-border-right-alt: solid windowtext .5pt; mso-border-themecolor: accent1; mso-border-themecolor: accent1; padding: 0in 5.4pt 0in 5.4pt; width: 169.25pt;" valign="top" width="226"><div class="MsoNormal" style="margin-bottom: 0.0001pt; text-indent: 4.5pt;">
<b><span style="font-size: 12.0pt;">Vysya Bank<o:p></o:p></span></b></div>
</td>
<td style="border-left: none; border: solid #4F81BD 1.0pt; height: 15.35pt; mso-border-left-alt: solid windowtext .5pt; mso-border-themecolor: accent1; padding: 0in 5.4pt 0in 5.4pt; width: 249.6pt;" valign="top" width="333"><div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span style="font-size: 12.0pt;">Experience relationship banking<b><o:p></o:p></b></span></div>
</td>
</tr>
<tr style="height: 15.35pt; mso-yfti-irow: 15;">
<td style="border-bottom: none; border-left: solid #4F81BD 1.0pt; border-right: solid windowtext 1.0pt; border-top: none; height: 15.35pt; mso-border-left-alt: solid #4F81BD 1.0pt; mso-border-left-themecolor: accent1; mso-border-left-themecolor: accent1; mso-border-right-alt: solid windowtext .5pt; padding: 0in 5.4pt 0in 5.4pt; width: 169.25pt;" valign="top" width="226"><div class="MsoNormal" style="margin-bottom: 0.0001pt; text-indent: 4.5pt;">
<b><span style="font-size: 12.0pt;">Canara Bank<o:p></o:p></span></b></div>
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<td style="border-right: solid #4F81BD 1.0pt; border: none; height: 15.35pt; mso-border-left-alt: solid windowtext .5pt; mso-border-right-themecolor: accent1; padding: 0in 5.4pt 0in 5.4pt; width: 249.6pt;" valign="top" width="333"><div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span style="font-size: 12.0pt;">Serving to grow. Growing to serve<b><o:p></o:p></b></span></div>
</td>
</tr>
<tr style="height: 15.95pt; mso-yfti-irow: 16;">
<td style="border-right: solid windowtext 1.0pt; border: solid #4F81BD 1.0pt; height: 15.95pt; mso-border-alt: solid #4F81BD 1.0pt; mso-border-right-alt: solid windowtext .5pt; mso-border-themecolor: accent1; mso-border-themecolor: accent1; padding: 0in 5.4pt 0in 5.4pt; width: 169.25pt;" valign="top" width="226"><div class="MsoNormal" style="margin-bottom: 0.0001pt; text-indent: 4.5pt;">
<b><span style="font-size: 12.0pt;">Aviva (Life Insurance Co.)<o:p></o:p></span></b></div>
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<td style="border-left: none; border: solid #4F81BD 1.0pt; height: 15.95pt; mso-border-left-alt: solid windowtext .5pt; mso-border-themecolor: accent1; padding: 0in 5.4pt 0in 5.4pt; width: 249.6pt;" valign="top" width="333"><div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span style="font-size: 12.0pt;">Kal par control<b><o:p></o:p></b></span></div>
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</tr>
<tr style="height: 15.35pt; mso-yfti-irow: 17;">
<td style="border-bottom: none; border-left: solid #4F81BD 1.0pt; border-right: solid windowtext 1.0pt; border-top: none; height: 15.35pt; mso-border-left-alt: solid #4F81BD 1.0pt; mso-border-left-themecolor: accent1; mso-border-left-themecolor: accent1; mso-border-right-alt: solid windowtext .5pt; padding: 0in 5.4pt 0in 5.4pt; width: 169.25pt;" valign="top" width="226"><div class="MsoNormal" style="margin-bottom: 0.0001pt; text-indent: 4.5pt;">
<b><span style="font-size: 12.0pt;">Iffco-Tokio<o:p></o:p></span></b></div>
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<td style="border-right: solid #4F81BD 1.0pt; border: none; height: 15.35pt; mso-border-left-alt: solid windowtext .5pt; mso-border-right-themecolor: accent1; padding: 0in 5.4pt 0in 5.4pt; width: 249.6pt;" valign="top" width="333"><div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span style="font-size: 12.0pt;">The life you deserve<b><o:p></o:p></b></span></div>
</td>
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<tr style="height: 15.35pt; mso-yfti-irow: 18;">
<td style="border-right: solid windowtext 1.0pt; border: solid #4F81BD 1.0pt; height: 15.35pt; mso-border-alt: solid #4F81BD 1.0pt; mso-border-right-alt: solid windowtext .5pt; mso-border-themecolor: accent1; mso-border-themecolor: accent1; padding: 0in 5.4pt 0in 5.4pt; width: 169.25pt;" valign="top" width="226"><div class="MsoNormal" style="margin-bottom: 0.0001pt; text-indent: 4.5pt;">
<b><span style="font-size: 12.0pt;">Oriental Bank of Commerce<o:p></o:p></span></b></div>
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<td style="border-left: none; border: solid #4F81BD 1.0pt; height: 15.35pt; mso-border-left-alt: solid windowtext .5pt; mso-border-themecolor: accent1; padding: 0in 5.4pt 0in 5.4pt; width: 249.6pt;" valign="top" width="333"><div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span style="font-size: 12.0pt;">Where every individual is committed<b><o:p></o:p></b></span></div>
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</tr>
<tr style="height: 15.35pt; mso-yfti-irow: 19;">
<td style="border-bottom: none; border-left: solid #4F81BD 1.0pt; border-right: solid windowtext 1.0pt; border-top: none; height: 15.35pt; mso-border-left-alt: solid #4F81BD 1.0pt; mso-border-left-themecolor: accent1; mso-border-left-themecolor: accent1; mso-border-right-alt: solid windowtext .5pt; padding: 0in 5.4pt 0in 5.4pt; width: 169.25pt;" valign="top" width="226"><div class="MsoNormal" style="margin-bottom: 0.0001pt; text-indent: 4.5pt;">
<b><span style="font-size: 12.0pt;">IL&FS (mutual fund)<o:p></o:p></span></b></div>
</td>
<td style="border-right: solid #4F81BD 1.0pt; border: none; height: 15.35pt; mso-border-left-alt: solid windowtext .5pt; mso-border-right-themecolor: accent1; padding: 0in 5.4pt 0in 5.4pt; width: 249.6pt;" valign="top" width="333"><div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span style="font-size: 12.0pt;">More than a fund. A friend<b><o:p></o:p></b></span></div>
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</tr>
<tr style="height: 15.35pt; mso-yfti-irow: 20;">
<td style="border-right: solid windowtext 1.0pt; border: solid #4F81BD 1.0pt; height: 15.35pt; mso-border-alt: solid #4F81BD 1.0pt; mso-border-right-alt: solid windowtext .5pt; mso-border-themecolor: accent1; mso-border-themecolor: accent1; padding: 0in 5.4pt 0in 5.4pt; width: 169.25pt;" valign="top" width="226"><div class="MsoNormal" style="margin-bottom: 0.0001pt; text-indent: 4.5pt;">
<b><span style="font-size: 12.0pt;">Ernst & Young<o:p></o:p></span></b></div>
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<td style="border-left: none; border: solid #4F81BD 1.0pt; height: 15.35pt; mso-border-left-alt: solid windowtext .5pt; mso-border-themecolor: accent1; padding: 0in 5.4pt 0in 5.4pt; width: 249.6pt;" valign="top" width="333"><div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span style="font-size: 12.0pt;">Quality in everything we do<b><o:p></o:p></b></span></div>
</td>
</tr>
<tr style="height: 15.35pt; mso-yfti-irow: 21;">
<td style="border-bottom: none; border-left: solid #4F81BD 1.0pt; border-right: solid windowtext 1.0pt; border-top: none; height: 15.35pt; mso-border-left-alt: solid #4F81BD 1.0pt; mso-border-left-themecolor: accent1; mso-border-left-themecolor: accent1; mso-border-right-alt: solid windowtext .5pt; padding: 0in 5.4pt 0in 5.4pt; width: 169.25pt;" valign="top" width="226"><div class="MsoNormal" style="margin-bottom: 0.0001pt; text-indent: 4.5pt;">
<b><span style="font-size: 12.0pt;">DSP Merrill Synch<o:p></o:p></span></b></div>
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<td style="border-right: solid #4F81BD 1.0pt; border: none; height: 15.35pt; mso-border-left-alt: solid windowtext .5pt; mso-border-right-themecolor: accent1; padding: 0in 5.4pt 0in 5.4pt; width: 249.6pt;" valign="top" width="333"><div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span style="font-size: 12.0pt;">The 100% money manager<b><o:p></o:p></b></span></div>
</td>
</tr>
<tr style="height: 15.95pt; mso-yfti-irow: 22; mso-yfti-lastrow: yes;">
<td style="border-right: solid windowtext 1.0pt; border: solid #4F81BD 1.0pt; height: 15.95pt; mso-border-alt: solid #4F81BD 1.0pt; mso-border-right-alt: solid windowtext .5pt; mso-border-themecolor: accent1; mso-border-themecolor: accent1; padding: 0in 5.4pt 0in 5.4pt; width: 169.25pt;" valign="top" width="226"><div class="MsoNormal" style="margin-bottom: 0.0001pt; text-indent: 4.5pt;">
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guruhttp://www.blogger.com/profile/01591375209227117455noreply@blogger.comtag:blogger.com,1999:blog-7850540640045081515.post-34038575546786184492016-01-04T18:05:00.001-08:002019-09-14T18:55:55.260-07:00MINERAL WEALTH OF INDIA <div dir="ltr" style="text-align: left;" trbidi="on">
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<a href="https://www.blogger.com/null" name="1"></a><span style="text-indent: 112pt;"><span style="line-height: 16px;"> </span><span style="font-family: 'Times New Roman', serif; font-size: 14pt; font-weight: bold; letter-spacing: -0.05pt; line-height: 16pt; text-decoration: underline;">MINERAL WEALTH OF INDIA</span></span></div>
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<u><span style="font-family: 'Arial Black', sans-serif; letter-spacing: -0.1pt;">METALLIC MINERALS</span></u><o:p></o:p></div>
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<div class="MsoNormal" style="line-height: 15.2pt; margin-bottom: .0001pt; margin-bottom: 0in; margin-left: 1.0in; margin-right: 0in; margin-top: 0in; mso-line-height-rule: exactly; tab-stops: 2.5in 289.0pt;">
<span style="font-family: 'Book Antiqua', serif; font-size: 9pt; letter-spacing: -0.05pt;">Antimony</span> <span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">(1)</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt; letter-spacing: -0.05pt;">Punjab</span> <span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">(2)</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Karnataka</span><o:p></o:p></div>
<div class="MsoNormal" style="line-height: 17.75pt; margin-bottom: .0001pt; margin-bottom: 0in; margin-left: 1.0in; margin-right: 0in; margin-top: 0in; mso-line-height-rule: exactly; tab-stops: 2.5in 289.0pt 397.05pt;">
<span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Bauxite</span> <span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">(1)</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Jharkhand</span> <span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">(2)</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Madhya</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt; letter-spacing: -0.05pt;">Pradesh</span> <span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">(3)</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Gujarat</span><o:p></o:p></div>
<div class="MsoNormal" style="line-height: 17.5pt; margin-bottom: .0001pt; margin-bottom: 0in; margin-left: 1.0in; margin-right: 0in; margin-top: 0in; mso-line-height-rule: exactly; tab-stops: 2.5in 289.0pt;">
<span style="font-family: 'Book Antiqua', serif; font-size: 9pt; letter-spacing: -0.05pt;">Chromite</span> <span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">(1)</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Orissa</span> <span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">(2)</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Maharashtra</span><o:p></o:p></div>
<div class="MsoNormal" style="line-height: 17.5pt; margin-bottom: .0001pt; margin-bottom: 0in; margin-left: 1.0in; margin-right: 0in; margin-top: 0in; mso-line-height-rule: exactly; tab-stops: 2.5in 289.0pt;">
<span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Coal</span> <span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">(1)</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Jharkhand</span> <span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">(2)</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt; letter-spacing: -0.05pt;">West</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Bengal</span><o:p></o:p></div>
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<span style="font-family: 'Book Antiqua', serif; font-size: 9pt; letter-spacing: -0.05pt;">Copper</span> <span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">(1)</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Jharkhand</span> <span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">(2)</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Rajasthan</span> <span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">(3)</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Madhya</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Pradesh</span><o:p></o:p></div>
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<span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Diaspore</span> <span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">(1)</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Uttar</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt; letter-spacing: -0.05pt;">Pradesh</span> <span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">(2)</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Madhya</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt; letter-spacing: -0.05pt;">Pradesh</span><o:p></o:p></div>
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<span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Gold</span> <span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">(1)</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Karnataka</span> <span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">(2)</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Andhra</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt; letter-spacing: -0.05pt;">Pradesh</span><o:p></o:p></div>
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<span style="font-family: 'Book Antiqua', serif; font-size: 9pt; letter-spacing: -0.05pt;">Iron</span> <span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">(1)</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Goa</span> <span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">(2)</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Madhya</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt; letter-spacing: -0.05pt;">Pradesh</span> <span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">(3)</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Jharkhand</span><o:p></o:p></div>
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<span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">(4)</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Orissa</span> <span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">(5)</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Andhra</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt; letter-spacing: -0.05pt;">Pradesh</span> <span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">(6)</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Tamil</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt; letter-spacing: -0.05pt;">Nadu</span><o:p></o:p></div>
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<span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Lead</span> <span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">(1)</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Rajasthan</span> <span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">(2)</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Andhra</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt; letter-spacing: -0.05pt;">Pradesh</span><o:p></o:p></div>
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<span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Lignite</span> <span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">(1)</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Tamil</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt; letter-spacing: -0.05pt;">Nadu</span> <span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">(2)</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Gujarat</span><o:p></o:p></div>
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<span style="font-family: 'Book Antiqua', serif; font-size: 9pt; letter-spacing: -0.05pt;">Manganese</span> <span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">(1)</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Orissa</span> <span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">(2)</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Madhya</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt; letter-spacing: -0.05pt;">Pradesh</span> <span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">(3)</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Karnataka</span><o:p></o:p></div>
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<span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">(4)</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Maharashtra</span><o:p></o:p></div>
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<span style="font-family: 'Book Antiqua', serif; font-size: 9pt; letter-spacing: -0.05pt;">Nickel</span> <span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">(1)</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Orissa</span><o:p></o:p></div>
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<span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Natural</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Gas</span> <span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">(1)</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Assam</span> <span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">(2)</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Gujarat</span> <span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">(3)</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Maharashtra</span><o:p></o:p></div>
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<span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">(4)</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Orissa</span> <span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">(5)</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Tamil</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt; letter-spacing: -0.05pt;">Nadu</span><o:p></o:p></div>
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<span style="font-family: 'Book Antiqua', serif; font-size: 9pt; letter-spacing: -0.05pt;">Petroleum</span> <span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">(1)</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Assam</span> <span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">(2)</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Gujarat</span><o:p></o:p></div>
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<br /></div>
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<span style="font-family: 'Book Antiqua', serif; font-size: 9pt; letter-spacing: -0.05pt;">Tungsten</span> <span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">(1)</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Rajasthan</span> <span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">(2)</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt; letter-spacing: -0.05pt;">West</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Bengal</span><o:p></o:p></div>
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<span style="font-family: 'Book Antiqua', serif; font-size: 9pt; letter-spacing: -0.05pt;">Uranium</span> <span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">(1)</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Kerala</span> <span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">(2)</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Jharkhand</span> <span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">(3)</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Rajasthan</span><o:p></o:p></div>
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<span style="font-family: 'Book Antiqua', serif; font-size: 9pt; letter-spacing: -0.05pt;">Zinc</span> <span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">(1)</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Rajasthan</span><o:p></o:p></div>
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<br /></div>
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<br /></div>
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<u><span style="font-family: 'Arial Black', sans-serif; letter-spacing: -0.1pt;">NON-METALLIC MINERALS</span></u><o:p></o:p></div>
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<span style="font-family: 'Book Antiqua', serif; font-size: 9pt; letter-spacing: -0.05pt;">Asbestos</span> <span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">(1)</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Andhra</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt; letter-spacing: -0.05pt;">Pradesh</span><o:p></o:p></div>
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<span style="font-family: 'Book Antiqua', serif; font-size: 9pt; letter-spacing: -0.05pt;">Ball</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">clay</span> <span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">(1)</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Andhra</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt; letter-spacing: -0.05pt;">Pradesh</span> <span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">(2)</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Rajasthan</span><o:p></o:p></div>
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<span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Barytes</span> <span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">(1)</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Andhra</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt; letter-spacing: -0.05pt;">Pradesh</span> <span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">(2)</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Maharashtra</span><o:p></o:p></div>
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<span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Calcite</span> <span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">(1)</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Rajasthan</span> <span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">(2)</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Gujarat</span><o:p></o:p></div>
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<span style="font-family: 'Book Antiqua', serif; font-size: 9pt; letter-spacing: -0.05pt;">China</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">clay</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">(Kaolin)</span> <span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">(1)</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Rajasthan</span> <span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">(2)</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt; letter-spacing: -0.05pt;">West</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Bengal</span><o:p></o:p></div>
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<span style="font-family: 'Book Antiqua', serif; font-size: 9pt; letter-spacing: -0.05pt;">Corundum</span> <span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">(1)</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Karnataka</span> <span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">(2)</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Maharashtra</span><o:p></o:p></div>
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<span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Diamond</span> <span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">(1)</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Madhya</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt; letter-spacing: -0.05pt;">Pradesh</span> <span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">(2)</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Andhra</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Pradesh</span><o:p></o:p></div>
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<span style="font-family: 'Book Antiqua', serif; font-size: 9pt; letter-spacing: -0.05pt;">Dolomite</span> <span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">(1)</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Madhya</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt; letter-spacing: -0.05pt;">Pradesh</span> <span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">(2)</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Orissa</span><o:p></o:p></div>
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<span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Feldspar</span> <span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">(1)</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Rajasthan</span> <span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">(2)</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Tamil</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt; letter-spacing: -0.05pt;">Nadu</span><o:p></o:p></div>
<div class="MsoNormal" style="line-height: 17.5pt; margin-bottom: .0001pt; margin-bottom: 0in; margin-left: 1.0in; margin-right: 0in; margin-top: 0in; mso-line-height-rule: exactly; tab-stops: 2.5in 289.0pt;">
<span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Fire</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">clay</span> <span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">(1)</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Bihar</span> <span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">(2)</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Gujarat</span><o:p></o:p></div>
<div class="MsoNormal" style="line-height: 17.75pt; margin-bottom: .0001pt; margin-bottom: 0in; margin-left: 1.0in; margin-right: 0in; margin-top: 0in; mso-line-height-rule: exactly; tab-stops: 2.5in 289.0pt;">
<span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Fluorite</span> <span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">(1)</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Gujarat</span> <span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">(2)</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Rajasthan</span><o:p></o:p></div>
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<span style="font-family: 'Book Antiqua', serif; font-size: 9pt; letter-spacing: -0.05pt;">Graphite</span> <span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">(1)</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Orissa</span> <span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">(2)</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Rajasthan</span><o:p></o:p></div>
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<span style="font-family: 'Book Antiqua', serif; font-size: 9pt; letter-spacing: -0.05pt; line-height: 17.35pt;"> Gypsum</span><span style="line-height: 17.35pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt; line-height: 17.35pt;">(1)</span><span style="font-size: 9pt; line-height: 17.35pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt; line-height: 17.35pt;">Rajasthan</span><span style="line-height: 17.35pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt; line-height: 17.35pt;">(2)</span><span style="font-size: 9pt; line-height: 17.35pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt; line-height: 17.35pt;">Andhra</span><span style="font-size: 9pt; line-height: 17.35pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt; letter-spacing: -0.05pt; line-height: 17.35pt;">Pradesh</span><br />
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<o:p></o:p></div>
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<span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Kyanite</span> <span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">(1)</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Bihar</span> <span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">(2)</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Maharashtra</span><o:p></o:p></div>
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<span style="font-family: 'Book Antiqua', serif; font-size: 9pt; letter-spacing: -0.05pt;">Limestone</span> <span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">(1)</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Madhya</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt; letter-spacing: -0.05pt;">Pradesh</span> <span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">(2)</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Tamil</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt; letter-spacing: -0.05pt;">Nadu</span><o:p></o:p></div>
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<span style="font-family: 'Book Antiqua', serif; font-size: 9pt; letter-spacing: -0.05pt;">Magnetite</span> <span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">(1)</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Tamil</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt; letter-spacing: -0.05pt;">Nadu</span> <span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">(2)</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Uttar</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt; letter-spacing: -0.05pt;">Pradesh</span><o:p></o:p></div>
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<span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Marble</span> <span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">(1)</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Rajasthan</span><o:p></o:p></div>
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<span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Mica</span> <span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">(1)</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Jharkhand</span> <span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">(2)</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Rajasthan</span> <span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">(3)</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Andhra</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Pradesh</span><o:p></o:p></div>
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<span style="font-family: 'Book Antiqua', serif; font-size: 9pt; letter-spacing: -0.05pt;">Ochre</span> <span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">(1)</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Rajasthan</span> <span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">(2)</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Madhya</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt; letter-spacing: -0.05pt;">Pradesh</span><o:p></o:p></div>
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<span style="font-family: 'Book Antiqua', serif; font-size: 9pt; letter-spacing: -0.05pt;">Pyrites</span> <span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">(1)</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Bihar</span><o:p></o:p></div>
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<span style="font-family: 'Book Antiqua', serif; font-size: 9pt; letter-spacing: -0.05pt;">Sulphur</span> <span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">(1)</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Tamil</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt; letter-spacing: -0.05pt;">Nadu</span><o:p></o:p></div>
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<span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Quartz</span> <span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">(1)</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Andhra</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt; letter-spacing: -0.05pt;">Pradesh</span> <span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">(2)</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Karnataka</span><o:p></o:p></div>
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<span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Quartzite</span> <span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">(1)</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Orissa</span> <span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">(2)</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Bihar</span><o:p></o:p></div>
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<span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Silica</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Sand</span> <span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">(1)</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Uttar</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt; letter-spacing: -0.05pt;">Pradesh</span> <span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">(2)</span><span style="font-size: 9pt;"> </span><span style="font-family: 'Book Antiqua', serif; font-size: 9pt;">Gujarat</span><o:p></o:p></div>
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guruhttp://www.blogger.com/profile/01591375209227117455noreply@blogger.comtag:blogger.com,1999:blog-7850540640045081515.post-90953651422672430422016-01-03T18:15:00.001-08:002019-09-14T18:55:55.405-07:00Top Scheme And Programme Launched By Narendra Modi Government<div dir="ltr" style="text-align: left;" trbidi="on">
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjB9rhERNc_xychwoV0bxBVr3WhYWHsLAap7qZk_znPqPSoUV0Lltiza6jd8Y1jkBYVvuYvFbzJvhZ8ne5WoxyuP7H_fJlQS1bMG1W3u8R5YuAHCUH_Wnvagt26MooVP0WKUcp8L-LdOYc/s1600/important_schemes_gov.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="334" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjB9rhERNc_xychwoV0bxBVr3WhYWHsLAap7qZk_znPqPSoUV0Lltiza6jd8Y1jkBYVvuYvFbzJvhZ8ne5WoxyuP7H_fJlQS1bMG1W3u8R5YuAHCUH_Wnvagt26MooVP0WKUcp8L-LdOYc/s640/important_schemes_gov.jpg" width="640" /></a></div>
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1. Make in India (Launched on 25th September 2014) – To make India a manufacturing hub.</div>
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2. Digital India (Launched on 1st July 2015)-To transform India’s economy</div>
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3. Skill India (Launched on 15th July 2015)-To create jobs for youth of the Country.</div>
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4. Smart Cities (Launched on 29th April 2015)list of schemes launched by indian government 2015</div>
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5. Unearthen Black Money (Bill Passed on 14th May 2015)</div>
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6. Namami Gange, Swachh Bharat Abhiyan (Launched on 2nd October 2014)-to have clean India by 2nd October 2019</div>
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7. Swadesh Darshan ,Sukanya Samridhi Account (Launched on 22nd January 2015)</div>
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8. Bal swachh mission (Launched on 14th November 2014)-awareness about the cleanliness of the children</div>
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9. Pradhan Mantri Jan Dhan Yojana (Launched on 28th August 2014 ) – To give financial services to weaker section of society.</div>
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10. Pradhan Mantri Suraksha Bima Yojana (Launched on 9th May 2015)</div>
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11. Beti Bachao Beti Padhao Yojana (Launched on 22nd January 2015)-To generate awareness of welfare service meants for girl child and women.</div>
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12. Atal Pension Scheme (Launched on 9th May 2015)-for unorganised sector ‘s workers</div>
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13. HRIDAY (National Heritage City Development and Augmentation Yojana) scheme (Launched on 21st January 2015)-To develop heritage cities.</div>
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14. Pradhan Mantri Jeevan Jyoti Bima Yojana (Launched on 9th May 2015)-Life insurance scheme by Government</div>
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15. MUDRA Bank Yojana (Launched on 8th April 2015) to provide up to 10 lakh to small entrepreneur.</div>
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16. Krishi Amani Bima Yojana-Scheme for farmer</div>
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17. Pradhan Mantri Gram Sinchai Yojana-To provide water to all field in the Country.</div>
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18. Pradhan Mantri Sansad Adarsh Gram Yojana (Launched on 11th October 2014)-Each MP to develop three villages by 2019.</div>
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19. Deen Dayal Upadhyaya Grameen Kaushalya Yojana (Launched on 25th September 2014)- To provide employment to youth residing in rural area.</div>
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20. Deendayal Upadhyaya Gram Jyoti Yojana -To provide power(electricity) to rural area of the country.</div>
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21. Mahatma Gandhi Pravasi Suraksha Yojana, Soil Health Card Scheme, Indradanush Scheme (Launched on 25th December 2014)</div>
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22. Rani Laxmi Bai Pension scheme-for victims of Muzzafar nagar riot.</div>
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23. Udaan Scheme-To provide skill to youth of India.</div>
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24. Shyama Prasad Mukherji Rurban Mission</div>
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25. Pandit Deendayal Upadhyay Shramev Jayate Karyakram (Launched on 16th October 2014)</div>
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26. Kisan Vikas Patra (Relaunched in 2014) – Saving certificate Scheme</div>
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27. AMRUT – Atal Mission for Rejuvenation and Urban Development (earlier name JNNURM)</div>
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28. PRASAD – Pilgrimage Rejuvenation and Spiritual Augmentation-To improve the infrastructure at pilgrimage places.</div>
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guruhttp://www.blogger.com/profile/01591375209227117455noreply@blogger.comtag:blogger.com,1999:blog-7850540640045081515.post-32030413976639784372016-01-03T07:47:00.000-08:002019-09-14T19:32:33.214-07:00List of important Committees in India<div dir="ltr" style="text-align: left;" trbidi="on">
<h2 style="text-align: left;">
List of important Committees in India </h2>
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<li>Ajay Shankar Committee : To review functioning of PPP Cell </li>
<li>Prof. NR Madhava Menon Panel : Reported guidelines for regulating expenditure and content of advertisement in govt. adds</li>
<li>H Devaraj Commitee : Reported most deemed university</li>
<li>H R Khan Panel : To evaluate unclaimed PPF and Post Office Savings</li>
<li>V V Daga Committee : To conduct forensic audit of NSEL</li>
<li>Sivaramakrishnan committee : Submit Report to build the capital city for Andhra Pradesh</li>
<li>Ramanujam committee : To avoid obsolete laws</li>
<li>Bimal Jalan : To head the Expenditure Management Commission </li>
<li>Hari Gautam Committee : To review the status of UGC</li>
<li>Justice SB Sinha (One Member Commission):To Probe 2006 Meerut Fire Tragedy </li>
<li>Suresh Prabhu Committee: To review gas pricing formula</li>
<li>R S Sharma Expert Committee : To review the Company (Cost Records and Cost Audits) Rules 2014</li>
<li>Justice MB Shah : On Black Money</li>
<li>Deepak Mohanty Committee : Data and Information Management in the RBI</li>
<li>Arvind Mayaram Committee : To clear definition to the FDI and FII</li>
<li>Nachiket Mor Committee : To permit NBFC's to work as Business correspondence </li>
<li>P J Nayak Committee : Governance of Boards of Bank in India</li>
<li>Bibek Debroy : for Restructuring the railway</li>
<li>Justice CS Dharmadhikari Committee: recommended complete ban on dance bars in hotels and restaurants.</li>
<li>Pratyush Sinha : To assess compensation for coal blocks.</li>
<li>Jairam Ramesh : On sustainable development </li>
<li>T.K. Vishwanathan committee: To provide Bankruptcy code for small and medium enterprises (SMEs).</li>
<li>K.V. Kamath panel: to examine the financial architecture for Micro, Small and Medium Enterprises (MSME) sector.</li>
<li>Gopalakrishna Committee: on Capacity Building in Banks and non-Banks </li>
<li>G N Bajpai Committee: Guidelines for national pension system (NPS) schemes in private sector. </li>
<li>Scientist Raghunath Anant Mashelkar panel: To recommend best technologies for Prime Minister Narendra Modi's "Swachh Bharat" national sanitation campaign. </li>
<li>T S R Subramanian Committee: To review five key green laws concerning protection and conservation of environment, forest, wildlife, water and air among others.</li>
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guruhttp://www.blogger.com/profile/01591375209227117455noreply@blogger.comtag:blogger.com,1999:blog-7850540640045081515.post-36548911956759626612015-12-28T21:12:00.000-08:002019-09-14T18:55:55.296-07:00Abbreviations <div dir="ltr" style="text-align: left;" trbidi="on">
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjxvwxIGt_AYnlRCHfp-5q5lLdIXeNtOyjO1vBJDSaUYVLBSsmLMAiDJxFiLBjkJUzu7n2oNR1HoIqCCVtLcOjIz0NXGJKxV8AqbRkH1UHV90l3bgzTNu2qmGvvtMAjFYbBRCrThJvKQhc/s1600/Abbreviations-and-acronyms.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="278" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjxvwxIGt_AYnlRCHfp-5q5lLdIXeNtOyjO1vBJDSaUYVLBSsmLMAiDJxFiLBjkJUzu7n2oNR1HoIqCCVtLcOjIz0NXGJKxV8AqbRkH1UHV90l3bgzTNu2qmGvvtMAjFYbBRCrThJvKQhc/s640/Abbreviations-and-acronyms.png" width="640" /></a></div>
<br /><br /><br /><br /><h1 class="entry-title" style="background: rgb(255, 255, 255); border: 0px; box-sizing: border-box; color: #333333; font-family: 'Droid Sans', Helvetica, Arial, Lucida, sans-serif; font-size: 30px; font-weight: 500; line-height: 1em; margin: 0px; outline: 0px; padding: 0px 0px 10px; vertical-align: baseline;">
Abbreviations Part -2</h1>
<br /><br /><br /><table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse; width: 526px;"><tbody>
<tr height="50" style="height: 38.1pt; mso-height-source: userset;"><td class="xl65" height="50" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 38.1pt; width: 82pt;" width="109">AAFI</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Amateur Athletics Federation of India</td>
</tr>
<tr height="26" style="height: 20.1pt; mso-height-source: userset;">
<td class="xl65" height="26" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 20.1pt; width: 82pt;" width="109">AAPSO</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Afro-Asian People's Solidarity Organisation</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">AASU</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">All Assam Students Union</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">ABM</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Anti Ballistic Missile</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">AC</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Alternate Current OR Air Conditioner</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">ACC</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Ancillary Cadet Core</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">AD</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Ano Domini (After the birth of Jesus)</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">ADB</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Asian Development Bank.</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">AERE</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Atomic Energy Research Establishment</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">AGOC</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Asian Games Organisation Committee</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">AICC</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">All India Congress Committee</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">AICTE</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">All India Council of Technical Education</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">AIDS</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Acquired Immuno Deficiency Syndrome</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">AIFE</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">All India Football Federation</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">AIIMS</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">All India Institute of Medical Sciences</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">AIL</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Aeronautics India Limited</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">AIMPLB</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">All India Muslim Personal Law Board</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">AIR</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">All India Radio (Broadcasting)</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">AITUE</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">All India Trade Union Congress</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">AM</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Anti Meridian (Before Noon)</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">ANC</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">African National Congress</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">APEC</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Asia Pacific Economic Cooperation</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">APSC</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Army Postal Services Core</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">ASEAN</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Association of South East Asian Nations</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">ASLV</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Augmented Satellite Launch Vehicle</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">ASI</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Archaeological Survey of India</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">ASSOCHAM</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Associated Chamber of Commerce and Industry (India)</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">ASWAC</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Airborne Surveillance Warning and Control</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">ATS</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Anti Tetanus Serum</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">BAMS</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Bachelor of Ayurvedic Medicine and Surgery</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">BARC</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Bhabha Atomic Research Centre</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">BBC</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">British Broadcasting Corporation</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">BC</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Before Christ (Before the birth of Jesus)</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">BCG</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Bacillus Calmette Guerin (Anti TB Vaccine)</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">BCCI</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Board of Control for Cricket in India</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">BEL</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Bharat Electronics Limited</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">BENELUX</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Belgium, Netherlands and Luxemburg</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">BHEL</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Bharat Heavy Electronics Limited</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">BIFR</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Board of Industrial Finance and Reconstruction
(Formerly Industrial Reconstruction Finance Board)</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">BIMSTEC</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Bangladesh, India, Myanmar, Sri Lanka, Thailand
Economic Cooperation</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">BIS</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Bureau of Indian Standards</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">B Pharma</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Bachelor of Pharmacy</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">BSF</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Border Security Force</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">CAD</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Command Area Development</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">CAG</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Comptroller and Auditor General</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">CARE</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Cooperative for American Relief Everywhere</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">CASE</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Commission for Alternative Sources of Energy</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">CBI</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Central Bureau of Investigation</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">CBSE</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Central. Board of Secondary Education</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">CCEA</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Cabinet Committee on Economic Affairs</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">CCS</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Cabinet Committee on Security</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">C-DAC</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Centre For Development of Advance Computing</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">CDMA</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Code Division Multiple Access</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">CDRI</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Central Drug Research Institute</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">CHOGM</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Commonwealth Heads of Government Meeting</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">CID</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Criminal Investigation Department</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">CIS</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Commonwealth of Independent States</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">CISF</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Central Industrial Security Force</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">CITU</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Centre of Indian Trade Unions</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">CLAT</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Common Law Admission Test (Started May 2008)</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">CNG</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Compressed Natural Gas</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">COD</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Central Ordnance Depot</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">COFEPOSA</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Conservation of Foreign Exchange and Prevention of
Smuggling Act</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">CPO</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Central Passport Organisation</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">CPRI</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Central Power Research Institute</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">CRPF</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Central Reserve Police Force</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">CRR</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Cash Reserve Ratio</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">CSIR</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Council of Scientific and Industrial Research</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">CSO</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Central Statistical Organisation</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">CTS</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Computerised Tomography Scanner</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">CVC</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Central Vigilance Commission</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">DDT</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Dichloro Diphenyle Tri-chloroethane</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">DFDR</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">'Digital Flight Data Recorder (Black box)'</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">DIG</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Deputy Inspector General</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">D. Lit.</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Doctor of Literature</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">DM</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">District Magistrate</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">DMK</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Dravida Munetra Kazhagam</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">DNA</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Di-oxyribo-Nucleic Acid</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">DPAP</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">'Drought Prone Area Programme</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">DPC</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Dabhol Power Company</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">DRDO</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Defence Research and Development Organisation</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">DTH</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Direct to Home</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">DVD</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Digital Versatile Disk</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">EAS</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Employment Assurance Scheme</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">ECD</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">European Central Bank</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">ECG</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Electro Cardiogram</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">EEC</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">European Economic Community</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">EEG</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Electro Encephalogram</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">ELISA</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Enzyme Linked Immuno Sorbent Assay</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">EMF</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Electromotive Force</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">EPABX</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Electronic Private Automatic Branch Exchange</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">EPZ</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Export Processing Zone</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">ERDA</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Energy Research and Development Administration</td>
</tr>
<tr height="33" style="height: 24.95pt; mso-height-source: userset;">
<td class="xl65" height="33" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 24.95pt; width: 82pt;" width="109">ESMA</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Essential Services Maintenance Act</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">EVM</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Electronic Voting Machine</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">EXIM Bank</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Export-Import Bank of India</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">FAO</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Food and Agriculture Organisation</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">FBI</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Federal Bureau of Investigation (USA)</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">FBTR</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Fast Breeder Test Reactor</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">FCI</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Food Corporation of India / Fertilizer Corporation of
India</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">FDR</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Flight Data Recorder (Black Box)</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">FERA</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Foreign Exchange Regulation Act</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">FEMA</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Foreign Exchange Management Act</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">FICCI</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Federation of India Chambers of Commerce and Industry</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">FIPB</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Foreign Investment Promotion Board</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">FIR</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">First Information Report</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">FRS</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Fellow of the Royal Society</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">FTII</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Films and Television Institute of India</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">FTZ</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Free Trade Zone</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">GAIL</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Gas Authority of India Limited</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">GATT</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">General Agreement on Tariff and Trade</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">GIC</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">General Insurance Corporation</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">GMT</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Greenwich Mean Time</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">GNLF</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Gorkha National Liberation Front</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">GNP</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Gross National Product</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">GPF</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">General Provident Fund</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">GPO</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">General Post Office</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">GPS</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Global Positioning System</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">GSI</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Geological Survey of India</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">HAC</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Hindustan Aluminium Corporation</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">HAL</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Hindustan Aeronautics Limited</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">HCF</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Highest Common Factor</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">HDFC</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Housing Development Finance Corporation</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">HIV</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Human Immuno-deficiency Virus</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">HMT</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Hindustan Machine Tools</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">HUDCO</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Housing and Urban Development Corporation</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">HYVS</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">High Yield Variety Seeds</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">IAAI</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">International Airport Authority of India</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">lAC</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Indian Airlines Corporation</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">IAEA</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">International Atomic Energy Agency</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">IARI</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Indian Agricultural Research Institute</td>
</tr>
<tr height="39" style="height: 29.25pt;">
<td class="xl65" height="39" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 29.25pt; width: 82pt;" width="109">IBRD</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">International Bank for Reconstruction and Development
(World Bank)</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">ICAR</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Indian Council of Agricultural Research</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">ICBM</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Inter Continental Ballistic Missile</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">ICC</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">International Cricket Council</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">ICFTU</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">International Confederation of Free Trade Unions</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">ICICI</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Industrial Credit and Investment Corporation of India
Limited</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">ICJ</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">International Court of Justice</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">ICMR</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Indian Council of Medical Research</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">ICSI</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Indian Company Secretaries Institute</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">IDA</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">International Development Agency</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">IDBI</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Industrial Development Bank of India</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">IDO</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">International Defence Organisation</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">IDPL</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Indian Drugs and Pharmaceuticals Limited</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">IFA</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Indian Football Association</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">IFCI</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Industrial Finance Corporation of India</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">IFFI</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">International Film Festival of India</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">IFFCO</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Indian Farmers Fertilizers Cooperative</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">IFTU</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">International Federation of Trade Unions</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">IIPA</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Indian Institute of Public Administration</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">IIS</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Indian Institute of Sciences</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">IISCO</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Indian Iron and Steel Company</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">IIT</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Indian Institute of Technology</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">ILO</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">International Labour Organisation</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">IMA</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Indian Military Academy</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">IMF</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">International Monetary Fund</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">INGCA</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Indira Gandhi Gallery for Culture and Art</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">INS</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Indian Naval Ship</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">INSAT</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Indian National Satellite</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">INTELSAT</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">International Telecommunication Satellite</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">INTERPOL</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">International Police Organisation</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">INTUC</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Indian National Trade Union Congress</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">IOC</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">International Olympic Committee / Indian Oil
Corporation</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">IPC</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Indian Penal Code</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">IPKF</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Indian Peace Keeping Force</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">IQ</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Intelligence Quotient</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">IRBM</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Intermediate Range Ballistic Missile</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">IRC</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">International Red Cross</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">IRDA</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Insurance Regulatory Development Authority</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">IRDP</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Integrated Rural Development Programme</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">ISB</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Indian Standard Bureau</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">ISM</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Indian School of Mines</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">ISO</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">International Organisation for Standardisation</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">ISP</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Internet Services Provider</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">ISRO</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Indian Space Research Organisation</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">IST</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Indian Standard Time</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">ITBP</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Indo-Tibet Border Police</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">ITDC</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Indian Tourism Development Corporation</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">ITPO</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Indian Trade Promotion Organisation</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">ITO</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">International Trade Organisation</td>
</tr>
<tr height="39" style="height: 29.25pt;">
<td class="xl65" height="39" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 29.25pt; width: 82pt;" width="109">IUCN</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">International Union for Conservation of Nature and
Natural Resource</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">ITUC</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Indian Trade Union Congress</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">JMM</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Jharkhand Mukti Morcha</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">KG</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Kinder Garten</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">LASER</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Light Amplification By Stimulated Emission of
Radiation</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">LIC</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Life Insurance Corporation of India</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">LLB</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Bachelor of Law</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">LLM</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Master of Law</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">LMG</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Light Machine Gum</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">LoC</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Line of Control (Pakistan)</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">LoAC</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Line of Actual Control (China)</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">LPG</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Liquefied Petroleum Gas</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">LSD</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Lysergic acid diethylamide</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">LTTE</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Liberation Tigers of Tamil Elam</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">MA</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Master of Arts</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">MASER</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Microwave Amplification by Stimulated Emission of
Radiation</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">MBA</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Master of Business Administration</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">MBBS</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Bachelor of Medicine and Bachelor of Surgery</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">MBT</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Main Battle Tank</td>
</tr>
<tr height="39" style="height: 29.25pt;">
<td class="xl65" height="39" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 29.25pt; width: 82pt;" width="109">MCA</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Monetary Compensatory Allowance / Master of Computer
Application</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">MCC</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Melbourne Cricket Club</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">MD</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Doctor of Medicine</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">MFN</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Most Favoured Nation</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">MI</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Military Intelligence</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">MISA</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Maintenance of Internal Security Act</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">MIT</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Mechachusates Institute of Technology (USA)</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">MLA</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Member of Legislative Assembly</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">MLC</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Member of Legislative Council</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">MNC</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Multi National Corporation</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">MRCP</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Member of Royal College of Physicians</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">MRCS</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Member of Royal College of Surgeons</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">MRTPC</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Monopoly and Restrictive Trade Practices Commission</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">MODVAT</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Modified Value Added Tax</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">NABARD</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">National Bank for Agricultural and Rural Development</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">NACO</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">National AIDS Control Organisation</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">NAEP</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">National Adult Education Programme</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">NAFED</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">National Agricultural and Marketing Federation</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">NAFTA</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">North American Free Trade Agreement</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">NAPP</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Narora Atomic Power Plant</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">NASA</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">National Aeronautics and Space Administration (USA)</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">NASDAQ</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">National Association of Security Dealer's Active
Quotation</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">NASSCOM</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">National Association of Software & Service
Companies</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">NATO</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">North Atlantic Treaty Organisation</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">NCW</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">National Commission for Women</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">NCCR</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">National Council for Civil Right</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">NCERT</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">National Council of Educational Research &
Training</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">NDA</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">National Defence Academy</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">NDDB</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">National Dairy Development Board</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">NDF</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">National Defence Fund.</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">NEERI</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">National Environment Engineering Research Institute</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">NEFA</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">North-East Frontier Agency</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">NEPA</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">National Environment Protection Authority</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">NFDC</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">National Film Development Corporation</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">NFL</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">National Fertilizer Limited</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">NHRC</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">National Human Rights Commission</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">NICO</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">New Information and Communication Order</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">NIDC</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">National Industrial Development Corporation</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">NIIT</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">National Institute of Information Technology</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">NIMHANS</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">National Institute of Mental Health and
Neuro-Sciences</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">NITIE</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">National Institute for Training in Industrial
Engineering</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">NMDS</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">National Missile Defence System (US)</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">NMEP</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">National Malaria Eradication Programme</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">NOIDA</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">New Okhla Industrial Development Authority</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">NPC</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">National Productivity Council</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">NPP</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">National Population Policy</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">NPT</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Nuclear Non-Proliferation Treaty</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">NRDC</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">National Research and Development Corporation</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">NREP</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">National Rural Employment Programme</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">NRI</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Non Resident Indian</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">NSC</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">National Security Council</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">NSSO</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">National Sample Survey Organisation</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">NTC</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">National Textile Corporation</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">NTPC</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">National Thermal Power Corporation</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">OGL</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Open General Licence</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">OIL</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Oil India Limited</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">OK</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">All Correct</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">ONGC</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Oil and Natural Gas Commission</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">OPEC</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Organisation of Petroleum Exporting Countries</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">PCI</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Press Council of India</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">PCS</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Provincial Civil Services</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">Ph. D</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Doctor of 'Philosophy</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">PIN</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Postal lndex Number</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">PLO</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Palestine Liberation Organisation</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">PM</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Post Meridian / Prime Minister</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">POTA</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Prevention of Terrorism Act</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">PSLV</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Polar. Satellite Launch Vehicle</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">PTI</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Press Trust of India</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">PRO</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Public Relations Officer</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">PTO</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Please Turn Over</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">PVC</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Poly Vinyl Chloride / Paramvir Chakra</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">PVSM</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Param Vishisht Seva Medal</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">PWD</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Public Work's Department</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">PWG</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">People's War Group</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">QED</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Quod Erat Demonstrandum (Which was to be proved)</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">QEF</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Quod Erat Faciendum (Which was to be done)</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">QEI</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Quod Erat Inveniendum (Which was to be found)</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">QMG</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Quarter Master General</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">RADAR</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Radio Angle Direction and Range</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">RAW</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Research and Analysis Wing</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">R & D</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Research and Development</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">RBI</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Reserve Bank of India</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">RCC</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Reinforced Cement Concrete</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">RDX</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Research Developed Explosive</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">RIMC</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Rashtriya Indian Military College</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">RMS</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Railway Mail Service</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">RLEGP</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Rural Landless Employment Guarantee Programme</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">RNA</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Ribonucleic Acid</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">RPM</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Revolutions Per Minute</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">RSS</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Rashtriya Swayamsevak Sangh</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">RTO</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Regional Transport Officer</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">SAARC</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">South Asian Association for Regional Cooperation</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">SAC</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Space Application Centre</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">SAFTA</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">South Asian Free Trade Agreement</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">SAI</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Sports Authority of India</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">SAIL</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Steel Authority of India Limited</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">SAPTA</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">South Asian Preferential Trade Arrangement</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">SARS</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Severe Acute Respiratory Syndrome</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">SC</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Security Council/Supreme Court</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">SCI</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Shipping Corporation of India</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">SCOPE</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Standing Conference of Public Enterprises</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">SCRA</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Special Class Railway Apprentice</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">SDR</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Special Drawing Rights</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">SEBI</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Security Exchange Board of India</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">SGPC</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Siromani Gurudwara Prabandhak Committee</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">SHAR</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Shri Harikota Range</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">SIDBI</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Small Industries Development Bank of India</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">SIS</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Secret Intelligence Service (U.K)</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">SITA</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Suppression of .Immoral Traffic in Women and Girls
Act</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">SLV</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Satellite Launch Vehicle</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">SPCA</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Society for the Prevention of Cruelty of Animals</td>
</tr>
<tr height="39" style="height: 29.25pt;">
<td class="xl65" height="39" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 29.25pt; width: 82pt;" width="109">SPICMC</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Society for the Promotion of Indian Classical music
and culture</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">STARS</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Satellite Tracking and Ranging Station</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">STD</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Subscribers Trunk Dialing</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">STPI</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Software Technology Parks of India</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">SWAPO</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">South West African People's Organisation</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">TA</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Travelling Aliowance / Territorial Anmy</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">TELCO</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Tata Engineering and Locomotive Company</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">TELEX</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Teleprinter Exchange</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">TISCO</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Tata Iron and Steel Company Limited</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">TNT</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Tri-nitro-toluene</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">TOEFL</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Test of English as a Foreign Language</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">TRAI</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Telecom Regulatory Authority of India</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">TRIPS</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Trade Related Intellectual Property Rights</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">TTE</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Travelling Ticket Examiner</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">TTFI</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Table Tennis Federation of India</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">TWA</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Trans World Airlines (USA)</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">UDC</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Upper Division Clerk</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">UFO</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Unidentified Flying Object</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">UGC</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">University Grants Commission</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">UHT</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Ultra High Temperature</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">ULFA</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">United Liberation Front of Assam</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">UNCTAD</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">United Nations Conference on Trade and Development</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">UNDP</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">United Nations Development Programme</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">UNEF</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">United Nations Emergency Force</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">UNEP</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">United Nations Environment Programme</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">UNESCO</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">United Nations Economic Social and Cultural
Organisation</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">UNFPA</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">United Nations for Population Activities</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">UNHCR</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">United Nations High Commission for Refugees</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">UNI</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">United News of India</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">UNICEF</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">United Nations International Children's Emergency
Fund</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">UNO</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">United Nations Organisation</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">UPS</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Uninterrupted Power Supply</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">UPSC</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Union Public Service Commission</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">USSR</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Union of Soviet Socialist Republic</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">UTI</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Unit Trust of India</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">VAT</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Value Added Tax</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">VDIS</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Voluntary Disclosure of Income Scheme</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">VC</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Vice-Chancellor / Victoria Cross</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">VIP</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Very Important Person</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">VPP</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Value Payable Post</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">VRS</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Voluntary Retirement Scheme</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">VSNL</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Videsh Sanchar Nigam Limited</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">VSSC</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Vikram Sarabhai Space Centre</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">WEF</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">World Economic Forum</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">WHO</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">World Health Organisation</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">WILL</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Wireless in Local Loop</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">WMO</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">World Meteorological Organisation</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">WWF</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">World Wild Life Fund</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">WPI</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Wholesale Price Index</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">WTO</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">World Trade Organisation</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">WWF</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">World Wild Life Fund for Nature</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">WWW</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">World Wide Web</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">YMCA</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Young Men's Christians Association</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">YWCA</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Young Women's Christians Association</td>
</tr>
<tr height="20" style="height: 15.0pt;">
<td class="xl65" height="20" style="background: white; border-color: rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 15pt; width: 82pt;" width="109">ZBB</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) black black; border-style: solid; border-width: 1pt 1pt 0.5pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Zero Based Budgeting</td>
</tr>
<tr height="19" style="height: 14.25pt;">
<td class="xl65" height="19" style="background: white; border-color: rgb(238, 238, 238) black; border-style: solid; border-width: 1pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; height: 14.25pt; width: 82pt;" width="109">ZSI</td>
<td class="xl65" style="background: white; border-color: rgb(238, 238, 238) rgb(238, 238, 238) rgb(238, 238, 238) black; border-style: solid; border-width: 1pt 1pt 1pt 0.5pt; color: #111111; font-family: Arial; font-size: 11pt; width: 313pt;" width="417">Zoological Survey of India</td>
</tr>
</tbody></table>
</div>
guruhttp://www.blogger.com/profile/01591375209227117455noreply@blogger.comtag:blogger.com,1999:blog-7850540640045081515.post-60736761315547718942015-12-28T07:01:00.001-08:002019-09-14T18:55:55.368-07:00Acronym and Abbreviation<div dir="ltr" style="text-align: left;" trbidi="on">
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjnkV2xZeiG4TYtf4QuncLCelwgBjtMIYwclKLGbpcyWAuxJnUhkL14gDOUZOdKnY9Wh5gebuTo9TtJ1sE6F9Tc9clXfJ1f4U6bsksWYTg3yQPT2BPBeK0XTlIgLvNTkngc_lr0Ij_rgj8/s1600/abbreviations.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="466" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjnkV2xZeiG4TYtf4QuncLCelwgBjtMIYwclKLGbpcyWAuxJnUhkL14gDOUZOdKnY9Wh5gebuTo9TtJ1sE6F9Tc9clXfJ1f4U6bsksWYTg3yQPT2BPBeK0XTlIgLvNTkngc_lr0Ij_rgj8/s640/abbreviations.png" width="640" /></a></div>
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<table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse; width: 510px;">
<colgroup><col style="mso-width-alt: 5162; mso-width-source: userset; width: 91pt;" width="121"></col>
<col style="mso-width-alt: 16597; mso-width-source: userset; width: 292pt;" width="389"></col>
</colgroup><tbody>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="height: 15.0pt; width: 91pt;" width="121">B.I.T.</td>
<td class="xl67" style="border-left: none; width: 292pt;" width="389">Birla Institute
of Technology.</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">I.I.T.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">Indian
Institute of Technology.</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">C.A.D.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">Computer
Aided Design.</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">H.T.T.P.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">Hyper
Text Transfer Protocol.</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">D.M.R.C.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">Delhi
Metro Rail Corporation.</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">A.D.B.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">Asian
Development Bank.</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">B.A.R.C.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">Bhabha
Atomic Research Centre.</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">B.B.C.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">British
Broadcasting Corporation.</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">C.S.I.R.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">Council
of Scientific and Industrial Research.</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">A.S.E.A.N.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">Association
of South East Asian Nations.</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">A.S.C.I.I.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">American
Standard Code for Information Interchange.</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">E.V.M.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">Electronic
Voting Machine.</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">L.I.C.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">Life
Insurance Corporation (of India).</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">I.G.N.O.U.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">Indira
Gandhi National Open University.</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">R.B.I.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">Reserve
Bank of India.</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">U.N.D.P.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">United
Nations Development Programme.</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">U.N.I.C.E.F.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">United
Nations International Children's Emergency Fund.</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">L.P.G.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">Liquified
Petroleum Gas.</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">M.B.B.S.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">Bachelor
of Medicine and Bachelor of Surgery.</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">N.C.E.R.T.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">National
Council of Educational Research and Training.</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">O.N.G.C.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">Oil
and Natural Gas Commission.</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">P.I.N. Code</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">Postal
Index Number Code.</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">V.A.T.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">Value
Added Tax.</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">W.H.O.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">World
Health Organization.</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">W.W.F.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">World
Wildlife Fund.</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">F.A.O.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">Food
and Agriculture Organization.</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">F.E.M.A.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">Foreign
Exchange Management Act.</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">G.A.T.T.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">General
Agreement on Tariff and Trade.</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">I.L.O.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">International
Labour Organisation.</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">I.M.F.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">International
Monetary Fund.</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">E.S.M.A.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">Essential
Services Maintenance Act.</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">I.S.R.O.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">Indian
Space Research Organisation.</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">W.W.W.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">World
Wide Web.</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">W.A.P.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">Wireless
Application Protocol.</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">C.I.A.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">Central
Intelligence Agency.</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">C.I.D.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">Criminal
Investigation Department.</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">I.T.D.C.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">Indian
Tourism Development Corporation.</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">MoU.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">Memorandum
of Understanding.</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">N.F.D.C.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">National
Film Development Corporation.</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">P.O.T.A.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">Prevention
of Terrorism Act.</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">R.T.I.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">Right
To Information.</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">P.T.I.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">Press
Trust of India.</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">U.N.O.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">United
Nation Organisation.</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">B.P.O.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">Business
Process Outsourcing.</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">C.I.C.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">Chief
Information Commissioner.</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">N.R.E.G.A.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">National
Rural Employment Guarantee Act.</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">P.A.N.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">Permanent
Account Number.</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">S.I.M.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">Subscriber
Identification Module.</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">N.D.A.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">National
Defence Academy.</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">O.P.E.C.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">Organisation
of Petroleum Exporting Countries.</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">RAM</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">Random
Access Memory.</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">RAW</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">Research
and Analysis Wing.</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">I.S.I.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">Inter
Services Intelligence.</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">I.I.M.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">Indian
Institute of Management.</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">N.A.S.A.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">National
Aeronautics Space Administration.</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">I.B.M.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">International
Business Machines.</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">N.G.O.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">Non-Governmental
Organization.</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">B.S.F.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">Border
Security Force.</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">C.E.O.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">Chief
Executive Officer.</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">C.B.I.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">Central
Bureau of Investigation.</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">P.W.D.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">Public
Works Department.</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">R.O.M.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">Read
Only Memory.</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">U.S.B.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">Universal
Serial Bus.</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">P.S.U.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">Public
Sector Undertaking.</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">R.T.G.S.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">Real
Time Gross Settlement.</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">N.E.F.T.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">National
Electronic Funds Transfer.</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">I.C.U.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">Intensive
Care Unit.</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">E.C.G.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">ElectroCardioGram.</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">C.A.T.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">Common
Admission Test.</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">G.A.T.E.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">Graduate
Aptitude Test in Engineering.</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">N.M.D.S.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">National
Missile Defence System.</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">A.I.R.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">All
India Radio.</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">H.T.M.L.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">HyperText
Markup Language.</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">U.R.L.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">Uniform
Resource Locator.</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">G.S.L.V.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">Geo-Synchronous
Satellite Launch Vehicle.</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">B.S.N.L.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">Bharat
Sanchar Nigam Limited.</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">S.A.R.S.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">Severe
Acute Respiratory Syndrome.</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">G.S.M.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">Global
System for Mobile Communications.</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">G.P.R.S.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">General
Packet Radio Service.</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">WiFi</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">Wireless
Fidelity.</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">I.M.E.I.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">International
Mobile Equipment Identity.</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">T.R.P.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">Television
Rating Point.</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">P.N.R.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">Passenger
Name Record.</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">M.L.A.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">Member
of Legislative Assembly.</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">S.S.C.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">Staff
Selection Commission.</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">U.P.S.C.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">Union
Public Service Commission.</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">N.R.I.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">Non
Resident Indian.</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">B.C.C.I.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">Board
of Control for Cricket in India.</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">F.I.R.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">First
Information Report.</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">P.D.F.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">Portable
Document Format.</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">S.Q.L.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">Structured
Query Language.</td>
</tr>
<tr height="33" style="height: 24.95pt; mso-height-source: userset;">
<td class="xl67" height="33" style="border-top: none; height: 24.95pt; width: 91pt;" width="121">J.P.E.G or J.P.G.</td>
<td class="xl68" style="border-left: none; border-top: none; width: 292pt;" width="389">Joint
Photographic Experts Group.</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">E.M.I.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">Equated
Monthly Installment.</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">I.F.S.C.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">Indian
Financial System Code.</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">T.D.S.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">Tax
Deducted at Source.</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">G.D.P.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">Gross
Domestic Product.</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">A.S.A.P.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">As
Soon As Possible.</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">U.A.E.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">United
Arab Emirates.</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">S.T.D.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">Subscriber
Trunk Dialing.</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">I.R.D.A.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">Insurance
Regulatory and Development Authority.</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">B.O.L.T.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">Build
Operate Lease Transfer</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">C.I.I.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">Confederation
of Indian Industry</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">C.R.I.S.I.L.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">Credit
Rating Information Services of India Limited</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">C.R.R.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">Cash
Reserve Ratio</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">E.P.Z.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">Export
Processing Zone</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">F.E.R.A.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">Foreign
Exchange Regulation Act</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">F.I.I.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">Foreign
Institutional Investors</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">G.D.P.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">Gross
Domestic Product</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">H.U.D.C.O.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">Housing
and Urban Development Corporation</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">N.A.B.A.R.D.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">National
Bank for Agriculture and Rural Development</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">O.G.L.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">Open
General License</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">V.A.T.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">Value
Added Tax</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">C.I.S.F.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">Central
Industrial Security Force</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">N.I.F.T.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">National
Institute of Fashion Technology</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">U.G.C.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">University
Grants Commission</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">I.O.C.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">International
Olympic Committee</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">D.D.T.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">Dichloro-Diphenyl-Trichloroethane</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">D.T.P.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">Desk
Top Publishing</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">I.C.B.M.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">Inter-Continental
Ballistic Missile</td>
</tr>
<tr height="20" style="height: 15.0pt; mso-height-source: userset;">
<td class="xl67" height="20" style="border-top: none; height: 15.0pt; width: 91pt;" width="121">N.A.T.O.</td>
<td class="xl67" style="border-left: none; border-top: none; width: 292pt;" width="389">North
Atlantic Treaty Organisation</td>
</tr>
</tbody></table>
</div>
guruhttp://www.blogger.com/profile/01591375209227117455noreply@blogger.comtag:blogger.com,1999:blog-7850540640045081515.post-10432353121876657532015-12-27T09:06:00.001-08:002019-09-14T18:55:55.043-07:00Newly Appointed Person<div dir="ltr" style="text-align: left;" trbidi="on">
<h2 style="background-color: white; font-family: helvetica, arial, sans-serif; font-size: 14px; line-height: 15.456px; margin-bottom: 6px; text-align: left;">
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<span style="color: red;"><br /></span></h2>
Newly Appointed Person</span></h2>
<div style="background-color: white; color: #141823; font-family: helvetica, arial, sans-serif; font-size: 14px; line-height: 15.456px; margin-bottom: 6px; margin-top: 6px;">
Post & Company</div>
<div style="background-color: white; color: #141823; font-family: helvetica, arial, sans-serif; font-size: 14px; line-height: 15.456px; margin-bottom: 6px; margin-top: 6px;">
1.</div>
<div class="text_exposed_show" style="background-color: white; color: #141823; display: inline; font-family: helvetica, arial, sans-serif; font-size: 14px; line-height: 15.456px;">
<div style="margin-bottom: 6px;">
Subir Vithal Gokarn</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
Executive Director on the board of the International Monetary Fund (IMF)</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
2.</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
Justice Ajit Prakash Shah</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
Ethics officer (ombudsman) at Board of Control for Cricket in India (BCCI).</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
3.</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
Navtej Singh Sarna</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
Indian High Commissioner to United Kingdom (UK)</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
4.</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
V-Sivaramakrishnan (Siva)</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
Managing Director, Oxford University Press India (OUPI)</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
5.</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
Justice T S Thakur</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
43rd Chief Justice of India</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
6.</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
Anil Kumar Jha</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
Chairman-cum-Managing Director of Mahanadi Coalfields Ltd (MCL)</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
7.</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
Deepak Singhal</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
Executive Director of Reserve Bank of India.</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
8.</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
Virender Mohan Khanna</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
Maintenance Head of IAF</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
9.</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
Harshavardhan Neotia</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
President of FICCI. Note: He will succeed Jyotsna Suri</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
10.</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
Nitish Kumar</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
Chief Minister of Bihar (Fifth Time)</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
11.</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
V Raja</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
Vice-Chairman and Managing Director, Philips India</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
12.</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
Vijay Keshav Gokhale</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
India’s Ambassador to China. Note: He will replace Ashok Kantha</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
13.</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
Zarin Daruwala</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
India’s Chief Executive Officer at Standard Chartered Bank</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
14.</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
Syed Akbaruddin</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
Permanent Representative of India to the United Nations</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
15.</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
Sandeep Girotra</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
Head of India, Nokia</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
16.</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
Mahesh Gupta</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
President,PHD Chamber of Commerce and Industry (PHDCCI)</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
17.</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
Rishi Gour</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
Country President of Sodexo India On-site Services.</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
18.</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
Sunil Kanoria</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
President, ASSOCHAM. Note: He succeeded Rana Kapoor.</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
19.</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
V Shanmuganthan</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
17th Governor of Manipur</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
20.</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
Dr Anup K Pujari</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
Chairman and Managing Director of Steel Authority of India Limited (SAIL)</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
21.</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
Sriram Venkataraman</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
Chief Finance Officer (CFO), Flipkart</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
22.</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
Ashish Bahuguna</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
Chief Executive Officer (CEO) of FSSAI</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
23.</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
Sekhar Basu</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
Chairman of Atomic Energy Commission (AEC)</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
24.</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
Ranganath D Mavinakere</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
Chief Financial Officer, Infosys</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
25.</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
N Venugopala Rao</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
Chief Executive Officer (CEO), Reliance Power</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
26.</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
Alkesh Sharma</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
CEO and MD of Delhi-Mumbai Industrial Corridor Development Corp</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
27.</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
Tapan Ray</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
Board of Securities and Exchange Board of India</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
28.</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
Sarat Kumar Acharya</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
CMD,Neyveli Lignite Corporation Limited.</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
29.</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
Alok Rawat</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
First male person to be appointed as member of National Commission for Women (NCW)</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
30.</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
Anup Vikal</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
Chief Financial Officer (CFO), Snapdeal</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
31.</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
Vijay Kumar Malhotra</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
President ,All India Council of Sports</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
32.</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
MK Surana</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
v CMD of Hindustan Petroleum Corp Ltd Note: will succeed Nishi Vasudeva.</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
33.</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
Shaktikanta Das</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
Central Board of Directors of the Reserve Bank of India (RBI)</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
34.</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
Najib Shah</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
Chairman of Central Board of Excise and Customs (CBEC), apex body on Indirect Taxes</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
35.</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
Samiran Chakraborty</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
Citibank, Chief Economist for India</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
36.</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
P K Gupta</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
Managing Director, State Bank of India</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
37.</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
Narendra Nayak</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
Managing Director(India), BlackBerry</div>
<div style="margin-bottom: 6px; margin-top: 6px;">
<br /></div>
<div style="margin-bottom: 6px; margin-top: 6px;">
<br /></div>
</div>
</div>
guruhttp://www.blogger.com/profile/01591375209227117455noreply@blogger.comtag:blogger.com,1999:blog-7850540640045081515.post-18573648083499858162015-12-27T09:02:00.004-08:002019-09-14T18:55:03.687-07:00G.K<div dir="ltr" style="text-align: left;" trbidi="on">
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4)<a href="http://mbaexamsprep.blogspot.in/2016/01/top-scheme-and-programme-launched-by.html">Top Scheme And Programme Launched By Narendra Modi Government</a></h1>
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5)<a href="http://mbaexamsprep.blogspot.in/2015/12/newly-appointed-person.html">Newly Appointed Person</a></h1>
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guruhttp://www.blogger.com/profile/01591375209227117455noreply@blogger.comtag:blogger.com,1999:blog-7850540640045081515.post-7584398922373106272015-12-16T00:36:00.000-08:002019-09-14T18:55:04.360-07:00Top MBA Colleges in India<div dir="ltr" style="text-align: left;" trbidi="on">
<table class="views-table cols-8" style="background-color: white; border-collapse: collapse; border-spacing: 0px; border: 1px solid rgb(204, 204, 204); color: black; font-family: robotoregular, sans-serif; font-size: 14px; line-height: 22px; margin-bottom: 1.5em; margin-top: 1.5em; padding: 5px; vertical-align: top; width: 700px;"><thead>
<tr><th class="views-field views-field-counter" style="border: 1px solid rgb(204, 204, 204); color: #111111; font-size: 15px; font-weight: normal; padding: 5px; vertical-align: top;">S.No.</th><th class="views-field views-field-title" style="border: 1px solid rgb(204, 204, 204); color: #111111; font-size: 15px; font-weight: normal; padding: 5px; vertical-align: top;">College Name</th><th class="views-field views-field-field-over-all-rank" style="border: 1px solid rgb(204, 204, 204); color: #111111; font-size: 15px; font-weight: normal; padding: 5px; vertical-align: top;">Overall Rank</th><th class="views-field views-field-field-overall-score" style="border: 1px solid rgb(204, 204, 204); color: #111111; font-size: 15px; font-weight: normal; padding: 5px; vertical-align: top;">Overall Score</th><th class="views-field views-field-field-select-national-rank" style="border: 1px solid rgb(204, 204, 204); color: #111111; font-size: 15px; font-weight: normal; padding: 5px; vertical-align: top;">National Rating</th><th class="views-field views-field-field-select-state-rank" style="border: 1px solid rgb(204, 204, 204); color: #111111; font-size: 15px; font-weight: normal; padding: 5px; vertical-align: top;">State Rating</th><th class="views-field views-field-php" style="border: 1px solid rgb(204, 204, 204); color: #111111; font-size: 15px; font-weight: normal; padding: 5px; vertical-align: top;">Cut off</th><th class="views-field views-field-field-college-level-fees" style="border: 1px solid rgb(204, 204, 204); color: #111111; font-size: 15px; font-weight: normal; padding: 5px; vertical-align: top;">Fees(In lakhs)</th></tr>
</thead><tbody>
<tr class="odd views-row-first" style="background-color: #eeeeee; border-bottom-color: rgb(204, 204, 204); border-bottom-style: solid; border-bottom-width: 1px; padding: 0.1em 0.6em;"><td class="views-field views-field-counter" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">1</td><td class="views-field views-field-title" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;"><a href="http://www.bschool.careers360.com/colleges/indian-institute-of-management-bangalore" style="color: #3b5998; font-size: 13px; outline: none; text-decoration: none;">Indian Institute of Management, Bangalore</a></td><td class="views-field views-field-field-over-all-rank" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">1</td><td class="views-field views-field-field-overall-score" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">433.20</td><td class="views-field views-field-field-select-national-rank" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">N.A</td><td class="views-field views-field-field-select-state-rank" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">AAAAA</td><td class="views-field views-field-php" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">CAT:85</td><td class="views-field views-field-field-college-level-fees" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">17.00</td></tr>
<tr class="even" style="border-bottom-color: rgb(204, 204, 204); border-bottom-style: solid; border-bottom-width: 1px; padding: 0.1em 0.6em;"><td class="views-field views-field-counter" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">2</td><td class="views-field views-field-title" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;"><a href="http://www.bschool.careers360.com/colleges/indian-institute-of-management-ahmedabad" style="color: #3b5998; font-size: 13px; outline: none; text-decoration: none;">Indian Institute of Management, Ahmedabad</a></td><td class="views-field views-field-field-over-all-rank" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">2</td><td class="views-field views-field-field-overall-score" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">420.80</td><td class="views-field views-field-field-select-national-rank" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">N.A</td><td class="views-field views-field-field-select-state-rank" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">AAAAA</td><td class="views-field views-field-php" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">CAT:90</td><td class="views-field views-field-field-college-level-fees" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">15.50</td></tr>
<tr class="odd" style="background-color: #eeeeee; border-bottom-color: rgb(204, 204, 204); border-bottom-style: solid; border-bottom-width: 1px; padding: 0.1em 0.6em;"><td class="views-field views-field-counter" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">3</td><td class="views-field views-field-title" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;"><a href="http://www.bschool.careers360.com/colleges/indian-institute-of-management-calcutta" style="color: #3b5998; font-size: 13px; outline: none; text-decoration: none;">Indian Institute of Management, Calcutta</a></td><td class="views-field views-field-field-over-all-rank" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">3</td><td class="views-field views-field-field-overall-score" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">410.72</td><td class="views-field views-field-field-select-national-rank" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">N.A</td><td class="views-field views-field-field-select-state-rank" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">AAAAA</td><td class="views-field views-field-php" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">CAT:90</td><td class="views-field views-field-field-college-level-fees" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">13.50</td></tr>
<tr class="even" style="border-bottom-color: rgb(204, 204, 204); border-bottom-style: solid; border-bottom-width: 1px; padding: 0.1em 0.6em;"><td class="views-field views-field-counter" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">4</td><td class="views-field views-field-title" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;"><a href="http://www.bschool.careers360.com/colleges/department-of-management-studies-indian-institute-of-technology-delhi" style="color: #3b5998; font-size: 13px; outline: none; text-decoration: none;">Department of Management Studies, Indian Institute of Technology, Delhi</a></td><td class="views-field views-field-field-over-all-rank" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">4</td><td class="views-field views-field-field-overall-score" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">382.04</td><td class="views-field views-field-field-select-national-rank" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">N.A</td><td class="views-field views-field-field-select-state-rank" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">AAAAA</td><td class="views-field views-field-php" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">CAT:95</td><td class="views-field views-field-field-college-level-fees" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">4.28</td></tr>
<tr class="odd" style="background-color: #eeeeee; border-bottom-color: rgb(204, 204, 204); border-bottom-style: solid; border-bottom-width: 1px; padding: 0.1em 0.6em;"><td class="views-field views-field-counter" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">5</td><td class="views-field views-field-title" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;"><a href="http://www.bschool.careers360.com/colleges/department-of-management-studies-indian-institute-of-technology-madras" style="color: #3b5998; font-size: 13px; outline: none; text-decoration: none;">Department of Management Studies, Indian Institute of Technology, Madras</a></td><td class="views-field views-field-field-over-all-rank" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">5</td><td class="views-field views-field-field-overall-score" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">372.66</td><td class="views-field views-field-field-select-national-rank" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">N.A</td><td class="views-field views-field-field-select-state-rank" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">AAAA+</td><td class="views-field views-field-php" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">CAT:95</td><td class="views-field views-field-field-college-level-fees" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">4.25</td></tr>
<tr class="even" style="border-bottom-color: rgb(204, 204, 204); border-bottom-style: solid; border-bottom-width: 1px; padding: 0.1em 0.6em;"><td class="views-field views-field-counter" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">6</td><td class="views-field views-field-title" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;"><a href="http://www.bschool.careers360.com/colleges/indian-institute-of-management-lucknow" style="color: #3b5998; font-size: 13px; outline: none; text-decoration: none;">Indian Institute of Management, Lucknow</a></td><td class="views-field views-field-field-over-all-rank" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">6</td><td class="views-field views-field-field-overall-score" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">357.33</td><td class="views-field views-field-field-select-national-rank" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">N.A</td><td class="views-field views-field-field-select-state-rank" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">AAAA+</td><td class="views-field views-field-php" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">CAT:90</td><td class="views-field views-field-field-college-level-fees" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">10.83</td></tr>
<tr class="odd" style="background-color: #eeeeee; border-bottom-color: rgb(204, 204, 204); border-bottom-style: solid; border-bottom-width: 1px; padding: 0.1em 0.6em;"><td class="views-field views-field-counter" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">7</td><td class="views-field views-field-title" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;"><a href="http://www.bschool.careers360.com/colleges/xlri-%E2%80%93-xavier-school-of-management-jamshedpur" style="color: #3b5998; font-size: 13px; outline: none; text-decoration: none;">XLRI – Xavier School of Management, Jamshedpur</a></td><td class="views-field views-field-field-over-all-rank" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">1</td><td class="views-field views-field-field-overall-score" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">352.57</td><td class="views-field views-field-field-select-national-rank" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">N.A</td><td class="views-field views-field-field-select-state-rank" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">AAAA+</td><td class="views-field views-field-php" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">XAT:92.25</td><td class="views-field views-field-field-college-level-fees" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">15.00</td></tr>
<tr class="even" style="border-bottom-color: rgb(204, 204, 204); border-bottom-style: solid; border-bottom-width: 1px; padding: 0.1em 0.6em;"><td class="views-field views-field-counter" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">8</td><td class="views-field views-field-title" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;"><a href="http://www.bschool.careers360.com/colleges/indian-institute-of-management-kozhikode" style="color: #3b5998; font-size: 13px; outline: none; text-decoration: none;">Indian Institute of Management, Kozhikode</a></td><td class="views-field views-field-field-over-all-rank" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">7</td><td class="views-field views-field-field-overall-score" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">348.03</td><td class="views-field views-field-field-select-national-rank" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">N.A</td><td class="views-field views-field-field-select-state-rank" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">AAAA+</td><td class="views-field views-field-php" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">CAT:90</td><td class="views-field views-field-field-college-level-fees" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">10.00</td></tr>
<tr class="odd" style="background-color: #eeeeee; border-bottom-color: rgb(204, 204, 204); border-bottom-style: solid; border-bottom-width: 1px; padding: 0.1em 0.6em;"><td class="views-field views-field-counter" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">9</td><td class="views-field views-field-title" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;"><a href="http://www.bschool.careers360.com/colleges/indian-institute-of-technology-roorkee" style="color: #3b5998; font-size: 13px; outline: none; text-decoration: none;">Indian Institute of Technology, Roorkee</a></td><td class="views-field views-field-field-over-all-rank" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">8</td><td class="views-field views-field-field-overall-score" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">346.68</td><td class="views-field views-field-field-select-national-rank" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">N.A</td><td class="views-field views-field-field-select-state-rank" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">AAAA+</td><td class="views-field views-field-php" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">CAT:90</td><td class="views-field views-field-field-college-level-fees" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">4.18</td></tr>
<tr class="even" style="border-bottom-color: rgb(204, 204, 204); border-bottom-style: solid; border-bottom-width: 1px; padding: 0.1em 0.6em;"><td class="views-field views-field-counter" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">10</td><td class="views-field views-field-title" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;"><a href="http://www.bschool.careers360.com/colleges/indian-institute-of-management-indore" style="color: #3b5998; font-size: 13px; outline: none; text-decoration: none;">Indian Institute of Management, Indore</a></td><td class="views-field views-field-field-over-all-rank" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">9</td><td class="views-field views-field-field-overall-score" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">345.12</td><td class="views-field views-field-field-select-national-rank" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">N.A</td><td class="views-field views-field-field-select-state-rank" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">AAAA+</td><td class="views-field views-field-php" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">CAT:90</td><td class="views-field views-field-field-college-level-fees" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">13.00</td></tr>
<tr class="odd" style="background-color: #eeeeee; border-bottom-color: rgb(204, 204, 204); border-bottom-style: solid; border-bottom-width: 1px; padding: 0.1em 0.6em;"><td class="views-field views-field-counter" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">11</td><td class="views-field views-field-title" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;"><a href="http://www.bschool.careers360.com/colleges/vinod-gupta-school-of-management-indian-institute-of-technology-kharagpur" style="color: #3b5998; font-size: 13px; outline: none; text-decoration: none;">Vinod Gupta School of Management, Indian Institute of Technology, Kharagpur</a></td><td class="views-field views-field-field-over-all-rank" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">10</td><td class="views-field views-field-field-overall-score" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">343.01</td><td class="views-field views-field-field-select-national-rank" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">N.A</td><td class="views-field views-field-field-select-state-rank" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">AAAA+</td><td class="views-field views-field-php" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">CAT:95.2</td><td class="views-field views-field-field-college-level-fees" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">6.72</td></tr>
<tr class="even" style="border-bottom-color: rgb(204, 204, 204); border-bottom-style: solid; border-bottom-width: 1px; padding: 0.1em 0.6em;"><td class="views-field views-field-counter" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">12</td><td class="views-field views-field-title" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;"><a href="http://www.bschool.careers360.com/colleges/faculty-of-management-studies-university-of-delhi-delhi" style="color: #3b5998; font-size: 13px; outline: none; text-decoration: none;">Faculty of Management Studies, University of Delhi, Delhi</a></td><td class="views-field views-field-field-over-all-rank" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">11</td><td class="views-field views-field-field-overall-score" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">341.01</td><td class="views-field views-field-field-select-national-rank" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">N.A</td><td class="views-field views-field-field-select-state-rank" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">AAAA+</td><td class="views-field views-field-php" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">CAT:95</td><td class="views-field views-field-field-college-level-fees" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">1.00</td></tr>
<tr class="odd" style="background-color: #eeeeee; border-bottom-color: rgb(204, 204, 204); border-bottom-style: solid; border-bottom-width: 1px; padding: 0.1em 0.6em;"><td class="views-field views-field-counter" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">13</td><td class="views-field views-field-title" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;"><a href="http://www.bschool.careers360.com/colleges/national-institute-of-industrial-engineering-mumbai" style="color: #3b5998; font-size: 13px; outline: none; text-decoration: none;">National Institute of Industrial Engineering, Mumbai</a></td><td class="views-field views-field-field-over-all-rank" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">12</td><td class="views-field views-field-field-overall-score" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">338.88</td><td class="views-field views-field-field-select-national-rank" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">N.A</td><td class="views-field views-field-field-select-state-rank" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">AAAA+</td><td class="views-field views-field-php" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">CAT:97</td><td class="views-field views-field-field-college-level-fees" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">8.10</td></tr>
<tr class="even" style="border-bottom-color: rgb(204, 204, 204); border-bottom-style: solid; border-bottom-width: 1px; padding: 0.1em 0.6em;"><td class="views-field views-field-counter" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">14</td><td class="views-field views-field-title" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;"><a href="http://www.bschool.careers360.com/colleges/shailesh-j-mehta-school-of-management-indian-institute-of-technology-bombay" style="color: #3b5998; font-size: 13px; outline: none; text-decoration: none;">Shailesh J Mehta School of Management, Indian Institute of Technology, Bombay</a></td><td class="views-field views-field-field-over-all-rank" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">13</td><td class="views-field views-field-field-overall-score" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">335.85</td><td class="views-field views-field-field-select-national-rank" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">N.A</td><td class="views-field views-field-field-select-state-rank" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">AAAA+</td><td class="views-field views-field-php" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">CAT:95</td><td class="views-field views-field-field-college-level-fees" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">8.98</td></tr>
<tr class="odd" style="background-color: #eeeeee; border-bottom-color: rgb(204, 204, 204); border-bottom-style: solid; border-bottom-width: 1px; padding: 0.1em 0.6em;"><td class="views-field views-field-counter" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">15</td><td class="views-field views-field-title" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;"><a href="http://www.bschool.careers360.com/colleges/management-development-institute-gurgaon" style="color: #3b5998; font-size: 13px; outline: none; text-decoration: none;">Management Development Institute, Gurgaon</a></td><td class="views-field views-field-field-over-all-rank" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">2</td><td class="views-field views-field-field-overall-score" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">332.49</td><td class="views-field views-field-field-select-national-rank" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">N.A</td><td class="views-field views-field-field-select-state-rank" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">AAAA+</td><td class="views-field views-field-php" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">CAT:92</td><td class="views-field views-field-field-college-level-fees" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">15.96</td></tr>
<tr class="even" style="border-bottom-color: rgb(204, 204, 204); border-bottom-style: solid; border-bottom-width: 1px; padding: 0.1em 0.6em;"><td class="views-field views-field-counter" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">16</td><td class="views-field views-field-title" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;"><a href="http://www.bschool.careers360.com/colleges/sp-jain-institute-of-management-and-research-mumbai" style="color: #3b5998; font-size: 13px; outline: none; text-decoration: none;">SP Jain Institute of Management and Research, Mumbai</a></td><td class="views-field views-field-field-over-all-rank" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">3</td><td class="views-field views-field-field-overall-score" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">326.13</td><td class="views-field views-field-field-select-national-rank" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">N.A</td><td class="views-field views-field-field-select-state-rank" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">AAAA+</td><td class="views-field views-field-php" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">CAT:85<br />XAT:85</td><td class="views-field views-field-field-college-level-fees" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">13.00</td></tr>
<tr class="odd" style="background-color: #eeeeee; border-bottom-color: rgb(204, 204, 204); border-bottom-style: solid; border-bottom-width: 1px; padding: 0.1em 0.6em;"><td class="views-field views-field-counter" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">17</td><td class="views-field views-field-title" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;"><a href="http://www.bschool.careers360.com/colleges/indian-institute-of-technology-kanpur" style="color: #3b5998; font-size: 13px; outline: none; text-decoration: none;">Indian Institute of Technology, Kanpur</a></td><td class="views-field views-field-field-over-all-rank" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">14</td><td class="views-field views-field-field-overall-score" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">320.07</td><td class="views-field views-field-field-select-national-rank" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">N.A</td><td class="views-field views-field-field-select-state-rank" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">AAAA+</td><td class="views-field views-field-php" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">CAT:95</td><td class="views-field views-field-field-college-level-fees" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">4.80</td></tr>
<tr class="even" style="border-bottom-color: rgb(204, 204, 204); border-bottom-style: solid; border-bottom-width: 1px; padding: 0.1em 0.6em;"><td class="views-field views-field-counter" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">18</td><td class="views-field views-field-title" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;"><a href="http://www.bschool.careers360.com/colleges/rajiv-gandhi-indian-institute-of-management-shillong" style="color: #3b5998; font-size: 13px; outline: none; text-decoration: none;">Rajiv Gandhi Indian Institute of Management, Shillong</a></td><td class="views-field views-field-field-over-all-rank" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">15</td><td class="views-field views-field-field-overall-score" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">312.95</td><td class="views-field views-field-field-select-national-rank" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">N.A</td><td class="views-field views-field-field-select-state-rank" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">AAAA+</td><td class="views-field views-field-php" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">CAT:90</td><td class="views-field views-field-field-college-level-fees" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">12.68</td></tr>
<tr class="odd" style="background-color: #eeeeee; border-bottom-color: rgb(204, 204, 204); border-bottom-style: solid; border-bottom-width: 1px; padding: 0.1em 0.6em;"><td class="views-field views-field-counter" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">19</td><td class="views-field views-field-title" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;"><a href="http://www.bschool.careers360.com/colleges/indian-institute-of-foreign-trade-delhi" style="color: #3b5998; font-size: 13px; outline: none; text-decoration: none;">Indian Institute of Foreign Trade, Delhi</a></td><td class="views-field views-field-field-over-all-rank" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">16</td><td class="views-field views-field-field-overall-score" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">309.71</td><td class="views-field views-field-field-select-national-rank" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">N.A</td><td class="views-field views-field-field-select-state-rank" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">AAAA+</td><td class="views-field views-field-php" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">N.A.</td><td class="views-field views-field-field-college-level-fees" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">13.50</td></tr>
<tr class="even" style="border-bottom-color: rgb(204, 204, 204); border-bottom-style: solid; border-bottom-width: 1px; padding: 0.1em 0.6em;"><td class="views-field views-field-counter" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">20</td><td class="views-field views-field-title" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;"><a href="http://www.bschool.careers360.com/node/127334" style="color: #3b5998; font-size: 13px; outline: none; text-decoration: none;">Department of Management Studies, Indian Institute of Science, Bangalore</a></td><td class="views-field views-field-field-over-all-rank" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">17</td><td class="views-field views-field-field-overall-score" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">304.74</td><td class="views-field views-field-field-select-national-rank" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">N.A</td><td class="views-field views-field-field-select-state-rank" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">AAAA+</td><td class="views-field views-field-php" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">N.A.</td><td class="views-field views-field-field-college-level-fees" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">N.A.</td></tr>
<tr class="odd" style="background-color: #eeeeee; border-bottom-color: rgb(204, 204, 204); border-bottom-style: solid; border-bottom-width: 1px; padding: 0.1em 0.6em;"><td class="views-field views-field-counter" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">21</td><td class="views-field views-field-title" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;"><a href="http://www.bschool.careers360.com/colleges/indian-institute-of-management-rohtak" style="color: #3b5998; font-size: 13px; outline: none; text-decoration: none;">Indian Institute of Management, Rohtak</a></td><td class="views-field views-field-field-over-all-rank" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">18</td><td class="views-field views-field-field-overall-score" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">303.83</td><td class="views-field views-field-field-select-national-rank" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">N.A</td><td class="views-field views-field-field-select-state-rank" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">AAAA+</td><td class="views-field views-field-php" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">CAT:85</td><td class="views-field views-field-field-college-level-fees" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">9.00</td></tr>
<tr class="even" style="border-bottom-color: rgb(204, 204, 204); border-bottom-style: solid; border-bottom-width: 1px; padding: 0.1em 0.6em;"><td class="views-field views-field-counter" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">22</td><td class="views-field views-field-title" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;"><a href="http://www.bschool.careers360.com/colleges/jamnalal-bajaj-institute-of-management-studies-mumbai" style="color: #3b5998; font-size: 13px; outline: none; text-decoration: none;">Jamnalal Bajaj Institute of Management Studies, Mumbai</a></td><td class="views-field views-field-field-over-all-rank" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">19</td><td class="views-field views-field-field-overall-score" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">297.35</td><td class="views-field views-field-field-select-national-rank" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">N.A</td><td class="views-field views-field-field-select-state-rank" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">AAAA</td><td class="views-field views-field-php" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">CMAT:350</td><td class="views-field views-field-field-college-level-fees" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">2.00</td></tr>
<tr class="odd" style="background-color: #eeeeee; border-bottom-color: rgb(204, 204, 204); border-bottom-style: solid; border-bottom-width: 1px; padding: 0.1em 0.6em;"><td class="views-field views-field-counter" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">23</td><td class="views-field views-field-title" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;"><a href="http://www.bschool.careers360.com/colleges/indian-institute-of-management-raipur" style="color: #3b5998; font-size: 13px; outline: none; text-decoration: none;">Indian Institute of Management, Raipur</a></td><td class="views-field views-field-field-over-all-rank" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">20</td><td class="views-field views-field-field-overall-score" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">297.11</td><td class="views-field views-field-field-select-national-rank" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">N.A</td><td class="views-field views-field-field-select-state-rank" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">AAAA</td><td class="views-field views-field-php" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">CAT:90</td><td class="views-field views-field-field-college-level-fees" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">9.78</td></tr>
<tr class="even" style="border-bottom-color: rgb(204, 204, 204); border-bottom-style: solid; border-bottom-width: 1px; padding: 0.1em 0.6em;"><td class="views-field views-field-counter" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">24</td><td class="views-field views-field-title" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;"><a href="http://www.bschool.careers360.com/colleges/ta-pai-management-institute-manipal" style="color: #3b5998; font-size: 13px; outline: none; text-decoration: none;">TA Pai Management Institute, Manipal</a></td><td class="views-field views-field-field-over-all-rank" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">4</td><td class="views-field views-field-field-overall-score" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">292.95</td><td class="views-field views-field-field-select-national-rank" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">N.A</td><td class="views-field views-field-field-select-state-rank" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">AAAA</td><td class="views-field views-field-php" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">CAT:86<br />XAT:86<br />MAT:700</td><td class="views-field views-field-field-college-level-fees" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">13.50</td></tr>
<tr class="odd" style="background-color: #eeeeee; border-bottom-color: rgb(204, 204, 204); border-bottom-style: solid; border-bottom-width: 1px; padding: 0.1em 0.6em;"><td class="views-field views-field-counter" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">25</td><td class="views-field views-field-title" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;"><a href="http://www.bschool.careers360.com/colleges/great-lakes-institute-of-management-chennai" style="color: #3b5998; font-size: 13px; outline: none; text-decoration: none;">Great Lakes Institute of Management, Chennai</a></td><td class="views-field views-field-field-over-all-rank" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">5</td><td class="views-field views-field-field-overall-score" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">292.66</td><td class="views-field views-field-field-select-national-rank" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">N.A</td><td class="views-field views-field-field-select-state-rank" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">AAAA</td><td class="views-field views-field-php" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">CAT:90<br />CMAT:280<br />XAT:90</td><td class="views-field views-field-field-college-level-fees" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">16.50</td></tr>
<tr class="even" style="border-bottom-color: rgb(204, 204, 204); border-bottom-style: solid; border-bottom-width: 1px; padding: 0.1em 0.6em;"><td class="views-field views-field-counter" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">26</td><td class="views-field views-field-title" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;"><a href="http://www.bschool.careers360.com/colleges/institute-of-management-technology-ghaziabad" style="color: #3b5998; font-size: 13px; outline: none; text-decoration: none;">Institute of Management Technology, Ghaziabad</a></td><td class="views-field views-field-field-over-all-rank" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">6</td><td class="views-field views-field-field-overall-score" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">291.24</td><td class="views-field views-field-field-select-national-rank" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">N.A</td><td class="views-field views-field-field-select-state-rank" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">AAAA</td><td class="views-field views-field-php" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">CAT:90<br />XAT:90</td><td class="views-field views-field-field-college-level-fees" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">15.00</td></tr>
<tr class="odd" style="background-color: #eeeeee; border-bottom-color: rgb(204, 204, 204); border-bottom-style: solid; border-bottom-width: 1px; padding: 0.1em 0.6em;"><td class="views-field views-field-counter" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">27</td><td class="views-field views-field-title" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;"><a href="http://www.bschool.careers360.com/colleges/indian-institute-of-management-tiruchirapalli" style="color: #3b5998; font-size: 13px; outline: none; text-decoration: none;">Indian Institute of Management, Tiruchirapalli</a></td><td class="views-field views-field-field-over-all-rank" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">21</td><td class="views-field views-field-field-overall-score" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">288.01</td><td class="views-field views-field-field-select-national-rank" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">N.A</td><td class="views-field views-field-field-select-state-rank" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">AAAA</td><td class="views-field views-field-php" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">CAT:94</td><td class="views-field views-field-field-college-level-fees" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">10.00</td></tr>
<tr class="even" style="border-bottom-color: rgb(204, 204, 204); border-bottom-style: solid; border-bottom-width: 1px; padding: 0.1em 0.6em;"><td class="views-field views-field-counter" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">28</td><td class="views-field views-field-title" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;"><a href="http://www.bschool.careers360.com/colleges/indian-institute-of-management-ranchi" style="color: #3b5998; font-size: 13px; outline: none; text-decoration: none;">Indian Institute of Management, Ranchi</a></td><td class="views-field views-field-field-over-all-rank" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">22</td><td class="views-field views-field-field-overall-score" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">283.91</td><td class="views-field views-field-field-select-national-rank" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">N.A</td><td class="views-field views-field-field-select-state-rank" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">AAAA</td><td class="views-field views-field-php" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">CAT:90</td><td class="views-field views-field-field-college-level-fees" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">N.A.</td></tr>
<tr class="odd" style="background-color: #eeeeee; border-bottom-color: rgb(204, 204, 204); border-bottom-style: solid; border-bottom-width: 1px; padding: 0.1em 0.6em;"><td class="views-field views-field-counter" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">29</td><td class="views-field views-field-title" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;"><a href="http://www.bschool.careers360.com/colleges/xavier-institute-of-management-bhubaneswar" style="color: #3b5998; font-size: 13px; outline: none; text-decoration: none;">Xavier Institute of Management, Bhubaneswar</a></td><td class="views-field views-field-field-over-all-rank" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">7</td><td class="views-field views-field-field-overall-score" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">281.90</td><td class="views-field views-field-field-select-national-rank" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">N.A</td><td class="views-field views-field-field-select-state-rank" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">AAAA</td><td class="views-field views-field-php" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">CAT:94<br />XAT:95</td><td class="views-field views-field-field-college-level-fees" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">12.00</td></tr>
<tr class="even views-row-last" style="border-bottom-color: rgb(204, 204, 204); border-bottom-style: solid; border-bottom-width: 1px; padding: 0.1em 0.6em;"><td class="views-field views-field-counter" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">30</td><td class="views-field views-field-title" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;"><a href="http://www.bschool.careers360.com/colleges/institute-for-financial-management-and-research-changambakkam" style="color: #3b5998; font-size: 13px; outline: none; text-decoration: none;">Institute for Financial Management and Research, Changambakkam</a></td><td class="views-field views-field-field-over-all-rank" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">8</td><td class="views-field views-field-field-overall-score" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">281.34</td><td class="views-field views-field-field-select-national-rank" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">N.A</td><td class="views-field views-field-field-select-state-rank" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">AAAA</td><td class="views-field views-field-php" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">CAT:87.5<br />XAT:85</td><td class="views-field views-field-field-college-level-fees" style="border: 1px solid rgb(204, 204, 204); padding: 5px; vertical-align: top;">10.00</td></tr>
</tbody></table>
</div>
guruhttp://www.blogger.com/profile/01591375209227117455noreply@blogger.comtag:blogger.com,1999:blog-7850540640045081515.post-279153389815148422015-11-29T22:45:00.001-08:002019-09-14T18:55:55.151-07:00NMAT Window 4 Questions <div dir="ltr" style="text-align: left;" trbidi="on">
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<u><span style="font-family: Arial, sans-serif; font-size: 14pt;">Quant :-</span></u><span style="font-family: 'Times New Roman', serif; font-size: 12pt;"><o:p></o:p></span></div>
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<span style="background: rgb(246, 247, 248); font-family: Arial, sans-serif; font-size: 12pt;">1)
Find trailing zeros</span><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
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<span style="background: rgb(246, 247, 248); font-family: Arial, sans-serif; font-size: 12pt;">In
100! In base 16- ans is 24olcapacity of tank??</span><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
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<span style="background: rgb(246, 247, 248); font-family: Arial, sans-serif; font-size: 12pt;">half
profit according their investments ..total profit 30,000 rs..what is B 's
share??</span><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
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<span style="background: rgb(246, 247, 248); font-family: Arial, sans-serif; font-size: 12pt;">Q3)
(10^0+10^1+10^2+..)mod 8</span><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
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<span style="background: rgb(246, 247, 248); font-family: Arial, sans-serif; font-size: 12pt;">Q4)
</span><span style="background: white; font-family: Arial, sans-serif; font-size: 10.5pt;">There
is one tank having hole @ bottom ...hole can empty tank in 6 hours..inlet valve
ws also there..it is filling tank..when both together open,tank ll empty in 30
hours...if inlet valve is filling tank by 3 liter per minute then what is the
capacity of tank??</span><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
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<span style="background: white; font-family: Arial, sans-serif; font-size: 10.5pt;">Q5)
Ratio of height of cone to radius height was given 12cm..& upto 9 cm felt
by chocolate ice cream given & remaining part was felt by vanila ice cream</span><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
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<span style="background: rgb(246, 247, 248); font-family: Arial, sans-serif; font-size: 12pt;">Q2)
a,b,c in DCvesting in such a way there investments form ap..nw they decided to
share their profit equally of total profit @ year end &nk having hrs..inlet
valve ws also there..it is filling tank..when both together open,tank ll empty
in 30 hours...if inlet valve is filling tank by 3 liter per minute then what is
the </span><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
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<br /></div>
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<span style="font-family: Arial, sans-serif; font-size: 12pt;">logx=log 24- log 2</span><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
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<span style="font-family: Arial, sans-serif; font-size: 12pt;">A can do a work in 6
hrs, B can do the same work in 8 hours. A,B,C together take 3 hours to do the
work. <br />
If they get 3000 from the work, how much should be C's share?<br />
<br />
How many 4 digit numbers can be formed using 0,1,2,3,4 which are divisible by
12. (Not repeated)</span><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
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<br /></div>
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<span style="font-family: Arial, sans-serif; font-size: 12pt;">4 intergers are in
proportion. the sum of the extremes is 7 and the ration of 1st 2 terms is 1:2.
if the sum of the squares of the numbers is 50. find the avrg of the numbers.??</span><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
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<span style="font-family: Arial, sans-serif; font-size: 12pt;">1. A borrows a sum of
10000 from B at q at a rate of 8% for 3yrs and lends the same amount to C
compounded annually at the same rate for 3 yrs. What amount does A gain at the
end of 3 yrs?<br />
2. Some number was given in binary format and they asked what's the value as
base 3.<br />
3. 5 eqs were given and they asked which eq has real roots<br />
4. Again 5 eqs and asked which has slope of 1/2 and some distance from a point.
( I don't rem the distance and point)</span><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
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<span style="font-family: Arial, sans-serif; font-size: 12pt;">Q8) the CP of 3 kinds of
sugar are 5,6,6.80. in what proportion should they be mixed so that the price
of the mixture be 6.5 per bag ?`<br />
1:3:10<br />
<br />
Q9) A man buys milk at rs5 a L and after adding water sells it at rs 6 a L,
thereby making a profit of 33 ⅓% . Find the proportion of water to milk in the
mixture.</span><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
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<br /></div>
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<span style="font-family: Arial, sans-serif; font-size: 12pt;">today's paper...the
probability that RCB will win is 3/5 and the probability that csk will win is
2/7 find the probability that non of the 2 teams win the tournament</span><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
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<span style="font-family: Arial, sans-serif; font-size: 12pt;">Q) total investment
60laks, 4 partners! Partner 1 invested 1/2 of others, partner 2 invested 1/3 of
others partner 3 invested 1/4 of others! How much did partner 4 invest<br />
A)1300000<br />
<br />
1*2*3*4...<br />
M has 58 zeroes<br />
Total no of factors of m</span><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
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<!--[if !supportLists]--><span style="font-family: Arial, sans-serif; font-size: 12pt;">A.<span style="font-family: 'Times New Roman'; font-size: 7pt; font-stretch: normal;">
</span></span><!--[endif]--><span style="font-family: Arial, sans-serif; font-size: 12pt;">20<br />
Q12) 1^2-3^2+5^2-7^2+............99^2-101^2<o:p></o:p></span></div>
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<br /></div>
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<span style="font-family: Arial, sans-serif; font-size: 12pt;">1. Raj writes numbers
from 1 to 200 on a sheet of paper how many are not divisible either by 3 or 7 .</span><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
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<span style="font-family: Arial, sans-serif; font-size: 12pt;">2 . A can do a work in 6
hours , B can do a work in 8 hours. A,B,C can do a work in 3 hours. In how many
hours can C alone do the work.</span><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
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<span style="font-family: Arial, sans-serif; font-size: 12pt;">3. 11 numbers are
written in series such that average of all the numbers is equal to the 6th
digit . Sum of first 6 is 435 and sum of last 6 is 335 . Find the 6th number .
( not sure about values of sum )</span><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
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<span style="font-family: Arial, sans-serif; font-size: 12pt;">4. A dice is thrown
probability that you get an even number after even number of throws.</span><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
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<span style="font-family: Arial, sans-serif; font-size: 12pt;">5. Probability of
success of an event is 1 in 10 what's the probability of 4 success in 5
attempts .</span><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
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<span style="font-family: Arial, sans-serif; font-size: 12pt;">6. Result of (remainder
of 9^100/8) - ( remainder of 7^99/8)</span><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
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<span style="font-family: Arial, sans-serif; font-size: 12pt;">7. Number of times digit
5 comes from 1 to 1000</span><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
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<span style="font-family: Arial, sans-serif; font-size: 12pt;">8. How many ways can you
arrange a group of 4,3,3 in a row such that all people ( of a group) stay
together .</span><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
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<span style="font-family: Arial, sans-serif; font-size: 12pt;">9. Sn =
1+1/6+1/6^2+1/6^3 .... 1/6^n-1)Find value of n for which 6/5 - Sn <= 1/100</span><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
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<span style="font-family: Arial, sans-serif; font-size: 12pt;">10. If a person takes
10000 at 8% interest rate for 3 years (SI) and lends the same at 7% compounded
annually for 3 years what's his profit/loss %.</span><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
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<span style="font-family: Arial, sans-serif; font-size: 12pt;">11. If A= 1+
1/2^2+1/3^2+1/4^2....</span><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
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<span style="font-family: Arial, sans-serif; font-size: 12pt;">B= 1+1/3^2+5^2+7^2 ....</span><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Arial, sans-serif; font-size: 12pt;">What's the relation
between A and B .</span><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Arial, sans-serif; font-size: 12pt;">12. In how many ways can
5 chocolates be divided among 3 children's if everyone gets at least 1 .<br />
<br />
<!--[if !supportLineBreakNewLine]--><br />
<!--[endif]--></span><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
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<br /></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<u><span style="font-family: Arial, sans-serif; font-size: 14pt;">Verbal:</span></u><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Arial, sans-serif;">1) LACONIC (antonym) - verbose<br />
2) BRIGHT (antonym) - dreary<br />
3) EXACERBATE - to make worse (pain/anger)<br />
4) Occidental - west <br />
5) Vindicate - clear of blame/guilt<br />
6) Malinger-pretend to be ill in order to escape duty or work.<br />
7) Guile- craft, cunning, slyness<br />
8) Salacious- lustful</span><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Arial, sans-serif;">9)Vindicate(synonym)-clear (someone) of blame or
suspicion.</span><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Arial, sans-serif;">10)Appease(synonym)-pacify or placate</span><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Arial, sans-serif;">11)Sartorial(syn)-relating to tailoring,
clothes, or style of dress.</span><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="background: white; font-family: Arial, sans-serif; font-size: 10.5pt;">12)
Vacuous- having or showing a lack of thought or intelligenc</span><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
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<span style="background: white; font-family: Arial, sans-serif; font-size: 10.5pt;">13)
Puerile-childishly silly and immature</span><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
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<span style="background: white; font-family: Arial, sans-serif; font-size: 10.5pt;">14) Insipid-tasteless</span><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
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<span style="background: white; font-family: Arial, sans-serif; font-size: 10.5pt;">15)
Bliss-perfect happiness/great joy</span><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
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<span style="background: white; font-family: Arial, sans-serif; font-size: 10.5pt;">16)
cor extravagant in language</span><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
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<span style="background: white; font-family: Arial, sans-serif; font-size: 10.5pt;"><br />
1) Accomplice<br />
2) Appease<br />
3) Bliss<br />
4) Bright<br />
5) Cantankerous<br />
6) Capricious<br />
7) Censure<br />
8) Clandestine<br />
9) Condemn<br />
10) Denial<br />
11) Exacerbate<br />
12) Execrate <br />
13) Fallacious<br />
14) Gargantuan<br />
15) Grandiloquent<br />
16) Guile<br />
17) Insipid<br />
18) Laconic<br />
19) Malevolent<br />
20) Malign<br />
21) Malinger<br />
22) Obloquy<br />
23) Occidental<br />
24) Onerous<br />
25) Opprobrium<br />
26) Puerile<br />
27) Revile<br />
28) Salacious<br />
29) Sartorial<br />
30) Surreptitious<br />
31) Tranquillity<br />
32) Vacuous<br />
33) Veracity<br />
34) Vilify<br />
35) Vindicate<br />
That's it...<br />
Share if you are having other words...</span><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
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guruhttp://www.blogger.com/profile/01591375209227117455noreply@blogger.comtag:blogger.com,1999:blog-7850540640045081515.post-16479569723680319012015-11-22T06:18:00.001-08:002019-09-14T18:55:55.115-07:00RIVERS IN INDIA <div dir="ltr" style="text-align: left;" trbidi="on">
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Important rivers of India are Ganga, Yamuna, Brahmaputra, Indus, Mahanadi, Godavari, Krishna, Kauery, Narmada and Tapti. A brief description of these rivers is given below:</div>
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<b style="border: 0px; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Ganga:</b> Ganga is the most holy river according to Hindu. It has started from the Gangotri in the Himalayas and poured into the Bay of Bengal by travelling of around 2525 km. It is ranked 3<span style="border: 0px; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">rd</span> largest river in the world. There are a number of cities had been developed across the banks of Ganga like Pataliputra, Kashi, Allahabad, Varanasi, Kolkata etc. It had also created the World’s largest delta in West Bengal named as Sundarban delta.</div>
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<b style="border: 0px; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Yamuna:</b> Yamuna River had originated from Yamunotri Glacier in the Himalayas, then travels across several states and merged into the Ganga at Triveni, Allahabad. Its total length is 1376 km. Yamuna River’s water contributed nearly 70% of Delhi’s water supply. The Taj Mahal is situated on the banks of Yamuna River.</div>
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<b style="border: 0px; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Brahmaputra:</b> Brahmaputra River is originated from Angsi Glacier, the northen Himalayas in Tibet, then entered into the Arunachal Pradesh to Assam and then merges with the Padma river in Bangladesh. Its length is around 2900 km and plays an important role in the irrigation and transportation. It emptied into the Bay of Bengal.</div>
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<b style="border: 0px; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Indus River:</b> The Indus River is a historically famous in Asia. It originated from the Tibetan Plateau, and then flows through the Ladakh then entered into Pakistan and finally merge into the Arabian Sea after travelling a distance of 3180 km.</div>
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<b style="border: 0px; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Mahanadi River:</b> Mahanadi is a major river in the state of Chhattisgarh and Odisha. Mahanadi’s water is used in the irrigation and drinking purpose, it is also called the ruin of Orissa due its devastating floods over the years but till Hirakud Dam was constructed. Its total length is 858 km.</div>
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<b style="border: 0px; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Godavari:</b> Godavari is the longest river in the southern India and 2<span style="border: 0px; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">nd</span> largest in India after Ganga. It is originated from Maharashtra and flows through Andhra Pradesh, then merges into the Bay of Bengal after travelling a distance of 1465 km.</div>
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<b style="border: 0px; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Krishna: </b>The 3<span style="border: 0px; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">rd</span> longest river in India after Ganga and Godavari, Krishna River which is originated from Mahabaleswar in Maharashtra and flows through the state of Karnataka, Andhra Pradesh and finally merges into the Bay of Bengal by travelling a distance of 1400 km.</div>
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<b style="border: 0px; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Kaveri: </b>The Kaveri River is one of the large rivers in India which is originated from Talakaveri in the Western Ghats of Karnataka and flows through the states of Karnataka & Tamilnadu, finally merges into Bay of Bengal by travelling a distance of 765 km.</div>
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<b style="border: 0px; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Narmada: </b>Narmada River is the 5<span style="border: 0px; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">th</span> Longest in the Indian sub-continent. It is also called the Life line of Madhya pradesh due to its huge contributions. Narmada River is originated from Narmada Kund, Amarkantak in Madhya Pradesh and merges into the Arabian Sea near Gujarat after travelling a distance of 1312 km.</div>
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<b style="border: 0px; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Tapti: </b>Tapti River is one of the major rivers in Central India. Its total length is around 724 km which flows through the states of Gujarat, Madhya Pradesh and Maharashtra and finally merges into the Gulf of Cambay of Arabian Sea near Gujarat.</div>
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Suggested External Readings:</div>
<div style="background-color: white; border: 0px; color: #666666; font-family: arial, sans-serif; font-size: 14px; font-stretch: inherit; line-height: 22px; margin-bottom: 20px; padding: 0px; vertical-align: baseline;">
1. <a href="http://en.wikipedia.org/wiki/Major_rivers_of_India" style="-webkit-tap-highlight-color: rgb(255, 94, 153); border: 0px; color: #526f94; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; padding: 0px; text-decoration: none; transition: all 0.2s linear; vertical-align: baseline;" target="_blank" title="Major Rivers of India in Wikipedia">Major Rivers of India in Wikipedia</a></div>
<div style="background-color: white; border: 0px; color: #666666; font-family: arial, sans-serif; font-size: 14px; font-stretch: inherit; line-height: 22px; margin-bottom: 20px; padding: 0px; vertical-align: baseline;">
2. <a href="http://en.wikipedia.org/wiki/List_of_rivers_of_India" style="-webkit-tap-highlight-color: rgb(255, 94, 153); border: 0px; color: #526f94; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; padding: 0px; text-decoration: none; transition: all 0.2s linear; vertical-align: baseline;" target="_blank" title="List of Rivers in India at Wikipedia">List of Rivers in India (Wikipedia)</a></div>
</div>
guruhttp://www.blogger.com/profile/01591375209227117455noreply@blogger.comtag:blogger.com,1999:blog-7850540640045081515.post-32087590402313746702015-11-19T08:07:00.002-08:002019-09-14T18:55:03.966-07:00Basic Formulae<div dir="ltr" style="text-align: left;" trbidi="on">
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<b><u><span style="color: #215d94; font-family: "trebuchet ms" , "sans-serif"; font-size: 15.5pt; font-variant: small-caps; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";"><a href="http://mbaexamsprep.blogspot.in/2015/11/basic-formulae.html">BasicFormulae</a><o:p></o:p></span></u></b></div>
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<span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;"><span style="font-size: 9.5pt; line-height: 14.4pt;">1. (a+b)</span><span style="font-size: 6.5pt; line-height: 14.4pt;">2</span><span style="font-size: 9.5pt; line-height: 14.4pt;">=a</span><span style="font-size: 6.5pt; line-height: 14.4pt;">2</span><span style="font-size: 9.5pt; line-height: 14.4pt;">+b</span><span style="font-size: 6.5pt; line-height: 14.4pt;">2</span><span style="font-size: 9.5pt; line-height: 14.4pt;">+2ab </span></span></div>
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<span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;"><span style="font-size: 9.5pt; line-height: 14.4pt;">2. (a−b)</span><span style="font-size: 6.5pt; line-height: 14.4pt;">2</span><span style="font-size: 9.5pt; line-height: 14.4pt;">=a</span><span style="font-size: 6.5pt; line-height: 14.4pt;">2</span><span style="font-size: 9.5pt; line-height: 14.4pt;">+b</span><span style="font-size: 6.5pt; line-height: 14.4pt;">2</span><span style="font-size: 9.5pt; line-height: 14.4pt;">−2ab</span></span></div>
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</span><span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;"><div style="text-align: left;">
<span style="font-size: 9.5pt; line-height: 14.4pt;">3. (a+b)</span><span style="font-size: 6.5pt; line-height: 14.4pt;">2</span><span style="font-size: 9.5pt; line-height: 14.4pt;">−(a−b)</span><span style="font-size: 6.5pt; line-height: 14.4pt;">2</span><span style="font-size: 9.5pt; line-height: 14.4pt;">=4ab </span></div>
</span><span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;"><div style="text-align: left;">
<span style="font-size: 9.5pt; line-height: 14.4pt;">4. (a+b)</span><span style="font-size: 6.5pt; line-height: 14.4pt;">2</span><span style="font-size: 9.5pt; line-height: 14.4pt;">+(a−b)</span><span style="font-size: 6.5pt; line-height: 14.4pt;">2</span><span style="font-size: 9.5pt; line-height: 14.4pt;">=2(a</span><span style="font-size: 6.5pt; line-height: 14.4pt;">2</span><span style="font-size: 9.5pt; line-height: 14.4pt;">+b</span><span style="font-size: 6.5pt; line-height: 14.4pt;">2</span><span style="font-size: 9.5pt; line-height: 14.4pt;">)</span></div>
</span><span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;"><div style="text-align: left;">
<span style="font-size: 9.5pt; line-height: 14.4pt;">5. (a</span><span style="font-size: 6.5pt; line-height: 14.4pt;">2</span><span style="font-size: 9.5pt; line-height: 14.4pt;">–b</span><span style="font-size: 6.5pt; line-height: 14.4pt;">2</span><span style="font-size: 9.5pt; line-height: 14.4pt;">)=(a+b)(a−b) </span></div>
</span><span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;"><div style="text-align: left;">
<span style="font-size: 9.5pt; line-height: 14.4pt;">6. (a+b+c)</span><span style="font-size: 6.5pt; line-height: 14.4pt;">2</span><span style="font-size: 9.5pt; line-height: 14.4pt;">=a</span><span style="font-size: 6.5pt; line-height: 14.4pt;">2</span><span style="font-size: 9.5pt; line-height: 14.4pt;">+b</span><span style="font-size: 6.5pt; line-height: 14.4pt;">2</span><span style="font-size: 9.5pt; line-height: 14.4pt;">+c</span><span style="font-size: 6.5pt; line-height: 14.4pt;">2</span><span style="font-size: 9.5pt; line-height: 14.4pt;">+2(ab+bc+ca)</span></div>
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<span style="font-size: 9.5pt; line-height: 14.4pt;">7. (a</span><span style="font-size: 6.5pt; line-height: 14.4pt;">3</span><span style="font-size: 9.5pt; line-height: 14.4pt;">+b</span><span style="font-size: 6.5pt; line-height: 14.4pt;">3</span><span style="font-size: 9.5pt; line-height: 14.4pt;">)=(a+b)(a</span><span style="font-size: 6.5pt; line-height: 14.4pt;">2</span><span style="font-size: 9.5pt; line-height: 14.4pt;">−ab+b</span><span style="font-size: 6.5pt; line-height: 14.4pt;">2</span><span style="font-size: 9.5pt; line-height: 14.4pt;">) </span></div>
</span><span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;"><div style="text-align: left;">
<span style="font-size: 9.5pt; line-height: 14.4pt;">8. (a</span><span style="font-size: 6.5pt; line-height: 14.4pt;">3</span><span style="font-size: 9.5pt; line-height: 14.4pt;">–b</span><span style="font-size: 6.5pt; line-height: 14.4pt;">3</span><span style="font-size: 9.5pt; line-height: 14.4pt;">)=(a−b)(a</span><span style="font-size: 6.5pt; line-height: 14.4pt;">2</span><span style="font-size: 9.5pt; line-height: 14.4pt;">+ab+b</span><span style="font-size: 6.5pt; line-height: 14.4pt;">2</span><span style="font-size: 9.5pt; line-height: 14.4pt;">)</span></div>
</span><span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;"><div style="text-align: left;">
<span style="font-size: 9.5pt; line-height: 14.4pt;">9. (a</span><span style="font-size: 6.5pt; line-height: 14.4pt;">3</span><span style="font-size: 9.5pt; line-height: 14.4pt;">+b</span><span style="font-size: 6.5pt; line-height: 14.4pt;">3</span><span style="font-size: 9.5pt; line-height: 14.4pt;">+c</span><span style="font-size: 6.5pt; line-height: 14.4pt;">3</span><span style="font-size: 9.5pt; line-height: 14.4pt;">−3abc)=(a+b+c)(a</span><span style="font-size: 6.5pt; line-height: 14.4pt;">2</span><span style="font-size: 9.5pt; line-height: 14.4pt;">+b</span><span style="font-size: 6.5pt; line-height: 14.4pt;">2</span><span style="font-size: 9.5pt; line-height: 14.4pt;">+c</span><span style="font-size: 6.5pt; line-height: 14.4pt;">2</span><span style="font-size: 9.5pt; line-height: 14.4pt;">−ab−bc−ca)</span></div>
</span><span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;"><div style="text-align: left;">
<span style="font-size: 9.5pt; line-height: 14.4pt;">10. If a+b+c=0, then a</span><span style="font-size: 6.5pt; line-height: 14.4pt;">3</span><span style="font-size: 9.5pt; line-height: 14.4pt;">+b</span><span style="font-size: 6.5pt; line-height: 14.4pt;">3</span><span style="font-size: 9.5pt; line-height: 14.4pt;">+c</span><span style="font-size: 6.5pt; line-height: 14.4pt;">3</span><span style="font-size: 9.5pt; line-height: 14.4pt;">=3abc.</span></div>
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<b><u><span style="color: #215d94; font-family: "trebuchet ms" , "sans-serif"; font-size: 15.5pt; font-variant: small-caps; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">Types
of Numbers:<o:p></o:p></span></u></b></div>
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<span style="color: #089ec6; font-family: "trebuchet ms" , sans-serif; font-size: 18px; font-weight: bold; text-decoration: underline;"><br /></span></div>
<b style="line-height: normal;"></b><br />
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<b style="line-height: normal;"><b><u><span style="color: #089ec6; font-family: "trebuchet ms" , "sans-serif"; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">I. Natural Numbers:</span></u></b></b></div>
<b style="line-height: normal;">
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<span style="font-family: "verdana" , "sans-serif"; font-size: 9.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">Counting
numbers 1,2,3,4,5,…… are called <i>natural numbers</i><o:p></o:p></span></div>
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<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";">
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<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";">
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<b><u><span style="color: #089ec6; font-family: "trebuchet ms" , "sans-serif"; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">II. Whole Numbers:<o:p></o:p></span></u></b></div>
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<span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;">All
counting numbers together with zero form the set of </span><i style="font-family: Verdana, sans-serif; font-size: 9.5pt; line-height: 14.4pt;">whole numbers</i><span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;">. </span></div>
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";"></span><br />
<div style="text-align: left;">
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";"><span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;">Thus, </span></span></div>
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</span>
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<span style="font-family: "verdana" , "sans-serif"; font-size: 9.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";"><span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;">(i) 0 is the only whole number which is not a natural number.</span></span></div>
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<div style="text-align: left;">
<span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;">(ii) Every natural number is a whole number.</span></div>
<o:p></o:p></span><br />
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<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";">
</span><br />
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<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";">
</span></div>
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<b><u><span style="color: #089ec6; font-family: "trebuchet ms" , "sans-serif"; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">III. Integers :<o:p></o:p></span></u></b></div>
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<span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;"> All natural numbers, 0 and negatives
of counting numbers i.e.,…,−3,−2,−1,0,1,2,3,….. together form
the set of </span><i style="font-family: Verdana, sans-serif; font-size: 9.5pt; line-height: 14.4pt;">integers</i><span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;">.</span></div>
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";"></span><br />
<div style="text-align: left;">
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";"><b style="font-family: Verdana, sans-serif; font-size: 9.5pt; line-height: 14.4pt;">(i) Positive Integers:</b><span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;"> 1,2,3,4,….. is the set of all </span><i style="font-family: Verdana, sans-serif; font-size: 9.5pt; line-height: 14.4pt;">positive
integers</i><span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;">.</span></span></div>
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">
<b></b></span>
<div style="text-align: left;">
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";"><b><b style="font-family: Verdana, sans-serif; font-size: 9.5pt; line-height: 14.4pt;">(ii) Negative Integers:</b><span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;"> −1,−2,−3,….. is the set of all </span><i style="font-family: Verdana, sans-serif; font-size: 9.5pt; line-height: 14.4pt;">negative
integers</i><span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;">.</span></b></span></div>
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</b>
<b><div style="text-align: left;">
<b style="font-family: Verdana, sans-serif; font-size: 9.5pt; line-height: 14.4pt;">(iii) Non-Positive and Non-Negative Integers:</b><span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;"> 0 is neither positive
nor negative. </span></div>
</b><div style="text-align: left;">
<span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;">So, 0,1,2,3,…. represents the set
of </span><i style="font-family: Verdana, sans-serif; font-size: 9.5pt; line-height: 14.4pt;">non-negative integers</i><span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;">, </span></div>
<div style="text-align: left;">
<span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;">while 0,−1,−2,−3,….. represents the set of </span><i style="font-family: Verdana, sans-serif; font-size: 9.5pt; line-height: 14.4pt;">non-positive
integers</i><span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;">.</span></div>
<o:p></o:p></span><br />
<div class="MsoNormal" style="line-height: normal; margin-bottom: 0.0001pt; text-align: right;">
<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: 0.0001pt; text-align: center;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";">
</span><br />
<hr noshade="" size="2" style="text-align: center; width: 421.6pt;" width="562" />
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";">
</span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: 0.0001pt; text-align: left;">
<b><u><span style="color: #089ec6; font-family: "trebuchet ms" , "sans-serif"; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">IV. Even Numbers:<o:p></o:p></span></u></b></div>
<div class="MsoNormal" style="line-height: 14.4pt; margin-bottom: 0.1in; text-align: left; vertical-align: middle;">
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">A
number divisible by 2 is called an <b>even number</b>, e.g.,2,4,6,8, etc.<o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: 0.0001pt; text-align: right;">
<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: 0.0001pt; text-align: center;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";">
</span><br />
<hr noshade="" size="2" style="text-align: center; width: 421.6pt;" width="562" />
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";">
</span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: 0.0001pt; text-align: left;">
<b><u><span style="color: #089ec6; font-family: "trebuchet ms" , "sans-serif"; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">V. Odd Numbers:<o:p></o:p></span></u></b></div>
<div class="MsoNormal" style="line-height: 14.4pt; margin-bottom: 0.1in; text-align: left; vertical-align: middle;">
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">A
number not divisible by 2 is called an odd number. e.g.,1,3,5,7,9,11, etc.<o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: 0.0001pt; text-align: right;">
<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: 0.0001pt; text-align: center;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";">
</span><br />
<hr noshade="" size="2" style="text-align: center; width: 421.6pt;" width="562" />
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";">
</span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: 0.0001pt; text-align: left;">
<b><u><span style="color: #089ec6; font-family: "trebuchet ms" , "sans-serif"; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">VI. Prime Numbers:<o:p></o:p></span></u></b></div>
<div class="MsoNormal" style="line-height: 14.4pt; margin-bottom: 0.1in; text-align: left; vertical-align: middle;">
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">A
number greater than 1 is called a prime number, if it has exactly two factors,
namely 1 and the number itself.<o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0.1in 11.25pt; text-align: left; text-indent: -0.25in;">
<!--[if !supportLists]--><span style="font-family: "symbol"; font-size: 10.0pt; mso-bidi-font-family: Symbol; mso-bidi-font-size: 11.0pt; mso-fareast-font-family: Symbol;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal;">
</span></span><!--[endif]--><span style="font-family: "georgia" , "serif"; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">Prime
numbers up to 100
are:2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97.<o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0.1in 11.25pt; text-align: left; text-indent: -0.25in;">
<!--[if !supportLists]--><span style="font-family: "symbol"; font-size: 10.0pt; mso-bidi-font-family: Symbol; mso-bidi-font-size: 11.0pt; mso-fareast-font-family: Symbol;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal;">
</span></span><!--[endif]--><span style="font-family: "georgia" , "serif"; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">Prime
numbers Greater than 100 : Let p be a given number greater than 100. To find
out whether it is prime or not, we use the following method :<o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: 14.4pt; margin-bottom: 0.1in; vertical-align: middle;">
</div>
<div style="text-align: left;">
<span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;">Find
a whole number nearly greater than the square root of p. Let k>*jp.
Test whether p is divisible by any prime number less than k. If yes, then p is
not prime. Otherwise, p is prime. </span><b style="font-family: Verdana, sans-serif; font-size: 9.5pt; line-height: 14.4pt;">Example:</b><span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;"> We have to find
whether 191 is a prime number or not. Now, 14>V191.</span></div>
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";"></span><br />
<div style="text-align: left;">
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";"><span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;">Prime numbers less than 14 are 2,3,5,7,11,13.</span></span></div>
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">
</span>
<div style="text-align: left;">
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";"><span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;">191 is not divisible by any of them. So, </span><i style="font-family: Verdana, sans-serif; font-size: 9.5pt; line-height: 14.4pt;">191 is a prime number</i><span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;">.</span></span></div>
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">
<o:p></o:p></span><br />
<div class="MsoNormal" style="line-height: normal; margin-bottom: 0.0001pt; text-align: right;">
<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: 0.0001pt; text-align: center;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";">
</span><br />
<hr noshade="" size="2" style="text-align: center; width: 421.6pt;" width="562" />
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";">
</span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: 0.0001pt; text-align: left;">
<b><u><span style="color: #089ec6; font-family: "trebuchet ms" , "sans-serif"; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">VII. Composite Numbers:<o:p></o:p></span></u></b></div>
<div class="MsoNormal" style="line-height: 14.4pt; margin-bottom: 0.1in; vertical-align: middle;">
</div>
<div style="text-align: left;">
<span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;">Numbers
greater than 1 which are not prime,
are known as </span><i style="font-family: Verdana, sans-serif; font-size: 9.5pt; line-height: 14.4pt;">composite numbers</i><span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;">,
e.g., 4,6,8,9,10,12.</span></div>
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";"></span><br />
<div style="text-align: left;">
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";"><b style="font-family: Verdana, sans-serif; font-size: 9.5pt; line-height: 14.4pt;">Note :</b><span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;"> </span></span></div>
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">
</span>
<div style="text-align: left;">
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";"><span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;">(i) 1 is neither prime nor composite.</span></span></div>
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">
<div style="text-align: left;">
<span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;">(ii) 2 is the only even number which is prime.</span></div>
<div style="text-align: left;">
<span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;">(iii) There are 25 prime numbers between 1 and 100.</span></div>
<o:p></o:p></span><br />
<div class="MsoNormal" style="line-height: normal; margin-bottom: 0.0001pt; text-align: right;">
<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: 0.0001pt; text-align: left;">
<b><u><span style="color: #215d94; font-family: "trebuchet ms" , "sans-serif"; font-size: 15.5pt; font-variant: small-caps; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">Remainder
& Quotient:<o:p></o:p></span></u></b></div>
<div class="MsoNormal" style="margin-bottom: 0.1in; vertical-align: middle;">
</div>
<div style="text-align: left;">
<span style="font-family: "verdana" , sans-serif; font-size: 12.6667px; line-height: 19.2px;"><br /></span></div>
<span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;"></span><br />
<div style="text-align: left;">
<span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;"><span style="font-size: 9.5pt; line-height: 14.4pt;">"The </span><i style="font-size: 9.5pt; line-height: 14.4pt;">remainder</i><span style="font-size: 9.5pt; line-height: 14.4pt;"> is r when p is divided by k"
means p=kq+r the integer q is called the </span><i style="font-size: 9.5pt; line-height: 14.4pt;">quotient</i><span style="font-size: 9.5pt; line-height: 14.4pt;">.</span></span></div>
<span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;">
</span>
<div style="text-align: left;">
<span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;"><span style="font-size: 9.5pt; line-height: 14.4pt;">For instance, "The remainder is 1 when 7 is divided by 3"
means 7=3*2+1. Dividing both sides of p=kq+r by k gives the
following alternative form </span></span></div>
<span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;">
<o:p></o:p></span><br />
<table border="0" cellpadding="0" class="MsoNormalTable" style="line-height: 25.2pt;">
<tbody>
<tr>
<td style="padding: .75pt .75pt .75pt .75pt;"></td>
</tr>
</tbody></table>
<div class="MsoNormal" style="line-height: 14.4pt; margin-bottom: 0.1in; text-align: left; vertical-align: middle;">
<b><u><span style="color: #003399; font-family: "verdana" , "sans-serif"; font-size: 9.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">Example:</span></u></b><span style="font-family: "verdana" , "sans-serif"; font-size: 9.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";"> The remainder is 57 when a number is divided by
10,000. What is the remainder when the same number is divided by 1,000?</span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: 14.4pt; margin-bottom: 0.1in; text-align: left; vertical-align: middle;">
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">(A)
5
(B)
7
(C)
43 (D)
57
(E)
570<o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: 14.4pt; margin-bottom: 0.1in; text-align: left; vertical-align: middle;">
<b><u><span style="color: #003399; font-family: "verdana" , "sans-serif"; font-size: 9.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">Solution:<o:p></o:p></span></u></b></div>
<div class="MsoNormal" style="line-height: 14.4pt; margin-bottom: 0.1in; vertical-align: middle;">
</div>
<div style="text-align: left;">
<span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;">Since
the remainder is 57 when the number
is divided by 10,000, the number can
be expressed as 10,000n+57, where n is an integer. </span></div>
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";"></span><br />
<div style="text-align: left;">
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";"><span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;">Rewriting 10,000 as 1,000*10 yields 10,000n+57=1,000(10n)+57</span></span></div>
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">
</span>
<div style="text-align: left;">
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";"><span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;">Now, since n is an integer, 10 n is an integer.
Letting 10n=q , we get</span></span></div>
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">
<div style="text-align: left;">
<span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;">10,000n+57=1,000*q+57</span></div>
<div style="text-align: left;">
<span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;">Hence, the remainder is still 57 (by the p=kq+r form) when the
number is divided by 1,000. The answer is</span><b style="font-family: Verdana, sans-serif; font-size: 9.5pt; line-height: 14.4pt;"> (D).</b></div>
<o:p></o:p></span><br />
<div class="MsoNormal" style="line-height: 14.4pt; margin-bottom: 0.1in; text-align: right; vertical-align: middle;">
<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: 0.0001pt; text-align: left;">
<b><u><span style="color: #215d94; font-family: "trebuchet ms" , "sans-serif"; font-size: 15.5pt; font-variant: small-caps; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">Even,
Odd Numbers:<o:p></o:p></span></u></b></div>
<div class="MsoNormal" style="margin-bottom: 0.1in; vertical-align: middle;">
</div>
<div style="text-align: left;">
<span style="font-family: "verdana" , sans-serif; font-size: 12.6667px; line-height: 19.2px;"><br /></span></div>
<span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;"></span><br />
<div style="text-align: left;">
<span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;"><span style="font-size: 9.5pt; line-height: 14.4pt;">A number n is even if the remainder is zero when n is divided by 2:n=2z+0,
orn=2z.</span></span></div>
<span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;">
</span>
<div style="text-align: left;">
<span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;"><span style="font-size: 9.5pt; line-height: 14.4pt;">A number n is odd if the remainder is one when n is divided by 2:n=2z+1.</span></span></div>
<span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;">
<div style="text-align: left;">
<span style="font-size: 9.5pt; line-height: 14.4pt;">The following properties for odd and even numbers are very useful—you should</span><i style="font-size: 9.5pt; line-height: 14.4pt;">memorize
them</i><span style="font-size: 9.5pt; line-height: 14.4pt;">:</span></div>
<o:p></o:p></span><br />
<div class="MsoNormal" style="line-height: 14.4pt; margin-bottom: 0.1in; vertical-align: middle;">
</div>
<div style="text-align: left;">
<span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;">even
* even = even</span></div>
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";"></span><br />
<div style="text-align: left;">
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";"><span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;">odd * odd = odd</span></span></div>
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">
</span>
<div style="text-align: left;">
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";"><span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;">even * odd = even</span></span></div>
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">
<div style="text-align: left;">
<span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;">even + even = even</span></div>
<div style="text-align: left;">
<span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;">odd + odd = even</span></div>
<div style="text-align: left;">
<span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;">even + odd = odd</span></div>
<o:p></o:p></span><br />
<div class="MsoNormal" style="line-height: normal; margin-bottom: 0.0001pt; text-align: center;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";">
</span><br />
<hr noshade="" size="2" style="text-align: center; width: 421.6pt;" width="562" />
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";">
</span></div>
<div class="MsoNormal" style="line-height: 14.4pt; margin-bottom: 0.1in; vertical-align: middle;">
</div>
<div style="text-align: left;">
<b style="line-height: 14.4pt;"><u><span style="color: #003399; font-family: "verdana" , "sans-serif"; font-size: 9.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">Example:</span></u></b><span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;"> If n is a positive integer and (n + 1)(n + 3) is odd,
then (n + 2)(n + 4) must be a multiple of which one of the following?</span></div>
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";"></span><br />
<div style="text-align: left;">
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";"><span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;">(A) 3
(B) 5
(C)
6 (D)
8
(E) 16</span></span></div>
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">
<o:p></o:p></span>
<div class="MsoNormal" style="line-height: 14.4pt; margin-bottom: 0.1in; text-align: left; vertical-align: middle;">
<b><u><span style="color: #003399; font-family: "verdana" , "sans-serif"; font-size: 9.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">Solution:<o:p></o:p></span></u></b></div>
<div class="MsoNormal" style="line-height: 14.4pt; margin-bottom: 0.1in; vertical-align: middle;">
</div>
<div style="text-align: left;">
<span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;">(n+1)(n+3) is
odd only when both (n+1) and (n+3) are odd. This is
possible only when n is even.</span></div>
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";"></span><br />
<div style="text-align: left;">
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";"><span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;">Hence, n = 2m, where m is a positive integer. Then,</span></span></div>
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">
</span>
<div style="text-align: left;">
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";"><span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;">(n+2)(n+4)=(2m+2)(2m+4)=2(m+1)2(m+2)=4(m+1)(m+2)=</span></span></div>
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">
<div style="text-align: left;">
<span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;">4 * (product of two consecutive positive integers, one which must be even)
= 4 * (an even number), and this equals a number that is at least a
multiple of 8</span></div>
<div style="text-align: left;">
<span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;">Hence, the answer is</span><b style="font-family: Verdana, sans-serif; font-size: 9.5pt; line-height: 14.4pt;"> (D).</b></div>
<o:p></o:p></span><br />
<div class="MsoNormal" style="line-height: normal; margin-bottom: 0.0001pt; text-align: right;">
<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: 0.0001pt; text-align: left;">
<b><u><span style="color: #215d94; font-family: "trebuchet ms" , "sans-serif"; font-size: 15.5pt; font-variant: small-caps; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">Tests
of Divisibility:<o:p></o:p></span></u></b></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
</div>
<div style="text-align: left;">
<span style="color: #089ec6; font-family: "trebuchet ms" , sans-serif; font-size: 18px; font-weight: bold; text-decoration: underline;"><br /></span></div>
<b style="line-height: normal;"></b><br />
<div style="text-align: left;">
<b style="line-height: normal;"><b><u><span style="color: #089ec6; font-family: "trebuchet ms" , "sans-serif"; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">1. Divisibility By 2:</span></u></b></b></div>
<b style="line-height: normal;">
</b>
<div class="MsoNormal" style="line-height: 14.4pt; margin-bottom: 0.1in; vertical-align: middle;">
</div>
<div style="text-align: left;">
<span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;">A
number is divisible by 2, if its unit's digit is any of 0,2,4,6,8.</span></div>
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";"></span><br />
<div style="text-align: left;">
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";"><b style="font-family: Verdana, sans-serif; font-size: 9.5pt; line-height: 14.4pt;">Ex.</b><span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;"> 84932 is divisible by 2, while 65935 is not.</span></span></div>
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">
<o:p></o:p></span>
<div class="MsoNormal" style="line-height: normal; margin-bottom: 0.0001pt; text-align: right;">
<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: 0.0001pt; text-align: center;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";">
</span><br />
<hr noshade="" size="2" style="text-align: center; width: 421.6pt;" width="562" />
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";">
</span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: 0.0001pt; text-align: left;">
<b><u><span style="color: #089ec6; font-family: "trebuchet ms" , "sans-serif"; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">2. Divisibility By
3:<o:p></o:p></span></u></b></div>
<div class="MsoNormal" style="line-height: 14.4pt; margin-bottom: 0.1in; vertical-align: middle;">
</div>
<div style="text-align: left;">
<span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;">A
number is divisible by 3, if the sum of its digits is divisible by 3.</span></div>
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";"></span><br />
<div style="text-align: left;">
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";"><b style="font-family: Verdana, sans-serif; font-size: 9.5pt; line-height: 14.4pt;">Ex.</b><span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;">592482 is divisible by 3, since sum of its
digits =(5+9+2+4+8+2)=30, which is divisible by 3.</span></span></div>
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">
</span>
<div style="text-align: left;">
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";"><span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;">But, 864329 is not divisible by 3, since sum of its digits=(8+6+4+3+2+9)=32,
which is not divisible by 3.</span></span></div>
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">
<o:p></o:p></span><br />
<div class="MsoNormal" style="line-height: normal; margin-bottom: 0.0001pt; text-align: right;">
<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: 0.0001pt; text-align: center;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";">
</span><br />
<hr noshade="" size="2" style="text-align: center; width: 421.6pt;" width="562" />
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";">
</span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: 0.0001pt; text-align: left;">
<b><u><span style="color: #089ec6; font-family: "trebuchet ms" , "sans-serif"; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">3. Divisibility By
4:<o:p></o:p></span></u></b></div>
<div class="MsoNormal" style="line-height: 14.4pt; margin-bottom: 0.1in; vertical-align: middle;">
</div>
<div style="text-align: left;">
<span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;">A
number is divisible by 4, if the number formed by the last two digits is
divisible by 4.</span></div>
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";"></span><br />
<div style="text-align: left;">
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";"><b style="font-family: Verdana, sans-serif; font-size: 9.5pt; line-height: 14.4pt;">Ex.</b><span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;"> 892648 is divisible by 4,
since the number formed by the last
two digits is 48, which is divisible by 4.
But, 749282 is not divisible by 4, since the number formed by the last two
digits is 82, which is not divisible by 4.</span></span></div>
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">
<o:p></o:p></span>
<div class="MsoNormal" style="line-height: normal; margin-bottom: 0.0001pt; text-align: right;">
<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: 0.0001pt; text-align: center;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";">
</span><br />
<hr noshade="" size="2" style="text-align: center; width: 421.6pt;" width="562" />
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";">
</span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: 0.0001pt; text-align: left;">
<b><u><span style="color: #089ec6; font-family: "trebuchet ms" , "sans-serif"; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">4. Divisibility By
5:<o:p></o:p></span></u></b></div>
<div class="MsoNormal" style="line-height: 14.4pt; margin-bottom: 0.1in; text-align: left; vertical-align: middle;">
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">A
number is divisible by 5, if its unit's digit is either 0 or 5. Thus, 20820 and
50345 are divisible by 5, while 30934 and 40946 are not.<o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: 0.0001pt; text-align: right;">
<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: 0.0001pt; text-align: center;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";">
</span><br />
<hr noshade="" size="2" style="text-align: center; width: 421.6pt;" width="562" />
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";">
</span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: 0.0001pt; text-align: left;">
<b><u><span style="color: #089ec6; font-family: "trebuchet ms" , "sans-serif"; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">5. Divisibility By
6:<o:p></o:p></span></u></b></div>
<div class="MsoNormal" style="line-height: 14.4pt; margin-bottom: 0.1in; vertical-align: middle;">
</div>
<div style="text-align: left;">
<span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;">A
number is divisible by 6, if it is divisible by both 2 and 3.</span></div>
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";"></span><br />
<div style="text-align: left;">
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";"><b style="font-family: Verdana, sans-serif; font-size: 9.5pt; line-height: 14.4pt;">Ex.</b><span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;"> The number 35256 is clearly divisible by 2.Sum of its
digits=(3+5+2+5+6)=21, which is divisible by 3. Thus, 35256 is divisible by 2
as well as 3. Hence, 35256 is divisible by 6.</span></span></div>
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">
<o:p></o:p></span>
<div class="MsoNormal" style="line-height: normal; margin-bottom: 0.0001pt; text-align: right;">
<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: 0.0001pt; text-align: center;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";">
</span><br />
<hr noshade="" size="2" style="text-align: center; width: 421.6pt;" width="562" />
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";">
</span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: 0.0001pt; text-align: left;">
<b><u><span style="color: #089ec6; font-family: "trebuchet ms" , "sans-serif"; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">6.
Divisibility By 8:<o:p></o:p></span></u></b></div>
<div class="MsoNormal" style="line-height: 14.4pt; margin-bottom: 0.1in; vertical-align: middle;">
</div>
<div style="text-align: left;">
<span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;">A
number is divisible by 8, if the number formed by the last Three digits of the
given number is divisible by 8.</span></div>
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";"></span><br />
<div style="text-align: left;">
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";"><b style="font-family: Verdana, sans-serif; font-size: 9.5pt; line-height: 14.4pt;">Ex.</b><span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;"> 953360 is divisible by 8,
since the number formed by last three
digits is 360, which is divisible by 8. But, 529418 is
not divisible by 8, since the number formed by last three digits is 418, which
is not divisible by 8.</span></span></div>
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">
<o:p></o:p></span>
<div class="MsoNormal" style="line-height: normal; margin-bottom: 0.0001pt; text-align: right;">
<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: 0.0001pt; text-align: center;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";">
</span><br />
<hr noshade="" size="2" style="text-align: center; width: 421.6pt;" width="562" />
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";">
</span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: 0.0001pt; text-align: left;">
<b><u><span style="color: #089ec6; font-family: "trebuchet ms" , "sans-serif"; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">7.
Divisibility By 9:<o:p></o:p></span></u></b></div>
<div class="MsoNormal" style="line-height: 14.4pt; margin-bottom: 0.1in; vertical-align: middle;">
</div>
<div style="text-align: left;">
<span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;">A
number is divisible by 9, if the sum of its digits is divisible by 9.</span></div>
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";"></span><br />
<div style="text-align: left;">
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";"><b style="font-family: Verdana, sans-serif; font-size: 9.5pt; line-height: 14.4pt;">Ex.</b><span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;"> 60732 is divisible by 9, since sum of digits =(6+0+7+3+2)=18,
which is divisible by 9.</span></span></div>
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">
</span>
<div style="text-align: left;">
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";"><span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;">But, 68956 is not divisible by 9, since sum of digits =(6+8+9+5+6)=34,
which is not divisible by 9.</span></span></div>
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">
<o:p></o:p></span><br />
<div class="MsoNormal" style="line-height: normal; margin-bottom: 0.0001pt; text-align: right;">
<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: 0.0001pt; text-align: center;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";">
</span><br />
<hr noshade="" size="2" style="text-align: center; width: 421.6pt;" width="562" />
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";">
</span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: 0.0001pt; text-align: left;">
<b><u><span style="color: #089ec6; font-family: "trebuchet ms" , "sans-serif"; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">8.
Divisibility By 10:<o:p></o:p></span></u></b></div>
<div class="MsoNormal" style="line-height: 14.4pt; margin-bottom: 0.1in; vertical-align: middle;">
</div>
<div style="text-align: left;">
<span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;">A
number is divisible by 10, if it ends with 0.</span></div>
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";"></span><br />
<div style="text-align: left;">
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";"><b style="font-family: Verdana, sans-serif; font-size: 9.5pt; line-height: 14.4pt;">Ex.</b><span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;"> 96410, 10480 are divisible by 10, while 96375 is not.</span></span></div>
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">
<o:p></o:p></span>
<div class="MsoNormal" style="line-height: normal; margin-bottom: 0.0001pt; text-align: right;">
<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: 0.0001pt; text-align: center;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";">
</span><br />
<hr noshade="" size="2" style="text-align: center; width: 421.6pt;" width="562" />
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";">
</span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: 0.0001pt; text-align: left;">
<b><u><span style="color: #089ec6; font-family: "trebuchet ms" , "sans-serif"; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">9.
Divisibility By 11:<o:p></o:p></span></u></b></div>
<div class="MsoNormal" style="line-height: 14.4pt; margin-bottom: 0.1in; vertical-align: middle;">
</div>
<div style="text-align: left;">
<span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;">A
number is divisible by 11, if the difference of the sum of its digits at odd
places and the sum of its digits at even places, is either 0 or a number
divisible by 11.</span></div>
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";"></span><br />
<div style="text-align: left;">
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";"><b style="font-family: Verdana, sans-serif; font-size: 9.5pt; line-height: 14.4pt;">Ex.</b><span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;"> The number 4832718 is divisible by 11, since :(sum of
digits at odd places) - (sum of digits at even places) = </span></span></div>
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">
</span>
<div style="text-align: left;">
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";"><span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;">=(8+7+3+4)−(1+2+8)=11, which is divisible by 11.</span></span></div>
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">
<o:p></o:p></span><br />
<div class="MsoNormal" style="line-height: normal; margin-bottom: 0.0001pt; text-align: right;">
<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: 0.0001pt; text-align: center;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";">
</span><br />
<hr noshade="" size="2" style="text-align: center; width: 421.6pt;" width="562" />
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";">
</span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: 0.0001pt; text-align: left;">
<b><u><span style="color: #089ec6; font-family: "trebuchet ms" , "sans-serif"; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">10. Divisibility By 12:<o:p></o:p></span></u></b></div>
<div class="MsoNormal" style="line-height: 14.4pt; margin-bottom: 0.1in; vertical-align: middle;">
</div>
<div style="text-align: left;">
<span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;">A
number is divisible by 12, if it is divisible by both 4 and3.</span></div>
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";"></span><br />
<div style="text-align: left;">
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";"><b style="font-family: Verdana, sans-serif; font-size: 9.5pt; line-height: 14.4pt;">Ex.</b><span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;"> Consider the number 34632.</span></span></div>
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">
<o:p></o:p></span>
<div class="MsoNormal" style="line-height: 14.4pt; margin-bottom: 0.1in; vertical-align: middle;">
</div>
<div style="text-align: left;">
<span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;">(i)
The number formed by last two digits is 32, which is divisible by 4,</span></div>
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";"></span><br />
<div style="text-align: left;">
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";"><span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;">(ii) Sum of digits =(3+4+6+3+2)=18, which is divisible by 3. Thus, 34632
is divisible by 4 as well as 3. Hence, 34632 is divisible by 12.</span></span></div>
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">
<o:p></o:p></span>
<div class="MsoNormal" style="line-height: normal; margin-bottom: 0.0001pt; text-align: right;">
<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: 0.0001pt; text-align: center;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";">
</span><br />
<hr noshade="" size="2" style="text-align: center; width: 421.6pt;" width="562" />
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";">
</span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: 0.0001pt; text-align: left;">
<b><u><span style="color: #089ec6; font-family: "trebuchet ms" , "sans-serif"; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">11. Divisibility By 14:<o:p></o:p></span></u></b></div>
<div class="MsoNormal" style="line-height: 14.4pt; margin-bottom: 0.1in; text-align: left; vertical-align: middle;">
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">A
number is divisible by 14, if it is divisible by 2 as well as 7.<o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: 0.0001pt; text-align: right;">
<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: 0.0001pt; text-align: center;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";">
</span><br />
<hr noshade="" size="2" style="text-align: center; width: 421.6pt;" width="562" />
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";">
</span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: 0.0001pt; text-align: left;">
<b><u><span style="color: #089ec6; font-family: "trebuchet ms" , "sans-serif"; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">12. Divisibility By 15:<o:p></o:p></span></u></b></div>
<div class="MsoNormal" style="line-height: 14.4pt; margin-bottom: 0.1in; text-align: left; vertical-align: middle;">
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">A
number is divisible by 15, if it is divisible by both 3 and 5.<o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: 0.0001pt; text-align: right;">
<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: 0.0001pt; text-align: center;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";">
</span><br />
<hr noshade="" size="2" style="text-align: center; width: 421.6pt;" width="562" />
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";">
</span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: 0.0001pt; text-align: left;">
<b><u><span style="color: #089ec6; font-family: "trebuchet ms" , "sans-serif"; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">13. Divisibility By 16:<o:p></o:p></span></u></b></div>
<div class="MsoNormal" style="line-height: 14.4pt; margin-bottom: 0.1in; vertical-align: middle;">
</div>
<div style="text-align: left;">
<span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;">A
number is divisible by 16, if the number formed by the last4 digits is
divisible by 16.</span></div>
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";"></span><br />
<div style="text-align: left;">
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";"><b style="font-family: Verdana, sans-serif; font-size: 9.5pt; line-height: 14.4pt;">Ex.</b><span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;">7957536 is divisible by 16, since the number formed by the last four
digits is 7536, which is divisible by 16.</span></span></div>
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">
<o:p></o:p></span>
<div class="MsoNormal" style="line-height: normal; margin-bottom: 0.0001pt; text-align: right;">
<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: 0.0001pt; text-align: center;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";">
</span><br />
<hr noshade="" size="2" style="text-align: center; width: 421.6pt;" width="562" />
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";">
</span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: 0.0001pt; text-align: left;">
<b><u><span style="color: #089ec6; font-family: "trebuchet ms" , "sans-serif"; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">14. Divisibility By
24:<o:p></o:p></span></u></b></div>
<div class="MsoNormal" style="line-height: 14.4pt; margin-bottom: 0.1in; text-align: left; vertical-align: middle;">
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">A
given number is divisible by 24, if it is divisible by both 3 and 8.<o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: 0.0001pt; text-align: right;">
<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: 0.0001pt; text-align: center;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";">
</span><br />
<hr noshade="" size="2" style="text-align: center; width: 421.6pt;" width="562" />
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";">
</span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: 0.0001pt; text-align: left;">
<b><u><span style="color: #089ec6; font-family: "trebuchet ms" , "sans-serif"; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">15. Divisibility By
40:<o:p></o:p></span></u></b></div>
<div class="MsoNormal" style="line-height: 14.4pt; margin-bottom: 0.1in; text-align: left; vertical-align: middle;">
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">A
given number is divisible by 40, if it is divisible by both 5 and 8.<o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: 0.0001pt; text-align: right;">
<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: 0.0001pt; text-align: center;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";">
</span><br />
<hr noshade="" size="2" style="text-align: center; width: 421.6pt;" width="562" />
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";">
</span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: 0.0001pt; text-align: left;">
<b><u><span style="color: #089ec6; font-family: "trebuchet ms" , "sans-serif"; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">16. Divisibility By
80:<o:p></o:p></span></u></b></div>
<div class="MsoNormal" style="line-height: 14.4pt; margin-bottom: 0.1in; vertical-align: middle;">
</div>
<div style="text-align: left;">
<span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;">A
given number is divisible by 80, if it is divisible by both 5 and 16.</span></div>
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";"></span><br />
<div style="text-align: left;">
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";"><b style="font-family: Verdana, sans-serif; font-size: 9.5pt; line-height: 14.4pt;">Note:</b><span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;"> If a number is divisible by p as well as q, where p and q are
co-primes, then the given number is divisible by pq. If p and
q are not co-primes, then the given
number need not be divisible by pq, even when it is divisible
by both p and q.</span></span></div>
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">
<o:p></o:p></span>
<div class="MsoNormal" style="line-height: 14.4pt; margin-bottom: 0.1in; text-align: left; vertical-align: middle;">
<b><span style="font-family: "verdana" , "sans-serif"; font-size: 9.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">Ex.</span></b><span style="font-family: "verdana" , "sans-serif"; font-size: 9.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";"> 36 is divisible
by both 4 and 6, but it is not divisible by (4*6)=24, since 4 and 6 are
not co- primes.<o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: 0.0001pt; text-align: right;">
<br /></div>
<div class="MsoNormal" style="background-attachment: initial; background-clip: initial; background-color: white; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; line-height: normal; margin-bottom: 0.0001pt; text-align: left;">
<b><u><span style="color: #215d94; font-family: "trebuchet ms" , "sans-serif"; font-size: 15.5pt; font-variant: small-caps; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">Progression:<o:p></o:p></span></u></b></div>
<div class="MsoNormal" style="background-attachment: initial; background-clip: initial; background-color: white; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; margin-bottom: 0.1in; vertical-align: middle;">
</div>
<div style="text-align: left;">
<span style="font-family: "verdana" , sans-serif; font-size: 12.6667px; line-height: 19.2px;"><br /></span></div>
<span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;"></span><br />
<div style="text-align: left;">
<span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;"><span style="font-size: 9.5pt; line-height: 14.4pt;">A succession of numbers formed and arranged in a definite order according to
certain definite rule, is called a progression.</span></span></div>
<span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;">
<o:p></o:p></span>
<div class="MsoNormal" style="background-attachment: initial; background-clip: initial; background-color: white; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; line-height: normal; margin-bottom: 0.0001pt; text-align: left;">
<b><u><span style="color: #089ec6; font-family: "trebuchet ms" , "sans-serif"; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">1. Arithmetic Progression
(A.P.):<o:p></o:p></span></u></b></div>
<div class="MsoNormal" style="background-attachment: initial; background-clip: initial; background-color: white; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; line-height: 14.4pt; margin-bottom: 0.1in; vertical-align: middle;">
</div>
<div style="text-align: left;">
<span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;">If each term of a progression differs from its preceding term
by a constant, then such a progression
is called an </span><i style="font-family: Verdana, sans-serif; font-size: 9.5pt; line-height: 14.4pt;">arithmetical progression</i><span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;">.
This constant difference is called the </span><i style="font-family: Verdana, sans-serif; font-size: 9.5pt; line-height: 14.4pt;">common difference</i><span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;"> of
the A.P.</span></div>
<span style="font-family: "verdana" , sans-serif; font-size: 9.5pt;"></span><br />
<div style="text-align: left;">
<span style="font-family: "verdana" , sans-serif; font-size: 9.5pt;"><span style="font-size: 9.5pt; line-height: 14.4pt;">An A.P. with first term a and common difference d is given
bya,(a+d),(a+2d),(a+3d),.....</span></span></div>
<span style="font-family: "verdana" , sans-serif; font-size: 9.5pt;">
</span>
<div style="text-align: left;">
<span style="font-family: "verdana" , sans-serif; font-size: 9.5pt;"><span style="font-size: 9.5pt; line-height: 14.4pt;">The<i> nth term </i>of this A.P. is given by T</span><span style="font-size: 6.5pt; line-height: 14.4pt;">n</span><span style="font-size: 9.5pt; line-height: 14.4pt;">=a(n−1)d. </span></span></div>
<span style="font-family: "verdana" , sans-serif; font-size: 9.5pt;">
</span><span style="font-family: "verdana" , sans-serif; font-size: 9.5pt;"><div style="text-align: left;">
<span style="font-size: 9.5pt; line-height: 14.4pt;">The <i>sum of n terms</i> of this A.P. S</span><span style="font-size: 6.5pt; line-height: 14.4pt;">n</span><span style="font-size: 9.5pt; line-height: 14.4pt;">=n/2(2a+(n-1)d)</span><span style="font-size: 9.5pt; line-height: 14.4pt;">.</span></div>
</span><br />
<div class="MsoNormal" style="background-attachment: initial; background-clip: initial; background-color: white; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; line-height: 14.4pt; margin-bottom: 0.1in; text-align: left; vertical-align: middle;">
<b><u><span style="color: #003399; font-family: "verdana" , "sans-serif"; font-size: 9.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">Some Important Results:<o:p></o:p></span></u></b></div>
<div class="MsoNormal" style="background-attachment: initial; background-clip: initial; background-color: white; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; line-height: 14.4pt; margin-bottom: 0.1in; text-align: left; vertical-align: middle;">
<span style="font-family: "verdana" , sans-serif; font-size: 9.5pt;">(i) (1+2+3+….+n)=</span><span style="font-family: "times new roman" , serif; font-size: 12pt;"><o:p></o:p></span></div>
<div class="MsoNormal" style="background-attachment: initial; background-clip: initial; background-color: white; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; margin-bottom: 0.1in; vertical-align: middle;">
</div>
<div style="text-align: left;">
<span style="font-family: "verdana" , sans-serif; font-size: 12.6667px; line-height: 19.2px;"><br /></span></div>
<span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;"></span><br />
<div style="text-align: left;">
<span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;"><span style="font-size: 9.5pt; line-height: 14.4pt;">(ii) (l</span><span style="font-size: 6.5pt; line-height: 14.4pt;">2</span><span style="font-size: 9.5pt; line-height: 14.4pt;">+2</span><span style="font-size: 6.5pt; line-height: 14.4pt;">2</span><span style="font-size: 9.5pt; line-height: 14.4pt;">+3</span><span style="font-size: 6.5pt; line-height: 14.4pt;">2</span><span style="font-size: 9.5pt; line-height: 14.4pt;">+...+n</span><span style="font-size: 6.5pt; line-height: 14.4pt;">2</span><span style="font-size: 9.5pt; line-height: 14.4pt;">)=</span></span></div>
<span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;">
</span>
<table border="0" cellpadding="0" class="MsoNormalTable" style="line-height: 25.2pt;">
<tbody>
<tr>
<td style="padding: .75pt .75pt .75pt .75pt;"><div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";">n(n+1)(2n+1)<o:p></o:p></span></div>
</td>
</tr>
<tr>
<td style="padding: .75pt .75pt .75pt .75pt;"><div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";">6<o:p></o:p></span></div>
</td>
</tr>
</tbody></table>
<div class="MsoNormal" style="background-attachment: initial; background-clip: initial; background-color: white; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; margin-bottom: 0.1in; vertical-align: middle;">
</div>
<div style="text-align: left;">
<span style="font-family: "verdana" , sans-serif; font-size: 12.6667px; line-height: 19.2px;"><br /></span></div>
<span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;"></span><br />
<div style="text-align: left;">
<span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;"><span style="font-size: 9.5pt; line-height: 14.4pt;">(iii) (1</span><span style="font-size: 6.5pt; line-height: 14.4pt;">3</span><span style="font-size: 9.5pt; line-height: 14.4pt;">+2</span><span style="font-size: 6.5pt; line-height: 14.4pt;">3</span><span style="font-size: 9.5pt; line-height: 14.4pt;">+3</span><span style="font-size: 6.5pt; line-height: 14.4pt;">3</span><span style="font-size: 9.5pt; line-height: 14.4pt;">+...+n</span><span style="font-size: 6.5pt; line-height: 14.4pt;">3</span><span style="font-size: 9.5pt; line-height: 14.4pt;">)=n</span><span style="font-size: 6.5pt; line-height: 14.4pt;">2</span><span style="font-size: 9.5pt; line-height: 14.4pt;">(n+1)</span><span style="font-size: 6.5pt; line-height: 14.4pt;">2</span></span></div>
<span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;">
</span>
<div class="MsoNormal" style="background-attachment: initial; background-clip: initial; background-color: white; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; line-height: normal; margin-bottom: 0.0001pt; text-align: right;">
<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: 0.0001pt;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";">
</span><br />
<hr noshade="" size="2" style="text-align: left; width: 421.6pt;" width="562" />
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";">
</span></div>
<div class="MsoNormal" style="background-attachment: initial; background-clip: initial; background-color: white; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; line-height: normal; margin-bottom: 0.0001pt; text-align: left;">
<b><u><span style="color: #089ec6; font-family: "trebuchet ms" , "sans-serif"; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">2. Geometrical
Progression (G.P.):<o:p></o:p></span></u></b></div>
<div class="MsoNormal" style="background-attachment: initial; background-clip: initial; background-color: white; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; line-height: 14.4pt; margin-bottom: 0.1in; text-align: left; vertical-align: middle;">
<span style="font-family: "verdana" , sans-serif; font-size: 9.5pt;">A progression of numbers in
which every term bears a constant ratio with its preceding
term, is called a geometrical progression. The constant ratio is called the
common ratio of the G.P. <o:p></o:p></span></div>
<div class="MsoNormal" style="background-attachment: initial; background-clip: initial; background-color: white; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; line-height: 14.4pt; margin-bottom: 0.1in; vertical-align: middle;">
</div>
<div style="text-align: left;">
<span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;">A G.P. with first term a and common ratio r is :a,ar,ar</span><span style="font-family: "verdana" , sans-serif; font-size: 6.5pt; line-height: 14.4pt;">2</span><span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;">,…..</span></div>
<span style="font-family: "verdana" , sans-serif; font-size: 9.5pt;"></span><br />
<div style="text-align: left;">
<span style="font-family: "verdana" , sans-serif; font-size: 9.5pt;"><span style="font-size: 9.5pt; line-height: 14.4pt;">In this G.P.n</span><span style="font-size: 6.5pt; line-height: 14.4pt;">th</span><span style="font-size: 9.5pt; line-height: 14.4pt;"> term, T</span><span style="font-size: 6.5pt; line-height: 14.4pt;">n</span><span style="font-size: 9.5pt; line-height: 14.4pt;">=ar</span><span style="font-size: 6.5pt; line-height: 14.4pt;">n−1</span></span></div>
<span style="font-family: "verdana" , sans-serif; font-size: 9.5pt;">
</span><span style="font-family: "verdana" , sans-serif; font-size: 9.5pt;"></span>
<div style="text-align: left;">
<span style="font-family: "verdana" , sans-serif; font-size: 9.5pt;"><span style="font-size: 9.5pt; line-height: 14.4pt;"><i>sum of n terms,</i> S</span><span style="font-size: 6.5pt; line-height: 14.4pt;">n</span><span style="font-size: 9.5pt; line-height: 14.4pt;">=ax [(r^n-1)-1]/r-1]</span></span></div>
<span style="font-family: "verdana" , sans-serif; font-size: 9.5pt;">
</span><br />
<div class="MsoNormal" style="background-attachment: initial; background-clip: initial; background-color: white; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; line-height: 14.4pt; margin-bottom: 0.1in; text-align: left; vertical-align: middle;">
<span style="font-family: "verdana" , sans-serif; font-size: 9.5pt; line-height: 14.4pt;"> </span><span style="font-family: "verdana" , sans-serif; font-size: 9.5pt;"> when r<1<o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: 12.75pt; margin-left: 0.5in; text-align: left; text-indent: -0.25in;">
<!--[if !supportLists]--><span style="font-family: "verdana" , "sans-serif"; font-size: 9.0pt; mso-bidi-font-family: Verdana; mso-fareast-font-family: Verdana;">1.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><b><span style="color: #555555; font-family: "verdana" , "sans-serif"; font-size: 9.0pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">Ratio:</span></b><span style="font-family: "verdana" , "sans-serif"; font-size: 9.0pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: 12.75pt; margin-left: 0.5in; text-align: left;">
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.0pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">The ratio of two quantities <i>a</i> and <i>b</i> in
the same units, is the fraction <!--[if gte vml 1]><v:shapetype
id="_x0000_t75" coordsize="21600,21600" o:spt="75" o:preferrelative="t"
path="m@4@5l@4@11@9@11@9@5xe" filled="f" stroked="f">
<v:stroke joinstyle="miter"/>
<v:formulas>
<v:f eqn="if lineDrawn pixelLineWidth 0"/>
<v:f eqn="sum @0 1 0"/>
<v:f eqn="sum 0 0 @1"/>
<v:f eqn="prod @2 1 2"/>
<v:f eqn="prod @3 21600 pixelWidth"/>
<v:f eqn="prod @3 21600 pixelHeight"/>
<v:f eqn="sum @0 0 1"/>
<v:f eqn="prod @6 1 2"/>
<v:f eqn="prod @7 21600 pixelWidth"/>
<v:f eqn="sum @8 21600 0"/>
<v:f eqn="prod @7 21600 pixelHeight"/>
<v:f eqn="sum @10 21600 0"/>
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<v:path o:extrusionok="f" gradientshapeok="t" o:connecttype="rect"/>
<o:lock v:ext="edit" aspectratio="t"/>
</v:shapetype><v:shape id="Picture_x0020_5" o:spid="_x0000_i1053" type="#_x0000_t75"
alt="http://www.indiabix.com/_files/images/aptitude/1-div-abyb.gif" style='width:7.5pt;
height:21pt;visibility:visible;mso-wrap-style:square'>
<v:imagedata src="file:///C:\Users\guru\AppData\Local\Temp\msohtmlclip1\01\clip_image001.png"
o:title="1-div-abyb"/>
</v:shape><![endif]--><!--[if !vml]--><img alt="http://www.indiabix.com/_files/images/aptitude/1-div-abyb.gif" border="0" height="28" src="file:///C:/Users/guru/AppData/Local/Temp/msohtmlclip1/01/clip_image001.png" v:shapes="Picture_x0020_5" width="10" /><!--[endif]--> and
we write it as <i>a</i> : <i>b</i>.<o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: 12.75pt; margin-left: 0.5in; text-align: left;">
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.0pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">In the ratio <i>a</i> : <i>b</i>, we
call <i>a</i> as the first term or <b><span style="color: #555555;">antecedent</span></b> and
b, the second term or <b><span style="color: #555555;">consequent</span></b>.<o:p></o:p></span></div>
<table border="0" cellpadding="0" cellspacing="0" class="MsoNormalTable" style="line-height: 25.2pt; margin-left: 0.5in;">
<tbody>
<tr>
<td rowspan="2" style="padding: 0in 5.25pt 0in 0in;"><div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;">
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.0pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">Eg. The ratio 5 : 9 represents<o:p></o:p></span></div>
</td>
<td style="border-bottom: solid black 1.0pt; border: none; mso-border-bottom-alt: solid black .75pt; padding: 0in 0in 1.5pt 0in;" valign="bottom"><div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;">
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.0pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">5<o:p></o:p></span></div>
</td>
<td rowspan="2" style="padding: 0in 5.25pt 0in 5.25pt;"><div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;">
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.0pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">with antecedent = 5, consequent = 9.<o:p></o:p></span></div>
</td>
</tr>
<tr>
<td style="padding: 1.5pt 0in 0in 0in;" valign="top"><div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;">
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.0pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">9<o:p></o:p></span></div>
</td>
</tr>
</tbody></table>
<div class="MsoNormal" style="line-height: 12.75pt; margin-left: 0.5in; text-align: left;">
<b><span style="color: #555555; font-family: "verdana" , "sans-serif"; font-size: 9.0pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">Rule:</span></b><span style="font-family: "verdana" , "sans-serif"; font-size: 9.0pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";"> The
multiplication or division of each term of a ratio by the same non-zero number
does not affect the ratio.<o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: 12.75pt; margin-left: 0.5in; text-align: left;">
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.0pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">Eg. 4 : 5 = 8 : 10 = 12 : 15. Also, 4 : 6 = 2 : 3.<o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: 12.75pt; margin-left: 0.5in; text-align: left; text-indent: -0.25in;">
<!--[if !supportLists]--><span style="font-family: "verdana" , "sans-serif"; font-size: 9.0pt; mso-bidi-font-family: Verdana; mso-fareast-font-family: Verdana;">2.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><b><span style="color: #555555; font-family: "verdana" , "sans-serif"; font-size: 9.0pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">Proportion:</span></b><span style="font-family: "verdana" , "sans-serif"; font-size: 9.0pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: 12.75pt; margin-left: 0.5in; text-align: left;">
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.0pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">The equality of two ratios is called proportion.<o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: 12.75pt; margin-left: 0.5in; text-align: left;">
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.0pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">If <i>a</i> : <i>b</i> = <i>c</i> : <i>d</i>,
we write <i>a</i> : <i>b</i> <b>::</b> <i>c</i> : <i>d</i> and
we say that <i>a, b, c, d</i> are in proportion.<o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: 12.75pt; margin-left: 0.5in; text-align: left;">
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.0pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">Here <i>a</i> and <i>d</i> are
called <b><span style="color: #555555;">extremes</span></b>, while <i>b</i> and <i>c</i> are
called <b><span style="color: #555555;">mean terms</span></b>.<o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: 12.75pt; margin-left: 0.5in; text-align: left;">
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.0pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">Product of means = Product of extremes.<o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: 12.75pt; margin-left: 0.5in; text-align: left;">
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.0pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">Thus, <i>a</i> : <i>b</i> <b>::</b> <i>c</i> : <i>d</i> <!--[if gte vml 1]><v:shape id="Picture_x0020_4"
o:spid="_x0000_i1052" type="#_x0000_t75" alt="http://www.indiabix.com/_files/images/aptitude/1-sym-bim.gif"
style='width:12.75pt;height:6.75pt;visibility:visible;mso-wrap-style:square'>
<v:imagedata src="file:///C:\Users\guru\AppData\Local\Temp\msohtmlclip1\01\clip_image002.png"
o:title="1-sym-bim"/>
</v:shape><![endif]--><!--[if !vml]--><img alt="http://www.indiabix.com/_files/images/aptitude/1-sym-bim.gif" border="0" height="9" src="file:///C:/Users/guru/AppData/Local/Temp/msohtmlclip1/01/clip_image002.png" v:shapes="Picture_x0020_4" width="17" /><!--[endif]--> (<i>b</i> x <i>c</i>)
= (<i>a</i> x <i>d</i>).<o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: 12.75pt; margin-left: 0.5in; text-align: left; text-indent: -0.25in;">
<!--[if !supportLists]--><span style="font-family: "verdana" , "sans-serif"; font-size: 9.0pt; mso-bidi-font-family: Verdana; mso-fareast-font-family: Verdana;">3.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><b><span style="color: #555555; font-family: "verdana" , "sans-serif"; font-size: 9.0pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">Fourth Proportional:</span></b><span style="font-family: "verdana" , "sans-serif"; font-size: 9.0pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: 12.75pt; margin-left: 0.5in; text-align: left;">
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.0pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">If <i>a</i> : <i>b</i> = <i>c</i> : <i>d</i>,
then <i>d</i> is called the fourth proportional to <i>a, b, c</i>.<o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: 12.75pt; margin-left: 0.5in; text-align: left;">
<b><span style="color: #555555; font-family: "verdana" , "sans-serif"; font-size: 9.0pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">Third Proportional:</span></b><span style="font-family: "verdana" , "sans-serif"; font-size: 9.0pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: 12.75pt; margin-left: 0.5in; text-align: left;">
<i><span style="font-family: "verdana" , "sans-serif"; font-size: 9.0pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">a</span></i><span style="font-family: "verdana" , "sans-serif"; font-size: 9.0pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";"> : <i>b</i> = <i>c</i> : <i>d</i>,
then <i>c</i> is called the third proportion to <i>a</i> and <i>b</i>.<o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: 12.75pt; margin-left: 0.5in; text-align: left;">
<b><span style="color: #555555; font-family: "verdana" , "sans-serif"; font-size: 9.0pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">Mean Proportional:</span></b><span style="font-family: "verdana" , "sans-serif"; font-size: 9.0pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: 12.75pt; margin-left: 0.5in; text-align: left;">
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.0pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">Mean proportional between <i>a</i> and <i>b</i> is <i>ab</i>.<o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: 12.75pt; margin-left: 0.5in; text-align: left; text-indent: -0.25in;">
<!--[if !supportLists]--><span style="font-family: "verdana" , "sans-serif"; font-size: 9.0pt; mso-bidi-font-family: Verdana; mso-fareast-font-family: Verdana;">4.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><b><span style="color: #555555; font-family: "verdana" , "sans-serif"; font-size: 9.0pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">Comparison of Ratios:</span></b><span style="font-family: "verdana" , "sans-serif"; font-size: 9.0pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
<table border="0" cellpadding="0" cellspacing="0" class="MsoNormalTable" style="line-height: 25.2pt; margin-left: 0.5in;">
<tbody>
<tr>
<td rowspan="2" style="padding: 0in 5.25pt 0in 0in;"><div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;">
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.0pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">We say that (<i>a</i> : <i>b</i>)
> (<i>c</i> : <i>d</i>) <!--[if gte vml 1]><v:shape id="Picture_x0020_3" o:spid="_x0000_i1051"
type="#_x0000_t75" alt="http://www.indiabix.com/_files/images/aptitude/1-sym-bim.gif"
style='width:12.75pt;height:6.75pt;visibility:visible;mso-wrap-style:square'>
<v:imagedata src="file:///C:\Users\guru\AppData\Local\Temp\msohtmlclip1\01\clip_image002.png"
o:title="1-sym-bim"/>
</v:shape><![endif]--><!--[if !vml]--><img alt="http://www.indiabix.com/_files/images/aptitude/1-sym-bim.gif" border="0" height="9" src="file:///C:/Users/guru/AppData/Local/Temp/msohtmlclip1/01/clip_image002.png" v:shapes="Picture_x0020_3" width="17" /><!--[endif]--> <o:p></o:p></span></div>
</td>
<td style="border-bottom: solid black 1.0pt; border: none; mso-border-bottom-alt: solid black .75pt; padding: 0in 0in 1.5pt 0in;" valign="bottom"><div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;">
<i><span style="font-family: "verdana" , "sans-serif"; font-size: 9.0pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">a</span></i><span style="font-family: "verdana" , "sans-serif"; font-size: 9.0pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
</td>
<td rowspan="2" style="padding: 0in 5.25pt 0in 5.25pt;"><div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;">
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.0pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">><o:p> </o:p></span></div>
</td>
<td style="border-bottom: solid black 1.0pt; border: none; mso-border-bottom-alt: solid black .75pt; padding: 0in 0in 1.5pt 0in;" valign="bottom"><div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;">
<i><span style="font-family: "verdana" , "sans-serif"; font-size: 9.0pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">c</span></i><span style="font-family: "verdana" , "sans-serif"; font-size: 9.0pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
</td>
<td nowrap="" rowspan="2" style="padding: 0in 0in 0in 0in;"><div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;">
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.0pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">.<o:p></o:p></span></div>
</td>
</tr>
<tr>
<td style="padding: 1.5pt 0in 0in 0in;" valign="top"><div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;">
<i><span style="font-family: "verdana" , "sans-serif"; font-size: 9.0pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">b</span></i><span style="font-family: "verdana" , "sans-serif"; font-size: 9.0pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
</td>
<td style="padding: 1.5pt 0in 0in 0in;" valign="top"><div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;">
<i><span style="font-family: "verdana" , "sans-serif"; font-size: 9.0pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">d</span></i><span style="font-family: "verdana" , "sans-serif"; font-size: 9.0pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
</td>
</tr>
</tbody></table>
<div class="MsoNormal" style="line-height: 12.75pt; margin-left: 0.5in; text-align: left; text-indent: -0.25in;">
<!--[if !supportLists]--><span style="font-family: "verdana" , "sans-serif"; font-size: 9.0pt; mso-bidi-font-family: Verdana; mso-fareast-font-family: Verdana;">5.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><b><span style="color: #555555; font-family: "verdana" , "sans-serif"; font-size: 9.0pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">Compounded Ratio:</span></b><span style="font-family: "verdana" , "sans-serif"; font-size: 9.0pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: 12.75pt; margin-left: 0.5in; text-align: left; text-indent: -0.25in;">
<!--[if !supportLists]--><span style="font-family: "verdana" , "sans-serif"; font-size: 9.0pt; mso-bidi-font-family: Verdana; mso-fareast-font-family: Verdana;">6.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span style="font-family: "verdana" , "sans-serif"; font-size: 9.0pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">The
compounded ratio of the ratios: (<i>a</i> : <i>b</i>), (<i>c</i> : <i>d</i>),
(<i>e</i> : <i>f</i>) is (<i>ace</i> : <i>bdf</i>).<o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: 12.75pt; margin-left: 0.5in; text-align: left; text-indent: -0.25in;">
<!--[if !supportLists]--><span style="font-family: "verdana" , "sans-serif"; font-size: 9.0pt; mso-bidi-font-family: Verdana; mso-fareast-font-family: Verdana;">7.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><b><span style="color: #555555; font-family: "verdana" , "sans-serif"; font-size: 9.0pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">Duplicate Ratios:</span></b><span style="font-family: "verdana" , "sans-serif"; font-size: 9.0pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: 12.75pt; margin-left: 0.5in; text-align: left;">
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.0pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">Duplicate ratio of (<i>a</i> : <i>b</i>) is (<i>a</i><sup>2</sup> : <i>b</i><sup>2</sup>).<o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: 12.75pt; margin-left: 0.5in; text-align: left;">
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.0pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">Sub-duplicate ratio of (<i>a</i> : <i>b</i>) is (<i>a</i> : <i>b</i>).<o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: 12.75pt; margin-left: 0.5in; text-align: left;">
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.0pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">Triplicate ratio of (<i>a</i> : <i>b</i>) is (<i>a</i><sup>3</sup> : <i>b</i><sup>3</sup>).<o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: 12.75pt; margin-left: 0.5in; text-align: left;">
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.0pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">Sub-triplicate ratio of (<i>a</i> : <i>b</i>) is (<i>a</i><sup>1/3</sup> : <i>b</i><sup>1/3</sup>).<o:p></o:p></span></div>
<table border="0" cellpadding="0" cellspacing="0" class="MsoNormalTable" style="line-height: 25.2pt; margin-left: 0.5in;">
<tbody>
<tr>
<td rowspan="2" style="padding: 0in 5.25pt 0in 0in;"><div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;">
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.0pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">If<o:p></o:p></span></div>
</td>
<td style="border-bottom: solid black 1.0pt; border: none; mso-border-bottom-alt: solid black .75pt; padding: 0in 0in 1.5pt 0in;" valign="bottom"><div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;">
<i><span style="font-family: "verdana" , "sans-serif"; font-size: 9.0pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">a</span></i><span style="font-family: "verdana" , "sans-serif"; font-size: 9.0pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
</td>
<td rowspan="2" style="padding: 0in 5.25pt 0in 5.25pt;"><div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;">
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.0pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">=<o:p></o:p></span></div>
</td>
<td style="border-bottom: solid black 1.0pt; border: none; mso-border-bottom-alt: solid black .75pt; padding: 0in 0in 1.5pt 0in;" valign="bottom"><div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;">
<i><span style="font-family: "verdana" , "sans-serif"; font-size: 9.0pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">c</span></i><span style="font-family: "verdana" , "sans-serif"; font-size: 9.0pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
</td>
<td rowspan="2" style="padding: 0in 5.25pt 0in 5.25pt;"><div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;">
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.0pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">, then<o:p></o:p></span></div>
</td>
<td style="border-bottom: solid black 1.0pt; border: none; mso-border-bottom-alt: solid black .75pt; padding: 0in 0in 1.5pt 0in;" valign="bottom"><div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;">
<i><span style="font-family: "verdana" , "sans-serif"; font-size: 9.0pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">a</span></i><span style="font-family: "verdana" , "sans-serif"; font-size: 9.0pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";"> + <i>b</i><o:p></o:p></span></div>
</td>
<td rowspan="2" style="padding: 0in 5.25pt 0in 5.25pt;"><div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;">
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.0pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">=<o:p></o:p></span></div>
</td>
<td style="border-bottom: solid black 1.0pt; border: none; mso-border-bottom-alt: solid black .75pt; padding: 0in 0in 1.5pt 0in;" valign="bottom"><div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;">
<i><span style="font-family: "verdana" , "sans-serif"; font-size: 9.0pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">c</span></i><span style="font-family: "verdana" , "sans-serif"; font-size: 9.0pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";"> + <i>d</i><o:p></o:p></span></div>
</td>
<td nowrap="" rowspan="2" style="padding: 0in 0in 0in 0in;"><div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;">
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.0pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">. [componendo and
dividendo]<o:p></o:p></span></div>
</td>
</tr>
<tr>
<td style="padding: 1.5pt 0in 0in 0in;" valign="top"><div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;">
<i><span style="font-family: "verdana" , "sans-serif"; font-size: 9.0pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">b</span></i><span style="font-family: "verdana" , "sans-serif"; font-size: 9.0pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
</td>
<td style="padding: 1.5pt 0in 0in 0in;" valign="top"><div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;">
<i><span style="font-family: "verdana" , "sans-serif"; font-size: 9.0pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">d</span></i><span style="font-family: "verdana" , "sans-serif"; font-size: 9.0pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
</td>
<td style="padding: 1.5pt 0in 0in 0in;" valign="top"><div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;">
<i><span style="font-family: "verdana" , "sans-serif"; font-size: 9.0pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">a</span></i><span style="font-family: "verdana" , "sans-serif"; font-size: 9.0pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";"> - <i>b</i><o:p></o:p></span></div>
</td>
<td style="padding: 1.5pt 0in 0in 0in;" valign="top"><div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;">
<i><span style="font-family: "verdana" , "sans-serif"; font-size: 9.0pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">c</span></i><span style="font-family: "verdana" , "sans-serif"; font-size: 9.0pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";"> - <i>d</i><o:p></o:p></span></div>
</td>
</tr>
</tbody></table>
<div class="MsoNormal" style="line-height: 12.75pt; margin-left: 0.5in; text-align: left; text-indent: -0.25in;">
<!--[if !supportLists]--><span style="font-family: "verdana" , "sans-serif"; font-size: 9.0pt; mso-bidi-font-family: Verdana; mso-fareast-font-family: Verdana;">8.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><b><span style="color: #555555; font-family: "verdana" , "sans-serif"; font-size: 9.0pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">Variations:</span></b><span style="font-family: "verdana" , "sans-serif"; font-size: 9.0pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: 12.75pt; margin-left: 0.5in; text-align: left;">
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.0pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">We say that <i>x</i> is directly proportional
to <i>y</i>, if <i>x</i> = <i>ky</i> for some constant <i>k</i> and
we write, <i>x</i> <!--[if gte vml 1]><v:shape
id="Picture_x0020_2" o:spid="_x0000_i1050" type="#_x0000_t75" alt="http://www.indiabix.com/_files/images/aptitude/1-sym-dpl.gif"
style='width:10.5pt;height:7.5pt;visibility:visible;mso-wrap-style:square'>
<v:imagedata src="file:///C:\Users\guru\AppData\Local\Temp\msohtmlclip1\01\clip_image003.png"
o:title="1-sym-dpl"/>
</v:shape><![endif]--><!--[if !vml]--><img alt="http://www.indiabix.com/_files/images/aptitude/1-sym-dpl.gif" border="0" height="10" src="file:///C:/Users/guru/AppData/Local/Temp/msohtmlclip1/01/clip_image003.png" v:shapes="Picture_x0020_2" width="14" /><!--[endif]--> <i>y</i>.<o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: 12.75pt; margin-left: 0.5in; text-align: left;">
<span style="font-family: "verdana" , "sans-serif"; font-size: 9.0pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">We say that <i>x</i> is inversely proportional
to <i>y</i>, if <i>xy</i> = k for some constant <i>k and</i><o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: 0.0001pt; text-align: center;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";">
</span><br />
<hr noshade="" size="1" style="color: #d4d4d4; text-align: center;" width="100%" />
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";">
</span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<b></b></div>
<div style="text-align: left;">
<b><b style="line-height: normal;"><span style="background: white; color: #5eac1a; font-family: "verdana" , "sans-serif"; font-size: 9.0pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">Read more:</span></b><span style="font-size: 12px; text-align: center;"><span style="color: #5eac1a; font-family: "verdana" , sans-serif;">https://mbaexamsprep.blogspot.com</span></span></b><br />
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<div class="MsoNormal" style="background-attachment: initial; background-clip: initial; background-color: white; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; line-height: 14.4pt; margin-bottom: 0.1in; text-align: left; vertical-align: middle;">
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<div style="line-height: 25.2pt;">
</div>
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<br /></div>
</div>
<div class="MsoNormal">
<!--[if gte vml 1]><v:shape id="_x0000_s1029" type="#_x0000_t75"
alt="" style='position:absolute;margin-left:0;margin-top:0;width:320.25pt;
height:470.25pt;z-index:251659264;mso-wrap-distance-left:0;
mso-wrap-distance-right:0;mso-position-horizontal:left;
mso-position-vertical-relative:line' o:allowoverlap="f">
<v:imagedata src="file:///C:\Users\guru\AppData\Local\Temp\msohtmlclip1\01\clip_image033.gif"
o:title="Image621"/>
<w:wrap type="square"/>
</v:shape><![endif]--><!--[if !vml]--><!--[endif]--><span lang="RU"><o:p></o:p></span></div>
</div>
guruhttp://www.blogger.com/profile/01591375209227117455noreply@blogger.comtag:blogger.com,1999:blog-7850540640045081515.post-18285411422213700282015-03-03T05:59:00.002-08:002019-09-14T18:55:03.648-07:00cord goemtry<div dir="ltr" style="text-align: left;" trbidi="on">
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<HTML
xmlns="http://www.w3.org/1999/xhtml"><HEAD><TITLE>Coordinate Geometry</TITLE>
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href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028.htm#Title">Contents</A></SPAN></P>
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href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028.htm#I">INTRODUCTION</A>
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href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028.htm#II">THE
COORDINATE SYSTEM</A>
<OL class=oTOCLevel2>
<LI><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028.htm#II_A">QUADRANTS,
AXES AND ORIGIN</A></LI></OL></LI>
<LI class=oTocLiClass><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028.htm#III">DISTANCE
FORMULA AND SECTION FORMULA</A>
<OL class=oTOCLevel2>
<LI><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028.htm#III_A">THE
DISTANCE FORMULA</A></LI>
<LI><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028.htm#III_B">THE
SECTION FORMULA</A></LI></OL></LI>
<LI class=oTocLiClass><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028.htm#IV">LINES</A>
<OL class=oTOCLevel2>
<LI><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028.htm#IV_A">SLOPE
OF A LINE</A></LI>
<LI><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028.htm#IV_B">EQUATION
OF A LINE</A></LI>
<LI><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028.htm#IV_C">POINT
OF INTERSECTION OF TWO LINES</A></LI>
<LI><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028.htm#IV_D">DISTANCE
OF A LINE FROM A POINT</A></LI>
<LI><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028.htm#IV_E">DISTANCE
BETWEEN TWO PARALLEL LINES</A></LI>
<LI><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028.htm#IV_F">ACUTE
ANGLE BETWEEN TWO LINES</A></LI></OL></LI>
<LI class=oTocLiClass><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028.htm#V">CHANGE
OF ORIGIN</A> </LI>
<LI class=oTocLiClass><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028.htm#VI">TRIANGLES</A>
</LI>
<LI class=oTocLiClass><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028.htm#VII">PARALLELOGRAM</A>
</LI>
<LI class=oTocLiClass><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028.htm#VIII">EQUATIONS
OF COMMON CURVES</A>
<OL class=oTOCLevel2>
<LI><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028.htm#VIII_A">EQUATION
OF A CIRCLE</A></LI>
<LI><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028.htm#VIII_B">EQUATION
OF AN ELLIPSE</A></LI></OL></LI></OL>
<P id=Title class=pTitle><SPAN class=Title>Coordinate Geometry</SPAN></P>
<HR>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<DIV id=028_I>
<TABLE id=028_I_bookmark class=tableh1 border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD
style="PADDING-BOTTOM: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 100%; PADDING-RIGHT: 0px; PADDING-TOP: 0px">
<OL class="oLevel1 Heading1">
<LI value=1><A id=I>INTRODUCTION</A></LI></OL></TD></TR></TBODY></TABLE>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal><B>Coordinate geometry</B> is a branch of
mathematics that uses the principles of algebra to study geometry. Figures like
lines and circles can be represented using algebraic equations and their
properties can be studied using these equations. Coordinate geometry also helps
us understand the behaviour of functions.</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>A one-dimensional coordinate system, used to
represent points, is the number line similar to the one that we have studied in
Number Systems.</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>A two-dimensional coordinate system is used
to represent two-dimensional figures like lines, circles and other
curves.</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P></DIV>
<DIV id=028_II>
<TABLE id=028_II_bookmark class=tableh1 border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD
style="PADDING-BOTTOM: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 100%; PADDING-RIGHT: 0px; PADDING-TOP: 0px">
<OL class="oLevel1 Heading1">
<LI value=2><A id=II>THE COORDINATE
SYSTEM</A></LI></OL></TD></TR></TBODY></TABLE>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>Two perpendicular number lines, XOX’ and
YOY’, intersecting at O form the coordinate system.</SPAN></P>
<P class=pLevel2><SPAN class=ListParagraph></SPAN></P>
<DIV id=028_II_A>
<TABLE id=028_II_A_bookmark class=tableh1 border=0 cellSpacing=0
cellPadding=0><TBODY>
<TR>
<TD
style="PADDING-BOTTOM: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 100%; PADDING-RIGHT: 0px; PADDING-TOP: 0px">
<OL class="oLevel2 Heading2">
<LI value=1><A id=II_A>QUADRANTS, AXES AND
ORIGIN</A></LI></OL></TD></TR></TBODY></TABLE>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pImage><SPAN><IMG alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_001.png"></SPAN></P>
<P class=pLevel3><SPAN class=ListParagraph>The line XOX’ is called the
<B><I>x-</I>axis</B> and the line YOY’ is called the
<B><I>y-</I>axis.</B></SPAN></P>
<P class=pLevel3><SPAN class=ListParagraph>The <I>x-</I>axis and the
<I>y-</I>axis are called the <B>coordinate axes.</B> The point O is called the
<B>origin.</B> The axes divide the plane on which they are drawn into four parts
known as <B>quadrants</B> and the plane is called the<I>x-y</I>
plane.</SPAN></P>
<P class=pLevel3><SPAN class=ListParagraph></SPAN></P>
<OL class="oLevel3 Heading3 ListStyleOL">
<LI value=1>COORDINATES OF A POINT</LI></OL>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pImage><SPAN><IMG alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_002.png"></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal>Any point on the <I>x-y</I> plane can be
identified by coordinates. The <I>x-</I>coordinate is the distance of the point
from the <I>y-</I>axis and the <I>y-</I>coordinate is the distance of the point
from the <I>x-</I>axis. The <I>x-</I>coordinate is also known as the
<B>abscissa</B> and the <I>y-</I>coordinate is also known as the
<B>ordinate.</B> Any point can be represented by an ordered pair of<I> x
</I>and<I> y </I>coordinates as (<I>x</I>, <I>y</I>). For example a point A with
<I>x-</I>coordinate 4 and <I>y-</I>coordinate 3 can be represented as A (4, 3).
The distance of this point A is 4 units from the <I>y-</I>axis and 3 units from
the <I>x-</I>axis.</SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal>The signs of the<I> x </I>and
<I>y-</I>coordinates of a point change depending on the quadrant in which it
lies.</SPAN></P>
<DIV class=tblCenter>
<TABLE class="visible_border_table tLevel4" cellSpacing=1 align=centre>
<TBODY>
<TR>
<TD class=visible_border_table><B>Quadrant</B></TD>
<TD class=visible_border_table><B><I>x</I>-coordinate</B></TD>
<TD class=visible_border_table><B><I>y</I>-coordinate</B></TD></TR>
<TR>
<TD class=visible_border_table>I</TD>
<TD class=visible_border_table>Positive</TD>
<TD class=visible_border_table>Positive</TD></TR>
<TR>
<TD class=visible_border_table>II</TD>
<TD class=visible_border_table>Negative</TD>
<TD class=visible_border_table>Positive</TD></TR>
<TR>
<TD class=visible_border_table>III</TD>
<TD class=visible_border_table>Negative</TD>
<TD class=visible_border_table>Negative</TD></TR>
<TR>
<TD class=visible_border_table>IV</TD>
<TD class=visible_border_table>Positive</TD>
<TD class=visible_border_table>Negative</TD></TR></TBODY></TABLE></DIV>
<P class=pLevel4><SPAN class=Normal><SPAN
style="MARGIN-LEFT: 25px"></SPAN></SPAN></P>
<P class=pImage><SPAN><IMG alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_003.png"></SPAN></P>
<P class=pLevel4><SPAN class=Note></SPAN></P>
<DIV class="divTextBox divColorNote divLevel4">
<P><SPAN class=Note>REMEMBER</SPAN></P>
<P><SPAN class=ListParagraph></SPAN></P>
<UL type=disc>
<LI>The points lying on the axes are not considered to be in any
quadrant.</LI></UL></DIV></DIV></DIV>
<DIV id=028_III>
<TABLE id=028_III_bookmark class=tableh1 border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD
style="PADDING-BOTTOM: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 100%; PADDING-RIGHT: 0px; PADDING-TOP: 0px">
<OL class="oLevel1 Heading1">
<LI value=3><A id=III>DISTANCE FORMULA AND SECTION
FORMULA</A></LI></OL></TD></TR></TBODY></TABLE>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<DIV id=028_III_A>
<TABLE id=028_III_A_bookmark class=tableh1 border=0 cellSpacing=0
cellPadding=0><TBODY>
<TR>
<TD
style="PADDING-BOTTOM: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 100%; PADDING-RIGHT: 0px; PADDING-TOP: 0px">
<OL class="oLevel2 Heading2">
<LI value=1><A id=III_A>THE DISTANCE
FORMULA</A></LI></OL></TD></TR></TBODY></TABLE>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>The distance between any two points lying on
a line parallel to the <I>x-</I>axis and having <I>x-</I>coordinates
<I>x</I><SUB>1</SUB> and <I>x</I><SUB>2</SUB> is |<I>x</I><SUB>2</SUB> –
<I>x</I><SUB>1</SUB>|.</SPAN></P>
<P class=pLevel3><SPAN class=Normal>Similarly, the distance between any two
points lying on a line parallel to the <I>y-</I>axis and having
<I>y-</I>coordinates <I>y</I><SUB>1</SUB> and <I>y</I><SUB>2</SUB> is
|<I>y</I><SUB>2</SUB> – <I>y</I><SUB>1</SUB>|.</SPAN></P>
<P class=pLevel3><SPAN class=Normal>Distance between two points
A(<I>x</I><SUB>1</SUB>, <I>y</I><SUB>1</SUB>) and B(<I>x</I><SUB>2</SUB>,
<I>y</I><SUB>2</SUB>) is given by</SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_004.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><B>Explanation:</B></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal></P></SPAN>
<P class=pImage><SPAN><IMG alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_005.png"></SPAN></P>
<P class=pLevel3><SPAN class=ListParagraph></SPAN></P>
<P class=pLevel3><SPAN class=Normal>Consider points A(<I>x</I><SUB>1</SUB>,
<I>y</I><SUB>1</SUB>), B(<I>x</I><SUB>2</SUB>, <I>y</I><SUB>2</SUB>),
C(<I>x</I><SUB>1</SUB>, <I>y</I><SUB>2</SUB>) as shown in the figure.</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>BC = (<I>x</I><SUB>1</SUB> –
<I>x</I><SUB>2</SUB>) and AC = (<I>y</I><SUB>1</SUB> <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/minus.png">
<I>y</I><SUB>2</SUB>)</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>Applying Pythagoras theorem in right angled
ΔABC,</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/therefore.png">
(AB)<SUP>2</SUP> = (BC)<SUP>2</SUP> + (AC)<SUP>2</SUP></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/therefore.png">
(AB)<SUP>2</SUP> = (<I>x</I><SUB>1</SUB> <IMG style="VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/minus.png">
<I>x</I><SUB>2</SUB>)<SUP>2</SUP> + (<I>y</I><SUB>1</SUB> <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/minus.png">
y<SUB>2</SUB>)<SUP>2</SUP></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>Thus,</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_006.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=028_solvedex_1>
<TABLE style="WIDTH: 100%" id=028_solvedex_1_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_1>Example
1:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>Find the distance between the points A(3, 4) and
B(7, <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/minus.png">3).</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>Using the formula for distance between two
points,</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_007.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_008.png">
<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_009.png">
units</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal></SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=028_solvedex_2>
<TABLE style="WIDTH: 100%" id=028_solvedex_2_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_2>Example
2:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>Distance between the points A(2, 3) and B(<I>x</I>,
<I>y</I>) is 5 units while the distance between points B(<I>x</I>,
<I>y</I>) and C(<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/minus.png">6,
1) is 13 units. Find the number of possible values for coordinates of B if
B lies in the first quadrant and its coordinates are integers.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>B lies in the first quadrant.</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/therefore.png">
Its coordinates are positive integers.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Distance between two points <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_010.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P><!--<p><span class="Normal"><span style="margin-left:25px"></span></span></p>--><SPAN
class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_011.png"></SPAN>
<P></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Squaring both sides,</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>(2 – <I>x</I>)<SUP>2</SUP> + (3 –
<I>y</I>)<SUP>2</SUP> = 25</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/therefore.png">
<I>x</I><SUP>2</SUP> + <I>y</I><SUP>2</SUP> – 4<I>x</I> – 6<I>y</I> =
12<SPAN style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN>…(i)</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Similarly,</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_012.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/therefore.png">
(<I>x</I> + 6)<SUP>2</SUP> + (<I>y</I> – 1)<SUP>2</SUP> = 169</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/therefore.png">
<I>x</I><SUP>2</SUP> + <I>y</I><SUP>2</SUP> + 12<I>x</I> – 2<I>y</I> =
132<SPAN style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN>…(ii)</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Subtracting (i) from (ii) we have,</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>16<I>x</I> + 4<I>y</I> = 120</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/therefore.png">
4<I>x</I> +<I> y </I>= 30</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/therefore.png"><I>
y </I>= 30 – 4<I>x</I></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>As<I> x </I>and<I> y </I>are positive integers,
substituting the values of<I> x </I>from 1 to 7 we have the possible
values of (<I>x</I>, <I>y</I>) as: (7, 2), (6, 6), (5, 10), (4, 14),(3,
18), (2, 22), (1, 26).</SPAN></P>
<P><SPAN class=Normal>But, since these values must satisfy equation (i)
and (ii) we get (6, 6) as the only possible pair satisfying the given
conditions.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Hence, possible number of values for coordinates of
B satisfying the given conditions is
1.</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV></DIV>
<DIV id=028_III_B>
<TABLE id=028_III_B_bookmark class=tableh1 border=0 cellSpacing=0
cellPadding=0><TBODY>
<TR>
<TD
style="PADDING-BOTTOM: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 100%; PADDING-RIGHT: 0px; PADDING-TOP: 0px">
<OL class="oLevel2 Heading2">
<LI value=2><A id=III_B>THE SECTION
FORMULA</A></LI></OL></TD></TR></TBODY></TABLE>
<OL class="oLevel3 Heading3 ListStyleOL">
<LI value=1>Internal Division</LI></OL>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pImage><SPAN><IMG alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_013.png"></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal>Coordinates of a point C(<I>x</I>,
<I>y</I>), which divides the line joining the points A(<I>x</I><SUB>1</SUB>,
<I>y</I><SUB>1</SUB>) and B(<I>x</I><SUB>2</SUB>, <I>y</I><SUB>2</SUB>)
internally in the ratio <I>a</I> : <I>b</I> are given by</SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_014.png"></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal>This is known as the section formula for
internal division.</SPAN></P>
<P class=pLevel4><SPAN class=ListParagraph></SPAN></P>
<OL class="oLevel3 Heading3 ListStyleOL">
<LI value=2>External Division</LI></OL>
<P class=pImage><SPAN><IMG alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_015.png"></SPAN></P>
<P class=pLevel4><SPAN class=ListParagraph></SPAN></P>
<P class=pLevel4><SPAN class=Normal>Coordinates of a point C(<I>x</I>,
<I>y</I>), which divides the line joining the points A(<I>x</I><SUB>1</SUB>,
<I>y</I><SUB>1</SUB>) and B(<I>x</I><SUB>2</SUB>, <I>y</I><SUB>2</SUB>)
externally in the ratio <I>a</I> : <I>b</I> is given by</SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_016.png"></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal>This is known as the <B>section formula for
external division.</B></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorNote divLevel4">
<P><SPAN class=Note>REMEMBER</SPAN></P>
<P><SPAN class=ListParagraph></SPAN></P>
<UL type=disc>
<LI>The formula for external division can be derived using the formula for
internal division by considering the fact that point A divides segment BC in
the ratio (<I>b</I> – <I>a</I>) : <I>a</I>.</LI></UL></DIV>
<P class=pLevel4><SPAN class=ListParagraph></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel4">
<DIV id=028_solvedex_3>
<TABLE style="WIDTH: 100%" id=028_solvedex_3_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_3>Example
3:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>Find the coordinates of the point M dividing line
segment AB internally in the ratio 3 : 7 if A <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/equal_equal.png">
(1, 6) and B <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/equal_equal.png">
(3, 11).</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>Let the coordinates of M be (<I>x</I>,
<I>y</I>).</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=pImage><IMG alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_017.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Applying section formula of internal
division,</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_018.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_019.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/therefore.png">
M <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/equal_equal.png">
(1.6, 7.5)</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<OL class="oLevel3 Heading3 ListStyleOL">
<LI value=3>The Midpoint Formula</LI></OL>
<P class=pLevel4><SPAN class=ListParagraph></SPAN></P>
<P class=pLevel4><SPAN class=Normal>If <I>a</I> : <I>b</I> <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/equal_equal.png">
1 : 1, C is the midpoint of line AB. Using the section formula, the coordinates
of C are:</SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_020.png"></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel4">
<DIV id=028_solvedex_4>
<TABLE style="WIDTH: 100%" id=028_solvedex_4_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_4>Example
4:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>If A <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/equal_equal.png">
(2, 3) and B <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/equal_equal.png">
(8, 9). Point C externally divides AB in the ratio 2 : 1. Find the
coordinates of C.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=pImage><IMG alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_021.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>As point C divides segment AB in the ratio 2 : 1, we
can say that B divides AC in the ratio 1 : 1.</SPAN></P><!--<p><span class="Normal">we can say that B divides AC in the ratio 1 : 1.</span></p>-->
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Thus, B is the mid-point of AC.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_022.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/therefore.png"><I>
x </I>= 14 and<I> y </I>= 15</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/therefore.png">
C <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/equal_equal.png">
(14, 15)</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel4><SPAN class=Normal></SPAN></P></DIV></DIV>
<DIV id=028_IV>
<TABLE id=028_IV_bookmark class=tableh1 border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD
style="PADDING-BOTTOM: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 100%; PADDING-RIGHT: 0px; PADDING-TOP: 0px">
<OL class="oLevel1 Heading1">
<LI value=4><A id=IV>LINES</A></LI></OL></TD></TR></TBODY></TABLE>
<P class=pLevel2><SPAN class=Normal><B>Positive and Negative
Angles</B></SPAN></P>
<P class=pLevel2><SPAN class=ListParagraph></SPAN></P>
<P class=pLevel2><SPAN class=Normal>The angles measured anti-clockwise from the
positive direction of the <I>x</I>-axis are considered to be positive. In the
following figure, the first two angles are positive.</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>The angles measured clockwise from the
positive direction of the <I>x</I>-axis are considered to be negative. The third
angle is a negative angle.</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN>
<P class=pImage><SPAN><IMG alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_023.png"></SPAN></P>
<DIV id=028_IV_A>
<TABLE id=028_IV_A_bookmark class=tableh1 border=0 cellSpacing=0
cellPadding=0><TBODY>
<TR>
<TD
style="PADDING-BOTTOM: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 100%; PADDING-RIGHT: 0px; PADDING-TOP: 0px">
<OL class="oLevel2 Heading2">
<LI value=1><A id=IV_A>SLOPE OF A LINE</A></LI></OL></TD></TR></TBODY></TABLE>
<P class=pLevel3><SPAN class=Normal>The <B>slope</B> of a line, generally
denoted by <I>m</I>, is the slant of a line. The slope of a horizontal line is
0. Its magnitude goes on increasing as we gradually make the line vertical as
shown.</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN>
<P class=pImage><SPAN><IMG alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_024.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>Clearly, the slope depends on the angle that
it makes with the horizontal. It can be calculated in two ways.</SPAN></P>
<OL class="oLevel3 ListParagraph ListStyleOL">
<LI value=1>If the angle made by the line with the positive direction of the
<I>x-</I>axis is known, and is say <I><IMG style="VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/theta.png"></I>,
the slope is given by:</LI></OL>
<P class=pLevel5><SPAN class=ListParagraph><I>m</I> = tan <I><IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/theta.png"></I></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel5><SPAN class=ListParagraph><I><IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/theta.png"></I>
is called the <B>inclination</B> of the line.</SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel5><SPAN class=ListParagraph>Slopes can be positive or negative
depending on <I><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/theta.png"></I>.
This is illustrated in the following table.</SPAN></P>
<P class=pLevel4><SPAN class=ListParagraph></SPAN></P>
<P class=pImage><SPAN><IMG alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_025.png"></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<OL class="oLevel3 ListParagraph ListStyleOL">
<LI value=2></LI></OL>
<P class=pImage><SPAN><IMG alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_026.png"></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal>If A(<I>x</I><SUB>1</SUB>,
<I>y</I><SUB>1</SUB>) and B(<I>x</I><SUB>2</SUB>, <I>y</I><SUB>2</SUB>) are any
two points on a line, the slope is given by:</SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_027.png"></SPAN></P>
<P class=pLevel4><SPAN class=ListParagraph></SPAN></P>
<P class=pLevel4><SPAN class=Normal><B>Slopes of Parallel Lines</B></SPAN></P>
<P class=pLevel4><SPAN class=ListParagraph></SPAN></P>
<P class=pLevel4><SPAN class=Normal>Parallel lines make equal angles with the
positive direction of the <I>x-</I>axis. Hence, the slopes of parallel lines are
always equal.</SPAN></P>
<P class=pLevel4><SPAN class=Normal><B>Slopes of Perpendicular
Lines</B></SPAN></P>
<P class=pLevel4><SPAN class=ListParagraph></SPAN></P>
<P class=pLevel4><SPAN class=Normal>Consider two perpendicular lines having
inclinations <I><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/theta.png"></I><SUB>1</SUB>
and <I><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/theta.png"></I><SUB>2</SUB>,
such that <I><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/theta.png"></I><SUB>2</SUB>
= (<I><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/theta.png"></I><SUB>1</SUB>
+ 90<SUP><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/degree.png"></SUP>).
Let their respective slopes be <I>m</I><SUB>1</SUB> and
<I>m</I><SUB>2</SUB>.</SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal>Then, <I>m</I><SUB>1</SUB> = tan <I><IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/theta.png"></I><SUB>1</SUB>
and <I>m</I><SUB>2</SUB> = tan <I><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/theta.png"></I><SUB>2</SUB></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/therefore.png">
<I>m</I><SUB>2 </SUB>= tan (<I><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/theta.png"></I><SUB>1</SUB>
+ 90) = –cot <I><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/theta.png"></I><SUB>1</SUB></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/therefore.png">
<I>m</I><SUB>1</SUB> <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/multiplication.png">
<I>m</I><SUB>2</SUB> = tan <I><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/theta.png"></I><SUB>1</SUB>
<IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/multiplication.png">
(–cot <I><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/theta.png"></I><SUB>1</SUB>)
= –1</SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P><!--<div class="divTextBox divColorListParagraph divLevel4">
<p><span class="ListParagraph"></span></p>
</div>-->
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorNote divLevel4">
<P><SPAN class=Note>REMEMBER</SPAN></P>
<P><SPAN class=ListParagraph></SPAN></P>
<UL type=disc>
<LI>Slope of the <I>x-</I>axis is 0 and the slope of the <I>y-</I>axis is
∞.</LI></UL></DIV>
<P class=pLevel4><SPAN class=ListParagraph></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel4">
<DIV id=028_solvedex_5>
<TABLE style="WIDTH: 100%" id=028_solvedex_5_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_5>Example
5:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>If the slope of a line is 1/2, then find the slope
of a line parallel to this line and a line perpendicular to this
line.</SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>Slope of two parallel lines is equal.</SPAN></P>
<P><SPAN class=Normal>Hence, slope of a line parallel to this line =
1/2.</SPAN></P>
<P><SPAN class=Normal>For two perpendicular lines,</SPAN></P>
<P><SPAN class=Normal><I>m</I><SUB>1</SUB> <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/multiplication.png">
<I>m</I><SUB>2</SUB> = <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/minus.png">1</SPAN></P>
<P><SPAN class=Normal>Here, <I>m</I><SUB>1</SUB> = 1/2</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/therefore.png">
1/2 <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/multiplication.png">
<I>m</I><SUB>2</SUB> = <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/minus.png">1</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/therefore.png">
<I>m</I><SUB>2</SUB> = <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/minus.png">2</SPAN></P>
<P><SPAN class=Normal>Hence, slope of a line perpendicular to the given
line is <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/minus.png">2.</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel4><SPAN class=ListParagraph></SPAN></P></DIV>
<DIV id=028_IV_B>
<TABLE id=028_IV_B_bookmark class=tableh1 border=0 cellSpacing=0
cellPadding=0><TBODY>
<TR>
<TD
style="PADDING-BOTTOM: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 100%; PADDING-RIGHT: 0px; PADDING-TOP: 0px">
<OL class="oLevel2 Heading2">
<LI value=2><A id=IV_B>EQUATION OF A
LINE</A></LI></OL></TD></TR></TBODY></TABLE>
<P class=pLevel3><SPAN class=ListParagraph></SPAN></P>
<P class=pLevel3><SPAN class=Normal>As stated earlier, coordinate geometry helps
us study geometry using algebraic principles. We can describe a line completely
using an algebraic equation. Now let us first try to understand how a line is
described.</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>We have seen the concept of slope. It
denotes the slant of a line. If we consider a line with slope 1, we know that it
makes an angle of tan<SUP>–1</SUP>(1) = 45<SUP><IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/degree.png"></SUP>
with the positive direction of the <I>x-</I>axis.</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN>
<P class=pImage><SPAN><IMG alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_028.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>All the above lines have slope 1. But all
the lines are different. If we know any one point on the line in addition to the
slope, we will know exactly where the line lies. Thus, to describe a line
completely, we need</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<OL class="oLevel3 ListParagraph ListStyleOL">
<LI value=1>A point on the line</LI></OL>
<OL class="oLevel3 ListParagraph ListStyleOL">
<LI value=2>Its slope</LI></OL>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>Using these two parameters, we can form the
equation of a line. The equation of a line can be stated in various forms as
described below.</SPAN></P>
<P class=pLevel4><SPAN class=ListParagraph></SPAN></P>
<OL class="oLevel3 Heading3 ListStyleOL">
<LI value=1>SLOPE-POINT FORM</LI></OL>
<P class=pLevel4><SPAN class=ListParagraph></SPAN></P>
<P class=pLevel4><SPAN class=Normal>The equation of a line having slope <I>m</I>
and passing through the point P(<I>x</I><SUB>1</SUB>, <I>y</I><SUB>1</SUB>) is
(<I>y</I> – <I>y</I><SUB>1</SUB>) = <I>m</I>(<I>x</I> –
<I>x</I><SUB>1</SUB>).</SPAN></P>
<P class=pLevel4><SPAN class=ListParagraph></SPAN></P>
<P class=pLevel4><SPAN class=Normal><B>Explanation:</B></SPAN></P>
<P class=pLevel4><SPAN class=Normal>If Q(<I>x</I>, <I>y</I>) is another point on
the line apart from P(<I>x</I><SUB>1</SUB>, <I>y</I><SUB>1</SUB>), the slope of
the line is</SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_029.png"></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal>But the slope is <I>m</I>.</SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_030.png"></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/therefore.png">
(<I>y</I> – <I>y</I><SUB>1</SUB>) = <I>m</I>(<I>x</I> –
<I>x</I><SUB>1</SUB>)</SPAN></P>
<P class=pLevel4><SPAN class=ListParagraph></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel4">
<DIV id=028_solvedex_6>
<TABLE style="WIDTH: 100%" id=028_solvedex_6_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_6>Example
6:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_031.png"></SPAN></P><!--<span class="Normal"><img style="vertical-align:middle" src="028_images/028_031.png" alt="" /></span></p>
<p><span class="Normal"> and passing through the point P(2, 3).</span></p>-->
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>The slope-point form of the equation of a line
is</SPAN></P>
<P><SPAN class=Normal>(<I>y</I> – <I>y</I><SUB>1</SUB>) =
<I>m</I>(<I>x</I> – <I>x</I><SUB>1</SUB>)</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/therefore.png">
The equation of the given line is</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_032.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/therefore.png">
4<I>y</I> – 12 = 3<I>x</I> – 6</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/therefore.png">
3<I>x</I> – 4<I>y</I> + 6 = 0</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel4><SPAN class=ListParagraph></SPAN></P>
<DIV class="divTextBox divColorNote divLevel4">
<P><SPAN class=Note>REMEMBER</SPAN></P>
<UL type=disc>
<LI>Any point on a line always satisfies the equation of the
line.</LI></UL></DIV>
<P class=pLevel4><SPAN class=ListParagraph></SPAN></P>
<OL class="oLevel3 Heading3 ListStyleOL">
<LI value=2>TWO POINT FORM</LI></OL>
<P class=pLevel4><SPAN class=ListParagraph></SPAN></P>
<P class=pLevel4><SPAN class=Normal>As stated earlier, we need a point on a line
and the slope to write the equation of the line. When we have two points, we can
find the slope. Using this slope and any one of the given points, the equation
of the line can be found.</SPAN></P>
<P class=pLevel4><SPAN class=Normal>The equation of a line passing through
points P(<I>x</I><SUB>1</SUB>, <I>y</I><SUB>1</SUB>) and Q(<I>x</I><SUB>2</SUB>,
<I>y</I><SUB>2</SUB>), when <I>x</I><SUB>1</SUB> ≠ <I>x</I><SUB>2</SUB>, is
given by</SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_033.png"></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal><B>Explanation:</B></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal>Using the slope-point form,</SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal>(<I>y</I> – <I>y</I><SUB>1</SUB>) =
<I>m</I>(<I>x</I> – <I>x</I><SUB>1</SUB>)</SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal>Using the definition of slope,</SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_034.png"></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_035.png"></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_036.png"></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel4">
<DIV id=028_solvedex_7>
<TABLE style="WIDTH: 100%" id=028_solvedex_7_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_7>Example
7:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>Find the equation of the line, which passes through
the points (3, 4) and (1, 2).</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>Two-point form of the equation of a line
is</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_037.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>This line passes through (3, 4) and (1,
2).</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Thus,</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_038.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><I>y</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/minus.png">
4 =<I> x </I><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/minus.png">
3</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><I>y</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/minus.png"><I>
x </I><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/minus.png">
1 = 0</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>This is the equation of the required
line.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>The equation of the line can be verified by
substituting points (3, 4) and (1, 2) in the </SPAN></P><!--<p><span class="Normal">substituting points (3, 4) and (1, 2) in the</span></p>-->
<P><SPAN class=Normal>Both these points lie on the line and hence, must
satisfy the equation of the
line.</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<OL class="oLevel3 Heading3 ListStyleOL">
<LI value=3>SLOPE-INTERCEPT FORM</LI></OL>
<P class=pLevel4><SPAN class=ListParagraph></SPAN></P>
<P class=pLevel4><SPAN class=Normal>The equation of a line having slope <I>m</I>
and making an intercept <I>c</I> on the <I>y-</I>axis is given by</SPAN></P>
<P class=pLevel4><SPAN class=Normal><I>y</I> = <I>mx</I> + <I>c</I></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal><B>Explanation:</B></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal>Let P(<I>x</I>, <I>y</I>) be any point on
the line. Let the line meet the <I>y-</I>axis at point Q. As the
<I>y-</I>intercept of the line is <I>c</I>, Q <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/equal_equal.png">
(0, <I>c</I>).</SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal>Using the slope-point form,</SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal>(<I>y</I> – <I>y</I><SUB>1</SUB>) =
<I>m</I>(<I>x</I> – <I>x</I><SUB>1</SUB>)</SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_039.png"></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/therefore.png"><I>
y </I>= <I>mx</I> + <I>c</I></SPAN></P>
<P class=pLevel4><SPAN class=ListParagraph></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel4">
<DIV id=028_solvedex_8>
<TABLE style="WIDTH: 100%" id=028_solvedex_8_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_8>Example
8:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=pImage><IMG alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_040.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>In the above figure, the angle made by the line AB
with the positive direction of the <I>x</I>-axis is 135<SUP><IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/degree.png"></SUP>
and OB = 3. Find equation of the line AB.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>Using the slope-intercept form,</SPAN></P>
<P><SPAN class=Normal><I>y</I> = <I>mx</I> + <I>c</I></SPAN></P>
<P><SPAN class=Normal><I>m</I> = tan <I><IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/theta.png"></I>
= tan 135<SUP><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/degree.png"></SUP>=
–tan 45<SUP><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/degree.png"></SUP>
= –1</SPAN></P>
<P><SPAN class=Normal><I>c</I> = OB = 3</SPAN></P>
<P><SPAN class=Normal>Hence, equation of the line is <I>y</I> = –<I>x</I>
+ 3</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel4><SPAN class=ListParagraph></SPAN></P>
<P class=pLevel4><SPAN class=Normal>The general form of the equation of a line
is given as <I>Ax</I> + <I>By</I> + <I>C</I> = 0.</SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal>This can also be written as <I>By</I> = <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/minus.png"><I>Ax</I>
<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/minus.png"><I>C</I></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal>Hence,</SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_041.png"></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal>Comparing with the slope-intercept form<I> y
</I>= <I>mx</I> + <I>c</I>, we get,</SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_042.png"></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_043.png"></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel4">
<DIV id=028_solvedex_9>
<TABLE style="WIDTH: 100%" id=028_solvedex_9_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_9>Example
9:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>Find the equation of a line parallel to 3<I>y</I> –
4<I>x</I> + 7 = 0 and passing through the point (1, 2).</SPAN></P><!--<p><span class="Normal">3<i>y</i> – 4<i>x</i> + 7 = 0 and passing through the point</span></p>
<p><span class="Normal">(1, 2).</span></p>-->
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>3<I>y</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/minus.png">
4<I>x</I> + 7 = 0 can be written as <IMG style="VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/minus.png">4<I>x</I>
+ 3<I>y</I> + 7 = 0.</SPAN></P>
<P><SPAN class=Normal>Comparing with <I>Ax</I> + <I>By</I> + <I>C</I> =
0,</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><I>A </I>= <IMG style="VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/minus.png">4,
<I>B</I> = 3 and <I>C</I> = 7</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Slope of the required line can be given
as:</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_044.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_045.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P><!--<p><span class="Normal"><img style="vertical-align:middle" src="028_images/028_045.png" alt="" /></span></p>-->
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>We are given a point (1, 2) lying on this
line.</SPAN></P>
<P><SPAN class=Normal>So, using the slope point form,</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>(<I>y</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/minus.png">
<I>y</I><SUB>1</SUB>) = <I>m</I>(<I>x</I> <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/minus.png">
<I>x</I><SUB>1</SUB>)</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_046.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/therefore.png">
3<I>y</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/minus.png">
6 = 4<I>x</I> – 4</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/therefore.png">
4<I>x</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/minus.png">
3<I>y</I> + 2 = 0</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>This is the equation of the required
line.</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel4">
<DIV id=028_solvedex_10>
<TABLE style="WIDTH: 100%" id=028_solvedex_10_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_10>Example
10:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>Find the slope of a line perpendicular to 2<I>y</I>
= 4<I>x</I> + 3.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>2<I>y</I> = 4<I>x</I> + 3</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/therefore.png">
4<I>x</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/minus.png">
2<I>y</I> + 3 = 0</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Comparing with <I>Ax </I>+ <I>By</I> + <I>C</I> = 0,
we get</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><I>A</I> = 4, <I>B</I> = <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/minus.png">2,
<I>C</I> = 3</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/therefore.png">
Slope (<I>m</I>) = <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/minus.png"><I>A</I>/<I>B</I>
= 2</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>The product of the slope of a line and that of a
line perpendicular to it is <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/minus.png">1.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_047.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal></SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel4><SPAN class=ListParagraph></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel4">
<DIV id=028_solvedex_11>
<TABLE style="WIDTH: 100%" id=028_solvedex_11_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_11>Example
11:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>A line passes through points (1, 3) and (7, 9). Find
the equation of a line perpendicular to this line and passing through (2,
5).</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>Slope of the line passing through (1, 3) and (7,
9),</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_048.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_049.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_050.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/therefore.png">
Slope of a line perpendicular to the given line is <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/minus.png">1.</SPAN></P>
<P><SPAN class=Normal>Also, we know that point (2, 5) lies on this
line.</SPAN></P>
<P><SPAN class=Normal>So, we now use the slope-point form,</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>(<I>y</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/minus.png">
<I>y</I><SUB>1</SUB>) = <I>m</I>(<I>x</I> <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/minus.png">
<I>x</I><SUB>1</SUB>)</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Here, <I>x</I><SUB>1</SUB> = 2, <I>y</I><SUB>1</SUB>
= 5 and <I>m</I> = <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/minus.png">1.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/therefore.png">
(<I>y</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/minus.png">
5) = <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/minus.png">1(<I>x</I>
<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/minus.png">
2)</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/therefore.png"><I>
y </I><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/minus.png">
5 = <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/minus.png"><I>x</I>
+ 2</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/therefore.png"><I>
x </I>+<I> y </I>= 7</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>This is the equation of the required
line.</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel4><SPAN class=ListParagraph></SPAN></P>
<OL class="oLevel3 Heading3 ListStyleOL">
<LI value=4>TWO-INTERCEPT FORM</LI></OL>
<P class=pLevel4><SPAN class=ListParagraph></SPAN></P>
<P class=pLevel4><SPAN class=Normal>The equation of a line forming intercepts
<I>a</I> and <I>b</I> (<I>a</I> ≠ 0 and <I>b</I> ≠ 0) on the <I>x-</I>axis and
<I>y-</I>axis respectively, is</SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_051.png"></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal><B>Explanation:</B></SPAN></P>
<P class=pLevel4><SPAN class=Normal>The line forms intercepts <I>a</I> and
<I>b</I> on the <I>x-</I>axis and <I>y-</I>axis respectively.</SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/therefore.png">
(<I>a</I>, 0) and (0, <I>b</I>) are two points on the line.</SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal>Using the two points form of the equation of
a line,</SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN>
<P class=pImage><SPAN><IMG alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_052.png"></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_053.png"></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_054.png"></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/therefore.png">
<I>ay</I> + <I>bx</I> = <I>ab</I></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_055.png"></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_056.png"></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P><!--<div class="divTextBox divColorNormal divLevel4">-->
<P><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=ListParagraph></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel4">
<DIV id=028_solvedex_12>
<TABLE style="WIDTH: 100%" id=028_solvedex_12_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_12>Example
12:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%"><!--<p><span class="ExampleHeading">Example 12:</span></p>-->
<P><SPAN class=Normal>Find the perimeter of a triangle formed by the
<I>x-</I>axis, <I>y-</I>axis and the line 3<I>x</I> + 4<I>y</I> =
12.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>3<I>x</I> + 4<I>y</I> = 12</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_057.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_058.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/therefore.png">
The <I>x</I>-intercept of the line is 4 and the <I>y</I>-intercept is 3.
The triangle can be constructed as shown.</SPAN></P>
<P><SPAN class=pImage><IMG alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_059.png"></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/therefore.png">
OB = 4 and OA = 3</SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_060.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/therefore.png">
Perimeter of the triangle = 3 + 4 + 5 = 12
units</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV></DIV>
<P class=pLevel4><SPAN class=ListParagraph></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel4">
<DIV id=028_solvedex_13>
<TABLE style="WIDTH: 100%" id=028_solvedex_13_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_13>Example
13:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>Consider a triangle drawn on the <I>x </I>- <I>y</I>
plane with its three vertices at (41, 0), (0, 41) and (0, 0), each vertex
being represented by its (<I>x</I>, <I>y</I>) coordinates. The number of
points with integer coordinates inside the triangle (excluding all the
points on the boundary) is</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><B>[CAT 2005]</B></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>(1) 780<SPAN style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN>(2) 800</SPAN></P>
<P><SPAN class=Normal>(3) 820<SPAN style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN>(4) 741</SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=pImage><IMG alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_061.png"></SPAN></P>
<P><SPAN class=Normal>The points satisfying the equations <I>x</I> +
<I>y</I> < 41, <I>y</I> > 0, <I>x</I> > 0 lie inside the
triangle.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Integer solutions of <I>x</I> + <I>y</I> < 41 can
be found as follows:</SPAN></P><!--<p><span class="Normal"> </span></p>-->
<P><SPAN class=Normal>If <I>x</I> + <I>y</I> = 40, then</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>(<I>x</I>, <I>y</I>) could be (1, 39), (2,
38), …, (39, 1) ...(39 solutions)</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal> If <I>x</I> + <I>y</I> = 39, then</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>(<I>x</I>, <I>y</I>) could be (1, 38), (2, 37), …,
(38, 1) ...(38 solutions)</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>If <I>x</I> + <I>y</I> = 38, we get 37 solutions and
so on till <I>x</I> + <I>y</I> = 2 ... (1 solution)</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Thus, there are 39 <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/multiplication.png">
40/2 = 780 integer solutions to <I>x</I> + <I>y</I> < 41.</SPAN></P>
<P><SPAN class=Normal>The number of points with integer coordinates lying
inside the triangle = 780</SPAN></P>
<P><SPAN class=Normal>Hence,<B> option
1.</B></SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel4><SPAN class=ListParagraph></SPAN></P>
<DIV class="divTextBox divColorNote divLevel4">
<P><SPAN class=Note>REMEMBER</SPAN></P>
<P><SPAN class=ListParagraph></SPAN></P>
<UL type=disc>
<LI>The equation of the <I>x-</I>axis is<I> y </I>= 0 and the equation of the
<I>y-</I>axis is<I> x </I>= 0.</LI></UL></DIV>
<P class=pLevel4><SPAN class=ListParagraph></SPAN></P></DIV>
<DIV id=028_IV_C>
<TABLE id=028_IV_C_bookmark class=tableh1 border=0 cellSpacing=0
cellPadding=0><TBODY>
<TR>
<TD
style="PADDING-BOTTOM: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 100%; PADDING-RIGHT: 0px; PADDING-TOP: 0px">
<OL class="oLevel2 Heading2">
<LI value=3><A id=IV_C>POINT OF INTERSECTION OF TWO
LINES</A></LI></OL></TD></TR></TBODY></TABLE>
<P class=pLevel3><SPAN class=ListParagraph></SPAN></P>
<P class=pLevel3><SPAN class=Normal>The point, at which two lines intersect, can
be found by solving the equations of the two lines simultaneously. The solution
satisfies the equations of both the lines and is hence the point of intersection
of the two lines.</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=028_solvedex_14>
<TABLE style="WIDTH: 100%" id=028_solvedex_14_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_14>Example
14:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>At what point do the lines 2<I>x</I> + 3<I>y</I> = 5
and <I>x </I>+<I> y </I>= 2 intersect?</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>2<I>x</I> + 3<I>y</I> = 5<SPAN
style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN><SPAN style="MARGIN-LEFT: 25px"></SPAN>…
(i)</SPAN></P>
<P><SPAN class=Normal><I>x</I>+<I> y </I>= 2 <SPAN
style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN><SPAN style="MARGIN-LEFT: 37px"></SPAN>…
(ii)</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Multiplying (ii) by 2 we get</SPAN></P>
<P><SPAN class=Normal>2<I>x</I> + 2<I>y</I> = 4 <SPAN
style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN><SPAN style="MARGIN-LEFT: 25px"></SPAN>…
(iii)</SPAN></P>
<P><SPAN class=Normal>Subtracting (iii) from (i) we get</SPAN></P>
<P><SPAN class=Normal><I>y</I> = 1</SPAN></P>
<P><SPAN class=Normal>Substituting<I> y </I>= 1 in (ii) we get</SPAN></P>
<P><SPAN class=Normal><I>x</I> + 1 = 2</SPAN></P>
<P><SPAN class=Normal>Hence,<I> x </I>= 1</SPAN></P>
<P><SPAN class=Normal>Thus the two lines intersect when<I> x </I>= 1
and<I> y </I>= 1, i.e. at (1,
1).</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel3><SPAN class=ListParagraph></SPAN></P>
<DIV class="divTextBox divColorNote divLevel3">
<P><SPAN class=Note>REMEMBER</SPAN></P>
<P><SPAN class=ListParagraph></SPAN></P>
<UL type=disc>
<LI>Three or more lines are said to be concurrent lines when all of them pass
through a common point.</LI></UL>
<UL type=disc>
<LI>If three points A, B, C are collinear (A <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/minus.png">
B <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/minus.png">
C) then any one of the following conditions must be true.</LI></UL>
<OL class=ListParagraph>
<LI value=1>Area of triangle ABC = 0</LI></OL>
<OL class=ListParagraph>
<LI value=2>Slope of AB = slope of BC = slope of AC</LI></OL>
<OL class=ListParagraph>
<LI value=3>AB + BC = AC</LI></OL></DIV>
<P class=pLevel4><SPAN class=ListParagraph></SPAN></P></DIV>
<DIV id=028_IV_D>
<TABLE id=028_IV_D_bookmark class=tableh1 border=0 cellSpacing=0
cellPadding=0><TBODY>
<TR>
<TD
style="PADDING-BOTTOM: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 100%; PADDING-RIGHT: 0px; PADDING-TOP: 0px">
<OL class="oLevel2 Heading2">
<LI value=4><A id=IV_D>DISTANCE OF A LINE FROM A
POINT</A></LI></OL></TD></TR></TBODY></TABLE>
<P class=pLevel3><SPAN class=ListParagraph></SPAN></P>
<P class=pLevel3><SPAN class=Normal>The distance of a point
P(<I>x</I><SUB>1</SUB>, <I>y</I><SUB>1</SUB>) from the line
<I>Ax</I> + <I>By</I> + <I>C</I> = 0 is</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_062.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN>
<P class=pImage><SPAN><IMG alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_063.png"></SPAN></P>
<P class=pLevel3><SPAN class=ListParagraph></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=028_solvedex_15>
<TABLE style="WIDTH: 100%" id=028_solvedex_15_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_15>Example
15:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>Find the distance of a point (1, 2) from the line
2<I>x</I> + 4<I>y</I> + 3 = 0.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>Comparing the given equation with <I>Ax</I> + <I>By
</I>+ <I>C</I> = 0 we get</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><I>A</I> = 2, <I>B</I> = 4, <I>C</I> = 3</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>P <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/equal_equal.png">
(1, 2)</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><I>x</I><SUB>1</SUB> = 1, <I>y</I><SUB>1</SUB> =
2</SPAN></P>
<P><SPAN class=Normal></SPAN></P><SPAN class=Normal><IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_064.png"></SPAN>
<P></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/therefore.png">
The distance of the point (1, 2) from the given line can be given
as</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_065.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_066.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_067.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_068.png"></SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=028_solvedex_16>
<TABLE style="WIDTH: 100%" id=028_solvedex_16_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_16>Example
16:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>An isosceles triangle has vertices of the unequal
side at (3, 5) and (–5, –1). The area of the triangle is 12.5 sq. units.
Find the third vertex of the triangle.</SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=pImage><IMG alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_069.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Let ΔABC be the isosceles triangle with AB =
AC.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Let B <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/equal_equal.png">
(3, 5) and C <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/equal_equal.png">
(–5, –1).</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_070.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>As area of the triangle is 12.5 sq.
units,</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_071.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/therefore.png">
AD = 2.5 units</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>As ΔABC is an isosceles triangle, D is the mid-point
of BC.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_072.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/therefore.png">
D <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/equal_equal.png">
(–1, 2)</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Let A <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/equal_equal.png">
(<I>m</I>, <I>n</I>).</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_073.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_074.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_075.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/therefore.png">
3<I>n</I> + 4<I>m</I> = 2<SPAN style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN>…(i)</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_076.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/therefore.png">
3<I>x</I> – 4<I>y</I> + 11 = 0</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Using the formula for the distance of a point from a
line, we have,</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_077.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/therefore.png">
<IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/Plus_minus.png">12.5
= 3<I>m</I> – 4<I>n</I> + 11</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/therefore.png">
3<I>m</I> – 4<I>n</I> = 1.5<SPAN style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN>…(ii) or</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>3<I>m</I> – 4<I>n</I> = –23.5<SPAN
style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 18px"></SPAN>…(iii)</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Solving (i) and (ii) simultaneously,</SPAN></P>
<P><SPAN class=Normal><I>m</I> = 0.5 and <I>n</I> = 0</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Solving (i) and (iii) simultaneously,</SPAN></P>
<P><SPAN class=Normal><I>m</I> = –2.5 and <I>n</I> = 4</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/therefore.png">
The third vertex of the triangle could be (0.5, 0) or (–2.5,
4).</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel3><SPAN class=ListParagraph></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=028_solvedex_17>
<TABLE style="WIDTH: 100%" id=028_solvedex_17_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_17>Example
17:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>ΔABC is a right triangle with <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/symb20.png">B
= 90<SUP><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/degree.png"></SUP>.
A <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/equal_equal.png">
(1, 10), B <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/equal_equal.png">
(1, 2) and C <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/equal_equal.png">
(7, 2). If BD <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/symb21.png">
AC, find the coordinates of D.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=pImage><IMG alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_078.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_079.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_080.png"></SPAN></P><!--<p><span class="Normal"><span style="margin-left:25px"></span></span></p>-->
<P><SPAN class=Normal>Let D <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/equal_equal.png">
(<I>h</I>, <I>k</I>)</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_081.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/therefore.png">
3<I>h</I> – 4<I>k</I> = –5<SPAN style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN>…(i)</SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_082.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/therefore.png">
4<I>x</I> + 3<I>y</I> = 34</SPAN></P>
<P><SPAN class=Normal>As D lies on this line,</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/therefore.png">
4<I>h</I> + 3<I>k</I> = 34<SPAN style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 45px"></SPAN>…(ii)</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Solving (i) and (ii) simultaneously,</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_083.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_084.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel3><SPAN class=ListParagraph></SPAN></P></DIV>
<DIV id=028_IV_E>
<TABLE id=028_IV_E_bookmark class=tableh1 border=0 cellSpacing=0
cellPadding=0><TBODY>
<TR>
<TD
style="PADDING-BOTTOM: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 100%; PADDING-RIGHT: 0px; PADDING-TOP: 0px">
<OL class="oLevel2 Heading2">
<LI value=5><A id=IV_E>DISTANCE BETWEEN TWO PARALLEL
LINES</A></LI></OL></TD></TR></TBODY></TABLE>
<P class=pLevel3><SPAN class=ListParagraph></SPAN></P>
<P class=pLevel3><SPAN class=Normal>The distance between two parallel lines
<I>Ax</I> + <I>By</I> + <I>C</I><SUB>1</SUB> = 0 and <I>Ax</I> + <I>By</I> +
<I>C</I><SUB>2</SUB> = 0 is given by</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_085.png"></SPAN></P>
<P class=pLevel3><SPAN class=ListParagraph></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=028_solvedex_18>
<TABLE style="WIDTH: 100%" id=028_solvedex_18_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_18>Example
18:</A></SPAN></P></TD></TR></TBODY></TABLE><!--<table cellpadding="0" cellspacing="0" border="0"><tr><td style="width:100%; background-color:Transparent">-->
<P><SPAN class=Normal>Opposite sides of a regular hexagon lie on the lines <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_086.png">
Find the area of the hexagon.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=pImage><IMG alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_087.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>From the figure, GH is the distance between the two
parallel lines and this is twice the height of ΔAOB.</SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_088.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_089.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Let <I>s</I> be the side of the hexagon.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_090.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_091.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_092.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_093.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P></TD></TR></TABLE></DIV></DIV>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=028_solvedex_19>
<TABLE style="WIDTH: 100%" id=028_solvedex_19_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_19>Example
19:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>Find the minimum distance between the lines
2<I>x</I> + 3<I>y</I> + 6 = 0 and 2<I>x </I>– 3<I>y</I> + 4 =
0.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>Slope of first line = <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/minus.png">2/3.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Slope of second line = 2/3.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/symb23.png">
The two lines are not parallel they will intersect and thus the minimum
distance between them is 0.</SPAN></P>
<P><SPAN class=Normal></SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel3><SPAN class=ListParagraph></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=028_solvedex_20>
<TABLE style="WIDTH: 100%" id=028_solvedex_20_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_20>Example
20:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_094.png">to
the given line and 4 units from it. A possible equation for L
is:</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><B>[FMS 2010]</B></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_095.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_096.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_097.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>The distance between (0, 6) and L should be 4
units.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Distance of a line <I>Ax</I> + <I>By</I> + <I>C</I>
= 0 from a point (<I>x</I><SUB>1</SUB>, <I>y</I><SUB>1</SUB>)</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_098.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Consider option 1.</SPAN></P>
<P><SPAN class=Normal>Distance of the given line from the point (0, 6)
<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_099.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_100.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_101.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/therefore.png">
Option 1 is correct.</SPAN></P>
<P><SPAN class=Normal>Hence<B>, option
1.</B></SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel3><SPAN class=ListParagraph></SPAN></P></DIV>
<DIV id=028_IV_F>
<TABLE id=028_IV_F_bookmark class=tableh1 border=0 cellSpacing=0
cellPadding=0><TBODY>
<TR>
<TD
style="PADDING-BOTTOM: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 100%; PADDING-RIGHT: 0px; PADDING-TOP: 0px">
<OL class="oLevel2 Heading2">
<LI value=6><A id=IV_F>ACUTE ANGLE BETWEEN TWO
LINES</A></LI></OL></TD></TR></TBODY></TABLE>
<P class=pLevel3><SPAN class=ListParagraph></SPAN></P>
<P class=pLevel3><SPAN class=Normal>Let <I><IMG style="VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/theta.png"></I>
be the acute angle between two lines with slopes <I>m</I><SUB>1</SUB> and
<I>m</I><SUB>2</SUB> respectively (<I>m</I><SUB>1</SUB> <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/multiplication.png">
<I>m</I><SUB>2</SUB> ≠ –1).</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_102.png"></SPAN></P>
<P class=pLevel3><SPAN class=ListParagraph></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=028_solvedex_21>
<TABLE style="WIDTH: 100%" id=028_solvedex_21_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_21>Example
21:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Find the acute angle between lines<I> x </I>+<I> y
</I>= 0 and<I> x </I><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/minus.png"><I>
y </I>= 0.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>Slope of first line = <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/minus.png">1</SPAN></P>
<P><SPAN class=Normal>Slope of second line = 1</SPAN></P>
<P><SPAN class=Normal>As the product of the slopes of the two lines is –1,
the lines are perpendicular to each
other.</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel3><SPAN class=ListParagraph></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=028_solvedex_22>
<TABLE style="WIDTH: 100%" id=028_solvedex_22_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_22>Example
22:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>ΔABC is an isosceles right angled triangle with <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/symb20.png">B
= 90<SUP><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/degree.png"></SUP>,
A <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/equal_equal.png">
(0, 3) and C <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/equal_equal.png">
(–2, 7). Find the coordinates of B if it lies in the first
quadrant.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=pImage><IMG alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_103.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_104.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>m<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/symb20.png">ACB
= 45<SUP><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/degree.png"></SUP></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Let slope of BC be <I>m</I><SUB>1</SUB>.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Using the formula for the angle between two lines we
have,</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_105.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_106.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_107.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_108.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Let the slope of AB be
<I>m</I><SUB>2</SUB>.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_109.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Now we can find the equations of lines BC and AB
using the slope-point form.</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/therefore.png">
Equation of BC is<I> x </I>+ 3<I>y</I> = 19 or<I> y </I>= 3<I>x</I> +
13</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Equation of AB is<I> y </I>= 3<I>x</I> + 3 or<I> x
</I>+ 3<I>y</I> = 9</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Solving simultaneously,</SPAN></P>
<P><SPAN class=Normal>B <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/equal_equal.png">
(1, 6) or B <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/equal_equal.png">
(–3, 4)</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>But as B lies in the first quadrant, B <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/equal_equal.png">
(1, 6).</SPAN></P>
<P><SPAN class=Normal></SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel3><SPAN class=ListParagraph></SPAN></P></DIV>
<DIV></DIV>
<DIV id=028_V>
<TABLE id=028_V_bookmark class=tableh1 border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD
style="PADDING-BOTTOM: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 100%; PADDING-RIGHT: 0px; PADDING-TOP: 0px">
<OL class="oLevel1 Heading1">
<LI value=5><A id=V>CHANGE OF ORIGIN</A></LI></OL></TD></TR></TBODY></TABLE>
<P class=pLevel2><SPAN class=ListParagraph></SPAN></P>
<P class=pLevel2><SPAN class=Normal>The origin of the coordinate system is (0,
0). If we wish to shift the origin to another location having coordinates
(<I>h</I>, <I>k</I>), axes remaining parallel, then the coordinates of a point
P(<I>x</I>, <I>y</I>) become (<I>x</I> – <I>h</I>,<I> y </I>– <I>k</I>) with
respect to the new origin.</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN>
<P class=pImage><SPAN><IMG alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_110.png"></SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel2">
<DIV id=028_solvedex_23>
<TABLE style="WIDTH: 100%" id=028_solvedex_23_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_23>Example
23:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>On shifting the origin to a point P, axes remaining
parallel, the coordinates of a point (5, 9) change to (1, –8). Find the
coordinates of P.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>Let P <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/equal_equal.png">
(<I>h</I>, <I>k</I>)</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/therefore.png">
By using the concept of change of origin,</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>1 = 5 – <I>h</I> and –8 = 9 – <I>k</I></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/therefore.png">
<I>h</I> = 4 and <I>k</I> = 17</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/therefore.png">
P <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/equal_equal.png">
(4, 17)</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV><!--<p class="pLevel2"><span class="Normal"></span></p>
<div class="divTextBox divColorNormal divLevel2">
<p><span class="Normal"></span></p>
</div>
<p class="pLevel2"><span class="Normal"></span></p>
<div class="divTextBox divColorNormal divLevel2">
<p><span class="Normal"></span></p>-->
<P class=pLevel3><SPAN class=ListParagraph></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel2">
<DIV id=028_solvedex_24>
<TABLE style="WIDTH: 100%" id=028_solvedex_24_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_24>Example
24:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%"><!--<p><span class="ExampleHeading">Example 24:</span></p>-->
<P><SPAN class=Normal>The equation of a curve is <I>x</I><SUP>2</SUP> +
<I>y</I><SUP>2</SUP> = 25. If the origin is shifted to point (5, 3), axes
remaining parallel, find the new equation of the curve.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>Let (<I>x</I>, <I>y</I>) change to (<I>X</I>,
<I>Y</I>) when the origin is shifted.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Then,<I> X </I>=<I> x </I>– 5 and<I> Y </I>=<I> y
</I>– 3</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/therefore.png"><I>
x </I>=<I> X </I>+ 5 and<I> y </I>=<I> Y </I>+ 3</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/therefore.png">
To find the new equation of the curve in the new system, we replace<I> x
</I>and<I> y </I>in the original equation with (<I>X</I> + 5) and
(<I>Y</I> + 3) respectively.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/therefore.png">
(<I>X</I> + 5)<SUP>2</SUP> + (<I>Y</I> + 3)<SUP>2</SUP> = 25</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/therefore.png">
<I>X</I><SUP>2</SUP> + <I>Y</I><SUP>2</SUP> + 10<I>X</I> + 6<I>Y</I> + 25
+ 9 = 25</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/therefore.png">
<I>X</I><SUP>2</SUP> + <I>Y</I><SUP>2</SUP> + 10<I>X</I> + 6<I>Y</I> + 9 =
0</SPAN></P>
<P><SPAN class=Normal>Replacing<I> X </I>and<I> Y </I>with<I> x </I>and
<I>y</I>,</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/therefore.png">
<I>x</I><SUP>2</SUP> + <I>y</I><SUP>2</SUP> + 10<I>x </I>+ 6<I>y</I> + 9 =
0 is the equation of the curve in the new
system.</SPAN></P></TD></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel2><SPAN class=ListParagraph></SPAN></P></DIV>
<DIV id=028_VI>
<TABLE id=028_VI_bookmark class=tableh1 border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD
style="PADDING-BOTTOM: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 100%; PADDING-RIGHT: 0px; PADDING-TOP: 0px">
<OL class="oLevel1 Heading1">
<LI value=6><A id=VI>TRIANGLES</A></LI></OL></TD></TR></TBODY></TABLE>
<P class=pLevel2><SPAN class=ListParagraph></SPAN></P>
<P class=pLevel2><SPAN class=Normal><B>The Centroid</B></SPAN></P>
<P class=pLevel2><SPAN class=Normal>The point of intersection of the three
medians of a triangle is its <B>centroid</B>.</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>In the following figure, G is the centroid
of ΔABC. It divides each median internally in the ratio 2:1.</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pImage><SPAN><IMG alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_111.png"></SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>If A <IMG style="VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/equal_equal.png">
(<I>x</I><SUB>1</SUB>, <I>y</I><SUB>1</SUB>), B <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/equal_equal.png">
(<I>x</I><SUB>2</SUB>, <I>y</I><SUB>2</SUB>) and C <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/equal_equal.png">
(<I>x</I><SUB>3</SUB>, <I>y</I><SUB>3</SUB>), the coordinates of the centroid G
can be found using the section formula and are given by</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_112.png"></SPAN></P>
<P class=pLevel2><SPAN class=ListParagraph></SPAN></P>
<P class=pLevel2><SPAN class=Normal><B>The Incentre</B></SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>The incentre of a triangle is the point of
intersection of the angle bisectors of its vertices. It is the centre of a
circle that has all the three sides of the triangle tangent to it.</SPAN></P>
<P class=pImage><SPAN><IMG alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_113.png"></SPAN></P>
<P class=pLevel2><SPAN class=Normal><SPAN
style="MARGIN-LEFT: 25px"></SPAN></SPAN></P>
<P class=pLevel2><SPAN class=Normal>The coordinates of <B>incentre</B>
(<I>x</I><SUB>i</SUB>, <I>y</I><SUB>i</SUB>) are given by</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_114.png"></SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_115.png"></SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_116.png"></SPAN></P>
<P class=pLevel2><SPAN class=ListParagraph></SPAN></P>
<P class=pLevel2><SPAN class=Normal><B>The Area of a Triangle</B></SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>The area of triangle with coordinates of
vertices (<I>x</I><SUB>1</SUB>, <I>y</I><SUB>1</SUB>), (<I>x</I><SUB>2</SUB>,
<I>y</I><SUB>2</SUB>), (<I>x</I><SUB>3</SUB>, <I>y</I><SUB>3</SUB>) is given
by</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_117.png"></SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_118.png"></SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>This can be proved using distance formula
and then applying Heron’s formula.</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel2">
<DIV id=028_solvedex_25>
<TABLE style="WIDTH: 100%" id=028_solvedex_25_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_25>Example
25:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>Find the area of the triangle with A (2, 3), B (4,
5) and C (6, 9) as its vertices.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_119.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_120.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_121.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/therefore.png">
<I>A</I> = 2 square units</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><I>Alternatively,</I></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_122.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>If BC is the base, we can find the length of BC
using the distance formula.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_123.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>The height of ΔABC, <I>h</I>, is the distance of A
from BC. This can be found using the formula for the distance of a point
from a line. The equation of BC can be found using the two-point
form.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/therefore.png">
The equation of BC is 2<I>x </I><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/minus.png">
<I>y </I>= 3</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_124.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_125.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_126.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/therefore.png">
Area of ΔABC = 2 square units</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV><!--<p class="pLevel2"><span class="Normal"></span></p>
<div class="divTextBox divColorNormal divLevel2">
<p><span class="Normal"></span></p>
</div>
<p class="pLevel2"><span class="Normal"></span></p>-->
<DIV class="divTextBox divColorExampleHeading divLevel2">
<DIV id=028_solvedex_26>
<TABLE style="WIDTH: 100%" id=028_solvedex_26_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_26>Example
26:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>The area of the triangle with the vertices
(<I>a</I>, <I>a</I>),(<I>a </I>+ 1, <I>a</I>) and (<I>a</I>, <I>a </I>+ 2)
is</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><B>[CAT 2002]</B></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>(1) <I>a</I><SUP>3</SUP><SUB><SPAN
style="MARGIN-LEFT: 25px"></SPAN></SUB><SPAN
style="MARGIN-LEFT: 25px"></SPAN>(2) 1<SPAN
style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN></SPAN></P>
<P><SPAN class=Normal>(3) 0<SPAN style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN> (4) None of these</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>Area of (Δ) = <IMG style="VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_127.png">
</SPAN></P><SPAN class=Normal></SPAN>
<P></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>where <I>x</I><SUB>1</SUB> = <I>y</I><SUB>1</SUB> =
<I>a</I>,<I> x</I><SUB>2</SUB> = <I>a</I> + 1, <I>y</I><SUB>2</SUB> =
<I>a</I>, <I>x</I><SUB>3</SUB> = <I>a</I> and <I>y</I><SUB>3</SUB> =
<I>a</I> + 2</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_128.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_129.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_130.png"></SPAN></P>
<P><SPAN class=Normal><SPAN style="MARGIN-LEFT: 70px"><IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_131.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Hence, <B>option
2.</B></SPAN></P></SPAN></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel2><SPAN class=ListParagraph></SPAN></P></DIV>
<DIV id=028_VII>
<TABLE id=028_VII_bookmark class=tableh1 border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD
style="PADDING-BOTTOM: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 100%; PADDING-RIGHT: 0px; PADDING-TOP: 0px">
<OL class="oLevel1 Heading1">
<LI value=7><A id=VII>PARALLELOGRAM</A></LI></OL></TD></TR></TBODY></TABLE>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>If a quadrilateral has vertices
P(<I>x</I><SUB>1</SUB>, <I>y</I><SUB>1</SUB>), Q(<I>x</I><SUB>2</SUB>,
<I>y</I><SUB>2</SUB>), R(<I>x</I><SUB>3</SUB>, <I>y</I><SUB>3</SUB>) and
S(<I>x</I><SUB>4</SUB>, <I>y</I><SUB>4</SUB>), and <I>x</I><SUB>1</SUB> +
<I>x</I><SUB>3</SUB> = <I>x</I><SUB>2</SUB> + <I>x</I><SUB>4</SUB> and
<I>y</I><SUB>1</SUB> + <I>y</I><SUB>3</SUB> = <I>y</I><SUB>2</SUB> +
<I>y</I><SUB>4</SUB>, the quadrilateral PQRS is a parallelogram. The converse of
this statement is also true.</SPAN></P>
<P class=pLevel2><SPAN class=ListParagraph></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel2">
<DIV id=028_solvedex_27>
<TABLE style="WIDTH: 100%" id=028_solvedex_27_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_27>Example
27:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>ABCD is a rhombus with the diagonals AC and BD
intersecting at the origin on the <I>x</I>-<I>y</I> plane. The equation of
the straight line AD is <I>x</I> + <I>y</I> = 1. What is the equation of
BC?</SPAN></P>
<P><SPAN class=Normal><B>[CAT 2000]</B></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>(1) <I>x</I> + <I>y</I> = –1<SPAN
style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN>(2) <I>x</I> – <I>y</I> = –1</SPAN></P>
<P><SPAN class=Normal>(3) <I>x</I> + <I>y</I> = 1<SPAN
style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 35px"></SPAN>(4) None of the above</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>Equation of line AD is <I>x</I> + <I>y</I> =
1.</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/therefore.png">
AD intersects <I>x</I>-axis at (1, 0) and <I>y</I>-axis at (0,
1).</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>ABCD is a rhombus, with diagonals AC and BD
intersecting at origin.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>The four vertices are (1, 0), (0, 1), (<IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/minus.png">1,
0) and (0, <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/minus.png">1).</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Since AD passes through (1, 0) and (0, 1), BC passes
through (<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/minus.png">1,
0) and (0, <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/minus.png">1).</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_132.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>i.e. <I>x</I> + <I>y</I> = <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/minus.png">1</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Hence, <B>option
1.</SPAN></P></B></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel2><SPAN class=ListParagraph></SPAN></P></DIV>
<DIV id=028_VIII>
<TABLE id=028_VIII_bookmark class=tableh1 border=0 cellSpacing=0
cellPadding=0><TBODY>
<TR>
<TD
style="PADDING-BOTTOM: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 100%; PADDING-RIGHT: 0px; PADDING-TOP: 0px">
<OL class="oLevel1 Heading1">
<LI value=8><A id=VIII>EQUATIONS OF COMMON
CURVES</A></LI></OL></TD></TR></TBODY></TABLE>
<DIV id=028_VIII_A>
<TABLE id=028_VIII_A_bookmark class=tableh1 border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD
style="PADDING-BOTTOM: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 100%; PADDING-RIGHT: 0px; PADDING-TOP: 0px">
<OL class="oLevel2 Heading2">
<LI value=1><A id=VIII_A>EQUATION OF A
CIRCLE</A></LI></OL></TD></TR></TBODY></TABLE>
<P class=pLevel3><SPAN class=Normal></B></SPAN>
<P class=pImage><SPAN><IMG alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_133.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal>Equation of a circle having its centre at
(0, 0) and radius of <I>a</I> units is</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_134.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN>
<P class=pImage><SPAN><IMG alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_135.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal>If the circle has its centre at (<I>h</I>,
<I>k</I>) and its radius is <I>a</I> units, its equation is</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_136.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>The general form of the equation of a circle
is</SPAN></P>
<P class=pLevel3><SPAN class=Normal><I>x</I><SUP>2</SUP> + <I>y</I><SUP>2</SUP>
+ 2<I>gx</I> + 2<I>fy</I> + <I>c</I> = 0,</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>where <I>g</I><SUP>2</SUP> +
<I>f</I><SUP>2</SUP> – <I>c</I> > 0</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>The centre of this circle is (–<I>g</I>,
–<I>f</I>) and radius is </SPAN><SPAN class=Normal><IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_137.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=028_solvedex_28>
<TABLE style="WIDTH: 100%" id=028_solvedex_28_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_28>Example
28:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>What is the length of the tangent drawn from the
point (2, −1) to the circle 3<I>x</I><SUP>2</SUP> + 3<I>y</I><SUP>2</SUP>
+ 4<I>x</I> + 2<I>y</I> + 6 = 0? </SPAN></P>
<P><SPAN class=Normal><B>[FMS 2009]</B></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>(1) 3<SPAN style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN>(2) 6</SPAN></P>
<P><SPAN class=Normal>(3) 9<SPAN style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN>(4) None of these</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>The standard equation a circle having centre
(<I>a</I>, <I>b</I>) and radius <I>r</I> is given by the following
equation:</SPAN></P>
<P><SPAN class=Normal>(<I>x </I><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/minus.png">
<I>a</I>)<SUP>2</SUP> + (<I>y </I><IMG style="VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/minus.png">
<I>b</I>)<SUP>2</SUP> = <I>r</I><SUP>2</SUP></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>The equation given in this case is</SPAN></P>
<P><SPAN class=Normal>3<I>x</I><SUP>2</SUP> + 3<I>y</I><SUP>2</SUP> +
4<I>x</I> + 2<I>y</I> + 6 = 0</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>To put it in the standard form, we divide the entire
equation by 3. Hence we get,</SPAN></P><SPAN class=Normal><IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_138.png"></SPAN>
<P></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_139.png">
<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_140.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_141.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_142.png">
<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_143.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/therefore.png">The
radius of the circle is an imaginary number, the circle is not
real.</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/therefore.png">
The length of the tangent from the point (2, <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/minus.png">1)
cannot be determined.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Hence, <B>option 4.</B></SPAN></P>
<P><SPAN class=Normal></SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV><!--<p class="pLevel2"><span class="Normal"></span></p>
<div class="divTextBox divColorNormal divLevel3">
<p><span class="Normal"></span></p>
</div>-->
<P class=pLevel3><SPAN class=ListParagraph></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=028_solvedex_29>
<TABLE style="WIDTH: 100%" id=028_solvedex_29_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_29>Example
29:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>The area of a circle whose centre is at (2, 0) is
<IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/pi.png">.
The circle does not pass through the following point:</SPAN></P>
<P><SPAN class=Normal><B>[JMET 2009]</B></SPAN></P>
<P><SPAN class=Normal>(1) (2, <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/minus.png">1)<SPAN
style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN>(2) (2, 1)</SPAN></P>
<P><SPAN class=Normal>(3) (3, 0)<SPAN
style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 35px"></SPAN>(4) (2, 2)</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>The circle whose centre is at (2, 0) and has area
<IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/pi.png">,
will have a radius of 1 unit.</SPAN></P>
<P><SPAN class=Normal>The equation of the circle with centre (<I>a</I>,
<I>b</I>) and radius <I>r</I> is (<I>x</I> <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/minus.png">
<I>a</I>)<SUP>2</SUP> + (<I>y</I> <IMG style="VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/minus.png">
<I>b</I>)<SUP>2</SUP> = <I>r</I><SUP>2</SUP></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/therefore.png">
Equation of the given circle is</SPAN></P>
<P><SPAN class=Normal>(<I>x</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/minus.png">
2)<SUP>2</SUP> + (<I>y</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/minus.png">
0)<SUP>2</SUP> = 1<SUP>2</SUP></SPAN></P>
<P><SPAN class=Normal>On substituting the points given in the options in
the equation of the circle, we get that (2, 2) does not satisfy the
equation.</SPAN></P>
<P><SPAN class=Normal>Hence, <B>option
4</B>.</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel3><SPAN class=ListParagraph></SPAN></P></DIV>
<DIV id=028_VIII_B>
<TABLE id=028_VIII_B_bookmark class=tableh1 border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD
style="PADDING-BOTTOM: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 100%; PADDING-RIGHT: 0px; PADDING-TOP: 0px">
<OL class="oLevel2 Heading2">
<LI value=2><A id=VIII_B>EQUATION OF AN
ELLIPSE</A></LI></OL></TD></TR></TBODY></TABLE>
<P class=pLevel3><SPAN class=Normal>Equation of a standard ellipse having its
centre at (0, 0) is given by</SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_144.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_145.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>where the lengths of its major and minor
axes are 2<I>a</I> and 2<I>b</I> respectively.</SPAN></P>
<P class=pLevel3><SPAN class=Normal>The area of an ellipse is <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/pi.png"><I>ab</I>
and its perimeter is <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/pi.png">(<I>a</I> +
<I>b</I>)</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=028_solvedex_30>
<TABLE style="WIDTH: 100%" id=028_solvedex_30_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_30>Example
30:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>Find the equation of circle having its centre at
origin and radius <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_146.png">
units.</SPAN></P><SPAN class=Normal></SPAN>
<P></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>Equation of the circle is</SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_147.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_148.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=028_solvedex_31>
<TABLE style="WIDTH: 100%" id=028_solvedex_31_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_31>Example
31:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>The graph of <I>x</I><SUP>2</SUP><I> </I>–
4<I>y</I><SUP>2</SUP><I> </I>= 0 is:</SPAN></P>
<P><SPAN class=Normal><B>[FMS 2010]</B></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>(1) a parabola</SPAN></P>
<P><SPAN class=Normal>(2) an ellipse</SPAN></P>
<P><SPAN class=Normal>(3) a pair of straight lines</SPAN></P>
<P><SPAN class=Normal>(4) none of these</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal><I>x</I><SUP>2</SUP> – 4<I>y</I><SUP>2</SUP> =
0</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/therefore.png">
(<I>x</I> + 2<I>y</I>) (<I>x</I> – 2<I>y</I>) = 0</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/therefore.png">
<I>x</I> = <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/Plus_minus.png">2<I>y</I></SPAN></P>
<P><SPAN class=Normal>Thus, the given equation is the equation of a pair
of straight lines.</SPAN></P>
<P><SPAN class=Normal>Hence, <B>option
3</B>.</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel3><SPAN class=ListParagraph></SPAN></P>
<DIV class="divTextBox divColorNote divLevel3">
<P><SPAN class=Note>REMEMBER</SPAN></P>
<P><SPAN class=ListParagraph></SPAN></P>
<UL type=disc>
<LI>Distance between two points A(<I>x</I><SUB>1</SUB>, <I>y</I><SUB>1</SUB>)
and B(<I>x</I><SUB>2</SUB>, <I>y</I><SUB>2</SUB>) is given by</LI></UL>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=ListParagraph><SPAN style="MARGIN-LEFT: 15px"><IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_149.png"></SPAN></P>
<P><SPAN class=ListParagraph></SPAN></P>
<UL type=disc>
<LI>Coordinates of a point C(<I>x</I>, <I>y</I>), which divides the line
joining the points A(<I>x</I><SUB>1</SUB>, <I>y</I><SUB>1</SUB>) and
B(<I>x</I><SUB>2</SUB>, <I>y</I><SUB>2</SUB>) internally in the ratio <I>a</I>
: <I>b</I> are given by</LI></UL>
<P><SPAN class=ListParagraph></SPAN></P>
<P><SPAN class=ListParagraph><SPAN style="MARGIN-LEFT: 20px"><IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_150.png"></SPAN></P>
<P><SPAN class=ListParagraph></SPAN></P>
<UL type=disc>
<LI>Coordinates of a point C(<I>x</I>, <I>y</I>), which divides the line
joining the points A(<I>x</I><SUB>1</SUB>, <I>y</I><SUB>1</SUB>) and
B(<I>x</I><SUB>2</SUB>, <I>y</I><SUB>2</SUB>) externally in the ratio <I>a</I>
: <I>b</I> is given by</LI></UL>
<P><SPAN class=ListParagraph><SPAN style="MARGIN-LEFT: 40px"><IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_151.png"></SPAN></P>
<UL type=disc>
<LI>Slope of a line = tan <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/theta.png">,
where <I><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/theta.png"></I>
i<I>s</I> the inclination of the line.</LI></UL>
<P><SPAN class=ListParagraph></SPAN></P>
<UL type=disc>
<LI>If A(<I>x</I><SUB>1</SUB>, <I>y</I><SUB>1</SUB>) and
B(<I>x</I><SUB>2</SUB>, <I>y</I><SUB>2</SUB>) are any two points on a line,
the slope is given by:</LI></UL>
<P><SPAN class=ListParagraph><SPAN style="MARGIN-LEFT: 37px"><IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_152.png"></SPAN></P>
<P><SPAN class=ListParagraph></SPAN></P>
<UL type=disc>
<LI>Slopes of parallel lines are always equal.</LI></UL>
<UL type=disc>
<LI>Product of the slopes of two perpendicular lines is –1.</LI></UL>
<UL type=disc>
<LI>The equation of a line having slope <I>m</I> and passing through the point
P(<I>x</I><SUB>1</SUB>, <I>y</I><SUB>1</SUB>) is:</LI></UL>
<P><SPAN class=ListParagraph><SPAN style="MARGIN-LEFT: 37px">(<I>y</I> –
<I>y</I><SUB>1</SUB>) = <I>m</I>(<I>x</I> – <I>x</I><SUB>1</SUB>).</SPAN></P>
<P><SPAN class=ListParagraph></SPAN></P>
<UL type=disc>
<LI>The equation of a line passing through points P(<I>x</I><SUB>1</SUB>,
<I>y</I><SUB>1</SUB>) and Q(<I>x</I><SUB>2</SUB>, <I>y</I><SUB>2</SUB>),
when<I> x</I><SUB>1</SUB> ≠ <I>x</I><SUB>2</SUB>, is given by</LI></UL>
<P><SPAN class=ListParagraph></SPAN></P>
<P><SPAN class=ListParagraph><SPAN style="MARGIN-LEFT: 37px"><IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/7eb090e7-d8ed-4701-b091-fa5b35c2d091/028_images/028_153.png"></SPAN></P>
<P><SPAN class=ListParagraph></SPAN></P>
<UL type=disc>
<LI>The equation of a line having slope <I>m</I> and making an intercept
<I>c</I> on the <I>y-</I>axis is given by:</LI></UL>
<P><SPAN class=ListParagraph><SPAN style="MARGIN-LEFT: 40px"><I>y </I>=
<I>mx</I> + <I>c</I>.</SPAN></P></DIV>
<P class=pLevel4><SPAN
class=Normal></SPAN></P></DIV></DIV></SPAN></SPAN></SPAN></SPAN></SPAN></SPAN></SPAN></BODY></HTML>
guruhttp://www.blogger.com/profile/01591375209227117455noreply@blogger.comtag:blogger.com,1999:blog-7850540640045081515.post-23618562130831841932015-03-03T05:50:00.003-08:002019-09-14T18:55:03.727-07:00seques ,prog,ser<div dir="ltr" style="text-align: left;" trbidi="on">
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<P class=oTOCTitle><SPAN class=oTOCTitleFont><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019.htm#Title">Contents</A></SPAN></P>
<OL class=oTOCLevel1>
<LI class=oTocLiClass><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019.htm#I">INTRODUCTION</A>
</LI>
<LI class=oTocLiClass><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019.htm#II">SEQUENCE</A>
<OL class=oTOCLevel2>
<LI><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019.htm#II_A">SUM
OR DIFFERENCE</A></LI>
<LI><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019.htm#II_B">CUMULATIVE
SEQUENCE</A></LI>
<LI><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019.htm#II_C">POWER
SERIES</A></LI>
<LI><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019.htm#II_D">ALTERNATE
SEQUENCE</A></LI></OL></LI>
<LI class=oTocLiClass><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019.htm#III">PROGRESSIONS</A>
</LI>
<LI class=oTocLiClass><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019.htm#IV">ARITHMETIC
PROGRESSION</A>
<OL class=oTOCLevel2>
<LI><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019.htm#IV_A">THE
<I>n</I><SUP>th</SUP> TERM OF AN A.P.</A></LI>
<LI><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019.htm#IV_B">SUM
OF <I>n</I> TERMS OF AN A.P.</A></LI>
<LI><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019.htm#IV_C">AVERAGE
OF THE TERMS OF AN A.P.</A></LI></OL></LI>
<LI class=oTocLiClass><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019.htm#V">GEOMETRIC
PROGRESSION</A>
<OL class=oTOCLevel2>
<LI><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019.htm#V_A">THE
<I>n</I><SUP>th</SUP> TERM OF A G.P.</A></LI>
<LI><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019.htm#V_B">SUM
OF <I>n</I> TERMS OF A G.P.</A></LI>
<LI><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019.htm#V_C">GEOMETRIC
MEAN</A></LI></OL></LI>
<LI class=oTocLiClass><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019.htm#VI">HARMONIC
PROGRESSION</A>
<OL class=oTOCLevel2>
<LI><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019.htm#VI_A">HARMONIC
MEAN</A></LI>
<LI><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019.htm#VI_B">RELATION
BETWEEN ARITHMETIC, GEOMETRIC AND HARMONIC MEANS</A></LI></OL></LI>
<LI class=oTocLiClass><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019.htm#VII">SERIES
OF NATURAL NUMBERS</A> </LI></OL>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P id=Title class=pTitle><SPAN class=Title>Sequences, Progressions and
Series</SPAN></P>
<HR>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN>
<DIV id=019_I>
<TABLE id=019_I_bookmark class=tableh1 border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD
style="PADDING-BOTTOM: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 100%; PADDING-RIGHT: 0px; PADDING-TOP: 0px">
<OL class="oLevel1 Heading1">
<LI value=1><A id=I>INTRODUCTION</A></LI></OL></TD></TR></TBODY></TABLE>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>Terms arranged in a definite order are
called <B>sequences</B>. Terms in a sequence may be numbers, letters, symbols or
words.</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>In this concept builder we will discuss
various types of sequences of numbers. The concepts covered here are also useful
in solving problems related to numbers.</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P></DIV>
<DIV id=019_II>
<TABLE id=019_II_bookmark class=tableh1 border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD
style="PADDING-BOTTOM: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 100%; PADDING-RIGHT: 0px; PADDING-TOP: 0px">
<OL class="oLevel1 Heading1">
<LI value=2><A id=II>SEQUENCE</A></LI></OL></TD></TR></TBODY></TABLE>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>A sequence is a logically ordered list of
elements related to each other. We can identify the patterns followed by terms
in a sequence and use the patterns to find the other terms of the sequence, sums
of the terms in the sequence or to identify properties of the
sequence.</SPAN></P>
<P class=pLevel2><SPAN class=Normal>Terms of a sequence are generally denoted by
<I>T</I><SUB>1</SUB>, <I>T</I><SUB>2</SUB>, <I>T</I><SUB>3</SUB>, …,
<I>T<SUB>n</SUB></I>.</SPAN></P>
<P class=pLevel2><SPAN class=Normal>Let us have a look at some basic patterns
followed by sequences. This is not an exhaustive list. However, it gives a fair
idea of how to analyse sequences.</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<DIV id=019_II_A>
<TABLE id=019_II_A_bookmark class=tableh1 border=0 cellSpacing=0
cellPadding=0><TBODY>
<TR>
<TD
style="PADDING-BOTTOM: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 100%; PADDING-RIGHT: 0px; PADDING-TOP: 0px">
<OL class="oLevel2 Heading2">
<LI value=1><A id=II_A>SUM OR DIFFERENCE</A></LI></OL></TD></TR></TBODY></TABLE>
<P class=pLevel3><SPAN class=Normal>In this type of sequence, each term is
determined by increasing or decreasing the previous term by some quantity. This
quantity may be fixed or it may follow some logical rule by itself.</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>For example, consider the sequence 1, 3, 7,
13, 21, 31,...</SPAN></P>
<P class=pLevel3><SPAN class=Normal>You will observe that the differences
between the successive terms are multiples of 2. Hence the sequence could be
broken up as</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><I>T</I><SUB>1</SUB> = 1 + 2 <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/multiplication.png">
0 = 1</SPAN></P>
<P class=pLevel3><SPAN class=Normal><I>T</I><SUB>2</SUB> = 1 + 2 <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/multiplication.png">
1 = 3</SPAN></P>
<P class=pLevel3><SPAN class=Normal><I>T</I><SUB>3</SUB> = 3 + 2 <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/multiplication.png">
2 = 7</SPAN></P>
<P class=pLevel3><SPAN class=Normal><I>T</I><SUB>4</SUB> = 7 + 2 <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/multiplication.png">
3 = 13</SPAN></P>
<P class=pLevel3><SPAN class=Normal><I>T</I><SUB>5</SUB> = 13 + 2 <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/multiplication.png">
4 = 21</SPAN></P>
<P class=pLevel3><SPAN class=Normal><I>T</I><SUB>6</SUB> = 21 + 2 <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/multiplication.png">
5 = 31</SPAN></P>
<P class=pLevel3><SPAN class=Normal>Thus the <I>n</I><SUP>th</SUP> term of the
sequence can be written as</SPAN></P>
<P class=pLevel3><SPAN class=Normal><I>T<SUB>n</I></SUB> = <I>T<SUB>n</I> <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/minus.png">
1</SUB> + 2(<I>n </I>– 1)</SPAN></P>
<P class=pLevel3><SPAN class=Normal>We can see that the <I>n</I><SUP>th</SUP>
term depends on the previous term as well as its position in the
sequence.</SPAN></P>
<P class=pLevel3><SPAN class=Normal>Hence,</SPAN></P>
<P class=pLevel3><SPAN class=Normal><I>T</I><SUB>7</SUB> = 31 + 2 <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/multiplication.png">
6 = 43</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=019_solvedex_1>
<TABLE style="WIDTH: 100%" id=019_solvedex_1_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_1>Example
1:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>Find the 7<SUP>th</SUP> term of the sequence 1, 2,
4, 7, 11, 16,…</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal><I>T</I><SUB>1</SUB><I> </I>= 1</SPAN></P>
<P><SPAN class=Normal><I>T</I><SUB>2</SUB><I> </I>= 1 + 1 = 2</SPAN></P>
<P><SPAN class=Normal><I>T</I><SUB>3</SUB><I> </I>= 2 + 2 = 4</SPAN></P>
<P><SPAN class=Normal><I>T</I><SUB>4</SUB><I> </I>= 4 + 3 = 7</SPAN></P>
<P><SPAN class=Normal><I>T</I><SUB>5</SUB> = 7 + 4 = 11</SPAN></P>
<P><SPAN class=Normal><I>T</I><SUB>6</SUB><I> </I>= 11 + 5 = 16</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>The <I>n</I><SUP>th</SUP> term of this sequence can
be expressed as</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><I>T<SUB>n</SUB></I> = <I>T<SUB>n </I><IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/minus.png">
1</SUB> + (<I>n</I> – 1)</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Hence the 7<SUP>th</SUP> term of this sequence would
be 16 + 6 = 22</SPAN></P>
<P><SPAN class=Normal></SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV></DIV>
<DIV id=019_II_B>
<TABLE id=019_II_B_bookmark class=tableh1 border=0 cellSpacing=0
cellPadding=0><TBODY>
<TR>
<TD
style="PADDING-BOTTOM: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 100%; PADDING-RIGHT: 0px; PADDING-TOP: 0px">
<OL class="oLevel2 Heading2">
<LI value=2><A id=II_B>CUMULATIVE
SEQUENCE</A></LI></OL></TD></TR></TBODY></TABLE>
<P class=pLevel3><SPAN class=ListParagraph></SPAN></P>
<P class=pLevel3><SPAN class=Normal>Consider the sequence 1, 2, 3, 5, 8, 13,
21,…</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>Analyzing the above sequence would reveal
that after the second term, the next term is the sum of the previous two terms.
Hence the sequence is broken up in the following manner.</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><I>T</I><SUB>1</SUB> = 1</SPAN></P>
<P class=pLevel3><SPAN class=Normal><I>T</I><SUB>2</SUB> = 2</SPAN></P>
<P class=pLevel3><SPAN class=Normal><I>T</I><SUB>3</SUB> = 1 + 2 = 3</SPAN></P>
<P class=pLevel3><SPAN class=Normal><I>T</I><SUB>4</SUB> = 3 + 2 = 5</SPAN></P>
<P class=pLevel3><SPAN class=Normal><I>T</I><SUB>5</SUB> = 5 + 3 = 8</SPAN></P>
<P class=pLevel3><SPAN class=Normal><I>T</I><SUB>6</SUB> = 8 + 5 = 13</SPAN></P>
<P class=pLevel3><SPAN class=Normal><I>T</I><SUB>7</SUB> = 13 + 8 =
21</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>As can be seen, the next term is the sum of
its previous two terms, hence,</SPAN></P>
<P class=pLevel3><SPAN class=Normal><I>T<SUB>n</SUB></I>= <I>T<SUB>n <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/minus.png">
</I>2</SUB> + <I>T<SUB>n – </I>1</SUB></SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/therefore.png">
<I>T</I><SUB>8</SUB> = 21 + 13 = 34</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>In these types of sequences the pattern is
formed with the help of its previous terms.</SPAN></P>
<P class=pLevel3><SPAN class=ListParagraph></SPAN></P></DIV>
<DIV id=019_II_C>
<TABLE id=019_II_C_bookmark class=tableh1 border=0 cellSpacing=0
cellPadding=0><TBODY>
<TR>
<TD
style="PADDING-BOTTOM: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 100%; PADDING-RIGHT: 0px; PADDING-TOP: 0px">
<OL class="oLevel2 Heading2">
<LI value=3><A id=II_C>POWER SERIES</A></LI></OL></TD></TR></TBODY></TABLE>
<P class=pLevel3><SPAN class=ListParagraph></SPAN></P>
<P class=pLevel3><SPAN class=Normal>Consider the sequence 0, 1, 7, 20, 44, 81,
135,…</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>Analyzing and breaking up the above sequence
we have,</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><I>T</I><SUB>1</SUB> = 0</SPAN></P>
<P class=pLevel3><SPAN class=Normal><I>T</I><SUB>2</SUB> = 1³ <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/minus.png">
0 = 1</SPAN></P>
<P class=pLevel3><SPAN class=Normal><I>T</I><SUB>3</SUB> = 2³ <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/minus.png">
1 = 7</SPAN></P>
<P class=pLevel3><SPAN class=Normal><I>T</I><SUB>4</SUB> = 3³ <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/minus.png">
7 = 20</SPAN></P>
<P class=pLevel3><SPAN class=Normal><I>T</I><SUB>5</SUB> = 4³ <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/minus.png">
20 = 44</SPAN></P>
<P class=pLevel3><SPAN class=Normal><I>T</I><SUB>6 </SUB>= 5³ <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/minus.png">
44 = 81</SPAN></P>
<P class=pLevel3><SPAN class=Normal><I>T</I><SUB>7 </SUB>= 6³ <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/minus.png">
81 = 135</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>Thus the sequence can be expressed
as</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><I>T<SUB>n</I> </SUB>= (<I>n</I> –
1)<SUP>3</SUP> – <I>T<SUB>n</I> – 1</SUB></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/therefore.png">
<I>T</I><SUB>8 </SUB>= 7³ <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/minus.png">
135 = 208</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=019_solvedex_2>
<TABLE style="WIDTH: 100%" id=019_solvedex_2_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_2>Example
2:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>If <I>a</I><SUB>1 </SUB>= 1 and <I>a<SUB>n</I>+1
</SUB>= 2<I>a<SUB>n</I></SUB> + 5, <I>n</I> = 1, 2,... , then
<I>a</I><SUB>100</SUB> is equal to</SPAN></P>
<P><SPAN class=Normal><B>[CAT 2000]</B></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>(1) (5 <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/multiplication.png">
2<SUP>99</SUP> – 6)<SPAN style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN>(2) (5 <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/multiplication.png">
2<SUP>99</SUP> + 6)</SPAN></P>
<P><SPAN class=Normal>(3) (6 <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/multiplication.png">
2<SUP>99</SUP> + 5)<SPAN style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN>(4) (6 <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/multiplication.png">
2<SUP>99</SUP> – 5)</SPAN></P>
<P><SPAN class=ListParagraph></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=ListParagraph><I>a</I><SUB>1 </SUB>= 1 = 6 – 5</SPAN></P>
<P><SPAN class=ListParagraph><I>a</I><SUB>2 </SUB>= 7 = 12 – 5 = 6 <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/multiplication.png">
2 – 5</SPAN></P>
<P><SPAN class=ListParagraph><I>a</I><SUB>3 </SUB>= 19 = 24 – 5 = 6 <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/multiplication.png">
2<SUP>2</SUP> – 5</SPAN></P>
<P><SPAN class=ListParagraph><I>a</I><SUB>4</SUB> = 43 = 48 – 5 = 6 <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/multiplication.png">
2<SUP>3</SUP> – 5 and so on.</SPAN></P>
<P><SPAN class=ListParagraph>Thus, <I>a</I><SUB>100 </SUB>= 6 <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/multiplication.png">
2<SUP>99</SUP> – 5</SPAN></P>
<P><SPAN class=ListParagraph>Hence, <B>option 4.</B></SPAN></P>
<P><SPAN
class=ListParagraph></SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV></DIV>
<DIV id=019_II_D>
<TABLE id=019_II_D_bookmark class=tableh1 border=0 cellSpacing=0
cellPadding=0><TBODY>
<TR>
<TD
style="PADDING-BOTTOM: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 100%; PADDING-RIGHT: 0px; PADDING-TOP: 0px">
<OL class="oLevel2 Heading2">
<LI value=4><A id=II_D>ALTERNATE
SEQUENCE</A></LI></OL></TD></TR></TBODY></TABLE>
<P class=pLevel3><SPAN class=ListParagraph></SPAN></P>
<P class=pLevel3><SPAN class=Normal>Consider the sequence 0, 6, 8, 16, 64, 116,
216,…</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>Analyzing and breaking up the sequence we
have</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><I>T</I><SUB>1 </SUB>= 0</SPAN></P>
<P class=pLevel3><SPAN class=Normal><I>T</I><SUB>2 </SUB>= 6</SPAN></P>
<P class=pLevel3><SPAN class=Normal><I>T</I><SUB>3 </SUB>= 2³ = 8</SPAN></P>
<P class=pLevel3><SPAN class=Normal><I>T</I><SUB>4 </SUB>= 6 + 10 =
16</SPAN></P>
<P class=pLevel3><SPAN class=Normal><I>T</I><SUB>5 </SUB>= 4³ = 64</SPAN></P>
<P class=pLevel3><SPAN class=Normal><I>T</I><SUB>6 </SUB>= 16 + 100 =
116</SPAN></P>
<P class=pLevel3><SPAN class=Normal><I>T</I><SUB>7 </SUB>= 6³ = 216</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>Thus the above sequence is a combination of
2 sequences, one being the cubes of even numbers</SPAN></P>
<P class=pLevel3><SPAN class=Normal>0, 8, 64, 216,…</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>and the other being the one with powers of
10 added to the previous terms</SPAN></P>
<P class=pLevel3><SPAN class=Normal>6, 16, 116,…</SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/therefore.png">
<I>T</I><SUB>8</SUB> = 10³ + 116 = 1116</SPAN></P>
<P class=pLevel3><SPAN class=ListParagraph></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=019_solvedex_3>
<TABLE style="WIDTH: 100%" id=019_solvedex_3_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_3>Example
3:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>Find the next term of the series 3, 4, 11, 24, 43,
…</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>Difference between the 1<SUP>st</SUP> and
2<SUP>nd</SUP> term = 1</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Difference between the 2<SUP>nd</SUP> and
3<SUP>rd</SUP> term = 7 = 1 + 6</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Difference between the 3<SUP>rd</SUP> and
4<SUP>th</SUP> term = 13 = 7 + 6</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Difference between the 4<SUP>th</SUP> and
5<SUP>th</SUP> term = 19 = 13 + 6</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Thus the difference between the 5<SUP>th</SUP> and
6<SUP>th</SUP> terms = 19 + 6 = 25</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Hence the next term is 43 + 25 =
68.</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel3><SPAN class=ListParagraph></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=019_solvedex_4>
<TABLE style="WIDTH: 100%" id=019_solvedex_4_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_4>Example
4:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ListParagraph></SPAN></P>
<P><SPAN class=Normal>Find the sum</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_001.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><B>[CAT 2008]</B></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_002.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_003.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_004.png"></SPAN></P>
<P><SPAN class=ListParagraph></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>Consider only first term,</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_005.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Now consider first two terms,</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_007.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_008.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_009.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Similarly,</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_001.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_011.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Hence,<B> option 1.</B></SPAN></P>
<P><SPAN
class=Normal></SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV></DIV></DIV>
<DIV id=019_III>
<TABLE id=019_III_bookmark class=tableh1 border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD
style="PADDING-BOTTOM: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 100%; PADDING-RIGHT: 0px; PADDING-TOP: 0px">
<OL class="oLevel1 Heading1">
<LI value=3><A id=III>PROGRESSIONS</A></LI></OL></TD></TR></TBODY></TABLE>
<P class=pLevel2><SPAN class=ListParagraph></SPAN></P>
<P class=pLevel2><SPAN class=Normal><B>Arithmetic Progression, Geometric
Progression </B>and <B>Harmonic Progression</B> are special types of sequences
which have a fixed relationship amongst their terms. We will study the
relationships and properties of these three types of progressions in
detail.</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P></DIV>
<DIV id=019_IV>
<TABLE id=019_IV_bookmark class=tableh1 border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD
style="PADDING-BOTTOM: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 100%; PADDING-RIGHT: 0px; PADDING-TOP: 0px">
<OL class="oLevel1 Heading1">
<LI value=4><A id=IV>ARITHMETIC
PROGRESSION</A></LI></OL></TD></TR></TBODY></TABLE>
<P class=pLevel2><SPAN class=Normal>Quantities are said to be in <B>Arithmetic
Progression (A.P.)</B> when they increase or decrease by a constant value known
as their <B>common difference</B>, denoted by <I>d</I>. In other words, the
difference between any two consecutive terms in an A.P. is constant. The first
term of an A.P. is generally denoted by <I>a</I>.</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>If every term of an A.P. is greater than the
previous term, the A.P. is said to be an <B>increasing A.P.</B></SPAN></P>
<P class=pLevel2><SPAN class=Normal>If every term of an A.P. is lesser than the
previous term, the A.P. is said to be a <B>decreasing A.P.</B></SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>For example,</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<OL class="oLevel3 ListParagraph ListStyleOL">
<LI value=1>1, 2, 3, 4,… where <I>a</I> = 1 and <I>d</I> = 1</LI></OL>
<OL class="oLevel3 ListParagraph ListStyleOL">
<LI value=2>3, 7, 11, 15,… where <I>a</I> = 3 and <I>d</I> = 4</LI></OL>
<OL class="oLevel3 ListParagraph ListStyleOL">
<LI value=3>8, 2, <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/minus.png">4,
<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/minus.png">10,…
where <I>a</I> = 8 and <I>d</I> = <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/minus.png">6</LI></OL>
<DIV id=019_IV_A>
<TABLE id=019_IV_A_bookmark class=tableh1 border=0 cellSpacing=0
cellPadding=0><TBODY>
<TR>
<TD
style="PADDING-BOTTOM: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 100%; PADDING-RIGHT: 0px; PADDING-TOP: 0px">
<OL class="oLevel2 Heading2">
<LI value=1><A id=IV_A>THE <I>n</I><SUP>th</SUP> TERM OF AN
A.P.</A></LI></OL></TD></TR></TBODY></TABLE>
<P class=pLevel3><SPAN class=Normal>The first, second, third,…<I>
n</I><SUP>th</SUP> terms of an A.P. are denoted by <I>T</I><SUB>1</SUB>,<SUB>
</SUB><I>T</I><SUB>2</SUB>,<SUB> </SUB><I>T</I><SUB>3</SUB>,<SUB> </SUB>…,
<I>T<SUB>n</I></SUB></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><I>T</I><SUB>1</SUB> = <I>a</I> = <I>a</I> +
(1 – 1)<I>d</I></SPAN></P>
<P class=pLevel3><SPAN class=Normal><I>T</I><SUB>2</SUB> = <I>a</I> + <I>d</I> =
<I>a</I> + (2 – 1)<I>d</I></SPAN></P>
<P class=pLevel3><SPAN class=Normal><I>T</I><SUB>3</SUB> = <I>a</I> + 2<I>d</I>
= <I>a</I> + (3 – 1)<I>d </I>and so on.</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>Continuing thus, the <I>n</I><SUP>th
</SUP>term of an A.P. is,</SPAN></P>
<P class=pLevel3><SPAN class=Normal><I>T<SUB>n</I></SUB> = <I>a</I> + (<I>n</I>
<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/minus.png">
1)<I>d</I></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=019_solvedex_5>
<TABLE style="WIDTH: 100%" id=019_solvedex_5_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_5>Example
5:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>Find the fifteenth term of the A.P. <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/minus.png">3,
<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/minus.png">9,
<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/minus.png">15,
…</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal><I>T<SUB>n</I></SUB> = <I>a</I> + (<I>n</I> <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/minus.png">
1)<I>d</I></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Here, <I>a</I> = <IMG style="VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/minus.png">3,
<I>d</I> = <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/minus.png">9
– (<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/minus.png">3)
= <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/minus.png">6
and <I>n</I> = 15</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/therefore.png">
<I>T</I><SUB>15 </SUB>= <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/minus.png">3
+ (14)(<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/minus.png">6)</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/therefore.png">
<I>T</I><SUB>15</SUB> = <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/minus.png">87</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=019_solvedex_6>
<TABLE style="WIDTH: 100%" id=019_solvedex_6_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_6>Example
6:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>The ninth term exceeds the fifth term of an A.P. by
32. The sum of the ninth and fifth terms is 114. Find the eighth term of
the A.P.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal><I>T</I><SUB>9</SUB> = <I>a</I> + (9 <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/minus.png">
1)<I>d</I> = <I>a</I> + 8<I>d</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>T</I><SUB>5</SUB> = <I>a</I> + (5 <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/minus.png">
1)<I>d</I> = <I>a</I> + 4<I>d</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>T</I><SUB>9</SUB><SUP> </SUP>–
<I>T</I><SUB>5</SUB><SUP> </SUP>= 32</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/therefore.png">
4<I>d</I> = 32</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/therefore.png">
<I>d</I> = 8</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/therefore.png">
<I>T</I><SUB>9</SUB><SUP> </SUP>+ <I>T</I><SUB>5</SUB> = 2<I>a</I> +
12<I>d</I></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/therefore.png">
114 = 2(<I>a</I> + 6<I>d</I>)</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/therefore.png">
<I>a</I> + 6<I>d</I> = 57</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><I>T</I><SUB>8</SUB> = <I>a</I> + 7<I>d</I> =
<I>a</I> + 6<I>d</I> + <I>d</I> = 57 + 8 =
65</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=019_solvedex_7>
<TABLE style="WIDTH: 100%" id=019_solvedex_7_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_7>Example
7:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>The 54<SUP>th</SUP> and the 4<SUP>th</SUP> terms of
an A.P. are <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/minus.png">61
and 64 respectively. Find the 23<SUP>rd</SUP> term.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal><I>a</I> + 53<I>d</I> = <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/minus.png">61<SPAN
style="MARGIN-LEFT: 35px"></SPAN> …(i)</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><I>a</I> + 3<I>d</I> = 64 <SPAN
style="MARGIN-LEFT: 50px"></SPAN>… (ii)</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Subtracting, we get,</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>50<I>d</I> = <IMG style="VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/minus.png">125</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><I>d</I> = <IMG style="VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/minus.png">5/2</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_012.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_013.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><SPAN style="MARGIN-LEFT: 145px"></SPAN><IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_014.png"></SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV></DIV>
<DIV id=019_IV_B>
<TABLE id=019_IV_B_bookmark class=tableh1 border=0 cellSpacing=0
cellPadding=0><TBODY>
<TR>
<TD
style="PADDING-BOTTOM: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 100%; PADDING-RIGHT: 0px; PADDING-TOP: 0px">
<OL class="oLevel2 Heading2">
<LI value=2><A id=IV_B>SUM OF <I>n</I> TERMS OF AN
A.P.</A></LI></OL></TD></TR></TBODY></TABLE>
<P class=pLevel3><SPAN class=ListParagraph></SPAN></P>
<P class=pLevel3><SPAN class=Normal>Let the first term and common difference of
an A.P. containing <I>n</I> terms be <I>a</I> and <I>d</I> respectively. Let
<I>T<SUB>n</I></SUB> be the <I>n</I><SUP>th</SUP> term of the A.P. Then, the sum
of <I>n</I> terms of the A.P. is</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_015.png"></SPAN></P>
<P class=pLevel3><SPAN class=ListParagraph></SPAN></P>
<P class=pLevel3><SPAN class=Normal><B>Explanation:</B></SPAN></P>
<P class=pLevel3><SPAN class=Normal><I>T<SUB>n</I></SUB> = <I>a</I> + (<I>n</I>
– 1)<I>d</I></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>Let the sum of <I>n</I> terms of the A.P. be
denoted by <I>S<SUB>n</I></SUB>. Then,</SPAN></P>
<P class=pLevel3><SPAN class=Normal><I>S<SUB>n</I></SUB> = <I>a</I> + (<I>a</I>
+ <I>d</I>) + (<I>a</I> + 2<I>d</I>) + … + (<I>T<SUB>n</I></SUB> – 2<I>d</I>) +
(<I>T<SUB>n</I></SUB> – <I>d</I>) + <I>T<SUB>n</SUB> </I><SPAN
style="MARGIN-LEFT: 30px"></SPAN>…(i)</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>The same equation can be written
as</SPAN></P>
<P class=pLevel3><SPAN class=Normal><I>S<SUB>n</I></SUB> = <I>T<SUB>n</I></SUB>
+ (<I>T<SUB>n</I></SUB> – <I>d</I>) + (<I>T<SUB>n</I></SUB> – 2<I>d</I>) + … +
(<I>a</I> + 2<I>d</I>) + (<I>a</I> + <I>d</I>) + <I>a</I><SPAN
style="MARGIN-LEFT: 30px"> </SPAN>…(ii)</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>Adding (i) and (ii),</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/therefore.png">
2<I>S<SUB>n</I></SUB> = (<I>a</I> + <I>T<SUB>n</I></SUB>) + (<I>a</I> +
<I>T<SUB>n</I></SUB>) + (<I>a</I> + <I>T<SUB>n</I></SUB>) … <I>n</I>
times</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/therefore.png">
2<I>S<SUB>n</I></SUB> = <I>n</I>(<I>a</I> + <I>T<SUB>n</I></SUB>)</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_017_.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_017.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_018.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>Equation (iii) gives us the sum of the
<I>n</I> terms of an A.P. in terms of its first and last terms.</SPAN></P>
<P class=pLevel3><SPAN class=Normal>Equation (iv) gives us the general formula
for finding the sum of <I>n</I> terms of an A.P.</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=019_solvedex_8>
<TABLE style="WIDTH: 100%" id=019_solvedex_8_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_8>Example
8:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>The sixth and eighth terms of an A.P. are 38 and 52,
respectively. Find the sum of the first twelve terms of the
A.P.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal><I>T</I><SUB>6 </SUB>= <I>a</I> + 5<I>d</I> =
38</SPAN></P>
<P><SPAN class=Normal><I>T</I><SUB>8</SUB> = <I>a</I> + 7<I>d</I> =
52</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Solving the two equations, <I>d</I> = 7 and <I>a</I>
= 3</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/therefore.png">
The sum of 12 terms of the A.P. is</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_019.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=019_solvedex_9>
<TABLE style="WIDTH: 100%" id=019_solvedex_9_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_9>Example
9:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>How many terms of the series <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/minus.png">12,
<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/minus.png">9,
<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/minus.png">6,
… must be taken so that the sum may become 78?</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_020.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_021.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><I>n</I> <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/multiplication.png">
(<I>n</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/minus.png">
9) = 52</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><I>n</I> =
13</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV></DIV>
<DIV id=019_IV_C>
<TABLE id=019_IV_C_bookmark class=tableh1 border=0 cellSpacing=0
cellPadding=0><TBODY>
<TR>
<TD
style="PADDING-BOTTOM: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 100%; PADDING-RIGHT: 0px; PADDING-TOP: 0px">
<OL class="oLevel2 Heading2">
<LI value=3><A id=IV_C>AVERAGE OF THE TERMS OF AN
A.P.</A></LI></OL></TD></TR></TBODY></TABLE>
<P class=pLevel3><SPAN class=ListParagraph></SPAN></P>
<P class=pLevel3><SPAN class=Normal>The <B>average</B> or the <B>arithmetic
mean</B> of <I>n</I> terms of an A. P.</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_022.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_023.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_024.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_023.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_025.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_023.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_026.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>Continuing thus, we see that the average of
the terms of an A.P. is equal to the average of its first and
<I>n</I><SUP>th</SUP> terms, second and (<I>n</I> – 1)<SUP>th</SUP> terms, third
and (<I>n</I> – 2)<SUP>th </SUP>terms and so on. In general, the average of the
terms of the A.P. is equal to the average of the <I>k</I><SUP>th</SUP> term from
the beginning and the <I>k<SUP>th</I></SUP> term from the end, or it is equal to
the average of any two terms of the A.P. that are equidistant from the beginning
and the end.</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>Also, if <I>n</I> is even the average of the
terms of the A.P. is equal to the average of its</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_027.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>If <I>n</I> is odd, the average of the terms
of the A.P. is equal to the</SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_028.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=019_solvedex_10>
<TABLE style="WIDTH: 100%" id=019_solvedex_10_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_10>Example
10:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>The sum of the first nine terms of an A.P. is 387.
Find the fifth term.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>The average of the first 9 terms of the A.P. = (9 +
1)/2<SUP>th</SUP> term</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/therefore.png">
The 5<SUP>th</SUP> term of the A.P. = 387/9 =
43</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel3><SPAN class=Normal>If any two consecutive terms of an
arithmetic progression are known, the series can be completely
determined.</SPAN></P>
<P class=pLevel3><SPAN class=ListParagraph></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=019_solvedex_11>
<TABLE style="WIDTH: 100%" id=019_solvedex_11_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_11>Example
11:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>The fourteenth and fifteenth terms of an A.P. are 25
and 32 respectively. Find the 30<SUP>th</SUP> term, sum of the first 30
terms and the first term of the A.P.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal><I>T</I><SUB>14</SUB> = <I>a</I> + 13<I>d</I> =
25</SPAN></P>
<P><SPAN class=Normal><I>T</I><SUB>15</SUB> = <I>a</I> + 14<I>d</I> =
32</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/therefore.png">
<I>d</I> = 7 and <I>a</I> = <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/minus.png">66</SPAN></P>
<P><SPAN class=Normal><I>T</I><SUB>30</SUB> = <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/minus.png">66
+ 29 <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/multiplication.png">
7 = 137</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>The sum of the first 30 terms <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_029.png"></SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=019_solvedex_12>
<TABLE style="WIDTH: 100%" id=019_solvedex_12_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_12>Example
12:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>A teacher observes that the marks that the students
in her class have scored are all different. She arranges her students in a
line in increasing order of their marks such that difference in marks
scored by any two students next to each other is 4. The lowest marks that
any student has scored are 11. The sum of the marks that all her students
have scored is 585. Find the marks scored by the student standing in the
middle of the line.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>The marks of the students standing in a line form an
A.P. with <I>a</I> = 11 and <I>d</I> = 4.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Let there be <I>n</I> students.</SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_030.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_031.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/therefore.png">
(<I>n</I> – 15)(2<I>n</I> + 39) = 0</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/therefore.png">
<I>n</I> = 15</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/therefore.png">
The 8<SUP>th</SUP> student stands in the middle of the line.</SPAN></P>
<P><SPAN class=Normal>His marks = 585/15 =
39</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=019_solvedex_13>
<TABLE style="WIDTH: 100%" id=019_solvedex_13_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_13>Example
13:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>A child has some marbles which he arranges into rows
such that every row has <I>n</I> marbles more than the row preceding it,
the first row having only 1 marble. There are 23 marbles in the fourth and
fifth rows put together. Which of 25, 35 and 45 could be the number of
marbles in the last row?</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>The number of marbles in the <I>r</I><SUP>th</SUP>
row = 1 + (<I>r</I> – 1)<I>n</I></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>By conditions,</SPAN></P>
<P><SPAN class=Normal>1 + 3<I>n</I> + 1 + 4<I>n</I> = 23</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/therefore.png">
<I>n</I> = 3</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/therefore.png">
The number of marbles in the <I>r</I><SUP>th</SUP> row = 1 + 3(<I>r</I> –
1) = 3<I>r</I> – 2</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/therefore.png">
The number of marbles in the last row has to be of the form 3<I>r</I> – 2,
where <I>r</I> is an integer.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>This is true only for 25.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/therefore.png">
The number of marbles in the last row could be
25.</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<DIV class="divTextBox divColorNote divLevel3">
<P><SPAN class=Note>REMEMBER:</SPAN></P>
<UL>
<LI>When three terms are in Arithmetic progression, the middle term is the
arithmetic mean of the other two. It is always convenient to take three terms
in an A.P. as (<I>a</I> – <I>d</I>), <I>a</I> and (<I>a</I> + <I>d</I>).
Similarly, four terms in an A.P. could be taken as <I>a</I> <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/minus.png">
3<I>d</I>, <I>a</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/minus.png">
<I>d</I>, <I>a</I> + <I>d</I> and <I>a </I>+ 3<I>d</I>; five terms could be
taken as <I>a</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/minus.png">
2<I>d</I>, <I>a</I> – <I>d</I>, <I>a</I>, <I>a</I> + <I>d</I>, <I>a</I> +
2<I>d</I>.</LI></UL>
<UL>
<P><SPAN class=Normal>The advantage of representing terms in this way is that
the sum of terms is then obtained in only one unknown.</SPAN></P></UL>
<P><SPAN class=Normal></SPAN></P></DIV>
<P class=pLevel4><SPAN class=Normal><B>SOME PROPERTIES OF AN A.P.</B></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<UL class="oLevel3 ListParagraph ListStyleUL5">
<LI value=1>If each term of an A.P. is increased, decreased, multiplied or
divided by the same non-zero number, then the resulting sequence is also an
A.P.</LI></UL>
<P class=pLevel5><SPAN class=Normal>In case the terms are increased or decreased
by some quantity, the common difference of the new A.P. remains equal to that of
the original A.P.</SPAN></P>
<P class=pLevel5><SPAN class=Normal>In case the terms are multiplied or divided
by a constant <I>c</I> (<I>c</I> ≠ 0), the common difference <I>d</I>
accordingly changes to <I>d</I> <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/multiplication.png">
<I>c</I> or <I>d</I>/<I>c</I> .</SPAN></P>
<UL class="oLevel3 ListParagraph ListStyleUL5">
<LI value=2>If the terms of a sequence are given in terms of <I>n</I>, then
whether or not the sequence is an A.P. will depend on the difference between
two consecutive terms. If this difference is independent of <I>n</I>, the
sequence is an A.P., otherwise not.</LI></UL>
<P class=pLevel5><SPAN class=Normal>For example,</SPAN></P>
<P class=pLevel5><SPAN class=Normal>If the <I>n</I><SUP>th</SUP> term of a
sequence is <I>T<SUB>n</I></SUB> = 3<I>n</I> + 2,</SPAN></P>
<P class=pLevel5><SPAN class=Normal>Then, <I>T<SUB>n</I> + 1</SUB> = 3<I>n</I> +
5</SPAN></P>
<P class=pLevel5><SPAN class=Normal><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/therefore.png">
<I>T<SUB>n</I> + 1</SUB> – <I>T<SUB>n</I></SUB> = 3</SPAN></P>
<P class=pLevel5><SPAN class=Normal><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/therefore.png">
The sequence of numbers given by <I>T<SUB>n</I></SUB> = 3<I>n</I> + 2 is an
A.P.</SPAN></P>
<P class=pLevel5><SPAN class=Normal>But, if <I>T<SUB>n</I></SUB> =
3<I>n</I><SUP>2</SUP> + 2<I>n</I> – 1, then the difference</SPAN></P>
<P class=pLevel5><SPAN class=Normal><I>T<SUB>n</I></SUB> <SUB>+ 1</SUB> –
<I>T<SUB>n</I></SUB> = 6<I>n</I> + 5, which is not independent of
<I>n</I>.</SPAN></P>
<P class=pLevel5><SPAN class=Normal><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/therefore.png">
The sequence of numbers given by <I>T<SUB>n</I></SUB> = 3<I>n</I><SUP>2</SUP> +
2<I>n</I> – 1 is not an A.P.</SPAN></P>
<UL class="oLevel3 ListParagraph ListStyleUL5">
<LI>The number of elements in an arithmetic series from <I>n</I><SUB>1</SUB>
to <I>n</I><SUB>2</SUB>, with a step size (or common difference) of <I>m</I>
is 1 + (<I>n</I><SUB>2</SUB> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/minus.png">
<I>n</I><SUB>1</SUB>)/<I>m</I></LI></UL>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=019_solvedex_14>
<TABLE style="WIDTH: 100%" id=019_solvedex_14_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_14>Example
14:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>How many multiples of 13 lie between 1000 and 5000?
What is the sum of all these multiples?</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>The lowest multiple of 13 that is greater than 1000
is 1001. The greatest multiple of 13 that is lesser than 5000 is
4992.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_032.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>= 308 multiples of 13 between 1000 and
5000.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>The sum of all these multiples is</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_033.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/therefore.png">
<I>S</I> = 922922</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<UL class="oLevel3 ListParagraph ListStyleUL5">
<LI value=4>If the sum of the first <I>p</I> terms of an A.P. is equal to the
sum of the first <I>q</I> terms of the A.P. such that <I>p</I> and <I>q</I>
are different, then the sum of (<I>p</I> + <I>q</I>) terms of the A.P. is
zero.</LI></UL>
<UL>
<P class=pLevel3><SPAN class=Normal><B>Explanation:</B></SPAN></P></UL>
<UL>
<P class=pLevel3><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_034.png"></SPAN></P></UL>
<UL>
<P class=pLevel3><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_035.png"></SPAN></P></UL>
<UL>
<P class=pLevel3><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_036.png"></SPAN></P></UL>
<UL>
<P class=pLevel3><SPAN class=Normal><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/therefore.png"></B>
<I>p</I> = <I>q</I> or [2<I>a </I>+ (<I>p </I>+ <I>q </I><IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/minus.png">
1)<I>d</I>] = 0</SPAN></P></UL>
<UL>
<P class=pLevel3><SPAN class=Normal>But <I>p</I> ≠ <I>q</I></SPAN></P></UL>
<UL>
<P class=pLevel3><SPAN class=Normal><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/therefore.png">
The sum of (<I>p</I> + <I>q</I>) terms of the A.P. is zero.</SPAN></P></UL>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=019_solvedex_15>
<TABLE style="WIDTH: 100%" id=019_solvedex_15_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_15>Example
15:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>The sum of the first 16 terms of an A.P. is equal to
the sum of the first 24 terms of the A.P. Find the sum of the first 40
terms of the A.P.</SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/symb23.png">
<I>S</I><SUB>16</SUB> = <I>S</I><SUB>24</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/therefore.png"></SUB>
<I>S</I><SUB>(16 + 24)</SUB> = 0</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/therefore.png">
<I>S</I><SUB>40</SUB> = 0</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=019_solvedex_16>
<TABLE style="WIDTH: 100%" id=019_solvedex_16_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_16>Example
16:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>The sum of 3<SUP>rd</SUP> and 15<SUP>th</SUP>
elements of an arithmetic progression is equal to the sum of
6<SUP>th</SUP>, 11<SUP>th</SUP> and 13<SUP>th</SUP> elements of the same
progression. Then which element of the series should necessarily be equal
to zero?</SPAN></P>
<P><SPAN class=Normal><B>[CAT 2003 Leaked Test]</B></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>(1) 1<SUP>st <SPAN
style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN></SUP>(2) 9<SUP>th</SUP></SPAN></P>
<P><SPAN class=Normal>(3) 12<SUP>th <SPAN
style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 15px"></SPAN></SUP>(4) None of the above</SPAN></P>
<P><SPAN class=ListParagraph></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>Assume that the first term of the progression is
<I>a</I> and the common difference is <I>d</I>.</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/symb23.png">
T<SUB>3</SUB> + T<SUB>15</SUB> = T<SUB>6</SUB> + T<SUB>11</SUB> +
T<SUB>13</SUB></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/therefore.png">
(<I>a</I> + 2<I>d</I>) + (<I>a</I> + 14<I>d</I>) = (<I>a</I> + 5<I>d</I>)
+ (<I>a</I> + 10<I>d</I>) + (<I>a</I> + 12<I>d</I>)</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/therefore.png">
<I>a</I> + 11<I>d</I> = 0</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>But, T<SUB>12</SUB> = <I>a</I> +
11<I>d</I></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/therefore.png">
The 12<SUP>th</SUP> term of an arithmetic progression is 0.</SPAN></P>
<P><SPAN class=Normal>Hence,<B> option
3.</B></SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=019_solvedex_17>
<TABLE style="WIDTH: 100%" id=019_solvedex_17_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_17>Example
17:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>Tanishka has 45 coins to be arranged in a line in
heaps such that every heap has a fixed number of coins more than the
preceding heap. There have to be at least three heaps and the first heap
has to have only one coin. In how many ways can Tanishka arrange the
coins?</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>As every heap has a fixed number of coins more than
that in the previous heap, the number of coins in the heaps form an
A.P.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>The sum of the terms in this A.P. = the total number
of coins = 45</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/therefore.png">
If there are <I>n</I> heaps and the last heap has <I>k</I>
coins,</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_037.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/therefore.png">
<I>n</I>(1 + <I>k</I>) = 90</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Also, <I>k</I> = 1 + (<I>n</I> – 1)<I>d</I> where
<I>d</I> is the common difference of the A.P.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>90 = 1 <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/multiplication.png">
90 = 2 <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/multiplication.png">
45 = 3 <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/multiplication.png">
30 = 5 <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/multiplication.png">
18 = 6 <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/multiplication.png">
15 = 9 <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/multiplication.png">
10</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>As the number of heaps <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/greater_equal.png">
3, we do not consider the first two cases. </SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>If <I>n</I>(1 + <I>k</I>) = 3 <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/multiplication.png">
30, the first heap has 1 coin, the second has 15 and the third has
29.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>If <I>n</I>(1 + <I>k</I>) = 5 <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/multiplication.png">
18, the first heap has 1 coin and the last has 17. The middle three have
5, 9 and 13 coins respectively.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>If <I>n</I>(1 + <I>k</I>) = 6 <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/multiplication.png">
15, the number of coins in all heaps cannot be integers.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>If <I>n</I>(1 + <I>k</I>) = 9 <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/multiplication.png">
10, the first heap has 1 coin, the second has 2, the third has 3 and so
on. The last heap has 9 coins.</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/therefore.png">
There are 3 ways in which she can arrange the coins.</SPAN></P>
<P><SPAN class=Normal></SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=019_solvedex_18>
<TABLE style="WIDTH: 100%" id=019_solvedex_18_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_18>Example
18:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>Thirty-one magazines are arranged from left to right
in order of increasing prices. The price of each magazine differs by Rs. 2
from that of each adjacent magazine. For the price of the magazine at the
extreme right a customer can buy the middle magazine and an adjacent one.
Then:</SPAN></P>
<P><SPAN class=Normal><B>[FMS 2010]</B></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>(1) The adjacent magazine referred to is at the left
of the middle magazine.</SPAN></P>
<P><SPAN class=Normal>(2) The middle magazine sells for Rs. 36.</SPAN></P>
<P><SPAN class=Normal>(3) The most expensive magazine sells for Rs.
64.</SPAN></P>
<P><SPAN class=Normal>(4) None of these is correct.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>Let the price of the cheapest magazine i.e. the one
at the extreme left be<I> x</I>.</SPAN></P>
<P><SPAN class=Normal>Difference, between the prices of two adjacent
magazines is Rs. 2</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/therefore.png">
Price of extreme right or the costliest magazine will be <I>x</I> + 30
<IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/multiplication.png">
2 = <I>x</I> + 60</SPAN></P>
<P><SPAN class=Normal>Now, the price of the magazine in middle (the
16<SUP>th</SUP> position) = <I>x</I> + 15 <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/multiplication.png">
2 </SPAN></P>
<P><SPAN class=Normal><SPAN style="MARGIN-LEFT: 415px"></SPAN>= <I>x</I> +
30</SPAN></P>
<P><SPAN class=Normal>The price of the magazines, adjacent to the one in
the middle is <I>x</I> + 28 or <I>x</I> + 32 depending on whether it is on
the left or right of the middle magazine respectively.</SPAN></P>
<P><SPAN class=Normal>Suppose, <I>x</I> + 60 = <I>x</I> + 28 + <I>x</I> +
30</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/therefore.png">
<I>x</I> + 60 = 2<I>x</I> + 58</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/therefore.png">
<I>x</I> = 2</SPAN></P>
<P><SPAN class=Normal>And if <I>x</I> + 60 = <I>x</I> + 30 + <I>x</I> +
32</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/therefore.png">
60 = <I>x</I> + 62</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/therefore.png">
<I>x</I> = <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/minus.png">2
(which is not possible)</SPAN></P>
<P><SPAN class=Normal>So the adjacent magazine is the one whose price is
<I>x</I> + 28 i.e. one to the left of the middle magazine.</SPAN></P>
<P><SPAN class=Normal>Hence, <B>option
1</B>.</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV></DIV></DIV>
<DIV id=019_V>
<TABLE id=019_V_bookmark class=tableh1 border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD
style="PADDING-BOTTOM: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 100%; PADDING-RIGHT: 0px; PADDING-TOP: 0px">
<OL class="oLevel1 Heading1">
<LI value=5><A id=V>GEOMETRIC
PROGRESSION</A></LI></OL></TD></TR></TBODY></TABLE>
<P class=pLevel2><SPAN class=ListParagraph></SPAN></P>
<P class=pLevel2><SPAN class=Normal>Quantities are said to be in <B>Geometric
Progression (G.P.)</B> when they increase or decrease by a constant factor. The
constant factor is called the <B>common ratio</B>, denoted by <I>r</I>, and it
is found by dividing any term by the preceding term.</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>If the first term is positive and common
ratio is greater than 1 (or if the first term is negative and the common ratio
is less than 1 and positive), the G.P. is an <B>increasing G.P.</B></SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>If the first term is positive and the common
ratio is less than 1 and positive (or if the first term is negative and the
common ratio is greater than 1), the G.P. is a <B>decreasing G.P.</B></SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>In other words, if all terms are greater
than the preceding terms, the G.P. is an increasing G.P. otherwise it is a
decreasing G.P.</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>If the first term is <I>a</I>, the terms of
the progression are <I>a, ar, ar</I><SUP>2</SUP><I>,
ar</I><SUP>3</SUP>,…</SPAN></P>
<P class=pLevel2><SPAN class=ListParagraph></SPAN></P>
<DIV id=019_V_A>
<TABLE id=019_V_A_bookmark class=tableh1 border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD
style="PADDING-BOTTOM: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 100%; PADDING-RIGHT: 0px; PADDING-TOP: 0px">
<OL class="oLevel2 Heading2">
<LI value=1><A id=V_A>THE <I>n</I><SUP>th</SUP> TERM OF A
G.P.</A></LI></OL></TD></TR></TBODY></TABLE>
<P class=pLevel3><SPAN class=ListParagraph></SPAN></P>
<P class=pLevel3><SPAN class=Normal>If <I>T</I><SUB>1</SUB>,<SUB>
</SUB><I>T</I><SUB>2</SUB>, <I>T</I><SUB>3</SUB>, …, <I>T<SUB>n</SUB> </I>denote
the terms of a G.P., then</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><I>T</I><SUB>1</SUB> = <I>a</I> =
<I>ar</I><SUP>1 – 1</SUP></SPAN></P>
<P class=pLevel3><SPAN class=Normal><I>T</I><SUB>2</SUB> = <I>ar</I> =
<I>ar</I><SUP>2 <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/minus.png">1</SUP></SPAN></P>
<P class=pLevel3><SPAN class=Normal><I>T</I><SUB>3</SUB> = <I>ar</I><SUP>2</SUP>
= <I>ar</I><SUP>3 <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/minus.png">
1</SUP></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>Continuing thus, the <I>n</I><SUP>th</SUP>
term of the geometric progression is given by,</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><I>T<SUB>n</I></SUB> = <I>ar<SUP>n</I> –
1</SUP></SPAN></P>
<P class=pLevel3><SPAN class=ListParagraph></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=019_solvedex_19>
<TABLE style="WIDTH: 100%" id=019_solvedex_19_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_19>Example
19:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>Find the fifth term of the G.P. whose first term is
3 and the common ratio is 1/3.</SPAN></P>
<P><SPAN class=ListParagraph></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal><I>a</I> = 3 and <I>r</I> = (1/3)</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>The 5<SUP>th</SUP> term = <I>ar</I><SUP>5 <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/minus.png">
1</SUP></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><SPAN style="MARGIN-LEFT: 90px"></SPAN><IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_038.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><SPAN style="MARGIN-LEFT: 90px"></SPAN>=
1/27</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=019_solvedex_20>
<TABLE style="WIDTH: 100%" id=019_solvedex_20_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_20>Example
20:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>The product of the first five terms of a G.P. is 28.
Find the third term.</SPAN></P>
<P><SPAN class=ListParagraph></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal><I>a</I> <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/multiplication.png">
<I>ar</I> <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/multiplication.png">
<I>ar</I><SUP>2</SUP> <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/multiplication.png">
<I>ar</I><SUP>3</SUP> <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/multiplication.png">
<I>ar</I><SUP>4 </SUP>= 28</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/therefore.png">
<I>a</I><SUP>5</SUP><I>r</I><SUP>10</SUP> = 28</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/therefore.png">
(<I>ar</I><SUP>2</SUP>)<SUP>5</SUP> = 28</SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_039.png"></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_040.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=019_solvedex_21>
<TABLE style="WIDTH: 100%" id=019_solvedex_21_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_21>Example
21:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>The product of the first three consecutive terms of
an increasing G.P. is 216 and their sum is 21. Find the fourth term of
this G.P.</SPAN></P>
<P><SPAN class=ListParagraph></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=ListParagraph>Let the three terms of the G.P. be
<I>a</I>/<I>r</I>, <I>a</I> and <I>ar</I> respectively.</SPAN></P>
<P><SPAN class=ListParagraph></SPAN></P>
<P><SPAN class=ListParagraph>Then,</SPAN></P>
<P><SPAN class=ListParagraph><I>a</I>/<I>r</I> <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/multiplication.png">
<I>a</I> <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/multiplication.png">
<I>ar</I> = 216</SPAN></P>
<P><SPAN class=ListParagraph></SPAN></P>
<P><SPAN class=ListParagraph><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/therefore.png">
<I>a</I><SUP>3</SUP> = 216</SPAN></P>
<P><SPAN class=ListParagraph></SPAN></P>
<P><SPAN class=ListParagraph><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/therefore.png">
The second term = <I>a </I>= 6</SPAN></P>
<P><SPAN class=ListParagraph></SPAN></P>
<P><SPAN class=ListParagraph>Also, <I>a</I>/<I>r </I>+ <I>a</I> +
<I>ar</I> = 21</SPAN></P>
<P><SPAN class=ListParagraph></SPAN></P>
<P><SPAN class=ListParagraph><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/therefore.png">
6<I>r</I><SUP>2</SUP> + 6<I>r</I> + 6 = 21<I>r</SPAN></P>
<P><SPAN class=ListParagraph></SPAN></P>
<P><SPAN class=ListParagraph><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/therefore.png"></I>
6<I>r</I><SUP>2</SUP> – 15<I>r</I> + 6 = 0</SPAN></P>
<P><SPAN class=ListParagraph></SPAN></P>
<P><SPAN class=ListParagraph><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/therefore.png">
(2<I>r</I> – 1)(<I>r</I> – 2) = 0</SPAN></P>
<P><SPAN class=ListParagraph></SPAN></P>
<P><SPAN class=ListParagraph><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_041.png"></SPAN></P>
<P><SPAN class=ListParagraph></SPAN></P>
<P><SPAN class=ListParagraph><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/symb23.png">
The G.P. is an increasing G.P.</SPAN></P>
<P><SPAN class=ListParagraph><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/therefore.png">
<I>r </I>= 2</SPAN></P>
<P><SPAN class=ListParagraph></SPAN></P>
<P><SPAN class=ListParagraph>The fourth term of this G.P. is
<I>ar</I><SUP>2</SUP> = 6 <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/multiplication.png">
2<SUP>2</SUP> = 24</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel3><SPAN class=ListParagraph></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=019_solvedex_22>
<TABLE style="WIDTH: 100%" id=019_solvedex_22_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_22>Example
22:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>The first two terms of a geometric progression add
up to 12. The sum of the third and the fourth terms is 48. If the terms of
the geometric progression are alternately positive and negative, then the
first term is</SPAN></P>
<P><SPAN class=Normal><B>[SNAP 2009]</B></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>(1) –2<SPAN style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN>(2) –4<SPAN
style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN></SPAN></P>
<P><SPAN class=Normal>(3) –12<SPAN style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 17px"></SPAN>(4) 8</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>Let the first term and the ratio of the Geometric
Progression be <I>a</I> and <I>r </I>respectively.</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/therefore.png">
<I>a</I> + <I>ar </I>= 12</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/therefore.png">
<I>a</I>(1 + <I>r</I>) = 12 <SPAN style="MARGIN-LEFT: 40px"></SPAN>…
(i)</SPAN></P>
<P><SPAN class=Normal>Also, <I>ar</I><SUP>2 </SUP>+ <I>ar</I><SUP>3
</SUP>= 48</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/therefore.png">
<I>ar</I><SUP>2</SUP>(1 + <I>r</I>) = 48<SPAN
style="MARGIN-LEFT: 30px"></SPAN>… (ii)</SPAN></P>
<P><SPAN class=Normal>Dividing (ii) by (i),</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_042.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/therefore.png">
<I>r</I><SUP>2 </SUP>= 4</SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_043.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Since the terms of the Geometric Progression are
alternately positive and negative, <I>r</I> = <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/minus.png">2</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/therefore.png">
From (i), <I>a</I>(1 <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/minus.png">
2) = 12</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/therefore.png">
<I>a </I>= <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/minus.png">12</SPAN></P>
<P><SPAN class=Normal>Hence, <B>option
3</B>.</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel3><SPAN class=ListParagraph></SPAN></P></DIV>
<DIV id=019_V_B>
<TABLE id=019_V_B_bookmark class=tableh1 border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD
style="PADDING-BOTTOM: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 100%; PADDING-RIGHT: 0px; PADDING-TOP: 0px">
<OL class="oLevel2 Heading2">
<LI value=2><A id=V_B>SUM OF <I>n</I> TERMS OF A
G.P.</A></LI></OL></TD></TR></TBODY></TABLE>
<P class=pLevel3><SPAN class=ListParagraph></SPAN></P>
<P class=pLevel3><SPAN class=Normal>The sum of <I>n </I>terms of a G.P. with
<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_044.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>The sum of <I>n </I>terms of a G.P. with
<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_045.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>The sum of an infinite number of terms of a
decreasing G.P.</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_046.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=ListParagraph><B>Explanation:</B></SPAN></P>
<P class=pLevel3><SPAN class=Normal>Let <I>S<SUB>n</I></SUB> be the sum of
<I>n</I> terms of a G.P.</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/therefore.png">
<I>S<SUB>n</I></SUB> = <I>a</I> + <I>ar</I> + <I>ar</I><SUP>2</SUP> +
<I>ar</I><SUP>3</SUP> + … + <I>ar<SUP>n</I> – 2 </SUP>+ <I>ar<SUP>n</I> –
1</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/therefore.png"></SUP>
<I>r</I> <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/multiplication.png">
<I>S<SUB>n</I></SUB> = <I>ar</I> + <I>ar</I><SUP>2</SUP> + <I>ar</I><SUP>3</SUP>
+ … + <I>ar<SUP>n</I> – 1</SUP> + <I>ar<SUP>n</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/therefore.png"></I></SUP>
<I>rS<SUB>n</I></SUB> – <I>S<SUB>n</SUB> </I>= <I>ar<SUP>n</I></SUP> –
<I>a</I></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/therefore.png">
<I>S<SUB>n</I></SUB>(<I>r</I> – 1) = <I>a</I>(<I>r<SUP>n</I></SUP> –
1)</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_047.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>If <I>r</I> < 0, (i) can be written
as</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_048.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_049.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>If <I>r</I> < 1 and <I>n</I> is very
large, then <I>ar<SUP>n</I></SUP> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/congru2.png">
0</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_050.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>Equation (ii) gives the sum of infinite
terms of a decreasing geometric series.</SPAN></P>
<P class=pLevel3><SPAN class=Normal>If <I>r</I> = 1, then <I>S<SUB>n</I></SUB> =
<I>n</I> <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/multiplication.png">
<I>a</I></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=019_solvedex_23>
<TABLE style="WIDTH: 100%" id=019_solvedex_23_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_23>Example
23:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_051.png"></SPAN></P>
<P><SPAN class=ListParagraph></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>The given series is a G.P. with <I>a</I> = 1/5 and
<I>r</I> = 5/2</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_052.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/therefore.png">
<I>S<SUB>n</I></SUB> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/congru2.png">
12.88</SPAN></P>
<P><SPAN class=ListParagraph></SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel3><SPAN class=ListParagraph></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=019_solvedex_24>
<TABLE style="WIDTH: 100%" id=019_solvedex_24_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_24>Example
24:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading></SPAN></P>
<P><SPAN class=Normal>Determine the sum of the infinite series in
Geometric Progression:</SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_053.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><B>[JMET 2009]</B></SPAN></P>
<P><SPAN class=Normal>(1) 1/4<SPAN style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 12px"></SPAN>(2) infinite</SPAN></P>
<P><SPAN class=Normal>(3) 2<SPAN style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN> (4) 4/3</SPAN></P>
<P><SPAN class=ListParagraph></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>The given series is an infinite Geometric
Progression with the common ratio <IMG style="VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/minus.png">1/3.</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/therefore.png">
The required sum ‘<I>s</I>’ would be given by,</SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_054.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_055.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><I>r</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_056.png"></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_057.png"></SPAN></P>
<P><SPAN class=Normal>Hence, <B>option
1</B>.</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=019_solvedex_25>
<TABLE style="WIDTH: 100%" id=019_solvedex_25_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_25>Example
25:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>Find the sum of <I>n</I> terms of the series 9.01 +
99.001 + 999.0001 + …</SPAN></P>
<P><SPAN class=ListParagraph></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal><I>S<SUB>n</I></SUB> = 9.01 + 99.001 + 999.0001 + …
upto <I>n</I> terms</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/therefore.png">
<I>S<SUB>n</I></SUB> = (9 + 99 + 999 + … upto <I>n</I> terms)</SPAN></P>
<P><SPAN class=Normal>+ (0.01 + 0.001 + 0.0001 + … upto <I>n</I>
terms)</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/therefore.png">
<I>S<SUB>n</I></SUB> = [(10 <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/minus.png">
1) + (100 <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/minus.png">
1) + (1000 <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/minus.png">
1) + … upto <I>n</I> terms]</SPAN></P>
<P><SPAN class=Normal>+ (0.01 + 0.001 + 0.0001 + … upto <I>n</I>
terms)</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Let <I>S</I><SUB>1 </SUB>= (10 <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/minus.png">
1) + (100 <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/minus.png">
1) + (1000 <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/minus.png">
1) + … upto <I>n</I> terms</SPAN></P>
<P><SPAN class=Normal>and <I>S</I><SUB>2 </SUB>= 0.01 + 0.001 + 0.0001 +
…upto <I>n</I> terms</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/therefore.png">
<I>S</I><SUB>1</SUB> = (10 + 100 + 1000 + …) – <I>n</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_058.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_059.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_060.png"></SPAN></P></I></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=019_solvedex_26>
<TABLE style="WIDTH: 100%" id=019_solvedex_26_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_26>Example
26:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>A</I> ping pong ball is dropped from a 45 metres
high multi-storey building. The ball bounces back three fifth of the
distance each time before coming to rest. The total distance traversed by
the ball is:</SPAN></P>
<P><SPAN class=Normal><B>[IIFT 2009]</B></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>(1) 150 m <SPAN
style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN>(2) 180 m</SPAN></P>
<P><SPAN class=Normal>(3) 175 m <SPAN
style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN>(4) None of the above</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=ListParagraph>The distance traversed by the ball
is</SPAN></P>
<P><SPAN class=ListParagraph><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_061.png"></SPAN></P>
<P><SPAN class=ListParagraph></SPAN></P>
<P><SPAN class=ListParagraph><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_062.png"></SPAN></P>
<P><SPAN class=ListParagraph></SPAN></P>
<P><SPAN class=ListParagraph><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_063.png"></SPAN></P>
<P><SPAN class=ListParagraph></SPAN></P>
<P><SPAN class=ListParagraph>= 180 m</SPAN></P>
<P><SPAN class=ListParagraph>Hence, <B>option
2</B>.</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel3><SPAN class=ListParagraph></SPAN></P>
<DIV class="divTextBox divColorNote divLevel3">
<P><SPAN class=Note>REMEMBER:</SPAN></P>
<P><SPAN class=ListParagraph></SPAN></P>
<UL class=Normal>
<LI value=1>During the exam, multiple choice questions based on the sums of
series in terms of <I>n</I> can be easily solved by evaluating
options.</LI></UL></DIV>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=019_solvedex_27>
<TABLE style="WIDTH: 100%" id=019_solvedex_27_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_27>Example
27:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>A man decides to save 10% of his income every year.
But from the second year onwards, he manages to save only 90% of his
savings percentage in the previous year. How much money does he save at
the end of five years if his income in the first year is Rs. 2,00,000 and
his income increases every year by 10%?</SPAN></P>
<P><SPAN class=ListParagraph></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>Savings in the first year = <I>x</I> = 200000 <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/multiplication.png">
0.1 = 20000</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Savings in the second year = (200000 <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/multiplication.png">
1.1) <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/multiplication.png">
(0.1 <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/multiplication.png">
0.9) = 0.99<I>x</I></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Savings in the third year =
(0.99)<SUP>2</SUP><I>x</I></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Similarly, savings in the fourth and fifth years are
(0.99)<SUP>3</SUP><I>x</I> and (0.99)<SUP>4</SUP><I>x</I>.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/therefore.png">
His savings at the end of five years</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>= <I>x</I>(1 + 0.99 + 0.99<SUP>2</SUP> +
0.99<SUP>3</SUP> + 0.99<SUP>4</SUP>)</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>= 20000(1 – 0.99<SUP>5</SUP>)/0.01</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/congru2.png">
2000000(1 – 0.95) <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/congru2.png">
98020</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/therefore.png">
The man saves approximately Rs. 98,020 at the end of five
years.</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=019_solvedex_28><!--<span class="ExampleHeading"><a id="ex_28">Example 28:</a></span></p>
<p><span class="ExampleHeading"></span></p>-->
<TABLE style="WIDTH: 100%" id=Table1 border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=A1>Example
28:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>The sum of the first, third, fifth, seventh, …,
thirteenth terms of a G.P. is 14.87 and the sum of the second, fourth,
sixth, …, fourteenth terms of a G.P. is 15.02. If the fifth term is 2.08,
find the common ratio.</SPAN></P>
<P><SPAN class=SolutionHeading></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>Let <I>a</I> and <I>r</I> be the first term and
common ratio of the G.P.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Let <I>S</I><SUB>1</SUB> = <I>a</I> +
<I>ar</I><SUP>2</SUP> + <I>ar</I><SUP>4</SUP> + … + <I>ar</I><SUP>12</SUP>
= <I>a</I>(1 + <I>r</I><SUP>2</SUP> + <I>r</I><SUP>4</SUP> +
<I>r</I><SUP>6</SUP> + …+<I> r</I><SUP>12</SUP>)</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>And <I>S</I><SUB>2</SUB> = <I>ar</I> +
<I>ar</I><SUP>3</SUP> + <I>ar</I><SUP>5</SUP> + … + <I>ar</I><SUP>13</SUP>
= <I>ar</I>(1 + <I>r</I><SUP>2</SUP> + <I>r</I><SUP>4</SUP> + …+<I>
r</I><SUP>12</SUP>)</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/therefore.png">
<I>S</I><SUB>2</SUB>/<I>S</I><SUB>1</SUB> = <I>r</I></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/therefore.png">
<I>r</I> = 15.02/14.87 <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/congru2.png">
1.01</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Here, the fact that the fifth term is 2.08 is not
used in the solution at all.</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel4><SPAN class=ListParagraph></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=019_solvedex_29>
<TABLE style="WIDTH: 100%" id=019_solvedex_29_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_29>Example
29:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>Find the sum of<I> </I>2<I>n</I> terms of the series
1, 1.5, 3, 2.25, 5, 3.375, 7,…</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=ListParagraph>The given series can be broken into two
different series:</SPAN></P>
<P><SPAN class=ListParagraph></SPAN></P>
<P><SPAN class=ListParagraph><I>S</I><SUB>1 </SUB>= 1, 3, 5, 7,
…</SPAN></P>
<P><SPAN class=ListParagraph><I>S</I><SUB>2 </SUB>= 1.5, 2.25, 3.375,
…</SPAN></P>
<P><SPAN class=ListParagraph></SPAN></P>
<P><SPAN class=ListParagraph><I>S</I><SUB>1 </SUB>is an A.P. with first
term = 1 and common difference = 2</SPAN></P>
<P><SPAN class=ListParagraph><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_064.png"></SPAN></P>
<P><SPAN class=ListParagraph></SPAN></P>
<P><SPAN class=ListParagraph><I>S</I><SUB>2 </SUB>is a G.P. with first
term = 1.5 and common ratio = 1.5</SPAN></P>
<P><SPAN class=ListParagraph><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_065.png"></SPAN></P>
<P><SPAN class=ListParagraph></SPAN></P>
<P><SPAN class=ListParagraph><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_066.png"></SPAN></P>
<P><SPAN
class=Normal><SPAN></SPAN><SPAN></SPAN>
= 3(1.5<I><SUP>n</I></SUP> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/minus.png">
1)</SPAN></P>
<P><SPAN class=ListParagraph></SPAN></P>
<P><SPAN class=ListParagraph><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/therefore.png">
Sum of series = <I>n</I><SUP>2</SUP> + 3(1.5<I><SUP>n</I></SUP> –
1)</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel4><SPAN class=ListParagraph></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=019_solvedex_30>
<TABLE style="WIDTH: 100%" id=019_solvedex_30_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_30>Example
30:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ListParagraph>What is the approximate sum of 50 terms of
the series given below?</SPAN></P>
<P><SPAN class=ListParagraph><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_067.png"></SPAN></P>
<P><SPAN class=ListParagraph></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=ListParagraph>We see that the numerators of the terms in
the given series are in A.P., with a common difference of 2. Thus the
numerator of the 50<SUP>th</SUP> term is 103.</SPAN></P>
<P><SPAN class=ListParagraph></SPAN></P>
<P><SPAN class=ListParagraph>The denominators are in G.P. with common
ratio = 1/2. The denominator of the 50<SUP>th</SUP> term is
2<SUP>50</SUP>.</SPAN></P>
<P><SPAN class=ListParagraph></SPAN></P>
<P><SPAN class=ListParagraph><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_068.png"></SPAN></P>
<P><SPAN class=ListParagraph></SPAN></P>
<P><SPAN class=ListParagraph><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_069.png"></SPAN></P>
<P><SPAN class=ListParagraph></SPAN></P>
<P><SPAN class=ListParagraph>Subtracting (ii) from (i),</SPAN></P>
<P><SPAN class=ListParagraph><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_070.png">
<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_071.png"></SPAN></P>
<P><SPAN class=ListParagraph></SPAN></P>
<P><SPAN class=ListParagraph><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_072.png"></SPAN></P>
<P><SPAN class=ListParagraph></SPAN></P>
<P><SPAN class=ListParagraph><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_073.png"></SPAN></P>
<P><SPAN class=ListParagraph></SPAN></P>
<P><SPAN class=ListParagraph><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_074.png"></SPAN></P>
<P><SPAN class=ListParagraph></SPAN></P>
<P><SPAN class=ListParagraph><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_075.png"></SPAN></P>
<P><SPAN class=ListParagraph></SPAN></P>
<P><SPAN class=ListParagraph><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/therefore.png">
<I>S</I><SUB>50</SUB> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/congru2.png">
5 + 2 = 7</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel4><SPAN class=ListParagraph></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=019_solvedex_31>
<TABLE style="WIDTH: 100%" id=019_solvedex_31_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_31>Example
31:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ListParagraph>Let <I>S</I> = 2<I>x</I> +
5<I>x</I><SUP>2</SUP> + 9<I>x</I><SUP>3</SUP> + 14<I>x</I><SUP>4</SUP> +
20<I>x</I><SUP>5</SUP> .... infinity</SPAN></P>
<P><SPAN class=ListParagraph></SPAN></P><SPAN class=ListParagraph><IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_079.png">
<P><SPAN class=ListParagraph><B>[CAT 2002]</B></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_077.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_078.png"></SPAN></P>
<P><SPAN class=ListParagraph></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=ListParagraph><I>S</I> = 2<I>x</I> + 5<I>x</I><SUP>2</SUP>
+ 9<I>x</I><SUP>3</SUP> +<SUP> </SUP>14<I>x</I><SUP>4</SUP> +
20<I>x</I><SUP>5</SUP> +... <SPAN
style="MARGIN-LEFT: 12px"> </SPAN> …(i)</SPAN></P>
<P><SPAN class=ListParagraph></SPAN></P>
<P><SPAN class=ListParagraph>Multiplying both sides by
<I>x</I>,</SPAN></P>
<P><SPAN class=ListParagraph></SPAN></P>
<P><SPAN class=ListParagraph><I>xS</I> = 2<I>x</I><SUP>2</SUP> +
5<I>x</I><SUP>3</SUP> +
9<I>x</I><SUP>4 </SUP> +... <SPAN
style="MARGIN-LEFT: 72px"> ...(ii)</SPAN></P>
<P><SPAN class=ListParagraph></SPAN></P>
<P><SPAN class=ListParagraph>Subtracting (ii) from (i),</SPAN></P>
<P><SPAN class=ListParagraph></SPAN></P>
<P><SPAN class=ListParagraph>(1 – <I>x</I>)<I>S</I> = 2<I>x</I> +
3<I>x</I><SUP>2</SUP> +
4<I>x</I><SUP>3</SUP>...
...(iii)</SPAN></P>
<P><SPAN class=ListParagraph><I>x</I>(1 – <I>x</I>)<I>S</I> =
2<I>x</I><SUP>2</SUP> + 3<I>x</I><SUP>3</SUP> +
4<I>x</I><SUP>4</SUP>...
...(iv)</SPAN></P>
<P><SPAN class=ListParagraph>Subtracting (iv) from (iii),</SPAN></P>
<P><SPAN class=ListParagraph></SPAN></P>
<P><SPAN class=ListParagraph>(1 – <I>x</I>)<SUP>2</SUP><I>S</I> =
2<I>x</I> + <I>x</I><SUP>2</SUP> + <I>x</I><SUP>3</SUP>...</SPAN></P>
<P><SPAN class=ListParagraph></SPAN></P><SPAN class=ListParagraph><IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_079_.png">
<P><SPAN class=ListParagraph></SPAN></P><SPAN class=ListParagraph><IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_080.png"></SPAN>
<P></P>
<P><SPAN class=ListParagraph></SPAN></P>
<P><SPAN class=ListParagraph>Hence, <B>option
1.</B></SPAN></P></SPAN></SPAN></SPAN></TD></TR></TBODY></TABLE></DIV></DIV>
<DIV id=019_V_C>
<TABLE id=019_V_C_bookmark class=tableh1 border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD
style="PADDING-BOTTOM: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 100%; PADDING-RIGHT: 0px; PADDING-TOP: 0px">
<OL class="oLevel2 Heading2">
<LI value=3><A id=V_C>GEOMETRIC MEAN</A></LI></OL></TD></TR></TBODY></TABLE>
<P class=pLevel3><SPAN class=ListParagraph></SPAN></P>
<P class=pLevel3><SPAN class=Normal>If <I>n</I> terms <I>a</I><SUB>1</SUB>,
<I>a</I><SUB>2</SUB>, …, <I>a<SUB>n</I></SUB> are in G.P., then the Geometric
Mean <I>G</I> of these <I>n</I> terms is given by</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=ListParagraph><IMG style="VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_081.png"></SPAN></P>
<P class=pLevel3><SPAN class=ListParagraph></SPAN></P>
<P class=pLevel3><SPAN class=ListParagraph>If three terms are in G.P. then the
middle term is the Geometric mean of the other two terms. If <I>a</I>, <I>b</I>
and <I>c</I> are in G.P. (<I>a</I>, <I>c</I> > 0 or <I>a</I>, <I>c</I> <
0), then <I>b</I> is the geometric mean of <I>a</I> and <I>c</I>, and is given
by <I>b</I> = <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_082.png">
or <I>b</I><SUP>2</SUP> = <I>ac</I>.</SPAN></P>
<P class=pLevel3><SPAN class=ListParagraph>If <I>n</I> is even, the geometric
mean of the terms of the G.P. is equal to the geometric mean of its</SPAN></P>
<P class=pLevel3><SPAN class=ListParagraph></SPAN></P>
<P class=pLevel3><SPAN class=ListParagraph><IMG style="VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_083.png"></SPAN></P>
<P class=pLevel3><SPAN class=ListParagraph></SPAN></P>
<P class=pLevel3><SPAN class=Normal>If <I>n</I> is odd, the geometric mean of
the terms of the G.P. is equal to the</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=ListParagraph><IMG style="VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_084.png"></SPAN></P>
<P class=pLevel3><SPAN class=ListParagraph></SPAN></P>
<DIV class="divTextBox divColorNote divLevel3">
<P><SPAN class=ListParagraph><B>REMEMBER:</B></SPAN></P>
<P><SPAN class=ListParagraph></SPAN></P>
<UL class=Normal>
<LI value=1>While solving problems, three terms in G.P. can be assumed to be
<I>a</I>/<I>r</I>, <I>a</I> and <I>ar</I>. Similarly, four terms in G.P. can
be assumed to be <I>a</I>/<I>r</I><SUP>3</SUP>, <I>a</I>/<I>r</I>, <I>ar</I>
and <I>ar</I><SUP>3</SUP>. The advantage of representing terms in this way is
that the product of terms is then obtained in only one unknown.</LI></UL></DIV>
<P class=pLevel4><SPAN class=ListParagraph><B>SOME PROPERTIES OF A
G.P.</B></SPAN></P>
<UL class="oLevel3 ListParagraph ListStyleUL5">
<LI value=3>If each term of a G.P. is multiplied or divided by the some
non-zero quantity, then the resulting sequence is also a G.P. with the common
ratio remaining the same.</LI></UL>
<UL class="oLevel3 ListParagraph ListStyleUL5">
<LI value=4>The reciprocals of the terms of a given G.P. also form a G.P.,
where the common ratio is the reciprocal of that of the earlier G.P.</LI></UL>
<UL class="oLevel3 ListParagraph ListStyleUL5">
<LI value=5>In a finite G.P., the product of two terms equidistant from the
first and the last terms is same as the product of the first and the last
term.</LI></UL>
<P class=pLevel4><SPAN class=Normal></SPAN></P></DIV></DIV>
<DIV id=019_VI>
<TABLE id=019_VI_bookmark class=tableh1 border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD
style="PADDING-BOTTOM: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 100%; PADDING-RIGHT: 0px; PADDING-TOP: 0px">
<OL class="oLevel1 Heading1">
<LI value=6><A id=VI>HARMONIC
PROGRESSION</A></LI></OL></TD></TR></TBODY></TABLE>
<P class=pLevel2><SPAN class=Normal>Quantities are said to be in <B>Harmonic
Progression (H.P.)</B> when their reciprocals are in A.P. In general if
<I>a</I>, <I>a</I> + <I>d</I>, <I>a</I> + 2<I>d</I>, <I>a</I> + 3<I>d</I>, … are
successive terms of an arithmetic progression, then 1/<I>a</I>, 1/(<I>a</I> +
<I>d</I>), 1/(<I>a</I> + 2<I>d</I>) and 1/(<I>a</I> + 3<I>d</I>), … are in
harmonic progression.</SPAN></P>
<P class=pLevel2><SPAN class=Normal>The <I>n</I><SUP>th</SUP> term
(<I>T<SUB>n</I></SUB>) of a harmonic progression is given by</SPAN></P>
<P class=pLevel2><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_085.png"></SPAN></P>
<P class=pLevel2><SPAN class=ListParagraph></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel2">
<DIV id=019_solvedex_32>
<TABLE style="WIDTH: 100%" id=019_solvedex_32_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_32>Example
32:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>The third term of a H.P. is 1/3 and the sixth term
is 1/9. Find the 31<SUP>st</SUP> term of the H.P.</SPAN></P>
<P><SPAN class=ListParagraph></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>For the corresponding A.P. the third term is 3 and
the sixth term is 9.</SPAN></P>
<P><SPAN class=Normal>Hence, <I>a</I> + 2<I>d</I> = 3 and <I>a</I> +
5<I>d</I> = 9</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/therefore.png">
<I>d</I> = 2 and <I>a</I> = <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/minus.png">1</SPAN></P>
<P><SPAN class=Normal>Hence the 31<SUP>st</SUP> term of this A.P. =
<I>a</I> + (31 – 1)<I>d</I> = <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/minus.png">1
+ 30 <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/multiplication.png">
2 = 59</SPAN></P>
<P><SPAN class=Normal>Hence for the corresponding H.P., the
31<SUP>st</SUP> term is 1/59.</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<DIV id=019_VI_A>
<TABLE id=019_VI_A_bookmark class=tableh1 border=0 cellSpacing=0
cellPadding=0><TBODY>
<TR>
<TD
style="PADDING-BOTTOM: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 100%; PADDING-RIGHT: 0px; PADDING-TOP: 0px">
<OL class="oLevel2 Heading2">
<LI value=1><A id=VI_A>HARMONIC MEAN</A></LI></OL></TD></TR></TBODY></TABLE>
<P class=pLevel3><SPAN class=ListParagraph>The <B>Harmonic</B> <B>Mean</B> of
<I>n</I> numbers is the reciprocal of the arithmetic mean of the reciprocals of
these <I>n</I> numbers.</SPAN></P>
<P class=pLevel3><SPAN class=ListParagraph>The harmonic mean can be derived
using the concept of arithmetic mean.</SPAN></P>
<P class=pLevel3><SPAN class=ListParagraph></SPAN></P>
<P class=pLevel3><SPAN class=ListParagraph>If <I>a</I>, <I>b</I> and <I>c</I>
are in H.P., <I>b</I> is the harmonic mean of <I>a</I> and <I>b</I>. 1/<I>a</I>,
1/<I>b</I> and 1/<I>c</I> are in A.P. Thus we have</SPAN></P>
<P class=pLevel3><SPAN class=ListParagraph><IMG style="VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_086.png"></SPAN></P>
<P class=pLevel3><SPAN class=ListParagraph><IMG style="VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_087.png"></SPAN></P>
<P class=pLevel3><SPAN class=ListParagraph><IMG style="VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_088.png"></SPAN></P>
<P class=pLevel3><SPAN class=ListParagraph>Hence the harmonic mean for two
numbers is given by</SPAN></P>
<P class=pLevel3><SPAN class=ListParagraph><IMG style="VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_089.png"></SPAN></P>
<P class=pLevel3><SPAN class=ListParagraph>In general, the harmonic mean of
<I>n</I> numbers <I>a</I><SUB>1</SUB>, <I>a</I><SUB>2</SUB>,
<I>a</I><SUB>3</SUB>, <I>a</I><SUB>4</SUB>, …, <I>a<SUB>n</I></SUB>
is</SPAN></P>
<P class=pLevel3><SPAN class=ListParagraph><IMG style="VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_090.png"></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=019_solvedex_33>
<TABLE style="WIDTH: 100%" id=019_solvedex_33_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_33>Example
33:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>If three positive real numbers <I>a</I>, <I>b</I>
and <I>c</I> (<I>c</I> > <I>a</I>) are in Harmonic Progression,
then</SPAN></P>
<P><SPAN class=Normal>log (<I>a</I> + <I>c</I>) + log (<I>a</I> –
2<I>b</I> + <I>c</I>) is equal to:</SPAN></P>
<P><SPAN class=Normal><B>[IIFT 2008]</B></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>(1) 2 log (<I>c</I> – <I>b</I>)<SPAN
style="MARGIN-LEFT: 25px"></SPAN>(2) 2 log (<I>a</I> –
<I>c</I>)</SPAN></P>
<P><SPAN class=Normal>(3) 2 log (<I>c</I> – <I>a</I>)<SPAN
style="MARGIN-LEFT: 25px"></SPAN>(4) log <I>a</I> + log<I> b </I>+ log
<I>c</I></SPAN></P>
<P><SPAN class=ListParagraph></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>As <I>a</I>, <I>b</I>, <I>c</I> are in
H.P.,</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_089.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/therefore.png">
log (<I>a</I> + <I>c</I>) + log (<I>a</I> – 2<I>b</I> + <I>c</I>) = log
[(<I>a</I> + <I>c</I>)<SUP>2</SUP> – 2<I>b</I>(<I>a</I> +
<I>c</I>)]</SPAN></P>
<P><SPAN class=Normal><SPAN style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 175px"></SPAN>= log [(<I>a</I> + <I>c</I>)<SUP>2</SUP>
– 4<I>ac</I>)]</SPAN></P>
<P><SPAN class=Normal><SPAN style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 175px"></SPAN>= log (<I>a</I> –
<I>c</I>)<SUP>2</SUP></SPAN></P>
<P><SPAN class=Normal><SPAN style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 175px"></SPAN>= 2 log(<I>c</I> – <I>a</I>)<SPAN
style="MARGIN-LEFT: 20px"></SPAN> … (<IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/symb23.png">
<I>c</I> > <I>a</I>)</SPAN></P>
<P><SPAN class=Normal>Hence, <B>option
3</B>.</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel3><SPAN class=ListParagraph></SPAN></P>
<DIV class="divTextBox divColorNote divLevel3">
<P><SPAN class=Note>REMEMBER:</SPAN></P>
<UL class=Normal>
<LI>There is no general formula for the sum of quantities in H.P. Questions on
H.P. are solved by making use of the properties of the corresponding
A.P.</LI></UL>
<UL class=Normal>
<LI value=1>The concept of harmonic mean is used when the quantities are
specified in rates like distance covered per hour or speed (km/hr, m/s) or
production per hour (20 units per hour, 56 units per hour), etc. Thus if
<I>a</I> and <I>b</I> are two rates, then the average rate is given by the
harmonic mean 2<I>ab</I>/(<I>a</I> + <I>b</I>). </LI></UL></DIV>
<P class=pLevel3><SPAN class=ListParagraph>A common application of harmonic mean
is to find the average speed. If a person travels a particular distance at speed
<I>a</I> and then the <I>same distance</I> at the speed <I>b</I>, then his
average speed is the <I>harmonic mean</I> of <I>a</I> and <I>b</I>.</SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=019_solvedex_34>
<TABLE style="WIDTH: 100%" id=019_solvedex_34_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_34>Example
34:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>Kishan travels a distance of 300 km from Rampur to
Lakhangaam at 30 km per hour in a truck and then to Shyamnagar at 50 km
per hour in a car. Find his average speed from Rampur to Shyamnagar, if
the distance between Lakhangaam and Shyamnagar is 300 km.</SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>As the distance between Rampur and Lakhangaam is
equal to the distance between Lakhangaam and Shyamnagar, Kishan’s average
speed is the harmonic mean of 30 and 50.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/therefore.png">
Kishan’s average speed</SPAN></P>
<P><SPAN class=Normal>= 2 <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/multiplication.png">
30 <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/multiplication.png">
50/80 = 37.5 km per hour.</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<DIV class="divTextBox divColorNote divLevel3">
<P><SPAN class=Note>IMPORTANT:</SPAN></P>
<P><SPAN class=ListParagraph>If a person travels at two different speeds for the
<I>same amount of time</I>, the average speed is the <I>arithmetic mean</I> of
the two different speeds.</SPAN></P></DIV></DIV>
<DIV id=019_VI_B>
<TABLE id=019_VI_B_bookmark class=tableh1 border=0 cellSpacing=0
cellPadding=0><TBODY>
<TR>
<TD
style="PADDING-BOTTOM: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 100%; PADDING-RIGHT: 0px; PADDING-TOP: 0px">
<OL class="oLevel2 Heading2">
<LI value=2><A id=VI_B>RELATION BETWEEN ARITHMETIC, GEOMETRIC AND
HARMONIC MEANS</A></LI></OL></TD></TR></TBODY></TABLE>
<P class=pLevel3><SPAN class=Normal>Let <I>A</I>, <I>G</I> and <I>H</I>
represent the Arithmetic, Geometric and Harmonic means of two positive or two
negative quantities <I>a</I> and <I>b</I>. Then,</SPAN></P>
<P class=pLevel3><SPAN class=Normal><I>A</I> = (<I>a</I> +
<I>b</I>)/2</SPAN></P>
<P class=pLevel3><SPAN class=Normal><I>G</I> = </SPAN><SPAN class=Normal><IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_091.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><I>H</I> = 2<I>ab</I>/(<I>a</I> +
<I>b</I>)</SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/therefore.png">
<I>A</I> <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/multiplication.png">
<I>H</I> = <I>ab</I> = <I>G</I><SUP>2</SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/therefore.png"></SUP>
<I>A</I> <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/multiplication.png">
<I>H</I> = <I>G</I><SUP>2</SUP></SPAN></P>
<P class=pLevel3><SPAN class=Normal>Thus <I>G</I> is the geometric mean of
<I>A</I> and <I>H</I>.</SPAN></P>
<P class=pLevel3><SPAN class=Normal>Also, we know that <I>A</I> > <I>G</I>
for two unequal quantities.</SPAN></P>
<P class=pLevel3><SPAN class=Normal>But as <I>G</I> is the geometric mean of
<I>A</I> and <I>H</I>, <I>G</I> lies between <I>A</I> and <I>H</I>.</SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/therefore.png">
Arithmetic Mean > Geometric Mean > Harmonic Mean</SPAN></P>
<P class=pLevel3><SPAN class=ListParagraph></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=019_solvedex_35>
<TABLE style="WIDTH: 100%" id=019_solvedex_35_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_35>Example
35:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>The arithmetic mean of two numbers exceeds its
geometric mean by 18. The geometric mean is 2.125 times the harmonic mean.
Find the difference between the two numbers.</SPAN></P>
<P><SPAN class=ListParagraph></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>Let <I>p</I> and <I>q</I> be the two
numbers.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><I>A</I> = <I>G</I> + 18</SPAN></P>
<P><SPAN class=Normal><I>G</I> = 2.125<I>H</I> = 17<I>H</I>/8</SPAN></P>
<P><SPAN class=Normal>But, <I>G</I><SUP>2</SUP> = <I>A</I> <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/multiplication.png">
<I>H</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_092.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_093.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/therefore.png"></I>
<I>A</I> = 34 and <I>G</I> = 16</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>(<I>p</I> – <I>q</I>)<SUP>2</SUP> = (<I>p</I> +
<I>q</I>)<SUP>2</SUP> – 4<I>pq</I></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/therefore.png">
(<I>p</I> – <I>q</I>)<SUP>2</SUP> = (2<I>A</I>)<SUP>2</SUP> –
4<I>G</I><SUP>2</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/therefore.png"></SUP>
(<I>p</I> – <I>q</I>)<SUP>2</SUP> = 68<SUP>2</SUP> – 4 <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/multiplication.png">
16<SUP>2</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/therefore.png"></SUP>
(<I>p</I> – <I>q</I>)<SUP>2</SUP> = 3600</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/therefore.png">
(<I>p</I> – <I>q</I>) = 60</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/therefore.png">
Difference between the two numbers =
60</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=019_solvedex_36>
<TABLE style="WIDTH: 100%" id=019_solvedex_36_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_36>Example
36:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>If the product of <I>n</I> positive real numbers is
unity, then their sum is necessarily</SPAN></P>
<P><SPAN class=Normal><B>[CAT 2003 Leaked Test]</B></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>(1) a multiple of <I>n</I></SPAN></P>
<P><SPAN class=Normal>(2) equal to <I>n</I> + 1/<I>n</I></SPAN></P>
<P><SPAN class=Normal>(3) never less than <I>n</I></SPAN></P>
<P><SPAN class=Normal>(4) a positive integer</SPAN></P>
<P><SPAN class=ListParagraph></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>Let <I>a</I><SUB>1</SUB>, <I>a</I><SUB>2</SUB>,
<I>a</I><SUB>3</SUB>, …, <I>a<SUB>n</I></SUB> be <I>n</I> positive real
numbers.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Now, <I>a</I><SUB>1</SUB> <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/multiplication.png">
<I>a</I><SUB>2</SUB> <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/multiplication.png">
<I>a</I><SUB>3</SUB> <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/multiplication.png">
… <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/multiplication.png">
<I>a<SUB>n</I></SUB> = 1</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>We know that, A.M. <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/greater_equal.png">
G.M.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><I>a</I><SUB>1</SUB> + <I>a</I><SUB>2</SUB> +
<I>a</I><SUB>3</SUB> + … + <I>a<SUB>n</I></SUB> <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/greater_equal.png">
<I>n</I> <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/multiplication.png">
(<I>a</I><SUB>1</SUB> <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/multiplication.png">
<I>a</I><SUB>2</SUB> <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/multiplication.png">
<I>a</I><SUB>3</SUB> <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/multiplication.png">
… <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/multiplication.png">
<I>a<SUB>n</I></SUB>)<SUP>1/<I>n</I></SUP></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/therefore.png">
<I>a</I><SUB>1</SUB> + <I>a</I><SUB>2</SUB> + <I>a</I><SUB>3</SUB> + … +
<I>a<SUB>n</I></SUB> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/greater_equal.png">
<I>n</I> <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/multiplication.png">
(1)<SUP>1/<I>n</I></SUP></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/therefore.png">
<I>a</I><SUB>1</SUB> + <I>a</I><SUB>2</SUB> + <I>a</I><SUB>3</SUB> + … +
<I>a<SUB>n</I></SUB> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/greater_equal.png">
<I>n</I></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Hence,<B> option
3.</SPAN></P></B></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel3><SPAN class=ListParagraph></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=019_solvedex_37>
<TABLE style="WIDTH: 100%" id=019_solvedex_37_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_37>Example
37:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal><I>p</I></B>, <I>q</I>, <I>r</I> and <I>s</I> are in
G.P. Are log<I><SUB>n</I></SUB>(<I>p<SUP>n</I></SUP> –
<I>q</I><SUP>n</SUP>), log<I><SUB>n</I></SUB>(<I>q<SUP>n</I> </SUP>–
<I>r<SUP>n</I></SUP>) and log<I><SUB>n</I></SUB>(<I>r<SUP>n</I></SUP> –
<I>s<SUP>n</I></SUP>) also terms of a progression?</SPAN></P>
<P><SPAN class=ListParagraph></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal><I>p</I>, <I>q</I>, <I>r</I> and <I>s</I> are in
G.P. Let the common ratio be <I>k</I>.</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/therefore.png">
<I>q</I> = <I>pk</I>, <I>r</I> = <I>pk</I><SUP>2</SUP>, <I>s</I> =
<I>pk</I><SUP>3</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/therefore.png"></SUP>
<I>p<SUP>n</I></SUP> – <I>q<SUP>n</I></SUP> = <I>p<SUP>n</I></SUP>(1 –
<I>k<SUP>n</I></SUP>),</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><I>q<SUP>n</I> </SUP>– <I>r<SUP>n</I></SUP><SUB>
</SUB>= <I>p<SUP>n</SUP>k<SUP>n</I></SUP>(1 –
<I>k<SUP>n</I></SUP>),</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><I>r<SUP>n</I></SUP> – <I>s<SUP>n</I></SUP><SUB>
</SUB>= <I>p<SUP>n</SUP>k</I><SUP>2<I>n</I></SUP>(1 –
<I>k<SUP>n</I></SUP>)</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/therefore.png">
(<I>p<SUP>n</I></SUP> – <I>q</I><SUP>n</SUP>), (<I>q<SUP>n</I> </SUP>–
<I>r<SUP>n</I></SUP>) and (<I>r<SUP>n</I></SUP> – <I>s<SUP>n</I></SUP>)
form a G.P. with common ratio <I>k<SUP>n</I></SUP>.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/therefore.png">
(<I>q<SUP>n</I> </SUP>– <I>r<SUP>n</I></SUP>)<SUP>2</SUP> =
(<I>p<SUP>n</I></SUP> – <I>q<SUP>n</I></SUP>)(<I>r<SUP>n</I></SUP> –
<I>s<SUP>n</I></SUP>)</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/therefore.png">
2 log<I><SUB>n</I></SUB>(<I>q<SUP>n</I> </SUP>– <I>r<SUP>n</I></SUP>) =
log<I><SUB>n</I></SUB>(<I>p<SUP>n</I></SUP> – <I>q</I><SUP>n</SUP>) +
log<I><SUB>n</I></SUB>(<I>r<SUP>n</I></SUP> –
<I>s<SUP>n</I></SUP>)</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/therefore.png">
(<I>p<SUP>n</I></SUP> – <I>q</I><SUP>n</SUP>), (<I>q<SUP>n</I> </SUP>–
<I>r<SUP>n</I></SUP>) and (<I>r<SUP>n</I></SUP> – <I>s<SUP>n</I></SUP>)
are in A.P.</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel3><SPAN class=ListParagraph></SPAN></P></DIV></DIV>
<DIV id=019_VII>
<TABLE id=019_VII_bookmark class=tableh1 border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD
style="PADDING-BOTTOM: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 100%; PADDING-RIGHT: 0px; PADDING-TOP: 0px">
<OL class="oLevel1 Heading1">
<LI value=7><A id=VII>SERIES OF NATURAL
NUMBERS</A></LI></OL></TD></TR></TBODY></TABLE>
<P class=pLevel2><SPAN class=ListParagraph></SPAN></P>
<P class=pLevel2><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_094.png">
<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_095.png"></SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>Sum of the squares of the first <I>n</I>
natural number is given by </P>
<P class=pLevel2><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_097_.png"></SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>Sum of the cubes of the first <I>n</I>
natural numbers is given by </P>
<P class=pLevel2><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_099_.png"></SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=ListParagraph></SPAN></P>
<P class=pLevel2><SPAN class=ListParagraph><B>Explanation:</B></SPAN></P>
<P class=pLevel2><SPAN class=ListParagraph></SPAN></P>
<P class=pLevel2><SPAN class=Normal>Sum of the first <I>n</I> natural numbers =
<I>S<SUB>n</SUB> </I>= 1 + 2 + 3 + 4 + … + <I>n</I></SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>This is an A.P. with first term = 1 and
common difference = 1</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_0.png"></SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_094.png">
<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_095.png"></SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>Sum of the squares of the first <I>n</I>
natural numbers = <I>S<SUB>n</I></SUB> = 1<SUP>2</SUP> + 2<SUP>2</SUP>
+3<SUP>2</SUP> + … + <I>n</I><SUP>2</SUP></SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>We know that (<I>x</I> + 1)<SUP>3</SUP> –
<I>x</I><SUP>3</SUP> = 3<I>x</I><SUP>2</SUP> + 3<I>x</I> + 1</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>For <I>x</I> = 1, 2<SUP>3</SUP> –
1<SUP>3</SUP> = 3 <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/multiplication.png">
1<SUP>2</SUP> + 3 <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/multiplication.png">
1 + 1<SPAN style="MARGIN-LEFT: 25px"></SPAN>… (i)</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>For <I>x</I> = 2, 3<SUP>3</SUP> –
2<SUP>3</SUP> = 3 <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/multiplication.png">
2<SUP>2</SUP> + 3 <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/multiplication.png">
2 + 1<SPAN style="MARGIN-LEFT: 25px"></SPAN>… (ii)</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>For <I>x</I> = 3, 4<SUP>3</SUP> –
3<SUP>3</SUP> = 3 <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/multiplication.png">
3<SUP>2</SUP> + 3 <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/multiplication.png">
3 + 1<SPAN style="MARGIN-LEFT: 25px"></SPAN>… (iii)</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>Continuing thus,</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>For <I>x</I> = <I>n</I>, (<I>n</I> +
1)<SUP>3</SUP> – <I>n</I><SUP>3</SUP> = 3<I>n</I><SUP>2</SUP> + 3<I>n</I> + 1
…(iv)</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>Adding all the equations from (i) to
(iv),</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/minus.png">1<SUP>3</SUP>
+ (<I>n</I> + 1)<SUP>3</SUP> = 3(1<SUP>2</SUP> + 2<SUP>2</SUP> + 3<SUP>2</SUP> +
… + <I>n</I><SUP>2</SUP>) + 3 (1 + 2 + 3 + … + <I>n</I>) + <I>n</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/therefore.png"></I>
(<I>n</I> + 1)<SUP>3</SUP> – 1 = 3<I>S<SUB>n</I></SUB> + 3<I>n</I>(<I>n</I> +
1)/2 + <I>n</I></SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_101.png"></SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>The sum of the cubes of the first <I>n</I>
natural numbers can be found in a similar way using (<I>x</I> + 1)<SUP>4</SUP> –
<I>x</I><SUP>4</SUP> = 4<I>x</I><SUP>3</SUP> + 6<I>x</I><SUP>2</SUP> + 4<I>x</I>
+ 1</SPAN></P>
<P class=pLevel2><SPAN class=ListParagraph></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel2">
<DIV id=019_solvedex_38>
<TABLE style="WIDTH: 100%" id=019_solvedex_38_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_38>Example
38:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>A student finds the sum 1 + 2 + 3 + ... as his
patience runs out. He found the sum as 575. When the teacher declared the
result wrong, the student realized that he missed a number. What was the
number the student missed?</SPAN></P>
<P><SPAN class=Normal><B>[CAT 2002]</B></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>(1) 16<SPAN style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN>(2) 18</SPAN></P>
<P><SPAN class=Normal>(3) 14<SPAN style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN>(4) 20</SPAN></P>
<P><SPAN class=ListParagraph></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=ListParagraph>Sum of first <I>n</I> natural<B> </B>numbers
= <I>S</I>(<I>n</I>)</SPAN></P>
<P><SPAN class=ListParagraph>Sum given by student = 575</SPAN></P>
<P><SPAN class=ListParagraph></SPAN></P>
<P><SPAN class=ListParagraph><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_107.png"></SPAN></P>
<P><SPAN class=ListParagraph></SPAN></P>
<P><SPAN class=ListParagraph><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_102.png"></SPAN></P>
<P><SPAN class=ListParagraph></SPAN></P>
<P><SPAN class=ListParagraph><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_103.png"></SPAN></P>
<P><SPAN class=ListParagraph></SPAN></P>
<P><SPAN class=ListParagraph><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_104.png"></SPAN></P>
<P><SPAN class=ListParagraph></SPAN></P>
<P><SPAN class=ListParagraph><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/therefore.png">
The student stopped counting somewhere between 30 and 40.</SPAN></P>
<P><SPAN class=ListParagraph></SPAN></P>
<P><SPAN class=ListParagraph><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_105.png"></SPAN></P>
<P><SPAN class=ListParagraph></SPAN></P>
<P><SPAN class=ListParagraph>The student stopped somewhere before
35.</SPAN></P>
<P><SPAN class=ListParagraph></SPAN></P>
<P><SPAN class=ListParagraph><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/therefore.png">
<I>S</I>(31) = 496, <I>S</I>(32) = 528, <I>S</I>(33) = 561 and
<I>S</I>(34) = 595</SPAN></P>
<P><SPAN class=ListParagraph></SPAN></P>
<P><SPAN class=ListParagraph>But the student gave 575 as the sum, so the
student missed the number 20.</SPAN></P>
<P><SPAN class=ListParagraph>Hence, <B>option
4.</B></SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel2><SPAN class=ListParagraph></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel2">
<DIV id=019_solvedex_39>
<TABLE style="WIDTH: 100%" id=019_solvedex_39_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_39>Example
39:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>Ginny is adding the numbers from 1 to 50 for her
homework. She makes a mistake and wrongly adds the reverse of a two-digit
number instead of the number. As a result, her sum is 1347. Which is the
number that she adds the reverse of?</SPAN></P>
<P><SPAN class=ListParagraph></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>Sum of numbers from 1 to 50 = 50 <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/multiplication.png">
51/2 = 1275</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Ginny’s sum = 1347</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Let the number that Ginny wrongly adds be 10<I>x</I>
+ <I>y</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/therefore.png"></I>
Ginny’s sum = 1275 – (10<I>x</I> + <I>y</I>) + (10<I>y</I> +
<I>x</I>)</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/therefore.png">
1347 = 1275 <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/minus.png">
9(<I>x</I> – <I>y</I>)</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/therefore.png">
9(<I>y</I> – <I>x</I>) = 72</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/therefore.png">
<I>y</I> – <I>x</I> = 8</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/therefore.png">
<I>y</I> = 9 and <I>x</I> = 1</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/therefore.png">
Ginny wrongly adds the reverse of
19.</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel2><SPAN class=ListParagraph></SPAN></P>
<DIV class="divTextBox divColorNote divLevel2">
<P><SPAN class=Note>REMEMBER:</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<UL class=Normal>
<LI value=1>For an Arithmetic Progression with the first term <I>a</I> and
common difference <I>d</I>,</LI></UL>
<UL>
<P><SPAN class=Normal>The <I>n</I><SUP>th</SUP> term <I>T<SUB>n</I></SUB> =
<I>a</I> + (<I>n</I> – 1)<I>d</I></SPAN></P></UL>
<UL>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_106.png"></SPAN></P></UL>
<P><SPAN class=Normal></SPAN></P>
<UL>
<P><SPAN class=Normal><SPAN style="MARGIN-LEFT: 40px"></SPAN><IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_108.png"></SPAN></P></UL>
<P><SPAN class=Normal></SPAN></P>
<UL class=Normal>
<LI>For a Geometric Progression with the first term <I>a</I> and common ratio
<I>r</I>,</LI></UL>
<P><SPAN class=Normal></SPAN></P>
<UL>
<P><SPAN class=Normal>The <I>n</I><SUP>th</SUP> term <I>T<SUB>n</I></SUB> =
<I>ar<SUP>n</SUP></I> – 1</SPAN></P></UL>
<P><SPAN class=Normal></SPAN></P>
<UL>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_109.png"><IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_110.png"></SPAN></P></UL>
<P><SPAN class=Normal></SPAN></P>
<UL>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_109.png"><IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_111.png"></SPAN></P></UL>
<UL>
<P><SPAN class=Normal>The sum of an infinite number of terms of a decreasing
G.P. <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_113.png"></SPAN></P></UL>
<P><SPAN class=Normal></SPAN></P>
<UL class=Normal>
<LI>The <I>n</I><SUP>th</SUP> term (<I>T<SUB>n</I></SUB>) of an harmonic
progression is given by</LI></UL>
<UL>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_114.png"></SPAN></P></UL>
<P><SPAN class=Normal></SPAN></P>
<UL>
<P><SPAN class=Normal>where <I>a</I> and <I>d</I> are the first term and
common difference of the corresponding A.P.</SPAN></P></UL>
<UL class=Normal>
<LI value=1>If <I>A, G </I>and<I> H </I>are the arithmetic mean, geometric
mean and harmonic mean of two unequal quantities,</LI></UL>
<P><SPAN class=Normal></SPAN></P>
<UL>
<P><SPAN class=Normal><I>A</I> <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/multiplication.png">
<I>H</I> = <I>G</I><SUP>2</SUP></SPAN></P></UL>
<P><SPAN class=Normal></SPAN></P>
<UL>
<P><SPAN class=Normal><I>A</I> > <I>G</I> > <I>H</I></SPAN></P></UL>
<P><SPAN class=Normal></SPAN></P>
<UL>
<LI>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_094.png">
<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_095.png"></SPAN></P></LI></UL>
<UL>
<LI>
<P><SPAN class=Normal>Sum of the squares of the first <I>n</I> natural number
is given by
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_097_.png"></SPAN></P></LI></UL>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<UL>
<LI><P<SPAN class=Normal>Sum of the cubes of the first <I>n</I> natural
numbers is given by
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/084d4ea2-163e-414d-8add-fc284658e3bf/019_images/019_099_.png"></SPAN></P></LI></UL>
<P><SPAN
class=Normal></SPAN></P></DIV></DIV></SPAN></SPAN></SPAN></DIV></BODY></HTML>
guruhttp://www.blogger.com/profile/01591375209227117455noreply@blogger.comtag:blogger.com,1999:blog-7850540640045081515.post-13628508303651095132015-03-03T05:46:00.003-08:002019-09-14T18:55:03.569-07:00prob<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
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<P class=oTOCTitle><SPAN class=oTOCTitleFont><A class=header
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<OL class=oTOCLevel1>
<LI class=oTocLiClass><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023.htm#I">INTRODUCTION</A>
<OL class=oTOCLevel2>
<LI><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023.htm#I_A">EVENTS</A></LI></OL></LI>
<LI class=oTocLiClass><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023.htm#II">PROBABILITY</A>
<OL class=oTOCLevel2>
<LI><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023.htm#II_A">THE
ADDITION THEOREM OF PROBABILITY</A></LI></OL></LI>
<LI class=oTocLiClass><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023.htm#III">CONDITIONAL
PROBABILITY</A> </LI></OL>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P id=Title class=pTitle><SPAN class=Title>PROBABILITY</SPAN></P>
<HR>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<DIV id=023_I>
<TABLE id=023_I_bookmark class=tableh1 border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD
style="PADDING-BOTTOM: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 100%; PADDING-RIGHT: 0px; PADDING-TOP: 0px">
<OL class="oLevel1 Heading1">
<LI value=1><A id=I>INTRODUCTION</A></LI></OL></TD></TR></TBODY></TABLE>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>It is very easy to understand the concept of
probability through the example of tossing a coin. A coin when tossed, gives
either heads or tails. The result of a toss cannot be certain. So, we estimate
the chances of a particular result. The chance of getting a head when a coin is
tossed is one in two or 50% or 1/2. This chance is known as
<B>probability</B>.</SPAN></P>
<P class=pLevel2><SPAN class=Normal>A <B>random experiment</B> is an action that
gives one or more results. Each result is called an <B>outcome</B>.</SPAN></P>
<P class=pLevel2><SPAN class=Normal>The <B>sample space</B> (<I>S</I>) is the
set of all possible outcomes of an event.</SPAN></P>
<P class=pLevel2><SPAN class=Normal>The number of elements in the sample space
is denoted by <I>n</I>(<I>S</I>).</SPAN></P>
<P class=pLevel2><SPAN class=Normal>Thus, tossing of a coin is a random
experiment. Getting heads or tails is the outcome. The sample space <I>S</I>
<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/equal_equal.png">
{Heads, Tails} and <I>n</I>(<I>S</I>) = 2.</SPAN></P>
<P class=pLevel2><SPAN class=Normal>Outcomes of a random experiment are
<B>equally likely</B> when the occurrence of any one result is not expected over
the other. A random experiment having equally likely outcomes is called an
<B>unbiased experiment.</B></SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel2">
<DIV id=023_solvedex_1>
<TABLE style="WIDTH: 100%" id=023_solvedex_1_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_1>Example
1:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>Find the number of elements in the sample space for
the following experiments:</SPAN></P>
<P><SPAN class=Normal>(i)<SPAN style="MARGIN-LEFT: 25px"></SPAN>Tossing
two coins together.</SPAN></P>
<P><SPAN class=Normal>(ii)<SPAN style="MARGIN-LEFT: 25px"></SPAN>Tossing
three coins together.</SPAN></P>
<P><SPAN class=Normal>(iii)<SPAN style="MARGIN-LEFT: 25px"></SPAN>Tossing
a coin four times.</SPAN></P>
<P><SPAN class=Normal>(iv)<SPAN style="MARGIN-LEFT: 25px"></SPAN>Rolling
an unbiased die.</SPAN></P>
<P><SPAN class=Normal>(v)<SPAN style="MARGIN-LEFT: 25px"></SPAN>Rolling
two unbiased dice and three unbiased dice.</SPAN></P>
<P><SPAN class=Normal>(vi)<SPAN style="MARGIN-LEFT: 25px"></SPAN>Drawing
two cards at random from a pack of 52 well-shuffled cards.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal><B>i.</B> Tossing one coin gives 2 outcomes—either
Heads (<I>H</I>) or Tails (<I>T</I>).</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/therefore.png">
Tossing two coins gives 2 <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/multiplication.png">
2 outcomes</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/therefore.png">
<I>n</I>(<I>S</I>) = 4</SPAN></P>
<P><SPAN class=Normal><I>S</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/equal_equal.png">
{<I>HH, HT, TH, TT</I>}</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><B>ii.</B> Tossing three coins gives 2 <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/multiplication.png">
2 <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/multiplication.png">
2 = 8 outcomes</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/therefore.png">
<I>n</I>(<I>S</I>) = 8</SPAN></P>
<P><SPAN class=Normal><I>S</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/equal_equal.png">
{<I>HHH, HHT, HTH, THH, HTT, THT, TTH, TTT</I>}</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><B>iii.</B> When a coin is tossed four times, the
possible outcomes for each toss = 2</SPAN></P>
<P><SPAN class=Normal>Total number of possible outcomes</SPAN></P>
<P><SPAN class=Normal>= 2 <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/multiplication.png">
2 <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/multiplication.png">
2 <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/multiplication.png">
2 = 16</SPAN></P>
<P><SPAN class=Normal>It is difficult to list the elements of the sample
space. But knowing the number of elements in the sample space is
enough.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><B>iv.</B> Rolling an unbiased die gives 6
outcomes.</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/therefore.png">
<I>n</I>(<I>S</I>) = 6</SPAN></P>
<P><SPAN class=Normal><I>S</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/equal_equal.png">
{1, 2, 3, 4, 5, 6}</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><B>v.</B> As rolling one die gives 6 outcomes,
rolling two dice gives 6 <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/multiplication.png">
6 = 36 outcomes.</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/therefore.png">
<I>n</I>(<I>S</I>) = 36</SPAN></P>
<P><SPAN class=Normal><I>S</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/equal_equal.png">
{(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6),</SPAN></P>
<P><SPAN class=Normal>(2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2,
6),</SPAN></P>
<P><SPAN class=Normal>(3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3,
6),</SPAN></P>
<P><SPAN class=Normal>(4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4,
6),</SPAN></P>
<P><SPAN class=Normal>(5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5,
6),</SPAN></P>
<P><SPAN class=Normal>(6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6,
6)}</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Consider the case where a die is rolled three
times.</SPAN></P>
<P><SPAN class=Normal>The possible outcomes for first, second and third
throw of the dice each = 6.</SPAN></P><SPAN class=Normal>∴ n(S) = 6 × 6 ×
6 = 216 </SPAN>
<P></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><B>vi.</B> Two cards can be drawn from a pack of 52
well shuffled cards in <SUP>52</SUP>C<SUB>2</SUB> ways.</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/therefore.png">
<I>n</I>(<I>S</I>) =
<SUP>52</SUP>C<SUB>2</SUB></SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<DIV id=023_I_A>
<TABLE id=023_I_A_bookmark class=tableh1 border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD
style="PADDING-BOTTOM: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 100%; PADDING-RIGHT: 0px; PADDING-TOP: 0px">
<OL class="oLevel2 Heading2">
<LI value=1><A id=I_A>EVENTS</A></LI></OL></TD></TR></TBODY></TABLE>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>Consider the sample space of the experiment
of rolling an unbiased die.</SPAN></P>
<P class=pLevel3><SPAN class=Normal><I>S</I> <IMG style="VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/equal_equal.png">
{1, 2, 3, 4, 5, 6}</SPAN></P>
<P class=pLevel3><SPAN class=Normal>Any subset of <I>S</I> is called an
<B>event</B>.</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>For example, the event <I>E</I> of getting
an odd number from the throw of a die is</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><I>E</I> <IMG style="VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/equal_equal.png">
{1, 3, 5}</SPAN></P>
<P class=pLevel3><SPAN class=Normal><I>n</I>(<I>E</I>) = 3</SPAN></P>
<P class=pLevel3><SPAN class=Normal>If <I>E</I> <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/equal_equal.png">
{ }, then the event is an <B>impossible event</B>.</SPAN></P>
<P class=pLevel3><SPAN class=Normal>For example, getting 7 by throwing a die is
an impossible event.</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>If <I>E</I> <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/equal_equal.png">
<I>S</I>, then the event is a <B>certain event</B>.</SPAN></P>
<P class=pLevel3><SPAN class=Normal>For example, getting a natural number less
than 7 by throwing a die is a certain event.</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><B>Occurrence of Two or More
Events:</B></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>The <B>union</B> of the events <I>A</I> and
<I>B</I> of a sample space (<I>A </I><IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/union.png">
<I>B</I>)<I> </I>is the event that either <I>A</I> or <I>B</I> or both take
place.</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pImage><SPAN><IMG alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/023_002.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>The <B>intersection</B> of the events
<I>A</I> and <I>B</I> of a sample space (<I>A </I><IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/intersect.png">
<I>B</I>)<I> </I>is the event that both <I>A</I> and <I>B</I> take
place.</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pImage><SPAN><IMG alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/023_003.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>Two events <I>A</I> and <I>B</I> of a sample
space are <B>mutually exclusive</B> if they cannot occur
simultaneously.</SPAN></P>
<P class=pImage><SPAN><IMG alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/023_004.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>If <I>A</I> and <I>B</I> are mutually
exclusive, <I>A</I> <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/intersect.png">
<I>B</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/equal_equal.png">
{ }</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/therefore.png">
<I>n</I>(<I>A</I> <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/intersect.png">
<I>B</I>) = 0</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>If two events <I>A</I> and <I>B</I> of a
sample space <I>S</I> are such that <I>A</I> <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/union.png">
<I>B</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/equal_equal.png">
<I>S</I>, then <I>A</I> and <I>B</I> are called <B>exhaustive</B>
events.</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pImage><SPAN><IMG alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/023_005.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>If <I>A</I> and <I>B</I> are exhaustive,
<I>A</I> <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/union.png">
<I>B</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/equal_equal.png">
<I>S</I></SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/therefore.png">
<I>n</I>(<I>A</I> <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/union.png">
<I>B</I>) = <I>n</I>(<I>S</I>)</SPAN></P>
<P class=pLevel3><SPAN class=Normal>An event<I> </I>(<I>A</I>) and its
complement (<I>A</I>’) are exhaustive events.</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P></DIV></DIV>
<DIV id=023_II>
<TABLE id=023_II_bookmark class=tableh1 border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD
style="PADDING-BOTTOM: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 100%; PADDING-RIGHT: 0px; PADDING-TOP: 0px">
<OL class="oLevel1 Heading1">
<LI value=2><A id=II>PROBABILITY</A></LI></OL></TD></TR></TBODY></TABLE>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>Probability is the chance of occurrence of
an event.</SPAN></P>
<P class=pLevel2><SPAN class=Normal>If <I>S</I> is the sample space of an
unbiased experiment and if <I>E</I> is an event, then the probability that
<I>E</I> takes place is</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/023_006.png"></SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>Consider tossing an unbiased coin. Let
<I>E</I> be the event that the result is heads. Let <I>E</I>’ be the event that
the result is tails. Then,</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal><I>n</I>(<I>S</I>) = 2, <I>n</I>(<I>E</I>) =
1 and <I>n</I>(<I>E</I>’) = 1</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/therefore.png">
<I>P</I>(<I>E</I>) = 1/2 and <I>P</I>(<I>E</I>’) = 1/2</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>So when a coin is tossed, the probability of
occurrence of heads is 1/2.</SPAN></P>
<P class=pLevel2><SPAN class=Normal>Similarly, the probability of occurrence of
tails is 1/2.</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>Finding the probability of any event
involves the following steps:</SPAN></P>
<P class=pLevel2><SPAN class=Normal><B>Step I:</B> Finding the number of
elements in the sample space.</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>Consider the event of rolling of a
die.</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal><I>S</I> <IMG style="VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/equal_equal.png">
{1, 2, 3, 4, 5, 6}</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal><I>n</I>(<I>S</I>) = 6</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal><B>Step II:</B> Finding the number of
outcomes satisfying the required event.</SPAN></P>
<P class=pLevel2><SPAN class=Normal>Let <I>A</I> be the event that the die shows
a multiple of 3. Then, the outcomes which satisfy this event <I>A</I> can be
written as</SPAN></P>
<P class=pLevel2><SPAN class=Normal><I>A</I> <IMG style="VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/equal_equal.png">
{3, 6}</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/therefore.png">
<I>n</I>(<I>A</I>) = 2</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal><B>Step III:</B> Finding the probability of
the occurrence of the event, P(<I>A</I>).</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/023_007.png"></SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>In this case, P(<I>A</I>) = 2/6 =
1/3</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel2">
<DIV id=023_solvedex_2>
<TABLE style="WIDTH: 100%" id=023_solvedex_2_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_2>Example
2:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>Find the probability that a number selected at
random from the first 10 natural numbers is prime.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>Since we are considering only the first 10 natural
numbers (1 to 10), <I>n</I>(<I>S</I>) = 10.</SPAN></P>
<P><SPAN class=Normal>There are 4 numbers less than 10, which are prime,
namely 2, 3, 5 and 7.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/therefore.png">
<I>n</I>(<I>A</I>) = 4</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/therefore.png">
The probability that a number selected at</SPAN></P>
<P><SPAN class=Normal>random from the first 10 natural numbers is prime
is</SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/023_008.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorNote divLevel2">
<P><SPAN class=Note>REMEMBER</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<UL type=disc>
<LI>0 <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/less_equal.png">
<I>P</I>(<I>E</I>)<I> </I><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/less_equal.png">
1</LI></UL>
<P><SPAN class=Normal></SPAN></P>
<UL type=disc>
<LI>If <I>P</I>(<I>E</I>) = 0, the event is an impossible event.</LI></UL>
<P><SPAN class=Normal></SPAN></P>
<UL type=disc>
<LI>If <I>P</I>(<I>E</I>) = 1, the event is a certain event.</LI></UL></DIV><!--<p class="pLevel4"><span class="ListParagraph"></span></p>
<div class="divTextBox divColorExampleHeading divLevel4">
<div id="023_solvedex_3">
<table cellpadding="0" cellspacing="0" border="0" style="width:100%;" id="023_solvedex_3_bookmark" ><tr><td style="width:100%; background-color:Transparent"><p><span class="ExampleHeading"><a id="ex_3">Example 3:</a></span></p></td></tr></table>
<table cellpadding="0" cellspacing="0" border="0"><tr><td style="width:100%; background-color:Transparent">-->
<DIV class="divTextBox divColorExampleHeading divLevel2">
<DIV id=023_solvedex_3>
<TABLE style="WIDTH: 100%" id=023_solvedex_3_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_3>Example
3:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>Two unbiased coins are tossed. What is the
probability that both the tosses give different results?</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>When a coin is tossed we get either heads or tails.
Let heads be denoted by <I>H</I><B> </B>and tails be denoted by
<I>T</I>.</SPAN></P>
<P><SPAN class=Normal>Tossing two coins gives 4 outcomes.</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/therefore.png">
<I>S</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/equal_equal.png">
{<I>HH</I>, <I>HT</I>, <I>TH</I>, <I>TT</I>}</SPAN></P>
<P><SPAN class=Normal>Hence, <I>n</I>(<I>S</I>) = 4</SPAN></P>
<P><SPAN class=Normal>Let <I>A</I> be the event that both the tosses give
different results.</SPAN></P>
<P><SPAN class=Normal>i.e. The first coin shows heads and the second shows
tails OR the first coin shows tails and the second shows heads.</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/therefore.png">
<I>A</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/equal_equal.png">
{<I>HT</I>, <I>TH</I>}</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/therefore.png">
<I>n</I>(<I>A</I>) = 2</SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/023_009.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV><!--<p class="pLevel2"><span class="Normal"></span></p>
<div class="divTextBox divColorNormal divLevel4">
<span class="Normal"></span></p>
</div>-->
<DIV class="divTextBox divColorExampleHeading divLevel2">
<DIV id=023_solvedex_4>
<TABLE style="WIDTH: 100%" id=023_solvedex_4_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_4>Example
4:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%"><!--<p class="pLevel2"><span class="Normal"></span></p>
<div class="divTextBox divColorExampleHeading divLevel4">
<div id="023_solvedex_4">
<table cellpadding="0" cellspacing="0" border="0" style="width:100%;" id="023_solvedex_4_bookmark" ><tr><td style="width:100%; background-color:Transparent"><p><span class="ExampleHeading"><a id="ex_4">Example 4:</a></span></p></td></tr></table>
<table cellpadding="0" cellspacing="0" border="0"><tr><td style="width:100%; background-color:Transparent">-->
<P><SPAN class=Normal>If a die is tossed two times, what is the
probability that the sum of the two throws is less than or equal to
4?</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>When a die is thrown twice, the sample space can be
given as</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><I>S</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/equal_equal.png">
{(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6),</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>(2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2,
6),</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>(3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3,
6),</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>(4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4,
6),</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>(5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5,
6),</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>(6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6,
6)}</SPAN></P>
<P><SPAN class=Normal>Hence,<I> n</I>(<I>S</I>) = 36.</SPAN></P>
<P><SPAN class=Normal>Let <I>B</I> be the event that sum of the two throws
is less than or equal to 4.</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/therefore.png">
<I>B</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/equal_equal.png">
{(1, 1), (1, 2), (2, 1), (2, 2), (1, 3), (3, 1)}</SPAN></P>
<P><SPAN class=Normal>Hence,<I> n</I>(<I>B</I>) = 6</SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/023_010.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel2">
<DIV id=023_solvedex_5>
<TABLE style="WIDTH: 100%" id=023_solvedex_5_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_5>Example
5:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>A coin is tossed 7 times. What is the probability of
getting exactly four heads?</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>Possible outcomes for each toss = 2</SPAN></P>
<P><SPAN class=Normal>Hence, total possible outcomes <I>n</I>(<I>S</I>) =
2<SUP>7</SUP> = 128</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Let <I>H</I> be the event of getting exactly 4
heads. The number of ways in which one can get 4 heads in 7 tosses =
<I>n</I>(<I>H</I>) = <SUP>7</SUP>C<SUB>4</SUB> = 35.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Hence, the probability of getting exactly 4 heads is
given by</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/023_011.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel2">
<DIV id=023_solvedex_6>
<TABLE style="WIDTH: 100%" id=023_solvedex_6_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_6>Example
6:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>There are 6 people who stand together for a
photograph. Find the probability that two of them stand
together.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal><I>n</I>(<I>S</I>) = The total number of ways in
which 6 people can be arranged together for a photograph =
<SUP>6</SUP>P<SUB>6</SUB> = 6! = 720</SPAN></P>
<P><SPAN class=Normal>Let <I>E</I> be the event that two people always
stand together.</SPAN></P>
<P><SPAN class=Normal>Let us assume that the two people who always stand
together are one entity.</SPAN></P>
<P><SPAN class=Normal>These 2 people can be arranged amongst themselves in
2 ways.</SPAN></P>
<P><SPAN class=Normal>Now we have 5 people who can be arranged amongst
themselves in <SUP>5</SUP>P<SUB>5</SUB> = 5! = 120 ways.</SPAN></P>
<P><SPAN class=Normal>Thus, the total number of ways in which 2 people
will stand together = 120 <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/multiplication.png">
2 = 240 ways.</SPAN></P>
<P><SPAN class=Normal>Hence, <I>n</I>(<I>E</I>) = 240</SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/023_012.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal></SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV><!--<p class="pLevel2"><span class="Normal"></span></p>
<div class="divTextBox divColorNormal divLevel4">
<p><span class="Normal"></span></p>
<p><span class="ExampleHeading">Example 7:</span></p>-->
<DIV class="divTextBox divColorExampleHeading divLevel2">
<DIV id=023_solvedex_7>
<TABLE style="WIDTH: 100%" id=023_solvedex_7_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_7>Example
7:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>A card is drawn at random from a pack of cards. Find
the probability that it is a/an:</SPAN></P>
<P><SPAN class=Normal>i. spade</SPAN></P>
<P><SPAN class=Normal>ii. ace</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>Total cards = 52</SPAN></P>
<P><SPAN class=Normal>One card can be drawn from 52 in
<SUP>52</SUP>C<SUB>1</SUB> = 52 ways.</SPAN></P>
<P><SPAN class=Normal>Hence, <I>n</I>(<I>S</I>)<I> </I>= 52.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><B>i.</B> Let <I>A</I> be the event that the card
drawn is a spade.</SPAN></P>
<P><SPAN class=Normal>Number of spades = 13</SPAN></P>
<P><SPAN class=Normal>One spade can be selected from 13 in
<SUP>13</SUP>C<SUB>1</SUB> = 13 ways.</SPAN></P>
<P><SPAN class=Normal>Hence, <I>n</I>(<I>A</I>) = 13.</SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/023_013.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><B>ii.</B> Let <I>B</I> be the event that the card
drawn is an ace.</SPAN></P>
<P><SPAN class=Normal>Number of aces = 4.</SPAN></P>
<P><SPAN class=Normal>One ace can be selected from 4 in
<SUP>4</SUP>C<SUB>1</SUB> = 4 ways.</SPAN></P>
<P><SPAN class=Normal>Hence<I>, n</I>(<I>B</I>) = 4.</SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/023_014.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel2">
<DIV id=023_solvedex_8>
<TABLE style="WIDTH: 100%" id=023_solvedex_8_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_8>Example
8:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>There is an urn containing 3 white and 4 red balls.
Two balls are drawn from this urn. Find the probability that:</SPAN></P>
<P><SPAN class=Normal>1) Both balls are red.</SPAN></P>
<P><SPAN class=Normal>2) Both balls are of different colours.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>Number of balls in the urn = 7</SPAN></P>
<P><SPAN class=Normal>Two balls can be selected from these in
<SUP>7</SUP>C<SUB>2</SUB> = 21 ways.</SPAN></P>
<P><SPAN class=Normal><I>n</I>(<I>S</I>) = 21</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><B>i.</B> Let <I>R</I> be the event that both balls
drawn are red.</SPAN></P>
<P><SPAN class=Normal>Number of red balls = 4</SPAN></P>
<P><SPAN class=Normal>Two red balls can be selected from these
in</SPAN></P>
<P><SPAN class=Normal><SUP>4</SUP>C<SUB>2</SUB> = 6 ways.</SPAN></P>
<P><SPAN class=Normal>Hence,<I> n</I>(<I>R</I>) = 6.</SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/023_015.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><B>ii. </B>Let <I>B</I> be the event that both balls
are of different colours.</SPAN></P>
<P><SPAN class=Normal>Total number of red balls = 4</SPAN></P>
<P><SPAN class=Normal>One red ball can be selected from these
in</SPAN></P>
<P><SPAN class=Normal><SUP>4</SUP>C<SUB>1</SUB> = 4 ways.</SPAN></P>
<P><SPAN class=Normal>Total number of white balls = 3</SPAN></P>
<P><SPAN class=Normal>One white ball can be selected from these
in</SPAN></P>
<P><SPAN class=Normal><SUP>3</SUP>C<SUB>1</SUB> = 3 ways.</SPAN></P>
<P><SPAN class=Normal>Thus, one red and one white ball can be selected
in</SPAN></P>
<P><SPAN class=Normal>4 <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/multiplication.png">
3 = 12 ways</SPAN></P>
<P><SPAN class=Normal>Thus, balls of different colours can be selected
in</SPAN></P>
<P><SPAN class=Normal>12 ways.</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/therefore.png">
<I>n</I>(<I>B</I>) = 12</SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/023_016.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel2">
<DIV id=023_solvedex_9>
<TABLE style="WIDTH: 100%" id=023_solvedex_9_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_9>Example
9:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>Four boys and eight girls are seated in a row. What
is the probability that no two boys sit next to each other?</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>12 people can be arranged among themselves in 12!
ways.</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/therefore.png">
<I>n</I>(<I>S</I>) = 12!</SPAN></P>
<P><SPAN class=Normal>Girls can be seated in 8! ways.</SPAN></P>
<P><SPAN class=Normal>__ G __ G __ G __ G __ G __ G __ G __ G
__</SPAN></P>
<P><SPAN class=Normal>There are nine places next to the girls in which the
boys can be seated.</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/therefore.png">
The boys can be seated in <SUP>9</SUP>P<SUB>4</SUB> ways so that no two
boys are next to each other.</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/therefore.png">
If <I>E</I> is the event that no two boys sit next to each
other,</SPAN></P>
<P><SPAN class=Normal><I>n</I>(<I>E</I>) = 8! <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/multiplication.png">
<SUP>9</SUP>P<SUB>4</SUB></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/therefore.png">
The required probability </SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/023_018.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/023_019.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/023_020.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/023_021.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<DIV id=023_II_A>
<TABLE id=023_II_A_bookmark class=tableh1 border=0 cellSpacing=0
cellPadding=0><TBODY>
<TR>
<TD
style="PADDING-BOTTOM: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 100%; PADDING-RIGHT: 0px; PADDING-TOP: 0px">
<OL class="oLevel2 Heading2">
<LI value=1><A id=II_A>THE ADDITION THEOREM OF
PROBABILITY</A></LI></OL></TD></TR></TBODY></TABLE>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>We know that for two sets <I>A</I> and
<I>B</I>,</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><I>n</I>(<I>A</I> <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/union.png">
<I>B</I>) = <I>n</I>(<I>A</I>) + <I>n</I>(<I>B</I>) – <I>n</I>(<I>A</I> <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/intersect.png">
<I>B</I>)</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>If <I>A</I> and <I>B</I> are events of a
sample space <I>S</I>, then dividing the equation by <I>n</I>(<I>S</I>), we
have</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/023_022.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/therefore.png">
<I>P</I>(<I>A</I> <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/union.png">
<I>B</I>) = <I>P</I>(<I>A</I>) + <I>P</I>(<I>B</I>) – <I>P</I>(<I>A</I> <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/intersect.png">
<I>B</I>)</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>If <I>A</I> and <I>B</I> are mutually
exclusive events,</SPAN></P>
<P class=pLevel3><SPAN class=Normal><I>n</I>(<I>A</I> <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/intersect.png">
<I>B</I>) = 0</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pImage><SPAN><IMG alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/023_023.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/therefore.png">
<I>P</I>(<I>A</I> <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/intersect.png">
<I>B</I>) = 0</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/therefore.png">
<I>P</I>(<I>A</I> <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/union.png">
<I>B</I>) = <I>P</I>(<I>A</I>) + <I>P</I>(<I>B</I>)</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>If <I>A</I> and <I>B</I> are exhaustive
events, <I>n</I>(<I>A</I> <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/union.png">
<I>B</I>) = <I>n</I>(<I>S</I>)</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pImage><SPAN><IMG alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/023_024.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/023_023..png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/therefore.png">
<I>P</I>(<I>A</I>) + <I>P</I>(<I>B</I>) = 1</SPAN></P>
<P class=pLevel3><SPAN class=Normal>An event and its complement are always
exhaustive events.</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/therefore.png">
<I>P</I>(<I>A</I>) + <I>P</I>(<I>A’</I>) = 1</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/therefore.png">
<I>P</I>(<I>A</I>) = 1 – <I>P</I>(<I>A’</I>)</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=023_solvedex_10>
<TABLE style="WIDTH: 100%" id=023_solvedex_10_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_10>Example
10:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>Three dice are thrown simultaneously. What is the
probability that the sum of the results of the three throws is more than
5?</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>As three dice are thrown simultaneously,</SPAN></P>
<P><SPAN class=Normal><I>n</I>(<I>S</I>) = 6<SUP>3</SUP> = 216.</SPAN></P>
<P><SPAN class=Normal>Rather than calculating the number of ways in which
the sum of the results of three throws is more than 5, it is simpler to
calculate the number of ways in which the sum of the results of three
throws is less than or equal to 5.</SPAN></P>
<P><SPAN class=Normal>Now, let <I>L</I> be the event that the throws give
a sum of less than or equal to 5 and <I>G</I> be the event that the throws
give a sum greater than 5.</SPAN></P>
<P><SPAN class=Normal>Clearly, <I>n</I>(<I>L</I>) + <I>n</I>(<I>G</I>) =
<I>n</I>(<I>S</I>)</SPAN></P>
<P><SPAN class=Normal>Or in other words, <I>L</I> and <I>G</I> are
exhaustive events.</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/therefore.png">
P(<I>G</I>) = 1– P(<I>L</I>)</SPAN></P>
<P><SPAN class=Normal><I>L</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/equal_equal.png">
{(1, 1, 1), (1, 1, 2), (1, 2, 1), (2, 1, 1), (1, 2, 2), (2, 1, 2), (2, 2,
1), (1, 1, 3), (1, 3, 1), (3, 1, 1)} </SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/therefore.png">
<I>n</I>(<I>L</I>) = 10</SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/023_024..png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/023_025.png">
<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/023_026.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=023_solvedex_11>
<TABLE style="WIDTH: 100%" id=023_solvedex_11_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_11>Example
11:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>A card is drawn at random from a well shuffled pack
of 52 cards.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>X. The card drawn is black or a king.</SPAN></P>
<P><SPAN class=Normal>Y. card drawn is a club or a heart or a
jack.</SPAN></P>
<P><SPAN class=Normal>Z. The card drawn is an ace or a diamond or a
queen.</SPAN></P>
<P><SPAN class=Normal>Then which of the following is correct?</SPAN></P>
<P><SPAN class=Normal><B>[IIFT 2009]</B></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>(1) P(<I>X</I>) > P(<I>Y</I>) >
P(<I>Z</I>)</SPAN></P>
<P><SPAN class=Normal>(2) P(<I>X</I>) >= P(<I>Y</I>) =
P(<I>Z</I>)</SPAN></P>
<P><SPAN class=Normal>(3) P(<I>X</I>) = P(<I>Y</I>) >
P(<I>Z</I>)</SPAN></P>
<P><SPAN class=Normal>(4) P(<I>X</I>) = P(<I>Y</I>) =
P(<I>Z</I>)</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal><I>X</I> is the event that the card drawn is black
or a king.</SPAN></P>
<P><SPAN class=Normal><I>Y</I> is the event that the card drawn is a club
or a heart or a jack.</SPAN></P>
<P><SPAN class=Normal><I>Z</I> is the event that the card drawn is an ace
or a diamond or a queen.</SPAN></P>
<P><SPAN class=Normal>Consider <I>X</I>.</SPAN></P>
<P><SPAN class=Normal>A black card can be drawn in 26 ways. A king can be
drawn in 4 ways. A black king can be drawn in 2 ways.</SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/023_027.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/023_028.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Consider <I>Y</I>.</SPAN></P>
<P><SPAN class=Normal>A club can be drawn in 13 ways. A heart can be drawn
in 13 ways. A jack can be drawn in 4 ways. A Jack of hearts can be drawn
in 1 way and a jack of club can be drawn in 1 way.</SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/023_029.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/023_030.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Consider <I>Z</I>.</SPAN></P>
<P><SPAN class=Normal>An ace can be drawn in 4 ways. A queen can be drawn
in 4 ways and a diamond can be drawn in 13 ways. Queen of Diamonds can be
drawn in 1 way and ace of diamonds can be drawn in 1 way.</SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/023_031.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/023_032.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/023_033.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Hence, <B>option
3.</B></SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=023_solvedex_12>
<TABLE style="WIDTH: 100%" id=023_solvedex_12_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_12>Example
12:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>In a survey conducted among 500 students in a class,
it was found that apart from 100 students, all the others watched at least
one of the three news channels I, II and III.</SPAN></P>
<P><SPAN class=Normal>75 watched only III, 270 watched II and 40 watched
only II and III. Out of the 190 students who watched more than one of the
three channels, 50 watched all the three and 70 watched I and II but not
III.</SPAN></P>
<P><SPAN class=Normal>What is the probability that a student picked at
random watched only one of the channels?</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=pImage><IMG alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/023_034.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>The given information is summarised in the venn
diagram.</SPAN></P>
<P><SPAN class=Normal><I>n</I>(<I>S</I>) = 500</SPAN></P>
<P><SPAN class=Normal><I>n</I>(<I>E</I>) = 25 + 110 + 75 = 210</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/023_035.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=023_solvedex_13>
<TABLE style="WIDTH: 100%" id=023_solvedex_13_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_13>Example
13:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>A shop sells 10 lamps out of which 3 are defective.
Sachin buys four lamps. Find the probability that all of Sachin’s lamps
work. Also find the probability that at least two of the lamps that he
buys work.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal><I>n</I>(<I>S</I>) = <SUP>10</SUP>C<SUB>4</SUB> =
210</SPAN></P>
<P><SPAN class=Normal>7 of the 10 lamps are not defective.</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/therefore.png">
If <I>L</I> is the event that all of Sachin’s lamps work,</SPAN></P>
<P><SPAN class=Normal><I>n</I>(<I>L</I>) = <SUP>7</SUP>C<SUB>4</SUB> =
35</SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/023_036.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>We need the probability that at least two of his
lamps work.</SPAN></P>
<P><SPAN class=Normal>The event that less than two of his lamps work, and
the event that at least two of his lamps work, are exhaustive.</SPAN></P>
<P><SPAN class=Normal>So we calculate the probability that less than two
of his lamps work and subtract it from 1.</SPAN></P>
<P><SPAN class=Normal>The probability that none of Sachin’s lamps work = 0
as there are only 3 defective lamps and he buys 4. If <I>K</I> is the
probability that only one of Sachin’s lamps works,</SPAN></P>
<P><SPAN class=Normal><I>n</I>(<I>K</I>) = <SUP>7</SUP>C<SUB>1</SUB> <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/multiplication.png">
<SUP>3</SUP>C<SUB>3</SUB> = 7</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/therefore.png">
Probability that less than two of Sachin's lamps work</SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/023_037.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/therefore.png">
Probability that at least two of Sachin's lamps work</SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/023_038.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=023_solvedex_14>
<TABLE style="WIDTH: 100%" id=023_solvedex_14_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_14>Example
14:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>Find the probability that a random arrangement of
the all letters of the word PARAGRAPH starts and ends with A.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>The word PARAGRAPH has 9 letters, out of which A is
repeated thrice, R is repeated twice and P is repeated twice.</SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/023_039.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/023_040.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>For the number of arrangements that start and end
with A, consider the following situation;</SPAN></P>
<P><SPAN class=Normal><U>A</U> __ __ __ __ __ __ __ <U>A</U></SPAN></P>
<P><SPAN class=Normal>The letters between the two As are P, A, R, G, R, P
and H in some order.</SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/023_041.png"></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/023_042.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/023_043.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV><!--<p class="pLevel2"><span class="Normal"></span></p>
<div class="divTextBox divColorNormal divLevel3">
<p><span class="Normal"></span></p>
</div>-->
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=023_solvedex_15>
<TABLE style="WIDTH: 100%" id=023_solvedex_15_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_15>Example
15:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>Amit, Sumit and seven other people sit around a
circular table. Find the probability that Amit and Sumit are not next to
each other.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>Nine people can be seated around a circular table in
8! ways.</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/therefore.png">
<I>n</I>(<I>S</I>) = 8!</SPAN></P>
<P><SPAN class=Normal>To find the number of arrangements with Amit and
Sumit sitting together, we consider them as one entity. Then there are 8
people who can be seated in 7! ways.</SPAN></P>
<P><SPAN class=Normal>Amit and Sumit can sit in 2 ways.</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/therefore.png">
The number of arrangements in which the two are not together = 8! – 2 <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/multiplication.png">
7!</SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/023_044.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><SPAN style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 37px"></SPAN><IMG style="VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/023_045.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=023_solvedex_16>
<TABLE style="WIDTH: 100%" id=023_solvedex_16_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_16>Example
16:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>A manufacturer claims that only 2% items are
defective in a shipment of 200 items sent by him. A random sample of two
items is drawn from the shipment of 200 items. What is the probability
that both the items drawn are defective?</SPAN></P>
<P><SPAN class=Normal><B>[FMS 2009]</B></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/023_046.png"></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/023_047.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>2 items out of 200 items are selected. Hence, the
Sample Space = <SUP>200</SUP>C<SUB>2</SUB></SPAN></P>
<P><SPAN class=Normal>Since only 2% of the items are defective in a
shipment of 200 items,</SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/023_048.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>We need to find the probability that both items
drawn are defective.</SPAN></P>
<P><SPAN class=Normal>2 defective items can be drawn from 4 defective
items in <SUP>4</SUP>C<SUB>2</SUB> ways.</SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/023_049.png">
<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/023_050.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Hence, <B>option
2.</B></SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=023_solvedex_17>
<TABLE style="WIDTH: 100%" id=023_solvedex_17_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_17>Example
17:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>Historical sales data of a retail store indicate
that 40 percent of all customers that enter the store make a purchase.
Determine the probability that exactly two of the next three customers
will make a purchase.</SPAN></P>
<P><SPAN class=Normal><B>[JMET 2009]</B></SPAN></P>
<P><SPAN class=Normal>(1) 0.388</SPAN></P>
<P><SPAN class=Normal>(2) 0.667</SPAN></P>
<P><SPAN class=Normal>(3) 0.400</SPAN></P>
<P><SPAN class=Normal>(4) 0.288</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>The probability that a customer who entered the
store makes a purchase is 0.4.</SPAN></P>
<P><SPAN class=Normal>The activity of any single customer making a
purchase is independent of the event of any other customer making a
purchase. The probability that 2 customers make a purchase and one does
not is (0.4)<SUP>2 </SUP><IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/multiplication.png">
(0.6).</SPAN></P>
<P><SPAN class=Normal>The two of the three people can be selected in
<SUP>3</SUP>C<SUB>2</SUB> i.e. 3 ways.</SPAN></P>
<P><SPAN class=Normal>Required probability = 3 <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/multiplication.png">
(0.4)<SUP>2 </SUP><IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/multiplication.png">
(0.6) = 0.288</SPAN></P>
<P><SPAN class=Normal>Hence, <B>option
4.</B></SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><B>Odds in favour and odds
against</B></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>Odds in Favour <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/023_051.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>Odds Against <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/023_052.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=023_solvedex_18>
<TABLE style="WIDTH: 100%" id=023_solvedex_18_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_18>Example
18:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>The odds in favour of a player winning a chess game
are 2/3. Find the probability that he loses the game and the probability
that he wins the game.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/023_053.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/therefore.png">
Total number of cases</SPAN></P>
<P><SPAN class=Normal>= favourable cases + unfavourable cases</SPAN></P>
<P><SPAN class=Normal>= 2 + 3</SPAN></P>
<P><SPAN class=Normal>= 5</SPAN></P>
<P><SPAN class=Normal>Let <I>A</I> be the event that the player wins the
game. Then <I>A</I>’ represents the event that he loses the
game.</SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/023_054.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/023_055.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/023_056.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=023_solvedex_19>
<TABLE style="WIDTH: 100%" id=023_solvedex_19_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_19>Example
19:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>The chance of India winning a cricket match against
Australia is 1/6. What is the minimum number of matches India should play
against Australia so that there is a fair chance of winning at least one
match?</SPAN></P>
<P><SPAN class=Normal><B>[XAT 2010]</B></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>(1) 3<SPAN style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN>(2) 4<SPAN
style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN></SPAN></P>
<P><SPAN class=Normal>(3) 5<SPAN style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN>(4) 6<SPAN
style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN></SPAN></P>
<P><SPAN class=Normal>(5) None of the above</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>A fair chance of winning is more than 50% chance of
winning.</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/therefore.png">
Probability = 0.5</SPAN></P>
<P><SPAN class=Normal>We need to find a fair chance winning atleast one
match.</SPAN></P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/023_057.png"></SPAN>
<P><SPAN class=Normal>By trial and error, <I>n</I> = 4</SPAN></P>
<P><SPAN class=Normal>Hence, <B>option 2.</B></SPAN></P>
<P><SPAN class=Normal></SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel3><SPAN class=Normal></SPAN></P></DIV></DIV>
<DIV id=023_III>
<TABLE id=023_III_bookmark class=tableh1 border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD
style="PADDING-BOTTOM: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 100%; PADDING-RIGHT: 0px; PADDING-TOP: 0px">
<OL class="oLevel1 Heading1">
<LI value=3><A id=III>CONDITIONAL
PROBABILITY</A></LI></OL></TD></TR></TBODY></TABLE>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>Let us consider the following two
cases.</SPAN></P>
<P class=pLevel2><SPAN class=Normal>1. We draw a card from a pack of 52 cards,
replace it, shuffle the cards and draw another card.</SPAN></P>
<P class=pLevel2><SPAN class=Normal>2. We draw a card from a pack of 52 cards,
keep it aside and draw another card.</SPAN></P>
<P class=pLevel2><SPAN class=Normal>Let us find the probability that the second
card drawn is a spade.</SPAN></P>
<P class=pLevel2><SPAN class=Normal>Let the event that the first card drawn is a
spade be called <I>A</I>, and let the event that the second card drawn is a
spade be called <I>B</I>.</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pImage><SPAN><IMG alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/023_058.png"></SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>As we replace the first card drawn in case
1, the probability that the second card drawn is a spade does not depend on what
the first card was.</SPAN></P>
<P class=pLevel2><SPAN class=Normal>However, in the second case, we do not
replace the first card drawn. So, the probability that the second card drawn is
a spade cannot be determined unless we know whether the first card was a spade
or not. In other words, event <I>B</I> depends on event <I>A</I>.</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pImage><SPAN><IMG alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/023_059.png"></SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>Therefore, the probability that the second
card drawn is a spade under the assumption that the first card was a spade,
denoted by P(<I>B</I>/<I>A</I>) = 12/51.</SPAN></P>
<P class=pLevel2><SPAN class=Normal>Similarly, the probability that the second
card drawn is a spade, under the assumption that the first card was not a spade,
denoted by</SPAN></P>
<P class=pLevel2><SPAN class=Normal>P(<I>B</I>/<I>A</I>’) = 13/51.</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>P(<I>B</I>/<I>A</I>) and
P(<I>B</I>/<I>A</I>’) are called <B>conditional</B>
<B>probabilities</B>.</SPAN></P>
<P class=pLevel2><SPAN class=Normal>P(<I>A</I> <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/intersect.png">
<I>B</I>) represents the probability that both the events <I>A</I> and <I>B</I>
occur and is given by:</SPAN></P>
<P class=pLevel2><SPAN class=Normal>P(<I>A</I> <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/intersect.png">
<I>B</I>) = P(<I>A</I>).P(<I>B</I>/<I>A</I>), where P(<I>A</I>) ≠ 0.</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>Similarly,</SPAN></P>
<P class=pLevel2><SPAN class=Normal>P(<I>A</I> <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/intersect.png">
<I>B</I>) = P(<I>B</I>).P(<I>A</I>/<I>B</I>), where P(<I>B</I>) ≠ 0.</SPAN></P>
<P class=pLevel2><SPAN class=Normal>If events <I>A</I> and <I>B </I>are
independent,</SPAN></P>
<P class=pLevel2><SPAN class=Normal>P(<I>B</I>/<I>A</I>) = P(<I>B</I>) and
P(<I>A</I>/<I>B</I>) = P(<I>A</I>)</SPAN></P>
<P class=pLevel2><SPAN class=Normal><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/therefore.png">
P(<I>A</I> <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/intersect.png">
<I>B</I>) = P(<I>A</I>) <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/multiplication.png">
P(<I>B</I>)</SPAN></P>
<P class=pLevel2><SPAN class=Normal>The concept of conditional probability can
be better understood from the following example.</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P></DIV>
<DIV class="divTextBox divColorExampleHeading divLevel2">
<DIV id=023_solvedex_20>
<TABLE style="WIDTH: 100%" id=023_solvedex_20_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_20>Example
20:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>Two cards are drawn at random from a pack of 52
cards. What is the probability that one card is an ace and the other is a
diamond?</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>Let <I>E</I> be the event that one card is an ace
and the other is a diamond.</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/therefore.png">
<I>E</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/equal_equal.png">
<I>A</I> <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/intersect.png">
<I>D</I></SPAN></P>
<P><SPAN class=Normal>Now two mutually exclusive cases arise.</SPAN></P>
<P><SPAN class=Normal><B>1.</B> The ace is a diamond.</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/therefore.png">
<I>n</I>(<I>A<SUB>D</I></SUB>) = <SUP>1</SUP>C<SUB>1 </SUB>= 1</SPAN></P>
<P><SPAN class=Normal>Now there are 12 diamonds left in 51
cards.</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/therefore.png">
<I>n</I>(<I>D</I>) = 12</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/therefore.png">
A diamond ace and a diamond card can be drawn in 1 <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/multiplication.png">
12 ways.</SPAN></P>
<P><SPAN class=Normal><B>2.</B> The ace is a non-diamond.</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/therefore.png">
<I>n</I>(<I>A<SUB>D</I></SUB>’) = <SUP>3</SUP>C<SUB>1 </SUB>= 3</SPAN></P>
<P><SPAN class=Normal>Now there are 13 diamonds left in 51
cards.</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/therefore.png">
<I>n</I>(<I>D</I>/<I>A<SUB>D</I></SUB>’) = 13</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/therefore.png">
A diamond ace and a diamond card can be drawn in 3 <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/multiplication.png">
13 ways.</SPAN></P>
<P><SPAN class=Normal>Also,</SPAN></P>
<P><SPAN class=Normal><I>n</I>(<I>S</I>) =
<SUP>52</SUP>C<SUB>2</SUB></SPAN></P>
<P><SPAN class=Normal>As the cases are mutually exclusive,</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/023_060.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/023_061.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel2">
<DIV id=023_solvedex_21>
<TABLE style="WIDTH: 100%" id=023_solvedex_21_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_21>Example
21:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>Two cards are drawn from a pack of cards one after
the other without replacement. What is the probability that the first card
picked up belongs to the suit of spades and the second one belongs to the
suit of diamonds?</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>Let <I>A</I> be the event that the first card drawn
is a spade and <I>B</I> be the event that the second card drawn is a
diamond.</SPAN></P>
<P><SPAN class=Normal>Number of spades = 13</SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/023_062.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/023_063.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Now, number of cards left = 51</SPAN></P>
<P><SPAN class=Normal>Number of diamonds = 13</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/therefore.png">
Probability that the second card is a diamond</SPAN></P>
<P><SPAN class=Normal>= P(<I>B/A</I>) = 13/51</SPAN></P>
<P><SPAN class=Normal>Thus, probability that the first card belongs to
spades and the next to diamonds</SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/023_064.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel2><SPAN class=Normal><B>Bayes’ Theorem</B></SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>P(<I>A </I>∩ <I>B</I> ) = P(<I>A</I>) <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/multiplication.png">
P(<I>B</I>/<I>A</I>)<SPAN style="MARGIN-LEFT: 25px"></SPAN>... (i)</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>Similarly,</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>P(<I>A</I> ∩ <I>B </I>) = P(<I>A</I>) <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/multiplication.png">
P(<I>A</I>/<I>B</I>)<SPAN style="MARGIN-LEFT: 25px"></SPAN>... (ii)</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>Equating (i) and (ii) we get</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>P(<I>A</I>) <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/multiplication.png">
P(<I>B</I>/<I>A</I>) = P(<I>B</I>) <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/multiplication.png">
P(<I>A</I>/<I>B</I>)</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/023_065.png"></SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>This is called the <B>Bayes’</B>
<B>Theorem</B>.</SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel2">
<DIV id=023_solvedex_22>
<TABLE style="WIDTH: 100%" id=023_solvedex_22_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_22>Example
22:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>An urn contains 3 red balls and 2 blue balls. Two
balls are picked from the urn one after the other without replacing the
first ball back in the urn. Find the probability that the second ball is
blue.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>The second ball has to be blue. The first ball can
be red or blue. So, we have two cases.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><B>Case 1:</B></SPAN></P>
<P><SPAN class=Normal>The first ball is red.</SPAN></P>
<P><SPAN class=Normal>The probability that first ball is red is
P(<I>R</I><SUB>1</SUB>) is 3/5.</SPAN></P>
<P><SPAN class=Normal>Now, there are 4 balls left—two red and two
blue.</SPAN></P>
<P><SPAN class=Normal>The probability that the second ball is blue is
P(<I>B</I><SUB>2</SUB>/<I>R</I><SUB>1</SUB>) = 2/4 = 1/2</SPAN></P>
<P><SPAN class=Normal>Thus, the probability that the first ball is red and
the second ball is blue</SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/023_066.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><B>Case 2:</B></SPAN></P>
<P><SPAN class=Normal>The first ball is blue.</SPAN></P>
<P><SPAN class=Normal>The probability that the first ball is blue
P(<I>B</I><SUB>1</SUB>) is 2/5.</SPAN></P>
<P><SPAN class=Normal>Now, there are 4 balls left—three red and one
blue.</SPAN></P>
<P><SPAN class=Normal>The probability that the second ball is blue
<I>P</I>(<I>B</I><SUB>2</SUB>/<I>B</I><SUB>1</SUB> )= 1/4.</SPAN></P>
<P><SPAN class=Normal>Thus, the probability that the first ball as well as
second ball is blue</SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/023_067.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Thus, the second ball is blue if either case 1 or
case 2 occurs.</SPAN></P>
<P><SPAN class=Normal>Thus, probability that the second ball is blue is
given by</SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/023_068.png"></SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel2">
<DIV id=023_solvedex_23>
<TABLE style="WIDTH: 100%" id=023_solvedex_23_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_23>Example
23:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>There are three identical boxes 1, 2 and 3. Box 1
contains 3 red balls and 2 blue balls. Box 2 contains 2 red and 5 white
balls and box 3 contains 2 blue, 1 red and 1 blue balls. One ball is drawn
from one of the boxes at random. Find the probability that it is
red.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>The ball is drawn from box 1 or box 2 or box
3.</SPAN></P>
<P><SPAN class=Normal>The probability that a ball drawn from box 1 is red
is 3/5.</SPAN></P>
<P><SPAN class=Normal>The probability that a ball drawn from box 2 is red
is 2/7.</SPAN></P>
<P><SPAN class=Normal>The probability that a ball drawn from box 3 is red
is 1/4.</SPAN></P>
<P><SPAN class=Normal>Probability of selecting any one of the three boxes
= 1/3.</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/therefore.png">
Probability that the ball is red is</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/023_069.png"></SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV><!--<p class="pLevel2"><span class="Normal"></span></p>
<div class="divTextBox divColorNormal divLevel2">
<span class="Normal"><span style="margin-left:25px"></span></span></p>
<p><span class="Normal"></span></p>
</div>--><!--<p class="pLevel2"><span class="Normal"></span></p>
<div class="divTextBox divColorExampleHeading divLevel2">
<div id="023_solvedex_23">-->
<DIV class="divTextBox divColorExampleHeading divLevel2">
<DIV id=023_solvedex_24>
<TABLE style="WIDTH: 100%" id=023_solvedex_24_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_24>Example
24:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>The chance of Geeta winning a competition is 1/9 and
that of Seema winning it is 2/15. Find the probability that one of Geeta
and Seema wins it.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>The probability that Seema or Geeta wins the
competition = 1 – the probability that both of them do not win
it.</SPAN></P>
<P><SPAN class=Normal>Probability that Geeta does not win the
competition</SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/023_070.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Probability that Seema does not win the
competition</SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/023_071.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/therefore.png">
The probability that neither Seema nor Geeta wins the
competition</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/023_072.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/therefore.png">
The probability that Seema or Geeta wins the competition</SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/023_073.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<DIV class="divTextBox divColorExampleHeading divLevel2">
<DIV id=023_solvedex_25>
<TABLE style="WIDTH: 100%" id=023_solvedex_25_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_25>Example
25:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>There are three similar boxes, containing (i) 6
black & 4 white balls; (ii) 3 black & 7 white balls and (iii) 5
black & 5 white balls, respectively. If you choose one of the three
boxes at random and from that particular box pick up a ball at random, and
find that to be black, what is the probability that the ball was picked up
from the second box?</SPAN></P>
<P><SPAN class=Normal><B>[JMET 2010]</B></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/023_074.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/023_075.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>We know that the ball picked up was
black.</SPAN></P>
<P><SPAN class=Normal>There are 6 + 3 + 5 = 14 black balls in
all.</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/therefore.png">
Probability that the black ball was from the second box = (Number of ways
in which one black ball can be picked from the second box)/(Total number
of ways in which a black ball can be picked)</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/023_076.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Hence, <B>option
1.</B></SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorNote divLevel2">
<P><SPAN class=Note>REMEMBER:</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<UL type=disc>
<LI>The sample space (<I>S</I>) is the set of all possible outcomes of an
event.</LI></UL>
<UL type=disc>
<LI>Any subset of <I>S</I> is called an event.</LI></UL>
<UL type=disc>
<LI>Probability is the chance of occurrence of an event.</LI></UL>
<P><SPAN class=Normal><SPAN style="MARGIN-LEFT: 25px"></SPAN><IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/023_077.png"></SPAN></P>
<UL type=disc>
<LI>If <I>P</I>(<I>E</I>) = 0, the event is an impossible event.</LI></UL>
<P><SPAN class=Normal></SPAN></P>
<UL type=disc>
<LI>If <I>P</I>(<I>E</I>) = 1, the event is a certain event.</LI></UL>
<P><SPAN class=Normal></SPAN></P>
<UL type=disc>
<LI><I>P</I>(<I>A</I> <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/union.png">
<I>B</I>) = <I>P</I>(<I>A</I>) + <I>P</I>(<I>B</I>) – <I>P</I>(<I>A</I> <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/intersect.png">
<I>B</I>)</LI></UL>
<P><SPAN class=Normal></SPAN></P>
<UL type=disc>
<LI>If <I>A</I> and <I>B</I> are mutually exclusive events, <I>P</I>(<I>A</I>
<IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/union.png">
<I>B</I>)</LI></UL>
<P><SPAN class=Normal><SPAN style="MARGIN-LEFT: 35px"></SPAN>=
<I>P</I>(<I>A</I>) + <I>P</I>(<I>B</I>)</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<UL type=disc>
<LI>If <I>A</I> and <I>B</I> are exhaustive events, <I>P</I>(<I>A</I>) +
<I>P</I>(<I>B</I>) = 1 </LI></UL>
<P><SPAN class=Normal><SPAN
style="MARGIN-LEFT: 35px"></SPAN>=<I>P</I>(<I>A</I>)=1 -
<I>P</I>(<I>A’</I>)</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<UL type=disc>
<LI>Odds in Favour <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/023_078.png"></LI></UL>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<UL type=disc>
<LI>Odds Against <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/023_079.png"></LI></UL>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<UL type=disc>
<LI>If <I>A</I> and <I>B</I> are dependent events,</LI></UL>
<P><SPAN class=Normal><SPAN style="MARGIN-LEFT: 35px"></SPAN>P(<I>A</I> <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/intersect.png">
<I>B</I>) = P(<I>A</I>).P(<I>B</I>/<I>A</I>), where P(<I>A</I>) ≠ 0 ,P (<I>A</I>
<IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/intersect.png">
<I>B</I>) = P(<I>B</I>).P(<I>A</I>/<I>B</I>), where P(<I>B</I>) ≠ 0</SPAN></P>
<UL type=disc>
<LI>If events <I>A</I> and <I>B </I>are independent,</LI></UL>
<P><SPAN class=Normal><SPAN style="MARGIN-LEFT: 35px"></SPAN>P(<I>A</I> <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/intersect.png">
<I>B</I>) = P(<I>A</I>) <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/647ae869-bae4-42ff-a5dd-5de0f35b0c5b/023_images/multiplication.png">
P(<I>B</I>)</SPAN></P></DIV></BODY></HTML>
guruhttp://www.blogger.com/profile/01591375209227117455noreply@blogger.comtag:blogger.com,1999:blog-7850540640045081515.post-36752486163006367762015-03-03T05:46:00.001-08:002019-09-14T18:55:04.124-07:00sets<div dir="ltr" style="text-align: left;" trbidi="on">
<br /></div>
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
<HTML xmlns="http://www.w3.org/1999/xhtml"><HEAD><TITLE>Sets</TITLE>
<META content="text/html; charset=unicode" http-equiv=Content-Type>
<SCRIPT type=text/javascript
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/CrossLink.js"></SCRIPT>
<SCRIPT type=text/javascript
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/Lesson.js"></SCRIPT>
<LINK rel=stylesheet type=text/css
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<META name=GENERATOR content="MSHTML 8.00.6001.19190"></HEAD>
<BODY onload="lesson_Function_1(); lesson_Function_2()">
<P class=oTOCTitle><SPAN class=oTOCTitleFont><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020.htm#Title">Contents</A></SPAN></P>
<OL class=oTOCLevel1>
<LI class=oTocLiClass><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020.htm#I">INTRODUCTION</A>
<OL class=oTOCLevel2>
<LI><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020.htm#I_A">CARDINALITY
OF SETS</A></LI>
<LI><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020.htm#I_B">CLASSIFICATION
OF SETS BASED ON CARDINALITY</A></LI>
<LI><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020.htm#I_C">EQUIVALENT
AND EQUAL SETS</A></LI></OL></LI>
<LI class=oTocLiClass><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020.htm#II">REPRESENTATION
OF A SET</A>
<OL class=oTOCLevel2>
<LI><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020.htm#II_A">ROSTER
NOTATION</A></LI>
<LI><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020.htm#II_B">SET
BUILDER NOTATION</A></LI></OL></LI>
<LI class=oTocLiClass><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020.htm#III">SUBSETS
AND SUPERSETS</A>
<OL class=oTOCLevel2>
<LI><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020.htm#III_A">POWER
SET</A></LI>
<LI><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020.htm#III_B">UNIVERSAL
SET</A></LI></OL></LI>
<LI class=oTocLiClass><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020.htm#IV">COMPLEMENT
OF A SET</A> </LI>
<LI class=oTocLiClass><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020.htm#V">UNION
AND INTERSECTION OF SETS</A> </LI>
<LI class=oTocLiClass><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020.htm#VI">VENN
DIAGRAMS</A> </LI>
<LI class=oTocLiClass><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020.htm#VII">PROPERTIES
OF SETS</A> </LI>
<LI class=oTocLiClass><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020.htm#VIII">CARTESIAN
PRODUCT</A> </LI></OL>
<P id=Title class=pTitle><SPAN class=Title>Sets</SPAN></P>
<HR>
<DIV id=020_I>
<TABLE id=020_I_bookmark class=tableh1 border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD
style="PADDING-BOTTOM: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 100%; PADDING-RIGHT: 0px; PADDING-TOP: 0px">
<OL class="oLevel1 Heading1">
<LI value=1><A id=I>INTRODUCTION</A></LI></OL></TD></TR></TBODY></TABLE>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>We have all come across the word
‘<B>set’</B> in our day-to-day lives. For example, “This is a set of the most
creative students in class.” or “This is the most beautiful set of diamonds I
have ever seen.” Hence, a set basically implies a group of certain similar
objects.</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>Mathematically, a set can be defined as a
group of certain similar and well defined objects. These objects are called the
<B>elements</B> of the set.</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>For example, consider the set of vowels in
the English alphabet. Let this set be represented by the letter <I>V</I>. This
set contains 5 elements – a, e, i, o, u. This can also be written as:</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal><I>V</I> <IMG style="VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/equal_equal.png">
{a, e, i, o, u}</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>The following two conditions should be taken
care of while writing a set:</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>A set must be denoted by a capital
letter.</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>The elements of the set must be denoted in
small letters.</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>Following are certain examples of
sets:</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal><I>A</I> <IMG style="VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/equal_equal.png">
{1, 2, 3, 4, 5}</SPAN></P>
<P class=pLevel2><SPAN class=Normal><I>B </I><IMG style="VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/equal_equal.png">
{a, b, c, d, e}</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>In the above examples, <I>A</I> and <I>B</I>
are sets while 1, 2, 3, 4, 5 and a, b, c, d, e are the elements belonging to
these sets respectively.</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>The symbol ϵ is used to denote that an
element belongs<I> </I>to a set. So, if we wish to write that 1 belongs to set
<I>A</I>, we write it as follows:</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>1 ϵ <I>A</I></SPAN></P>
<P class=pLevel2><SPAN class=Normal>ϵ is a Greek letter called
<I>epsilon</I>.</SPAN></P>
<P class=pLevel2><SPAN class=Normal>Certain sets are used very frequently, like
the set of all natural numbers. Following are certain standard letters used to
denote commonly used sets:</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>N <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/equal_equal.png">
Set of all natural numbers <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/equal_equal.png">
{1, 2, 3, 4, …}</SPAN></P>
<P class=pLevel2><SPAN class=Normal>W <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/equal_equal.png">
Set of all whole numbers <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/equal_equal.png">
{0, 1, 2, 3, …}</SPAN></P>
<P class=pLevel2><SPAN class=Normal>I <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/equal_equal.png">
Set of all integers <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/equal_equal.png">
{…, <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/minus.png">3,
<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/minus.png">2,
<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/minus.png">1,
0, 1, 2, 3, …}</SPAN></P>
<P class=pLevel2><SPAN class=Normal>Q <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/equal_equal.png">
Set of all rational numbers</SPAN></P>
<P class=pLevel2><SPAN class=Normal>R <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/equal_equal.png">
Set of all real numbers</SPAN></P>
<P class=pLevel2><SPAN class=Normal>P <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/equal_equal.png">
Set of all prime numbers</SPAN></P>
<P class=pLevel2><SPAN class=Normal>C <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/equal_equal.png">
Set of all complex numbers</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<DIV id=020_I_A>
<TABLE id=020_I_A_bookmark class=tableh1 border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD
style="PADDING-BOTTOM: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 100%; PADDING-RIGHT: 0px; PADDING-TOP: 0px">
<OL class="oLevel2 Heading2">
<LI value=1><A id=I_A>CARDINALITY OF
SETS</A></LI></OL></TD></TR></TBODY></TABLE>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><B>Cardinality</B> of a set refers to the
total number of elements that are present in the set.</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>For example, <I>V</I> <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/equal_equal.png">
{a, e, i, o, u}</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>The cardinality of this set of vowels
<I>V</I> is 5 as the total number of elements in the set is 5.</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>This can be represented as:</SPAN></P>
<P class=pLevel3><SPAN class=Normal>|<I>V</I>| = 5 or <I>n</I>(<I>V</I>) =
5</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P></DIV>
<DIV id=020_I_B>
<TABLE id=020_I_B_bookmark class=tableh1 border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD
style="PADDING-BOTTOM: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 100%; PADDING-RIGHT: 0px; PADDING-TOP: 0px">
<OL class="oLevel2 Heading2">
<LI value=2><A id=I_B>CLASSIFICATION OF SETS BASED ON
CARDINALITY</A></LI></OL></TD></TR></TBODY></TABLE>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>Based on the cardinality of sets, sets can
be classified into the following types.</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><B>Null Set</B><I>/</I> Void Set<I>/
</I>Empty Set</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>Consider the set <I>B</I> <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/equal_equal.png">
{ }</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>This set has no elements. Hence, the
cardinality of this set is zero.</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>This can be represented as:</SPAN></P>
<P class=pLevel3><SPAN class=Normal>|<I>B</I>| = 0 or <I>n</I>(<I>B</I>) =
0</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>Such a set is called a <B>null set</B> or an
<B>empty set</B>. A null set can be represented as:</SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/020_015.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><B>Infinite Set</B></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>Certain sets have infinite elements. Such
sets are called <B>infinite</B> sets.</SPAN></P>
<P class=pLevel3><SPAN class=Normal>For example, the set of natural numbers N,
set of integers I, etc.</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><B>Finite Set</B></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>Sets which have a finite number of elements
are called <B>finite</B> sets.</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>For example, the set of vowels</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><B>Singleton Set/Singlet</B></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>A set which has one and only one element in
it is called a <B>singleton</B> set.</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>For example,</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>1. <I>A</I> <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/equal_equal.png">
{a}</SPAN></P>
<P class=pLevel3><SPAN class=Normal>2. <I>B </I><IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/equal_equal.png">
{0}</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>In the above cases, <I>A</I> and <I>B</I>
are both singleton sets.</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorNote divLevel2">
<P><SPAN class=Note>REMEMBER:</SPAN></P>
<UL class=ListParagraph>
<LI class=ListStyleUL5>In the second example, <I>B</I> is not an empty set. It
has the element 0 in it.</LI></UL>
<P><SPAN class=ListParagraph></SPAN></P>
<UL><SPAN class=normal><I>B</I> would have been an empty set if it did not
contain any elements, i.e. <I>B</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/equal_equal.png">
{ }.</SPAN></UL></DIV>
<P class=pLevel4><SPAN class=Normal></SPAN></P></DIV>
<DIV id=020_I_C>
<TABLE id=020_I_C_bookmark class=tableh1 border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD
style="PADDING-BOTTOM: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 100%; PADDING-RIGHT: 0px; PADDING-TOP: 0px">
<OL class="oLevel2 Heading2">
<LI value=3><A id=I_C>EQUIVALENT AND EQUAL
SETS</A></LI></OL></TD></TR></TBODY></TABLE>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><B>Equivalent Sets</B></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>Two finite sets are said to be equivalent if
their cardinality is the same,</SPAN></P>
<P class=pLevel3><SPAN class=Normal>i.e. <I>n</I>(<I>A</I>) =
<I>n</I>(<I>B</I>)</SPAN></P>
<P class=pLevel3><SPAN class=Normal>For example, if <I>A</I> <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/equal_equal.png">
{1, 2, 3} and <I>B</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/equal_equal.png">
{4, 5, 6}</SPAN></P>
<P class=pLevel3><SPAN class=Normal>Then, <I>n</I>(<I>A</I>) =
<I>n</I>(<I>B</I>) = 3</SPAN></P>
<P class=pLevel3><SPAN class=Normal>Hence, <I>A </I>and <I>B</I> are equivalent
sets.</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><B>Equal Sets</B></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>Two sets are said to be equal if and only if
they contain the same elements.</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>This means that every element of set
<I>B</I> is present in set <I>A</I> and every element of set <I>A</I> is present
in set <I>B</I>.</SPAN></P>
<P class=pLevel3><SPAN class=Normal>For example, if <I>A</I> <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/equal_equal.png">
{a, b, c} and <I>B</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/equal_equal.png">
{c, b, a}, then <I>A</I> and <I>B</I> are said to be equal sets.</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>Equality is denoted as <I>A</I> = <I>B</I>.
Note that the order in which the elements of a set are written is not
important.</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P></DIV></DIV>
<DIV id=020_II>
<TABLE id=020_II_bookmark class=tableh1 border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD
style="PADDING-BOTTOM: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 100%; PADDING-RIGHT: 0px; PADDING-TOP: 0px">
<OL class="oLevel1 Heading1">
<LI value=2><A id=II>REPRESENTATION OF A
SET</A></LI></OL></TD></TR></TBODY></TABLE>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>There are two ways of representing a set.
They are</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P><B>
<OL class="oLevel3 ListParagraph ListStyleOL">
<LI value=1>Roster notation</LI></OL>
<OL class="oLevel3 ListParagraph ListStyleOL">
<LI value=2>Set Builder notation</B> (or <B>Rule Method</B>)</LI></OL>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<DIV id=020_II_A>
<TABLE id=020_II_A_bookmark class=tableh1 border=0 cellSpacing=0
cellPadding=0><TBODY>
<TR>
<TD
style="PADDING-BOTTOM: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 100%; PADDING-RIGHT: 0px; PADDING-TOP: 0px">
<OL class="oLevel2 Heading2">
<LI value=1><A id=II_A>ROSTER NOTATION</A></LI></OL></TD></TR></TBODY></TABLE>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>Let us consider the set of
vowels:</SPAN></P>
<P class=pLevel3><SPAN class=Normal><I>V</I> <IMG style="VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/equal_equal.png">
{a, e, i, o, u}</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>Here, each vowel is individually listed down
in the set <I>V</I>.</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>This method in which each element of a set
is specified within the set is called the Roster method.</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorNote divLevel2">
<P><SPAN class=Note>REMEMBER:</SPAN></P>
<P><SPAN class=ListParagraph></SPAN></P>
<UL class=ListParagraph>
<LI class=ListStyleUL5>To denote an infinite set using the Roster method, like
the set of natural numbers, we write the first few elements followed by ‘…’ to
indicate continuation up to infinity.</LI></UL>
<UL><SPAN class=Normal>N <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/equal_equal.png">
{1, 2, 3, …}</SPAN></UL></DIV>
<P class=pLevel4><SPAN class=Normal></SPAN></P></DIV>
<DIV id=020_II_B>
<TABLE id=020_II_B_bookmark class=tableh1 border=0 cellSpacing=0
cellPadding=0><TBODY>
<TR>
<TD
style="PADDING-BOTTOM: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 100%; PADDING-RIGHT: 0px; PADDING-TOP: 0px">
<OL class="oLevel2 Heading2">
<LI value=2><A id=II_B>SET BUILDER
NOTATION</A></LI></OL></TD></TR></TBODY></TABLE>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>In the Set Builder notation, instead of
writing down every single element of a set, we describe the common property of
the elements of the set.</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>For example, in the Roster notation the set
of the first five natural numbers is written as follows:</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><I>D</I> <IMG style="VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/equal_equal.png">
{1, 2, 3, 4, 5}</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>Now, if we were to describe this set in
words, we would say, “Set <I>D</I> is a set of all elements <I>x</I> such that
<I>x</I> is a natural number and <I>x</I> lies between 1 and 5
(inclusive).”</SPAN></P>
<P class=pLevel3><SPAN class=Normal>This is written in Set Builder notation as
follows:</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><I>D</I> <IMG style="VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/equal_equal.png">
{<I>x</I> | <I>x</I> ( N and 1 <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/less_equal.png">
<I>x</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/less_equal.png">
5}</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorNote divLevel2">
<P><SPAN class=Note>REMEMBER:</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<UL class=ListParagraph>
<LI class=ListStyleUL5>The symbol | is read as “such that”.</LI></UL>
<P><SPAN class=Normal></SPAN></P>
<UL><SPAN class=Normal>Let us take another example,</SPAN></UL>
<UL><SPAN class=Normal><I>V</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/equal_equal.png">
{a, e, i, o, u}</SPAN></UL>
<UL><SPAN class=Normal>Here, we can say, “the set <I>V</I> is a set of
elements <I>x</I>, such that <I>x</I> is a vowel from the English
alphabet.”</SPAN></UL>
<P><SPAN class=Normal></SPAN></P>
<UL><SPAN class=Normal>This can be written in Set Builder notation
as:</SPAN></UL>
<UL><SPAN class=Normal><I>V</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/equal_equal.png">
{<I>x</I> | <I>x</I> is a vowel from the English alphabet}</SPAN></UL>
<P><SPAN class=Normal></SPAN></P>
<UL><SPAN class=Normal>Hence, in general, if all the elements of the set
<I>A</I> have a common property <I>B</I>, then it can be expressed in the set
builder form in the following way:</SPAN></UL>
<UL><SPAN class=Normal><I>A</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/equal_equal.png">
{<I>x</I> | <I>x</I> has property <I>B</I>}</SPAN></UL></DIV>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel2">
<DIV id=020_solvedex_1>
<TABLE style="WIDTH: 100%" id=020_solvedex_1_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_1>Example
1:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>Convert the following sets from the Set Builder
notation to the Roster notation:</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>i) <I>P</I> <IMG style="VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/equal_equal.png">
{<I>y</I> | <I>y</I> ( W and <I>y</I> < 3}</SPAN></P>
<P><SPAN class=Normal>ii) <I>M</I> <IMG style="VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/equal_equal.png">
{<I>x</I> | <I>x</I><SUP>2 </SUP>– 5<I>x</I> + 6 = 0}</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>i) Here, <I>P</I> is a set of all numbers <I>y</I>,
such that <I>y</I> is a whole number less than 3. The whole numbers less
than 3 are 0, 1 and 2. Hence, set <I>P</I> can be written in Roster
notation as:</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><I>P</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/equal_equal.png">
{0, 1, 2}</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>ii) Here, <I>M</I> is a set of all such numbers
<I>x</I> where</SPAN></P>
<P><SPAN class=Normal><I>x</I><SUP>2 </SUP>– 5<I>x</I> + 6 = 0. To find
the values of <I>x</I>, we need to solve this equation.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><I>x</I><SUP>2 </SUP>–<SUP> </SUP>5<I>x</I> + 6 =
0</SPAN></P>
<P><SPAN class=Normal><I>x</I><SUP>2 </SUP>– 3<I>x</I> – 2<I>x</I> + 6 =
0</SPAN></P>
<P><SPAN class=Normal><I>x</I>(<I>x</I> – 3) – 2(<I>x</I> – 3) =
0</SPAN></P>
<P><SPAN class=Normal>(<I>x</I> – 3)(<I>x</I> – 2) = 0</SPAN></P>
<P><SPAN class=Normal><I>x</I> = 3 or <I>x</I> = 2</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Hence, we can write set <I>M</I> in the Roster form
as:</SPAN></P>
<P><SPAN class=Normal><I>M</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/equal_equal.png">
{2, 3}</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel4><SPAN class=Normal></SPAN></P></DIV></DIV>
<DIV id=020_III>
<TABLE id=020_III_bookmark class=tableh1 border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD
style="PADDING-BOTTOM: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 100%; PADDING-RIGHT: 0px; PADDING-TOP: 0px">
<OL class="oLevel1 Heading1">
<LI value=3><A id=III>SUBSETS AND
SUPERSETS</A></LI></OL></TD></TR></TBODY></TABLE>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>Let us consider the set of all students in a
college. Now, the set of first year students is a part of this bigger set of all
students. Such a set, all of whose elements are present in a bigger set, is
called a <B>subset</B> of the bigger set. The bigger set is called the
<B>superset</B> of the smaller set.</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>Let us consider the following
example:</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal><I>A</I> <IMG style="VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/equal_equal.png">
{1, 2, 3, 4, 5, 6} and</SPAN></P>
<P class=pLevel2><SPAN class=Normal><I>B</I> <IMG style="VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/equal_equal.png">
{1, 2, 3}</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>We can see that every element of the set
<I>B</I> is an element of the set <I>A</I>.</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>Hence, we can say that <I>B</I> is a subset
of <I>A</I>. We can also say that <I>B</I> is contained in<I> A</I>.</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal><I>A</I> is said to be the superset of
<I>B</I>.</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorNote divLevel2">
<P><SPAN class=Note>REMEMBER:</SPAN></P>
<P><SPAN class=ListParagraph></SPAN></P>
<UL class=ListParagraph>
<LI class=ListStyleUL5>A Null Set is a subset of every set.</LI></UL>
<UL class=ListParagraph>
<LI class=ListStyleUL5>Every set is a subset of itself.</LI></UL></DIV>
<P class=pLevel1><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal><B>Definition of a Subset</B></SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>We say that <I>A</I> is a subset of <I>B</I>
if and only if every element of <I>A</I> is an element of <I>B</I>.</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>This is denoted as follows:</SPAN></P>
<P class=pLevel2><SPAN class=Normal><I>A</I> <IMG style="VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/symb30.png">
<I>B</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>F</I>or example,</SPAN></P>
<P class=pLevel2><SPAN class=Normal><I>A </I><IMG style="VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/equal_equal.png">
{1, 2}, <I>B</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/equal_equal.png">
{1, 2}</SPAN></P>
<P class=pLevel2><SPAN class=Normal><I>A</I> <IMG style="VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/equal_equal.png">
{1, 2, 3}, <I>B</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/equal_equal.png">
{1, 2, 3, 4}</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>In both these examples, <I>A</I> <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/symb30.png">
<I>B</I>.</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>In the first example, <I>B</I> is also a
subset of <I>A</I>. Hence,</SPAN></P>
<P class=pLevel2><SPAN class=Normal><I>A</I> <IMG style="VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/symb30.png"><I>B</I>
and <I>B</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/symb30.png">
<I>A</I>.</SPAN></P>
<P class=pLevel2><SPAN class=Normal><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/therefore.png">
Set <I>B</I> = Set <I>A</I></SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>Such a subset is called an <B>improper
subset</B>.</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>Subset <I>A</I> is an improper subset of
<I>B</I> if <I>A</I> = <I>B</I>.</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>In the second example, <I>B</I> is not a
subset of <I>A</I>, which can be denoted by <I>B</I> <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/symb18.png">
<I>A</I>.</SPAN></P>
<P class=pLevel2><SPAN class=Normal>Such a subset, where the superset also
contains other elements, is called a <B>proper subset</B>.</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>Set <I>A</I> is said to be a proper subset
of <I>B</I> if <I>A</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/symb30.png">
<I>B</I> but <I>B</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/symb18.png">
<I>A</I>.</SPAN></P>
<P class=pLevel2><SPAN class=Normal>A proper subset is denoted by the symbol
<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/symb28.png">.
Thus, in the second example, <I>A</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/symb28.png">
<I>B</I>.</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<DIV id=020_III_A>
<TABLE id=020_III_A_bookmark class=tableh1 border=0 cellSpacing=0
cellPadding=0><TBODY>
<TR>
<TD
style="PADDING-BOTTOM: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 100%; PADDING-RIGHT: 0px; PADDING-TOP: 0px">
<OL class="oLevel2 Heading2">
<LI value=1><A id=III_A>POWER SET</A></LI></OL></TD></TR></TBODY></TABLE>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>A <B>power set</B> is the set of all subsets
of a set.</SPAN></P>
<P class=pLevel3><SPAN class=Normal>Consider a set <I>A</I> <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/equal_equal.png">
{1, 2, 3}</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>Now, consider a set {1}. This set is a
subset of <I>A</I>. Similarly, {1, 2} is also a subset of <I>A</I>. So, let us
try and list down all the subsets of <I>A</I>. They would be:</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>{1}</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>{2}</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>{3}</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>{1, 2}</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>{1, 3}</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>{2, 3}</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>{1, 2, 3} … (Every set is a subset of
itself)</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>{ } … (Null set is a subset of every
set)</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>Total number of subsets = 8</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>Hence, a set of these subsets is called the
power set of <I>A</I>.</SPAN></P>
<P class=pLevel3><SPAN class=Normal><I>P</I> <IMG style="VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/equal_equal.png">
{{1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}, { }}</SPAN></P>
<P class=pLevel3><SPAN class=Normal>If the number of elements (cardinality) in a
set is <I>n</I>, then the cardinality of the power set is given by
2<I><SUP>n</I></SUP>.</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><B>Explanation:</B></SPAN></P>
<P class=pLevel3><SPAN class=Normal>To form a subset <I>B</I> of a set <I>A</I>
containing <I>n</I> elements, each element of <I>A</I> can either be chosen or
not chosen to be a part of <I>B</I>. Thus, the first, second, third…,
<I>n</I><SUP>th</SUP> elements of <I>A</I> can either be chosen or not chosen in
2 ways each.</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/therefore.png">
The total number of ways in which a subset can be formed = 2 <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/multiplication.png">
2 <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/multiplication.png">
2 … <I>n</I> times = 2<I><SUP>n</I></SUP></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/therefore.png">
The power set of a set containing <I>n</I> elements, has 2<I><SUP>n</I></SUP>
elements.</SPAN></P>
<P class=pLevel3><SPAN class=Normal>For example, in the above case, the
cardinality of set <I>A</I>, <I>n</I>(<I>A</I>) = 3</SPAN></P>
<P class=pLevel3><SPAN class=Normal>So, the cardinality of power set <I>P</I>,
<I>n</I>(<I>P</I>) = 2<SUP>3</SUP> = 8</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>Let us consider another example,</SPAN></P>
<P class=pLevel3><SPAN class=Normal>If <I>B</I> <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/equal_equal.png">
{a, b}, then <I>n</I>(<I>B</I>) = 2</SPAN></P>
<P class=pLevel3><SPAN class=Normal>( Cardinality of the power set = 2<SUP>2
</SUP>= 4</SPAN></P>
<P class=pLevel3><SPAN class=Normal>Verifying the above statement, power set
<I>P</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/equal_equal.png">
{{ }, {a}, {b}, {a, b}}</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P></DIV>
<DIV id=020_III_B>
<TABLE id=020_III_B_bookmark class=tableh1 border=0 cellSpacing=0
cellPadding=0><TBODY>
<TR>
<TD
style="PADDING-BOTTOM: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 100%; PADDING-RIGHT: 0px; PADDING-TOP: 0px">
<OL class="oLevel2 Heading2">
<LI value=2><A id=III_B>UNIVERSAL SET</A></LI></OL></TD></TR></TBODY></TABLE>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>Let us consider a college, consisting of
students from three streams—Arts (set <I>A</I>), Commerce (set <I>C</I>) and
Science (set <I>S</I>). All these students together form the set of all the
students in the entire college. Such a set is called a <B>universal set</B> (set
<I>U</I>).</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>A universal set is a set containing all the
elements under consideration. It is represented by the capital letter
U.</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>The universal set for the set of all numbers
is as follows:</SPAN></P>
<P class=pLevel3><SPAN class=Normal><I>U</I> <IMG style="VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/equal_equal.png">
{<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/minus.png">∞
… ∞}. It includes all natural numbers, integers, rational numbers, irrational
numbers, etc.</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P></DIV></DIV>
<DIV id=020_IV>
<TABLE id=020_IV_bookmark class=tableh1 border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD
style="PADDING-BOTTOM: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 100%; PADDING-RIGHT: 0px; PADDING-TOP: 0px">
<OL class="oLevel1 Heading1">
<LI value=4><A id=IV>COMPLEMENT OF A SET</A></LI></OL></TD></TR></TBODY></TABLE>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>Once we understand the idea of a universal
set, it becomes easy for us to understand what the <B>complement</B> of a set
is.</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>The complement of a set <I>A</I>, which is a
subset of a universal set <I>U</I>, includes all the elements present in
<I>U</I> which are not present in <I>A</I>. It is denoted by
<I>A</I>’.</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>Suppose,</SPAN></P>
<P class=pLevel2><SPAN class=Normal><I>U</I> <IMG style="VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/equal_equal.png">
{a, b, c, d, e, f, g, h, i} and</SPAN></P>
<P class=pLevel2><SPAN class=Normal><I>A</I> <IMG style="VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/equal_equal.png">
{a, b, c, d, e},</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>Then <I>A</I>’, or complement of the set
<I>A</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/equal_equal.png">
{f, g, h, i}</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P></DIV>
<DIV id=020_V>
<TABLE id=020_V_bookmark class=tableh1 border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD
style="PADDING-BOTTOM: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 100%; PADDING-RIGHT: 0px; PADDING-TOP: 0px">
<OL class="oLevel1 Heading1">
<LI value=5><A id=V>UNION AND INTERSECTION OF
SETS</A></LI></OL></TD></TR></TBODY></TABLE>
<P class=pLevel2><SPAN class=Normal>Let us consider the following example. In a
room, there are 5 people a, b, c, d, e. Out of them, a, b and c are men while d
and e are women. Also, a and e study science while b, c and d study
commerce.</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/therefore.png">
The set of males is:</SPAN></P>
<P class=pLevel2><SPAN class=Normal><I>M</I> <IMG style="VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/equal_equal.png">
{a, b, c}</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>The set of females is:</SPAN></P>
<P class=pLevel2><SPAN class=Normal><I>F</I> <IMG style="VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/equal_equal.png">
{d, e}</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>The set of science students is</SPAN></P>
<P class=pLevel2><SPAN class=Normal><I>S</I> <IMG style="VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/equal_equal.png">
{a, e}</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>The set of commerce students is</SPAN></P>
<P class=pLevel2><SPAN class=Normal><I>C</I> <IMG style="VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/equal_equal.png">
{b, c, d}</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>If we wish to find out all female students
who have taken science, we need to find out what is common in set <I>F</I> and
set <I>S</I>. This is called an <B>intersection</B> of set <I>F</I> and set
<I>S</I> and is denoted by <I>F</I> <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/intersect.png">
<I>S</I>.</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>Here, <I>F</I> <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/intersect.png">
<I>S</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/equal_equal.png">
{e}</SPAN></P>
<P class=pLevel2><SPAN class=Normal>Thus, an intersection of two sets is formed
by the elements which are common to both the sets.</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>Similarly, if we consider sets <I>M</I> and
<I>F</I>, there is no common element between them.</SPAN></P>
<P class=pLevel2><SPAN class=Normal>Hence, <I>M</I> <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/intersect.png">
<I>F</I> = ∅</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>Such sets which have no element in common
are called <B>disjoint sets.</B></SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>Now, let us find out those females who have
not taken science. Here, we have to check the set <I>F</I> and remove all
elements of set <I>S</I> present in this set. This is called the
<B>difference</B> between two sets.</SPAN></P>
<P class=pLevel2><SPAN class=Normal><I>F</I> <IMG style="VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/minus.png">
<I>S</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/equal_equal.png">
{d}</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>Thus, difference of set <I>A</I> and set
<I>B</I> is defined as the elements present in <I>A</I> but not present in
<I>B</I>.</SPAN></P>
<P class=pLevel2><SPAN class=Normal>A <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/minus.png">
B <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/equal_equal.png">
{<I>x</I> | <I>x</I> ( A and <I>x</I> ∉ B}</SPAN></P>
<P class=pLevel2><SPAN class=Normal>Now, suppose you want to represent a set
containing “either males or commerce students or both”. This would mean taking
all the elements from set <I>M</I> and set <I>C</I> together into one set. This
is called the <B>union</B> of set <I>M</I> and set <I>C</I> and is denoted by
<I>M</I> <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/union.png">
<I>C</I>.</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>Thus, <I>M</I> <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/union.png">
<I>C</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/equal_equal.png">
{a, b, c, d}</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorNote divLevel2">
<P><SPAN class=Note>IMPORTANT:</SPAN></P>
<P><SPAN class=ListParagraph></SPAN></P>
<P><SPAN class=ListParagraph>Though b and c exist in both sets, they are written
only once while writing the union. This is because no element is ever written
twice while writing a set.</SPAN></P></DIV>
<P class=pLevel2><SPAN class=ListParagraph></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel2">
<DIV id=020_solvedex_2>
<TABLE style="WIDTH: 100%" id=020_solvedex_2_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_2>Example
2:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>Consider the set <I>S </I>= {2, 3, 4, ..., 2<I>n</I>
+ 1}, where <I>n </I> is a positive integer larger than 2007. Define
<I>X</I> as the average of the odd integers in <I>S </I>and <I>Y</I> as
the average of the even integers in <I>S</I>. What is the value of
<I>X</I> – <I>Y</I>?</SPAN></P>
<P><SPAN class=Strong>[<SPAN class=Strong>CAT 2007]</SPAN></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>(1) 0<SPAN style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN>(2) 1</SPAN></P>
<P><SPAN class=Normal>(3) <I>n</I>/2<SPAN
style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 34px"></SPAN>(4) <I>n</I> + 1<I>/2n</I></SPAN></P>
<P><SPAN class=Normal>(5) 2008</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal><I>Y</I> = (2 + 4 + 6 + 8 + … +
2<I>n</I>)/<I>n</I></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><I>X</I> = (3 + 5 + 7 + 9 + … + (2<I>n </I>+
1))/<I>n</I></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>= [(2 + 1) + (4 + 1) + (6 + 1) + (8 + 1) + … +
(2<I>n </I>+ 1)]/<I>n</I></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>= (2 + 4 + 6 + 8 + … + 2<I>n</I>)/<I>n</I> + (1 + 1
+ 1 + 1 + … <I>n</I> times)/<I>n</I></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>= <I>Y </I>+ 1</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal> <IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/therefore.png">
<I>X</I> – <I>Y</I> = 1</SPAN></P>
<P><SPAN class=Normal>Hence, <B>option 2.</B></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Note: The information that '<I>n</I> is a positive
integer larger than 2007' does not affect the answer in any
way.</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel2">
<DIV id=020_solvedex_3>
<TABLE style="WIDTH: 100%" id=020_solvedex_3_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_3>Example
3:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>Let the set consisting of the squares of the
positive integers be called <I>u; </I>thus <I>u </I>is the set 1, 4, 9,
..... If a certain operation on one or more members of the set always
yields a member of the set, we say that the set is closed under that
operation. Then <I>u </I>is closed under :</SPAN></P>
<P><SPAN class=Normal><B>[FMS 2010]</B></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>(1) addition</SPAN></P>
<P><SPAN class=Normal>(2) multiplication</SPAN></P>
<P><SPAN class=Normal>(3) division</SPAN></P>
<P><SPAN class=Normal>(4) none of these</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>Let <I>x</I> and<I> y</I> be any two elements of set
<I>u</I>.</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/symb23.png">
Every element of set <I>u</I> is a perfect square,</SPAN></P>
<P><SPAN class=Normal><I>x</I> = <I>a</I><SUP>2</SUP> and <I>y</I> =
<I>b</I><SUP>2</SUP> for some positive integers <I>a</I> and
<I>b</I></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/therefore.png"><I>
xy</I> = <I>a</I><SUP>2</SUP><I>b</I><SUP>2</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/therefore.png"></SUP>
<I>xy </I>is also a perfect square.</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/therefore.png">
<I>xy</I> is also an element of <I>u</I>.</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/therefore.png">
Set <I>u</I> is closed with respect to multiplication.</SPAN></P>
<P><SPAN class=Normal>Hence, <B>option
2</B>.</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel2><SPAN class=Normal></SPAN></P></DIV>
<DIV id=020_VI>
<TABLE id=020_VI_bookmark class=tableh1 border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD
style="PADDING-BOTTOM: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 100%; PADDING-RIGHT: 0px; PADDING-TOP: 0px">
<OL class="oLevel1 Heading1">
<LI value=6><A id=VI>VENN DIAGRAMS</A></LI></OL></TD></TR></TBODY></TABLE>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>Sets can be represented using diagrams known
as Venn Diagrams. Sets can be represented in the form of circles, triangles,
rectangles, etc.</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>Let us take an example to understand Venn
diagrams.</SPAN></P>
<P class=pLevel2><SPAN class=Normal>Let <I>U</I> be the universal set containing
all the natural numbers between (and not including) 0 and 11.</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>Hence, <I>U</I> <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/equal_equal.png">
{1, 2, 3, 4, 5, 6, 7, 8, 9, 10}</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>Let <I>P</I> be the set containing all the
prime numbers between 0 and 11.</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>Thus, <I>P</I> <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/equal_equal.png">
{2, 3, 5, 7}</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>Let <I>E</I> be the set containing all the
even numbers between 0 and 11.</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>Hence, <I>E</I> <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/equal_equal.png">
{2, 4, 6, 8, 10}</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>Hence, <I>P</I> <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/union.png">
<I>E</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/equal_equal.png">
{2, 3, 4, 5, 6, 7, 8, 10}</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>Also, <I>P</I> <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/intersect.png">
<I>E</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/equal_equal.png">
{2}</SPAN></P>
<P class=pLevel2><SPAN class=Normal>This can be represented using a Venn diagram
in the following manner.</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pImage><SPAN><IMG alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/020_001.png"></SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>In the above diagram, the universal set is
represented by the rectangle while all other sets are represented by circles.
The common portion between the two sets <I>P</I> and <I>E</I> is the
intersection of these two sets.</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>The universal set contains two numbers, 1
and 9, which do not belong to the set of prime or even numbers. Hence, they are
written outside both circles but within the universal set.</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>Now, let us look at the cardinality of these
sets.</SPAN></P>
<P class=pLevel2><SPAN class=Normal><I>n</I>(<I>P</I>) = 4, <I>n</I>(<I>E</I>) =
5 and <I>n</I>(<I>P</I> <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/union.png">
<I>E</I>) = 8</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>Thus if we observe, <I>n</I>(<I>P</I> <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/union.png">
<I>E</I>) ≠ <I>n</I>(<I>P</I>) + <I>n</I>(<I>E</I>)</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>This is because the element 2 is included in
both <I>P</I> and <I>E</I> but while taking the union of the two sets, we need
to count it only once.</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>Therefore, if we subtract <I>n</I>(<I>P</I>
<IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/intersect.png">
<I>E</I>) from <I>n</I>(<I>P</I>) + <I>n</I>(<I>E</I>), then we will get
<I>n</I>(<I>P</I> <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/union.png">
<I>E</I>).</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal><I>n</I>(<I>P</I> <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/union.png">
<I>E</I>) = <I>n</I>(<I>P</I>) + <I>n</I>(<I>E</I>) – <I>n</I>(<I>P</I> <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/intersect.png">
<I>E</I>)</SPAN></P>
<P class=pLevel2><SPAN class=Normal>i.e. <I>n</I>(<I>P</I> <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/union.png">
<I>E</I>) = 4 + 5 – 1 = 8</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>To generalise this, for any two sets A and
B,</SPAN></P>
<P class=pLevel2><SPAN class=Normal><I>n</I>(<I>A</I> <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/union.png">
<I>B</I>) = <I>n</I>(<I>A</I>) + <I>n</I>(<I>B</I>) <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/minus.png">
<I>n</I>(<I>A</I> <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/intersect.png">
<I>B</I>)</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel2">
<DIV id=020_solvedex_4>
<TABLE style="WIDTH: 100%" id=020_solvedex_4_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_4>Example
4:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>In a class of 100 students, 60 play football, 40
play hockey while 10 do not play any sport. Find the number of students
who play both hockey and football.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal><I>n</I>(<I>U</I>) = 100</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Since 10 do not play anything,</SPAN></P>
<P><SPAN class=Normal><I>n</I>(<I>F</I> <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/union.png">
<I>H</I>) = 100 – 10 = 90</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/symb23.png">
<I>n</I>(<I>F</I>) = 60</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/symb23.png">
<I>n</I>(<I>H</I>) = 40</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/therefore.png">
<I>n</I>(<I>F</I> <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/union.png">
<I>H</I>) = <I>n</I>(<I>F</I>) + <I>n</I>(<I>H</I>) – <I>n</I>(<I>F</I>
<IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/intersect.png">
<I>H</I>)</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/therefore.png">
90 = 60 + 40 – <I>n</I>(<I>F</I> <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/intersect.png">
<I>H</I>)</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Hence, <I>n</I>(<I>F</I> <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/intersect.png">
<I>H</I>) = 100 – 90 = 10</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/therefore.png">
10 students play both football and
hockey.</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>Similarly, now let us consider a universal
set as follows:</SPAN></P>
<P class=pLevel2><SPAN class=Normal><I>U</I> <IMG style="VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/equal_equal.png">
{1, 2, 3, 4, 5, 6, 7, 8, 9, 15}</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>Let <I>A</I> be the set of all even
numbers.</SPAN></P>
<P class=pLevel2><SPAN class=Normal>Hence, <I>A</I> <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/equal_equal.png">
{2, 4, 6, 8}</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>Let <I>B</I> be the set of all perfect
squares.</SPAN></P>
<P class=pLevel2><SPAN class=Normal>Hence, <I>B</I> <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/equal_equal.png">
{1, 4, 9}</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>Let <I>C</I> be the set of all composite
numbers.</SPAN></P>
<P class=pLevel2><SPAN class=Normal>Hence, <I>C</I> <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/equal_equal.png">
{4, 6, 8, 9, 15}</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>This can be represented using a Venn diagram
as follows:</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pImage><SPAN><IMG alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/020_002.png"></SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>For a three level diagram, the formula
is</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal><I>n</I>(<I>A</I> <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/union.png">
<I>B</I> <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/union.png"><I>
C</I>) = <I>n</I>(<I>A</I>) + <I>n</I>(<I>B</I>) + <I>n</I>(<I>C</I>) <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/minus.png">
<I>n</I>(<I>A</I> <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/intersect.png">
<I>B</I>)</SPAN></P>
<P class=pLevel2><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/minus.png">
<I>n</I>(<I>A</I> <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/intersect.png">
<I>C</I>) – <I>n</I>(<I>B</I> <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/intersect.png">
<I>C</I>) + <I>n</I>(<I>A</I> <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/intersect.png">
<I>B</I> <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/intersect.png">
<I>C</I>)</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>So in this case,</SPAN></P>
<P class=pLevel2><SPAN class=Normal><I>n</I>(<I>A</I> <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/union.png">
<I>B</I> <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/union.png">
<I>C</I>) = 4 + 3 + 5 <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/minus.png">
1 <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/minus.png">
3 <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/minus.png">
2 + 1 = 7</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel2">
<DIV id=020_solvedex_5>
<TABLE style="WIDTH: 100%" id=020_solvedex_5_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_5>Example
5:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>Draw Venn diagrams for:</SPAN></P>
<P><SPAN class=Normal><I>A</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/minus.png">
<I>B</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/minus.png">
<I>C</I></SPAN></P>
<P><SPAN class=Normal><I>A</I> <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/intersect.png">
<I>B</I> <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/union.png">
<I>C</I>’</SPAN></P>
<P><SPAN class=Normal><I>A</I> <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/union.png">
<I>B</I> <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/intersect.png">
<I>C</I>’</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>i) <I>A</I> <IMG style="VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/minus.png">
<I>B</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/minus.png">
<I>C</I></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=pImage><IMG alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/020_005.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>ii) <I>A</I> <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/intersect.png">
<I>B</I> <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/union.png">
<I>C</I>’</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=pImage><IMG alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/020_004.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>iii) <I>A</I> <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/union.png">
<I>B</I> <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/intersect.png">
<I>C</I>’</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=pImage><IMG alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/020_003.png">
</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel2">
<DIV id=020_solvedex_6>
<TABLE style="WIDTH: 100%" id=020_solvedex_6_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_6>Example
6:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>Express the following as Venn diagrams:</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<OL class=ListParagraph>
<LI value=1>Every bull is an animal</LI></OL>
<OL class=ListParagraph>
<LI value=2>Some bulls are in the stock market</LI></OL>
<OL class=ListParagraph>
<LI value=3>No animal is a bull</LI></OL>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>Let <I>A</I> = Set of Animals, <I>B</I> = Set of
bulls, <I>S </I>= Set of things in the stock market</SPAN></P>
<OL class=ListParagraph>
<LI value=1>Every bull is an animal.</LI></OL>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=pImage><IMG alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/020_006.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<OL class=ListParagraph>
<LI value=2>Some bulls are in the stock market.</LI></OL>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=pImage><IMG alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/020_007.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<OL class=ListParagraph>
<LI value=3>No animal is a bull</LI></OL>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=pImage><IMG alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/020_008.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Here, <I>A</I> and <I>B</I> are disjoint
sets.</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel4><SPAN class=Normal></SPAN></P></DIV>
<DIV id=020_VII>
<TABLE id=020_VII_bookmark class=tableh1 border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD
style="PADDING-BOTTOM: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 100%; PADDING-RIGHT: 0px; PADDING-TOP: 0px">
<OL class="oLevel1 Heading1">
<LI value=7><A id=VII>PROPERTIES OF SETS</A></LI></OL></TD></TR></TBODY></TABLE>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>Let us now look at some properties of
sets:</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<OL class="oLevel3 ListParagraph ListStyleOL">
<LI value=1>Union of any set with a null set is equal to that set.</LI></OL>
<P class=pLevel4><SPAN class=Normal><SPAN style="MARGIN-LEFT: 20px"></SPAN><IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/020_016.png">
</SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<OL class="oLevel3 ListParagraph ListStyleOL">
<LI value=2>Intersection of any set with a null set is equal to a null
set.</LI></OL>
<P class=pLevel4><SPAN class=Normal><SPAN style="MARGIN-LEFT: 20px"></SPAN><IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/020_017.png"></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<OL class="oLevel3 ListParagraph ListStyleOL">
<LI value=3>The intersection of any set with its complement is a null
set.</LI></OL>
<P class=pLevel4><SPAN class=Normal><SPAN style="MARGIN-LEFT: 20px"></SPAN><IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/020_018.png">
</SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<OL class="oLevel3 ListParagraph ListStyleOL">
<LI value=4>The union of any set with its complement is the universal
set.</LI></OL>
<P class=pLevel4><SPAN class=Normal><SPAN
style="MARGIN-LEFT: 20px"></SPAN><I>A</I> <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/union.png">
<I>A</I>’ = <I>U</I></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<OL class="oLevel3 ListParagraph ListStyleOL">
<LI value=5>(<I>A</I> <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/intersect.png">
<I>B</I>)’ = <I>A</I>’ <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/union.png">
<I>B’</I></LI></OL>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<OL class="oLevel3 ListParagraph ListStyleOL">
<LI value=6>(<I>A</I> <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/union.png">
<I>B</I>)’ = <I>A</I>’ <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/intersect.png">
<I>B</I>’</LI></OL>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal>The last two properties are called <B>De
Morgan’s laws</B>.</SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal><B>EXPLANATION:</B></SPAN></P>
<P class=pLevel4><SPAN class=Normal>We can prove De Morgan’s laws using Venn
Diagrams.</SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pImage><SPAN><IMG alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/020_009.png"></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal><B>Law 1</B>: (<I>A</I> <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/union.png">
<I>B</I>)’ = <I>A</I>’ <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/intersect.png">
<I>B</I>’</SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal>In the above diagram,</SPAN></P>
<P class=pLevel4><SPAN class=Normal>(<I>A</I> <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/union.png">
<I>B</I>) = <I>a</I> + <I>b</I> + <I>c</I></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal>Hence,</SPAN></P>
<P class=pLevel4><SPAN class=Normal>(<I>A</I> <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/union.png">
<I>B</I>)’ = <I>U</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/minus.png">
(<I>A</I> <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/union.png">
<I>B</I>) = <I>a</I> + <I>b</I> + <I>c </I>+ <I>d</I> <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/minus.png">
(<I>a</I> + <I>b</I> + <I>c</I>) = <I>d</I></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/therefore.png">
(<I>A</I> <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/union.png">
<I>B</I>)’ = <I>d</I></SPAN></P>
<P class=pLevel4><SPAN class=Normal><I>A</I>’ = <I>c </I>+ <I>d</I></SPAN></P>
<P class=pLevel4><SPAN class=Normal><I>B</I>’ = <I>a</I> + <I>d</I></SPAN></P>
<P class=pLevel4><SPAN class=Normal><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/therefore.png">
<I>A</I>’ <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/intersect.png">
<I>B</I>’ = <I>d</I></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/therefore.png">
(<I>A</I> <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/union.png">
<I>B</I>)’ = <I>A</I>’ <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/intersect.png">
<I>B</I>’</SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal><B>Law 2</B>:<B> </B>(<I>A</I> <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/intersect.png">
<I>B</I>)’ = <I>A</I>’ <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/union.png">
<I>B</I>’</SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal>(<I>A</I> <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/intersect.png">
<I>B</I>) = <I>b</I></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/therefore.png">
(<I>A</I> <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/intersect.png">
<I>B</I>)’ = <I>a</I> + <I>c </I>+ <I>d</I></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal><I>A</I>’ = <I>c</I> + <I>d</I></SPAN></P>
<P class=pLevel4><SPAN class=Normal><I>B</I>’ = <I>a</I> + <I>d</I></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal><I>A</I>’ <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/union.png">
<I>B’</I> = <I>a</I> + <I>c</I> + <I>d</I></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/therefore.png">
(<I>A</I> <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/intersect.png">
<I>B</I>)’ = <I>A</I>’ <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/union.png">
<I>B</I>’</SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<OL class="oLevel3 ListParagraph ListStyleOL">
<LI value=7>Associative Property</LI></OL>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal><SPAN
style="MARGIN-LEFT: 17px"></SPAN>(<I>A</I> <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/union.png">
<I>B</I>) <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/union.png">
<I>C</I> = <I>A</I> <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/union.png">
(<I>B</I> <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/union.png">
<I>C</I>)</SPAN></P>
<P class=pLevel4><SPAN class=Normal><SPAN
style="MARGIN-LEFT: 17px"></SPAN>(<I>A</I> <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/intersect.png">
<I>B</I>) <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/intersect.png">
<I>C</I> = <I>A</I> <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/intersect.png">
(<I>B</I> <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/intersect.png">
<I>C</I>)</SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<OL class="oLevel3 ListParagraph ListStyleOL">
<LI value=8>Distributive Property</LI></OL>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal><SPAN
style="MARGIN-LEFT: 17px"></SPAN><I>A</I> <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/union.png">
(<I>B</I> <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/intersect.png">
<I>C</I>) = (<I>A</I> <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/union.png">
<I>B</I>) <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/intersect.png">
(<I>A</I> <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/union.png">
<I>C</I>)</SPAN></P>
<P class=pLevel4><SPAN class=Normal><SPAN
style="MARGIN-LEFT: 17px"></SPAN><I>A</I> <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/intersect.png">
(<I>B</I> <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/union.png">
<I>C</I>) = (<I>A</I> <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/intersect.png">
<I>B</I>) <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/union.png">
(<I>A</I> <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/intersect.png">
<I>C</I>)</SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P></DIV>
<DIV id=020_VIII>
<TABLE id=020_VIII_bookmark class=tableh1 border=0 cellSpacing=0
cellPadding=0><TBODY>
<TR>
<TD
style="PADDING-BOTTOM: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 100%; PADDING-RIGHT: 0px; PADDING-TOP: 0px">
<OL class="oLevel1 Heading1">
<LI value=8><A id=VIII>CARTESIAN PRODUCT</A></LI></OL></TD></TR></TBODY></TABLE>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>An <B>ordered</B> <B>pair</B> (<I>x</I>,
<I>y</I>) represents two objects <I>x</I> and <I>y</I> taken in that order. Thus
(<I>x</I>, <I>y</I>) is different from (<I>y</I>, <I>x</I>).</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>The <B>Cartesian</B> <B>product</B> of any
two sets <I>A</I> and <I>B</I> is given by</SPAN></P>
<P class=pLevel2><SPAN class=Normal><I>A</I> <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/multiplication.png">
<I>B</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/equal_equal.png">
{(<I>x</I>, <I>y</I>) | <I>x</I> ( <I>A</I> and <I>y</I> ( <I>B</I>}</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>For example, consider two sets <I>A</I> and
<I>B</I> as follows:</SPAN></P>
<P class=pLevel2><SPAN class=Normal><I>A</I> <IMG style="VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/equal_equal.png">
{1, 2, 3}</SPAN></P>
<P class=pLevel2><SPAN class=Normal><I>B</I> <IMG style="VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/equal_equal.png">
{a, b, c}</SPAN></P>
<P class=pLevel2><SPAN class=Normal>Then the Cartesian product of these sets is
given by</SPAN></P>
<P class=pLevel2><SPAN class=Normal><I>A</I> <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/multiplication.png">
<I>B</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/equal_equal.png">
{(1, a), (1, b), (1, c), (2, a), (2, b), (2, c), (3, a), (3, b), (3,
c)}</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>Note that <I>A</I> <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/multiplication.png">
<I>B</I> is different from <I>B</I> <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/multiplication.png">
<I>A</I></SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P></DIV>
<DIV class="divTextBox divColorExampleHeading divLevel2">
<DIV id=020_solvedex_7>
<TABLE style="WIDTH: 100%" id=020_solvedex_7_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_7>Example
7:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>Find the Cartesian product <I>A</I> <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/multiplication.png">
<I>B</I> where</SPAN></P>
<P><SPAN class=Normal><I>A</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/equal_equal.png">
{m, n, o} and <I>B</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/equal_equal.png">
{13, 14, 15, 16}.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal><I>A</I> <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/multiplication.png">
<I>B</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/equal_equal.png">
{(m, 13), (m, 14), (m, 15), (m, 16), (n, 13), (n, 14), (n, 15), (n, 16),
(o, 13), (o, 14), (o, 15), (o,
16)}</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel2">
<DIV id=020_solvedex_8>
<TABLE style="WIDTH: 100%" id=020_solvedex_8_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_8>Example
8:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>A teacher tested 80 students of class V for their
abilities in four areas - sports, academics, music and art. She found that
all her students had at least one of these four abilities strongly
developed. 15, 24 and 45 students respectively were good at music, art and
sports.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<OL class="ListParagraph ListStyleOL2">
<LI value=1>There were 3 students who excelled in all the four
areas.</LI></OL>
<OL class="ListParagraph ListStyleOL2">
<LI value=2>There were four students who showed a musical ability as
well as an artistic ability. Only one of these did not show any other
ability.</LI></OL>
<OL class="ListParagraph ListStyleOL2">
<LI value=3>8 students were good academically and also showed a musical
ability, but were not good at art or sports.</LI></OL>
<OL class="ListParagraph ListStyleOL2">
<LI value=4>Out of the 30 students who were good at academics, 9 were
also good at sports.</LI></OL>
<OL class="ListParagraph ListStyleOL2">
<LI value=5>3 students excelled in academics and were good artists, but
were not good at sports or music.</LI></OL>
<OL class="ListParagraph ListStyleOL2">
<LI value=6>The number of students who were good at only art was exactly
one more than the number of students who were good only
academically.</LI></OL>
<OL class="ListParagraph ListStyleOL2">
<LI value=7>The number of students good at only sports was thrice the
number of students good at only academics.</LI></OL>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>How many students were good only in
academics?</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>What was the difference between the number of
students who were not good at music but good at the other three activities
and the number of students who were good only at sports and
music?</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>Here we need a four level Venn diagram.</SPAN></P>
<P><SPAN class=Normal>From statements (i) and (ii), there was 1 student
who showed only musical and artistic abilities.</SPAN></P>
<P><SPAN class=Normal>From statements (iii), (iv) and (v), (30 – 9 – 3 –
8) = 10 students were good only in academics.</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/therefore.png">
From statement (vi), 11 students were good only at art and from statement
(vii), 30 students were good only at sports.</SPAN></P>
<P><SPAN class=Normal>All this information can be presented in the
following Venn diagram.</SPAN></P>
<P><SPAN class=pImage><IMG alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/020_010.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>From (iv), <I>x</I> + <I>y</I> + <I>z</I> + 3 =
9</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/therefore.png">
<I>x</I> + <I>y</I> + <I>z</I> = 6</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/therefore.png">
<I>m</I> + <I>n</I> = 45 – 30 – 9 = 6</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Also, <I>m</I> + <I>y</I> + <I>p</I> = 3 and
<I>n</I> + <I>z</I> = 6</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/therefore.png">
<I>n</I> +<I> z</I> = <I>m</I> + <I>n</I></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/therefore.png">
<I>z</I> = <I>m</I></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/therefore.png">
<I>z</I> – <I>m</I> = 0</SPAN></P>
<P><SPAN class=Normal>Hence, 10 students were good only at
academics.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><I>z</I> represents the number of students who were
not good at music but good at the other three activities and <I>m</I>
represents the number of students who were good only at sports and music.
The difference between <I>z</I> and <I>m</I> is
zero.</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel2">
<DIV id=020_solvedex_9>
<TABLE style="WIDTH: 100%" id=020_solvedex_9_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_9>Example
9:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>ABC coaching classes conducted a study in a locality
to find out the number of students of class eighth who took tuitions for
English, Maths and Science. 36 students took Maths tuitions, out of which
20 did not take tuitions for any other subject. 27 students took English
tuitions. The number of students who took only English tuitions and the
number of students who took English as well as Science tuitions but not
Maths tuitions was equal. Also, an equal number of students took tuitions
for Maths as well as English and Maths as well as Science.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Find the maximum possible number of students who
took only English tuitions, the maximum possible number of students who
took tuitions for all the three subjects and the minimum possible number
of students who took tuitions for Math and English but not
Science.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=pImage><IMG alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/020_011.png"></SPAN></P>
<P><SPAN class=Normal>From the figure,</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>2<I>y</I> + <I>z</I> + 20 = 36</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/therefore.png">
2<I>y</I> + <I>z</I> = 16 <SPAN
style="MARGIN-LEFT: 30px"></SPAN>…(i)</SPAN></P>
<P><SPAN class=Normal>2<I>x</I> + <I>y</I> + <I>z</I> = 27 <SPAN
style="MARGIN-LEFT: 15px"></SPAN>…(ii)</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/therefore.png">
27 – 2<I>x</I> – <I>y</I> = 16 – 2<I>y</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/therefore.png"></I>
2<I>x</I> – <I>y </I>= 11</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/therefore.png">
<I>y</I> is an odd number.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/therefore.png">
Minimum possible value of <I>y</I> = 1</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/therefore.png">
Minimum possible value of <I>x</I> = 6</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>When <I>y</I> = 3, <I>x</I> = 7</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>When <I>y</I> = 5, <I>x</I> = 8</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>When <I>y</I> = 7, <I>x</I> = 9</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>When <I>y</I> = 9, <I>x</I> = 10</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>But from (ii), 2<I>x</I> + <I>y</I> <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/less_equal.png">
27</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/therefore.png">
Maximum value of <I>x</I> = 9</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>From (ii), <I>z</I> is maximum when <I>x</I> and
<I>y</I> are minimum.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/therefore.png">
Maximum value of <I>z</I> = 27 – <I>y</I><SUB>min</SUB> –
2<I>x</I><SUB>min</SPAN></P>
<P><SPAN class=Normal></SUB><SPAN style="MARGIN-LEFT: 65px"></SPAN><SPAN
style="MARGIN-LEFT: 90px"></SPAN>= 27 – 1 – 12 = 14</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/therefore.png">
<I>z</I><SUB>max</SUB> = 14, <I>x</I><SUB>max</SUB> = 9 and
<I>y</I><SUB>min</SUB> = 1</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>Number theory questions often appear
disguised as Set Theory questions. Also, number theory questions can be solved
easily using concepts in set theory.</SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel2">
<DIV id=020_solvedex_10>
<TABLE style="WIDTH: 100%" id=020_solvedex_10_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_10>Example
10:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>Let <I>S</I> be the set of prime numbers greater
than or equal to 2 and less than 100. Multiply all elements of S. With how
many consecutive zeros will the product end?</SPAN></P>
<P><SPAN class=Normal><B>[CAT 2000]</B></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>(1) 1<SPAN style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN>(2) 4</SPAN></P>
<P><SPAN class=Normal>(3) 5<SPAN style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN>(4) 10</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>There is only one<B> </B>even<B> </B>prime number
and one prime number ending with 5 (i.e 5 itself) in set
<I>S</I>.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>When all elements of set <I>S</I> are multiplied,
there will be only 1 zero at the end of the product.</SPAN></P>
<P><SPAN class=Normal>Hence, <B>option
1.</B></SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel2">
<DIV id=020_solvedex_11>
<TABLE style="WIDTH: 100%" id=020_solvedex_11_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_11>Example
11:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>How many numbers lying in the range 5000 to 10000
are divisible by 7, 11 or 13?</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>Let <I>A</I>, <I>B</I> and<I> C</I> be the set of
numbers divisible by 7, 11 and 13 respectively.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Lowest number in <I>A</I> = 5005. Highest number in
<I>A</I> = 9996</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/020_012.png">
= 714</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Lowest number in <I>B</I> = 5005. Highest number in
<I>B</I> = 9999</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/therefore.png">
Number of Elements in <I>B</I> = 455</SPAN></P>
<P><SPAN class=Normal>Lowest number in <I>C</I> = 5005. Highest number in
<I>C</I> = 9997</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/therefore.png">
Number of Elements in <I>C</I> = 385</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Similarly,</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Number of elements divisible by 7 and 11</SPAN></P>
<P><SPAN class=Normal>= Number of elements in the set <I>A</I> <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/intersect.png">
<I>B</I> = 65</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Number of elements in the set <I>A</I> <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/intersect.png">
<I>C</I> = 55</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Number of elements in the set <I>B</I> <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/intersect.png">
<I>C</I> = 35</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Number of elements in the set <I>A</I> <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/intersect.png">
<I>B </I><IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/intersect.png">
<I>C</I> = 5</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/therefore.png">
Number of elements in the set <I>A</I> <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/union.png">
<I>B</I> <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/union.png">
<I>C</I></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><I>n</I>(<I>A</I> <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/union.png">
<I>B</I> <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/union.png">
<I>C</I> ) = <I>n</I>(<I>A</I>) + <I>n</I>(<I>B</I>) + <I>n</I>(<I>C</I>)
<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/minus.png">
<I>n</I>(<I>A</I> <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/intersect.png">
<I>B</I>) <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/minus.png">
<I>n</I>(<I>A</I> <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/intersect.png">
<I>C</I>) <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/minus.png">
<I>n</I>(<I>B</I> <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/intersect.png">
<I>C</I>) + <I>n</I>(<I>A</I> <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/intersect.png">
<I>B</I> <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/intersect.png">
<I>C</I>)</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/therefore.png">
<I>n</I>(<I>A</I> <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/union.png">
<I>B</I> <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/union.png">
<I>C</I> ) = 714 + 455 + 385 – 65 – 55 – 35 + 5 =
1404</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel2">
<DIV id=020_solvedex_12>
<TABLE style="WIDTH: 100%" id=020_solvedex_12_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_12>Example
12:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>A survey on a sample of 25 new cars being sold at a
local auto dealer was conducted to see which of the three popular options-
air conditioning, radio and power windows- were already installed. The
survey found</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<UL class=ListParagraph>
<LI value=1>15 had air conditioning,</LI></UL>
<UL class=ListParagraph>
<LI value=2>2 had air conditioning and power windows but no
radios,</LI></UL>
<UL class=ListParagraph>
<LI value=3>12 had radio,</LI></UL>
<UL class=ListParagraph>
<LI value=4>6 had air conditioning and radio, but no power
windows,</LI></UL>
<UL class=ListParagraph>
<LI value=5>11 had power windows,</LI></UL>
<UL class=ListParagraph>
<LI value=6>4 had radio and power windows and</LI></UL>
<UL class=ListParagraph>
<LI value=7>3 had all three options.</LI></UL>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>What is the number of cars that had none of the
options?</SPAN></P>
<P><SPAN class=Normal><B>[CAT 2003 Re-Test]</B></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>(1) 4<SPAN style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN>(2) 3</SPAN></P>
<P><SPAN class=Normal>(3) 1<SPAN style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN>(4) 2</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=pImage><IMG alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/020_013.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Here, AC – Air Conditioning, R – Radio
and</SPAN></P>
<P><SPAN class=Normal>PW – Power Windows</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>From the given conditions, we have the above Venn
diagram.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>When we add up all the values in the Venn diagram,
we get (15 + 5 + 1 + 2) = 23 cars.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/therefore.png">
2 (i.e. 25 <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/minus.png">
23) cars don’t have any of the three options.</SPAN></P>
<P><SPAN class=Normal>Hence, <B>option
4</B>.</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel2">
<DIV id=020_solvedex_13>
<TABLE style="WIDTH: 100%" id=020_solvedex_13_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_13>Example
13:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>Three of a popular actor’s films, <I>A</I>, <I>B</I>
and <I>C</I>, were nominated for the best film award. 60 persons were
asked which film deserved the award the most.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>16 people thought only <I>C</I> deserved
it.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>The number of people who thought that all the three
films equally deserved the award was twice the number of people who
thought that <I>B</I> and <I>C</I> deserved the award but <I>A</I> did
not.</SPAN></P>
<P><SPAN class=Normal>The number of people who thought that <I>B</I>
deserved the award was equal to twice the number of people who thought all
the three equally deserved it.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>The number of people who thought that <I>A</I> and
<I>B</I>, but not <I>C</I> deserved the award was equal to the number of
people who thought that only <I>A</I> or <I>A</I> and <I>C</I> but not<I>
B</I> deserved it.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Find the maximum number of people who thought that
all the three films deserved the award?</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>Let <I>x</I> be the number of people who thought
that <I>B</I> and <I>C</I> deserved the award but <I>A</I> did
not.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Then the number of people who thought all the three
equally deserved it is 2<I>x</I>.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Let <I>m</I> be the number of people who thought
that <I>A</I> and <I>B</I>, but not <I>C</I> deserved the
award.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>The number of people who thought that <I>B</I>
deserved the award = 4<I>x</I></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>The number of people who thought that only <I>B</I>
deserved the award = 4<I>x</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/minus.png">
(<I>m</I> + 2<I>x</I> + <I>x</I>) = <I>x</I> <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/minus.png">
<I>m</I></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>From the given data, we can draw the following Venn
diagram.</SPAN></P>
<P><SPAN class=pImage><IMG alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/020_014.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Thus,</SPAN></P>
<P><SPAN class=Normal>2<I>m</I> + <I>x</I> – <I>m</I> + 2<I>x</I> +
<I>x</I> + 16 = 60</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/therefore.png">
<I>m</I> + 4<I>x</I> = 44</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/therefore.png">
<I>x</I> is maximum when <I>m</I> = 0</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/therefore.png">
<I>x</I> = 11</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/therefore.png">
2<I>x</I> = 22</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/therefore.png">
The maximum number of people who thought that all the three films deserved
the award = 22.</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorNote divLevel2">
<P><SPAN class=Note>REMEMBER:</SPAN></P>
<UL class=ListParagraph>
<LI class=ListStyleUL5>For any two sets <I>A</I> and <I>B</I>,</LI></UL>
<P><SPAN class=ListParagraph></SPAN></P>
<UL><SPAN class=Normal><I>n</I>(<I>A</I> <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/union.png">
<I>B</I>) = <I>n</I>(<I>A</I>) + <I>n</I>(<I>B</I>) – <I>n</I>(<I>A</I> <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/intersect.png">
<I>B</I>)</SPAN></UL>
<P><SPAN class=ListParagraph></SPAN></P>
<UL class=ListParagraph>
<LI class=ListStyleUL5>For any three sets <I>A</I>, <I>B</I> and
<I>C</I>,</LI></UL>
<UL><SPAN class=Normal><I>n</I>(<I>A</I> <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/union.png">
<I>B</I> <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/union.png"><I>
C</I>) = <I>n</I>(<I>A</I>) + <I>n</I>(<I>B</I>) + <I>n</I>(<I>C</I>) <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/minus.png">
<I>n</I>(<I>A</I> <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/intersect.png">
<I>B</I>) – <I>n</I>(<I>A</I> <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/intersect.png">
<I>C</I>) <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/minus.png">
<I>n</I>(<I>B</I> <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/intersect.png">
<I>C</I>) + <I>n</I>(<I>A</I> <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/intersect.png">
<I>B</I> <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/intersect.png">
<I>C</I>)</SPAN></UL>
<P><SPAN class=ListParagraph></SPAN></P>
<UL class=ListParagraph>
<LI class=ListStyleUL5><B>De</B> <B>Morgan’s</B> <B>Laws</B></LI></UL>
<UL><SPAN class=Normal>(<I>A</I> <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/intersect.png">
<I>B</I>)’ = <I>A</I>’ <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/union.png">
<I>B’</SPAN></UL>
<UL><SPAN class=ListParagraph>(A</I> <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/union.png">
<I>B</I>)’ = <I>A</I>’ <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/d2a103f3-d594-4e8f-a2b5-f5856136759e/020_images/intersect.png">
<I>B</I>’</SPAN></UL></DIV></BODY></HTML>
guruhttp://www.blogger.com/profile/01591375209227117455noreply@blogger.comtag:blogger.com,1999:blog-7850540640045081515.post-32375473603476349802015-03-03T05:44:00.001-08:002019-09-14T18:55:04.006-07:00p anc <div dir="ltr" style="text-align: left;" trbidi="on">
<br />
<html xmlns="http://www.w3.org/1999/xhtml"><head><title>Probability Distributions and Statistics</title>
<script src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/CrossLink.js" type="text/javascript"></script>
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</head>
<body onload="lesson_Function_1(); lesson_Function_2()">
<div class="oTOCTitle">
<span class="oTOCTitleFont"><a class="header" href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202.htm#Title">Contents</a></span></div>
<ol class="oTOCLevel1">
<li class="oTocLiClass"><a class="header" href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202.htm#I">INTRODUCTION</a>
</li>
<li class="oTocLiClass"><a class="header" href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202.htm#II">BASIC
STATISTICS</a>
<ol class="oTOCLevel2">
<li><a class="header" href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202.htm#II_A">STANDARD
DEVIATION</a></li>
<li><a class="header" href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202.htm#II_B">VARIANCE</a></li>
</ol>
</li>
<li class="oTocLiClass"><a class="header" href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202.htm#III">RANDOM
VARIABLE </a>
<ol class="oTOCLevel2">
<li><a class="header" href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202.htm#III_A">DEFINITION
</a></li>
<li><a class="header" href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202.htm#III_B">TYPES
OF RANDOM VARIABLES</a></li>
<li><a class="header" href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202.htm#III_C">PROBABILITY
MASS FUNCTION </a></li>
<li><a class="header" href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202.htm#III_D">PROBABILITY
DENSITY FUNCTION </a></li>
<li><a class="header" href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202.htm#III_E">CUMULATIVE
DISTRIBUTION FUNCTION </a></li>
<li><a class="header" href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202.htm#III_F">EXPECTATION
</a></li>
<li><a class="header" href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202.htm#III_G">VARIANCE</a></li>
<li><a class="header" href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202.htm#III_H">STANDARD
DEVIATION </a></li>
</ol>
</li>
<li class="oTocLiClass"><a class="header" href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202.htm#IV">DISCRETE
PROBABILITY DISTRIBUTIONS</a>
<ol class="oTOCLevel2">
<li><a class="header" href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202.htm#IV_A">BINOMIAL
DISTRIBUTION</a></li>
<li><a class="header" href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202.htm#IV_B">POISSON
DISTRIBUTION</a></li>
<li><a class="header" href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202.htm#IV_C">HYPERGEOMETRIC
DISTRIBUTION</a></li>
</ol>
</li>
<li class="oTocLiClass"><a class="header" href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202.htm#V">CONTINUOUS
PROBABILITY DISTRIBUTIONS</a>
<ol class="oTOCLevel2">
<li><a class="header" href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202.htm#V_A">UNIFORM
DISTRIBUTION </a></li>
<li><a class="header" href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202.htm#V_B">NORMAL
DISTRIBUTION </a></li>
</ol>
</li>
<li class="oTocLiClass"><a class="header" href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202.htm#VI">DETERMINING
DISTRIBUTIONS</a> </li>
<li class="oTocLiClass"><a class="header" href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202.htm#VII">SUMMARY
OF DISTRIBUTIONS</a> </li>
<li class="oTocLiClass"><a class="header" href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202.htm#VIII">COEFFICIENT
OF CORRELATION AND LINEARITY</a>
<ol class="oTOCLevel2">
<li><a class="header" href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202.htm#VIII_A">DEFINITION</a></li>
<li><a class="header" href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202.htm#VIII_B">INTERPRETATION
</a></li>
</ol>
</li>
<li class="oTocLiClass"><a class="header" href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202.htm#IX">COEFFICIENT
OF VARIATION</a>
<ol class="oTOCLevel2">
<li><a class="header" href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202.htm#IX_A">DEFINITION</a></li>
<li><a class="header" href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202.htm#IX_B">APPLICATIONS
</a></li>
</ol>
</li>
</ol>
<div class="pTitle" id="Title">
<span class="Title">Probability Distributions and
Statistics</span></div>
<hr />
<div class="pLevel2">
<span class="Normal"></span>
<div id="202_I">
<table border="0" cellpadding="0" cellspacing="0" class="tableh1" id="202_I_bookmark">
<tbody>
<tr>
<td style="margin: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; width: 100%;">
<ol class="oLevel1 Heading1">
<li value="1"><a href="https://www.blogger.com/null" id="I">INTRODUCTION</a></li>
</ol>
</td></tr>
</tbody></table>
<div class="pLevel2">
<span class="Normal"></span></div>
<div class="pLevel2">
<span class="Normal">Statistics is a science that deals with the
collection, classification, analysis, and interpretation of data or numerical
facts.</span></div>
<div class="pLevel2">
<span class="Normal">Statistics is extensively used in business
for market research and risk analysis. Probability forms an important part of
statistics. We can use probability to forecast results based on past
data.</span></div>
<div class="pLevel2">
<span class="Normal">Questions on probability distributions and
statistics appear in exams like FMS, IIFT and JMET. Every year there are at
least two or three questions from this topic, thereby making it an important
one. It is also an important part of the MBA curriculum.</span></div>
</div>
<div id="202_II">
<table border="0" cellpadding="0" cellspacing="0" class="tableh1" id="202_II_bookmark">
<tbody>
<tr>
<td style="margin: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; width: 100%;">
<ol class="oLevel1 Heading1">
<li value="2"><a href="https://www.blogger.com/null" id="II">BASIC STATISTICS</a></li>
</ol>
</td></tr>
</tbody></table>
<div id="202_II_A">
<table border="0" cellpadding="0" cellspacing="0" class="tableh1" id="202_II_A_bookmark"><tbody>
<tr>
<td style="margin: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; width: 100%;">
<ol class="oLevel2 Heading2">
<li value="1"><a href="https://www.blogger.com/null" id="II_A">STANDARD
DEVIATION</a></li>
</ol>
</td></tr>
</tbody></table>
<div class="pLevel3">
<span class="Normal">The standard deviation is a “measure of
dispersion” from the mean of a population, a data set, or a probability
distribution i.e. it measures how the values are scattered from the
mean.</span></div>
<div class="pLevel3">
<span class="Normal">For example, if the average SAT score of
students in Stanford University is 1400, with a standard deviation of 20. This
means that most students have scores within 20 marks of the mean (1380 – 1420),
while almost all students have scores within 40 marks of the mean (1360 – 1440).
If the standard deviation is zero, then the score of all students would be
exactly 1400.</span></div>
<div class="pLevel3">
<span class="Normal">For <i>n</i> values, <i>X</i><sub>1</sub>,
<i>X</i><sub>2</sub>, <i>X</i><sub>3</sub>, …, <i>X<sub>n</sub></i> the standard
deviation, <i><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_373.png" style="vertical-align: middle;" /></i>,
can be calculated as:</span></div>
<div class="pLevel3">
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_001.png" style="vertical-align: middle;" />
or <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_100.png" style="vertical-align: middle;" /></span></div>
<div class="pLevel3">
<span class="Normal"></span><span class="Normal"></span></div>
<div class="pLevel3">
<span class="Normal"></span></div>
<div class="pLevel3">
<span class="Normal">Where <i>μ</i> is the mean of the sample of
size <i>n</i> and is defined as:</span></div>
<div class="pLevel3">
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_002.png" style="vertical-align: middle;" /></span></div>
<div class="pLevel3">
<span class="Normal"></span></div>
<div class="divTextBox divColorExampleHeading divLevel3">
<div id="202_solvedex_1">
<table border="0" cellpadding="0" cellspacing="0" id="202_solvedex_1_bookmark" style="width: 100%;">
<tbody>
<tr>
<td style="background-color: transparent; width: 100%;">
<span class="ExampleHeading"><a href="https://www.blogger.com/null" id="ex_1">Example
1:</a></span></td></tr>
</tbody></table>
<table border="0" cellpadding="0" cellspacing="0">
<tbody>
<tr>
<td style="background-color: transparent; width: 100%;">
<span class="Normal">Recently, smoking at public places was declared as
an offence. Delhi Police has started imposing a penalty against smoking in
public and has eight raid teams in place. In a surprise check, the raid
team caught 40 people smoking in the Connaught Place area of Delhi. The
standard deviation and sum of squares of the amount found in their pockets
were Rs. 10 and Rs. 40.000, respectively. If the total fine imposed on
these offenders is equal to the total amount found in their pockets and if
the fine imposed is uniform, what is the amount that each offender will
have to pay as fine?</span><br />
<span class="Normal"><b>[FMS 2009]</b></span><br />
<span class="Normal"></span><br />
<span class="Normal">(1) Rs. 90<span style="margin-left: 25px;"></span><span style="margin-left: 25px;"></span>(2) Rs. 60</span><br />
<span class="Normal">(3) Rs. 30<span style="margin-left: 25px;"></span><span style="margin-left: 25px;"></span>(4) Rs. 15</span><br />
<span class="Normal"></span><br />
<span class="SolutionHeading">Solution:</span><br />
<span class="Normal">The standard deviation for <i>n</i> items having
mean value (i.e. arithmetic mean)<img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/204_337.png" style="vertical-align: middle;" />
is given as:</span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_003.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"></span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_004.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"></span><br />
<span class="Normal">We have also been given that each offender was fined
an equal amount.</span><br />
<span class="Normal">Let each person be fined Rs. <i>y</i>.</span><br />
<span class="Normal">Then, the total fine collected from all 40 offenders
= 40<i>y</i></span><br />
<span class="Normal">Which is given to be equal to the sum of amounts
collected from all the offender’s pockets.</span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_005.png" style="vertical-align: middle;" />
<img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_006.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"></span><br />
<span class="Normal"></span><br />
<span class="Normal">Hence, <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/204_337.png" style="vertical-align: middle;" />
= <i>y</i></span><br />
<span class="Normal">Substituting all these values in
equation (i), we get , </span>
<br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_007.png" style="vertical-align: middle;" /></span><br />
<span class="Normal">Squaring both sides , </span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_008.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"><!--<img style="vertical-align:middle" src="202_images/202_010.png" alt="" />--><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/therefore.png" style="margin-bottom: 0px; padding-bottom: 1px;" />
<i>y</i>² = 1000 – 100 = 900</span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/therefore.png" style="margin-bottom: 0px; padding-bottom: 1px;" />
<i>y</i> = 30</span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/therefore.png" style="margin-bottom: 0px; padding-bottom: 1px;" />
Every person has to pay a fine of Rs. 30.</span><br />
<span class="Normal">Hence, <b>option
3</b>.</span></td></tr>
</tbody></table>
</div>
</div>
<!--<p class="pLevel2">
<span class="Normal"></span></p>
<div class="divTextBox divColorNormal divLevel3">
<p>
<span class="Normal"></span></p>
</div>
-->
<div class="pLevel2">
<span class="Normal"></span></div>
<div class="divTextBox divColorExampleHeading divLevel3">
<div id="202_solvedex_2">
<table border="0" cellpadding="0" cellspacing="0" id="202_solvedex_2_bookmark" style="width: 100%;">
<tbody>
<tr>
<td style="background-color: transparent; width: 100%;">
<span class="ExampleHeading"><a href="https://www.blogger.com/null" id="ex_2">Example
2:</a></span></td></tr>
</tbody></table>
<table border="0" cellpadding="0" cellspacing="0">
<tbody>
<tr>
<td style="background-color: transparent; width: 100%;">
<span class="Normal">The mean salary in ICM Ltd. was Rs. 1,500 and the
standard deviation was Rs. 400. A year later each employee got a Rs. 100
raise. After another year each employee’s salary (including the above
mentioned raise) was increased by 20%. The standard deviation of the
current salary is:</span><br />
<span class="Normal"><b>[IIFT 2008]</b></span><br />
<span class="Normal"></span><br />
<span class="Normal">(1) 460<span style="margin-left: 25px;"></span><span style="margin-left: 25px;"></span>(2) 480</span><br />
<span class="Normal">(3) 580<span style="margin-left: 25px;"></span><span style="margin-left: 25px;"></span>(4) None of the above</span><br />
<span class="Normal"></span><br />
<span class="SolutionHeading">Solution:</span><br />
<span class="Normal">The mean salary was Rs. 1,500 and standard deviation
(SD) was Rs. 400. Since after 1 year each employee got a raise of Rs. 100,
it would mean that SD would still remain same. </span><br />
<span class="Normal">Now in the next year, since the increase in salary
was 20% for all of the employees, the SD would have gone up by 20%. (<img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/symb23.png" style="margin-bottom: 0px; padding-bottom: 1px;" />
The increase in salary will depend on the salary of the employee and might
not be uniform)</span><br />
<span class="Normal">Standard deviation after 20% raise = 400 + 20% of
400 = 400 + 80 = Rs. 480</span><br />
<span class="Normal"></span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/therefore.png" style="margin-bottom: 0px; padding-bottom: 1px;" />
The new standard deviation would be 480.</span><br />
<span class="Normal">Hence, <b>option 2</b>.<span style="margin-left: 25px;"></span></span></td></tr>
</tbody></table>
</div>
</div>
</div>
<div id="202_II_B">
<table border="0" cellpadding="0" cellspacing="0" class="tableh1" id="202_II_B_bookmark"><tbody>
<tr>
<td style="margin: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; width: 100%;">
<ol class="oLevel2 Heading2">
<li value="2"><a href="https://www.blogger.com/null" id="II_B">VARIANCE</a></li>
</ol>
</td></tr>
</tbody></table>
<div class="pLevel3">
<span class="Normal">The variance of a random variable or a
sample tells us how widely spread the values of the random variable or sample
are likely to be from the mean or the average. The larger the variance, the more
scattered the observations from the average.</span></div>
<div class="pLevel3">
<span class="Normal">Variance is the square of the standard
deviation and hence, is always a non negative number.</span></div>
<div class="pLevel3">
<span class="Normal">It is denoted by <i><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_373.png" style="vertical-align: middle;" /></i><sup>2</sup>.</span></div>
<div class="pLevel3">
<span class="Normal">The variance is <i><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_373.png" style="vertical-align: middle;" /></i><sup>2</sup>
means, maximum values of the random variable or sample lie in the
interval</span></div>
<div class="pLevel3">
<span class="Normal">[<img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/204_337.png" style="vertical-align: middle;" />
<img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/minus.png" style="vertical-align: middle;" />
<i><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_373.png" style="vertical-align: middle;" /></i><sup>2</sup>,
<img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/204_337.png" style="vertical-align: middle;" />
+ <i><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_373.png" style="vertical-align: middle;" /></i>
<sup>2</sup>]</span></div>
<div class="pLevel3">
<span class="Normal">For example,</span></div>
<div class="pLevel3">
<span class="Normal">For numbers 1, 2, 3, 4 and 5 the mean is
3.</span></div>
<div class="pLevel3">
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_009.png" style="vertical-align: middle;" />
</span></div>
<div class="pLevel3">
<span class="Normal"></span><span class="Normal"><span style="margin-left: 15px;"></span><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_011.png" style="vertical-align: middle;" /></span></div>
<div class="pLevel3">
<span class="Normal"></span></div>
<!--<p class="pLevel3">
<span class="Normal"> <span style="margin-left:45px"></span><img style="vertical-align:middle" src="202_images/202_012.png" alt="" /></span></p>
-->
<div class="pLevel3">
<span class="Normal"></span></div>
<div class="divTextBox divColorExampleHeading divLevel3">
<div id="202_solvedex_3">
<table border="0" cellpadding="0" cellspacing="0" id="202_solvedex_3_bookmark" style="width: 100%;">
<tbody>
<tr>
<td style="background-color: transparent; width: 100%;">
<span class="ExampleHeading"><a href="https://www.blogger.com/null" id="ex_3">Example
3:</a></span></td></tr>
</tbody></table>
<table border="0" cellpadding="0" cellspacing="0">
<tbody>
<tr>
<td style="background-color: transparent; width: 100%;">
<span class="Normal">The mean of the numbers <i>a</i>, <i>b</i>, 8, 5, 10
is 6 and the variance is 6.80. Then which one of the following gives
possible values of <i>a</i> and <i>b</i>?</span><br />
<span class="Normal"><b>[SNAP 2009]</b></span><br />
<span class="Normal">(1) <i>a</i> = 0, <i>b</i> = 7<span style="margin-left: 25px;"></span>(2) <i>a </i>= 5, <i>b</i> = 2</span><br />
<span class="Normal">(3) <i>a</i> = 3, <i>b</i> = 4<span style="margin-left: 25px;"></span>(4) <i>a</i> = 2, <i>b</i> = 4</span><br />
<span class="Normal"></span><br />
<span class="SolutionHeading">Solution:</span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_012.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"></span><br />
<span class="Normal">Mean is given as 6.</span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_013.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"></span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/therefore.png" style="margin-bottom: 0px; padding-bottom: 1px;" />
<i>a</i> + <i>b</i> = 30 – 23 = 7</span><br />
<span class="Normal">So, option 4 can be eliminated.</span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_014.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_015.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"></span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_016.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"></span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/therefore.png" style="margin-bottom: 0px; padding-bottom: 1px;" />
(<i>a</i> <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/minus.png" style="vertical-align: middle;" />
6)<sup>2 </sup>+ (<i>b</i> <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/minus.png" style="vertical-align: middle;" />
6)<sup>2 </sup>+ 21 = 34</span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/therefore.png" style="margin-bottom: 0px; padding-bottom: 1px;" />
(<i>a</i> <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/minus.png" style="vertical-align: middle;" />
6)<sup>2 </sup>+ (<i>b</i> <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/minus.png" style="vertical-align: middle;" />
6)<sup>2 </sup>= 13</span><br />
<span class="Normal">Only option 3 fits into the above
equation.</span><br />
<span class="Normal">Hence, <b>option
3.</b></span></td></tr>
</tbody></table>
</div>
</div>
<div class="pLevel3">
<span class="Normal"></span></div>
</div>
</div>
<div id="202_III">
<table border="0" cellpadding="0" cellspacing="0" class="tableh1" id="202_III_bookmark">
<tbody>
<tr>
<td style="margin: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; width: 100%;">
<ol class="oLevel1 Heading1">
<li value="3"><a href="https://www.blogger.com/null" id="III">RANDOM VARIABLE</a></li>
</ol>
</td></tr>
</tbody></table>
<div id="202_III_A">
<table border="0" cellpadding="0" cellspacing="0" class="tableh1" id="202_III_A_bookmark"><tbody>
<tr>
<td style="margin: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; width: 100%;">
<ol class="oLevel2 Heading2">
<li value="1"><a href="https://www.blogger.com/null" id="III_A">DEFINITION</a></li>
</ol>
</td></tr>
</tbody></table>
<div class="pLevel3">
<span class="Normal">A random variable is a function, which
assigns unique numerical values to all possible outcomes of a random experiment
subject to fixed conditions.</span></div>
<div class="pLevel3">
<span class="Normal"></span></div>
<div class="pLevel3">
<span class="Normal">For example, if a coin is tossed the outcome
can either be ‘heads’ or ‘tails’. If the random variable <i>X</i> is defined as
the number of heads then <i>X</i> can take only two values, 0 and 1 as there are
only two outcomes of the experiment either ‘heads’ or ‘not
heads’(‘tails’).</span></div>
<div class="pLevel3">
<span class="Normal">A random variable is a function that maps
events to numbers.</span></div>
<div class="pLevel3">
<span class="Normal"></span></div>
<!--<p class="pLevel3">
<span class="ExampleHeading">Example 4:</span></p>
-->
<div class="divTextBox divColorExampleHeading divLevel3">
<div id="202_solvedex_4">
<table border="0" cellpadding="0" cellspacing="0" id="202_solvedex_4_bookmark" style="width: 100%;">
<tbody>
<tr>
<td style="background-color: transparent; width: 100%;">
<span class="ExampleHeading"><a href="https://www.blogger.com/null" id="ex_4">Example
4:</a></span></td></tr>
</tbody></table>
<table border="0" cellpadding="0" cellspacing="0">
<tbody>
<tr>
<td style="background-color: transparent; width: 100%;">
<span class="Normal">A coin is tossed three times and the sequence of
heads and tails are noted. The random variable <i>X</i> is defined as the
number of heads in three coin tosses. What are the values that <i>X</i>
can take?</span><br />
<span class="SolutionHeading">Solution:</span><br />
<span class="Normal">When three coins are tossed the total number of
outcomes = 2 <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/multiplication.png" style="margin-bottom: 0px; padding-bottom: 0px;" />
2 <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/multiplication.png" style="margin-bottom: 0px; padding-bottom: 0px;" />
2 = 8</span><br />
<span class="Normal">The table below lists the all possible elements of
the sample space and the corresponding values of <i>X</i>.</span><br />
<div class="tblCenter">
<table align="center" cellspacing="1" class="visible_border_table tLevel3">
<tbody>
<tr>
<td class="visible_border_table"><b>Outcome</b></td>
<td class="visible_border_table" style="padding-left: 25px; padding-right: 25px;"><b><i>X</i></b></td></tr>
<tr>
<td class="visible_border_table">HHH</td>
<td class="visible_border_table">3</td></tr>
<tr>
<td class="visible_border_table">HHT</td>
<td class="visible_border_table">2</td></tr>
<tr>
<td class="visible_border_table">HTH</td>
<td class="visible_border_table">2</td></tr>
<tr>
<td class="visible_border_table">HTT</td>
<td class="visible_border_table">1</td></tr>
<tr>
<td class="visible_border_table">THH</td>
<td class="visible_border_table">2</td></tr>
<tr>
<td class="visible_border_table">THT</td>
<td class="visible_border_table">1</td></tr>
<tr>
<td class="visible_border_table">TTH</td>
<td class="visible_border_table">1</td></tr>
<tr>
<td class="visible_border_table">TTT</td>
<td class="visible_border_table">0</td></tr>
</tbody></table>
</div>
<span class="Normal"></span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/therefore.png" style="margin-bottom: 0px; padding-bottom: 1px;" />
The random variable <i>X</i> can take values 0, 1, 2 and 3.</span><br />
<span class="Normal"></span></td></tr>
</tbody></table>
</div>
</div>
<div class="divTextBox divColorExampleHeading divLevel3">
<div id="202_solvedex_5">
<table border="0" cellpadding="0" cellspacing="0" id="202_solvedex_5_bookmark" style="width: 100%;">
<tbody>
<tr>
<td style="background-color: transparent; width: 100%;">
<span class="ExampleHeading"><a href="https://www.blogger.com/null" id="ex_5">Example
5:</a></span></td></tr>
</tbody></table>
<table border="0" cellpadding="0" cellspacing="0">
<tbody>
<tr>
<td style="background-color: transparent; width: 100%;">
<span class="Normal">Two dice are thrown and the sum of numbers on the
two faces is noted. The random variable <i>X</i> is defined as the sum of
numbers on the faces of the two dice. What are the values that <i>X</i>
can take?</span><br />
<span class="SolutionHeading">Solution:</span><br />
<span class="Normal">The minimum value that <i>X</i> can take is 2 and
the maximum value that <i>X</i> can take is 12 achieving all the values in
between.</span><br />
<span class="pImage"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_017.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/therefore.png" style="margin-bottom: 0px; padding-bottom: 1px;" />
The random variable <i>X</i> can take values 2, 3, 4, 5, 6, 7, 8, 9, 10,
11 and 12.</span>
</td></tr>
</tbody></table>
</div>
</div>
<div class="pLevel3">
<span class="Normal"></span></div>
</div>
<div id="202_III_B">
<table border="0" cellpadding="0" cellspacing="0" class="tableh1" id="202_III_B_bookmark"><tbody>
<tr>
<td style="margin: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; width: 100%;">
<ol class="oLevel2 Heading2">
<li value="2"><a href="https://www.blogger.com/null" id="III_B">TYPES OF RANDOM
VARIABLES</a></li>
</ol>
</td></tr>
</tbody></table>
<div class="pLevel3">
<span class="Normal"></span></div>
<ol class="oLevel3 Heading3 ListStyleOL">
<li value="1">DISCRETE</li>
</ol>
<div class="pLevel4">
<span class="Normal">A discrete random variable takes only
integer values and is usually the result of counting.</span></div>
<div class="pLevel4">
<span class="Normal">For example, if we flip a coin and count the
number of heads then the number of heads results from a random process -
flipping a coin. Also, the number of heads will always be an integral value - a
number between 0 and <i>n</i> (both inclusive), where <i>n</i> is the number of
times the coin is tossed. Therefore, the number of heads is a discrete random
variable.</span></div>
<div class="pLevel4">
<span class="Normal">A discrete random variable can take only a
finite number of distinct values. Examples of discrete random variables include
the number of students in a class, the Saturday night attendance at a movie
theatre, the number of patients in a doctor's surgery, the number of defective
goods in a box of a consignment.</span></div>
<ol class="oLevel3 Heading3 ListStyleOL">
<li value="2">CONTINUOUS</li>
</ol>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="pLevel4">
<span class="Normal">A continuous random variable can take any
value within a range of values. In this case the outcomes are measured, not
counted. Cricket batting averages, IQ scores, the length of time a long distance
telephone call lasts and the length of time a computer chip lasts are just a few
examples of continuous random variables.</span></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="pLevel4">
<span class="Normal">The continuous random variable is not
defined at specific values. The probability of observing any single value is
equal to 0, as the continuous random variable can take infinite
values.</span></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="pLevel4">
<span class="Normal">It is defined over an interval of values,
and is represented by the area under a curve (in higher mathematics, this is
known as an integral).</span></div>
<div class="pLevel4">
<span class="Normal"></span></div>
</div>
<div id="202_III_C">
<table border="0" cellpadding="0" cellspacing="0" class="tableh1" id="202_III_C_bookmark"><tbody>
<tr>
<td style="margin: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; width: 100%;">
<ol class="oLevel2 Heading2">
<li value="3"><a href="https://www.blogger.com/null" id="III_C">PROBABILITY MASS
FUNCTION</a></li>
</ol>
</td></tr>
</tbody></table>
<div class="pLevel3">
<span class="Normal">If the discrete random variable <i>X</i> can
take values <i>r</i><sub>1</sub>, <i>r</i><sub>2</sub>, <i>r</i><sub>3,
</sub>...., <i>r<sub>n</sub></i> then the function that gives the probability
when <i>X</i> = <i>r</i><sub>i</sub> for all values <i>r</i><sub>1</sub>,
<i>r</i><sub>2</sub>, <i>r</i><sub>3, </sub>....,
<i>r<sub>n</sub></i></span></div>
</div>
</div>
</div>
</body></html>
<br />
<div class="pLevel3">
<span class="Normal">That is, <i>f</i> <i><sub>X</sub></i></span></div>
(<i>r</i><sub>i</sub>) = P(<i>X</i> = <i>r</i><sub>i</sub>) , i = 1,
2,3,…,<i>n</i> is the probability mass function.
<br />
<div class="pLevel3">
<span class="Normal"></span></div>
<div class="divTextBox divColorExampleHeading divLevel3">
<div id="202_solvedex_6">
<table border="0" cellpadding="0" cellspacing="0" id="202_solvedex_6_bookmark" style="width: 100%;">
<tbody>
<tr>
<td style="background-color: transparent; width: 100%;"><span class="ExampleHeading">Example
6:</span></td></tr>
</tbody></table>
<table border="0" cellpadding="0" cellspacing="0">
<tbody>
<tr>
<td style="background-color: transparent; width: 100%;"><span class="Normal">If we throw a die what is the sum of the probability
mass function corresponding to each outcome?</span><br />
<span class="SolutionHeading">Solution:</span><br />
<span class="Normal">Total number of outcomes = 6</span><br />
<span class="Normal">Let the random variable <i>X</i> denote the score of
the die.</span><br />
<span class="Normal">Then <i>X</i> can take values 1, 2, 3, 4, 5 and
6.</span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_192.png" style="vertical-align: middle;" /></span><br />
<!--<div class="tblCenter">
<table align="center" cellspacing="1" style="border-collapse:collapse">
<tr><td class="visible_border_table" style="padding-left:25px;padding-right:25px">X</td><td class="visible_border_table">Value of the probability mass function f x(x)</td></tr>
<tr><td class="visible_border_table">1</td><td class="visible_border_table">16</td></tr>
<tr><td class="visible_border_table">2</td><td class="visible_border_table">16</td></tr>
<tr><td class="visible_border_table">3</td><td class="visible_border_table">16</td></tr>
<tr><td class="visible_border_table">4</td><td class="visible_border_table">16</td></tr>
<tr><td class="visible_border_table">5</td><td class="visible_border_table">16</td></tr>
<tr><td class="visible_border_table">6</td><td class="visible_border_table">16</td></tr>
</table>
</div>
-->
<br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_019.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_020.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"></span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_021.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_022.png" style="vertical-align: middle;" /></span><br />
<!--<p>
<span class="Normal"></span></p>
<p>
<span class="Normal"><img style="vertical-align:middle" src="202_images/202_023.png" alt="" /></span></p>
<p>
<span class="Normal"></span></p>
<p>
<span class="Normal"><img style="vertical-align:middle" src="202_images/202_024.png" alt="" /></span></p>
<p>
<span class="Normal"></span></p>
-->
<br />
<span class="Normal">This means the probability of throwing a die and
getting a score of 1 or 2 or 3 or 4 or 5 or 6 is
1.</span></td></tr>
</tbody></table>
</div>
</div>
<!--<p class="pLevel2">
<span class="Normal"></span></p>
<div class="divTextBox divColorNormal divLevel3">
<p>
<span class="Normal"></span></p>
<p>
<span class="Normal"></span></p>
</div>
-->
<br />
<div class="pLevel2">
<span class="Normal"></span></div>
<div class="divTextBox divColorNote divLevel3">
<span class="Note">REMEMBER</span><br />
<ul type="disc">
<li>The sum of probabilities (or the probability mass functions) corresponding
to each element of the sample space is always 1.</li>
</ul>
<ul>
<span class="Normal">We can also understand this concept with a simple
example of flipping a coin.</span></ul>
<ul>
<span class="Normal">On flipping a coin we either get heads (<i>H</i>) or
tails (<i>T</i>).</span></ul>
<span class="Normal"></span><br />
<ul>
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_023.png" style="vertical-align: middle;" /></span></ul>
<ul><span class="Note"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_025.png" style="vertical-align: middle;" /></span>
</ul>
<span class="Normal"></span><br />
<!--<p>
<span class="Normal"><img style="vertical-align:middle" src="202_images/202_027.png" alt="" /></span></p>
-->
<span class="Normal"></span><br />
<ul>
<span class="Normal">Probability of flipping a coin and getting ‘heads’ or
‘tails’ is 1.</span></ul>
<ul>
<span class="Normal">In general, </span></ul>
<ul>
<span class="Normal">If the event <i>A</i> has outcomes <i>A</i><sub>1</sub>,
<i>A</i><sub>2</sub>, <i>A</i><sub>3</sub>, …,
<i>A<sub>n</sub></i></span></ul>
</div>
.
<br />
<ul>
<span class="Normal">Then P(<i>A</i><sub>1</sub>) + P(<i>A</i><sub>2</sub>) +
… + P(<i>A<sub>n</sub></i></span></ul>
) = 1
<br />
<div id="202_III_D">
<table border="0" cellpadding="0" cellspacing="0" class="tableh1" id="202_III_D_bookmark"><tbody>
<tr>
<td style="margin: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; width: 100%;"><ol class="oLevel2 Heading2">
<li value="4">PROBABILITY DENSITY
FUNCTION</li>
</ol>
</td></tr>
</tbody></table>
<div class="pLevel3">
<span class="Normal"></span></div>
<div class="pImage">
<img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_028.png" /></div>
<div class="pLevel3">
<span class="Normal"><span style="margin-left: 125px;"></span><b>Figure (i)</b></span></div>
<div class="pLevel3">
<span class="Normal"></span></div>
<div class="pLevel3">
<span class="Normal">If <i>X</i> is a continuous random variable,
then the probability density function (pdf) of <i>X</i>, represented by
<i>f<sub>X</sub></i></span></div>
</div>
(<i>x</i>) is a function such that for two numbers, <i>a</i>
and <i>b</i>, <i>a</i> <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/less_equal.png" style="vertical-align: middle;" />
<i>b</i>, the probability that the random variable <i>X</i>, takes a value in
the interval [<i>a</i>, <i>b</i>] is the area under the curve of the density
function, <i>f<sub>X</sub></i>(<i>x</i>), from <i>a</i> to <i>b</i>.
<br />
<div class="pLevel3">
<span class="Normal">That is, P( <i>a</i> <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/less_equal.png" style="vertical-align: middle;" />
<i>X</i> <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/less_equal.png" style="vertical-align: middle;" />
<i>b</i>) = Area of the shaded region in <b>Figure (i)</b></span></div>
<div class="pLevel3">
<span class="Normal">Also, total area under the probability
density curve is 1 square unit.</span></div>
<div class="pLevel3">
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/therefore.png" style="margin-bottom: 0px; padding-bottom: 1px;" />
Area of the shaded region in <b>Figure (ii)</b> = 1 square unit</span></div>
<div class="pImage">
<img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_027.png" /></div>
<div class="pLevel3">
<span class="Normal"><span style="margin-left: 125px;"></span><b>Figure (ii)</b></span></div>
<div id="202_III_E">
<table border="0" cellpadding="0" cellspacing="0" class="tableh1" id="202_III_E_bookmark"><tbody>
<tr>
<td style="margin: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; width: 100%;"><ol class="oLevel2 Heading2">
<li value="5">CUMULATIVE DISTRIBUTION
FUNCTION</li>
</ol>
</td></tr>
</tbody></table>
<div class="pLevel3">
<span class="ListParagraph"></span></div>
<div class="pImage">
<img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_026.png" /></div>
<div class="pLevel3">
<span class="Normal">The cumulative distribution function, cdf,
is function <i>f<sub>X</sub></i></span></div>
</div>
(<i>x</i>) of a random variable, <i>X</i>, and
is defined for a number <i>x</i> as the probability that the observed value of
<i>X</i> will be at most <i>x</i>. It is the area under the probability density
curve from <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/minus.png" style="vertical-align: middle;" />∞
to <i>x</i>.
<br />
<div id="202_III_F">
<table border="0" cellpadding="0" cellspacing="0" class="tableh1" id="202_III_F_bookmark"><tbody>
<tr>
<td style="margin: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; width: 100%;"><ol class="oLevel2 Heading2">
<li value="6">EXPECTATION</li>
</ol>
</td></tr>
</tbody></table>
<div class="pLevel3">
<span class="Normal">The expected value (or mean) of a random
variable is its central or average value.</span></div>
<div class="pLevel3">
<span class="Normal"></span></div>
<div class="pLevel3">
<span class="Normal">Stating the e<i>x</i>pected value gives a
general impression of the behaviour of random variables without having all the
details of its probability mass function (if it is discrete) or its probability
density function (if it is continuous). The expected value is based on the past
data and is used to forecast future results.</span></div>
<div class="pLevel3">
<span class="Normal">The expected value of a random variable
<i>X</i> is represented by E(<i>X</i>) or <i><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/204_337.png" style="vertical-align: middle;" /></i>.</span></div>
<ol class="oLevel3 Heading3 ListStyleOL">
<li value="1">EXPECTATION FOR A DISCRETE RANDOM VARIABLE</li>
</ol>
<div class="pLevel4">
<span class="Normal">If <i>X</i> is a discrete random variable
that can take values <i>x</i><sub>1</sub>, <i>x</i><sub>2</sub>,
<i>x</i><sub>3</sub>, ..., <i>x<sub>n</sub></i></span></div>
</div>
and P(<i>x<sub>i</sub></i>)
represents
<br />
<div class="pLevel4">
<span class="Normal">P(<i>X</i> = <i>x<sub>i</sub></i>), then the
expected value<i> X</i> is given by:</span></div>
<div class="pLevel4">
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_029.png" style="vertical-align: middle;" /></span></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="pLevel4">
<span class="Normal">All we are doing is computing the weighted
average of the random variable over its range, using probabilities as
weights.</span></div>
<div class="pLevel4">
<span class="Normal">For example,</span></div>
<div class="pLevel4">
<span class="Normal">If there are three people in the room of
which 2 are 5 ft tall and 1 is 6 ft tall.</span></div>
<div class="pLevel4">
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_030.png" style="vertical-align: middle;" />
<img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_031.png" style="vertical-align: middle;" /></span></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="pLevel4">
<span class="Normal">The random variable <i>X</i> denotes the
height of a person in the room.</span></div>
<div class="pLevel4">
<span class="Normal">Then the only values <i>X</i> can take are 5
or 6.</span></div>
<div class="pLevel4">
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_032.png" style="vertical-align: middle;" />
</span></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="pLevel4">
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_034.png" style="vertical-align: middle;" />
</span></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="pLevel4">
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_036.png" style="vertical-align: middle;" /></span></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="pLevel4">
<span class="Normal">Expectation for a continuous random
variable</span></div>
<div class="pLevel4">
<span class="Normal">If <i>X</i> is a continuous random variable
with probability density function <i>f<sub>X</sub></i>(<i>x</i>), then the
expected value of <i>X</i> is given by:</span></div>
<div class="pLevel4">
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_037.png" style="vertical-align: middle;" /></span></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="pLevel4">
<span class="Normal">Where <i>f<sub>X</sub></i>(<i>x</i>) is the
probability density function of the continuous random variable
<i>X</i>.</span></div>
<div class="pLevel4">
<span class="Normal">General Expectation</span></div>
<div class="pLevel4">
<span class="Normal">In general, for discrete random variable
<i>X</i>,</span></div>
<div class="pLevel4">
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_038.png" style="vertical-align: middle;" /></span></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="pLevel4">
<span class="Normal">For example,</span></div>
<div class="pLevel4">
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_039.png" style="vertical-align: middle;" /></span></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="pLevel4">
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_040.png" style="vertical-align: middle;" /></span></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="divTextBox divColorExampleHeading divLevel3">
<div id="202_solvedex_7">
<table border="0" cellpadding="0" cellspacing="0" id="202_solvedex_7_bookmark" style="width: 100%;">
<tbody>
<tr>
<td style="background-color: transparent; width: 100%;"><span class="ExampleHeading">Example
7:</span></td></tr>
</tbody></table>
<table border="0" cellpadding="0" cellspacing="0">
<tbody>
<tr>
<td style="background-color: transparent; width: 100%;"><span class="Normal">A and B throw a die for a stake of Rs.11, which is
to be won by the player who first throws a six. The game ends when the
stake is won by A or B. If A has the first throw, what are their
respective expectations?</span><br />
<span class="Normal"><b>[SNAP 2008]</b></span><br />
<span class="Normal">(1) 5 and 6<span style="margin-left: 25px;"></span><span style="margin-left: 25px;"></span>(2) 6 and 5</span><br />
<span class="Normal">(3) 11 and 10<span style="margin-left: 25px;"></span><span style="margin-left: 7px;"></span>(4)
10 and 1</span><br />
<span class="SolutionHeading">Solution:</span><br />
<span class="Normal">A wins either in the first throw of the game, or in
the third throw after A and B lose in the first and second throws,
respectively, or in the fifth throw after A loses in the first and third
throws and B loses in the second and fourth throws and so on. The
probability of a six being thrown is 1/6 and the probability of 6 not
being thrown is 5/6.</span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/therefore.png" style="margin-bottom: 0px; padding-bottom: 1px;" />
The probability of A winning</span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_041.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"></span><br />
<span class="Normal"><span style="margin-left: 20px;"></span><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_042.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"></span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_043.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"></span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_044.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"></span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_045.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"></span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_046.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"></span><br />
<span class="Normal">Hence, <b>option
2</b>.</span></td></tr>
</tbody></table>
</div>
</div>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="divTextBox divColorExampleHeading divLevel3">
<div id="202_solvedex_8">
<table border="0" cellpadding="0" cellspacing="0" id="202_solvedex_8_bookmark" style="width: 100%;">
<tbody>
<tr>
<td style="background-color: transparent; width: 100%;"><span class="ExampleHeading">Example
8:</span></td></tr>
</tbody></table>
<table border="0" cellpadding="0" cellspacing="0">
<tbody>
<tr>
<td style="background-color: transparent; width: 100%;"><span class="Normal">A medical clinic tests blood for certain disease
from which approximately one person in a hundred suffers. People come to
the clinic in group of 50. The operator of the clinic wonders whether he
can increase the efficiency of the testing procedure by conducting pooled
tests. In the pooled tests, the operator would pool the 50 blood samples
and test them altogether. If the pooled test was negative, he could
pronounce the whole group healthy. If not, he could then test each
person’s blood individually. The expected number of tests the operator
will have to perform if he pools the blood samples are:</span><br />
<span class="Normal"><b>[IIFT 2008]</b></span><br />
<span class="Normal">(1) 47<span style="margin-left: 25px;"></span><span style="margin-left: 25px;"></span>(2) 25</span><br />
<span class="Normal">(3) 21<span style="margin-left: 25px;"></span><span style="margin-left: 25px;"></span>(4) None of the above</span><br />
<span class="SolutionHeading">Solution:</span><br />
<span class="Normal">Probability that a random person has the disease is
1/100.</span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/therefore.png" style="margin-bottom: 0px; padding-bottom: 1px;" />
Probability that a random person does not have the disease is
99/100.</span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/therefore.png" style="margin-bottom: 0px; padding-bottom: 1px;" />
Out of a sample of 50 people, the probability that no one has the disease
is (99/100)<sup>50</sup></span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/therefore.png" style="margin-bottom: 0px; padding-bottom: 1px;" />
Out of a sample of 50 people, the probability that at least one person has
the disease is </span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_047.png" style="vertical-align: middle;" /></span>
<br />
<span class="Normal"></span><br />
<span class="Normal">If no one in the sample of 50 people has the
disease, then only 1 test will have to be conducted.</span><br />
<span class="Normal">If at least one person in the sample of 50 people
has the disease then 51 tests will have to be conducted.</span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/therefore.png" style="margin-bottom: 0px; padding-bottom: 1px;" />
The expected number of tests that the operator will have to
conduct</span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_048.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"></span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_049.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"></span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_050.png" style="vertical-align: middle;" />
<img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_051.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"></span><br />
<span class="Normal"></span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/congru2.png" style="vertical-align: middle;" />
51 – 30 <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/congru2.png" style="vertical-align: middle;" />
21</span><br />
<span class="Normal">Hence, <b>option
3.</b></span></td></tr>
</tbody></table>
</div>
</div>
<div id="202_III_G">
<table border="0" cellpadding="0" cellspacing="0" class="tableh1" id="202_III_G_bookmark"><tbody>
<tr>
<td style="margin: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; width: 100%;"><ol class="oLevel2 Heading2">
<li value="7">VARIANCE</li>
</ol>
</td></tr>
</tbody></table>
<div class="pLevel3">
<span class="Normal">If a random variable <i>X</i> has expected
value (mean)</span></div>
<div class="pLevel3">
<span class="Normal"><i><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/204_337.png" style="vertical-align: middle;" /></i>
= E(<i>X</i>), then the variance Var(<i>X</i>) of <i>X</i> is given by:
Var(<i>X</i>) = E(<i>X</i> <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/minus.png" style="vertical-align: middle;" />
<i><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/204_337.png" style="vertical-align: middle;" /></i>)<sup>2</sup></span></div>
<div class="pLevel3">
<span class="Normal">If random variable <i>X</i> is discrete
then, </span><span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_052.png" style="vertical-align: middle;" /></span></div>
<div class="pLevel3">
<span class="Normal"></span></div>
<div class="pLevel3">
<span class="Normal">If the random variable is continuous then,
</span><span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_053.png" style="vertical-align: middle;" />
</span></div>
<div class="pLevel3">
<span class="Normal">Where <i>f<sub>X</sub></i>(<i>x</i>) is the
probability density function of the random variable <i>X</i>.</span></div>
<div class="pLevel3">
<span class="Normal"></span><!--<span class="Normal"><img style="vertical-align:middle" src="202_images/202_054.png" alt="" /></span></p>
-->
</div>
<ol class="oLevel3 Heading3 ListStyleOL">
<li value="1">RELATION BETWEEN THE EXPECTED VALUE AND VARIANCE</li>
</ol>
<div class="pLevel4">
<span class="Normal">Var(<i>X</i>) = E(<i>X</i><sup>2</sup>) <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/minus.png" style="vertical-align: middle;" />
{E(<i>X</i>)}<sup>2</sup></span></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="divTextBox divColorExampleHeading divLevel3">
<div id="202_solvedex_9">
<table border="0" cellpadding="0" cellspacing="0" id="202_solvedex_9_bookmark" style="width: 100%;">
<tbody>
<tr>
<td style="background-color: transparent; width: 100%;"><span class="ExampleHeading">Example
9:</span></td></tr>
</tbody></table>
<table border="0" cellpadding="0" cellspacing="0">
<tbody>
<tr>
<td style="background-color: transparent; width: 100%;"><div class="Normal">
<span class="Normal">The return levels and associated
probabilities of two securities are given below:</span></div>
<div class="pImage">
<img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_054.png" /></div>
<span class="Normal">What are their respective expected values and
variances of returns?</span><br />
<span class="Normal"><b>[JMET 2008]</b></span><br />
<div class="pLevel4">
<span class="Normal"></span></div>
<ol>
<li value="1"><i>X</i>: E(<i>X</i>) = 12 and Var(<i>X</i>) = 0</li>
</ol>
<div class="pLevel2">
<span class="Normal"><i>Y</i>: E(<i>Y</i>) = 13 and
Var(<i>Y</i>) = 2.4</span></div>
<ol>
<li value="2"><i>X</i>: E(<i>X</i>) = 12 and Var(<i>X</i>) = 5.4</li>
</ol>
<div class="pLevel2">
<span class="Normal"><i>Y</i>: E(<i>Y</i>) = 13 and
Var(<i>Y</i>) = 2.4</span></div>
<ol>
<li value="3"><i>X</i>: E(<i>X</i>) = 12 and Var(<i>X</i>) =3</li>
</ol>
<div class="pLevel2">
<span class="Normal"><i>Y</i>: E(<i>Y</i>) = 14.3 and
Var(<i>Y</i>) = 4.13</span></div>
<ol>
<li value="4"><i>X</i>: E(<i>X</i>) = 12 and Var(<i>X</i>) = 4.5</li>
</ol>
<div class="pLevel2">
<span class="Normal"><i>Y</i>: E(<i>Y</i>) = 14.3 and
Var(<i>Y</i>) = 6.19</span></div>
<span class="SolutionHeading">Solution:</span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_055.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"></span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_056.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_057.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"></span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_058.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"></span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_059.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_060.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"></span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_061.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"></span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_062.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_063.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"></span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_064.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"></span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_065.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_200.png" style="vertical-align: middle;" /></span><br />
<span class="Normal">Hence, <b>option
2.</b></span></td></tr>
</tbody></table>
</div>
</div>
<div id="202_III_H">
<table border="0" cellpadding="0" cellspacing="0" class="tableh1" id="202_III_H_bookmark"><tbody>
<tr>
<td style="margin: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; width: 100%;"><ol class="oLevel2 Heading2">
<li value="8">STANDARD
DEVIATION</li>
</ol>
</td></tr>
</tbody></table>
<div class="pLevel3">
<span class="Normal">For a random variable <i>X</i>, the standard
deviation <i><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_373.png" style="vertical-align: middle;" /></i>
is</span></div>
<div class="pLevel3">
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_066.png" style="vertical-align: middle;" /></span></div>
<div class="pLevel3">
<span class="Normal"></span></div>
<div class="divTextBox divColorExampleHeading divLevel3">
<div id="202_solvedex_10">
<table border="0" cellpadding="0" cellspacing="0" id="202_solvedex_10_bookmark" style="width: 100%;">
<tbody>
<tr>
<td style="background-color: transparent; width: 100%;"><span class="ExampleHeading">Example
10:</span></td></tr>
</tbody></table>
<table border="0" cellpadding="0" cellspacing="0">
<tbody>
<tr>
<td style="background-color: transparent; width: 100%;"><span class="Normal">Find the expectation, variance and standard
deviation of the number of heads when three coins are tossed.</span><br />
<span class="SolutionHeading">Solution:</span><br />
<span class="Normal">The random variable <i>X</i> is defined as the
number of heads in three coin tosses.</span><br />
<span class="Normal"><i>X</i> can take values 0, 1, 2 and 3 as shown in
<b>Example 1</b>.</span><br />
<span class="Normal"></span><br />
<!--<div class="tblCenter">
<table align="center" cellspacing="1">
<tr><td class="visible_border_table">X=xi</td><td class="visible_border_table">Favourable Outcomes</td><td class="visible_border_table">P(xi)</td><td class="visible_border_table">xi P(xi)</td><td class="visible_border_table">xi2 P(xi)</td></tr>
<tr><td class="visible_border_table"></td><td class="visible_border_table"></td><td class="visible_border_table">Favourable outcomesTotal outcomes</td><td class="visible_border_table">First column<img src="202_images/multiplication.png" style="padding-bottom:0px; margin-bottom:0px;" alt=""/> Third column</td><td class="visible_border_table">(First column)2<img src="202_images/multiplication.png" style="padding-bottom:0px; margin-bottom:0px;" alt=""/> Third column</td></tr>
<tr><td class="visible_border_table">0</td><td class="visible_border_table">TTT</td><td class="visible_border_table">18</td><td class="visible_border_table">0</td><td class="visible_border_table">0</td></tr>
<tr><td class="visible_border_table">1</td><td class="visible_border_table">HTT, THT, TTH</td><td class="visible_border_table">38</td><td class="visible_border_table">38</td><td class="visible_border_table">38</td></tr>
<tr><td class="visible_border_table">2</td><td class="visible_border_table">HHT, HTH, THH</td><td class="visible_border_table">38</td><td class="visible_border_table">68</td><td class="visible_border_table">128</td></tr>
<tr><td class="visible_border_table">3</td><td class="visible_border_table">HHH</td><td class="visible_border_table">18</td><td class="visible_border_table">38</td><td class="visible_border_table">98</td></tr>
<tr><td class="visible_border_table"></td><td class="visible_border_table"></td><td class="visible_border_table">𝑓(𝑥) = 1</td><td class="visible_border_table">𝑥𝑓(𝑥)=128</td><td class="visible_border_table">𝑥2𝑓(𝑥)=248</td></tr>
</table>
</div>
-->
<br />
<span class="-1"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_101.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_080.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"></span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_081.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"></span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_082.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"></span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_083.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"></span></td></tr>
</tbody></table>
</div>
</div>
</div>
</div>
<div id="202_IV">
<table border="0" cellpadding="0" cellspacing="0" class="tableh1" id="202_IV_bookmark">
<tbody>
<tr>
<td style="margin: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; width: 100%;"><ol class="oLevel1 Heading1">
<li value="4">DISCRETE PROBABILITY
DISTRIBUTIONS</li>
</ol>
</td></tr>
</tbody></table>
<div id="202_IV_A">
<table border="0" cellpadding="0" cellspacing="0" class="tableh1" id="202_IV_A_bookmark"><tbody>
<tr>
<td style="margin: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; width: 100%;"><ol class="oLevel2 Heading2">
<li value="1">BINOMIAL
DISTRIBUTION</li>
</ol>
</td></tr>
</tbody></table>
<ol class="oLevel3 Heading3 ListStyleOL">
<li value="1">DEFINITION AND APPLICATION</li>
</ol>
<div class="pLevel4">
<span class="Normal">When a coin is flipped, the outcome is
either a head or a tail; when a student solves a sum, the answer can either be
correct or incorrect; when a baby is born, the baby is either born in the month
of October or is not. Each of these events has two possible independent
outcomes. One of the outcomes can be labelled "success" and the other,
"failure". These are few examples of experiments that follow binomial
distribution.</span></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="pLevel4">
<span class="Normal">From the above examples we can generalize
that an experiment that follows binomial distribution has the following
properties:</span></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<ul class="oLevel3">
<li value="1">The experiment consists of <i>n</i> repeated trials.</li>
</ul>
<ul class="oLevel3">
<li value="2">Each trial results in just two possible outcomes. One of these
outcomes can be termed as a success and the other, a failure.</li>
</ul>
<ul class="oLevel3">
<li value="3">The probability of success, <i>p</i>, is constant
throughout.</li>
</ul>
<ul class="oLevel3 ">
<li value="4">The outcome of one trial is not affected by the outcome of other
trials, that is, each trial is independent.</li>
</ul>
<div class="pLevel4">
<span class="Normal">If an event occurs <i>n</i> times (for
example, a die is tossed <i>n</i> times), then the binomial distribution can be
used to determine the probability of obtaining exactly <i>x</i> successes in the
<i>n</i> outcomes.</span></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="pLevel4">
<span class="Normal">The binomial probability for obtaining
<i>x</i> successes in <i>n</i> trials, is given by the following
function:</span></div>
<div class="pLevel4">
<span class="Normal"><i>f</i>(<i>x</i>) =
<i><sup>n</sup></i>C<i><sub>x</sub></i> <i>p<sup>x</sup></i> <i>q<sup>(n <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/minus.png" style="vertical-align: middle;" />
x)</sup></i></span></div>
</div>
</div>
, <i>x</i> = 1, 2, 3, …, <i>n</i>
<br />
<div class="pLevel4">
<span class="Normal">We say that random variable <i>X</i> follows
binomial distribution with parameters <i>n</i> and <i>p</i>.</span></div>
<div class="pLevel4">
<span class="Normal">Where,</span></div>
<div class="pLevel4">
<span class="Normal"><i>x</i> = Number of successes that result
from a binomial experiment.</span></div>
<div class="pLevel4">
<span class="Normal"><i>p</i> = Probability of success on an
individual trial.</span></div>
<div class="pLevel4">
<span class="Normal"><i>q</i> = Probability of failure in an
individual trial.</span></div>
<div class="pLevel4">
<span class="Normal"><i>n</i> = Number of trials.</span></div>
<div class="pLevel4">
<span class="Normal">Also, <i>p</i> + <i>q</i> = 1</span></div>
<div class="divTextBox divColorExampleHeading divLevel3">
<div id="202_solvedex_11">
<table border="0" cellpadding="0" cellspacing="0" id="202_solvedex_11_bookmark" style="width: 100%;">
<tbody>
<tr>
<td style="background-color: transparent; width: 100%;"><span class="ExampleHeading">Example
11:</span></td></tr>
</tbody></table>
<table border="0" cellpadding="0" cellspacing="0">
<tbody>
<tr>
<td style="background-color: transparent; width: 100%;"><span class="Normal">Two dice are tossed 5 times. Let <i>X</i> denote the
number of throws in which the score of the 1<sup>st</sup> die exceeds the
score of the 2<sup>nd</sup> die. What is the distribution of the
experiment and the probability of its success?</span><br />
<span class="Normal"></span><br />
<span class="SolutionHeading">Solution:</span><br />
<span class="Normal">Let the number of successes to be obtained in 5
trials be <i>x</i>.</span><br />
<span class="Normal">Number of trials = <i>n</i> = 5</span><br />
<span class="Normal">In each trial, success = score on the 1<sup>st</sup>
die > score of the 2<sup>nd</sup> die</span><br />
<span class="Normal"></span><br />
<span class="Normal">Favourable outcomes = {(2, 1), (3, 1), (3, 2),(4,
1), (4, 2), (4, 3), (5, 1), (5, 2), (5, 3), (5, 4),(6, 1), (6, 2), (6, 3),
(6, 4), (6, 5)}</span><br />
<span class="Normal"></span><br />
<span class="Normal"></span><br />
<span class="Normal">Number of favourable outcomes = 15</span><br />
<span class="Normal">Total outcomes = 6 <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/multiplication.png" style="margin-bottom: 0px; padding-bottom: 0px;" />
6 = 36</span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_084.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"></span><br />
<span class="Normal">The experiment consists of repeated trials. The die
is tossed 5 times.</span><br />
<span class="Normal">Either, Score on the 1<sup>st</sup> die > Score
of the 2<sup>nd</sup> die or Score on the 1<sup>st</sup> die < Score of
the 2<sup>nd</sup> die</span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/therefore.png" style="margin-bottom: 0px; padding-bottom: 1px;" />
Each trial can result in just two possible outcomes</span><br />
<span class="Normal">The probability of success is 0.416(constant) on
every trial.</span><br />
<span class="Normal">The trials are independent; that is, the outcome of
one trial does not affect the outcome of the second trial.</span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/therefore.png" style="margin-bottom: 0px; padding-bottom: 1px;" />
The random variable <i>X</i> follows binomial distribution with parameters
<i>n</i> and <i>p</i>.</span><br />
<span class="Normal">Here, <i>n</i> = 5 and <i>p</i> = 0.416.</span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/therefore.png" style="margin-bottom: 0px; padding-bottom: 1px;" />
<i>q</i> = 1 <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/minus.png" style="vertical-align: middle;" />
<i>p</i> = 0.584</span><br />
<span class="Normal">The binomial probability for obtaining <i>x</i>
successes in 5 trials are given by the following function:</span><br />
<span class="Normal"><i>f</i>(<i>x</i>) =
<sup>5</sup>C<i><sub>x</sub></i></span></td></tr>
</tbody></table>
</div>
</div>
(0.416)<i><sup>x</sup></i> (0.584)<sup>(5
<img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/minus.png" style="vertical-align: middle;" />
<i>x</i>)</sup>
<br />
<ol class="oLevel3 Heading3 ListStyleOL">
<li value="2">PROPERTIES</li>
</ol>
<ol class="oLevel3 Heading4 ListStyleOL">
<li value="1">MEAN</li>
</ol>
<div class="pLevel4">
<span class="Normal">The mean or expected value of the random
variable <i>X</i> that follows binomial distribution with parameters <i>n</i>
and <i>p</i> is <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/204_337.png" style="vertical-align: middle;" />
= E(<i>X</i>) = <i>np</i></span></div>
<ol class="oLevel3 Heading4 ListStyleOL">
<li value="2">VARIANCE</li>
</ol>
<div class="pLevel4">
<span class="Normal">The variance of the random variable <i>X</i>
that follows binomial distribution with parameters <i>n</i> and <i>p</i> is
Var(<i>X</i>) = <i>npq</i></span></div>
<ol class="oLevel3 Heading4 ListStyleOL">
<li value="3">STANDARD DEVIATION</li>
</ol>
<div class="pLevel4">
<span class="Normal">The standard deviation of the random
variable <i>X </i>that follows binomial distribution with parameters <i>n</i>
and <i>p</i> is </span><span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_085.png" style="vertical-align: middle;" /></span></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="divTextBox divColorExampleHeading divLevel3">
<div id="202_solvedex_12">
<table border="0" cellpadding="0" cellspacing="0" id="202_solvedex_12_bookmark" style="width: 100%;">
<tbody>
<tr>
<td style="background-color: transparent; width: 100%;"><span class="ExampleHeading">Example
12:</span></td></tr>
</tbody></table>
<table border="0" cellpadding="0" cellspacing="0">
<tbody>
<tr>
<td style="background-color: transparent; width: 100%;"><span class="Normal">The random variable <i>X</i> follows binomial
distribution with parameters <i>n</i> and <i>p</i>. If E(<i>X</i>) = 5 and
Var(<i>X</i>) = 4 then find <i>n</i> and <i>p</i>.</span><br />
<span class="SolutionHeading">Solution:</span><br />
<span class="Normal">For a binomial distribution,</span><br />
<span class="Normal">E(<i>X</i>) = <i>n</i>p</span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/therefore.png" style="margin-bottom: 0px; padding-bottom: 1px;" />
<i>np</i> = 5<span style="margin-left: 25px;"></span>...(i)</span><br />
<span class="Normal">Also, Var(<i>X</i>) = <i>npq</i></span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/therefore.png" style="margin-bottom: 0px; padding-bottom: 1px;" />
<i>npq</i> = 4<span style="margin-left: 15px;"></span>...(ii)</span><br />
<span class="Normal">Dividing (ii) by (i) we get,</span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_086.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"></span><br />
<span class="Normal">We know that <i>p</i> = 1 <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/minus.png" style="vertical-align: middle;" />
<i>q</i></span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_087.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"></span><br />
<span class="Normal">Substituting the value of <i>p</i> in (i) we
get,</span><br />
<span class="Normal"><i>n</i> =
25</span></td></tr>
</tbody></table>
</div>
</div>
<div id="202_IV_B">
<table border="0" cellpadding="0" cellspacing="0" class="tableh1" id="202_IV_B_bookmark"><tbody>
<tr>
<td style="margin: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; width: 100%;"><ol class="oLevel2 Heading2">
<li value="2">POISSON
DISTRIBUTION</li>
</ol>
</td></tr>
</tbody></table>
<ol class="oLevel3 Heading3 ListStyleOL">
<li value="1">DEFINITION AND APPLICATION</li>
</ol>
<div class="pLevel4">
<span class="Normal">Poisson distribution is generally used when
the sample size is very large.</span></div>
<div class="pLevel4">
<span class="Normal">An experiment that follows Poisson
distribution has the following properties:</span></div>
<ul class="oLevel3">
<li value="1">Each outcome of the experiment can be classified distinctly as a
success or a failure.</li>
</ul>
<ul class="oLevel3">
<li value="2">The average number of successes (<i>λ</i>) that occur in a
specified region is known.</li>
</ul>
<ul class="oLevel3">
<li value="3">The probability of success is proportional to the size of the
region.</li>
</ul>
<ul class="oLevel3">
<li value="4">The probability of success in an extremely small region is
approximately equal to zero.</li>
</ul>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="pLevel4">
<span class="Normal"><b>Note</b>: The specified region could be a
length, an area, a volume, a period of time, etc.</span></div>
<div class="pLevel4">
<span class="Normal">In Mumbai on an average 20 car accidents
happen in a month:</span></div>
<ul class="oLevel3">
<li value="1">On a particular day either car accidents happen or they
don’t.</li>
</ul>
<ul class="oLevel3">
<li value="2">The average number of car accidents (success) that occur in a
specified region (month) is known.</li>
</ul>
<ul class="oLevel3">
<li value="3">The probability that no accident takes place reduces as the number
of months increase.</li>
</ul>
<ul class="oLevel3">
<li value="4">The probability of a car accident in a minute (extremely small
region as compared to a month) will be virtually zero.</li>
</ul>
<div class="pLevel4">
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/therefore.png" style="margin-bottom: 0px; padding-bottom: 1px;" />
The number of car accidents taking place over a given period is an example of a
random variable that follows Poisson distribution.</span></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="pLevel4">
<span class="Normal">Some other examples of experiments that
follow Poisson distribution are:</span></div>
<ul class="oLevel3">
<li value="1">Birth defects and genetic mutations.</li>
</ul>
<ul class="oLevel3">
<li value="2">Traffic flow.</li>
</ul>
<ul class="oLevel3">
<li value="3">Mortality of infants in a city.</li>
</ul>
<ul class="oLevel3">
<li value="4">The number of misprints in a book.</li>
</ul>
<ul class="oLevel3">
<li value="5">The number of bacteria on a plate.</li>
</ul>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="pLevel4">
<span class="Normal">The average number of successes in a
specified region is represented by λ and the probability of <i>x </i>successes
is given by the following function:</span></div>
<div class="pLevel4">
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_088.png" style="vertical-align: middle;" /></span></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="pLevel4">
<span class="Normal">In this case we say that the random variable
<i>X</i> follows Poisson distribution with parameter λ.</span></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="divTextBox divColorExampleHeading divLevel3">
<div id="202_solvedex_13">
<table border="0" cellpadding="0" cellspacing="0" id="202_solvedex_13_bookmark" style="width: 100%;">
<tbody>
<tr>
<td style="background-color: transparent; width: 100%;"><span class="ExampleHeading">Example
13:</span></td></tr>
</tbody></table>
<table border="0" cellpadding="0" cellspacing="0">
<tbody>
<tr>
<td style="background-color: transparent; width: 100%;"><span class="Normal">Suppose that flaws in the plywood occur at an
average rate of 1 per 50 square feet. What is the probability that a 4
<img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/multiplication.png" style="margin-bottom: 0px; padding-bottom: 0px;" />
8 feet sheet will have:</span><br />
<span class="Normal">(i) No flaws</span><br />
<span class="Normal">(ii) At most one flaw</span><br />
<span class="Normal"></span><br />
<span class="SolutionHeading">Solution (i):</span><br />
<span class="Normal">The average number of flaws per 50 sq ft is
1.</span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_089.png" style="vertical-align: middle;" /></span>
<br />
<span class="Normal"></span><br />
<span class="Normal">Let the random variable <i>X</i> be the number of
flaws.</span><br />
<span class="Normal">Let us term success as finding a flaw and failure as
otherwise.</span><br />
<span class="Normal">This is an example of an experiment that follows
Poisson distribution as the experiment results in outcomes that can be
classified as successes (flaw) or failures (no flaw).</span><br />
<span class="Normal"></span><br />
<span class="Normal">The average number of flaws (successes) that occur
per 32 sq feet (specified region) is known.</span><br />
<span class="Normal">As the area of the plywood increases the probability
of finding at least one flaw increases.</span><br />
<span class="Normal">The probability of success (flaw) in an extremely
small region (0.01 sq foot) is 0.0064 i.e. virtually zero.</span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/therefore.png" style="margin-bottom: 0px; padding-bottom: 1px;" />
The probability that a success will occur is proportional to the size of
the region.</span><br />
<span class="Normal">Hence, the random variable <i>X</i> follows Poisson
distribution with parameter λ = 0.64.</span><br />
<span class="Normal">Probability that 32 sq feet of the plywood will have
<i>x</i> flaws is given by the following function</span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_091.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"></span><br />
<span class="Normal">If the 32 sq ft plywood has no flaw then<i> x</i> =
0.</span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_092.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"></span><br />
<span class="SolutionHeading">Solution (ii):</span><br />
<span class="Normal">If the 32 sq ft plywood has at the most one flaw
then <i>x</i> <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/less_equal.png" style="vertical-align: middle;" />
1 i.e. <i>x</i> = 0, 1</span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_093.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"></span><br />
<span class="Normal"><span style="margin-left: 10px;"></span><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_094.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"></span><br />
<span class="Normal"><span style="margin-left: 10px;"></span><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_095.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"></span></td></tr>
</tbody></table>
</div>
</div>
<ol class="oLevel3 Heading3 ListStyleOL">
<li value="2">PROPERTIES</li>
</ol>
<ol class="oLevel3 Heading4 ListStyleOL">
<li value="1">MEAN</li>
</ol>
<div class="pLevel4">
<span class="Normal">The mean or expected value of the random
variable <i>X</i> that follows Poisson distribution with parameter λ
is</span></div>
<div class="pLevel4">
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/204_337.png" style="vertical-align: middle;" />
= E(<i>X</i>) = λ</span></div>
<ol class="oLevel3 Heading4 ListStyleOL">
<li value="2">VARIANCE</li>
</ol>
<div class="pLevel4">
<span class="Normal">The variance of the random variable <i>X</i>
that follows Poisson distribution with parameter λ is Var(<i>X</i>) = λ which is
same as its MEAN.</span></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<ol class="oLevel3 Heading4 ListStyleOL">
<li value="3">STANDARD DEVIATION</li>
</ol>
<div class="pLevel4">
<span class="Normal">The standard deviation of the random
variable <i>X</i> that follows Poisson distribution with parameter λ is
</span><span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_096.png" style="vertical-align: middle;" /></span></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="divTextBox divColorExampleHeading divLevel3">
<div id="202_solvedex_14">
<table border="0" cellpadding="0" cellspacing="0" id="202_solvedex_14_bookmark" style="width: 100%;">
<tbody>
<tr>
<td style="background-color: transparent; width: 100%;"><span class="ExampleHeading">Example
14:</span></td></tr>
</tbody></table>
<table border="0" cellpadding="0" cellspacing="0">
<tbody>
<tr>
<td style="background-color: transparent; width: 100%;"><span class="Normal">Random variable <i>X</i> follows Poisson
distribution with parameter λ such that P(<i>X</i> = 0) = P(<i>X</i> = 1).
Find E(<i>X</i>).</span><br />
<span class="Normal"></span><br />
<span class="SolutionHeading">Solution:</span><br />
<span class="Normal">By Poisson distribution the probability of <i>x</i>
successes is given by the following function</span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_088.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"></span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_098.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"></span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_099.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"></span><br />
<span class="Normal">Solving (i) and (ii) for λ we get, λ = 1</span><br />
<span class="Normal">For Poisson distribution, E(<i>X</i>) = λ</span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/therefore.png" style="margin-bottom: 0px; padding-bottom: 1px;" />
E(<i>X</i>) = 1</span><br />
<!--<span class="Normal"><img style="vertical-align:middle" src="202_images/202_091.png" alt="" /></span>--></td></tr>
</tbody></table>
</div>
</div>
<ol class="oLevel3 Heading4 ListStyleOL">
<li value="4">LIMITING CASE OF BINOMIAL DISTRIBUTION</li>
</ol>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="pLevel4">
<span class="Normal">For <i>n</i> <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/greater_equal.png" style="vertical-align: middle;" />
20 and <i>p</i> <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/less_equal.png" style="vertical-align: middle;" />
0.05, or <i>n</i> <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/greater_equal.png" style="vertical-align: middle;" />
100 and <i>np</i> <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/less_equal.png" style="vertical-align: middle;" />
10,the binomial distribution with parameters <i>n</i> and <i>p</i> can be
approximated by Poisson distribution by taking λ = <i>np</i>.</span></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="divTextBox divColorExampleHeading divLevel3">
<div id="202_solvedex_15">
<table border="0" cellpadding="0" cellspacing="0" id="202_solvedex_15_bookmark" style="width: 100%;">
<tbody>
<tr>
<td style="background-color: transparent; width: 100%;"><span class="ExampleHeading">Example
15:</span></td></tr>
</tbody></table>
<table border="0" cellpadding="0" cellspacing="0">
<tbody>
<tr>
<td style="background-color: transparent; width: 100%;"><span class="Normal">If 1% of Nike t-shirts are defective, what is the
probability that a carton of 50 Nike t-shirts has at least 2 defective
t-shirts?</span><br />
<span class="Normal"></span><br />
<span class="SolutionHeading">Solution:</span><br />
<span class="Normal">Let the random variable <i>X</i> represent the
number of defective t-shirts.</span><br />
<span class="Normal"><b>1</b>. The experiment consists of repeated
trials.</span><br />
<span class="Normal">All the t-shirts have to be checked for a
defect.</span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/therefore.png" style="margin-bottom: 0px; padding-bottom: 1px;" />
<i>n</i> = 50</span><br />
<span class="Normal"><b>2</b>. Either a t-shirt is defective or it’s not
defective.</span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/therefore.png" style="margin-bottom: 0px; padding-bottom: 1px;" />
Each trial can result in just two possible outcomes.</span><br />
<span class="Normal"><b>3.</b> The probability of the t-shirt being
defective is 0.01 on every trial.</span><br />
<span class="Normal"><b>4.</b> The trials are independent; that is,
getting a defective t-shirt in one trial does not affect whether we get
defective t-shirts in other trials.</span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/therefore.png" style="margin-bottom: 0px; padding-bottom: 1px;" />
<i>X</i> follows the binomial distribution with parameters <i>n</i> and
<i>p</i>.</span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/symb23.png" style="margin-bottom: 0px; padding-bottom: 1px;" />
<i>n</i> <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/greater_equal.png" style="vertical-align: middle;" />
50 and <i>p</i> <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/less_equal.png" style="vertical-align: middle;" />
0.01</span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/therefore.png" style="margin-bottom: 0px; padding-bottom: 1px;" />
B(<i>n</i>, <i>p</i>) <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/congru2.png" style="vertical-align: middle;" />
P(λ) where λ = <i>np</i> = 0.5</span><br />
<span class="Normal">The probability of finding <i>x</i> defective items
out of 50 is given by the function</span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_097.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"></span><br />
<span class="Normal">The probability of finding at least 2 defective
items out of 50 is given by</span><br />
<span class="Normal"><!--<img style="vertical-align:middle" src="202_images/202_093.png" alt="" />-->P(<i>X</i>
<img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/greater_equal.png" style="vertical-align: middle;" />
2) = 1 <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/minus.png" style="vertical-align: middle;" />
P(<i>X</i> < 2)</span><br />
<span class="Normal"><span style="margin-left: 60px;"></span><!--<img style="vertical-align:middle" src="202_images/202_094.png" alt="" /> -->=
1<img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/minus.png" style="vertical-align: middle;" />
{<i>f</i>(0) + <i>f</i>(1)}</span><br />
<span class="Normal"></span><br />
<span class="Normal"><span style="margin-left: 57px;"></span><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_201.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"><span style="margin-left: 60px;"></span><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/congru2.png" style="vertical-align: middle;" />
0.09<span style="margin-left: 25px;"></span></span></td></tr>
</tbody></table>
</div>
</div>
</div>
<div id="202_IV_C">
<table border="0" cellpadding="0" cellspacing="0" class="tableh1" id="202_IV_C_bookmark"><tbody>
<tr>
<td style="margin: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; width: 100%;"><ol class="oLevel2 Heading2">
<li value="3">HYPERGEOMETRIC
DISTRIBUTION</li>
</ol>
</td></tr>
</tbody></table>
<ol class="oLevel3 Heading3 ListStyleOL">
<li value="1">DEFINITION AND APPLICATION</li>
</ol>
<div class="pLevel4">
<span class="Normal">Hypergeometric distribution is used for
calculating probabilities for samples drawn from relatively small population and
without replacement. This means that an item's chance of being selected
increases on each trial.</span></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="pLevel4">
<span class="Normal">An experiment that follows hypergeometric
distribution should have the following properties:</span></div>
<ul class="oLevel3">
<li value="1">A sample of size <i>n</i> is randomly selected one at a time
(<i>n</i> trials) without replacement from <i>M</i> items.</li>
</ul>
<ul class="oLevel3">
<li value="2">Out of <i>M</i> items, <i>k</i> items are of one type and can be
classified as successes and <i>M</i> <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/minus.png" style="vertical-align: middle;" />
<i>k</i> items are of one type and can be classified as failures.</li>
</ul>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="pLevel4">
<span class="Normal">The discrete random variable <i>X</i> is
used to count the number of successes in the sample.</span></div>
<div class="pLevel4">
<span class="Normal">Choosing a team of 8 from a group of 12 boys
and 6 girls or a council of 5 senators from the legislature of 22 Democrats and
18 Republicans are examples of hypergeometric experiments.</span></div>
<div class="pLevel4">
<span class="Normal">If we have to choose a team of 8 with 4 boys
from a group of 10 boys and 7 girls then:</span></div>
<ul class="oLevel3">
<li value="1">A sample of size 8 is randomly selected one at a time (8 trials)
without replacement from a population of 17 students.</li>
</ul>
<ul class="oLevel3">
<li value="2">In the population, 10 (<i>k</i>) items are boys and can be
classified as successes and 7 students are girls and can be classified as
failures.</li>
</ul>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="pLevel4">
<span class="Normal">Note that it would not be an experiment that
follows binomial distribution as the probability of success is not constant on
every trial. In the beginning, the probability of selecting a boy is 10/17. If
you select a boy on the first trial, the probability of selecting a boy on the
second trial is 9/16. And if you select a girl on the first trial, the
probability of selecting a boy on the second trial is 10/16.</span></div>
<div class="pLevel4">
<span class="Normal">In a population of <i>M</i> items if
<i>k</i> items are of one type and <i>M</i> <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/minus.png" style="vertical-align: middle;" />
<i>k</i> items are of one type then the probability of selecting a sample of
size <i>n</i> (<i>n</i> trials) without replacement where <i>x</i> are of one
type and (<i>n</i> – <i>x</i>) are of the other is given by the following
function:</span></div>
<div class="pLevel4">
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_104.png" style="vertical-align: middle;" /></span></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="pLevel4">
<span class="Normal">We say that the random variable <i>X</i>
follows hypergeometric distribution with parameters <i>M</i>, <i>k</i> and
<i>n</i>.</span></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="divTextBox divColorExampleHeading divLevel4">
<div id="202_solvedex_16">
<table border="0" cellpadding="0" cellspacing="0" id="202_solvedex_16_bookmark" style="width: 100%;">
<tbody>
<tr>
<td style="background-color: transparent; width: 100%;"><span class="ExampleHeading">Example
16:</span></td></tr>
</tbody></table>
<table border="0" cellpadding="0" cellspacing="0">
<tbody>
<tr>
<td style="background-color: transparent; width: 100%;"><span class="Normal">An urn has 10 balls out of which 4 are defective. If
a sample of 5 balls is drawn, find the probability of getting exactly 2
defective balls when the sampling is done:</span><br />
<span class="Normal">(i) Without replacement</span><br />
<span class="Normal">(ii) With replacement</span><br />
<span class="Normal"></span><br />
<span class="SolutionHeading">Solution (i):</span><br />
<span class="Normal">Let the random variable <i>X</i> denote the number
of defective balls selected.</span><br />
<span class="Normal">If the sample is drawn without replacement then the
experiment follows hypergeometric distribution as </span><br />
<span class="Normal">A sample of size 5(<i>n</i>) is randomly selected
one at a time (5 trials) without replacement from a population of
10(<i>M</i>) items.</span><br />
<span class="Normal">In the population, 4 items (balls) are defective and
can be classified as successes and 10 <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/minus.png" style="vertical-align: middle;" />
4 = 6 items (balls) are not defective and can be classified as
failures.</span><br />
<span class="Normal">The probability of getting exactly <i>x</i>
defective ball’s is </span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_105.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/therefore.png" style="margin-bottom: 0px; padding-bottom: 1px;" />
The probability of getting exactly 2 defective balls is </span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_106.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"></span><br />
<span class="SolutionHeading">Solution (ii):</span><br />
<span class="Normal">If the balls are drawn with replacement then the
experiment follows binomial distribution as</span><br />
<ul class="Normal">
<li value="1">The experiment consists of repeated trials. 5 balls are
drawn.</li>
</ul>
<ul class="Normal">
<li value="2">Either the ball drawn is defective or its not. Therefore,
each trial can result in just two possible outcomes.</li>
</ul>
<ul class="Normal">
<li value="3">The probability of success is 0.4 on every trial.</li>
</ul>
<ul class="Normal">
<li value="4">The trials are independent; that is, a defective ball in one
draw does not affect whether we get defective balls in other
draws.</li>
</ul>
<span class="Normal">Here, <i>n</i> = 5 and <i>p</i> = 0.4</span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/therefore.png" style="margin-bottom: 0px; padding-bottom: 1px;" />
<i>q</i> = 1<img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/minus.png" style="vertical-align: middle;" />
<i>p</i> = 0.6</span><br />
<span class="Normal">The binomial probability for obtaining <i>x</i>
successes in <i>n</i> trials, is given by the following
function:</span><br />
<span class="Normal"><i>f</i>(<i>x</i>) =
<i><sup>n</sup></i></span></td></tr>
</tbody></table>
</div>
</div>
</div>
C<i><sub>x</sub></i> <i>p<sup>x</sup></i>
<i>q</i><sup>(<i>n</i> <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/minus.png" style="vertical-align: middle;" />
<i>x</i>)</sup> , <i>x</i> = 1, 2, 3, …., <i>n</i>
<br />
<span class="Normal">The binomial probability for obtaining 2 successes
in 5 trials is given by,</span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/therefore.png" style="margin-bottom: 0px; padding-bottom: 1px;" />
<i>f</i>(2) = <sup>5</sup>C<sub>2</sub> (0.4)<sup>2</sup>(0.6)<sup>(5 <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/minus.png" style="vertical-align: middle;" />
2) </sup>= 0.3456</span><br />
<ol class="oLevel3 Heading3 ListStyleOL">
<li value="2">PROPERTIES</li>
</ol>
<ol class="oLevel3 Heading4 ListStyleOL">
<li value="1">MEAN</li>
</ol>
<div class="pLevel4">
<span class="Normal">The mean or expected value of the random
variable <i>X</i> that follows hypergeometric distribution with parameters
<i>M</i>, <i>k</i> and <i>n</i> is </span></div>
<div class="pLevel4">
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_107.png" style="vertical-align: middle;" /></span></div>
<ol class="oLevel3 Heading4 ListStyleOL">
<li value="2">VARIANCE</li>
</ol>
<div class="pLevel4">
<span class="Normal">The variance of the random variable <i>X</i>
that follows hypergeometric distribution with parameters <i>M</i>, <i>k</i> and
<i>n </i>is </span></div>
<div class="pLevel4">
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_108.png" style="vertical-align: middle;" /></span></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<ol class="oLevel3 Heading4 ListStyleOL">
<li value="3">STANDARD DEVIATION</li>
</ol>
<div class="pLevel4">
<span class="Normal">The standard deviation of the random
variable <i>X</i> that follows hypergeometric distribution with parameters<i>
M</i>, <i>k</i> and <i>n</i> is </span></div>
<div class="pLevel4">
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_109.png" style="vertical-align: middle;" /></span></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<div id="202_V">
<table border="0" cellpadding="0" cellspacing="0" class="tableh1" id="202_V_bookmark">
<tbody>
<tr>
<td style="margin: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; width: 100%;"><ol class="oLevel1 Heading1">
<li value="5">CONTINOUS PROBABILITY
DISTRIBUTIONS</li>
</ol>
</td></tr>
</tbody></table>
<div id="202_V_A">
<table border="0" cellpadding="0" cellspacing="0" class="tableh1" id="202_V_A_bookmark">
<tbody>
<tr>
<td style="margin: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; width: 100%;"><ol class="oLevel2 Heading2">
<li value="1">UNIFORM
DISTRIBUTION</li>
</ol>
</td></tr>
</tbody></table>
<ol class="oLevel3 Heading3 ListStyleOL">
<li value="1">DEFINITION AND APPLICATION</li>
</ol>
<div class="pLevel4">
<span class="Normal">In a uniform distribution, the probability
of success is the same for all values of <i>X</i>. It is also called a
rectangular distribution.</span></div>
<div class="pLevel4">
<span class="Normal">For example, if a die is tossed we can get
the score as either 1 or 2 or 3 or 4 or 5 or 6. The probability of obtaining any
one of the six possible outcomes is 1/6. Since the probability of getting any of
the outcomes is the same, it’s a uniform distribution. This is an example of
discrete uniform distribution.</span></div>
<div class="pLevel4">
<span class="Normal">Also, if buses arrive at a given bus stop
every 15 minutes and you arrive at the bus stop at a random time, the time you
wait for the next bus to arrive follows uniform distribution in the interval [0,
15]. In this case the random variable, <i>X</i>, is continuous as it can take
any value between 0 and 15 and is not necessarily an integer. Whether a person
arrives at the bus stop 5 minutes after the last bus left or 12.34 minutes after
the last bus left, his probability of boarding the next bus will be
same.</span></div>
<div class="pLevel4">
<span class="Normal">The probability of occurrence of any event
<i>x</i> is given by the following function</span></div>
<div class="pLevel4">
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_110.png" style="vertical-align: middle;" /></span></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="pLevel4">
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_111.png" style="vertical-align: middle;" /></span></div>
<div class="pImage">
<img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_112.png" /></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="divTextBox divColorExampleHeading divLevel3">
<div id="202_solvedex_17">
<table border="0" cellpadding="0" cellspacing="0" id="202_solvedex_17_bookmark" style="width: 100%;">
<tbody>
<tr>
<td style="background-color: transparent; width: 100%;"><span class="ExampleHeading">Example
17:</span></td></tr>
</tbody></table>
<table border="0" cellpadding="0" cellspacing="0">
<tbody>
<tr>
<td style="background-color: transparent; width: 100%;"><span class="Normal">If we toss a die then what is the probability that
the die will show a number that is smaller than 4?</span><br />
<span class="SolutionHeading">Solution:</span><br />
<span class="Normal">The probability function of the experiment is
defined in <b>Example 5</b>. Each possible outcome is equally likely to
occur. Thus, we have a uniform distribution.</span><br />
<span class="Normal">The probability that the die will land on a number
smaller than 4 is equal to:</span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_113.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"></span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_114.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"></span></td></tr>
</tbody></table>
</div>
</div>
<ol class="oLevel3 Heading3 ListStyleOL">
<li value="2">PROPERTIES</li>
</ol>
<ol class="oLevel3 Heading4 ListStyleOL">
<li value="1">MEAN</li>
</ol>
<div class="pLevel4">
<span class="Normal">The mean or expected value of the random
variable <i>X</i> that follows uniform distribution in the interval [<i>a</i>,
<i>b</i>] is</span></div>
<div class="pLevel4">
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_115.png" style="vertical-align: middle;" /></span></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<ol class="oLevel3 Heading4 ListStyleOL">
<li value="2">VARIANCE</li>
</ol>
<div class="pLevel4">
<span class="Normal">The variance of the random variable <i>X</i>
that follows uniform distribution in the interval [<i>a</i>, <i>b</i>]
is</span></div>
<div class="pLevel4">
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_116.png" style="vertical-align: middle;" /></span></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<ol class="oLevel3 Heading4 ListStyleOL">
<li value="3">STANDARD DEVIATION</li>
</ol>
<div class="pLevel4">
<span class="Normal">The standard deviation of the random
variable <i>X</i> that follows uniform distribution in the interval
[<i>a</i>,<i> b</i>] is </span></div>
<div class="pLevel4">
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_117.png" style="vertical-align: middle;" />
</span><span class="Normal"></span></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="divTextBox divColorExampleHeading divLevel3">
<div id="202_solvedex_18">
<table border="0" cellpadding="0" cellspacing="0" id="202_solvedex_18_bookmark" style="width: 100%;">
<tbody>
<tr>
<td style="background-color: transparent; width: 100%;"><span class="ExampleHeading">Example
18:</span></td></tr>
</tbody></table>
<table border="0" cellpadding="0" cellspacing="0">
<tbody>
<tr>
<td style="background-color: transparent; width: 100%;"><span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_118.png" style="vertical-align: middle;" /></span>
<br />
<span class="Normal"></span><br />
<span class="SolutionHeading">Solution:</span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_119.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"></span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_120.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"></span><br />
<span class="Normal">Solving (i) and (ii) for <i>a </i>and <i>b</i> we
get,</span><br />
<span class="Normal"><i>a</i> = <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/minus.png" style="vertical-align: middle;" />1
and <i>b</i> = 3</span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_121.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"></span><br />
<span class="Normal"><span style="margin-left: 20px;"></span><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_123.png" style="vertical-align: middle;" /></span><br />
<!--<p>
<span class="Normal"><span style="margin-left:95px"></span>0<span style="margin-left:75px"></span> elsewhere</span></p>
-->
<br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_122.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"></span><br />
<span class="Normal"><span style="margin-left: 20px;"></span><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_123.png" style="vertical-align: middle;" /></span><br />
<span class="Normal">The graph of the given function is,</span><br />
<span class="pImage"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_124.png" /></span><br />
<span class="Normal">P(<i>X</i> < 0) is the area under the curve from
<img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/minus.png" style="vertical-align: middle;" />1
to 0, that is, the area under the curve shaded with forward slanting
lines.</span><br />
<span class="pImage"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_125.png" /></span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_126.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"></span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_127.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"></span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_128.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"></span></td></tr>
</tbody></table>
</div>
</div>
<!--<p class="pLevel2">
<span class="Normal"></span></p>
<div class="divTextBox divColorNormal divLevel3">
<span class="Normal"></span></p>
</div>
-->
<br />
<div class="pLevel2">
<span class="Normal"></span></div>
<div class="divTextBox divColorExampleHeading divLevel3">
<div id="202_solvedex_19">
<table border="0" cellpadding="0" cellspacing="0" id="202_solvedex_19_bookmark" style="width: 100%;">
<tbody>
<tr>
<td style="background-color: transparent; width: 100%;"><span class="ExampleHeading">Example
19:</span><br />
<span class="ExampleHeading"><br /></span></td></tr>
</tbody></table>
<table border="0" cellpadding="0" cellspacing="0">
<tbody>
<tr>
<td style="background-color: transparent; width: 100%;"><span class="Normal">The following probability distribution can be used
to represent the waiting time (<i>w</i>) of the customer in a
bank:</span><br />
<span class="Normal"><br /></span>
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_129.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"></span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_130.png" style="vertical-align: middle;" /></span><br />
<span class="Normal">The expected waiting time of the customer in the
bank E(<i>w</i>) is:</span><br />
<span class="Normal"><b>[JMET 2009]</b></span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_131.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"></span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_132.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"></span><br />
<span class="SolutionHeading">Solution:</span><br />
<span class="SolutionHeading"><br /></span>
<span class="Normal">The probability density function given in the
question is that of a random variable which follows uniform
distribution.</span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_133.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"></span><br />
<span class="Normal">Hence, <b>option 1.</b></span><br />
<br />
<span class="Normal"></span></td></tr>
</tbody></table>
</div>
</div>
</div>
<div id="202_V_B">
<table border="0" cellpadding="0" cellspacing="0" class="tableh1" id="202_V_B_bookmark">
<tbody>
<tr>
<td style="margin: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; width: 100%;"><ol class="oLevel2 Heading2">
<li value="2">NORMAL
DISTRIBUTION</li>
</ol>
</td></tr>
</tbody></table>
<ol class="oLevel3 Heading3 ListStyleOL">
<li value="1">DEFINITION AND APPLICATION</li>
</ol>
<div class="pLevel4">
<span class="Normal">The normal distribution is used to
approximately describe a random variable that tends to cluster around the mean.
For example, the weight of adult males in the Africa is normally distributed,
with a mean of 50 kilograms. This means that the weight of most men is close to
the mean, though a small number of outliers weigh significantly above or below
the mean. A histogram of male weights will appear similar to a normal curve,
with the correspondence becoming closer as the sample size increases.</span></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="pLevel4">
<span class="Normal">If the random variable <i>X</i> follows
normal distribution, then the probability of occurrence of any event <i>x</i> is
given by:</span></div>
<div class="pLevel4">
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_144.png" style="vertical-align: middle;" /></span></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="pLevel4">
<span class="Normal">Where <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/204_337.png" style="vertical-align: middle;" />
is the mean and <i><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_373.png" style="vertical-align: middle;" /></i>
is the standard deviation of the random variable <i>X</i>.</span></div>
<div class="pLevel4">
<span class="Normal">We say that the random variable <i>X</i>
follows normal distribution with parameters <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/204_337.png" style="vertical-align: middle;" />
and <i><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_373.png" style="vertical-align: middle;" /></i>.</span></div>
<div class="pLevel4">
<span class="Normal">By definition the mean, standard deviation
and variance of a random variable that follows normal distribution is <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/204_337.png" style="vertical-align: middle;" />,
<i><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_373.png" style="vertical-align: middle;" /></i>
and <i><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_373.png" style="vertical-align: middle;" /></i><sup>2</sup>.</span></div>
<ol class="oLevel3 Heading3 ListStyleOL">
<li value="2">NORMAL CURVE</li>
</ol>
<div class="pLevel4">
<span class="Normal">The graph of the probability density
function of the normal distribution is bell-shaped and is known as the bell
curve. It attains maximum value at <i>X</i> = <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/204_337.png" style="vertical-align: middle;" />.</span></div>
<div class="pLevel4">
<span class="Normal"><b>Figure 1:</b></span></div>
<div class="pImage">
<img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_145.png" />
</div>
<div class="pLevel4">
<span class="Normal"><b>Figure 2:</b></span></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="pImage">
<img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_146.png" />
</div>
<div class="pLevel4">
<span class="Normal">The graph of the normal distribution depends
on the mean and the standard deviation. The mean of the distribution determines
maxima of the graph, and the standard deviation determines its width and
height.</span></div>
<div class="pLevel4">
<span class="Normal">For a large standard deviation, the curve is
short and wide (See <b>Figure 1</b>). For a small standard deviation, the curve
is tall and narrow (See <b>Figure 2</b>). All normal distribution curves/bell
curves are symmetric irrespective of their mean or standard
deviation.</span></div>
<ol class="oLevel3 Heading4 ListStyleOL">
<li value="1">PROPERTIES OF THE NORMAL CURVE</li>
</ol>
<ul class="oLevel3">
<li value="1">The total area under the bell curve is equal to 1.</li>
</ul>
<ul class="oLevel3">
<li value="2">The probability that a normal random variable equals any
particular value is 0, that is,</li>
</ul>
<div class="pLevel5">
<span class="Normal">P(<i>X</i> = <i>a</i>) = 0.</span></div>
<div class="pLevel5">
<span class="Normal">Therefore, P(<i>X</i> < <i>a</i>) =
P(<i>X</i> <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/less_equal.png" style="vertical-align: middle;" />
<i>a</i>)</span></div>
<div class="pLevel5">
<span class="Normal">This is true in case of all continuous
random variables. </span></div>
<ul class="oLevel3">
<li value="1">The probability that <i>X</i> is greater than <i>a</i> equals the
area under the normal curve between <i>X</i> = <i>a</i> and <i>X</i> = ∞
(non-shaded area in the figure below).</li>
</ul>
<ul class="oLevel3">
<li value="2">The probability that <i>X</i> is less than <i>a</i> equals the
area under the normal curve between <i>X</i> = <i>a</i> and <i>X</i> = <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/minus.png" style="vertical-align: middle;" />
∞ (the shaded area in the figure below).</li>
</ul>
<div class="pImage">
<img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_147.png" /></div>
<ol class="oLevel3 Heading3 ListStyleOL">
<li value="3">EMPIRICAL RULE</li>
</ol>
<div class="pLevel4">
<span class="Normal">This rule holds true for every normal curve
and is independent of the standard deviation and mean of the normal
distribution.</span></div>
<ul class="oLevel3">
<li value="1">Approximately 68% of the area under the normal curve lies between
<i>X</i> = <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/204_337.png" style="vertical-align: middle;" />
– <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_373.png" style="vertical-align: middle;" />
and <i>X </i>= <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/204_337.png" style="vertical-align: middle;" />
+ <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_373.png" style="vertical-align: middle;" />
i.e. within 1 standard deviation of the mean.</li>
</ul>
<div class="pLevel4">
<span class="Normal"></span></div>
<ul class="oLevel3">
<li value="2">Approximately 95% of the area under the normal curve lies between
<i>X</i> = <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/204_337.png" style="vertical-align: middle;" />
– 2 <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_373.png" style="vertical-align: middle;" />
and <i>X</i> = <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/204_337.png" style="vertical-align: middle;" />
+ 2 <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_373.png" style="vertical-align: middle;" />
i.e. within 2 standard deviations of the mean.</li>
</ul>
<div class="pLevel4">
<span class="Normal"></span></div>
<ul class="oLevel3">
<li value="3">Approximately 99.7% of the area under the curve lies between
<i>X</i> = <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/204_337.png" style="vertical-align: middle;" />
– 3 <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_373.png" style="vertical-align: middle;" />
and <i>X</i> = <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/204_337.png" style="vertical-align: middle;" />
+ 3 <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_373.png" style="vertical-align: middle;" />
i.e. within 3 standard deviations of the mean.</li>
</ul>
<div class="pLevel4">
<span class="Normal">This rule is also known as the 68-95-99.7
rule.</span></div>
<div class="pLevel4">
<span class="Normal">For example,</span></div>
<div class="pLevel4">
<span class="Normal">Let the random variable <i>X</i> be defined
as the weight of females in London with mean as 55 kg and standard deviation as
5 kg. Then <i>X</i> is a continuous random variable as its value is not
necessarily an integer. Moreover, weight is measured not counted.</span></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="pLevel4">
<span class="Normal">The average weight of females in London is
55 kg, with a standard deviation of 5 kg. This means that 68% of the females
weigh within 5 kg of the mean (50 kg–60 kg), while 95% of females weigh within
10 kg of the mean (45–65) and 99.7% of the females weigh within 15 kg of the
mean (40 kg–70 kg).</span></div>
<ol class="oLevel3 Heading3 ListStyleOL">
<li value="4">STANDARD NORMAL DISTRIBUTION</li>
</ol>
<div class="pLevel4">
<span class="Normal">A continuous random variable <i>Z</i>
follows standard normal distribution if it follows normal distribution with mean
0 and standard deviation 1.</span></div>
<div class="pLevel4">
<span class="Normal">Probability of occurrence of any event
<i>z</i> is given by,</span></div>
<div class="pLevel4">
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_148.png" style="vertical-align: middle;" /></span></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="pLevel2">
<img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_149.png" />
</div>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="pLevel2">
<img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_150.png" /></div>
<ol class="oLevel3 Heading4 ListStyleOL">
<li value="1">READING THE STANDARD NORMAL TABLE</li>
</ol>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="pLevel4">
<span class="Normal">Using the standard normal table we can
determine the probability that P(<i>Z</i> <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/less_equal.png" style="vertical-align: middle;" />
<i>z</i>) or simply the area under the bell curve to the left of <i>Z</i> =
<i>z</i>.</span></div>
<div class="pLevel4">
<span class="Normal">In order to calculate P(<i>Z</i> <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/less_equal.png" style="vertical-align: middle;" />
<i>z</i>) using the standard normal table we follow the given steps :</span></div>
<div class="pLevel4">
<span class="Normal">Calculate the value of <i>Z</i> to 2 decimal
places. Round off the value if necessary.</span></div>
<div class="pLevel4">
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/therefore.png" style="margin-bottom: 0px; padding-bottom: 1px;" />
<i>Z</i> = <i>a</i>.<i>bc</i>, for some integer <i>a</i> and non negative
integers <i>b</i> and <i>c</i>.</span></div>
<div class="pLevel4">
<span class="Normal">Find <i>a</i>.<i>b</i> from the column
headed <i>z</i>.</span></div>
<div class="pLevel4">
<span class="Normal">Find 0.0<i>c</i> from the row whose first
element is <i>z</i>.</span></div>
<div class="pLevel4">
<span class="Normal">P(<i>Z</i> <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/less_equal.png" style="vertical-align: middle;" />
<i>z</i>) = value found at the intersection of the row and column.</span></div>
<div class="pLevel4">
<span class="Normal">For example,</span></div>
<div class="pLevel4">
<span class="Normal">P(<i>Z</i> <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/less_equal.png" style="vertical-align: middle;" />
0.45) = 0.6736</span></div>
<div class="pLevel2">
<img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_151.png" /></div>
<div class="pLevel4">
<span class="Normal">The following examples show how to use the
standard normal tables to find the probability of <i>Z</i> for different values
of its range.</span></div>
<ol class="oLevel3 ListParagraph ListStyleOL">
<li value="1">P(<i>Z</i> <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/less_equal.png" style="vertical-align: middle;" />
0.57) = 0.7157</li>
</ol>
<div class="pLevel4">
<span class="Normal">Also, P(<i>Z</i> <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/less_equal.png" style="vertical-align: middle;" />
0.57) = Area under the normal curve to the left of <i>Z</i> = 0.57 (As shown in
<b>Figure a)</b>)</span></div>
<div class="pLevel4">
<span class="Normal"><b>Figure (a):</b></span></div>
<div class="pImage">
<img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_152.png" /></div>
<div class="pLevel4">
<span class="ListParagraph"></span></div>
<ol class="oLevel3 ListParagraph ListStyleOL">
<li value="2">P(<i>Z</i> <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/less_equal.png" style="vertical-align: middle;" />
1.23) = 0.8907</li>
</ol>
<div class="pLevel4">
<span class="Normal">Also, P(<i>Z</i> <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/less_equal.png" style="vertical-align: middle;" />
1.23) = Area under the normal curve to the left of <i>Z</i> = 1.23 (As shown in
<b>Figure (b)</b>)</span></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="pLevel4">
<span class="Normal"><b>Figure (b):</b></span></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="pImage">
<img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_153.png" /></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<ol class="oLevel3 ListParagraph ListStyleOL">
<li value="3">P(<i>Z</i> > 1.23) = 1 <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/minus.png" style="vertical-align: middle;" />
P(<i>Z</i> <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/less_equal.png" style="vertical-align: middle;" />
1.23)</li>
</ol>
<div class="pLevel4">
<span class="Normal"><span style="margin-left: 75px;"></span><span style="margin-left: 25px;"></span>= 1 <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/minus.png" style="vertical-align: middle;" />
0.8907</span></div>
<div class="pLevel4">
<span class="Normal"><span style="margin-left: 95px;"></span>=
0.1093</span></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="pLevel4">
<span class="Normal">Area of the shaded region in <b>Figure
(c)</b></span></div>
<div class="pLevel4">
<span class="Normal">= Total area under the normal curve <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/minus.png" style="vertical-align: middle;" />
Area of the shaded region in <b>Figure (b)</b></span></div>
<div class="pLevel4">
<span class="Normal">= 1 <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/minus.png" style="vertical-align: middle;" />
Area of the shaded region in <b>Figure (b)</b></span></div>
<div class="pLevel4">
<span class="Normal">Also, P(<i>Z</i> > 1.23)</span></div>
<div class="pLevel4">
<span class="Normal">= Area under the normal curve to the right
of <i>Z</i></span></div>
<div class="pLevel4">
<span class="Normal">= 1.23 (As shown in <b>Figure
(c)</b>).</span></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="pLevel4">
<span class="Normal"><b>Figure (c):</b></span></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="pLevel4">
<span class="Normal"></span>
</div>
<div class="pImage">
<img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_154.png" /></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<ol class="oLevel3 ListParagraph ListStyleOL">
<li value="4">P(<i>Z</i> <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/less_equal.png" style="vertical-align: middle;" />
<img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/minus.png" style="vertical-align: middle;" />1.23)
= 1 <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/minus.png" style="vertical-align: middle;" />
P(<i>Z</i> <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/less_equal.png" style="vertical-align: middle;" />
1.23)</li>
</ol>
<div class="pLevel4">
<span class="Normal"><span style="margin-left: 115px;"></span>= 1
<img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/minus.png" style="vertical-align: middle;" />
0.8907</span></div>
<div class="pLevel4">
<span class="Normal"><span style="margin-left: 115px;"></span>=
0.1093</span></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="pLevel4">
<span class="Normal">Since, the normal curve is symmetrical about
the mean</span></div>
<div class="pLevel4">
<span class="Normal">Area of the shaded region in <b>Figure
(c)</b></span></div>
<div class="pLevel4">
<span class="Normal">= Area of the shaded region in <b>Figure
(d)</b></span></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="pLevel4">
<span class="Normal"><b>Figure (d):</b></span></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="pImage">
<img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_155.png" /></div>
<div class="pLevel4">
<span class="Normal"><i>Alternatively</i>,</span></div>
<div class="pLevel4">
<span class="Normal">We can directly determine the value of P(
<i>Z</i> < <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/minus.png" style="vertical-align: middle;" />1.23)
from the normal table that gives cumulative probability for negative <i>Z</i>
values.</span></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="pLevel4">
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/therefore.png" style="margin-bottom: 0px; padding-bottom: 1px;" />
P(<i>Z</i> < <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/minus.png" style="vertical-align: middle;" />1.23)
= 0.1093</span></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<ol class="oLevel3">
<li value="5">P (<i>Z</i> > <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/minus.png" style="vertical-align: middle;" />
1.23) = 1 <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/minus.png" style="vertical-align: middle;" />
P(<i>Z</i> <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/less_equal.png" style="vertical-align: middle;" />
<img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/minus.png" style="vertical-align: middle;" />1.23)
<span style="margin-left: 33px;"></span>…[As done in <b>3</b>]</li>
</ol>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="pLevel4">
<span class="Normal"><span style="margin-left: 120px;"></span>= 1
<img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/minus.png" style="vertical-align: middle;" />
{1 <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/minus.png" style="vertical-align: middle;" />
P(<i>Z</i> <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/less_equal.png" style="vertical-align: middle;" />
1.23)} …[As done in <b>4</b>] </span></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="pLevel4">
<span class="Normal"><span style="margin-left: 120px;"></span>=
P(<i>Z</i> <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/less_equal.png" style="vertical-align: middle;" />
1.23)</span></div>
<div class="pLevel4">
<span class="Normal"><span style="margin-left: 120px;"></span>=
0.8907</span></div>
<div class="pLevel4">
<span class="Normal">By the above example we have
that,</span></div>
<div class="pLevel4">
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/symb23.png" style="margin-bottom: 0px; padding-bottom: 1px;" />
Area of the shaded region in <b>Figure (c)</b></span></div>
<div class="pLevel4">
<span class="Normal">= Area of the shaded region in <b>Figure
(d)</b></span></div>
<div class="pLevel4">
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/therefore.png" style="margin-bottom: 0px; padding-bottom: 1px;" />
1 <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/minus.png" style="vertical-align: middle;" />
Area of the shaded region in <b>Figure (c)</b></span></div>
<div class="pLevel4">
<span class="Normal">= 1 <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/minus.png" style="vertical-align: middle;" />
Area of the shaded region in <b>Figure (d)</b></span></div>
<div class="pLevel4">
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/therefore.png" style="margin-bottom: 0px; padding-bottom: 1px;" />
Area of the shaded region in <b>Figure (b)</b></span></div>
<div class="pLevel4">
<span class="Normal">= Area of the shaded region in <b>Figure
(e)</b></span></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="pLevel4">
<span class="Normal"><b>Figure (e):</b></span></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="pImage">
<img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_157.png" /></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="pLevel4">
<span class="Normal"><i>Alternatively</i>,</span></div>
<div class="pLevel4">
<span class="Normal">P(<i>Z</i> > <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/minus.png" style="vertical-align: middle;" />
1.23) = 1 <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/minus.png" style="vertical-align: middle;" />
P(<i>Z</i> <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/less_equal.png" style="vertical-align: middle;" />
<img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/minus.png" style="vertical-align: middle;" />1.23)
…[As done in 3]</span></div>
<div class="pLevel4">
<span class="Normal"><span style="margin-left: 95px;"></span>= 1
<img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/minus.png" style="vertical-align: middle;" />
0.1093</span></div>
<div class="pLevel4">
<span class="Normal"><span style="margin-left: 95px;"></span>=
0.8907</span></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<ol class="oLevel3 ListParagraph ListStyleOL">
<li value="6">P (0.57 < <i>Z</i> < 1.72) = P(<i>Z</i> <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/less_equal.png" style="vertical-align: middle;" />
1.72) <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/minus.png" style="vertical-align: middle;" />
P(<i>Z</i> <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/less_equal.png" style="vertical-align: middle;" />
0.57)</li>
</ol>
<div class="pLevel4">
<span class="Normal"><span style="margin-left: 155px;"></span>=
0.9573 <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/minus.png" style="vertical-align: middle;" />
0.7157</span></div>
<div class="pLevel4">
<span class="Normal"><span style="margin-left: 155px;"></span>=
0.2416</span></div>
<div class="pLevel4">
<span class="Normal">Area of the shaded region in <b>Figure
(g)</b></span></div>
<div class="pLevel4">
<span class="Normal">= Area of the shaded region in <b>Figure
(f)</b> <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/minus.png" style="vertical-align: middle;" />
Area of the shaded region <b>Figure (a)</b></span></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="pLevel4">
<span class="Normal"><b>Figure (f):</b></span></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="pLevel4">
<span class="Normal"></span>
</div>
<div class="pImage">
<img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_158.png" /></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="pLevel4">
<span class="Normal"><b>Figure (g):</b></span></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="pLevel4">
<span class="Normal"></span>
</div>
<div class="pImage">
<img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_159.png" /></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<ol class="oLevel3 ListParagraph ListStyleOL">
<li value="7">P(<img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/minus.png" style="vertical-align: middle;" />0.57
< <i>Z</i> < 1.72) = P(<i>Z</i> <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/less_equal.png" style="vertical-align: middle;" />
1.72) <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/minus.png" style="vertical-align: middle;" />
P(<i>Z</i> <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/less_equal.png" style="vertical-align: middle;" />
<img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/minus.png" style="vertical-align: middle;" />0.57)</li>
</ol>
<div class="pLevel4">
<span class="Normal"><span style="margin-left: 155px;"></span>=
P(<i>Z</i> <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/less_equal.png" style="vertical-align: middle;" />
1.72) <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/minus.png" style="vertical-align: middle;" />
{1 <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/minus.png" style="vertical-align: middle;" />
P(<i>Z</i> <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/less_equal.png" style="vertical-align: middle;" />
0.57)}</span></div>
<div class="pLevel4">
<span class="Normal"><span style="margin-left: 155px;"></span>=
0.9573 <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/minus.png" style="vertical-align: middle;" />
0.2843</span></div>
<div class="pLevel4">
<span class="Normal"><span style="margin-left: 155px;"></span>=
0.673</span></div>
<div class="pLevel4">
<span class="Normal">Area of the shaded region in <b>Figure
(i)</b></span></div>
<div class="pLevel4">
<span class="Normal">= Area of the shaded region in <b>Figure
(f)</b> <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/minus.png" style="vertical-align: middle;" />
Area of the shaded region in <b>Figure (h)</b></span></div>
<div class="pLevel4">
<span class="Normal">= Area of the shaded region in <b>Figure
(f)</b> <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/minus.png" style="vertical-align: middle;" />
{1 <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/minus.png" style="vertical-align: middle;" />
Area of the shaded region in <b>Figure (a)</b>}</span></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="pLevel4">
<span class="Normal"><b>Figure (h):</b></span></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="pLevel4">
<span class="Normal"></span>
</div>
<div class="pImage">
<img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_160.png" /></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="pLevel4">
<span class="Normal"><b>Figure (i):</b></span></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="pLevel4">
<span class="Normal"></span>
</div>
<div class="pImage">
<img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_161.png" /></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<ol class="oLevel3 ListParagraph ListStyleOL">
<li value="8">P (<img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/minus.png" style="vertical-align: middle;" />1.72
< <i>Z</i> < <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/minus.png" style="vertical-align: middle;" />0.57)
= P(<i>Z</i> <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/less_equal.png" style="vertical-align: middle;" />
<img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/minus.png" style="vertical-align: middle;" />0.57)
<img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/minus.png" style="vertical-align: middle;" />
P(<i>Z</i> <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/less_equal.png" style="vertical-align: middle;" />
<img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/minus.png" style="vertical-align: middle;" />1.72)</li>
</ol>
<div class="pLevel4">
<span class="ListParagraph"></span></div>
<div class="pLevel4">
<span class="Normal"><span style="margin-left: 170px;"></span>=
0.2843 <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/minus.png" style="vertical-align: middle;" />
0.0427</span></div>
<div class="pLevel4">
<span class="Normal"><span style="margin-left: 170px;"></span>=
0.2416</span></div>
<div class="pLevel4">
<span class="Normal">Area of the shaded region in <b>Figure
(k)</b></span></div>
<div class="pLevel4">
<span class="Normal">= Area of the shaded region in <b>Figure
(j)</b> <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/minus.png" style="vertical-align: middle;" />
Area of the shaded region in <b>Figure (h)</b></span></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="pLevel4">
<span class="Normal"><b>Figure (j):</b></span></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="pImage">
<img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_162.png" /></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="pLevel4">
<span class="Normal"><b>Figure (k):</b></span></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="pLevel4">
<span class="Normal"></span>
</div>
<div class="pImage">
<img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_163.png" /></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<ol class="oLevel3 ListParagraph ListStyleOL">
<li value="9">P</li>
</ol>
<br />
<br />
<br />
<br />
<br /></div>
</div>
(<img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/minus.png" style="vertical-align: middle;" />1.72
< <i>Z</i> and <i>Z</i> > 0.57) = P(<i>Z</i> > 0.57) + P(<i>Z</i>
<img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/less_equal.png" style="vertical-align: middle;" />
<img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/minus.png" style="vertical-align: middle;" />1.72)
<br />
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="pLevel4">
<span class="Normal"><span style="margin-left: 200px;"></span>= {1
<img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/minus.png" style="vertical-align: middle;" />
P(<i>Z</i> <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/less_equal.png" style="vertical-align: middle;" />
0.57)} + {1 <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/minus.png" style="vertical-align: middle;" />
P(<i>Z</i> <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/less_equal.png" style="vertical-align: middle;" />
1.72)}</span></div>
<div class="pLevel4">
<span class="Normal"><span style="margin-left: 200px;"></span>= 1
<img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/minus.png" style="vertical-align: middle;" />
0.7157 + ( 1 <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/minus.png" style="vertical-align: middle;" />
0.9573)</span></div>
<div class="pLevel4">
<span class="Normal"><span style="margin-left: 200px;"></span>=
0.2843 + 0.0427</span></div>
<div class="pLevel4">
<span class="Normal"><span style="margin-left: 200px;"></span>=
0.327</span></div>
<div class="pLevel4">
<span class="Normal">Area of the shaded region in <b>Figure
(m)</b></span></div>
<div class="pLevel4">
<span class="Normal">= Area of the shaded region in <b>Figure
(l)</b> + Area of the shaded region in <b>Figure (j)</b></span></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="pLevel4">
<span class="Normal"><b>Figure (l):</b></span></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="pLevel4">
<span class="Normal"></span>
</div>
<div class="pImage">
<img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_164.png" /></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="pLevel4">
<span class="Normal"><b>Figure (m)</b></span></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="pLevel4">
<span class="Normal"></span>
</div>
<div class="pImage">
<img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_165.png" /></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="divTextBox divColorNote divLevel3">
<span class="Note">REMEMBER</span><br />
<span class="Normal"></span><br />
<ul type="disc">
<li><b>For <i>Z</i> < 0</b>,</li>
</ul>
<ul>The area under the bell curve < 0.5</ul>
<ul type="disc">
<li><b>For <i>Z</i> > 0,</b></li>
</ul>
<ul>The area under the bell curve > 0.5</ul>
</div>
<div class="pLevel4">
<span class="Normal"></span></div>
<ol class="oLevel3 Heading4 ListStyleOL">
<li value="2">CONVERTING NORMAL VARIABLE TO STANDARD NORMAL VARIABLE</li>
</ol>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="pLevel4">
<span class="Normal">If a continuous random variable <i>X</i>
follows normal distribution with parameters <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/204_337.png" style="vertical-align: middle;" />
and <i><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_373.png" style="vertical-align: middle;" /></i>
then <i>X</i> can be converted to a standard normal variable, <i>Z</i>, by
substituting the value of <i>Z</i> as</span></div>
<div class="pLevel4">
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_166.png" style="vertical-align: middle;" /></span></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="pLevel4">
<span class="Normal">Using (i) we get the following:</span></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="pLevel4">
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_167.png" style="vertical-align: middle;" /></span></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="pLevel4">
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_168.png" style="vertical-align: middle;" /></span></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="pLevel4">
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_169.png" style="vertical-align: middle;" /></span></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="divTextBox divColorExampleHeading divLevel3">
<div id="202_solvedex_20">
<table border="0" cellpadding="0" cellspacing="0" id="202_solvedex_20_bookmark" style="width: 100%;">
<tbody>
<tr>
<td style="background-color: transparent; width: 100%;"><span class="ExampleHeading">Example
20:</span></td></tr>
</tbody></table>
<table border="0" cellpadding="0" cellspacing="0">
<tbody>
<tr>
<td style="background-color: transparent; width: 100%;"><span class="Normal">If <i>X</i> follows a normal distribution with mean
50 and standard deviation 5, then find the values of the following using
standard normal tables.</span><br />
<span class="Normal">(i) P(<i>X</i> < 45)</span><br />
<span class="Normal">(ii) P(<i>X</i> > 60)</span><br />
<span class="Normal">(iii) P(48 < <i>X</i> < 55)</span><br />
<span class="Normal"></span><br />
<span class="SolutionHeading">Solution:</span><br />
<span class="Normal">(i) We will first find the <i>Z</i> value of
45.</span><br />
<span class="Normal"></span><br />
<ul>
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_170.png" style="vertical-align: middle;" />
<img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_171.png" style="vertical-align: middle;" /></span></ul>
<span class="Normal"></span><br />
<ul>
<span class="Normal">P(<i>X <</i> 45) = P(<i>Z <</i> <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/minus.png" style="vertical-align: middle;" />1)
= 0.1587</span></ul>
<ul>
<span class="Normal"></span></ul>
<span class="Normal"></span><br />
<span class="Normal">(ii) We will first find the <i>Z</i> value of
60.</span><br />
<ul>
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_172.png" style="vertical-align: middle;" />
<img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_173.png" style="vertical-align: middle;" /></span></ul>
<span class="Normal"></span><br />
<span class="Normal"></span><br />
<ul>
<span class="Normal">P(<i>X</i> > 60) = 1 <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/minus.png" style="vertical-align: middle;" />
P (<i>X</i> <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/less_equal.png" style="vertical-align: middle;" />
60) </span></ul>
<ul>
<span class="Normal"><span style="margin-left: 68px;"></span>= 1 <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/minus.png" style="vertical-align: middle;" />
P(<i>Z</i> <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/less_equal.png" style="vertical-align: middle;" />
2)</span></ul>
<ul>
<span class="Normal"><span style="margin-left: 68px;"></span>= 1 <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/minus.png" style="vertical-align: middle;" />
0.9772</span></ul>
<ul>
<span class="Normal"><span style="margin-left: 68px;"></span>=
0.0228</span></ul>
<div class="Normal" span="">
</div>
</td></tr>
</tbody></table>
</div>
</div>
<br />
<br />
<span class="Normal">(iii)We will first find the <i>Z</i> value of 48 and
55.</span><br />
<span class="Normal"></span><br />
<ul>
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_174.png" style="vertical-align: middle;" /></span></ul>
<span class="Normal"></span><br />
<ul>
<span class="Normal" style="margin-left: 55px;"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_175.png" style="vertical-align: middle;" /></span></ul>
<span class="Normal"></span><br />
<ul>
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_176.png" style="vertical-align: middle;" /></span></ul>
<span class="Normal"></span><br />
<ul>
<span class="Normal">P(48 < <i>X</i> < 55) = P(<img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/minus.png" style="vertical-align: middle;" />0.4
< <i>Z</i> < 1)</span></ul>
<ul>
<span class="Normal"></span></ul>
<ul>
<span class="Normal"><span style="margin-left: 105px;"></span>=
P(<i>Z</i> <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/less_equal.png" style="vertical-align: middle;" />
1) <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/minus.png" style="vertical-align: middle;" />
P(<i>Z</i> <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/less_equal.png" style="vertical-align: middle;" />
<img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/minus.png" style="vertical-align: middle;" />0.4)</span></ul>
<ul>
<span class="Normal"><span style="margin-left: 105px;"></span>= 0.8413
<img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/minus.png" style="vertical-align: middle;" />
0.3446</span></ul>
<ul>
<span class="Normal"><span style="margin-left: 105px;"></span>=
0.4967</span></ul>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="divTextBox divColorExampleHeading divLevel3">
<div id="202_solvedex_21">
<table border="0" cellpadding="0" cellspacing="0" id="202_solvedex_21_bookmark" style="width: 100%;">
<tbody>
<tr>
<td style="background-color: transparent; width: 100%;"><span class="ExampleHeading">Example
21:</span></td></tr>
</tbody></table>
<table border="0" cellpadding="0" cellspacing="0">
<tbody>
<tr>
<td style="background-color: transparent; width: 100%;"><span class="Normal">The diameter of the toothpaste tube manufactured by
HUL is a random variable with mean as 2 cm and standard deviation as 0.1
cm. For a particular order it is required that the diameter of the tube
should not be less than or equal to 1.95 and it should not be greater than
or equal to 2.3 cm. What proportion of the toothpaste tubes made would
meet the requirement?</span><br />
<span class="SolutionHeading">Solution:</span><br />
<span class="Normal">Let the random variable <i>X</i> be defined as the
diameter of the toothpaste tube.</span><br />
<span class="Normal">The random variable <i>X</i> follows normal
distribution with mean 2 and standard deviation 0.1.</span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/therefore.png" style="margin-bottom: 0px; padding-bottom: 1px;" />
Proportion of the toothpaste tubes with diameter between 1.95 and 2.3 = P
( 1.95 < <i>X</i> < 2.3)</span><br />
<span class="Normal"></span><br />
<span class="Normal">We will first find the <i>Z</i> value of 1.95 and
2.3.</span><br />
<span class="Normal"></span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_177.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"></span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_178.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"></span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_179.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"></span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/therefore.png" style="margin-bottom: 0px; padding-bottom: 1px;" />
P( 1.95 < <i>X</i> < 2.3)</span><br />
<span class="Normal">= P(<i>Z</i> < 3) <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/minus.png" style="vertical-align: middle;" />
P(<i>Z</i> < 0.5)</span><br />
<span class="Normal">= 0.9987 <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/minus.png" style="vertical-align: middle;" />
0.6915 ... [From the standard normal tables]</span><br />
<span class="Normal">=
0.3072</span></td></tr>
</tbody></table>
</div>
</div>
<div id="202_VI">
<table border="0" cellpadding="0" cellspacing="0" class="tableh1" id="202_VI_bookmark">
<tbody>
<tr>
<td style="margin: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; width: 100%;"><ol class="oLevel1 Heading1">
<li value="6">DETERMINING
DISTRIBUTIONS</li>
</ol>
</td></tr>
</tbody></table>
<div class="pLevel2">
<span class="Normal"></span></div>
<div class="divTextBox divColorExampleHeading divLevel3">
<div id="202_solvedex_22">
<table border="0" cellpadding="0" cellspacing="0" id="202_solvedex_22_bookmark" style="width: 100%;">
<tbody>
<tr>
<td style="background-color: transparent; width: 100%;"><span class="ExampleHeading">Example
22:</span></td></tr>
</tbody></table>
<table border="0" cellpadding="0" cellspacing="0">
<tbody>
<tr>
<td style="background-color: transparent; width: 100%;"><span class="Normal"><b>Question 1</b>:</span><br />
<span class="Normal">The 5’s, 10's and 20's biscuit packets are pro duced
in lot sizes of 100 each. Three packets each of 5's, 10's and 20's are
inspected at random. If even one biscuit is found broken in any of the
three, the respective lot is rejected, the probability that 1 broken
biscuit will be found in a 5's, 10's or 20's packet is estimated to be
0.10, 0.20 and 0.30 respectively. If a sample of three packets each of
5's, 10's and 20's is inspected, which probability distribution should we
use to estimate the probability that all nine packets will be
accepted?</span><br />
<span class="Normal"></span><br />
<span class="Normal">(1) Normal<span style="margin-left: 25px;"></span><span style="margin-left: 28px;"></span>(2) Binomial</span><br />
<span class="Normal">(3) Poisson <span style="margin-left: 25px;"></span><span style="margin-left: 25px;"></span>(4) Hypergeometric</span><br />
<span class="Normal"><b>Question 2</b>:</span><br />
<span class="Normal">What is the probability that all nine packets will
be accepted?</span><br />
<span class="Normal">(1) 0.749 <span style="margin-left: 25px;"></span><span style="margin-left: 25px;"></span>(2) 0.006 <span style="margin-left: 25px;"></span><span style="margin-left: 25px;"></span></span><br />
<span class="Normal">(3) 0.128 <span style="margin-left: 25px;"></span><span style="margin-left: 25px;"></span>(4) 0.504</span><br />
<span class="Normal"><b>[JMET 2008]</b></span><br />
<span class="Normal"></span><br />
<span class="SolutionHeading">Solution 1:</span><br />
<span class="Normal">Normal distribution is a continuous distribution
that clutters around a mean or average.</span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/therefore.png" style="margin-bottom: 0px; padding-bottom: 1px;" />
We can eliminate option 1.</span><br />
<span class="Normal">The hypergeometric distribution is a discrete
probability distribution that describes the number of successes in a
sequence of <i>n</i> draws from a finite population without replacement,
just as the binomial distribution describes the number of successes for
draws with replacement.</span><br />
<span class="Normal">In the question, the probability that a biscuit is
broken is same for each of the 3 packets picked up from the same lot. And
here we assume that the packets are picked with replacement. Therefore, we
may use either Binomial or Poisson distribution to evaluate the
probability.</span><br />
<span class="Normal">The binomial distribution converges towards the
Poisson distribution for <i>n</i> <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/greater_equal.png" style="vertical-align: middle;" />
20 and <i>p</i> <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/less_equal.png" style="vertical-align: middle;" />
0.05.</span><br />
<span class="Normal">Here, <i>n</i> = 3 and <i>n</i> < 20.</span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/therefore.png" style="margin-bottom: 0px; padding-bottom: 1px;" />
We can also eliminate option 3.</span><br />
<span class="Normal">Hence, <b>option 2</b>.</span><br />
<span class="Normal"></span><br />
<span class="SolutionHeading">Solution 2:</span><br />
<span class="Normal">Probability of not having any defective packs
containing 5 biscuits = (1 <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/minus.png" style="vertical-align: middle;" />
0.1)<sup>3</sup> = 0.93 = 0.729</span><br />
<span class="Normal">Similarly, probability of not having any defective
packs containing 10 and 20 biscuits is 0.512 and 0.343.</span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/therefore.png" style="margin-bottom: 0px; padding-bottom: 1px;" />
Total probability of all packets being accepted</span><br />
<span class="Normal">= 0.729 <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/multiplication.png" style="margin-bottom: 0px; padding-bottom: 0px;" />
0.512 <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/multiplication.png" style="margin-bottom: 0px; padding-bottom: 0px;" />
0.343 = 0.128</span><br />
<span class="Normal">Hence, <b>option
3</b>.</span></td></tr>
</tbody></table>
</div>
</div>
<!--<p class="pLevel2">
<span class="Normal"></span></p>
<div class="divTextBox divColorNormal divLevel2">
<p>
<span class="Normal"></span></p>
<p>
<span class="Normal"></span></p>
</div>
-->
<br />
<div class="pLevel2">
<span class="Normal"></span></div>
<div class="divTextBox divColorExampleHeading divLevel3">
<div id="202_solvedex_23">
<table border="0" cellpadding="0" cellspacing="0" id="202_solvedex_23_bookmark" style="width: 100%;">
<tbody>
<tr>
<td style="background-color: transparent; width: 100%;"><span class="ExampleHeading">Example
23:</span></td></tr>
</tbody></table>
<table border="0" cellpadding="0" cellspacing="0">
<tbody>
<tr>
<td style="background-color: transparent; width: 100%;"><span class="Normal">The number of shoppers entering a retail store in an
hour is a good example of a random variable that follows:</span><br />
<span class="Normal"><b>[JMET 2009]</b></span><br />
<span class="Normal">(1) Normal distribution</span><br />
<span class="Normal">(2) Binomial distribution</span><br />
<span class="Normal">(3) Poisson distribution</span><br />
<span class="Normal">(4) None of the above</span><br />
<span class="Normal"></span><br />
<span class="SolutionHeading">Solution:</span><br />
<span class="Normal">If a random variable <i>X</i> is defined as the
number of shoppers entering a retail store in an hour, then <i>X</i> is a
discrete random variable as the number of shoppers entering the store per
hour, though large, can be counted and will always take an integral
value.</span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/therefore.png" style="margin-bottom: 0px; padding-bottom: 1px;" />
We can eliminate option 1 as it’s a continuous probability
distribution.</span><br />
<span class="Normal">Here we don’t know the probability of the shopper
entering the store but we know the number of shoppers entering the shop
per hour i.e. λ.</span><br />
<span class="Normal">Moreover,</span><br />
<span class="Normal">The average number of shoppers (successes) that
enter per hour (specified region) is known.</span><br />
<span class="Normal">The probability of “no shopper” entering the retail
store decreases as the number of hours increase.</span><br />
<span class="Normal">Also, the probability of shoppers entering the store
per 5 seconds (very small region) will be very small or virtually
zero.</span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/therefore.png" style="margin-bottom: 0px; padding-bottom: 1px;" />
The number of shoppers entering in a retail store in an hour is a good
example of the random variable that follows Poisson
distribution.</span><br />
<span class="Normal">Hence, <b>option
3</b>.</span></td></tr>
</tbody></table>
</div>
</div>
<div id="202_VII">
<table border="0" cellpadding="0" cellspacing="0" class="tableh1" id="202_VII_bookmark">
<tbody>
<tr>
<td style="margin: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; width: 100%;"><ol class="oLevel1 Heading1">
<li value="7">SUMMARY OF
DISTRIBUTIONS</li>
</ol>
</td></tr>
</tbody></table>
<div class="pLevel2">
<img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_180.png" /></div>
<div class="pLevel2">
<span class="ListParagraph"></span></div>
<div class="pImage">
<img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_181.png" /></div>
<div class="pLevel2">
<span class="Normal"></span></div>
</div>
<div id="202_VIII">
<table border="0" cellpadding="0" cellspacing="0" class="tableh1" id="202_VIII_bookmark"><tbody>
<tr>
<td style="margin: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; width: 100%;"><ol class="oLevel1 Heading1">
<li value="8">COEFFICIENT OF CORRELATION AND
LINEARITY</li>
</ol>
</td></tr>
</tbody></table>
<div id="202_VIII_A">
<table border="0" cellpadding="0" cellspacing="0" class="tableh1" id="202_VIII_A_bookmark">
<tbody>
<tr>
<td style="margin: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; width: 100%;"><ol class="oLevel2 Heading2">
<li value="1">DEFINITION</li>
</ol>
</td></tr>
</tbody></table>
<div class="pLevel3">
<span class="ListParagraph"></span></div>
<div class="pLevel3">
<span class="Normal">Two variables <i>p</i> and <i>q</i> are said
to share a linear relationship if the graph of <i>p</i> when plotted against
<i>q</i> or <i>q</i> when plotted against <i>p</i> is a straight
line.</span></div>
<div class="pLevel3">
<span class="Normal"></span></div>
<div class="pLevel3">
<span class="Normal">Coefficient of correlation is used to
measure the degree of the linear dependence between any two random variables
<i>X</i> and <i>Y</i>.</span></div>
<div class="pLevel3">
<span class="Normal">It is represented by ρ and its value ranges
between <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/minus.png" style="vertical-align: middle;" />1
and 1.</span></div>
<div class="pLevel3">
<span class="Normal"></span></div>
<div class="pLevel3">
<span class="Normal">The co-efficient of correlation,
<i>ρ<sub>x</sub></i></span></div>
</div>
</div>
</div>
, <i>Y</i> between two random variables <i>X</i> and
<i>Y</i> is defined as
<br />
<div class="pLevel3">
<span class="Normal"></span></div>
<div class="pLevel3">
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_182.png" style="vertical-align: middle;" /></span></div>
<div class="pLevel3">
<span class="Normal"></span></div>
<div class="pLevel3">
<span class="Normal">Where <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/204_337.png" style="vertical-align: middle;" /><i><sub>X</sub></i>
and <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_373.png" style="vertical-align: middle;" /><i><sub>X</sub></i>
are the mean and standard deviation respectively, of random variable <i>X</i>,
and <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/204_337.png" style="vertical-align: middle;" /><sub>Y</sub>
and <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_373.png" style="vertical-align: middle;" /><i><sub>Y</sub></i></span></div>
are the mean and standard deviation respectively, of random variable
<i>Y</i>.
<br />
<div class="pLevel3">
<span class="Normal">or</span></div>
<div class="pLevel3">
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_183.png" style="vertical-align: middle;" /></span></div>
<div class="pLevel3">
<span class="Normal"></span></div>
<div class="pLevel3">
<span class="Normal"><b>Note</b>: The coefficient of correlation
for two random variables is defined only if their standard deviations are non
zero.</span></div>
<div class="pLevel3">
<span class="Normal"></span></div>
<div id="202_VIII_B">
<table border="0" cellpadding="0" cellspacing="0" class="tableh1" id="202_VIII_B_bookmark">
<tbody>
<tr>
<td style="margin: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; width: 100%;"><ol class="oLevel2 Heading2">
<li value="2">INTERPRETATION</li>
</ol>
</td></tr>
</tbody></table>
<div class="pLevel3">
<span class="ListParagraph"></span></div>
<div class="pLevel3">
<span class="Normal">The greater the absolute value of the
coefficient of correlation (i.e. the closest the value is to <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/minus.png" style="vertical-align: middle;" />1
or 1), the greater is the linear dependence between the variables.</span></div>
<div class="pLevel3">
<span class="Normal">Coefficient of correlation equal to <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/minus.png" style="vertical-align: middle;" />1
or 1 indicates the strongest linear relationship. In this case data points fall
exactly in a straight line as shown in the following scatter plots.</span></div>
<div class="pLevel3">
<span class="Normal">For <i>ρ</i> = 1</span></div>
<div class="pLevel3">
<span class="Normal"></span></div>
<div class="pImage">
<img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_185.png" /></div>
<div class="pLevel3">
<span class="Normal">For <i>ρ</i> = <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/minus.png" style="vertical-align: middle;" />1</span></div>
<div class="pLevel3">
<span class="Normal"></span></div>
<div class="pImage">
<img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_186.png" /></div>
<div class="pLevel3">
<span class="Normal"></span></div>
<div class="pLevel3">
<span class="Normal">If 0 < <i>ρ</i> <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/less_equal.png" style="vertical-align: middle;" />
1, then there exists an increasing linear relation between random variables
<i>X</i> and Y. In this case, the value of <i>Y</i> increases as <i>X</i>
increases as shown in the scatter plot below.</span></div>
<div class="pLevel3">
<span class="Normal"></span></div>
<div class="pLevel3">
<span class="Normal">For <i>ρ</i> = 0.46</span></div>
<div class="pLevel3">
<span class="Normal"></span></div>
<div class="pImage">
<img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_184.png" /></div>
<div class="pLevel3">
<span class="Normal"></span></div>
<div class="pLevel3">
<span class="Normal">If <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/minus.png" style="vertical-align: middle;" />1
<img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/less_equal.png" style="vertical-align: middle;" />
<i>ρ</i> < 0, then there exists a decreasing linear relation between random
variables <i>X</i> and <i>Y</i>. In this case, the value of <i>Y</i> decreases
as <i>X</i> increases as shown in the following scatter plot.</span></div>
<div class="pLevel3">
<span class="Normal"></span></div>
<div class="pLevel3">
<span class="Normal">For <i>ρ</i> = <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/minus.png" style="vertical-align: middle;" />0.54</span></div>
<div class="pLevel3">
<span class="Normal"></span></div>
<div class="pImage">
<img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_187.png" /></div>
<div class="pLevel3">
<span class="Normal"></span></div>
<div class="pLevel3">
<span class="Normal">If <i>ρ</i> = 0 then the variables are
linearly independent.</span></div>
<div class="pLevel3">
<span class="Normal"></span></div>
<div class="pLevel3">
<span class="Normal">For <i>ρ</i> = 0</span></div>
<div class="pImage">
<img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_188.png" /></div>
<div class="pLevel3">
<span class="Normal"></span></div>
<div class="divTextBox divColorNote divLevel3">
<span class="Note">REMEMBER</span><br />
<span class="Normal"></span><br />
<ul type="disc">
<li>The coefficient of correlation measures the linear relationship between
variables.</li>
</ul>
<ul>For example, </ul>
<ul>A random variable <i>X</i> is uniformly distributed over the interval
[<img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/minus.png" style="vertical-align: middle;" />1,
1] and <i>Y</i> = <i>X</i><sup>3</sup>.</ul>
<ul>The value of <i>Y</i> is totally dependent on <i>X</i> but the value of
the coefficient of correlation of <i>X</i> and <i>Y</i> is zero as <i>X</i>
and <i>Y</i> share a curvilinear relationship.</ul>
<ul type="disc">
<li>Correlation is affected by outliers. One outlier can considerably reduce
the value of the coefficient of correlation as shown in the scatter
plot.</li>
</ul>
<span class="Normal"></span><br />
<ul>For <i>ρ</i> = 0.71</ul>
<span class="pImage"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_189.png" />
</span></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="divTextBox divColorExampleHeading divLevel3">
<div id="202_solvedex_24">
<table border="0" cellpadding="0" cellspacing="0" id="202_solvedex_24_bookmark" style="width: 100%;">
<tbody>
<tr>
<td style="background-color: transparent; width: 100%;"><span class="ExampleHeading">Example
24:</span></td></tr>
</tbody></table>
<table border="0" cellpadding="0" cellspacing="0">
<tbody>
<tr>
<td style="background-color: transparent; width: 100%;"><span class="Normal">An investment consultant has plotted the graph
between BSE Sensex and Indian inflation rate as shown in the graph to
better understand their relationship.</span><br />
<span class="Normal"></span><br />
<span class="pImage"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_190.png" /></span><br />
<span class="Normal"></span><br />
<span class="Normal">What is the coefficient of correlation between BSE
Sensex and the Indian inflation rate?</span><br />
<span class="Normal"><b>[JMET 2009]</b></span><br />
<span class="Normal"></span><br />
<span class="Normal">(1) <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/minus.png" style="vertical-align: middle;" />1000<span style="margin-left: 25px;"></span><span style="margin-left: 25px;"></span>(2) <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/minus.png" style="vertical-align: middle;" />0.05</span><br />
<span class="Normal">(3) <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/minus.png" style="vertical-align: middle;" />1<span style="margin-left: 25px;"></span><span style="margin-left: 50px;"></span>(4) <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/minus.png" style="vertical-align: middle;" />2000</span><br />
<span class="Normal"></span><br />
<span class="SolutionHeading">Solution:<span style="margin-left: 25px;"></span></span><br />
<span class="Normal">Since the BSE sensex decreases linearly when the
inflation rate increases linearly, the coefficient of correlation has to
be less than 0. Also, the points plotted fall exactly in a straight line
indicating a strong linear relationship.</span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/therefore.png" style="margin-bottom: 0px; padding-bottom: 1px;" />
The coefficient of correlation is <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/minus.png" style="vertical-align: middle;" />1.</span><br />
<span class="Normal">Hence, <b>option
3</b>.</span></td></tr>
</tbody></table>
</div>
</div>
<div class="pLevel4">
<span class="Normal"></span></div>
</div>
<div id="202_IX">
<table border="0" cellpadding="0" cellspacing="0" class="tableh1" id="202_IX_bookmark">
<tbody>
<tr>
<td style="margin: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; width: 100%;"><ol class="oLevel1 Heading1">
<li value="9">COEFFICIENT OF
VARIATION</li>
</ol>
</td></tr>
</tbody></table>
<div class="pLevel2">
<span class="Normal"></span></div>
<div id="202_IX_A">
<table border="0" cellpadding="0" cellspacing="0" class="tableh1" id="202_IX_A_bookmark"><tbody>
<tr>
<td style="margin: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; width: 100%;"><ol class="oLevel2 Heading2">
<li value="1">DEFINITION</li>
</ol>
</td></tr>
</tbody></table>
<div class="pLevel3">
<span class="Normal">Coefficient of variation is defined as the
ratio of mean and standard deviation of a data set or a random
variable.</span></div>
<div class="pLevel3">
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_191.png" style="vertical-align: middle;" /></span></div>
<div class="pLevel3">
<span class="Normal"></span></div>
<div class="pLevel3">
<span class="Normal">For example,</span></div>
<div class="pLevel3">
<span class="Normal">A random variable <i>X</i> is defined as the
share price of company P and the mean of <i>X </i>is Rs. 5,000/month. If the
coefficient of variation is 0.02 then the standard deviation will be Rs.
100.</span></div>
<div class="pLevel3">
<span class="Normal"></span></div>
<div class="pLevel3">
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/therefore.png" style="margin-bottom: 0px; padding-bottom: 1px;" />
By the empirical rule, 95% of the time the price of the share ranges between Rs.
4,800 and Rs 5,200(See figure below).</span></div>
<div class="pImage">
<img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_193.png" /></div>
<div class="pLevel3">
<span class="Normal"></span></div>
<div class="pLevel3">
<span class="Normal">If the coefficient of variation is 0.2 then
the standard deviation will be Rs. 1,000.</span></div>
<div class="pLevel3">
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/therefore.png" style="margin-bottom: 0px; padding-bottom: 1px;" />
By the empirical rule, 95% of the time the price of the share ranges between Rs.
3,000 and Rs 7,000(See figure below).</span></div>
<div class="pLevel3">
<span class="Normal"></span></div>
<div class="pImage">
<img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/300f0de9-6243-44da-8da8-6ddcaeed0cce/202_images/202_194.png" /></div>
<div class="pLevel3">
<span class="Normal"></span></div>
<div class="pLevel3">
<span class="Normal">Thus, a smaller coefficient of variation
means lesser fluctuations, and therefore a more stable trading
environment.</span></div>
<div class="pLevel3">
<span class="Normal"></span></div>
</div>
<div id="202_IX_B">
<table border="0" cellpadding="0" cellspacing="0" class="tableh1" id="202_IX_B_bookmark"><tbody>
<tr>
<td style="margin: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; width: 100%;"><ol class="oLevel2 Heading2">
<li value="2">APPLICATIONS</li>
</ol>
</td></tr>
</tbody></table>
<div class="pLevel3">
<span class="Normal">It helps in comparing one data series to the
other even if their means are different.</span></div>
<div class="pLevel3">
<span class="Normal">It helps to analyse the amount of risk taken
as compared to the return expected on the investment.</span></div>
<div class="pLevel3">
<span class="Normal"></span></div>
<div class="divTextBox divColorExampleHeading divLevel3">
<div id="202_solvedex_25">
<table border="0" cellpadding="0" cellspacing="0" id="202_solvedex_25_bookmark" style="width: 100%;">
<tbody>
<tr>
<td style="background-color: transparent; width: 100%;"><span class="ExampleHeading">Example
25:</span></td></tr>
</tbody></table>
<table border="0" cellpadding="0" cellspacing="0">
<tbody>
<tr>
<td style="background-color: transparent; width: 100%;"><span class="Normal">Coefficient of variation is useful to
study</span><br />
<span class="Normal"><b>[SNAP 2008]</b></span><br />
<span class="Normal">(1) Risk<span style="margin-left: 25px;"></span><span style="margin-left: 50px;"></span> (2) Disparity</span><br />
<span class="Normal">(3) Consistency<span style="margin-left: 25px;"></span>(4) All of the above</span><br />
<span class="Normal"></span><br />
<span class="SolutionHeading">Solution:</span><br />
<span class="Normal">The coefficient of variation is useful for studying
risk, disparity as well as consistency. </span><br />
<span class="Normal">Hence, <b>option 4.</b></span><br />
<div>
</div>
<div>
</div>
</td></tr>
</tbody></table>
</div>
</div>
</div>
</div>
</div>
guruhttp://www.blogger.com/profile/01591375209227117455noreply@blogger.comtag:blogger.com,1999:blog-7850540640045081515.post-51909524989827040772015-03-03T03:51:00.002-08:002019-09-14T18:55:04.203-07:00surds and indices<div dir="ltr" style="text-align: left;" trbidi="on">
<br /></div>
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xmlns="http://www.w3.org/1999/xhtml"><HEAD><TITLE>Surds and Indices</TITLE>
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<P class=oTOCTitle><SPAN class=oTOCTitleFont><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013.htm#Title">Contents</A></SPAN></P>
<OL class=oTOCLevel1>
<LI class=oTocLiClass><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013.htm#I">INTRODUCTION</A>
</LI>
<LI class=oTocLiClass><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013.htm#II">SURDS</A>
<OL class=oTOCLevel2>
<LI><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013.htm#II_A">PURE
SURDS AND MIXED SURDS</A></LI>
<LI><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013.htm#II_B">SIMILAR
AND DISSIMILAR SURDS</A></LI>
<LI><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013.htm#II_C">MULTIPLYING
SURDS</A></LI>
<LI><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013.htm#II_D">RATIONALIZATION
OF SURDS</A></LI>
<LI><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013.htm#II_E">COMPARISON
OF SURDS</A></LI></OL></LI>
<LI class=oTocLiClass><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013.htm#III">INDICES</A>
<OL class=oTOCLevel2>
<LI><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013.htm#III_A">RULES
OF INDICES</A></LI>
<LI><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013.htm#III_B">COMPARISON
OF INDICES</A></LI></OL></LI></OL>
<P id=Title class=pTitle><SPAN class=Title>Surds and Indices</SPAN></P>
<HR>
<P class=pLevel2><SPAN class=Normal></SPAN>
<DIV id=013_I>
<TABLE id=013_I_bookmark class=tableh1 border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD
style="PADDING-BOTTOM: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 100%; PADDING-RIGHT: 0px; PADDING-TOP: 0px">
<OL class="oLevel1 Heading1">
<LI value=1><A id=I>INTRODUCTION</A></LI></OL></TD></TR></TBODY></TABLE>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>We have come across numbers like <IMG
style="VERTICAL-ALIGN: bottom" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_001.png">,
and 5<SUP>4</SUP> in the concept of Number Systems. We read there that numbers
such as</SPAN> <SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_001_1.png"><IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_073.png">…
are irrational numbers. We also learnt about powers and exponents in Number
Systems and Number Theory. In this chapter, we will focus on the rules that
govern the mathematics of these numbers, and will learn how to manipulate them
using the Laws of Indices.</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P></DIV>
<DIV id=013_II>
<TABLE id=013_II_bookmark class=tableh1 border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD
style="PADDING-BOTTOM: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 100%; PADDING-RIGHT: 0px; PADDING-TOP: 0px">
<OL class="oLevel1 Heading1">
<LI value=2><A id=II>SURDS</A></LI></OL></TD></TR></TBODY></TABLE>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>A <B>surd</B> is an ‘unresolved’ (meaning it
still contains the </SPAN><SPAN class=Normal><IMG style="VERTICAL-ALIGN: bottom"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_002.png">
symbol) mathematical expression of an <I>n</I><SUP>th</SUP> root. It can also be
defined as an irrational number which is represented as the
<I>n</I><SUP>th</SUP> root of a rational number.</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>For example, an irrational number which
represents the <I>n</I><SUP>th</SUP> root of a positive rational number <I>a</I>
is a surd and is represented as</SPAN> <SPAN class=Normal><IMG
style="VERTICAL-ALIGN: bottom" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_003.png">
.</SPAN></P>
<P class=pLevel2><SPAN class=Normal>An alternate way of representing it is <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_004.png">
(this representation uses indices – we will discuss indices later in this
chapter).</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>The symbol </SPAN><SPAN class=Normal><IMG
style="VERTICAL-ALIGN: bottom" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_002.png">
is called a <B>radical</B>, the base number <I>a</I> is called the
<B>radicand</B> and <I>n</I> is the <B>index</B> of the radical, also known as
the <B>order</B> of the surd.</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>Therefore </SPAN><SPAN class=Normal><IMG
style="VERTICAL-ALIGN: bottom" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_005.png">
are surds of order 2, 3 and 4 respectively.</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal><IMG style="VERTICAL-ALIGN: bottom" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_006.png">
is read as the third root (or the cube root) of 6 and </SPAN><SPAN
class=Normal><IMG style="VERTICAL-ALIGN: bottom" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_007.png">
is read as the fourth root of 7. If the index of the radical is not given, it is
assumed to be 2.</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>For example, </SPAN><SPAN class=Normal><IMG
style="VERTICAL-ALIGN: bottom" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_008.png">
is read as the square root (or the second root) of 5.</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorNote divLevel2">
<P><SPAN class=Note>IMPORTANT:</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<UL class=ListParagraph>
<LI class=ListStyleUL5>Conditions for a number to be a surd are:</LI></UL>
<P><SPAN class=ListParagraph></SPAN></P>
<OL class="oLevel1 ListParagraph ListStyleOL">
<LI value=1>It is a positive irrational number</LI></OL>
<OL class="oLevel1 ListParagraph ListStyleOL">
<LI value=2>It is of the type<IMG
style="MARGIN-TOP: 4px; VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_003.png">,
where a is a positive rational number</LI></OL>
<P><SPAN class=ListParagraph></SPAN></P>
<UL><SPAN class=ListParagraph>Hence, rational numbers like <IMG
style="MARGIN-TOP: 4px; VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_010.png">
are not surds, because these are not unresolved. <IMG
style="MARGIN-TOP: 4px; VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_011.png">
is equal to +2 and <IMG style="MARGIN-TOP: 4px; VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_012.png">
is equal to 3.</SPAN></UL>
<P><SPAN class=ListParagraph></SPAN></P>
<UL><SPAN class=ListParagraph><SPAN class=Normal>Again, <IMG
style="MARGIN-TOP: 4px; VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_013.png">
is equal to <IMG style="MARGIN-TOP: 4px; VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_014.png">
or +20, hence it is not a surd.</SPAN></UL>
<P><SPAN class=ListParagraph></SPAN></P>
<UL><SPAN class=ListParagraph>Also, <IMG
style="MARGIN-TOP: 7px; VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_015.png">
is not a surd, as the radicand itself is not a rational number.
</SPAN></UL></DIV>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<DIV id=013_II_A>
<TABLE id=013_II_A_bookmark class=tableh1 border=0 cellSpacing=0
cellPadding=0><TBODY>
<TR>
<TD
style="PADDING-BOTTOM: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 100%; PADDING-RIGHT: 0px; PADDING-TOP: 0px">
<OL class="oLevel2 Heading2">
<LI value=1><A id=II_A>PURE SURDS AND MIXED
SURDS</A></LI></OL></TD></TR></TBODY></TABLE>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>In the surd <IMG
style="MARGIN-TOP: 7px; VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_016.png">,
5 is called the <B>coefficient</B> of the surd. When there is no coefficient in
a surd, it is assumed that the coefficient is unity. Surds with unit
coefficients are known as <B>pure surds</B> and surds with coefficients other
than unity are known as <B>mixed surds</B>.</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>For example: </SPAN><SPAN class=Normal><IMG
style="MARGIN-TOP: 7px; VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_017.png">
are pure surds whereas <IMG style="MARGIN-TOP: 7px; VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_018.png">
are mixed surds.</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>The following rules of radicals are useful
to simplify surds:</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_019.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_020.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_021.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel2">
<DIV id=013_solvedex_1>
<TABLE style="WIDTH: 100%" id=013_solvedex_1_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_1>Example
1:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>Convert the following into pure surds:</SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_022.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_023.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>We need to convert the given mixed surds into surds
where the coefficient is 1. In other words, we have to ‘take the
coefficient’ inside the root sign.</SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_024.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_025.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel2">
<DIV id=013_solvedex_2>
<TABLE style="WIDTH: 100%" id=013_solvedex_2_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_2>Example
2:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>Convert the following into mixed surds:</SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_026.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_027.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>To convert a pure surd into a mixed surd, we have to
see if there is any factor of the form <I>a<SUP>n</I></SUP> inside the
<I>n</I><SUP>th</SUP> root sign.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_028.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_029.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel2">
<DIV id=013_solvedex_3>
<TABLE style="WIDTH: 100%" id=013_solvedex_3_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_3>Example
3:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>What is the square root of</SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_030.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_031.png">
<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_032.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_033.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_034.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_035.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_036.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_037.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_038.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_039.png">
<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_040.png"></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_041.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Now,</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_042.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_043.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_044.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Also, <I>a</I> + <I>b</I> + <I>c</I> = 3 + 45 + 25 =
73</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_045.png">
<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_046.png"><IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_047.png"></SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel3><SPAN class=Normal><IMG
style="MARGIN-TOP: 7px; VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_048.png">
and vice-versa</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><B>Explanation:</B></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_049.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_050.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><SPAN style="MARGIN-LEFT: 67px"></SPAN><IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_051.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><SPAN style="MARGIN-LEFT: 67px"></SPAN><IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_052.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_053.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_054.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_055.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_056.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>The same procedure can be followed to prove
that</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_057.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel2">
<DIV id=013_solvedex_4>
<TABLE style="WIDTH: 100%" id=013_solvedex_4_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_4>Example
4:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_058.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_059.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_060.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Multiplying the above two equations, we
get,</SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_061.png">
<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_062.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_063.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_064.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_065.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>By comparing, we get,</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>100 = <I>a</I><SUP>3</SUP> + 3<I>ab</I> =
<I>a</I><SUP>3</SUP> + 3<I>a</I>(<I>a</I><SUP>2</SUP> – 13)</SPAN></P>
<P><SPAN class=Normal><SPAN style="MARGIN-LEFT: 32px"></SPAN>=
<I>a</I><SUP>3</SUP> + 3<I>a</I><SUP>3</SUP> – 39<I>a</I> =
4<I>a</I><SUP>3</SUP> – 39<I>a</I></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>i.e. <I>a</I>(4<I>a</I><SUP>2</SUP> – 39) =
100</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>By trial and error, we get <I>a</I> = 4 and thus
<I>b</I> = 4<SUP>2</SUP> – 13 = 3</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Hence, the cube root of (100 – 51<IMG
style="MARGIN-TOP: 7px; VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_066.png">)
is (4 –<IMG style="MARGIN-TOP: 7px; VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_066.png">)</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel3><SPAN class=Normal></SPAN></P></DIV>
<DIV id=013_II_B>
<TABLE id=013_II_B_bookmark class=tableh1 border=0 cellSpacing=0
cellPadding=0><TBODY>
<TR>
<TD
style="PADDING-BOTTOM: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 100%; PADDING-RIGHT: 0px; PADDING-TOP: 0px">
<OL class="oLevel2 Heading2">
<LI value=2><A id=II_B>SIMILAR AND DISSIMILAR
SURDS</A></LI></OL></TD></TR></TBODY></TABLE>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>Surds which have the same irrational part
are known as <B>similar surds</B> (even if their coefficients are
different).</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>Surds with different irrational parts are
known as <B>dissimilar surds</B>.</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>For example, <IMG
style="MARGIN-TOP: 7px; VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_067.png"></SPAN>
and <IMG style="MARGIN-TOP: 7px; VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_068.png">
are similar surds, whereas <IMG style="MARGIN-TOP: 7px; VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_069.png">
and <IMG style="MARGIN-TOP: 7px; VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_070.png">
are dissimilar surds.</SPAN></P>
<P class=pLevel3><SPAN class=Normal>We can add or subtract similar
surds.</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>For example,</SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_071.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_072.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P></DIV>
<DIV id=013_II_C>
<TABLE id=013_II_C_bookmark class=tableh1 border=0 cellSpacing=0
cellPadding=0><TBODY>
<TR>
<TD
style="PADDING-BOTTOM: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 100%; PADDING-RIGHT: 0px; PADDING-TOP: 0px">
<OL class="oLevel2 Heading2">
<LI value=3><A id=II_C>MULTIPLYING SURDS</A></LI></OL></TD></TR></TBODY></TABLE>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>Surds which have the same index can be
multiplied without changing their index. The result of multiplication or
division of surds having the same index will also have the same
index.</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>For example, </SPAN><SPAN class=Normal><IMG
style="MARGIN-TOP: 7px; VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_066.png">
and <IMG style="MARGIN-TOP: 7px; VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_073.png">
can be multiplied, and the result will also have an index of 2. Similarly, <IMG
style="MARGIN-TOP: 7px; VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_074.png">
and <IMG style="MARGIN-TOP: 7px; VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_075.png">
can be multiplied with the result having index 3. However, </SPAN><SPAN
class=Normal><IMG style="MARGIN-TOP: 7px; VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_076.png">
and <IMG style="MARGIN-TOP: 7px; VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_075.png">
cannot be multiplied directly as their indices are different; i.e. 2 and 3
respectively.</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>We multiply or divide such surds without
changing the index. For example:</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_077.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_078.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>We can also multiply similar surds (since
similar surds will have the same index). The result of multiplying/dividing two
similar surds will always be an integer value.</SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_079.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel2">
<DIV id=013_solvedex_5>
<TABLE style="WIDTH: 100%" id=013_solvedex_5_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_5>Example
5:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>Simplify the following:</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_080.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_081.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_082.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_083.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_084.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_085.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_086.png">
<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_087.png"></SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel3><SPAN class=Normal></SPAN></P></DIV>
<DIV id=013_II_D>
<TABLE id=013_II_D_bookmark class=tableh1 border=0 cellSpacing=0
cellPadding=0><TBODY>
<TR>
<TD
style="PADDING-BOTTOM: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 100%; PADDING-RIGHT: 0px; PADDING-TOP: 0px">
<OL class="oLevel2 Heading2">
<LI value=4><A id=II_D>RATIONALIZATION OF
SURDS</A></LI></OL></TD></TR></TBODY></TABLE>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>The process of converting a surd to a
rational number by multiplying it with a suitable number is called
<B>rationalization</B>. To rationalize, we multiply the surd with a
rationalizing factor. When the rationalizing factor is multiplied with the surd,
we get a rational number.</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>For example, to rationalize </SPAN><SPAN
class=Normal><IMG style="MARGIN-TOP: 7px; VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_088.png">
we multiply it with </SPAN><SPAN class=Normal><IMG
style="MARGIN-TOP: 7px; VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_088.png">
(the rationalizing factor) to get <IMG
style="MARGIN-TOP: 7px; VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_089.png">
and to rationalize </SPAN><SPAN class=Normal><IMG
style="MARGIN-TOP: 7px; VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_090.png">
we multiply it with <IMG style="MARGIN-TOP: 7px; VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_066.png">
to get <IMG style="MARGIN-TOP: 7px; VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_091.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal>It is to be noted that there are always
multiple rationalizing factors available. In the above example, we could have
also multiplied </SPAN><SPAN class=Normal><IMG
style="MARGIN-TOP: 7px; VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_008.png">
with <IMG style="MARGIN-TOP: 7px; VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_092.png">
to rationalize it,</SPAN></P>
<P class=pLevel3><SPAN class=Normal>since <IMG
style="MARGIN-TOP: 7px; VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_093.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<OL class="oLevel3 Heading3 ListStyleOL">
<LI value=1>RATIONALIZING FACTOR OF A SUM OF SURDS</LI></OL>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal>Consider the term </SPAN><SPAN
class=Normal><IMG style="MARGIN-TOP: 7px; VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_094.png">
which is the sum of two surds. The rationalizing factor for this is </SPAN><SPAN
class=Normal><IMG style="MARGIN-TOP: 7px; VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_095.png">.
When we multiply these two terms we get,</SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal><IMG
style="MARGIN-TOP: 7px; VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_096.png"></SPAN></P>
<P class=pLevel4><SPAN class=Normal>This brings us to the concept of the
conjugate. The conjugate of</SPAN> <SPAN class=Normal><IMG
style="MARGIN-TOP: 7px; VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_097.png">
is <IMG style="MARGIN-TOP: 7px; VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_098.png">.</SPAN></P>
<P class=pLevel4><SPAN class=Normal>Similarly, the conjugate of</SPAN> <SPAN
class=Normal><IMG style="MARGIN-TOP: 7px; VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_099.png">
is <IMG style="MARGIN-TOP: 7px; VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_097.png">.
The conjugate is the rationalizing factor (RF) for a sum of surds of the form
</SPAN><SPAN class=Normal><IMG style="MARGIN-TOP: 7px; VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image102.png">.</SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<OL class="oLevel3 Heading3 ListStyleOL">
<LI value=2>USE OF RATIONALIZATION</LI></OL>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal>Rationalization is mostly used to
rationalize the denominator of an expression, if the denominator features a sum
of surds.</SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel2">
<DIV id=013_solvedex_6>
<TABLE style="WIDTH: 100%" id=013_solvedex_6_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_6>Example
6:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image103.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image104.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image105.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image106.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image107.png"></SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal>Rationalizing surds can often be simplified
by converting the surds to their indices form.</SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel2">
<DIV id=013_solvedex_7>
<TABLE style="WIDTH: 100%" id=013_solvedex_7_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_7>Example
7:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image108.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image109.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image110.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image111.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image112.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image113.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel4><SPAN class=Normal>To rationalize a fraction such as
</SPAN><SPAN class=Normal><IMG style="MARGIN-TOP: 7px; VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image114.png">
the following steps are carried out:</SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<OL class="oLevel3 ListParagraph ListStyleOL">
<LI value=1>Multiply the Numerator & Denominator by <IMG
style="MARGIN-TOP: 7px; VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image115.png"></LI></OL>
<OL class="oLevel3 ListParagraph ListStyleOL">
<LI value=2>Then multiply the Numerator & Denominator by [(<I>x</I> +
<I>y</I> – <I>z</I>) – 2 <IMG style="MARGIN-TOP: 7px; VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image116.png">]</LI></OL>
<P class=pLevel4><SPAN class=Normal><B>Explanation:</B></SPAN></P>
<P class=pLevel4><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image117.png"></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image118.png"></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image119.png"></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image120.png"></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image121.png"></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal>Thus, the expression is rationalized (since
the denominator is a rational quantity).</SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel2">
<DIV id=013_solvedex_8>
<TABLE style="WIDTH: 100%" id=013_solvedex_8_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_8>Example
8:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image122.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image123.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image124.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image125.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image126.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Now,</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image127.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image128.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image129.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image130.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image131.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image132.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV></DIV>
<DIV id=013_II_E>
<TABLE id=013_II_E_bookmark class=tableh1 border=0 cellSpacing=0
cellPadding=0><TBODY>
<TR>
<TD
style="PADDING-BOTTOM: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 100%; PADDING-RIGHT: 0px; PADDING-TOP: 0px">
<OL class="oLevel2 Heading2">
<LI value=5><A id=II_E>COMPARISON OF
SURDS</A></LI></OL></TD></TR></TBODY></TABLE>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>It is difficult to compare two surds of
different indices. The strategy we follow is to change both surds to the same
order. We can then compare them by the value of their radicands.</SPAN></P>
<P class=pLevel3><SPAN class=Normal>The new index of the two surds is the LCM of
the original indices of the surds. The following example should make it
clear.</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel2">
<DIV id=013_solvedex_9>
<TABLE style="WIDTH: 100%" id=013_solvedex_9_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_9>Example
9:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image133.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>The order of the surds is 2 and 3, hence we convert
both the surds to surds of order 6 (which is the LCM of 2 and
3).</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image134.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image135.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P><SPAN class=Normal>Now we can say that
<IMG style="MARGIN-TOP: 7px; VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image136.png"></SPAN>
<P></P>
<P><SPAN class=Normal></SPAN></P><SPAN class=Normal>Hence, <IMG
style="MARGIN-TOP: 7px; VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image137.png"></SPAN>
<P></P>
<P><SPAN class=Normal></SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorNote divLevel2">
<P><SPAN class=Note>REMEMBER:</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<UL class=ListParagraph>
<LI class=ListStyleUL5>The conversions of <IMG
style="MARGIN-TOP: 7px; VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/013_076.png">
to <IMG style="MARGIN-TOP: 7px; VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image138.png">
and <IMG style="MARGIN-TOP: 7px; VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image139.png">
to <IMG style="MARGIN-TOP: 7px; VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image140.png">
is done using the rules of indices.</LI></UL></DIV>
<P class=pLevel4><SPAN class=Normal></SPAN></P></DIV></DIV>
<DIV id=013_III>
<TABLE id=013_III_bookmark class=tableh1 border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD
style="PADDING-BOTTOM: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 100%; PADDING-RIGHT: 0px; PADDING-TOP: 0px">
<OL class="oLevel1 Heading1">
<LI value=3><A id=III>INDICES</A></LI></OL></TD></TR></TBODY></TABLE>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>Exponential notation is a convenient
notation for repeated multiplication. It is generally used when a number is
multiplied by itself several times.</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal><B>Notation:</B> If <I>a</I> is any rational
number, then <I>a</I> <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/multiplication.png">
<I>a</I> <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/multiplication.png">
<I>a</I> <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/multiplication.png">
... <I>n</I> times = <I>a<SUP>n</I></SUP>. Here, <I>a</I> is called the
<B>base</B> and <I>n</I> is called the <B>exponent</B> or <B>index</B>. The
plural of the word index is indices.</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>For example,</SPAN></P>
<P class=pLevel2><SPAN class=Normal>4 <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/multiplication.png">
4 = 4<SUP>2</SUP> is read as ‘the square of 4’ or ‘4 to the power 2’,
and</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>5 <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/multiplication.png">
5 <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/multiplication.png">
5 = 5<SUP>3</SUP> is read as ‘the cube of 5’ or ‘5 to the power 3’
and</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>6 <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/multiplication.png">
6 <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/multiplication.png">
6 <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/multiplication.png">
6 = 6<SUP>4</SUP> is read as ‘6 raised to the power 4’.</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>In general, <I>a<SUP>n</I></SUP> is read as
‘<I>a</I> raised to the power <I>n</I>’.</SPAN></P>
<DIV id=013_III_A>
<TABLE id=013_III_A_bookmark class=tableh1 border=0 cellSpacing=0
cellPadding=0><TBODY>
<TR>
<TD
style="PADDING-BOTTOM: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 100%; PADDING-RIGHT: 0px; PADDING-TOP: 0px">
<OL class="oLevel2 Heading2">
<LI value=1><A id=III_A>RULES OF INDICES</A></LI></OL></TD></TR></TBODY></TABLE>
<P class=pLevel3><SPAN class=Normal>If <I>a</I> and <I>b</I> are non-zero
rational numbers and <I>m</I> and <I>n</I> are rational numbers, then</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image141.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image142.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image143.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image144.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image145.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image146.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image147.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image148.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image149.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal>Also, we can observe the following
properties of indices.</SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image150.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image151.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image152.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>We would now see a few applications of these
rules.</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel2">
<DIV id=013_solvedex_10>
<TABLE style="WIDTH: 100%" id=013_solvedex_10_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_10>Example
10:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>Find the value of or simplify the following using
the rules of indices:</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image153.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image154.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image155.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image156.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image157.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image158.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image159.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image160.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image161.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image162.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image163.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image164.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image165.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image166.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image167.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image168.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image169.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image170.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel2">
<DIV id=013_solvedex_11>
<TABLE style="WIDTH: 100%" id=013_solvedex_11_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_11>Example
11:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>Solve the following equations:</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image171.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image172.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image173.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image174.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image171.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image175.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image176.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image172.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image177.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image178.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image173.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image179.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image180.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image174.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image181.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image182.png"></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image183.png"></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image184.png"></SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel2">
<DIV id=013_solvedex_12>
<TABLE style="WIDTH: 100%" id=013_solvedex_12_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_12>Example
12:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>Calculate the values of <I>x</I> in the following
equation:</SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image185.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><B>[JMET 2009]</B></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>(1) <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/minus.png">4,
9<SPAN style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN>(2) 4, 9</SPAN></P>
<P><SPAN class=Normal>(3) 4, 8<SPAN style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 35px"></SPAN>(4) 7, 6</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>Since we have,</SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image185.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image186.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image187.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image188.png"></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/therefore.png">
<I>x</I><SUP>2 </SUP><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/minus.png">
5<I>x</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/minus.png">
36 = 0</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/therefore.png">
(<I>x</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/minus.png">
9)(<I>x</I> + 4) = 0</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/therefore.png">
<I>x</I> = 9 or <I>x</I> = <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/minus.png">4</SPAN></P>
<P><SPAN class=Normal>Hence, <B>option
1.</B></SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel2">
<DIV id=013_solvedex_13>
<TABLE style="WIDTH: 100%" id=013_solvedex_13_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_13>Example
13:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>Given 2<I><SUP>x</I></SUP> = 8<I><SUP>y</I> +
1</SUP> and 9<I><SUP>y</I></SUP> = 3<I><SUP>x</I> – 9</SUP>; the value of
<I>x </I>+<I> y </I>is :</SPAN></P>
<P><SPAN class=Normal><B>[FMS 2010]</B></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>(1) 18<SPAN style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN>(2) 21</SPAN></P>
<P><SPAN class=Normal>(3) 24<SPAN style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN>(4) 27</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>2<I><SUP>x</I></SUP> = 8<I><SUP>y</I> +
1</SUP></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/therefore.png">
2<I><SUP>x </I></SUP>= (2<SUP>3</SUP>)<I><SUP> y </I>+ 1</SUP></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/therefore.png">
2<I><SUP>x</I> </SUP>= 2<SUP>3(<I>y</I> + 1)</SUP></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/therefore.png">
<I>x</I> = 3<I>y</I> + 3 <SPAN style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN>…(i)</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Consider,</SPAN></P>
<P><SPAN class=Normal>9<I><SUP>y</I> </SUP>= 3<I><SUP>x</I></SUP> <SUP>–
9</SUP></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/therefore.png">
(3<SUP>2</SUP>)<I><SUP>y</I> </SUP>=<SUP> </SUP>3<I><SUP> x</I></SUP>
<SUP>– 9</SUP></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/therefore.png">
3<SUP>2<I>y</I> </SUP>= 3<I><SUP> x</I></SUP> <SUP>– 9</SUP></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/therefore.png">
2<I>y</I> = <I>x</I> – 9 <SPAN style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN>…(ii)</SPAN></P>
<P><SPAN class=Normal>Solving (i) and (ii) simultaneousl<I>y</I>, we
get,</SPAN></P>
<P><SPAN class=Normal><I>x</I> = 21 and <I>y</I> = 6</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/therefore.png">
<I>x</I> + <I>y</I> = 21 + 6 = 27</SPAN></P>
<P><SPAN class=Normal>Hence, <B>option
4.</B></SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV></DIV></DIV>
<DIV id=013_III_B>
<TABLE id=013_III_B_bookmark class=tableh1 border=0 cellSpacing=0
cellPadding=0><TBODY>
<TR>
<TD
style="PADDING-BOTTOM: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 100%; PADDING-RIGHT: 0px; PADDING-TOP: 0px">
<OL class="oLevel2 Heading2">
<LI value=2><A id=III_B>COMPARISON OF
INDICES</A></LI></OL></TD></TR></TBODY></TABLE>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>To compare, we must try to equate the bases
or the indices of the two numbers.</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel2">
<DIV id=013_solvedex_14>
<TABLE style="WIDTH: 100%" id=013_solvedex_14_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_14>Example
14:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>Which is greater?</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>(i) 324<SUP>5</SUP> or 63<SUP>10</SUP></SPAN></P>
<P><SPAN class=Normal>(ii) (22.5)<SUP>27</SUP> or
(7.5)<SUP>54</SUP></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>(i) Now,</SPAN></P>
<P><SPAN class=Normal>324<SUP>5</SUP> = (3<SUP>4</SUP> <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/multiplication.png">
2<SUP>2</SUP>)<SUP>5</SUP> = 3<SUP>20</SUP> <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/multiplication.png">
2<SUP>10</SUP></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Also, 63<SUP>10</SUP> = (3<SUP>2</SUP> <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/multiplication.png">
7)<SUP>10</SUP> = 3<SUP>20</SUP> <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/multiplication.png">
7<SUP>10</SUP></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Since 7<SUP>10</SUP> > 2<SUP>10</SUP>, we have
63<SUP>10</SUP> > 324<SUP>5</SUP></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>(ii) Here, let <I>n</I> = 27; then 2<I>n</I> =
54</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image189.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image190.png"><IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image191.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image192.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image193.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image194.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image195.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/734fd766-a946-4a21-877f-90a694dc34d5/013_images/image196.png"></SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV></DIV></BODY></HTML>
guruhttp://www.blogger.com/profile/01591375209227117455noreply@blogger.comtag:blogger.com,1999:blog-7850540640045081515.post-852341016162649232015-03-03T03:50:00.004-08:002019-09-14T18:55:04.401-07:00quadratic eq<div dir="ltr" style="text-align: left;" trbidi="on">
<br /></div>
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<HTML xmlns="http://www.w3.org/1999/xhtml"><HEAD><TITLE>Quadratic and Higher Order Equations</TITLE>
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<P class=oTOCTitle><SPAN class=oTOCTitleFont><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017.htm#Title">Contents</A></SPAN></P>
<OL class=oTOCLevel1>
<LI class=oTocLiClass><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017.htm#I">INTRODUCTION</A>
</LI>
<LI class=oTocLiClass><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017.htm#II">SOLVING
A QUADRATIC EQUATION</A>
<OL class=oTOCLevel2>
<LI><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017.htm#II_A">METHOD
OF COMPLETING SQUARES</A></LI>
<LI><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017.htm#II_B">THE
QUADRATIC FORMULA</A></LI>
<LI><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017.htm#II_C">FACTORISATION</A></LI></OL></LI>
<LI class=oTocLiClass><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017.htm#III">NATURE
OF THE ROOTS OF A QUADRATIC EQUATION</A> </LI>
<LI class=oTocLiClass><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017.htm#IV">GRAPHICAL
REPRESENTATION OF A QUADRATIC EQUATION</A> </LI>
<LI class=oTocLiClass><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017.htm#V">HIGHER
ORDER EQUATIONS</A>
<OL class=oTOCLevel2>
<LI><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017.htm#V_A">SOLVING
HIGHER ORDER EQUATIONS</A></LI>
<LI><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017.htm#V_B">DESCARTES’
RULE OF SIGNS</A></LI></OL></LI>
<LI class=oTocLiClass><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017.htm#VI">FINDING
THE COMMON ROOT</A> </LI></OL>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P id=Title class=pTitle><SPAN class=Title>Quadratic and Higher Order
Equations</SPAN></P>
<HR>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN>
<DIV id=017_I>
<TABLE id=017_I_bookmark class=tableh1 border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD
style="PADDING-BOTTOM: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 100%; PADDING-RIGHT: 0px; PADDING-TOP: 0px">
<OL class="oLevel1 Heading1">
<LI value=1><A id=I>INTRODUCTION</A></LI></OL></TD></TR></TBODY></TABLE>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>A second degree polynomial is called a
<B>quadratic polynomial</B>. An equation of the form <I>ax</I><SUP>2</SUP> +
<I>bx</I> + <I>c</I> = 0, where <I>a</I>, <I>b</I>, <I>c</I> are real numbers
and <I>a</I> ≠ 0, is called a <B>quadratic</B> equation in variable <I>x</I>.
The values of <I>x</I> for which the equation holds true are called the
<B>roots</B> of the equation. Since a quadratic equation is of degree 2, it can
have at most two roots.</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>Here are some examples of quadratic
equations.</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>5<I>x</I><SUP>2</SUP> = 0</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>3<I>x</I><SUP>2</SUP> + 2<I>x</I> =
0</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><I>x</I><SUP>2</SUP> – 4<I>x</I> + 7 =
0</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P></DIV>
<DIV id=017_II>
<TABLE id=017_II_bookmark class=tableh1 border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD
style="PADDING-BOTTOM: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 100%; PADDING-RIGHT: 0px; PADDING-TOP: 0px">
<OL class="oLevel1 Heading1">
<LI value=2><A id=II>SOLVING A QUADRATIC
EQUATION</A></LI></OL></TD></TR></TBODY></TABLE>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>Solving a quadratic equation is finding all
its roots. This can be done in the following ways.</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<DIV id=017_II_A>
<TABLE id=017_II_A_bookmark class=tableh1 border=0 cellSpacing=0
cellPadding=0><TBODY>
<TR>
<TD
style="PADDING-BOTTOM: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 100%; PADDING-RIGHT: 0px; PADDING-TOP: 0px">
<OL style="MARGIN-LEFT: 12%" class="oLevel2 Heading2">
<LI value=1><A id=II_A>METHOD OF COMPLETING
SQUARES</A></LI></OL></TD></TR></TBODY></TABLE>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>In this method, we express the given
quadratic <I>ax</I><SUP>2</SUP> + <I>bx</I> + <I>c</I> = 0 as (<I>mx</I> <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/Plus_minus.png">
<I>n</I>)<SUP>2</SUP> = <I>k</I><SUP>2</SUP>, where <I>m</I>, <I>n</I> and
<I>k</I> are numbers and then find the value of <I>x</I> by taking the square
root.</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>The steps to be followed while using this
method are as explained in the following example.</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>Consider the quadratic equation
5<I>x</I><SUP>2</SUP> + 20<I>x</I> + 13 = 0. Here <I>a</I> = 5, <I>b</I> = 20
and <I>c</I> = 13.</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><B>Step 1:</B> Express the equation
as</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/017_001.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/017_002.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/017_003.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/017_004.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/017_005.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><B>Step 3:</B> Express the equation
as</SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/017_006.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/017_007.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/017_008.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><B>Step 4:</B> Take the square root and find
the value of <I>x</I>.</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/017_009.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/017_010.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/017_011.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=017_solvedex_1>
<TABLE style="WIDTH: 100%" id=017_solvedex_1_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_1>Example
1:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>Find the roots of the equation, <I>x</I><SUP>2</SUP>
+ 12<I>x</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">
20 = 0.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal><I>x</I><SUP>2</SUP> + 12<I>x</I> <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">
20 = 0</SPAN></P>
<P><SPAN class=Normal><I>x</I><SUP>2</SUP> + 2 <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/multiplication.png">
6<I>x</I> + 36 = 20 + 36</SPAN></P>
<P><SPAN class=Normal>(<I>x</I> + 6)<SUP>2</SUP> = 56</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/017_012.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/017_013.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Hence, the roots of the equation are</SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/017_014.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=017_solvedex_2>
<TABLE style="WIDTH: 100%" id=017_solvedex_2_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_2>Example
2:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>Find the roots of the equation,
9<I>x</I><SUP>2</SUP> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">
42<I>x</I> + 24 = 0.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>9<I>x</I><SUP>2</SUP> <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">
42<I>x</I> + 24 = 0</SPAN></P>
<P><SPAN class=Normal>(3<I>x</I>)<SUP>2</SUP> <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">
2 <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/multiplication.png">
3 <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/multiplication.png">
7<I>x</I> + 49 = 49 – 24</SPAN></P>
<P><SPAN class=Normal>(3<I>x</I> <IMG style="VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">
7)<SUP>2</SUP> = 25</SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/017_015.png"></SPAN></P>
<P><SPAN class=Normal>3<I>x</I> = 7 <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/Plus_minus.png">
5</SPAN></P>
<P><SPAN class=Normal>3<I>x</I> = 2 or 12</SPAN></P>
<P><SPAN class=Normal>Hence, the roots of the equation are 2/3 and
4.</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Note>Note: </SPAN><SPAN class=Normal>The method of
completing squares can be used to find the minimum value of an algebraic
expression without using concepts of higher mathematics.</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=017_solvedex_3>
<TABLE style="WIDTH: 100%" id=017_solvedex_3_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_3>Example
3:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>If <I>x</I> and <I>y</I> are real numbers, then the
minimum value of <I>x</I><SUP>2</SUP> + 4<I>xy</I> + 6<I>y</I><SUP>2</SUP>
– 4<I>y</I> + 4 is</SPAN></P>
<P><SPAN class=Normal><B>[XAT 2010]</B></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>(1) ‒4<SPAN style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN>(2) 0</SPAN></P>
<P><SPAN class=Normal>(3) 2<SPAN style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN>(4) 4</SPAN></P>
<P><SPAN class=Normal>(5) None of the above</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal><I>x</I><SUP>2</SUP> + 4<I>xy</I> +
6<I>y</I><SUP>2</SUP> – 4<I>y</I> + 4</SPAN></P>
<P><SPAN class=Normal>= <I>x</I><SUP>2</SUP> + 4<I>xy</I> +
4<I>y</I><SUP>2</SUP> + 2<I>y</I><SUP>2</SUP> – 4<I>y</I> + 4</SPAN></P>
<P><SPAN class=Normal>= (<I>x</I> + 2<I>y</I>)<SUP>2</SUP> +
2(<I>y</I><SUP>2</SUP> – 2<I>y</I> + 1 + 1)</SPAN></P>
<P><SPAN class=Normal>= (<I>x</I> + 2<I>y</I>)<SUP>2</SUP> + 2(<I>y</I> –
1)<SUP>2</SUP> + 2</SPAN></P>
<P><SPAN class=Normal>This expression will take the minimum value when
both the square terms are zero</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
(<I>y</I> – 1)<SUP>2</SUP> = 0</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
<I>y</I> = 1</SPAN></P>
<P><SPAN class=Normal>and</SPAN></P>
<P><SPAN class=Normal>(<I>x</I> + 2<I>y</I>)<SUP>2</SUP> = 0</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
<I>x</I> = –2</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
The minimum value of this expression is 2.</SPAN></P>
<P><SPAN class=Normal>Hence, <B>option
3.</B></SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel3><SPAN class=Normal></SPAN></P></DIV>
<DIV id=017_II_B>
<TABLE id=017_II_B_bookmark class=tableh1 border=0 cellSpacing=0
cellPadding=0><TBODY>
<TR>
<TD
style="PADDING-BOTTOM: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 100%; PADDING-RIGHT: 0px; PADDING-TOP: 0px">
<OL style="MARGIN-LEFT: 12%" class="oLevel2 Heading2">
<LI value=2><A id=II_B>THE QUADRATIC
FORMULA</A></LI></OL></TD></TR></TBODY></TABLE>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>The method of completing squares can be used
to find a general formula for finding the roots of a quadratic
equation.</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>The two roots of the quadratic equation,
<I>ax<SUP>2</I></SUP> + <I>bx</I> + <I>c</I> = 0 are given by:</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/017_016.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><B>Explanation</B></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><I>ax<SUP>2</I></SUP> + <I>bx</I> + <I>c</I>
= 0</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>Dividing both the sides by<I> a</I> we
get</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/017_017.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/017_018.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/017_019.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/017_020.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/017_021.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/017_022.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=017_solvedex_4>
<TABLE style="WIDTH: 100%" id=017_solvedex_4_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_4>Example
4:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>Find the roots of the equation, <I>x</I><SUP>2</SUP>
<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">
8<I>x</I> + 11 = 0.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>In the equation <I>a</I> = 1, <I>b</I> = –8,
<I>c</I> = 11</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Hence, the roots of the equation are:</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/017_023.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/017_024.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Hence, the two roots of the equation are</SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/017_025.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=017_solvedex_5>
<TABLE style="WIDTH: 100%" id=017_solvedex_5_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_5>Example
5:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>Find the roots of the equation,
7<I>x</I><SUP>2</SUP> + 12<I>x</I> <IMG style="VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">
2 = 0.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>In the quadratic equation, 7<I>x</I><SUP>2</SUP> +
12<I>x</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">
2 = 0, we have <I>a</I> = 7, <I>b</I> = 12, <I>c</I> = <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">2.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Hence, the roots of the equation are:</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/017_026.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/017_027.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/017_028.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/017_029.png"></SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=017_solvedex_6>
<TABLE style="WIDTH: 100%" id=017_solvedex_6_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_6>Example
6:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>Find the value of <I>x</I> if,</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/017_030.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/017_031.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Multiplying both the sides by (<I>x</I> +
2)(<I>x</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">
1),</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>(<I>x </I>+ 1)(<I>x </I><IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">
1) + (2<I>x</I> + 1)(<I>x</I> + 2) = 0</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
<I>x</I><SUP>2</SUP> – 1 + 2<I>x</I><SUP>2</SUP> + 4<I>x </I>+ <I>x</I> +
2 = 0</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
3<I>x</I><SUP>2</SUP> + 5<I>x</I> + 1 = 0</SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/017_032.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/017_033.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/017_034.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel3><SPAN class=Normal><B>Sum and Product of Roots of a Quadratic
Equation</B></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>Consider the equation, <I>ax<SUP>2</I></SUP>
+ <I>bx</I> + <I>c</I> = 0<SPAN
style="MARGIN-LEFT: 25px"></SPAN>...(i)</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>This equation can be written as</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/017_035.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>Let <I><IMG style="VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/alpha.png"></I>
and <I><IMG style="PADDING-BOTTOM: 2px; VERTICAL-ALIGN: bottom" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/beta.png"></I>
be the two roots of this equation.</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>We know that the two roots of this equation
are</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/017_036.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/017_037.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/017_038.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/017_039.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>and</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/017_040.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>Using (ii), (iii) and (iv), we can rewrite
(i) as</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><I>x</I><SUP>2</SUP> <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">
(<I><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/alpha.png"></I>
+ <I><IMG style="PADDING-BOTTOM: 2px; VERTICAL-ALIGN: bottom" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/beta.png"></I>)<I>x</I>
+ (<I><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/alpha.png"><IMG
style="PADDING-BOTTOM: 2px; VERTICAL-ALIGN: bottom" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/beta.png"></I>)
= 0<SPAN style="MARGIN-LEFT: 120px"></SPAN>...(v)</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>Thus,</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><I>x</I><SUP>2</SUP> <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">
(Sum of the roots)<I>x</I> + (Product of the roots) = 0</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>Equation (v) can be written as (<I>x</I> –
<I><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/alpha.png"></I>)(<I>x</I>
– <I><IMG style="PADDING-BOTTOM: 2px; VERTICAL-ALIGN: bottom" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/beta.png"></I>
) = 0</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=017_solvedex_7>
<TABLE style="WIDTH: 100%" id=017_solvedex_7_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_7>Example
7:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>Ujakar and Keshab attempted to solve a quadratic
equation. Ujakar made a mistake in writing down the constant term. He
ended up with the roots (4, 3). Keshab made a mistake in writing down the
coefficient of <I>x</I>. He got the root as (3, 2). What will be the exact
roots of the original quadratic equation?</SPAN></P>
<P><SPAN class=Normal><B>[CAT 2001]</B></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>(1) (6, 1)<SPAN
style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN>(2) (–3, –4)</SPAN></P>
<P><SPAN class=Normal>(3) (4, 3)<SPAN
style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN>(4) (–4, –3)</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>In Ujakar’s<B> </B>case, the constant term is
wrong.</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
Product of roots will be wrong but the sum of roots will be
correct.</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
Correct Sum of the roots = 4 + 3 = 7</SPAN></P>
<P><SPAN class=Normal>In Keshab’s case, coefficient of <I>x</I> is
wrong.</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
Sum of roots will be wrong but the product of roots will be
correct.</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
Correct product of roots = 3 <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/multiplication.png">
2 = 6</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
Correct quadratic equation is <I>x</I><SUP>2</SUP> – 7<I>x</I> + 6 =
0</SPAN></P>
<P><SPAN class=Normal>(<I>x</I> – 6)(<I>x</I> – 1) = 0</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png"><I>
x</I> = 6 or <I>x</I> = 1</SPAN></P>
<P><SPAN class=Normal>Hence, <B>option
1</B>.</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=017_solvedex_8>
<TABLE style="WIDTH: 100%" id=017_solvedex_8_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_8>Example
8:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>If the roots of the equation
<I>x</I><SUP>3</SUP><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">
<I>ax</I><SUP>2 </SUP>+<I>bx </I>– <I>c = </I>0 are three consecutive
integers, then what is the smallest possible value of <I>b</I>?</SPAN></P>
<P><SPAN class=Normal><B>[CAT 2008]</B></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/017_041.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>(3) 0<SPAN style="MARGIN-LEFT: 78px"></SPAN>(4)
1</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/017_042.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>If <I>p</I>, <I>q</I> and <I>r</I> are the roots of
a cubic equation <I>ax</I><SUP>3</SUP> + <I>bx</I><SUP>2</SUP> + <I>cx</I>
+ <I>d</I> = 0, then <I>pq</I> + <I>pr</I> + <I>qr</I> =
<I>c</I>/<I>a</I></SPAN></P>
<P><SPAN class=Normal>If <I>a</I> is 1, then <I>pq</I> + <I>qr</I> +
<I>pr</I> = <I>c</I></SPAN></P>
<P><SPAN class=Normal>Let the three roots of the given equation be
(<I>n</I> – 1), <I>n</I> and (<I>n</I> + 1).</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
(<I>n </I>– 1)<I>n </I>+ <I>n</I>(<I>n </I>+ 1) + (<I>n </I>- 1)(<I>n
</I>+ 1) = <I>b</I></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
<I>n</I><SUP>2</SUP> – <I>n </I>+ <I>n</I><SUP>2</SUP> + <I>n </I>+
<I>n</I><SUP>2</SUP> – 1 = <I>b</I></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
3<I>n</I><SUP>2</SUP> – 1 = <I>b</I></SPAN></P>
<P><SPAN class=Normal>For <I>b</I> to take minimum value, the LHS must
take minimum value. This is possible for <I>n</I> = 0.</SPAN></P>
<P><SPAN class=Normal>From this, we can see that minimum value of <I>b</I>
= – 1</SPAN></P>
<P><SPAN class=Normal>Hence, <B>option
2.</B></SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=017_solvedex_9>
<TABLE style="WIDTH: 100%" id=017_solvedex_9_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_9>Example
9:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>If one root of the equation <I>ax</I><SUP>2</SUP>
+<I> bx </I>+<I> c</I> = 0 is double of the other, then
2<I>b</I><SUP>2</SUP> =</SPAN></P>
<P><SPAN class=Normal><B>[IIFT 2008]</B></SPAN></P>
<P><SPAN class=Normal></SPAN></P><SPAN class=Normal>(1) <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/017_043.png">
<SPAN style="MARGIN-LEFT: 39px"></SPAN>(2) <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/017_044.png"></SPAN>
<P><SPAN class=Normal></SPAN></P><SPAN class=Normal>(3) <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/017_045.png"><SPAN
style="MARGIN-LEFT: 25px"></SPAN>(4) None of the above</SPAN>
<P></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal><I>ax</I><SUP>2</SUP> +<I> bx </I>+<I> c</I> =
0</SPAN></P>
<P><SPAN class=Normal>Let the roots of the given equation are <I><IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/alpha.png"></I>
and 2<I><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/alpha.png"></I>.</SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/017_046.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/017_047.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/017_048.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>2<I>b</I><SUP>2</SUP> = 9<I>ac</I></SPAN></P>
<P><SPAN class=Normal>Hence, <B>option
1</B>.</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel3><SPAN class=Normal></SPAN></P></DIV>
<DIV id=017_II_C>
<TABLE id=017_II_C_bookmark class=tableh1 border=0 cellSpacing=0
cellPadding=0><TBODY>
<TR>
<TD
style="PADDING-BOTTOM: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 100%; PADDING-RIGHT: 0px; PADDING-TOP: 0px">
<OL style="MARGIN-LEFT: 12%" class="oLevel2 Heading2">
<LI value=3><A id=II_C>FACTORISATION</A></LI></OL></TD></TR></TBODY></TABLE>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>Consider the equation, <I>ax<SUP>2</I></SUP>
+ <I>bx</I> + <I>c</I> = 0</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>In this method we write <I>b</I> as a sum of
two terms in such a way that the product of the two terms is equal to
<I>ac</I>.</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=017_solvedex_10>
<TABLE style="WIDTH: 100%" id=017_solvedex_10_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_10>Example
10:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>Find the roots of the equation 2<I>x</I><SUP>2</SUP>
+ <I>x</I> – 6 = 0.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>In the given equation <I>a</I> = 2, <I>b</I> = 1 and
<I>c</I> = <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">6</SPAN></P>
<P><SPAN class=Normal>Hence <I>ac</I> = 2 <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/multiplication.png">
(<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">6)
= <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">12</SPAN></P>
<P><SPAN class=Normal>The numbers whose sum is 1 (<I>b</I>) and product is
<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">12
(<I>ac</I>) are 4 and <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">3.</SPAN></P>
<P><SPAN class=Normal>Hence the given equation can be split as</SPAN></P>
<P><SPAN class=Normal>2<I>x</I><SUP>2</SUP> + 4<I>x</I> <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">
3<I>x </I><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">
6 = 0</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
2<I>x</I>(<I>x</I> + 2) <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">
3(<I>x</I> + 2) = 0</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
(<I>x</I> + 2)(2<I>x</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">
3) = 0<SPAN style="MARGIN-LEFT: 25px"></SPAN></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
<I>x</I> + 2 = 0 or 2<I>x</I> – 3 = 0</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
<I>x</I> = <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">2
or <I>x </I>= 3/2</SPAN></P>
<P><SPAN class=Normal>Hence, the roots of the equation are <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">2
and 3/2.</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=017_solvedex_11>
<TABLE style="WIDTH: 100%" id=017_solvedex_11_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_11>Example
11:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>Find the roots of the equation,
3<I>x</I><SUP>2</SUP> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">
23<I>x</I> + 44 = 0.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>In the given equation <I>a</I> = 3, <I>b</I> = <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">23
and <I>c</I> = 44</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Hence <I>ac</I> = 3 <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/multiplication.png">
44 = 132 = 12 <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/multiplication.png">
11</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Thus the factors of 132 whose sum is equal to <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">23
are <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">11
and <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">12.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
3<I>x</I><SUP>2</SUP> – 11<I>x </I>– 12<I>x</I> + 44 = 0</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
<I>x</I>(3<I>x</I> – 11) – 4(3<I>x</I> – 11) = 0</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
(<I>x</I> – 4)(3<I>x</I> – 11) = 0</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
<I>x</I> – 4 = 0 or 3<I>x</I> – 11 = 0</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
<I>x</I> = 4 or 11/3</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Hence, the roots of the equation are 4 and
11/3.</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=017_solvedex_12>
<TABLE style="WIDTH: 100%" id=017_solvedex_12_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_12>Example
12:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/017_049.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/017_050.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
3<I>x</I><SUP>2</SUP> + 4 = 7<I>x</I></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
3<I>x</I><SUP>2</SUP> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">
7<I>x</I> + 4 = 0</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
3<I>x</I><SUP>2</SUP> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">
4<I>x</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">
3<I>x</I> + 4 = 0</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
<I>x</I>(3<I>x</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">
4) <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">
1(3<I>x</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">
4) = 0</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
(<I>x</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">
1)(3<I>x</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">
4) = 0</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
<I>x</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">
1 = 0 or 3<I>x</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">
4 = 0</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
<I>x</I> = 1 or 4/3</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
Value of <I>x</I> can be 1 or
4/3.</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=017_solvedex_13>
<TABLE style="WIDTH: 100%" id=017_solvedex_13_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_13>Example
13:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>There are as many mangoes in a crate as there are
crates in a fruit shop. Pablo takes away one crate and then the shop has
only 90 mangoes left. How many mangoes does each crate have?</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>Let there be <I>x</I> mangoes in each crate. Then
there are <I>x</I> crates in the shop.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Total number of mangoes in the shop =
<I>x</I><SUP>2</SUP></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Pablo takes away one crate i.e. <I>x</I>
mangoes.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
<I>x</I><SUP>2</SUP> – <I>x</I> = 90</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
<I>x</I><SUP>2</SUP> – <I>x</I> – 90 = 0</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
<I>x</I><SUP>2</SUP> – 10<I>x</I> + 9<I>x </I>– 90 = 0</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
(<I>x</I> – 10)(<I>x </I>+ 9) = 0</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
<I>x</I> = 10 or <I>x</I> = –9</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>As the number of mangoes cannot be
negative,</SPAN></P>
<P><SPAN class=Normal><I>x</I> = 10.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
Each crate has 10 mangoes.</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=017_solvedex_14>
<TABLE style="WIDTH: 100%" id=017_solvedex_14_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_14>Example
14:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>Find the value of <I>x</I> if</SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/017_051.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/017_051.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>This can be rewritten as</SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/017_052.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
<I>x</I><SUP>2</SUP> – 110 = <I>x</I></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
<I>x</I><SUP>2</SUP> – <I>x</I> – 110 = 0</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
(<I>x</I> – 11)(<I>x</I> + 10) = 0</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
<I>x </I>= 11 or <I>x</I> = –10</SPAN></P>
<P><SPAN class=Normal>But <I>x</I> is positive.</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
<I>x</I> = 11</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorNote divLevel3">
<P><SPAN class=Note>REMEMBER:</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<UL class=Normal>
<LI value=1>The Method of Factorisation is very commonly used if <I>b</I> can
be expressed as a sum of two integers whose product is <I>ac</I>.</LI></UL>
<P><SPAN class=Normal></SPAN></P>
<UL>After sufficient practice in solving quadratic equations by factorisation,
some quadratic equations can be solved just by observation.</UL></DIV>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=017_solvedex_15>
<TABLE style="WIDTH: 100%" id=017_solvedex_15_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_15>Example
15:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>Find the roots of the equation, <I>x</I><SUP>2</SUP>
<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">
11<I>x</I> + 28 = 0</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>The sum of the roots of this quadratic equation is
11, and the product of the roots is 28. Hence, the roots are 4 and
7.</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=017_solvedex_16>
<TABLE style="WIDTH: 100%" id=017_solvedex_16_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_16>Example
16:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>Find the roots of the equation, <I>x</I><SUP>2</SUP>
+ <I>x</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">
30 = 0</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><B>Solution:</B></SPAN></P>
<P><SPAN class=Normal>The sum of the roots of this quadratic equation is
<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">1
and the product of the roots is <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">30.
Hence, the roots are <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">6
and 5.</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorNote divLevel3">
<P><SPAN class=Note>REMEMBER:</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<UL class=Normal>
<LI value=1>The expressions for the sum and product of the roots of a
quadratic equation can be used to find the values of symmetric expressions
involving the roots. If <I><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/alpha.png"></I>
and <I><IMG style="PADDING-BOTTOM: 2px; VERTICAL-ALIGN: bottom" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/beta.png"></I>
are the roots of a quadratic equation then (<I><IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/alpha.png"></I><SUP>2</SUP>
<IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/Plus_minus.png">
<I><IMG style="PADDING-BOTTOM: 2px; VERTICAL-ALIGN: bottom" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/beta.png"></I><SUP>2</SUP>),
(<I><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/alpha.png"></I><SUP>3</SUP>
<IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/Plus_minus.png">
<I><IMG style="PADDING-BOTTOM: 2px; VERTICAL-ALIGN: bottom" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/beta.png"></I><SUP>3</SUP>),
(<I><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/alpha.png"></I>/<I><IMG
style="PADDING-BOTTOM: 2px; VERTICAL-ALIGN: bottom" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/beta.png"></I>
+ <I><IMG style="PADDING-BOTTOM: 2px; VERTICAL-ALIGN: bottom" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/beta.png"></I>/<I><IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/alpha.png"></I>)
etc. are symmetric expressions in <I><IMG style="VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/alpha.png"></I>
and <I><IMG style="PADDING-BOTTOM: 2px; VERTICAL-ALIGN: bottom" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/beta.png"></I>.</LI></UL></DIV>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=017_solvedex_17>
<TABLE style="WIDTH: 100%" id=017_solvedex_17_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_16>Example
17:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>If <I><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/alpha.png"></I>
and <I><IMG style="PADDING-BOTTOM: 2px; VERTICAL-ALIGN: bottom" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/beta.png"></I>
are the roots of the equation 3<I>x</I><SUP>2</SUP> <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">
7<I>x</I> + 11 = 0, then find the value of <I><IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/alpha.png"></I><SUP>3</SUP>
+ <I><IMG style="PADDING-BOTTOM: 2px; VERTICAL-ALIGN: bottom" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/beta.png"></I><SUP>3</SUP>.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>In the quadratic equation 3<I>x</I><SUP>2</SUP> <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">
7<I>x</I> + 11 = 0, we have <I>a</I> = 3, <I>b</I> = <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">7,
<I>c</I> = 11</SPAN></P>
<P><SPAN class=Normal>Therefore,</SPAN></P>
<P><SPAN class=Normal><I><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/alpha.png"></I>
+ <I><IMG style="PADDING-BOTTOM: 2px; VERTICAL-ALIGN: bottom" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/beta.png"></I>
= (<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png"><I>b</I>/<I>a</I>)
= (7/3) and <I><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/alpha.png"><IMG
style="PADDING-BOTTOM: 2px; VERTICAL-ALIGN: bottom" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/beta.png"></I>
= (<I>c</I>/<I>a</I>) = (11/3)</SPAN></P>
<P><SPAN class=Normal>As we know, <I><IMG style="VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/alpha.png"></I><SUP>3</SUP>
+ <I><IMG style="PADDING-BOTTOM: 2px; VERTICAL-ALIGN: bottom" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/beta.png"></I><SUP>3</SUP>
= (<I><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/alpha.png"></I>
+ <I><IMG style="PADDING-BOTTOM: 2px; VERTICAL-ALIGN: bottom" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/beta.png"></I>)<SUP>3</SUP>
<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">
3<I><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/alpha.png"><IMG
style="PADDING-BOTTOM: 2px; VERTICAL-ALIGN: bottom" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/beta.png"></I>(<I><IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/alpha.png"></I>
+ <I><IMG style="PADDING-BOTTOM: 2px; VERTICAL-ALIGN: bottom" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/beta.png"></I>)</SPAN></P>
<P><SPAN class=Normal>Substituting the values that we have, we
get</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/017_053.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/017_054.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
<I><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/alpha.png"></I><SUP>3</SUP>
+ <I><IMG style="PADDING-BOTTOM: 2px; VERTICAL-ALIGN: bottom" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/beta.png"></I><SUP>3</SUP>
= (343 <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">
693)/27 = <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">350/27</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=017_solvedex_18>
<TABLE style="WIDTH: 100%" id=017_solvedex_18_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_18>Example
18:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>If one root of the equation, 4<I>x</I><SUP>2</SUP>
<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">
17<I>x</I> + <I>k</I> = 0 is 16 times the other, find <I>k</I>.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>In the quadratic equation, 4<I>x</I><SUP>2</SUP>
<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">
17<I>x</I> + <I>k</I> = 0</SPAN></P>
<P><SPAN class=Normal><I>a</I> = 4, <I>b</I> = <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">17,
<I>c</I> = k</SPAN></P>
<P><SPAN class=Normal>Let <I><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/alpha.png"></I>
and <I><IMG style="PADDING-BOTTOM: 2px; VERTICAL-ALIGN: bottom" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/beta.png"></I>
be the roots of the equation, and <I><IMG
style="PADDING-BOTTOM: 2px; VERTICAL-ALIGN: bottom" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/beta.png"></I>
= 16<I><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/alpha.png"></I></SPAN></P>
<P><SPAN class=Normal>Now, <I><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/alpha.png"></I>
+ <I><IMG style="PADDING-BOTTOM: 2px; VERTICAL-ALIGN: bottom" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/beta.png">
</I>= (<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png"><I>b</I>/<I>a</I>)
= (17/4)</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
<I><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/alpha.png"></I>
+ 16<I><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/alpha.png"></I>
= 17/4</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
17<I><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/alpha.png"></I>
= 17/4</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
<I><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/alpha.png"></I>
= (1/4)</SPAN></P>
<P><SPAN class=Normal>Now, <I><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/alpha.png"><IMG
style="PADDING-BOTTOM: 2px; VERTICAL-ALIGN: bottom" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/beta.png"></I>
= (<I>c</I>/<I>a</I>) = (<I>k</I>/4)</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
16<I><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/alpha.png"></I><SUP>2</SUP>
= (<I>k</I>/4)</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
16(1/4)<SUP>2</SUP> = <I>k</I>/4</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
<I>k</I> = 4</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=017_solvedex_19>
<TABLE style="WIDTH: 100%" id=017_solvedex_19_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_19>Example
19:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>Arjun and Abhi are together told to find the roots
of the equation <I>x</I><SUP>2</SUP> + <I>mx</I> + <I>n</I> = 0. They are
given two clues.</SPAN></P>
<OL class=Normal>
<LI value=1>If <I>m</I> is replaced by some other number, the roots of
the new equation are 3 and –6.</LI></OL>
<OL class=Normal>
<LI value=2>If <I>n</I> is replaced by some other number, the roots of
the resulting equation are –2 and –5.</LI></OL>
<P><SPAN class=Normal>Find the roots of the equation.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>Let <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/alpha.png">
and <IMG style="PADDING-BOTTOM: 2px; VERTICAL-ALIGN: bottom" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/beta.png">
be the roots of the equation <I>x</I><SUP>2</SUP> + <I>mx</I> + <I>n</I> =
0</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
<I><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/alpha.png"><IMG
style="PADDING-BOTTOM: 2px; VERTICAL-ALIGN: bottom" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/beta.png"></I>
= <I>n</I> and <I><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/alpha.png">
</I>+ <I><IMG style="PADDING-BOTTOM: 2px; VERTICAL-ALIGN: bottom" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/beta.png"></I>
= –<I>m</I></SPAN></P>
<P><SPAN class=Normal>When <I>n</I> is replaced, only the value of <I><IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/alpha.png"><IMG
style="PADDING-BOTTOM: 2px; VERTICAL-ALIGN: bottom" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/beta.png"></I>
gets affected, where as <I><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/alpha.png"></I>
+ <I><IMG style="PADDING-BOTTOM: 2px; VERTICAL-ALIGN: bottom" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/beta.png">
</I>is correct.</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
<I><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/alpha.png"></I>
+ <I><IMG style="PADDING-BOTTOM: 2px; VERTICAL-ALIGN: bottom" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/beta.png">
</I>= –2 – 5 = –7<SPAN style="MARGIN-LEFT: 12%"></SPAN>...(i)</SPAN></P>
<P><SPAN class=Normal>When <I>m</I> is replaced, only the value of <I><IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/alpha.png"></I>
+ <I><IMG style="PADDING-BOTTOM: 2px; VERTICAL-ALIGN: bottom" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/beta.png"></I>
gets affected, where as <I><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/alpha.png"><IMG
style="PADDING-BOTTOM: 2px; VERTICAL-ALIGN: bottom" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/beta.png"></I>
is correct.</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
<I><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/alpha.png"><IMG
style="PADDING-BOTTOM: 2px; VERTICAL-ALIGN: bottom" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/beta.png"></I>
= –18<SPAN style="MARGIN-LEFT: 24%"></SPAN>...(ii)</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
The actual equation is <I>x</I><SUP>2</SUP> + 7<I>x</I> – 18 = 0 and its
roots are –9 and 2.</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=017_solvedex_20>
<TABLE style="WIDTH: 100%" id=017_solvedex_20_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_20>Example
20:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>The roots of the equation <I>x</I><SUP>2</SUP> +
10<I>x</I> + 2 = 0 are <I><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/alpha.png"></I>
and <I><IMG style="PADDING-BOTTOM: 2px; VERTICAL-ALIGN: bottom" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/beta.png"></I>.
Find the equation with roots <I><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/alpha.png"></I>/2
and <I><IMG style="PADDING-BOTTOM: 2px; VERTICAL-ALIGN: bottom" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/beta.png"></I>/2.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal><I>x</I><SUP>2</SUP> + 10<I>x</I> + 2 = 0</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
<I><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/alpha.png"></I>
+ <I><IMG style="PADDING-BOTTOM: 2px; VERTICAL-ALIGN: bottom" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/beta.png">
</I>= –10 and <I><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/alpha.png"><IMG
style="PADDING-BOTTOM: 2px; VERTICAL-ALIGN: bottom" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/beta.png"></I>
= 2</SPAN></P>
<P><SPAN class=Normal>The equation with roots <I><IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/alpha.png"></I>/2
and <I><IMG style="PADDING-BOTTOM: 2px; VERTICAL-ALIGN: bottom" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/beta.png"></I>/2
is</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/017_055.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/017_056.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/017_057.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=017_solvedex_21>
<TABLE style="WIDTH: 100%" id=017_solvedex_21_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_21>Example
21:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>One root of the equation <I>ax</I><SUP>2</SUP> +
2<I>px</I> + <I>b</I><SUP>2</SUP> = 0 is five times the other. Find the
relation between <I>p</I>, <I>a</I> and <I>b</I>.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>Let <I><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/alpha.png"></I>
and 5<I><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/alpha.png"></I>
be the roots of the equation <I>ax</I><SUP>2</SUP> + 2<I>px</I> +
<I>b</I><SUP>2</SUP> = 0</SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/017_058.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>and</SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/017_059.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/017_060.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
5<I>p</I><SUP>2</SUP> =
9<I>ab</I><SUP>2</SUP></SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=017_solvedex_22>
<TABLE style="WIDTH: 100%" id=017_solvedex_22_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_22>Example
22:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>The sum of two numbers is 2/3 and the sum of their
cubes is 7/54. Find the quadratic equation, the roots of which are
reciprocals of these two numbers.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>Let the two numbers be <I>u</I> and
<I>v</I>.</SPAN></P>
<P><SPAN class=Normal>Then, <I>u</I> + <I>v</I> = 2/3 and
<I>u</I><SUP>3</SUP> + <I>v</I><SUP>3</SUP> = 7/54</SPAN></P>
<P><SPAN class=Normal>(<I>u</I> + <I>v</I>)<SUP>3</SUP> =
<I>u</I><SUP>3</SUP> + <I>v</I><SUP>3</SUP> + 3<I>uv</I>(<I>u</I> +
<I>v</I>)</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
<I>uv</I> = 1/12</SPAN></P>
<P><SPAN class=Normal>The required equation is</SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/017_061.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/017_062.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/017_063.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
<I>x</I><SUP>2</SUP> – 8<I>x</I> + 12 =
0</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=017_solvedex_23>
<TABLE style="WIDTH: 100%" id=017_solvedex_23_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_23>Example
23:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/017_064.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/017_065.png"><IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/017_066.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><I>x</I><SUP>2</SUP> + 8<I>x</I> + 20 = (<I>x</I> +
4)<SUP>2</SUP> + 4</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>This value will be minimum when (<I>x</I> +
4)<SUP>2</SUP> = 0</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Thus the minimum value of this expression will be
4.</SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/017_067.png">
= 0.25</SPAN></P>
<P><SPAN class=Normal></SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=017_solvedex_24>
<TABLE style="WIDTH: 100%" id=017_solvedex_24_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_24>Example
24:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/017_068.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/017_069.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/017_070.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/017_071.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
<I>x</I>(<I>x</I> + 5) = 36</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
<I>x</I><SUP>2</SUP> + 5<I>x </I>– 36 = 0</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
(<I>x </I>+ 9)(<I>x</I> – 4) = 0</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
<I>x</I> = –9 or <I>x</I> = 4</SPAN></P>
<P><SPAN class=Normal></SPAN></P><SPAN class=Normal>As logarithm of a
negative number is not defined, </SPAN>
<P></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/017_072.png"></SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<DIV id=017_III>
<TABLE id=017_III_bookmark class=tableh1 border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD
style="PADDING-BOTTOM: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 100%; PADDING-RIGHT: 0px; PADDING-TOP: 0px">
<OL class="oLevel1 Heading1">
<LI value=3><A id=III>NATURE OF THE ROOTS OF A QUADRATIC
EQUATION</A></LI></OL></TD></TR></TBODY></TABLE>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>The factor <I>b</I><SUP>2 </SUP>– 4<I>ac</I>
in the quadratic formula is called the <B>discriminant</B> and is denoted by ∆.
The nature of the roots of the equation depends on ∆.</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/017_073.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
The equation is a perfect square and has equal roots.</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>If ∆ > 0 and is a perfect square, roots
are real and rational.</SPAN></P>
<P class=pLevel3><SPAN class=Normal>If ∆ > 0 but is not a perfect square,
roots are real and irrational.</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>If ∆ < 0, the roots are complex in
nature.</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>Complex roots and irrational roots always
occur as conjugate pairs.</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
If </SPAN><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/017_074.png">
is one root of a quadratic equation, the other root is <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/017_075.png">.</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=017_solvedex_25>
<TABLE style="WIDTH: 100%" id=017_solvedex_25_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_25>Example
25:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>Find the nature of the roots of the following
quadratic equations.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>(1) <I>x</I><SUP>2</SUP> + <I>x</I> <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">
6 = 0</SPAN></P>
<P><SPAN class=Normal>(2) 3<I>x</I><SUP>2</SUP> <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">
8<I>x</I> + 1 = 0</SPAN></P>
<P><SPAN class=Normal>(3) 4<I>x</I><SUP>2</SUP> <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">
12<I>x</I> + 9 = 0</SPAN></P>
<P><SPAN class=Normal>(4) <I>x</I><SUP>2</SUP> <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">
4<I>x</I> + 13 = 0</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><B>Solution:</B></SPAN></P>
<P><SPAN class=Normal>1) ∆ = <I>b</I><SUP>2</SUP> <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">
4<I>ac</I> = 1 + 24 = 25</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Since 25 > 0 the roots are real.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Further, 25 is a perfect square and hence the roots
are rational.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>2) ∆ = <I>b</I><SUP>2</SUP> <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">
4<I>ac</I> = 64 <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">12
= 52</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Since 52 > 0, the roots are real and since 52 is
not a perfect square, the roots are irrational and are conjugates of each
other.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>3) ∆ = <I>b</I><SUP>2</SUP> <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">
4<I>ac</I> = 144 <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">
144 = 0</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Since ∆ = 0, the roots are real and
equal.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>4) ∆ = <I>b</I><SUP>2</SUP> <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">
4<I>ac</I> = 16 <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">
52 = <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">36</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Since ∆ < 0, the roots are imaginary and are
conjugates of each other.</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=017_solvedex_26>
<TABLE style="WIDTH: 100%" id=017_solvedex_26_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_26>Example
26:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>If the equation <I>x</I><SUP>3 </SUP>–
<I>ax</I><SUP>2</SUP> + <I>bx</I> – <I>a</I> = 0 has three real roots,
then it must be the case that,</SPAN></P>
<P><SPAN class=Normal><B>[CAT 2000]</B></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>(1) <I>b</I> = 1 <SPAN
style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN>(2) <I>b</I> ≠ 1</SPAN></P>
<P><SPAN class=Normal>(3) <I>a</I> = 1 <SPAN
style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN>(4) <I>a</I> ≠ 1</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal><I>x</I><SUP>3</SUP> – <I>ax</I><SUP>2</SUP> +<SUP>
</SUP><I>bx</I> – <I>a</I></SPAN></P>
<P><SPAN class=Normal>= <I>x</I><SUP>3</SUP> +<SUP> </SUP><I>bx</I> –
<I>ax</I><SUP>2 </SUP>– <I>a</I></SPAN></P>
<P><SPAN class=Normal>= <I>x</I>(<I>x</I><SUP>2</SUP> +<SUP>
</SUP><I>b</I>) – <I>a</I>(<I>x</I><SUP>2 </SUP>+ 1)</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
<I>x</I>(<I>x</I><SUP>2</SUP> +<SUP> </SUP><I>b</I>) –
<I>a</I>(<I>x</I><SUP>2 </SUP>+ 1) = 0</SPAN></P>
<P><SPAN class=Normal>If <I>b</I> = 1 then,</SPAN></P>
<P><SPAN class=Normal><I>x</I>(<I>x</I><SUP>2</SUP> +<SUP> </SUP>1) –
<I>a</I>(<I>x</I><SUP>2 </SUP>+ 1) = (<I>x</I> – <I>a</I>)(<I>x</I><SUP>2
</SUP>+ 1) = 0</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
(<I>x</I> – <I>a</I>) = 0 or (<I>x</I><SUP>2 </SUP>+ 1) = 0</SPAN></P>
<P><SPAN class=Normal>But then (<I>x</I><SUP>2 </SUP>+ 1) = 0 does not
have real roots.</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
<I>b</I> ≠ 1</SPAN></P>
<P><SPAN class=Normal>Hence, <B>option
2</B>.</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=017_solvedex_27>
<TABLE style="WIDTH: 100%" id=017_solvedex_27_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_27>Example
27:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/017_076.png">
<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/017_077.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><B>[XAT 2008]</B></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>(1) Both 0</SPAN></P>
<P><SPAN class=Normal>(2) Imaginary</SPAN></P>
<P><SPAN class=Normal>(3) Real and both positive</SPAN></P>
<P><SPAN class=Normal>(4) Real and of opposite sign</SPAN></P>
<P><SPAN class=Normal>(5) Real and both negative</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>We have,</SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/017_078.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/017_079.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/017_080.png"></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
Roots are real and after checking for the arbitrary values of <I>p</I> say
0.5, we get both the roots will have negative roots.</SPAN></P>
<P><SPAN class=Normal>Hence, <B>option
5.</B></SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=017_solvedex_28>
<TABLE style="WIDTH: 100%" id=017_solvedex_28_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_28>Example
28:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>If the roots of the equation 2<I>ax</I><SUP>2</SUP>
+ 2<I>ax</I> + 1 = 0 are real and distinct, then the number of possible
integer value(s) a can take between 1 and 4 (inclusive) is:</SPAN></P>
<P><SPAN class=Normal><B>[JMET 2010]</B></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>(1) 1 <SPAN style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN>(2) 4 <SPAN
style="MARGIN-LEFT: 25px"></SPAN></SPAN></P>
<P><SPAN class=Normal>(3) 3 <SPAN style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN>(4) 2</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>Roots of equation 2<I>ax</I><SUP>2</SUP> +
2<I>ax</I> + 1 = 0 are real and distinct</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
∆ > 0</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
4<I>a</I><SUP>2</SUP> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">8<I>a</I>
> 0</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
2<I>a</I>(<I>a</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">
2) > 0</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
<I>a</I> > 0 and <I>a</I> > 2 or <I>a</I> < 0 and <I>a</I> <
2</SPAN></P>
<P><SPAN class=Normal>But <I>a</I> can take values between 1 and 4
(inclusive) only.</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
<I>a</I> > 2</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
<I>a</I> = 3, 4</SPAN></P>
<P><SPAN class=Normal>So, <I>a</I> can take 2 values.</SPAN></P>
<P><SPAN class=Normal>Hence, <B>option
4</B>.</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel2><SPAN class=Normal></SPAN></P></DIV>
<DIV id=017_IV>
<TABLE id=017_IV_bookmark class=tableh1 border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD
style="PADDING-BOTTOM: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 100%; PADDING-RIGHT: 0px; PADDING-TOP: 0px">
<OL class="oLevel1 Heading1">
<LI value=4><A id=IV>GRAPHICAL REPRESENTATION OF A QUADRATIC
EQUATION</A></LI></OL></TD></TR></TBODY></TABLE>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>The quadratic function <I>y</I> =
<I>ax</I><SUP>2</SUP> + <I>bx</I> + <I>c</I> is a parabola whose axis is
parallel to the <I>y</I>-axis.</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>If <I>a</I> > 0, then the parabola opens
upwards. If <I>a</I> < 0, then the parabola opens downwards.</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pImage><SPAN><IMG alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/017_081.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><B>Nature of Roots</B></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pImage><SPAN><IMG alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/017_085.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>When ∆ > 0, the parabola intersects the
<I>x</I>-axis in two points and hence the roots are real and distinct in nature.
This condition is shown by the curve 1. </SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>When ∆ = 0, the parabola intersects the
<I>x</I>-axis at only one point and hence we get only one root which is given by
<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png"><I>b</I>/2<I>a</I>.
This condition is shown by the curve 2.</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>When ∆ < 0, then the parabola does not
intersect the <I>x</I>-axis and hence we do not have any real roots. The roots
are complex in nature. This condition is shown by the curve 3.</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P></DIV>
<DIV id=017_V>
<TABLE id=017_V_bookmark class=tableh1 border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD
style="PADDING-BOTTOM: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 100%; PADDING-RIGHT: 0px; PADDING-TOP: 0px">
<OL class="oLevel1 Heading1">
<LI value=5><A id=V>HIGHER ORDER
EQUATIONS</A></LI></OL></TD></TR></TBODY></TABLE>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>An expression of the form
<I>a</I><SUB>1</SUB><I>x<SUP>n</I></SUP> +
<I>a</I><SUB>2</SUB><I>x<SUP>n</I>–1</SUP> +
<I>a</I><SUB>3</SUB><I>x<SUP>n</I>-2</SUP> + … +
<I>a<SUB>n</I>-1</SUB><I>x</I><SUP>2</SUP> + <I>a<SUB>n</SUB>x </I>+
<I>a<SUB>n</I>+1</SUB>, where <I>a</I><SUB>1</SUB>, <I>a</I><SUB>2</SUB>,
<I>a</I><SUB>3</SUB>, ..., <I>a<SUB>n</I>+1</SUB> are real numbers is called an
<B><I>n</I><SUP>th</SUP> degree polynomial</B> in variable <I>x</I>. An equation
of the form <I>a</I><SUB>1</SUB><I>x<SUP>n</I></SUP> +
<I>a</I><SUB>2</SUB><I>x<SUP>n</I>-1</SUP> +
<I>a</I><SUB>3</SUB><I>x<SUP>n</I>-2</SUP> + … +
<I>a<SUB>n</I>-1</SUB><I>x</I><SUP>2</SUP> + <I>a<SUB>n</SUB>x</I> +
<I>a<SUB>n</I>+1 </SUB>= 0 is called an <B><I>n</I><SUP>th</SUP> degree
equation</B> in variable <I>x</I>. An <I>n</I><SUP>th</SUP> degree equation has
<I>n</I> roots.</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>If the roots of an <I>n</I><SUP>th</SUP>
order equation are <I><IMG
style="PADDING-BOTTOM: 2px; VERTICAL-ALIGN: text-bottom" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/beta.png"></I><SUB>1</SUB>,
<I><IMG style="PADDING-BOTTOM: 2px; VERTICAL-ALIGN: text-bottom" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/beta.png"></I><SUB>2</SUB>,
<I><IMG style="PADDING-BOTTOM: 2px; VERTICAL-ALIGN: text-bottom" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/beta.png"></I><SUB>3</SUB>,
…, <I><IMG style="PADDING-BOTTOM: 2px; VERTICAL-ALIGN: text-bottom" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/beta.png"><SUB>n</I>-1</SUB>,
<I><IMG style="PADDING-BOTTOM: 2px; VERTICAL-ALIGN: text-bottom" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/beta.png"><SUB>n</I></SUB>
then the equation can be rewritten as follows.</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><I>a</I><SUB>1</SUB>(<I>x </I><IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">
<I><IMG style="PADDING-BOTTOM: 2px; VERTICAL-ALIGN: text-bottom" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/beta.png"></I><SUB>1</SUB>)(<I>x</I>
<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">
<I><IMG style="PADDING-BOTTOM: 2px; VERTICAL-ALIGN: text-bottom" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/beta.png"></I><SUB>2</SUB>)(<I>x</I>
– <I><IMG style="PADDING-BOTTOM: 2px; VERTICAL-ALIGN: text-bottom" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/beta.png"></I><SUB>3</SUB>)…(<I>x</I>
<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">
<I><IMG style="PADDING-BOTTOM: 2px; VERTICAL-ALIGN: text-bottom" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/beta.png"><SUB>n</I>-1</SUB>)(<I>x</I>
<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">
<I><IMG style="PADDING-BOTTOM: 2px; VERTICAL-ALIGN: text-bottom" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/beta.png"><SUB>n</I></SUB>)
= 0</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><B>Explanation:</B></SPAN></P>
<P class=pLevel3><SPAN class=Normal>The root of an equation is the value of
<I>x</I> that satisfies the equation. Since <I><IMG
style="PADDING-BOTTOM: 2px; VERTICAL-ALIGN: text-bottom" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/beta.png"></I><SUB>1</SUB>
is a root of the equation the equation should be satisfied when the value of
<I>x</I> is substituted as <I><IMG
style="PADDING-BOTTOM: 2px; VERTICAL-ALIGN: text-bottom" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/beta.png"></I><SUB>1</SUB>
in the equation. Hence (<I>x</I> – <I><IMG
style="PADDING-BOTTOM: 2px; VERTICAL-ALIGN: text-bottom" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/beta.png"></I><SUB>1</SUB>)
has to be one of the factors of the LHS of the equation. Similarly using the
same logic for the other roots the equation can be rewritten as shown</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><B>Relation between Roots and
Coefficients</B></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>Let <I>S</I><SUB>1</SUB> = Sum of roots
taken one at a time </SPAN></P>
<P class=pLevel3><SPAN class=Normal>i.e. <I><IMG
style="PADDING-BOTTOM: 2px; VERTICAL-ALIGN: text-bottom" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/beta.png"></I><SUB>1</SUB>
+ <I><IMG style="PADDING-BOTTOM: 2px; VERTICAL-ALIGN: text-bottom" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/beta.png"></I><SUB>2</SUB>
+ <I><IMG style="PADDING-BOTTOM: 2px; VERTICAL-ALIGN: text-bottom" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/beta.png"></I><SUB>3
</SUB>+…</SPAN></P>
<P class=pLevel3><SPAN class=Normal><I>S</I><SUB>2</SUB> = Sum of roots taken
two at a time</SPAN></P>
<P class=pLevel3><SPAN class=Normal>i.e. <I><IMG
style="PADDING-BOTTOM: 2px; VERTICAL-ALIGN: text-bottom" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/beta.png"></I><SUB>1</SUB><I><IMG
style="PADDING-BOTTOM: 2px; VERTICAL-ALIGN: text-bottom" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/beta.png"></I><SUB>2</SUB>
+ <I><IMG style="PADDING-BOTTOM: 2px; VERTICAL-ALIGN: text-bottom" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/beta.png"></I><SUB>2</SUB><I><IMG
style="PADDING-BOTTOM: 2px; VERTICAL-ALIGN: text-bottom" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/beta.png"></I><SUB>3
</SUB>+ <I><IMG style="PADDING-BOTTOM: 2px; VERTICAL-ALIGN: text-bottom" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/beta.png"></I><SUB>3</SUB><I><IMG
style="PADDING-BOTTOM: 2px; VERTICAL-ALIGN: text-bottom" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/beta.png"></I><SUB>4
</SUB>+…</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><I>S</I><SUB>3</SUB> = Sum of roots taken
three at a time</SPAN></P>
<P class=pLevel3><SPAN class=Normal>i.e. <I><IMG
style="PADDING-BOTTOM: 2px; VERTICAL-ALIGN: text-bottom" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/beta.png"></I><SUB>1</SUB><I><IMG
style="PADDING-BOTTOM: 2px; VERTICAL-ALIGN: text-bottom" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/beta.png"></I><SUB>2</SUB><I><IMG
style="PADDING-BOTTOM: 2px; VERTICAL-ALIGN: text-bottom" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/beta.png"></I><SUB>3</SUB>
+ <I><IMG style="PADDING-BOTTOM: 2px; VERTICAL-ALIGN: text-bottom" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/beta.png"></I><SUB>2</SUB><I><IMG
style="PADDING-BOTTOM: 2px; VERTICAL-ALIGN: text-bottom" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/beta.png"></I><SUB>3</SUB><I><IMG
style="PADDING-BOTTOM: 2px; VERTICAL-ALIGN: text-bottom" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/beta.png"></I><SUB>4
</SUB>+…</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>In the equation <I>ax</I><SUP>2</SUP> +
<I>bx</I> + <I>c</I> = 0, we have</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><I>S</I><SUB>1</SUB> = <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png"><I>b</I>/<I>a</I>
and <I>S</I><SUB>2</SUB> = <I>c</I>/<I>a</I><SPAN
style="MARGIN-LEFT: 25px"></SPAN></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>In the equation <I>ax</I><SUP>3</SUP> +
<I>bx</I><SUP>2</SUP> + <I>cx</I> + <I>d</I> = 0</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><I>S</I><SUB>1</SUB> = <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png"><I>b</I>/<I>a</I>
and <I>S</I><SUB>2</SUB> = <I>c</I>/<I>a</I> and <I>S</I><SUB>3</SUB> = <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png"><I>d</I>/<I>a</I></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>In equation <I>ax</I><SUP>4</SUP> +
<I>bx</I><SUP>3</SUP> + <I>cx</I><SUP>2</SUP> + <I>dx</I> + <I>e</I> =
0</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><I>S</I><SUB>1</SUB> = <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png"><I>b</I>/<I>a</I>
and <I>S</I><SUB>2</SUB> = <I>c</I>/<I>a</I> and <I>S</I><SUB>3</SUB> = <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png"><I>d</I>/<I>a</I>
and <I>S</I><SUB>4 </SUB>= <I>e</I>/<I>a</I></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>This can be generalised for any equation of
the form</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><I>a</I><SUB>1</SUB><I>x<SUP>n</I></SUP> +
<I>a</I><SUB>2</SUB><I>x<SUP>n</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">
1</SUP> + <I>a</I><SUB>3</SUB><I>x<SUP>n </I><IMG style="VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">
2</SUP> + … + <I>a<SUB>n</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">
1</SUB><I>x</I><SUP>2</SUP> + <I>a<SUB>n</SUB>x</I> + <I>a<SUB>n</I> + 1 </SUB>=
0</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>Thus,</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><I>S</I><SUB>1</SUB><SUP> </SUP>=
–<I>a</I><SUB>2</SUB>/<I>a</I><SUB>1</SPAN></P>
<P class=pLevel3><SPAN class=Normal><I>S</I>2</SUB> =
<I>a</I><SUB>3</SUB>/<I>a</I><SUB>1</SPAN></P>
<P class=pLevel3><SPAN class=Normal><I>S</I>3</SUB> =
–<I>a</I><SUB>4</SUB>/<I>a</I><SUB>1</SUB> and so on till</SPAN></P>
<P class=pLevel3><SPAN class=Normal><I>S<SUB>n</I></SUB> =
(–1)<I><SUP>n</I></SUP> <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/multiplication.png">
(<I>a<SUB>n</I>+1</SUB>/<I>a</I><SUB>1</SUB>)</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=017_solvedex_29>
<TABLE style="WIDTH: 100%" id=017_solvedex_29_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_29>Example
29:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>If <I>a</I>, <I>b</I>, <I>c</I>, <I>d</I> are the
roots of the equation <I>x</I><SUP>4</SUP> – 3<I>x</I><SUP>3</SUP> +
2<I>x</I><SUP>2</SUP> – 5<I>x</I> + 1 = 0,</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/017_082.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>For the equation, <I>x</I><SUP>4</SUP> –
3<I>x</I><SUP>3</SUP> + 2<I>x</I><SUP>2</SUP> – 5<I>x</I> + 1 =
0,</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><I>S</I><SUB>1</SUB> = 3, <I>S</I><SUB>2</SUB> = 2,
<I>S</I><SUB>3</SUB> = 5 and <I>S</I><SUB>4 </SUB>= 1</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/017_083.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><I>bcd</I> + <I>acd</I> + <I>abd</I> + <I>abc</I> =
<I>S</I><SUB>3</SUB> = 5</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><I>abcd</I> = <I>S</I><SUB>4 </SUB>= 1</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/017_084.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=017_solvedex_30>
<TABLE style="WIDTH: 100%" id=017_solvedex_30_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_30>Example
30:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>If <I>a</I>, <I>b</I> and <I>c</I> are roots of
<I>x</I><SUP>3</SUP> – 6<I>x</I><SUP>2</SUP> + 11<I>x</I> – 6 = 0 and the
roots of the equation <I>x</I><SUP>3</SUP> – <I>px</I><SUP>2</SUP> +
<I>qx</I> – <I>r </I>= 0 are <I>a</I> + <I>b</I>, <I>b</I> + <I>c</I> and
<I>c</I> + <I>a</I>, then <I>r</I> equals:</SPAN></P>
<P><SPAN class=Normal><B>[JMET 2010]</B></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>(1) 40<SPAN style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN>(2) 50<SPAN
style="MARGIN-LEFT: 25px"></SPAN></SPAN></P>
<P><SPAN class=Normal>(3) 60<SPAN style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN>(4) 70</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>Here <I>x</I><SUP>3 </SUP>– 6<I>x</I><SUP>2</SUP> +
11<I>x</I> – 6 = 0</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
(<I>x</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">
1)(<I>x</I><SUP>2 </SUP>– 5<I>x</I> + 6) = 0</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
(<I>x </I><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">
1)(<I>x</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">
3) (<I>x</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">
2) = 0</SPAN></P>
<P><SPAN class=Normal><I>x</I> = 1, 2, 3</SPAN></P>
<P><SPAN class=Normal>Roots of the equations <I>x</I><SUP>3</SUP> –
<I>px</I><SUP>2</SUP> + <I>qx</I> <IMG style="VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">
<I>r</I> = 0 are <I>a</I> + <I>b</I>, <I>b</I> + <I>c</I> and <I>c</I> +
<I>a</SPAN></P>
<P><SPAN class=Normal></I><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
Its roots will be 1 + 2 = 3, 2 + 3 = 5 and 1 + 3 = 4</SPAN></P>
<P><SPAN class=Normal>So the equation will be (<I>x</I> <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">
3)(<I>x</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">
4 )(<I>x</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">
5) = 0</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
<I>r</I> = 3 <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/multiplication.png">
4 <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/multiplication.png">
5 = 60</SPAN></P>
<P><SPAN class=Normal>Hence, <B>option 3.</B>
</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<DIV id=017_V_A>
<TABLE id=017_V_A_bookmark class=tableh1 border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD
style="PADDING-BOTTOM: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 100%; PADDING-RIGHT: 0px; PADDING-TOP: 0px">
<OL style="MARGIN-LEFT: 12%" class="oLevel2 Heading2">
<LI value=1><A id=V_A>SOLVING HIGHER ORDER
EQUATIONS</A></LI></OL></TD></TR></TBODY></TABLE>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>In this section we would restrict ourselves
to the equations solvable by using simple tips and tricks.</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<OL class="oLevel3 Heading3 ListStyleOL">
<LI value=1>EQUATIONS THAT CAN BE CONVERTED TO A QUADRATIC EQUATION</LI></OL>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal>Please refer to the following
example.</SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel4">
<DIV id=017_solvedex_31>
<TABLE style="WIDTH: 100%" id=017_solvedex_31_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_31>Example
31:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>Find the roots of the equation, <I>x</I><SUP>4</SUP>
<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">
5<I>x</I><SUP>2</SUP> + 4 = 0.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>We have to find out roots of the
equation,</SPAN></P>
<P><SPAN class=Normal><I>x</I><SUP>4</SUP> <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">
5<I>x</I><SUP>2</SUP> + 4 = 0</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Substituting <I>x</I><SUP>2</SUP> = <I>y</I> in this
equation,</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><I>y</I><SUP>2</SUP> <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">
5<I>y</I> + 4 = 0, which is nothing but a simple quadratic
equation</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
(<I>y</I> – 1)(<I>y</I> – 4) = 0</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
<I>y</I> = 1 or <I>y</I> = 4</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
<I>x</I><SUP>2</SUP> = 4 or <I>x</I><SUP>2</SUP> = 1</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
<I>x</I> = <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/Plus_minus.png">2
or <I>x</I> = <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/Plus_minus.png">1</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
Roots of the equation <I>x</I><SUP>4</SUP> <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">
5<I>x</I><SUP>2</SUP> + 4 = 0 are:</SPAN></P>
<P><SPAN class=Normal>2, <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">2,
1 and <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">1.</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<OL class="oLevel3 Heading3 ListStyleOL">
<LI value=2>USING SYNTHETIC DIVISION FOR FACTORISATION</LI></OL>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal>Please refer to the following
example.</SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel4">
<DIV id=017_solvedex_32>
<TABLE style="WIDTH: 100%" id=017_solvedex_32_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_32>Example
32:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>Find the roots of 5<I>x</I><SUP>3</SUP> +
4<I>x</I><SUP>2</SUP> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">
5<I>x</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">
4 = 0.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>On observation we can see that if we substitute
<I>x</I> = 1 the equation is satisfied. Hence one of the roots is 1 and
consequently one of the factors of the LHS is (<I>x</I> <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">
1).</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Using <B>synthetic division</B> we can factorise the
equation.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=pImage><IMG alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/017_087.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
5<I>x</I><SUP>3</SUP> + 4<I>x</I><SUP>2</SUP> <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">
5<I>x</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">
4 = (<I>x</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">
1)(5<I>x</I><SUP>2</SUP> + 9<I>x</I> + 4)</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
(<I>x</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">
1)(5<I>x</I><SUP>2</SUP> + 9<I>x</I> + 4) = 0</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
(<I>x</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">
1)(5<I>x</I> + 4)(<I>x</I> + 1) = 0</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
<I>x</I> = 1 or <I>x</I> = –4/5 or <I>x</I> = –1</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><I>Alternatively,</I></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>If <I><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/alpha.png"></I><SUB>1</SUB>,
<I><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/alpha.png"></I><SUB>2</SUB>
and <I><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/alpha.png"></I><SUB>3</SUB>
are the roots of the equation, we have,</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><I><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/alpha.png"></I><SUB>1</SUB>
+ <I><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/alpha.png"></I><SUB>2</SUB>
+ <I><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/alpha.png"></I><SUB>3
</SUB>= <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">4/5</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><I><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/alpha.png"></I><SUB>1</SUB><I><IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/alpha.png"></I><SUB>2</SUB>
+ <I><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/alpha.png"></I><SUB>2</SUB><I><IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/alpha.png"></I><SUB>3</SUB>
+ <I><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/alpha.png"></I><SUB>3</SUB><IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/alpha.png"><SUB>1</SUB>
= <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">5/5
= –1</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><I><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/alpha.png"></I><SUB>1</SUB><I><IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/alpha.png"></I><SUB>2</SUB><I><IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/alpha.png"></I><SUB>3</SUB>
= 4/5</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>We have three equations and three variables using
which the value of roots can be found
out.</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<OL class="oLevel3 Heading3 ListStyleOL">
<LI value=3>USING ALGEBRAIC FORMULAE</LI></OL>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal>Equations may be simplified using algebraic
formulae.</SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel4">
<DIV id=017_solvedex_33>
<TABLE style="WIDTH: 100%" id=017_solvedex_33_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_33>Example
33:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>Find the values of <I>x</I> and <I>y</I>
if</SPAN></P>
<P><SPAN class=Normal><I>x</I><SUP>3</SUP> + <I>y</I><SUP>3</SUP> = 169
and <I>x</I> + <I>y</I> = 1.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal><I>x</I><SUP>3</SUP> + <I>y</I><SUP>3</SUP> = 169
<SPAN style="MARGIN-LEFT: 80px"></SPAN>...(i)</SPAN></P>
<P><SPAN class=Normal><I>x</I> + <I>y</I> = 1<SPAN
style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 66px"></SPAN>...(ii)</SPAN></P>
<P><SPAN class=Normal>We know that <I>x</I><SUP>3</SUP> +
<I>y</I><SUP>3</SUP> = (<I>x</I> + <I>y</I>)(<I>x</I><SUP>2</SUP> <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">
<I>xy</I> + <I>y</I><SUP>2</SUP>)</SPAN></P>
<P><SPAN class=Normal>Hence the first equation can be written
as:</SPAN></P>
<P><SPAN class=Normal>(<I>x </I>+ <I>y</I>)(<I>x</I><SUP>2</SUP> <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">
<I>xy</I> + <I>y</I><SUP>2</SUP>) = 169</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
(<I>x</I><SUP>2 </SUP><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">
<I>xy</I> + <I>y</I><SUP>2</SUP>) = 169 <SPAN
style="MARGIN-LEFT: 25px"></SPAN>...(iii)</SPAN></P>
<P><SPAN class=Normal>Squaring (ii),</SPAN></P>
<P><SPAN class=Normal><I>x</I><SUP>2</SUP> + 2<I>xy</I> +
<I>y</I><SUP>2</SUP> = 1<SPAN
style="MARGIN-LEFT: 61px"></SPAN>...(iv)</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Subtracting equation (i) from equation (ii) we
get,</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>3<I>xy</I> = <IMG style="VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">168</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
<I>x</I> = <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">56/<I>y</I></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>From (ii),</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">56/<I>y</I>
+ <I>y</I> = 1</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">56
<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">
<I>y</I> + <I>y</I><SUP>2</SUP> = 0</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
<I>y</I> = 8 or <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">7</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
<I>x</I> = <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">7
or 8</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel4">
<DIV id=017_solvedex_34>
<TABLE style="WIDTH: 100%" id=017_solvedex_34_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_34>Example
34:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>(<I>x</I><SUP>2</SUP> – 5<I>x</I> + 6) completely
divides <I>x</I><SUP>3</SUP> + <I>px</I> + <I>q</I>. Find the quadratic
equation having roots <I>p</I> and <I>q</I>.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal><I>x</I><SUP>2</SUP> – 5<I>x</I> + 6 = (<I>x</I> –
2)(<I>x</I> – 3)</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
<I>x</I> = 2 and <I>x</I> = 3 satisfy the equation <I>x</I><SUP>3</SUP> +
<I>px</I> + <I>q</I> = 0</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
8 + 2<I>p</I> + <I>q</I> = 0<SPAN
style="MARGIN-LEFT: 70px"></SPAN>...(i)</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
27 + 3<I>p</I> + <I>q</I> = 0<SPAN style="MARGIN-LEFT: 58px"></SPAN>
...(ii)</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Solving (i) and (ii), <I>p</I> = –19 and <I>q</I> =
30</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
The quadratic equation having roots <I>p</I> and <I>q</I> is</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><I>x</I><SUP>2</SUP> – (<I>p</I> + <I>q</I>)<I>x</I>
+ <I>pq</I> = 0</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
<I>x</I><SUP>2</SUP> – 11<I>x</I> – 570 = 0</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><I>Alternatively,</I></SPAN></P>
<P><SPAN class=Normal><I>x</I><SUP>3</SUP> + <I>px</I> + <I>q</I> =
(<I>x</I> – 2)(<I>x</I> – 3)(<I>x</I> – <I>k</I>)</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
<I>x</I><SUP>3</SUP> + <I>px</I> + <I>q</I> = <I>x</I><SUP>3</SUP> +
(<I>k</I> – 5)<I>x</I><SUP>2</SUP> + (6 – 5<I>k</I>)<I>x</I> +
6<I>k</I></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
<I>k</I> – 5 = 0, 6 – 5<I>k</I> = <I>p</I> and 6<I>k</I> = q</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
<I>p</I> = –19 and <I>q</I> = 30</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
The quadratic equation having roots <I>p</I> and <I>q</I> is</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><I>x</I><SUP>2</SUP> – (<I>p</I> + <I>q</I>)<I>x</I>
+ <I>pq</I> = 0</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
<I>x</I><SUP>2</SUP> – 11<I>x</I> – 570 =
0</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel4><SPAN class=Normal></SPAN></P></DIV>
<DIV id=017_V_B>
<TABLE id=017_V_B_bookmark class=tableh1 border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD
style="PADDING-BOTTOM: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 100%; PADDING-RIGHT: 0px; PADDING-TOP: 0px">
<OL style="MARGIN-LEFT: 12%" class="oLevel2 Heading2">
<LI value=2><A id=V_B>DESCARTES’ RULE OF
SIGNS</A></LI></OL></TD></TR></TBODY></TABLE>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>If the coefficients of all the powers of
<I>x</I> of an equation are positive, the equation does not have any positive
root.</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>For example, the equation
<I>x</I><SUP>4</SUP> + 4<I>x</I><SUP>3</SUP> + 5<I>x</I><SUP>2</SUP> + 9<I>x</I>
+ 1 = 0 will have no positive root.</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>If the coefficients of all even powers of
<I>x</I> in an equation are of one sign and the coefficients of all the odd
powers of <I>x</I> are of the opposite sign, the equation does not have any
negative root.</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>For example, the equation
<I>x</I><SUP>5</SUP> – 2<I>x</I><SUP>4</SUP> + 9<I>x</I><SUP>3</SUP> <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">
5<I>x</I><SUP>2</SUP> + 7<I>x</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">
20 = 0 will not have any negative root.</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=017_solvedex_35>
<TABLE style="WIDTH: 100%" id=017_solvedex_35_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_35>Example
35:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>If <I>a</I> > 3, how many positive roots does the
following equation have?</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><I>x</I><SUP>9</SUP> + (2<I>a</I> +
7)<I>x</I><SUP>7</SUP> + (2<I>a</I> – 5)<I>x</I><SUP>5</SUP> + (2<I>a</I>
+ 3)<I>x</I><SUP>3</SUP> + (2<I>a</I> – 1)<I>x </I>+ 2<I>a</I> =
0.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>As <I>a</I> > 3, (2<I>a</I> + 7), (2<I>a</I> –
5), (2<I>a</I> + 3), (2<I>a</I> – 1) and 2<I>a</I> are all positive. Thus,
by Descartes’ rule of signs, the equation has no positive
roots.</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorNote divLevel3">
<P><SPAN class=Note>REMEMBER:</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<UL class=Normal>
<LI value=1>Consider a quadratic equation <I>ax</I><SUP>2</SUP> + <I>bx</I> +
c = 0</LI></UL>
<UL>The roots of this equation are given by</UL>
<P><SPAN class=Normal></SPAN></P>
<UL><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/017_086.png"></UL>
<P><SPAN class=Normal></SPAN></P>
<UL>If <I><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/alpha.png"></I>
and <I><IMG style="PADDING-BOTTOM: 2px; VERTICAL-ALIGN: bottom" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/beta.png"></I>
are the roots of the quadratic equation,</UL>
<UL class=Normal>
<LI value=2>Sum of the roots = <I><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/alpha.png"></I>
+ <I><IMG style="PADDING-BOTTOM: 2px; VERTICAL-ALIGN: bottom" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/beta.png"></I>
= –<I>b</I>/<I>a</I></LI></UL>
<P><SPAN class=Normal></SPAN></P>
<UL class=Normal>
<LI value=3>Product of the roots = <I><IMG style="VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/alpha.png"><IMG
style="PADDING-BOTTOM: 2px; VERTICAL-ALIGN: bottom" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/beta.png"></I>
= <I>c</I>/<I>a</I></LI></UL>
<P><SPAN class=Normal></SPAN></P>
<UL class=Normal>
<LI value=4>The equation can be expressed as:</LI></UL>
<P><SPAN class=Normal></SPAN></P>
<UL><I>x</I><SUP>2</SUP> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">
(Sum of the roots)<I>x</I> + (Product of the roots) = 0</UL>
<UL></UL>
<UL class=Normal>
<LI value=5>The discriminant ∆ of a quadratic equation is given by</LI></UL>
<UL></UL>
<UL>∆ = <I>b</I><SUP>2</SUP> – 4<I>ac</I></UL>
<UL></UL>
<UL>If ∆ = 0, the equation has real roots.</UL>
<UL></UL>
<UL>If ∆ > 0 and is a perfect square, roots are real and rational.</UL>
<UL></UL>
<UL>If ∆ > 0 but is not a perfect square, roots are real and irrational.</UL>
<UL></UL>
<UL>If ∆ < 0, the roots are complex in nature.</UL></DIV>
<P class=pLevel4><SPAN class=Normal></SPAN></P></DIV></DIV>
<DIV id=017_VI>
<TABLE id=017_VI_bookmark class=tableh1 border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD
style="PADDING-BOTTOM: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 100%; PADDING-RIGHT: 0px; PADDING-TOP: 0px">
<OL class="oLevel1 Heading1">
<LI value=6><A id=VI>FINDING THE COMMON
ROOT</A></LI></OL></TD></TR></TBODY></TABLE>
<P class=pLevel3><SPAN class=Normal>Let <I>f</I>(<I>x</I>) = 0 and
<I>g</I>(<I>x</I>) = 0 be two quadratic equations in <I>x </I>then their common
roots will also be the roots of the equation <I>f</I>(<I>x</I>) <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">
<I>g</I>(<I>x</I>) = 0.</SPAN></P>
<P class=pLevel3><SPAN class=Normal>The roots of <I>f</I>(<I>x</I>) <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">
<I>g</I>(<I>x</I>) = 0 may or may not be the roots of <I>f</I>(<I>x</I>) = 0 and
<I>g</I>(<I>x</I>) = 0. Therefore, to find the common roots of
<I>f</I>(<I>x</I>) = 0 and <I>g</I>(<I>x</I>) = 0 follow the following
procedure.</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal><B>Step 1:</B></SPAN></P>
<P class=pLevel3><SPAN class=Normal>Find <I>f</I>(<I>x</I>) <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">
<I>g</I>(<I>x</I>)</SPAN></P>
<P class=pLevel3><SPAN class=Normal><B>Step 2:</B></SPAN></P>
<P class=pLevel3><SPAN class=Normal>Find the root(s) of <I>f</I>(<I>x</I>) <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">
<I>g</I>(<I>x</I>) = 0</SPAN></P>
<P class=pLevel3><SPAN class=Normal><B>Step 3:</B></SPAN></P>
<P class=pLevel3><SPAN class=Normal>Substitute the roots of <I>f</I>(<I>x</I>)
<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/minus.png">
<I>g</I>(<I>x</I>) = 0 in <I>f</I>(<I>x</I>) and <I>g</I>(<I>x</I>). If the
roots satisfy both the equations then they are the common roots.</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=017_solvedex_36>
<TABLE style="WIDTH: 100%" id=017_solvedex_34_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_36>Example
36:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>The number of roots common between the two equations
<I>x</I><SUP>3 </SUP>+ 3<I>x</I><SUP>2 </SUP>+ 4<I>x</I> + 5 = 0 and
<I>x</I><SUP>3 </SUP>+ 2<I>x</I><SUP>2 </SUP>+ 7<I>x</I> + 3 = 0
is:</SPAN></P>
<P><SPAN class=Normal><B>[CAT 2003 Re-Test]</B></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>(1) 0 <SPAN style="MARGIN-LEFT: 50px"></SPAN>(2)
1</SPAN></P>
<P><SPAN class=Normal>(3) 2 <SPAN style="MARGIN-LEFT: 50px"></SPAN>(4)
3</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal><I>x</I><SUP>3 </SUP>+ 3<I>x</I><SUP>2</SUP> +
4<I>x</I> + 5 = 0<SPAN style="MARGIN-LEFT: 50px"></SPAN>... (i)</SPAN></P>
<P><SPAN class=Normal><I>x</I><SUP>3</SUP> + 2<I>x</I><SUP>2</SUP> +
7<I>x</I> + 3 = 0<SPAN style="MARGIN-LEFT: 49px"></SPAN>...
(ii)</SPAN></P>
<P><SPAN class=Normal>Equating equations (i) and (ii), we get,</SPAN></P>
<P><SPAN class=Normal><I>x</I><SUP>2 </SUP>– 3<I>x</I> + 2 = 0</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
(<I>x</I> – 1)(<I>x</I> – 2) = 0</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/5a6cd06e-8365-46f6-8892-d17b3b298fa6/017_images/therefore.png">
<I>x</I> = 1 or <I>x</I> = 2</SPAN></P>
<P><SPAN class=Normal>Since both <I>x</I> = 1 and <I>x</I> = 2, do not
satisfy either (i) or (ii), there exists no common roots for equations (i)
and (ii).</SPAN></P>
<P><SPAN class=Normal>Hence, <B>option
1</B>.</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel3><SPAN class=Normal></SPAN></P></DIV></DIV></DIV></BODY></HTML>
guruhttp://www.blogger.com/profile/01591375209227117455noreply@blogger.comtag:blogger.com,1999:blog-7850540640045081515.post-38458837872350540142015-03-03T03:50:00.002-08:002019-09-14T18:55:04.164-07:00log<div dir="ltr" style="text-align: left;" trbidi="on">
<html xmlns="http://www.w3.org/1999/xhtml"><head><title>Logarithms</title>
<script src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/CrossLink.js" type="text/javascript"></script>
<script src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/Lesson.js" type="text/javascript"></script>
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</head>
<body onload="lesson_Function_1(); lesson_Function_2()">
<div class="oTOCTitle">
<span class="oTOCTitleFont"><a class="header" href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014.htm#Title">Contents</a></span></div>
<ol class="oTOCLevel1">
<li class="oTocLiClass"><a class="header" href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014.htm#I">INTRODUCTION</a>
</li>
<li class="oTocLiClass"><a class="header" href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014.htm#II">COMMON
LOGARITHMS</a> </li>
<li class="oTocLiClass"><a class="header" href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014.htm#III">NATURAL
LOGARITHMS</a> </li>
<li class="oTocLiClass"><a class="header" href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014.htm#IV">LAWS
AND PROPERTIES OF LOGARITHMS</a> </li>
<li class="oTocLiClass"><a class="header" href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014.htm#V">THE
CHARACTERISTIC AND THE MANTISSA</a> </li>
<li class="oTocLiClass"><a class="header" href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014.htm#VI">SOLVED
EXAMPLES</a> </li>
</ol>
<div class="pLevel2">
<span class="Normal"></span></div>
<div class="pTitle" id="Title">
<span class="Title">Logarithms</span></div>
<hr />
<div class="pLevel2">
<span class="Normal"></span></div>
<div class="pLevel2">
<span class="Normal"></span>
<div id="014_I">
<table border="0" cellpadding="0" cellspacing="0" class="tableh1" id="014_I_bookmark">
<tbody>
<tr>
<td style="margin: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; width: 100%;">
<ol class="oLevel1 Heading1">
<li value="1"><a href="https://www.blogger.com/null" id="I">INTRODUCTION</a></li>
</ol>
</td></tr>
</tbody></table>
<div class="pLevel3">
<span class="Normal"></span></div>
<div class="pLevel3">
<span class="Normal">Long before the invention of calculators or
any other computing device, mathematicians used logarithms to make complex
calculations.</span></div>
<div class="pLevel3">
<span class="Normal"></span></div>
<div class="pLevel3">
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/014_001.png" style="vertical-align: middle;" /></span></div>
<div class="pLevel3">
<span class="Normal"></span></div>
<div class="pLevel3">
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/014_002.png" style="vertical-align: middle;" /></span></div>
<div class="pLevel3">
<span class="Normal"></span></div>
<div class="pLevel3">
<span class="Normal">In words, <i>p</i> is referred to as the
<b>logarithm</b> of <i>N</i> to the base <i>a</i>. Here, <i>a</i> cannot be
negative, 0 or 1 because logarithm of negative numbers and 0 is not defined and
the logarithm of 1 (to the base 10) is equal to 0 (<img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/symb23.png" style="margin-bottom: 0px; padding-bottom: 1px;" />
<i>a</i><sup>0</sup> = 1).</span></div>
<div class="pLevel3">
<span class="Normal"></span></div>
<div class="pLevel3">
<span class="Normal">In other words, the logarithm of any number
to a given base is the power to which the base must be raised to get that
number.</span></div>
<div class="pLevel3">
<span class="Normal"></span></div>
<div class="pLevel3">
<span class="Normal">The reverse is also true. That is, if
log<i><sub>a</sub></i></span></div>
</div>
</div>
</body></html> <i>N</i> = <i>p</i>, then we can write <i>N =
a<sup>p</sup></i>
<br />
<div class="pLevel3">
<span class="Normal"></span></div>
<div class="pLevel3">
<span class="Normal">For example,</span></div>
<div class="pLevel3">
<span class="Normal">Since 81 = 3<sup>4</sup>,</span></div>
<div class="pLevel3">
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/therefore.png" style="margin-bottom: 0px; padding-bottom: 1px;" />
log<sub>3</sub> 81 = 4</span></div>
<div class="pLevel3">
<span class="Normal"></span></div>
<div class="divTextBox divColorNote divLevel3">
<span class="Note">REMEMBER:</span><br />
<span class="Normal"></span><br />
<ul class="ListParagraph ListStyleUL5">
<li value="1">Logarithms are defined only for positive numbers. It is not
defined for zero or negative numbers.</li>
</ul>
<ul class="ListParagraph ListStyleUL5">
<li value="2">Logarithms can be expressed in any base.</li>
</ul>
<ul class="ListParagraph ListStyleUL5">
<li value="3">Logarithms expressed in one base can be converted to logarithms
expressed in any other base.</li>
</ul>
<ul class="ListParagraph ListStyleUL5">
<li value="4">Logarithms are generally expressed to the base 10. These are
called <b>common logarithms</b>.</li>
</ul>
</div>
<div class="pLevel3">
<span class="Normal">We will now study them in detail.</span></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="divTextBox divColorExampleHeading divLevel3">
<div id="014_solvedex_1">
<table border="0" cellpadding="0" cellspacing="0" id="014_solvedex_1_bookmark" style="width: 100%;">
<tbody>
<tr>
<td style="background-color: transparent; width: 100%;"><span class="ExampleHeading"><a href="https://www.blogger.com/null" id="ex_1">Example
1:</a></span></td></tr>
</tbody></table>
<table border="0" cellpadding="0" cellspacing="0">
<tbody>
<tr>
<td style="background-color: transparent; width: 100%;"><span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/014_003.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"></span><br />
<span class="Normal">Then what is the value of <i>x</i> ?</span><br />
<span class="Normal"><b>[FMS 2009]</b></span><br />
<span class="Normal"></span><br />
<span class="Normal">(1) 2 <span style="margin-left: 25px;"></span><span style="margin-left: 25px;"></span>(2) 3</span><br />
<span class="Normal">(3) 4 <span style="margin-left: 25px;"></span><span style="margin-left: 25px;"></span>(4) 5</span><br />
<span class="Normal"></span><br />
<span class="SolutionHeading">Solution:</span><br />
<span class="Normal">If log₇ <i>a</i> = 0, then <i>a</i> = 7⁰. Thus
<i>a</i> = 1.</span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/014_004.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"></span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/014_005.png" style="vertical-align: middle;" /></span><br />
<span class="Normal">On substituting the values of <i>x</i> from the
given options, we see that <i>x</i> = 4 satisfies the above
equation.</span><br />
<span class="Normal">Hence, <b>option
3</b>.</span></td></tr>
</tbody></table>
</div>
</div>
<div class="pLevel4">
<span class="Normal"></span></div>
<div id="014_II">
<table border="0" cellpadding="0" cellspacing="0" class="tableh1" id="014_II_bookmark">
<tbody>
<tr>
<td style="margin: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; width: 100%;"><ol class="oLevel1 Heading1">
<li value="2"><a href="https://www.blogger.com/null" id="II">COMMON LOGARITHMS</a></li>
</ol>
</td></tr>
</tbody></table>
<div class="pLevel3">
<span class="Normal"></span></div>
<div class="pLevel3">
<span class="Normal">Common logarithms are expressed in the base
10. Most problems in logarithms deal with common logarithms. If no base is
mentioned, it is assumed that the base is 10.</span></div>
<div class="pLevel3">
<span class="Normal">For example, the common logarithm of 100
would be expressed as log<sub>10</sub> 100 and is equal to 2.</span></div>
<div class="pLevel3">
<span class="Normal">(<img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/symb23.png" style="margin-bottom: 0px; padding-bottom: 1px;" />
100 = 10<sup>2</sup>)</span></div>
<div class="pLevel3">
<span class="Normal"></span></div>
<div class="pLevel3">
<span class="Normal">The following table gives the values of
common logarithms for some select values.</span></div>
<div class="pLevel3">
<span class="Normal"></span></div>
<div class="tblCenter" style="margin-left: 2%; text-align: left;">
<table cellspacing="1" class="visible_border_table tLevel2" style="text-align: center;">
<tbody>
<tr>
<td class="visible_border_table"><i>N</i></td>
<td class="visible_border_table">log<sub>10</sub> <i>N</i></td></tr>
<tr>
<td class="visible_border_table">0.001</td>
<td class="visible_border_table">–3</td></tr>
<tr>
<td class="visible_border_table">0.01</td>
<td class="visible_border_table">–2</td></tr>
<tr>
<td class="visible_border_table">0.1</td>
<td class="visible_border_table">–1</td></tr>
<tr>
<td class="visible_border_table">1</td>
<td class="visible_border_table">0</td></tr>
<tr>
<td class="visible_border_table">10</td>
<td class="visible_border_table">1</td></tr>
<tr>
<td class="visible_border_table">100</td>
<td class="visible_border_table">2</td></tr>
<tr>
<td class="visible_border_table">1000</td>
<td class="visible_border_table">3</td></tr>
<tr>
<td class="visible_border_table">10000</td>
<td class="visible_border_table">4</td></tr>
</tbody></table>
</div>
<div class="pLevel3">
<span class="Normal"></span></div>
<div class="pLevel3">
<span class="Normal">Shown below is a graph of the function
<i>y</i> = log<sub>10</sub> <i>x</i></span></div>
<div class="pLevel3">
<span class="Normal"></span></div>
</div>
<div id="014_III">
<table border="0" cellpadding="0" cellspacing="0" class="tableh1" id="014_III_bookmark">
<tbody>
<tr>
<td style="margin: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; width: 100%;"><ol class="oLevel1 Heading1">
<li value="3"><a href="https://www.blogger.com/null" id="III">NATURAL LOGARITHMS</a></li>
</ol>
</td></tr>
</tbody></table>
<div class="pLevel3">
<span class="Normal"></span></div>
<div class="pLevel3">
<img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/014_006.png" /></div>
<div class="pLevel3">
<span class="Normal">Logarithms are also expressed in the base
‘<i>e</i>’. Such logarithms are called <b>Natural logarithms</b> or Napierian
logarithms.</span></div>
<div class="pLevel3">
<span class="Normal"><i>e</i> is an irrational constant
approximately equal to 2.718281828.</span></div>
<div class="pLevel3">
<span class="Normal">The natural logarithm of a positive real
number <i>x</i>, is represented by ln(<i>x</i>) or
log<sub><i>e</i></sub>(<i>x</i>).</span></div>
<div class="pLevel3">
<span class="Normal">Like common logarithm, natural logarithm is
also not defined for zero and negative real numbers .</span></div>
<div class="pLevel3">
<span class="Normal"></span></div>
</div>
<div id="014_IV">
<table border="0" cellpadding="0" cellspacing="0" class="tableh1" id="014_IV_bookmark">
<tbody>
<tr>
<td style="margin: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; width: 100%;"><ol class="oLevel1 Heading1">
<li value="4"><a href="https://www.blogger.com/null" id="IV">LAWS AND PROPERTIES OF
LOGARITHMS</a></li>
</ol>
</td></tr>
</tbody></table>
<div class="pLevel3">
<span class="Normal"></span></div>
<div class="pLevel3">
<span class="Normal">Most problems involving logarithms can be
solved by applying the following laws/properties:</span></div>
<div class="pLevel3">
<span class="Normal"></span></div>
<ol class="oLevel2 ListParagraph ListStyleOL">
<li value="1">The logarithm of 1 to any base is always 0.</li>
</ol>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="pLevel4">
<span class="Normal">i.e. log<i><sub>b</sub></i></span></div>
</div>
1 = 0 (provided
that <i>b</i> is positive and ≠ 1)
<br />
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="pLevel4">
<span class="Normal"><b>Explanation:</b></span></div>
<div class="pLevel4">
<span class="Normal">For any real number <i>b</i>,
<i>b</i><sup>0</sup> is defined as 1.</span></div>
<div class="pLevel4">
<span class="Normal">Hence, log<i><sub>b</sub></i></span></div>
1 =
0
<br />
<div class="pLevel4">
<span class="Normal"></span></div>
<ol class="oLevel2 ListParagraph ListStyleOL">
<li value="2">The logarithm of a number to a base which is equal to that number
is 1.</li>
</ol>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="pLevel4">
<span class="Normal">i.e. log<i><sub>b</sub> b</i> = 1
</span></div>
<div class="pLevel4">
<span class="Normal">(provided that <i>b</i> is positive and ≠
1)</span></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="pLevel4">
<span class="Normal"><b>Explanation:</b></span></div>
<div class="pLevel4">
<span class="Normal">For any real number <i>b, b<sup>1</sup></i></span></div>
= <i>b</i>
<br />
<div class="pLevel4">
<span class="Normal">Hence, log<i><sub>b</sub> b</i> =
1</span></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="divTextBox divColorExampleHeading divLevel3">
<div id="014_solvedex_2">
<table border="0" cellpadding="0" cellspacing="0" id="014_solvedex_2_bookmark" style="width: 100%;">
<tbody>
<tr>
<td style="background-color: transparent; width: 100%;"><span class="ExampleHeading"><a href="https://www.blogger.com/null" id="ex_2">Example
2:</a></span></td></tr>
</tbody></table>
<table border="0" cellpadding="0" cellspacing="0">
<tbody>
<tr>
<td style="background-color: transparent; width: 100%;"><span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/014_007.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"></span><br />
<span class="Normal"><b>[FMS 2010]</b></span><br />
<span class="Normal"></span><br />
<span class="Normal">(1) 25<span style="margin-left: 25px;"></span><span style="margin-left: 25px;"></span>(2) 10</span><br />
<span class="Normal">(3) 2 <span style="margin-left: 25px;"></span><span style="margin-left: 25px;"></span>(4) 1</span><br />
<span class="Normal"></span><br />
<span class="SolutionHeading">Solution:</span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/014_008.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"></span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/014_009.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"></span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/014_010.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"></span><br />
<span class="Normal">= 1<sup>2</sup></span><br />
<span class="Normal">= 1</span><br />
<span class="Normal">Hence, <b>option
4</b>.</span></td></tr>
</tbody></table>
</div>
</div>
<div class="pLevel4">
<span class="Normal"></span></div>
<ol class="oLevel2 ListParagraph ListStyleOL">
<li value="3">log<i><sub>a</sub> b</i> <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/multiplication.png" style="margin-bottom: 0px; padding-bottom: 0px;" />
log<i><sub>b</sub> a</i> = 1</li>
</ol>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="pLevel4">
<span class="Normal"><b>Explanation:</b></span></div>
<div class="pLevel4">
<span class="Normal">Let log<i><sub>a</sub> b</i> =
<i>x</i></span></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="pLevel4">
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/therefore.png" style="margin-bottom: 0px; padding-bottom: 1px;" />
<i>a<sup>x</sup> </i>=<i> b</i></span></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="pLevel4">
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/therefore.png" style="margin-bottom: 0px; padding-bottom: 1px;" />
<i>a </i>=<i> b<sup>1</sup></i>/<i>x</i></span></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="pLevel4">
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/therefore.png" style="margin-bottom: 0px; padding-bottom: 1px;" />
log<i><sub>b</sub> a</i> = 1/<i>x</i> = 1/(log<i><sub>a</sub> b</i>)</span></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="pLevel4">
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/therefore.png" style="margin-bottom: 0px; padding-bottom: 1px;" />
log<i><sub>a</sub> b</i> <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/multiplication.png" style="margin-bottom: 0px; padding-bottom: 0px;" />
log<i><sub>b</sub> a</i> = 1</span></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<ol class="oLevel2 ListParagraph ListStyleOL">
<li value="4">The logarithm of the product of two numbers is equal to the sum of
the logarithms of the individual numbers.</li>
</ol>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="pLevel4">
<span class="Normal">i.e. log<i><sub>b</sub></i></span></div>
(<i>m</i> <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/multiplication.png" style="margin-bottom: 0px; padding-bottom: 0px;" />
<i>n</i>) = log<i><sub>b</sub></i> <i>m</i> + log<i><sub>b</sub></i>
<i>n</i>
<br />
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="divTextBox divColorExampleHeading divLevel3">
<div id="014_solvedex_3">
<table border="0" cellpadding="0" cellspacing="0" id="014_solvedex_3_bookmark" style="width: 100%;">
<tbody>
<tr>
<td style="background-color: transparent; width: 100%;"><span class="ExampleHeading"><a href="https://www.blogger.com/null" id="ex_3">Example
3:</a></span></td></tr>
</tbody></table>
<table border="0" cellpadding="0" cellspacing="0">
<tbody>
<tr>
<td style="background-color: transparent; width: 100%;"><span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/014_011.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"></span><br />
<span class="Normal"><b>[CAT 2002]</b></span><br />
<span class="Normal"></span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/014_012.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"></span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/014_013.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"></span><br />
<span class="SolutionHeading">Solution:</span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/014_014.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"></span><br />
<span class="Normal"><span style="margin-left: 85px;"></span><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/014_015.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"></span><br />
<span class="Normal"><span style="margin-left: 85px;"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/014_016.png" style="vertical-align: middle;" /></span></span><br />
<span class="Normal"></span><br />
<span class="Normal"><span style="margin-left: 85px;"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/014_017.png" style="vertical-align: middle;" /></span></span><br />
<span class="Normal"></span><br />
<span class="Normal"><span style="margin-left: 85px;"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/014_018.png" style="vertical-align: middle;" /></span></span><br />
<span class="Normal"></span><br />
<span class="Normal">Hence, <b>option
4.</b></span></td></tr>
</tbody></table>
</div>
</div>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="divTextBox divColorExampleHeading divLevel3">
<div id="014_solvedex_4">
<table border="0" cellpadding="0" cellspacing="0" id="014_solvedex_4_bookmark" style="width: 100%;">
<tbody>
<tr>
<td style="background-color: transparent; width: 100%;"><span class="ExampleHeading"><a href="https://www.blogger.com/null" id="ex_4">Example
4:</a></span></td></tr>
</tbody></table>
<table border="0" cellpadding="0" cellspacing="0">
<tbody>
<tr>
<td style="background-color: transparent; width: 100%;"><span class="Normal">What is the sum of<i> n</i> terms in the
series:</span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/014_019.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"></span><br />
<span class="Normal"><b>[CAT 2003 Re-test]</b></span><br />
<span class="Normal"></span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/014_020.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"></span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/014_021.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"></span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/014_022.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"></span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/014_023.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"></span><br />
<span class="SolutionHeading">Solution:</span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/014_024.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"></span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/014_025.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"></span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/014_026.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"></span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/014_027.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"></span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/014_028.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"></span><br />
<span class="Normal">Hence, <b>option
4</b>.</span></td></tr>
</tbody></table>
</div>
</div>
<div class="pLevel4">
<span class="Normal"></span></div>
<ol class="oLevel2 ListParagraph ListStyleOL">
<li value="5">The logarithm of the ratio of numbers is equal to the difference
obtained when the logarithm of the denominator is subtracted from that of the
numerator.</li>
</ol>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="pLevel4">
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/014_029.png" style="vertical-align: middle;" /></span></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="pLevel4">
<span class="Normal"><b>Explanation:</b></span></div>
<div class="pLevel4">
<span class="Normal">Let log<i><sub>b</sub></i></span></div>
<i>m</i> =
<i>x</i> and log<i><sub>b</sub></i> <i>n</i> = <i>y</i>
<br />
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="pLevel4">
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/therefore.png" style="margin-bottom: 0px; padding-bottom: 1px;" />
<i>m </i>=<i> b<sup>x</sup></i></span></div>
and <i>n </i>=<i> b<sup>y</sup></i>
<br />
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="pLevel4">
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/therefore.png" style="margin-bottom: 0px; padding-bottom: 1px;" />
<i>mn</i> = <i>b<sup>x</sup></i></span></div>
<img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/multiplication.png" style="margin-bottom: 0px; padding-bottom: 0px;" />
<i>b<sup>y</sup></i> = <i>b<sup>x </sup></i>+<i> y</i>
<br />
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="pLevel4">
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/therefore.png" style="margin-bottom: 0px; padding-bottom: 1px;" />
log<i><sub>b</sub></i></span></div>
(<i>mn</i>) = <i>x </i>+ <i>y</i> =
log<i><sub>b</sub></i> <i>m</i> + log<i><sub>b</sub></i> <i>n</i>
<br />
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="pLevel4">
<span class="Normal">Similarly, we can prove that,</span></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="pLevel4">
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/014_029.png" style="vertical-align: middle;" /></span></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="divTextBox divColorExampleHeading divLevel3">
<div id="014_solvedex_5">
<table border="0" cellpadding="0" cellspacing="0" id="014_solvedex_5_bookmark" style="width: 100%;">
<tbody>
<tr>
<td style="background-color: transparent; width: 100%;"><span class="ExampleHeading"><a href="https://www.blogger.com/null" id="ex_5">Example
5:</a></span></td></tr>
</tbody></table>
<table border="0" cellpadding="0" cellspacing="0">
<tbody>
<tr>
<td style="background-color: transparent; width: 100%;"><span class="Normal">Find the value of log<sub>9</sub> 171 –
log<sub>9</sub> 19?</span><br />
<span class="Normal"></span><br />
<span class="SolutionHeading">Solution:</span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/014_030.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"></span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/014_031.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"></span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/014_032.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"></span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/014_033.png" style="vertical-align: middle;" /><span style="margin-left: 41px;"></span>… using law 2</span><br />
<span class="Normal"></span></td></tr>
</tbody></table>
</div>
</div>
<div class="pLevel4">
<span class="Normal"></span></div>
<ol class="oLevel2 ListParagraph ListStyleOL">
<li value="6">The logarithm of a number raised to a power is equal to the
product of the power and the logarithm of the number.</li>
</ol>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="pLevel4">
<span class="Normal">i.e. log<i><sub>b</sub></i></span></div>
(<i>m<sup>n</sup></i>) = <i>n</i> log<i><sub>b</sub></i> <i>m</i>
<br />
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="pLevel4">
<span class="Normal"><b>Explanation:</b></span></div>
<div class="pLevel4">
<span class="Normal">Let log<i><sub>a</sub></i></span></div>
<i>m<sup>n</sup></i> = log<i><sub>a</sub></i> (<i>m</i> <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/multiplication.png" style="margin-bottom: 0px; padding-bottom: 0px;" />
<i>m</i> <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/multiplication.png" style="margin-bottom: 0px; padding-bottom: 0px;" />
<i>m</i> <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/multiplication.png" style="margin-bottom: 0px; padding-bottom: 0px;" />
… <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/multiplication.png" style="margin-bottom: 0px; padding-bottom: 0px;" />
<i>m</i>)
<br />
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="pLevel4">
<span class="Normal">Using the law 4 above, the RHS can be
expressed as,</span></div>
<div class="pLevel4">
<span class="Normal">log<i><sub>a</sub></i></span></div>
(<i>m</i> <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/multiplication.png" style="margin-bottom: 0px; padding-bottom: 0px;" />
<i>m</i> <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/multiplication.png" style="margin-bottom: 0px; padding-bottom: 0px;" />
<i>m</i> <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/multiplication.png" style="margin-bottom: 0px; padding-bottom: 0px;" />
… <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/multiplication.png" style="margin-bottom: 0px; padding-bottom: 0px;" />
<i>m</i>) = log<i><sub>a</sub></i> <i>m</i> + log<i><sub>a</sub></i> <i>m</i> +
log<i><sub>a</sub></i> <i>m</i> + … + log<i><sub>a</sub></i> <i>m</i> = <i>n</i>
log<i><sub>a</sub></i> <i>m</i>
<br />
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="pLevel4">
<span class="Normal">Hence, log<i><sub>a</sub></i></span></div>
<i>m<sup>n</sup></i> = <i>n </i>log<i><sub>a</sub></i> <i>m</i>
<br />
<div class="pLevel4">
<span class="Normal"></span></div>
<ol class="oLevel2 ListParagraph ListStyleOL">
<li value="7">Logarithms expressed in one base can be converted to logarithms
expressed in any other base.</li>
</ol>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="pLevel4">
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/014_034.png" style="vertical-align: middle;" /></span></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="pLevel4">
<span class="Normal"><b>Explanation:</b></span></div>
<div class="pLevel4">
<span class="Normal">Let log<i><sub>b</sub></i></span></div>
<i>m</i> =
<i>x</i>
<br />
<div class="pLevel4">
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/therefore.png" style="margin-bottom: 0px; padding-bottom: 1px;" />
<i>m</i> = <i>b<sup>x</sup></i></span></div>
<div class="pLevel4">
<span class="Normal">Let <i>a<sup>y</sup></i></span></div>
=
<i>b</i>
<br />
<div class="pLevel4">
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/therefore.png" style="margin-bottom: 0px; padding-bottom: 1px;" />
<i>y</i> = log<i><sub>a</sub></i></span></div>
<i>b</i>
<br />
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="pLevel4">
<span class="Normal">If we substitute the value of <i>b</i> in
the expression for <i>m</i>, we get,</span></div>
<div class="pLevel4">
<span class="Normal"><i>m</i> =
(<i>a<sup>y</sup></i></span></div>
)<i><sup>x</sup></i> = <i>a<sup>xy</sup></i>
<br />
<div class="pLevel4">
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/therefore.png" style="margin-bottom: 0px; padding-bottom: 1px;" />
log<i><sub>a</sub></i></span></div>
<i>m</i> = <i>xy</i> = log<i><sub>b</sub></i> <i>m</i>
<img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/multiplication.png" style="margin-bottom: 0px; padding-bottom: 0px;" />
log<i><sub>a</sub></i> <i>b</i>
<br />
<div class="pLevel4">
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/therefore.png" style="margin-bottom: 0px; padding-bottom: 1px;" />
log<i><sub>b</sub></i></span></div>
<i>m</i> = log<i><sub>a</sub></i>
<i>m</i>/log<i><sub>a</sub></i> <i>b</i>
<br />
<div class="pLevel4">
<span class="Normal"><span style="margin-left: 60px;"></span>=
log<i><sub>a</sub></i></span></div>
<i>m</i> <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/multiplication.png" style="margin-bottom: 0px; padding-bottom: 0px;" />
log<i><sub>b</sub></i> <i>a</i> ...(from law 3)
<br />
<div class="pLevel4">
<span class="Normal"></span></div>
<ol class="oLevel2 ListParagraph ListStyleOL">
<li value="8"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/014_035.png" style="vertical-align: middle;" /></li>
</ol>
<div class="pLevel4">
<span class="Normal"><b>Explanation:</b></span></div>
<div class="pLevel4">
<span class="Normal">Let log<i><sub>b</sub></i></span></div>
<i>n</i> =
<i>x</i>
<br />
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="pLevel4">
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/therefore.png" style="margin-bottom: 0px; padding-bottom: 1px;" />
<i>n = b<sup>x</sup></i></span></div>
<div class="pLevel4">
<span class="Normal">Substituting the value of <i>x</i> in the
above expression, we get,</span></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="pLevel4">
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/014_036.png" style="vertical-align: middle;" /></span></div>
<div class="pLevel4">
<span class="Normal"></span></div>
<div class="divTextBox divColorExampleHeading divLevel3">
<div id="014_solvedex_6">
<table border="0" cellpadding="0" cellspacing="0" id="014_solvedex_6_bookmark" style="width: 100%;">
<tbody>
<tr>
<td style="background-color: transparent; width: 100%;"><span class="ExampleHeading"><a href="https://www.blogger.com/null" id="ex_6">Example
6:</a></span></td></tr>
</tbody></table>
<table border="0" cellpadding="0" cellspacing="0">
<tbody>
<tr>
<td style="background-color: transparent; width: 100%;"><span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/014_037.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"></span><br />
<span class="SolutionHeading">Solution:</span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/014_038.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"></span></td></tr>
</tbody></table>
</div>
</div>
<ol class="oLevel2 ListParagraph ListStyleOL">
<li value="9">Given an equation, log<i><sub>a</sub></i></li>
</ol>
<i>M</i> =
log<i><sub>b</sub></i> <i>N</i>:
<br />
<ol class="oLevel3 ListParagraph ListStyleOL2">
<li value="1">If <i>M</i> = <i>N</i>, then <i>a</i> will be equal to
<i>b</i>.</li>
</ol>
<ol class="oLevel3 ListParagraph ListStyleOL2">
<li value="2">If <i>a</i> = <i>b</i>, then <i>M</i> will be equal to
<i>N</i>.</li>
</ol>
<div class="pLevel4">
<span class="Normal"></span></div>
<ol class="oLevel2 ListParagraph ListStyleOL">
<li value="10">In order to compare two numbers, when the comparison between
their respective logarithms is given:</li>
</ol>
<ol class="oLevel3 ListParagraph ListStyleOL2">
<li value="1">If <i>b</i> > 1 and log<i><sub>b</sub></i></li>
</ol>
<i>m</i> >
log<i><sub>b</sub></i> <i>n</i>, then <i>m</i> > <i>n</i>
<br />
<ol class="oLevel3 ListParagraph ListStyleOL2">
<li value="2">Also, if <i>b</i> < 1 and log<i><sub>b</sub></i></li>
</ol>
<i>m</i> >
log<i><sub>b</sub></i> <i>n</i>, then <i>m</i> < <i>n</i>.
<br />
<div class="pLevel4">
<span class="Normal"></span></div>
<div id="014_V">
<table border="0" cellpadding="0" cellspacing="0" class="tableh1" id="014_V_bookmark">
<tbody>
<tr>
<td style="margin: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; width: 100%;"><ol class="oLevel1 Heading1">
<li value="5"><a href="https://www.blogger.com/null" id="V">THE CHARACTERISTIC AND THE
MANTISSA</a></li>
</ol>
</td></tr>
</tbody></table>
<div class="pLevel3">
<span class="Normal"></span></div>
<div class="pLevel3">
<span class="Normal">The logarithm of a number has an integral
part and a decimal part. The integral part is called the <b>characteristic</b>
and the decimal part is called the <b>mantissa</b>.</span></div>
<div class="pLevel3">
<span class="Normal"></span></div>
<div class="pLevel3">
<span class="Normal">For example, log<sub>10</sub>
9100</span></div>
<div class="pLevel3">
<span class="Normal">= log<sub>10</sub> (10<sup>3</sup> <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/multiplication.png" style="margin-bottom: 0px; padding-bottom: 0px;" />
9.1)</span></div>
<div class="pLevel3">
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/congru2.png" style="vertical-align: middle;" />
log<sub>10</sub> 10<sup>3</sup> + log<sub>10</sub> 9.1</span></div>
<div class="pLevel3">
<span class="Normal">= 3 + 0.959</span></div>
<div class="pLevel3">
<span class="Normal"></span></div>
<div class="pLevel3">
<span class="Normal">Here, 3 is the characteristic and 0.959 is
the mantissa.</span></div>
<div class="pLevel3">
<span class="Normal"></span></div>
<div class="pLevel3">
<span class="Normal">An important property of the characteristic
(for common logarithms) is that (characteristic + 1) will give the number of
digits of the integral part of the number whose logarithm you determined. In the
above example, 9100 contains (3 + 1) = 4 digits.</span></div>
<div class="pLevel3">
<span class="Normal">This is explained as follows:</span></div>
<div class="pLevel3">
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/symb23.png" style="margin-bottom: 0px; padding-bottom: 1px;" />
log<sub>10</sub> 9100 = 3.959,</span></div>
<div class="pLevel3">
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/therefore.png" style="margin-bottom: 0px; padding-bottom: 1px;" />
9100 = 10<sup>3.959</sup> = 10<sup>3 + 0.959</sup> = 10<sup>3</sup> <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/multiplication.png" style="margin-bottom: 0px; padding-bottom: 0px;" />
10<sup>0.959</sup></span></div>
<div class="pLevel3">
<span class="Normal">= 10<i><sup>x</sup></i></span></div>
</div>
<img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/multiplication.png" style="margin-bottom: 0px; padding-bottom: 0px;" />
10<i><sup>y</sup></i>
<br />
<div class="pLevel3">
<span class="Normal">Since <i>y</i> will always be < 1,
10<i><sup>y</sup></i></span></div>
will always be < 10. Hence, the number of digits in the
number will be equal to the number of digits in 10<i><sup>x</sup></i>, which in
turn is equal to (<i>x</i> + 1).
<br />
<div class="pLevel3">
<span class="Normal"></span></div>
<div class="pLevel3">
<span class="Normal">Also, the characteristic of a logarithm
could be either positive or negative; however, the mantissa should always be
positive.</span></div>
<div class="pLevel3">
<span class="Normal">For example, log<sub>10</sub> 0.091 =
log<sub>10</sub> (10<sup>–2</sup> <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/multiplication.png" style="margin-bottom: 0px; padding-bottom: 0px;" />
9.1)</span></div>
<div class="pLevel3">
<span class="Normal"></span></div>
<div class="pLevel3">
<span class="Normal"><span class="Normal"><span style="margin-left: 175px;"></span><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/congru2.png" style="vertical-align: middle;" />
–2 + 0.959 = <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/014_039.png" style="vertical-align: middle;" /></span></span></div>
<div class="pLevel3">
<span class="Normal"></span></div>
<div class="pLevel3">
<span class="Normal">If the calculated value of the logarithm of
some number is negative (usually, calculators generate values containing a
negative mantissa), then we make the mantissa positive as follows:</span></div>
<div class="pLevel3">
<span class="Normal"></span></div>
<div class="pLevel3">
<span class="Normal">–1.041 = –(1 + 0.041) = –1 –
0.041</span></div>
<div class="pLevel3">
<span class="Normal"><span style="margin-left: 50px;"></span>= (–1
– 1) + (–0.041 + 1)</span></div>
<div class="pLevel3">
<span class="Normal"><span style="margin-left: 50px;"></span>= –2
+ 0.959</span></div>
<div class="pLevel3">
<span class="Normal"><span style="margin-left: 49px;"></span>=
</span><span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/014_039.png" style="vertical-align: middle;" /></span></div>
<div class="pLevel3">
<span class="Normal"></span></div>
<div class="divTextBox divColorExampleHeading divLevel3">
<div id="014_solvedex_7">
<table border="0" cellpadding="0" cellspacing="0" id="014_solvedex_7_bookmark" style="width: 100%;">
<tbody>
<tr>
<td style="background-color: transparent; width: 100%;"><span class="ExampleHeading"><a href="https://www.blogger.com/null" id="ex_7">Example
7:</a></span></td></tr>
</tbody></table>
<table border="0" cellpadding="0" cellspacing="0">
<tbody>
<tr>
<td style="background-color: transparent; width: 100%;"><span class="Normal">If the logarithm of a number is <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/minus.png" style="vertical-align: middle;" />3.153,
what are Characteristic and Mantissa?</span><br />
<span class="Normal"><b>[FMS 2009]</b></span><br />
<span class="Normal"></span><br />
<span class="Normal">(1) Characteristic = <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/minus.png" style="vertical-align: middle;" />4,
mantissa = 0.847</span><br />
<span class="Normal">(2) Characteristic = <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/minus.png" style="vertical-align: middle;" />3,
mantissa = <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/minus.png" style="vertical-align: middle;" />0.153</span><br />
<span class="Normal">(3) Characteristic = 4, mantissa = <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/minus.png" style="vertical-align: middle;" />0.847</span><br />
<span class="Normal">(4) Characteristic = 3, mantissa = <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/minus.png" style="vertical-align: middle;" />0.153</span><br />
<span class="Normal"></span><br />
<span class="SolutionHeading">Solution:</span><br />
<span class="Normal">The characteristic can either be positive or
negative; however, the mantissa always remains positive. In the given
answer options, only option 1 has a positive mantissa.</span><br />
<span class="Normal">Here,</span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/minus.png" style="vertical-align: middle;" />3.153
= <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/minus.png" style="vertical-align: middle;" />(3
+ 0.153) = <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/minus.png" style="vertical-align: middle;" />3
<img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/minus.png" style="vertical-align: middle;" />
0.153</span><br />
<span class="Normal"><span style="margin-left: 52px;"></span>= (<img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/minus.png" style="vertical-align: middle;" />3
<img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/minus.png" style="vertical-align: middle;" />
1) + (1 <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/minus.png" style="vertical-align: middle;" />
0.153) = <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/minus.png" style="vertical-align: middle;" />4
+ 0.847</span><br />
<span class="Normal"><span style="margin-left: 52px;"></span><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/014_040.png" style="vertical-align: middle;" /></span>
<br />
<span class="Normal"></span><br />
<span class="Normal">Hence, <b>option
1</b>.</span></td></tr>
</tbody></table>
</div>
</div>
<div class="pLevel2">
<span class="Normal"></span></div>
<div class="divTextBox divColorExampleHeading divLevel3">
<div id="014_solvedex_8">
<table border="0" cellpadding="0" cellspacing="0" id="014_solvedex_8_bookmark" style="width: 100%;">
<tbody>
<tr>
<td style="background-color: transparent; width: 100%;"><span class="ExampleHeading"><a href="https://www.blogger.com/null" id="ex_8">Example
8:</a></span></td></tr>
</tbody></table>
<table border="0" cellpadding="0" cellspacing="0">
<tbody>
<tr>
<td style="background-color: transparent; width: 100%;"><span class="Normal">Find the number of digits in 1625<sup>21</sup>,
given that log 2 = 0.301 and log 1.3 = 0.114</span><br />
<span class="Normal"></span><br />
<span class="SolutionHeading">Solution:</span><br />
<span class="Normal">log (1625<sup>21</sup>)</span><br />
<span class="Normal">= 21 <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/multiplication.png" style="margin-bottom: 0px; padding-bottom: 0px;" />
log 1625</span><br />
<span class="Normal"></span><br />
<span class="Normal">= 21 <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/multiplication.png" style="margin-bottom: 0px; padding-bottom: 0px;" />
log (125 <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/multiplication.png" style="margin-bottom: 0px; padding-bottom: 0px;" />
13)</span><br />
<span class="Normal"></span><br />
<span class="Normal">= 21 <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/multiplication.png" style="margin-bottom: 0px; padding-bottom: 0px;" />
(log 125 + log 13)</span><br />
<span class="Normal"></span><br />
<span class="Normal">= 21 <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/multiplication.png" style="margin-bottom: 0px; padding-bottom: 0px;" />
(log 5<sup>3</sup> + log 13)</span><br />
<span class="Normal"></span><br />
<span class="Normal">= 21 <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/multiplication.png" style="margin-bottom: 0px; padding-bottom: 0px;" />
(3 log 5 + log 13)</span><br />
<span class="Normal"></span><br />
<span class="Normal">= 21 <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/multiplication.png" style="margin-bottom: 0px; padding-bottom: 0px;" />
[3 log (10/2) + log (10 <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/multiplication.png" style="margin-bottom: 0px; padding-bottom: 0px;" />
1.3)]</span><br />
<span class="Normal"></span><br />
<span class="Normal">= 21 <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/multiplication.png" style="margin-bottom: 0px; padding-bottom: 0px;" />
[3 (log 10 – log 2) + log 10 + log 1.3]</span><br />
<span class="Normal"></span><br />
<span class="Normal">= 21 <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/multiplication.png" style="margin-bottom: 0px; padding-bottom: 0px;" />
[3 (1 – log 2) + 1 + log 1.3]</span><br />
<span class="Normal"></span><br />
<span class="Normal">= 21 <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/multiplication.png" style="margin-bottom: 0px; padding-bottom: 0px;" />
[(3 (1 – 0.301) + 1 + 0.114]</span><br />
<span class="Normal"></span><br />
<span class="Normal">= 21 <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/multiplication.png" style="margin-bottom: 0px; padding-bottom: 0px;" />
3.211</span><br />
<span class="Normal"></span><br />
<span class="Normal">= 67.431</span><br />
<span class="Normal"></span><br />
<span class="Normal">Therefore, we can say that 10<sup>67</sup> <
1625<sup>21</sup> < 10<sup>68</sup></span><br />
<span class="Normal"></span><br />
<span class="Normal">Hence, the number of digits in the number will be
68.</span></td></tr>
</tbody></table>
</div>
</div>
<div class="pLevel2">
<span class="Normal"></span></div>
<div id="014_VI">
<table border="0" cellpadding="0" cellspacing="0" class="tableh1" id="014_VI_bookmark">
<tbody>
<tr>
<td style="margin: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; width: 100%;"><ol class="oLevel1 Heading1">
<li value="6"><a href="https://www.blogger.com/null" id="VI">SOLVED EXAMPLES</a></li>
</ol>
</td></tr>
</tbody></table>
<div class="pLevel2">
<span class="Normal"></span></div>
<div class="divTextBox divColorExampleHeading divLevel3">
<div id="014_solvedex_9">
<table border="0" cellpadding="0" cellspacing="0" id="014_solvedex_9_bookmark" style="width: 100%;">
<tbody>
<tr>
<td style="background-color: transparent; width: 100%;"><span class="ExampleHeading"><a href="https://www.blogger.com/null" id="ex_9">Example
9:</a></span></td></tr>
</tbody></table>
<table border="0" cellpadding="0" cellspacing="0">
<tbody>
<tr>
<td style="background-color: transparent; width: 100%;"><span class="Normal">Simplify log<sub>3</sub> 9 + log<sub>3</sub>
3</span><br />
<span class="Normal"></span><br />
<span class="SolutionHeading">Solution:</span><br />
<span class="Normal">log<sub>3</sub> 9 + log<sub>3</sub> 3</span><br />
<span class="Normal">= log<sub>3</sub> 27<span style="margin-left: 25px;"></span><span style="margin-left: 25px;"></span>…
using law 4</span><br />
<span class="Normal">= log<sub>3</sub> 3<sup>3</sup></span><br />
<span class="Normal">=</span></td></tr>
</tbody></table>
</div>
</div>
</div>
3log<sub>3</sub> 3<span style="margin-left: 25px;"></span><span style="margin-left: 25px;"></span>…
using law 6
<br />
<span class="Normal">= 3<span style="margin-left: 64px;"></span><span style="margin-left: 25px;"></span>… using law
2</span><br />
<div class="pLevel2">
<span class="Normal"><span style="margin-left: 25px;"></span><span style="margin-left: 25px;"></span></span></div>
<div class="divTextBox divColorExampleHeading divLevel3">
<div id="014_solvedex_10">
<table border="0" cellpadding="0" cellspacing="0" id="014_solvedex_10_bookmark" style="width: 100%;">
<tbody>
<tr>
<td style="background-color: transparent; width: 100%;"><span class="ExampleHeading"><a href="https://www.blogger.com/null" id="ex_10">Example
10:</a></span></td></tr>
</tbody></table>
<table border="0" cellpadding="0" cellspacing="0">
<tbody>
<tr>
<td style="background-color: transparent; width: 100%;"><span class="Normal">Find the value of (log 3000 + log 400 – 3 log 25),
if log 3 = 0.4771, log 2 = 0.301 and log 5 = 0.699.</span><br />
<span class="Normal"></span><br />
<span class="SolutionHeading">Solution:</span><br />
<span class="Normal">The base of the logarithms is 10.</span><br />
<span class="Normal">log 3000 + log 400 – 3 log 25</span><br />
<span class="Normal"></span><br />
<span class="Normal">= log (3 <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/multiplication.png" style="margin-bottom: 0px; padding-bottom: 0px;" />
10<sup>3</sup>) + log (2<sup>2</sup> <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/multiplication.png" style="margin-bottom: 0px; padding-bottom: 0px;" />
10<sup>2</sup>) – 3 log 5<sup>2</sup></span><br />
<span class="Normal"></span><br />
<span class="Normal">=</span></td></tr>
</tbody></table>
</div>
</div>
log 3 + log 10<sup>3</sup> + log
2<sup>2</sup> + log 10<sup>2</sup> – 6 log 5
<br />
<span class="Normal"></span><br />
<span class="Normal">= log 3 + 3 log 10 + 2 log 2 + 2 log 10 – 6 log
5</span><br />
<span class="Normal"></span><br />
<span class="Normal">= log 3 + 3 + 2 log 2 + 2 – 6 log 5</span><br />
<span class="Normal"></span><br />
<span class="Normal">= log 3 + 2 log 2 – 6 log 5 + 5</span><br />
<span class="Normal"></span><br />
<span class="Normal">= 0.4771 + 2 <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/multiplication.png" style="margin-bottom: 0px; padding-bottom: 0px;" />
0.301 – 6 <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/multiplication.png" style="margin-bottom: 0px; padding-bottom: 0px;" />
0.699 + 5</span><br />
<span class="Normal"></span><br />
<span class="Normal">= 1.8851</span><br />
<div class="pLevel2">
<span class="Normal"></span></div>
<div class="divTextBox divColorExampleHeading divLevel3">
<div id="014_solvedex_11">
<table border="0" cellpadding="0" cellspacing="0" id="014_solvedex_11_bookmark" style="width: 100%;">
<tbody>
<tr>
<td style="background-color: transparent; width: 100%;"><span class="ExampleHeading"><a href="https://www.blogger.com/null" id="ex_11">Example
11:</a></span></td></tr>
</tbody></table>
<table border="0" cellpadding="0" cellspacing="0">
<tbody>
<tr>
<td style="background-color: transparent; width: 100%;"><span class="Normal">Solve for <i>x</i>: log [(<i>x</i> – 1)(<i>x</i> +
1)] = log 8</span><br />
<span class="Normal"></span><br />
<span class="SolutionHeading">Solution:</span><br />
<span class="Normal">log [(<i>x</i> – 1)(<i>x</i> + 1)] = log
8</span><br />
<span class="Normal"></span><br />
<span class="Normal">(<i>x</i> + 1)(<i>x </i>– 1) = 8</span><br />
<span class="Normal"></span><br />
<span class="Normal"><i>x</i><sup>2</sup> – 1 = 8</span><br />
<span class="Normal"></span><br />
<span class="Normal"><i>x</i><sup>2</sup> = 9</span><br />
<span class="Normal"></span><br />
<span class="Normal"><i>x</i> = +3 or <i>x</i> =
–3</span></td></tr>
</tbody></table>
</div>
</div>
<div class="pLevel2">
<span class="Normal"></span></div>
<div class="divTextBox divColorExampleHeading divLevel3">
<div id="014_solvedex_12">
<table border="0" cellpadding="0" cellspacing="0" id="014_solvedex_12_bookmark" style="width: 100%;">
<tbody>
<tr>
<td style="background-color: transparent; width: 100%;"><span class="ExampleHeading"><a href="https://www.blogger.com/null" id="ex_12">Example
12:</a></span></td></tr>
</tbody></table>
<table border="0" cellpadding="0" cellspacing="0">
<tbody>
<tr>
<td style="background-color: transparent; width: 100%;"><span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/014_041.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/014_042.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"></span><br />
<span class="SolutionHeading">Solution:</span><br />
<span class="Normal"></span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/014_043.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"></span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/014_044.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"></span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/014_045.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"></span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/014_046.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"></span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/014_047.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"></span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/014_048.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"></span><br />
<span class="Normal">= 3</span></td></tr>
</tbody></table>
</div>
</div>
<div class="pLevel2">
<span class="Normal"></span></div>
<div class="divTextBox divColorExampleHeading divLevel3">
<div id="014_solvedex_13">
<table border="0" cellpadding="0" cellspacing="0" id="014_solvedex_13_bookmark" style="width: 100%;">
<tbody>
<tr>
<td style="background-color: transparent; width: 100%;"><span class="ExampleHeading"><a href="https://www.blogger.com/null" id="ex_13">Example
13:</a></span></td></tr>
</tbody></table>
<table border="0" cellpadding="0" cellspacing="0">
<tbody>
<tr>
<td style="background-color: transparent; width: 100%;"><span class="Normal">If <i>a</i>, <i>b</i>, <i>c</i> are in G.P. and <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/014_049.png" style="vertical-align: middle;" />
= <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/014_050.png" style="vertical-align: middle;" />
= <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/014_051.png" style="vertical-align: middle;" />
then:</span>
<br />
<span class="Normal"><b>[FMS 2009]</b></span><br />
<span class="Normal"></span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/014_052.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"></span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/014_053.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"></span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/014_054.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"></span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/014_055.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"></span><br />
<span class="SolutionHeading">Solution:</span><br />
<span class="Normal">In this problem, the answer option does not contain
<i>a</i>, <i>b</i> or <i>c</i>. Hence we will need to eliminate <i>a</i>,
<i>b</i> and <i>c</i> and get the answers in terms of <i>x</i>,<i> y</i>
and<i> z</i>. This is done easily using logarithms.</span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/symb23.png" style="margin-bottom: 0px; padding-bottom: 1px;" />
<i>a</i>, <i>b</i> and <i>c</i> are in G.P</span><br />
<span class="Normal"></span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/014_056.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"></span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/therefore.png" style="margin-bottom: 0px; padding-bottom: 1px;" />
<i>b</i>² = <i>a</i> <img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/multiplication.png" style="margin-bottom: 0px; padding-bottom: 0px;" />
<i>c</i></span><br />
<span class="Normal">Taking log on both sides, we get,</span><br />
<span class="Normal">2 log <i>b</i> = log <i>a</i> + log <i>c</i> …
(i)</span><br />
<span class="Normal"></span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/014_057.png" style="vertical-align: middle;" /></span><br />
<span class="Normal">Taking log throughout, we get,</span><br />
<span class="Normal"><i>x </i>log <i>a</i> = <i>y</i> log <i>b </i>= <i>z
</i>log <i>c</i></span><br />
<span class="Normal"></span><br />
<span class="Normal">To eliminate <i>a</i>, <i>b</i> and <i>c</i>, we
should express any 2 of <i>a, b</i> or <i>c</i> in terms of the
3<sup>rd</sup> variable. Here, we will express <i>a</i> and <i>c</i> in
terms of <i>b</i>.</span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/014_058.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"></span><br />
<span class="Normal">Substituting the same in equation (i), we
get,</span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/014_059.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/014_060.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"></span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/014_061.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"></span><br />
<span class="Normal"><img alt="" src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c81f3cf2-db89-41a0-8690-e30334c7a0bd/014_images/014_062.png" style="vertical-align: middle;" /></span><br />
<span class="Normal"></span><br />
<span class="Normal">Hence, <b>option 1</b>.</span><br />
<div>
</div>
</td></tr>
</tbody></table>
</div>
</div>
</div>
guruhttp://www.blogger.com/profile/01591375209227117455noreply@blogger.comtag:blogger.com,1999:blog-7850540640045081515.post-88568267475123791792015-03-03T03:49:00.003-08:002019-09-14T18:55:04.440-07:00linear equtions<div dir="ltr" style="text-align: left;" trbidi="on">
<br /></div>
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<P class=oTOCTitle><SPAN class=oTOCTitleFont><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016.htm#Title">Contents</A></SPAN></P>
<OL class=oTOCLevel1>
<LI class=oTocLiClass><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016.htm#I">INTRODUCTION</A>
</LI>
<LI class=oTocLiClass><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016.htm#II">TYPES
OF EQUATIONS</A>
<OL class=oTOCLevel2>
<LI><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016.htm#II_A">LINEAR
EQUATIONS IN ONE VARIABLE</A></LI>
<LI><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016.htm#II_B">GRAPHICAL
FORM OF LINEAR EQUATIONS</A></LI>
<LI><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016.htm#II_C">LINEAR
EQUATIONS IN TWO VARIABLES</A></LI>
<LI><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016.htm#II_D">SIMULTANEOUS
EQUATIONS</A></LI>
<LI><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016.htm#II_E">LINEAR
EQUATIONS IN THREE OR MORE VARIABLES</A></LI>
<LI><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016.htm#II_F">OTHER
EXAMPLES</A></LI></OL></LI></OL>
<P id=Title class=pTitle><SPAN class=Title>Linear Equations</SPAN></P>
<HR>
<P class=pLevel2><SPAN class=Normal></SPAN>
<DIV id=016_I>
<TABLE id=016_I_bookmark class=tableh1 border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD
style="PADDING-BOTTOM: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 100%; PADDING-RIGHT: 0px; PADDING-TOP: 0px">
<OL class="oLevel1 Heading1">
<LI value=1><A id=I>INTRODUCTION</A></LI></OL></TD></TR></TBODY></TABLE>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>An <B>equation</B> is a mathematical
statement which implies that two quantities separated by the symbol ‘=’ are
equal.</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>For example,</SPAN></P>
<P class=pLevel2><SPAN class=Normal>3<I>x</I> + 6 = 0,</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal><I>x</I><SUP>2 </SUP>+ 5<I>x</I> = <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/minus.png">
6</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>The terms on the left hand side and the
right hand side of the ‘=’ sign are called the <B>LHS</B> and <B>RHS</B>
respectively.</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>Equations contain variables, coefficients
and constants. A <B>variable</B> is the unknown quantity in the equation.
Variables are generally denoted by letters like <I>x, y, z, a, b, c</I> etc. An
equation can contain one or more variables. <B>Coefficient</B> is the number
which multiplies the variable and a <B>constant</B> is an independent numerical
value in the equation. For instance, in the equation 3<I>x</I> + 6 = 0, <I>x</I>
is the variable, 3 is the coefficient of the variable and 6 is the
constant.</SPAN></P></DIV>
<DIV id=016_II>
<TABLE id=016_II_bookmark class=tableh1 border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD
style="PADDING-BOTTOM: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 100%; PADDING-RIGHT: 0px; PADDING-TOP: 0px">
<OL class="oLevel1 Heading1">
<LI value=2><A id=II>TYPES OF EQUATIONS</A></LI></OL></TD></TR></TBODY></TABLE>
<P class=pLevel2><SPAN class=ListParagraph></SPAN></P>
<P class=pLevel2><SPAN class=Normal>Depending on the highest power of the
variable(s), equations are classified as linear, quadratic, cubic etc. The
highest power of the variable(s) in a <B>linear equation</B> is 1 and that in a
<B>quadratic equation</B> is 2, for a <B>cubic</B> equation is 3 and so on. In
this chapter, we will study linear equations in detail.</SPAN></P>
<DIV id=016_II_A>
<TABLE id=016_II_A_bookmark class=tableh1 border=0 cellSpacing=0
cellPadding=0><TBODY>
<TR>
<TD
style="PADDING-BOTTOM: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 100%; PADDING-RIGHT: 0px; PADDING-TOP: 0px">
<OL class="oLevel2 Heading2">
<LI value=1><A id=II_A>LINEAR EQUATIONS IN ONE
VARIABLE</A></LI></OL></TD></TR></TBODY></TABLE>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>To solve an equation is to find a numerical
value that the variable can take so that the equation holds true. Such a value
of the variable is called the <B>solution</B>.</SPAN></P>
<P class=pLevel3><SPAN class=Normal>For instance, consider <I>x</I> + 4 = 8.
This equation can hold true only if <I>x</I> = 4. Thus, <I>x </I>= 4 is a
solution of the equation.</SPAN></P>
<P class=pLevel3><SPAN class=Normal>Similarly, if 3<I>x</I> + 6 = 0, then
<I>x</I> = <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/minus.png">2
is a solution of the equation.</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorNote divLevel3">
<P><SPAN class=Note>REMEMBER</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<UL type=disc>
<LI value=1>The solution of a given equation does not change if we add or
subtract the same number from both sides of the equation, nor does it change
if we multiply both the sides with the same number or divide both the sides
with the same non-zero number.</LI></UL>
<UL type=disc>
<LI value=2>To solve a linear equation in one variable the following steps
should be followed:</LI></UL>
<P><SPAN class=Normal></SPAN></P>
<UL>
<P><SPAN class=Normal><B>Step 1:</B> Simplify the LHS and the RHS by removing
brackets.</SPAN></P></UL>
<P><SPAN class=Normal></SPAN></P>
<UL>
<P><SPAN class=Normal><B>Step 2:</B> Shift all the terms containing the
variable to the LHS and the constant terms to the RHS with appropriate change
of sign and then simplify.</SPAN></P></UL>
<P><SPAN class=Normal></SPAN></P>
<UL>
<P><SPAN class=Normal>While shifting terms to the opposite side, a positive
sign changes to negative (and vice versa) and a multiplication changes to a
division.</SPAN></P></UL></DIV>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=016_solvedex_1>
<TABLE style="WIDTH: 100%" id=016_solvedex_1_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_1>Example
1:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>Find the value of <I>x</I>, if 7<I>x</I> + 5(2 <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/minus.png">
<I>x</I>) + 10 = 4<I>x</I> – 6</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><B>Step 1:</B></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/therefore.png">
7<I>x</I> + 10 <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/minus.png">
5<I>x</I> + 10 = 4<I>x</I> – 6</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/therefore.png">
2<I>x</I> + 20 = 4<I>x</I> – 6</SPAN></P>
<P><SPAN class=Normal><B>Step 2:</B></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/therefore.png">
2<I>x</I> – 4<I>x</I> = –20 – 6</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/therefore.png">
–2<I>x</I> = –26</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/therefore.png"><I>
x</I> = 13</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV><!--<p class="pLevel2"><span class="Normal"></span></p>
<div class="divTextBox divColorNormal divLevel4">
<p><span class="Normal"></span></p>
<p><span class="Normal"></span></p>
</div>-->
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=016_solvedex_2>
<TABLE style="WIDTH: 100%" id=Table1 border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=A1>Example
2:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%"><!--<span class="ExampleHeading"><a id="ex_2">Example 2:</a></span></p>-->
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/016_001.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal><B>Step 1:</B></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/016_032.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/therefore.png">7<I>y</I>
– 4 = 5 (<I>y</I> + 2)</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/therefore.png">
7<I>y</I> – 4 = 5<I>y</I> + 10</SPAN></P>
<P><SPAN class=Normal><B>Step 2:</B></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/therefore.png">
7<I>y</I> – 5<I>y</I> = 4 + 10</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/therefore.png">
2<I>y</I> = 14</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/therefore.png">
<I>y</I> = 7</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel3><SPAN class=Normal>Forming a linear equation from the given
information is as important a step as solving it. Let us look at a few examples.
</SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorNote divLevel3">
<P><SPAN class=Note>REMEMBER</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<UL type=disc>
<LI value=1>During the exam, some multiple choice questions can be solved by
substituting the variable with the values given in the options or substituting
values in place of variables like in the following example.</LI></UL></DIV>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=016_solvedex_3>
<TABLE style="WIDTH: 100%" id=016_solvedex_3_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_3>Example
3:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/016_002.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><B>[CAT 2002]</B></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/016_003.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/016_004.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/symb23.png"></B>
<I>pqr </I>= 1</SPAN></P>
<P><SPAN class=Normal><B>Case 1: </B></SPAN></P>
<P><SPAN class=Normal>Let <I>p</I> = 2/3, <I>q</I> = 3/2 and <I>r</I> =
1</SPAN></P>
<P><SPAN class=Normal>Substituting the values of <I>p</I>, <I>q</I>,
<I>r</I>, we get,</SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/016_005.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/016_006.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><B>Case 2: </B></SPAN></P>
<P><SPAN class=Normal>Let <I>p</I> = 1, <I>q</I> = 1 and <I>r</I> =
1</SPAN></P>
<P><SPAN class=Normal>Substituting the values of <I>p</I>, <I>q</I>,
<I>r</I>, we get,</SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/016_005.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/016_007.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Hence, <B>option 3.
</B></SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV><!--<p class="pLevel2"><span class="Normal"></span></p>
<div class="divTextBox divColorNormal divLevel4">
<p><span class="Normal"></span></p>
<p><span class="Normal"></span></p>
</div>-->
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=016_solvedex_4>
<TABLE style="WIDTH: 100%" id=Table2 border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=A2>Example
4:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>The sum of three consecutive numbers is 84. Find the
numbers.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>Let <I>x</I> be the first number. Then the other two
consecutive numbers are <I>x</I> + 1 and <I>x</I> + 2.</SPAN></P>
<P><SPAN class=Normal>Hence we have,</SPAN></P>
<P><SPAN class=Normal><I>x </I>+ (<I>x</I> + 1) + (<I>x</I> + 2) =
84</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/therefore.png">
3<I>x </I>+ 3 = 84</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/therefore.png">
<I>x</I> = 27</SPAN></P>
<P><SPAN class=Normal>The numbers are 27, 28 and
29.</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV></DIV></DIV><!--<p class="pLevel2"><span class="Normal"></span></p>
<div class="divTextBox divColorNormal divLevel4">
<p><span class="Normal"></span></p>
<p><span class="Normal"></span></p>
</div>-->
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=016_solvedex_5>
<TABLE style="WIDTH: 100%" id=Table3 border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=A3>Example
5:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>After purchasing four mangoes, a man commented:
"Well, if the price of a mango was Rs. 4 less, I would have got two more
mangoes for the same amount of money I have now paid". What is the price
of one mango?</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><B>[JMET 2009] </B></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>(1) Rs. 8<SPAN
style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 35px"></SPAN>(2) Rs. 10</SPAN></P>
<P><SPAN class=Normal>(3) Rs. 15<SPAN
style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN>(4) Rs. 12</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>Let the price of mango be <I>x</I>.</SPAN></P>
<P><SPAN class=Normal>If the price of mango was Rs. 4 less, I would have
got two more mangoes for the same amount of money</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/therefore.png">
4<I>x</I> = 6(<I>x</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/minus.png">
4)</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/therefore.png">
4<I>x</I> = 6<I>x</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/minus.png">
24</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/therefore.png">
2<I>x</I> = 24</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/therefore.png">
<I>x</I> = 12</SPAN></P>
<P><SPAN class=Normal>Hence, <B>option 4.
</B></SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV><!--<p class="pLevel2"><span class="Normal"></span></p>
<div class="divTextBox divColorNormal divLevel4">
<p><span class="Normal"></span></p>
<p><span class="Normal"></span></p>
</div>-->
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=016_solvedex_6>
<TABLE style="WIDTH: 100%" id=Table4 border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=A4>Example
6:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>Tickets for a concert were sold at Rs. 5, Rs. 3 and
Re. 1 each. Thirty more tickets were sold at Rs. 5 than at Rs. 3, and
twice as many at Re. 1 as at Rs. 3. If total receipts from the sale of
tickets were Rs. 950, then how many tickets of each kind were
sold?</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>Let the number of Rs. 3 tickets =
<I>x</I></SPAN></P>
<P><SPAN class=Normal>Number of Rs. 5 tickets = 30 + <I>x</I></SPAN></P>
<P><SPAN class=Normal>Number of Re. 1 tickets = 2<I>x</I></SPAN></P>
<P><SPAN class=Normal>Total amount = Rs. 950</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/therefore.png">
3<I>x </I>+ 5(30 + <I>x</I>) + 2<I>x</I> = 950</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/therefore.png">
10<I>x</I> = 950 – 150</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/therefore.png">
<I>x</I> = 80</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/therefore.png">
The number of tickets sold for Rs. 3 is 80, the number of tickets sold for
Rs. 5 is 110 and the number of tickets sold for Re. 1 is
160.</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV><!--<p class="pLevel2"><span class="Normal"></span></p>
<div class="divTextBox divColorNormal divLevel4">
<p><span class="Normal"></span></p>
<p><span class="Normal"></span></p>
</div>-->
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=016_solvedex_7>
<TABLE style="WIDTH: 100%" id=Table5 border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=A5>Example
7:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>A string of length 40 metres is divided into three
parts of different lengths. The first part is three times the second part,
and the last part is 23 metres smaller than the first part. Find the
length of the largest part.</SPAN></P>
<P><SPAN class=Normal><B>[CAT 2002] </B></SPAN></P>
<P><SPAN class=Normal>(1) 27<SPAN style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN>(2) 4</SPAN></P>
<P><SPAN class=Normal>(3) 5<SPAN style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 35px"></SPAN>(4) 9</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>Let <I>l</I>, <I>m</I> and <I>s</I> be the longest,
medium and the shortest lengths of the strings.</SPAN></P>
<P><SPAN class=Normal><I>l</I> = 3<I>m</I> and <I>s</I> = <I>l</I> –
23</SPAN></P>
<P><SPAN class=Normal><I>l</I> + <I>s</I> + <I>m</I> = 40</SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/016_008.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/016_009.png"></SPAN></P>
<P><SPAN class=Normal><I>l</I> = 27</SPAN></P>
<P><SPAN class=Normal>Hence, <B>option 1</B>.
</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV><!--<p class="pLevel2"><span class="Normal"></span></p>
<div class="divTextBox divColorNormal divLevel4">
<p><span class="Normal"></span></p>
<p><span class="Normal"></span></p>
</div>-->
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=016_solvedex_8>
<TABLE style="WIDTH: 100%" id=Table6 border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=A6>Example
8:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>A man received a cheque. The amount in Rs. has been
transposed for paise and vice versa. After spending Rs. 5 and 42 paise, he
discovered he now had exactly 6 times the value of the correct cheque
amount. What amount he should have received?</SPAN></P>
<P><SPAN class=Normal><B>[CAT 2002]</B></SPAN></P>
<P><SPAN class=Normal>(1) Rs. 5.30<SPAN
style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 35px"></SPAN>(2) Rs. 6.44</SPAN></P>
<P><SPAN class=Normal>(3) Rs. 60.44<SPAN
style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN>(4) Rs.6.44</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>Let the man has received a cheque of <I>x</I> rupees
and <I>y</I> paise.</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/therefore.png">
The amount on cheque = (100<I>x</I> + <I>y</I>)<SPAN
style="MARGIN-LEFT: 265px"></SPAN>...(i)</SPAN></P>
<P><SPAN class=Normal>The amount actually received by him = 100<I>y</I> +
<I>x</I></SPAN></P>
<P><SPAN class=Normal>After spending Rs. 5 and 42 paise, the remaining
amount = (100<I>y</I> + <I>x</I> – 542 …(ii)</SPAN></P>
<P><SPAN class=Normal></SPAN>But, (100<I>y</I> + <I>x</I> – 542) = 6 <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/multiplication.png">
(100<I>x</I> + <I>y</I>)<SPAN
style="MARGIN-LEFT: 263px"></SPAN>...(iii)</P>
<P><SPAN class=normal>Substituting the values from the given options,
<I>x</I> = 6 and <I>y</I> = 44</SPAN></P>
<P><SPAN class=Normal>Hence, <B>option
2.</B></SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<DIV id=016_II_B>
<TABLE id=016_II_B_bookmark class=tableh1 border=0 cellSpacing=0
cellPadding=0><TBODY>
<TR>
<TD
style="PADDING-BOTTOM: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 100%; PADDING-RIGHT: 0px; PADDING-TOP: 0px">
<OL class="oLevel2 Heading2">
<LI value=2><A id=II_B>GRAPHICAL FORM OF LINEAR
EQUATIONS</A></LI></OL></TD></TR></TBODY></TABLE>
<P class=pLevel3><SPAN class=ListParagraph></SPAN></P>
<P class=pLevel3><SPAN class=Normal>Now let us try to understand a linear
equation graphically. A linear equation of the form <I>ax + by</I> = <I>c</I>,
where <I>x</I> and <I>y </I>are variables, <I>a</I> and <I>b</I> are
coefficients and <I>c</I> is a constant, represents a line on the <I>X–Y</I>
plane. Hence, any equation of this form can be plotted on the <I>X–Y</I> plane.
A linear equation when plotted on a graph will give a straight line. Graphical
representation of linear equations would enable us to understand linear
equations in a better way.</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>A linear equation with only one variable
gives a line which is parallel to the <I>x</I>-axis or the
<I>y</I>-axis.</SPAN></P>
<P class=pLevel3><SPAN class=Normal>For example, <I>x</I> – 2 = 0 can be
represented as:</SPAN></P>
<P class=pImage><SPAN><IMG alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/016_012.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>The solution of the linear equation is the
point where the line intersects the axis. In this case the solution is <I>x</I>
= 2.</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P></DIV>
<DIV id=016_II_C>
<TABLE id=016_II_C_bookmark class=tableh1 border=0 cellSpacing=0
cellPadding=0><TBODY>
<TR>
<TD
style="PADDING-BOTTOM: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 100%; PADDING-RIGHT: 0px; PADDING-TOP: 0px">
<OL class="oLevel2 Heading2">
<LI value=3><A id=II_C>LINEAR EQUATIONS IN TWO
VARIABLES</A></LI></OL></TD></TR></TBODY></TABLE>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>An equation of the form <I>ax</I> +
<I>by</I> = <I>c</I>, where the highest power of the variables <I>x</I> and
<I>y</I> is unity is called a linear equation in two variables. Here <I>a</I> ≠
0, <I>b</I> ≠ 0 and <I>a</I>,<I> b</I> and <I>c</I> are real numbers. The values
of <I>x</I> and <I>y</I> for which the equation holds are called the solution of
the equation.</SPAN></P>
<P class=pLevel3><SPAN class=Normal>A line representing a linear equation with
two variables cannot be parallel to the co-ordinate axes.</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>For example, let us represent the
equation</SPAN></P>
<P class=pLevel3><SPAN class=Normal>3<I>x </I>+ 4<I>y </I>= 12 on the <I>X–Y</I>
plane.</SPAN></P>
<P class=pLevel3><SPAN class=Normal>We have,</SPAN></P>
<P class=pLevel3><SPAN class=Normal>3<I>x</I> + 4<I>y</I> = 12<SPAN
style="MARGIN-LEFT: 25px"></SPAN> …(i)</SPAN></P>
<P class=pLevel3><SPAN class=Normal>Put <I>x</I> = 0, then equation (i)
becomes</SPAN></P>
<P class=pLevel3><SPAN class=Normal>0 + 4<I>y</I> = 12</SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/therefore.png">
4<I>y</I> = 12</SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/therefore.png">
<I>y</I> = 3</SPAN></P>
<P class=pLevel3><SPAN class=Normal>Thus (0, 3) are the co-ordinates of a point
on the line.</SPAN></P>
<P class=pLevel3><SPAN class=Normal>Put <I>y</I> = 0, then equation (i)
becomes</SPAN></P>
<P class=pLevel3><SPAN class=Normal>3<I>x</I> + 0 = 12</SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/therefore.png">
3<I>x</I> = 12</SPAN></P>
<P class=pLevel3><SPAN class=Normal><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/therefore.png">
<I>x </I>= 4</SPAN></P>
<P class=pLevel3><SPAN class=Normal>Hence (4, 0) are the co-ordinates of another
point on the same line.</SPAN></P>
<P class=pLevel3><SPAN class=Normal>Hence we can say that (0, 3) and (4, 0) are
the <I>Y</I> - intercept and <I>X</I> - intercept of the line respectively. We
join these two points on the <I>X–Y</I> plane as shown and the line thus formed
is the graphical representation of the linear equation 3<I>x</I> + 4<I>y</I> =
12.</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pImage><SPAN><IMG alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/016_011.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>All the points on the line shown satisfy
this equation. A line extends in both directions infinitely. Hence such an
equation has infinite solutions.</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorNote divLevel3">
<P><SPAN class=Note>REMEMBER</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<UL type=disc>
<LI value=1>A linear equation with two or more variables does not have a
unique solution. It has an infinite number of solutions.</LI></UL></DIV>
<P class=pLevel4><SPAN class=ListParagraph></SPAN></P>
<P class=pLevel3><SPAN class=Normal><B>EQUATIONS IN TWO VARIABLES WITH POSITIVE
INTEGER SOLUTIONS</B></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>Any one linear equation in two variables can
have an infinite number of solutions, but if the solutions are known to be
positive integers, the number of solutions can be finite and can be determined
as shown in the following example.</SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=016_solvedex_9>
<TABLE style="WIDTH: 100%" id=016_solvedex_9_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_9>Example
9:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>Find how many positive integers <I>x</I> and
<I>y</I> satisfy the equation 5<I>x </I>+ 3<I>y</I> = 59.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>5<I>x</I> + 3<I>y</I> = 59</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/therefore.png">
<I>x</I> = (59 – 3<I>y</I>)/5</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/therefore.png">
<I>x</I> = (55 + 4 – 3<I>y</I>)/5</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/therefore.png">
<I>x</I> = 11 + (4 – 3<I>y</I>)/5</SPAN></P>
<P><SPAN class=Normal>As <I>x</I> and <I>y</I> are positive integers,
<I>y</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/greater_equal.png">
1 and (4 – 3<I>y</I>)/5 is an integer.</SPAN></P>
<P><SPAN class=Normal>Thus the smallest possible value of <I>y</I> is 3.
Corresponding value of <I>x</I> is 10.</SPAN></P>
<P><SPAN class=Normal>The other possible values of <I>y</I> are 8, 13 and
18, and the corresponding values of <I>x</I> are 7, 4 and 1. <I>x</I> is
negative for higher possible values of <I>y</I>. Thus there are 4 possible
solutions.</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV><!--<p class="pLevel2"><span class="Normal"></span></p>
<div class="divTextBox divColorNormal divLevel4">
<p><span class="Normal"></span></p>
<p><span class="Normal"></span></p>
</div>-->
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=016_solvedex_10>
<TABLE style="WIDTH: 100%" id=Table7 border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=A7>Example
10:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>A test has 50 questions. A student scores 1 mark for
a correct answer, <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/minus.png">1/3
for a wrong answer, and <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/minus.png">1/6
for not attempting a question. If the net score of a student is 32, the
number of questions answered wrongly by that student cannot be less
than</SPAN></P>
<P><SPAN class=Normal><B>[CAT 2003 Leaked Test]</B></SPAN></P>
<P><SPAN class=Normal>(1) 6<SPAN style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN>(2) 12</SPAN></P>
<P><SPAN class=Normal>(3) 3<SPAN style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN>(4) 9</SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>Let <I>R</I>, <I>W</I> and <I>N</I> be the number of
questions with right answers, questions with wrong answers and not
attempted questions respectively.</SPAN></P>
<P><SPAN class=Normal>From the conditions given in the question, we
have,</SPAN></P>
<P><SPAN class=Normal><I>R</I> + <I>W</I> + <I>N</I> = 50<SPAN
style="MARGIN-LEFT: 57px"></SPAN>… (i)</SPAN></P>
<P><SPAN class=Normal><I>R</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/minus.png">
<I>W</I>/3 <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/minus.png">
<I>N</I>/6 = 32 <SPAN style="MARGIN-LEFT: 15px"></SPAN>… (ii)</SPAN></P>
<P><SPAN class=Normal>Solving equations (i) and (ii), we get,</SPAN></P>
<P><SPAN class=Normal>7<I>R</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/minus.png">
242 = <I>W</I></SPAN></P>
<P><SPAN class=Normal></SPAN><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/therefore.png">
<I>W</I> will be minimum for <I>R</I> = 35, i.e. <I>W</I> = 3</P>
<P><SPAN class=Normal>Hence, <B>option
3.</B></SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorNote divLevel3">
<P><SPAN class=Note>REMEMBER</SPAN></P>
<UL type=disc>
<LI value=1>For an equation <I>ax</I> – <I>by</I> = <I>c</I> (<I>a</I>, <I>b
</I>and <I>c </I>are positive integers), having positive integer solutions,
the values of <I>x</I> form an arithmetic progression with common difference
<I>b</I> and the values of<I> y</I> form an arithmetic progression with common
difference <I>a</I>.</LI></UL></DIV>
<P class=pLevel4><SPAN class=Normal></SPAN></P></DIV>
<DIV id=016_II_D>
<TABLE id=016_II_D_bookmark class=tableh1 border=0 cellSpacing=0
cellPadding=0><TBODY>
<TR>
<TD
style="PADDING-BOTTOM: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 100%; PADDING-RIGHT: 0px; PADDING-TOP: 0px">
<OL class="oLevel2 Heading2">
<LI value=4><A id=II_D>SIMULTANEOUS
EQUATIONS</A></LI></OL></TD></TR></TBODY></TABLE>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>Linear equations in two variables, which are
both satisfied by the same unique solution, are called <B>simultaneous
equations</B>. Graphically, these equations are two non parallel lines which
intersect at a particular point (<I>x</I>, <I>y</I>), which is the solution of
both the equations.</SPAN></P>
<P class=pLevel3><SPAN class=Normal>For example, 3<I>x</I> + <I>y</I> = 5 and
5<I>x</I> – <I>y</I> = 3 intersect at <I>x</I> = 1 and <I>y</I> = 2.</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pImage><SPAN><IMG alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/016_010.png"></SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>Thus <I>x</I> = 1 and <I>y</I> = 2 lies on
both the lines and we say that the solution of the equations 3<I>x</I> +
<I>y</I> = 5 and 5<I>x</I> – <I>y</I> = 3 is (1, 2). In other words (1, 2) is a
point that lies on both the lines. Hence the point should satisfy both the
equations.</SPAN></P>
<P class=pLevel3><SPAN class=Normal>Substituting <I>x</I> = 1 and <I>y</I> = 2
in 3<I>x</I> + <I>y</I> = 5 we have,</SPAN></P>
<P class=pLevel3><SPAN class=Normal>LHS = 3<I>x</I> + <I>y</I></SPAN></P>
<P class=pLevel3><SPAN class=Normal><SPAN style="MARGIN-LEFT: 33px"></SPAN>=
3(1) + 2</SPAN></P>
<P class=pLevel3><SPAN class=Normal><SPAN style="MARGIN-LEFT: 33px"></SPAN>=
5</SPAN></P>
<P class=pLevel3><SPAN class=Normal><SPAN style="MARGIN-LEFT: 33px"></SPAN>=
RHS</SPAN></P>
<P class=pLevel3><SPAN class=Normal>Similarly, substituting <I>x</I> = 1 and
<I>y</I> = 2 in the equation 5<I>x</I> – <I>y</I> = 3 we have,</SPAN></P>
<P class=pLevel3><SPAN class=Normal>LHS = 5<I>x</I> – <I>y</I></SPAN></P>
<P class=pLevel3><SPAN class=Normal><SPAN style="MARGIN-LEFT: 33px"></SPAN>=
5(1) – 2</SPAN></P>
<P class=pLevel3><SPAN class=Normal><SPAN style="MARGIN-LEFT: 33px"></SPAN>=
3</SPAN></P>
<P class=pLevel3><SPAN class=Normal><SPAN style="MARGIN-LEFT: 33px"></SPAN>=
RHS</SPAN></P>
<P class=pLevel3><SPAN class=Normal>In the above example, we can see that 5 –
3<I>x</I> > 5<I>x</I> – 3, for<I> x</I> < 1 and 5 – 3<I>x</I> <
5<I>x</I> – 3, for <I>x</I> > 1. Also, we note that the minimum possible
value that the greater value of (5 – 3<I>x</I>) and (5<I>x</I> – 3) can take is
at <I>x</I> = 1 and this value is 2.</SPAN></P>
<P class=pLevel3><SPAN class=Normal>Two or more linear equations in the same
number of variables having a common solution are called a system of simultaneous
equations. The common solutions to these equations can be found out by graphical
method as explained earlier or by the algebraic method discussed
below.</SPAN></P>
<OL class="oLevel3 Heading3 ListStyleOL">
<LI value=1>SOLVING SIMULTANEOUS EQUATIONS</LI></OL>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal>It is impractical to plot equations on a
graph paper to find common solutions. Hence, we use the algebraic method to
obtain the same common solutions of simultaneous equations.</SPAN></P>
<P class=pLevel4><SPAN class=Normal>Two general methods of solving simultaneous
equations are described below:</SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal><B>SUBSTITUTION METHOD</B></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal>Consider the equations 3<I>y</I> <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/minus.png">
<I>x</I> = 1 and 7<I>y </I><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/minus.png">
2<I>x</I> = 4 to be solved simultaneously. Thus we have,</SPAN></P>
<P class=pLevel4><SPAN class=Normal>3<I>y </I>– <I>x </I>= 1<SPAN
style="MARGIN-LEFT: 50px"></SPAN><SPAN
style="MARGIN-LEFT: 33px"></SPAN>…(i)</SPAN></P>
<P class=pLevel4><SPAN class=Normal><I>7y</I> – 2<I>x </I>= 4<SPAN
style="MARGIN-LEFT: 50px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN>…(ii)</SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal>From equation(i),</SPAN></P>
<P class=pLevel4><SPAN class=Normal>3<I>y</I> = 1 + <I>x</I></SPAN></P>
<P class=pLevel4><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/016_013.png"></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal>Substituting this value of <I>y</I> in
equation (ii),</SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/016_014.png"></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/016_015.png"></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/therefore.png">
7 + 7<I>x</I> – 6<I>x</I> = 12</SPAN></P>
<P class=pLevel4><SPAN class=Normal><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/therefore.png">
<I>x</I> = 5</SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal>On substituting <I>x</I> = 5 in equation
(iii), we have</SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/016_016.png"></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/therefore.png">
<I>y</I> = 2</SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/therefore.png">
<I>x</I> = 5 and <I>y</I> = 2 satisfy the two equations and (5, 2) is a solution
of the two equations.</SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal><B>ELIMINATION METHOD</B></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal>Consider the equations 2<I>x</I> + 3<I>y</I>
= 4 and 3<I>x</I> + 4<I>y</I> = 5 to be solved simultaneously. Thus we
have,</SPAN></P>
<P class=pLevel4><SPAN class=Normal>2<I>x</I> + 3<I>y</I> = 4<SPAN
style="MARGIN-LEFT: 25px"></SPAN><SPAN style="MARGIN-LEFT: 37px"></SPAN>
…(i)</SPAN></P>
<P class=pLevel4><SPAN class=Normal>3<I>x</I> + 4<I>y</I> = 5<SPAN
style="MARGIN-LEFT: 25px"></SPAN><SPAN style="MARGIN-LEFT: 35px"></SPAN>
…(ii)</SPAN></P>
<P class=pLevel4><SPAN class=Normal>Multiply each equation by the coefficient of
<I>x</I> (or <I>y</I>) in the other equation.</SPAN></P>
<P class=pLevel4><SPAN class=Normal>Multiplying equation (i) by 3 and equation
(ii) by 2</SPAN></P>
<P class=pLevel4><SPAN class=Normal>6<I>x</I> + 9<I>y</I> = 12<SPAN
style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN>…(iii)</SPAN></P>
<P class=pLevel4><SPAN class=Normal>6<I>x</I> + 8<I>y</I> = 10<SPAN
style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN>…(iv)</SPAN></P>
<P class=pLevel4><SPAN class=Normal>Subtracting equation (iv) from equation
(iii),</SPAN></P>
<P class=pLevel4><SPAN class=Normal><I>y</I> = 2</SPAN></P>
<P class=pLevel4><SPAN class=Normal>Substituting <I>y</I> = 2 in equation
(i)</SPAN></P>
<P class=pLevel4><SPAN class=Normal>2<I>x</I> + 3 <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/multiplication.png">
2 = 4</SPAN></P>
<P class=pLevel4><SPAN class=Normal><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/therefore.png">
<I>x</I> = –2/2 = –1</SPAN></P>
<P class=pLevel4><SPAN class=Normal><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/therefore.png">
<I>x</I> = –1 and <I>y</I> = 2 satisfy both the equations.</SPAN></P>
<P class=pLevel4><SPAN class=Normal><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/therefore.png">
(–1, 2) is a solution of the two equations.</SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=016_solvedex_11>
<TABLE style="WIDTH: 100%" id=016_solvedex_11_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_11>Example
11:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>In a particular jungle which only had deer and human
visitors, there were 70 heads and 188 legs. How many deer and visitors
were there?</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>Let the number of deer be <I>x</I> and the number of
visitors be <I>y</I>.</SPAN></P>
<P><SPAN class=Normal><I>x</I> + <I>y</I> = 70<SPAN
style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 53px"></SPAN>...(i)</SPAN></P>
<P><SPAN class=Normal>Since deer have 4 legs and human visitors have 2
legs we have,</SPAN></P>
<P><SPAN class=Normal>4<I>x</I> + 2<I>y</I> = 188<SPAN
style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN>...(ii)</SPAN></P>
<P><SPAN class=Normal>Multiplying equation (i) by 2 and subtracting the
same from equation (ii), we get</SPAN></P>
<P><SPAN class=Normal><I>x</I> = 24 and <I>y</I> =
46</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV><!--<p class="pLevel2"><span class="Normal"></span></p>
<div class="divTextBox divColorNormal divLevel4">
<p><span class="Normal"></span></p>
<p><span class="Normal"></span></p>
</div>-->
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=016_solvedex_12>
<TABLE style="WIDTH: 100%" id=Table8 border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=A8>Example
12:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>The sum of the digits of a two digit number is 7. If
the digits are reversed, the number so obtained when increased by 3 equals
4 times the original number. Find the original number.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>Let the digit in the ten’s place =
<I>x</I></SPAN></P>
<P><SPAN class=Normal>Let the digit in the unit’s place =
<I>y</I></SPAN></P>
<P><SPAN class=Normal>So the number = 10<I>x</I> + <I>y</I></SPAN></P>
<P><SPAN class=Normal>If the digits are reversed, the new number =
10<I>y</I> + <I>x</I></SPAN></P>
<P><SPAN class=Normal>According to given conditions,</SPAN></P>
<P><SPAN class=Normal><I>x </I>+ <I>y</I> = 7<SPAN
style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 45px"></SPAN>...(i)</SPAN></P>
<P><SPAN class=Normal>(10<I>y</I> + <I>x</I>) + 3 = 4 (10<I>x</I> +
<I>y</I>)</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/therefore.png">
10<I>y</I> + <I>x</I> + 3 = 40<I>x</I> + 4<I>y</I></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/therefore.png">
–39<I>x</I> + 6<I>y </I>= –3</SPAN></P>
<P><SPAN class=Normal>Dividing the above equation by –3</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/therefore.png">
13<I>x</I> – 2<I>y</I> = 1<SPAN style="MARGIN-LEFT: 25px"></SPAN>
...(ii)</SPAN></P>
<P><SPAN class=Normal>Multiplying equation (i) by 2 and adding equation
(ii)</SPAN></P>
<P><SPAN class=Normal>15<I>x</I> = 15</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/therefore.png">
<I>x</I> = 1</SPAN></P>
<P><SPAN class=Normal>Substituting the value of <I>x</I> in equation (i),
we get <I>y</I> = 6</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/therefore.png">
The original number is 16.</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<OL class="oLevel3 Heading3 ListStyleOL">
<LI value=2>INDETERMINATE SIMULTANEOUS EQUATIONS</LI></OL>
<P class=pLevel4><SPAN class=Normal>Two given equations that can be
algebraically derived from each other are called dependent equations. Such
equations have infinitely many solutions and are called Indeterminate
Simultaneous Equations. For example,</SPAN></P>
<P class=pLevel4><SPAN class=Normal>4<I>x</I> + 5<I>y</I> = 18<SPAN
style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 35px"></SPAN>…(i)</SPAN></P>
<P class=pLevel4><SPAN class=Normal>8<I>x</I> + 10<I>y</I> = 36<SPAN
style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN>…(ii)</SPAN></P>
<P class=pLevel4><SPAN class=Normal>The above two equations are dependent
equations as equation (ii) can be obtained by multiplying equation (i) with
2.</SPAN></P>
<P class=pLevel4><SPAN class=Normal>In general, simultaneous equations <I>ax</I>
+ <I>by</I> + <I>c</I> = 0 and <I>lx</I> + <I>my</I> + <I>n</I> = 0 are called
indeterminate if and only if</SPAN></P>
<P class=pLevel4><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/016_017.png"></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=016_solvedex_13>
<TABLE style="WIDTH: 100%" id=016_solvedex_13_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_13>Example
13:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>For what value of <I>k</I> will the following system
of equations be indeterminate?</SPAN></P>
<P><SPAN class=Normal>2<I>x</I> – 3<I>y</I> = 5</SPAN></P>
<P><SPAN class=Normal>6<I>x</I> + <I>ky</I> = 15</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>Here, the ratio of the coefficients of <I>x</I>,
which is 1/3, is the same as the ratio of the constant terms in the two
equations. Hence, for the equations to be indeterminate, the ratio of
coefficients of <I>y</I> also would be 1/3.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Hence, the value of <I>k</I> for which the equations
are indeterminate is – 9.</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorNote divLevel3">
<P><SPAN class=Note>REMEMBER</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<UL type=disc>
<LI value=1>Indeterminate simultaneous equations have infinite number of
solutions. Graphically, they are two lines superimposed on each
other.</LI></UL></DIV>
<P class=pImage><SPAN><IMG alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/016_020.png"></SPAN></P>
<OL class="oLevel3 Heading3 ListStyleOL">
<LI value=3>INCONSISTENT SIMULTANEOUS EQUATIONS</LI></OL>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal>Inconsistent equations are those, for which
there is no common solution, i.e. there is not a single pair of numbers that
satisfies both the equations simultaneously, although each of the equations has
infinite solutions. For example,</SPAN></P>
<P class=pLevel4><SPAN class=Normal>13<I>x</I> + 32<I>y</I> = 19<SPAN
style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN>…(i)</SPAN></P>
<P class=pLevel4><SPAN class=Normal>26<I>x </I>+ 64<I>y</I> = 34<SPAN
style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN>…(ii)</SPAN></P>
<P class=pLevel4><SPAN class=Normal>Multiplying equation (i) by 2, we get
26<I>x</I> + 64<I>y</I> = 38, which contradicts equation (ii). Thus equation (i)
and equation (ii) are not indeterminate but are inconsistent.</SPAN></P>
<P class=pLevel4><SPAN class=Normal>In general two equations of the form
<I>ax</I> + <I>by</I> + <I>c</I> = 0 and <I>lx</I> + <I>my</I> + <I>n</I> = 0
are inconsistent if and only if</SPAN></P>
<P class=pLevel4><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/016_019.png"></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pImage><SPAN><IMG alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/016_018.png"></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorNote divLevel3">
<P><SPAN class=Note>REMEMBER</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<UL type=disc>
<LI value=1>Graphically, inconsistent simultaneous equations are two parallel
lines in a plane, which can never intersect each other.</LI></UL></DIV>
<P class=pLevel4><SPAN class=Normal></SPAN></P></DIV>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=0_solvedex_1>
<TABLE style="WIDTH: 100%" id=0_solvedex_1_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=A13>Example
14:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>Which one of the following conditions must <I>p</I>,
<I>q</I> and <I>r</I> satisfy so that the following system of linear
simultaneous equations has at least one solution, such that <I>p</I> +
<I>q</I> + <I>r</I> ≠ 0?</SPAN></P>
<P><SPAN class=Normal><I>x</I> + 2<I>y</I> <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/minus.png">
3<I>z</I> = <I>p</I></SPAN></P>
<P><SPAN class=Normal>2<I>x</I> + 6<I>y</I> <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/minus.png">
11<I>z</I> = <I>q</I></SPAN></P>
<P><SPAN class=Normal><I>x</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/minus.png">
2<I>y</I> + 7<I>z</I> = <I>r</I></SPAN></P>
<P><SPAN class=Normal><B>[</I>CAT 2003 Leaked Test]</B></SPAN></P>
<P><SPAN class=Normal>(1) 5<I>p</I> <IMG style="VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/minus.png">
2<I>q</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/minus.png">
<I>r</I> = 0<SPAN style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN>(2) 5<I>p</I> + 2<I>q</I> + <I>r</I> =
0</SPAN></P>
<P><SPAN class=Normal>(3) 5<I>p</I> + 2<I>q</I> <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/minus.png">
<I>r</I> = 0<SPAN style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN>(4) 5<I>p</I> <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/minus.png">
2<I>q</I> + <I>r</I> = 0</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>On substituting the values of <I>p</I>, <I>q</I> and
<I>r</I> in the options we see that the values of <I>p</I>, <I>q</I> and
<I>r </I>satisfy only the equation 5<I>p</I> <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/minus.png">
2<I>q</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/minus.png">
<I>r</I> = 0.</SPAN></P>
<P><SPAN class=Normal>Hence, <B>option
1.</B></SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<DIV id=016_II_E>
<TABLE id=016_II_E_bookmark class=tableh1 border=0 cellSpacing=0
cellPadding=0><TBODY>
<TR>
<TD
style="PADDING-BOTTOM: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 100%; PADDING-RIGHT: 0px; PADDING-TOP: 0px">
<OL class="oLevel2 Heading2">
<LI value=5><A id=II_E>LINEAR EQUATIONS IN THREE OR MORE
VARIABLES</A></LI></OL></TD></TR></TBODY></TABLE>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>A set of <I>n</I> unique equations (those
which cannot be algebraically derived from one another) in <I>n</I> variables
also can be solved simultaneously for a common solution.</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>For example,</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<P class=pLevel3><SPAN class=Normal>We have the following set of
equations,</SPAN></P>
<P class=pLevel3><SPAN class=Normal><I>x</I> + <I>y</I> + 2<I>z</I> = 9<SPAN
style="MARGIN-LEFT: 25px"></SPAN><SPAN style="MARGIN-LEFT: 25px"></SPAN>
...(i)</SPAN></P>
<P class=pLevel3><SPAN class=Normal>2<I>x</I> + 4<I>y</I> + 6<I>z</I> = 28<SPAN
style="MARGIN-LEFT: 25px"></SPAN>...(ii)</SPAN></P>
<P class=pLevel3><SPAN class=Normal>3<I>x</I> + 4<I>y</I> + 5<I>z</I> = 26<SPAN
style="MARGIN-LEFT: 25px"></SPAN>...(iii)</SPAN></P>
<P class=pLevel3><SPAN class=Normal>The above set of equations is a system of
three equations with three variables.</SPAN></P>
<P class=pLevel3><SPAN class=Normal>Both substitution and elimination methods
can be used to solve these equations.</SPAN></P>
<P class=pLevel3><SPAN class=Normal>If we multiply (i) by 2 and subtract it from
(ii), we get</SPAN></P>
<P class=pLevel3><SPAN class=Normal><I>y</I> + <I>z</I> = 5<SPAN
style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 55px"></SPAN>...(iv)</SPAN></P>
<P class=pLevel3><SPAN class=Normal>If we multiply (i) by 3 and subtract it from
(iii), we get</SPAN></P>
<P class=pLevel3><SPAN class=Normal><I>z</I> – <I>y</I> = 1<SPAN
style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 55px"></SPAN>...(v)</SPAN></P>
<P class=pLevel3><SPAN class=Normal>Equation (iv) and equation (v) can be added
to get <I>z</I> = 3. </SPAN></P>
<P class=pLevel3><SPAN class=Normal>Using this value of <I>z</I> we can easily
find <I>y</I> and <I>x</I>.</SPAN></P>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorNote divLevel3">
<P><SPAN class=Note>REMEMBER</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<UL type=disc>
<LI value=2>To solve a system of simultaneous equations, the number of
independent equations must be at least equal to the number of
variables.</LI></UL></DIV>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=016_solvedex_14>
<TABLE style="WIDTH: 100%" id=016_solvedex_14_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_14>Example
15:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>Solve the system of equations for <I>x</I>, <I>y</I>
and <I>z</I>.</SPAN></P>
<P><SPAN class=Normal><I>x</I> – <I>y</I> + <I>z</I> = 6</SPAN></P>
<P><SPAN class=Normal>2<I>x</I> + 2<I>y</I> + 3<I>z</I> = 11</SPAN></P>
<P><SPAN class=Normal>2<I>x</I> – 3<I>y</I> – 2<I>z</I> = 1</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal><I>x </I>– <I>y</I> + <I>z</I> = 6<SPAN
style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 55px"></SPAN>...(i)</SPAN></P>
<P><SPAN class=Normal>2<I>x</I> + 2<I>y</I> + 3<I>z</I> = 11<SPAN
style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 43px"></SPAN>...(ii)</SPAN></P>
<P><SPAN class=Normal>2<I>x</I> – 3<I>y</I> – 2<I>z</I> = 1<SPAN
style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 50px"></SPAN>...(iii)</SPAN></P>
<P><SPAN class=Normal>Multiplying equation (i) by 2 and subtracting from
equation (ii) we get,</SPAN></P>
<P><SPAN class=Normal>4<I>y</I> +<I> z</I> = –1<SPAN
style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 60px"></SPAN>...(iv)</SPAN></P>
<P><SPAN class=Normal>Subtracting (iii) from (ii),</SPAN></P>
<P><SPAN class=Normal>5<I>y</I> + 5<I>z</I> = 10 or <I>y</I> + <I>z</I> =
2<SPAN style="MARGIN-LEFT: 25px"></SPAN>...(v)</SPAN></P>
<P><SPAN class=Normal>Now equation (iv) and (v) are simultaneous equations
in<I> y</I> and <I>z</I>.</SPAN></P>
<P><SPAN class=Normal>Solving, <I>y</I> = –1 and <I>z </I>= 3</SPAN></P>
<P><SPAN class=Normal>Substituting the value of <I>y</I> and <I>z</I> in
(i), we get <I>x </I>= 2</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Thus <I>x</I> = 2, <I>y</I> = –1 and <I>z</I> =
3.</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV><!--<p class="pLevel2"><span class="Normal"></span></p>
<div class="divTextBox divColorNormal divLevel4">
<p><span class="Normal"></span></p>
<p><span class="Normal"></span></p>
</div>-->
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=016_solvedex_15>
<TABLE style="WIDTH: 100%" id=Table9 border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=A9>Example
16:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>Find the values of <I>x</I>, <I>y</I> and
<I>z</I>.</SPAN></P>
<P><SPAN class=Normal><I>z</I> + 4<I>y</I> + 3<I>x</I> = 33</SPAN></P>
<P><SPAN class=Normal>4<I>x</I> + <I>y</I> – <I>z</I> = 6</SPAN></P>
<P><SPAN class=Normal>2<I>y</I> + 8<I>x</I> – 2<I>z</I> = 12</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal><I>z</I> + 4<I>y</I> + 3<I>x</I> = 33<SPAN
style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 35px"></SPAN>...(i)</SPAN></P>
<P><SPAN class=Normal>4<I>x</I> + <I>y</I> – <I>z</I> = 6 <SPAN
style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN>...(ii)</SPAN></P>
<P><SPAN class=Normal>2<I>y</I> + 8<I>x</I> – 2<I>z</I> = 12<SPAN
style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN>...(iii)</SPAN></P>
<P><SPAN class=Normal>Dividing equation (iii) by 2,</SPAN></P>
<P><SPAN class=Normal><I>y</I> + 4<I>x</I> – <I>z</I> = 6, which is the
same as equation (ii). Thus we have only two equations but three
variables. Hence, the values of <I>x</I>, <I>y</I> and <I>z</I> cannot be
found.</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV></DIV><!--<p class="pLevel2"><span class="Normal"></span></p>
<div class="divTextBox divColorNormal divLevel4">
<p><span class="Normal"></span></p>
<p><span class="Normal"></span></p>
</div>-->
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=016_solvedex_16>
<TABLE style="WIDTH: 100%" id=Table10 border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=A10>Example
17:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>Ina bought 2 candies, 3 cans of soft drink and 9
pieces of cake for Rs. 109. Sona bought 4 candies, 5 pieces of cake and a
can of soft drink for Rs. 67. How much did Koena pay for 3 candies, 2 cans
of soft drink and 7 pieces of cake, if all the three bought candies, cakes
and soft drinks of the same kind?</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>Let <I>a</I>, <I>b</I> and <I>c</I> be the price of
a candy, a can of soft drink and a piece of cake respectively.</SPAN></P>
<P><SPAN class=Normal>2<I>a </I>+ 3<I>b</I> + 9<I>c</I> = 109<SPAN
style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 37px"></SPAN>...(i)</SPAN></P>
<P><SPAN class=Normal>4<I>a</I> + <I>b</I> + 5<I>c</I> = 67<SPAN
style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 53px"></SPAN>...(ii)</SPAN></P>
<P><SPAN class=Normal>Adding (i) and (ii),</SPAN></P>
<P><SPAN class=Normal>6<I>a</I> + 4<I>b</I> + 14<I>c</I> = 176<SPAN
style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN>...(iii)</SPAN></P>
<P><SPAN class=Normal>Dividing equation (iii) by 2,</SPAN></P>
<P><SPAN class=Normal>3<I>a</I> + 2<I>b</I> + 7<I>c</I> = 88</SPAN></P>
<P><SPAN class=Normal>Thus, Koena paid Rs. 88 for 3 candies, 2 cans of
soft drink and 7 pieces of
cake.</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV><!--<p class="pLevel2"><span class="Normal"></span></p>
<div class="divTextBox divColorNormal divLevel4">
<p><span class="Normal"><span style="margin-left:25px"></span></span></p>
<p><span class="Normal"></span></p>
</div>-->
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=016_solvedex_17>
<TABLE style="WIDTH: 100%" id=Table11 border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=A11>Example
18:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>A change making machine contains 1 rupee, 2 rupee
and 5 rupee coins. The total number of coins is 300. The amount is Rs.
960. If the number of 1 rupee coins and the number of 2 rupee coins are
interchanged, the value comes down by Rs. 40. The total number of 5 rupee
coins is</SPAN></P>
<P><SPAN class=Normal><B>[CAT 2001]</B></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>(1) 100<SPAN style="MARGIN-LEFT: 43px"></SPAN>(2)
140</SPAN></P>
<P><SPAN class=Normal>(3) 60<SPAN style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN>(4) 150</SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>Let the number of Re. 1, Re. 2 and Re. 5 coins be
<I>x</I>, <I>y</I> and <I>z</I> respectively.</SPAN></P>
<P><SPAN class=Normal><I>x</I> + <I>y</I> + <I>z </I>= 300<SPAN
style="MARGIN-LEFT: 75px"></SPAN>...(i)</SPAN></P>
<P><SPAN class=Normal><I>x</I> + 2<I>y</I> + 5<I>z</I> = 960<SPAN
style="MARGIN-LEFT: 55px"></SPAN>...(ii)</SPAN></P>
<P><SPAN class=Normal>2<I>x</I> + <I>y</I> + 5<I>z</I> = 920<SPAN
style="MARGIN-LEFT: 53px"></SPAN>...(iii)</SPAN></P>
<P><SPAN class=Normal>Adding equation (ii) and equation (iii), we
get,</SPAN></P>
<P><SPAN class=Normal>3<I>x</I> + 3<I>y</I> + 10<I>z</I> = 1880<SPAN
style="MARGIN-LEFT: 25px"></SPAN>...(iv)</SPAN></P>
<P><SPAN class=Normal>Multiply equation (i) by 3, we get,</SPAN></P>
<P><SPAN class=Normal></SPAN>3<I>x</I> + 3<I>y</I> + 3<I>z</I> = 900<SPAN
style="MARGIN-LEFT: 45px"></SPAN>...(v)</P>
<P><SPAN class=Normal>Subtracting equation (v) from equation (iv), we
get,</SPAN></P>
<P><SPAN class=Normal></SPAN>7<I>z</I> = 980</P>
<P><SPAN class=Normal></SPAN><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/therefore.png">
<I>z</I> = 140</P>
<P><SPAN class=Normal>Hence, <B>option
2</B>.</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV><!--<p class="pLevel2"><span class="Normal"></span></p>
<div class="divTextBox divColorNormal divLevel4">
<p><span class="Normal"></span></p>
<p><span class="Normal"></span></p>
</div>-->
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=016_solvedex_18>
<TABLE style="WIDTH: 100%" id=Table12 border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=A12>Example
19:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>In a cricket match, Team A scored 232 runs without
losing a wicket. The score consisted of byes, wides and runs scored by two
opening batsmen: Ram and Shyam. The runs scored by the two batsmen are 26
times wides. There are 8 more byes than wides. If the ratio of the runs
scored by Ram and Shyam is 6 : 7, then the runs scored by Ram
is</SPAN></P>
<P><SPAN class=Normal><B>[XAT 2008]</B></SPAN></P>
<P><SPAN class=Normal>(1) 88<SPAN style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 37px"></SPAN>(2) 96</SPAN></P>
<P><SPAN class=Normal>(3) 102<SPAN style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN>(4) 112</SPAN></P>
<P><SPAN class=Normal>(5) None of these</SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>Let the number of runs scored by byes, wides and
runs be<I> x</I>, <I>y</I> and <I>z</I> respectively.</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/therefore.png">
<I>x</I> + <I>y</I> + <I>z</I> = 232<SPAN
style="MARGIN-LEFT: 25px"></SPAN>…(i)</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/symb23.png">
The runs scored by the two batsmen are 26 times the wides</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/therefore.png">
<I>z</I> = 26<I>y</I><SPAN
style="MARGIN-LEFT: 70px"></SPAN>…(ii)</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/symb23.png">
There are 8 more byes than wides</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/therefore.png">
<I>x</I> = <I>y</I> + 8<SPAN
style="MARGIN-LEFT: 60px"></SPAN>…(iii)</SPAN></P>
<P><SPAN class=Normal>Substituting equations (iii) and (ii) in equation
(i), we get,</SPAN></P>
<P><SPAN class=Normal><I>y</I> = 8</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/therefore.png">
<I>z</I> = 208</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/symb23.png">
The runs scored by Ram and Shyam were in the ratio 6 : 7</SPAN></P>
<P><SPAN class=Normal>Let the runs scored by Ram be 6<I>r</I> and by Shyam
be 7<I>r</I>.</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/therefore.png">
13<I>r</I> = 208</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/therefore.png">
<I>r</I> = 16</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/therefore.png">
Runs scored by Ram is 96.</SPAN></P>
<P><SPAN class=Normal>Hence, <B>option
2.</B></SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<DIV id=016_II_F>
<TABLE id=016_II_F_bookmark class=tableh1 border=0 cellSpacing=0
cellPadding=0><TBODY>
<TR>
<TD
style="PADDING-BOTTOM: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 100%; PADDING-RIGHT: 0px; PADDING-TOP: 0px">
<OL class="oLevel2 Heading2">
<LI value=6><A id=II_F>OTHER EXAMPLES</A></LI></OL></TD></TR></TBODY></TABLE>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=016_solvedex_19>
<TABLE style="WIDTH: 100%" id=016_solvedex_19_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_19>Example
20:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>An artist has completed one fourth of a rectangular
oil painting. When he will paint another 100 square centimeters of the
painting, he would complete three quarters of the painting. If the height
of the oil painting is 10 centimeters, determine the length (in
centimeters) of the oil painting.</SPAN></P>
<P><SPAN class=Normal><B>[JMET 2009]</B></SPAN></P>
<P><SPAN class=Normal>(1)15<SPAN style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN>(2) 20</SPAN></P>
<P><SPAN class=Normal>(3) 10<SPAN style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN>(4) 25</SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>Let the total area of the painting be <I>x</I>
cm<SUP>2</SUP>.</SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/016_021.png">
<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/016_022.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal></SPAN></P><SPAN class=Normal><IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/016_023.png"></SPAN>
<P></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/therefore.png">
<I>x</I> = 200 cm<SUP>2</SUP></SPAN></P>
<P><SPAN class=Normal>As the canvas is rectangular in shape,</SPAN></P>
<P><SPAN class=Normal>Area of the canvas = Length <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/multiplication.png">
(Breadth i.e. Height)</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>But, Height of the painting = 10 cm</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/therefore.png">
200 = 10 <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/multiplication.png">
Length</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/therefore.png">
Length = 20 cm</SPAN></P>
<P><SPAN class=Normal>Hence, <B>option
2.</B></SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=016_solvedex_20>
<TABLE style="WIDTH: 100%" id=016_solvedex_20_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_20>Example
21:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>A shop stores <I>x</I> kg of rice. The first
customer buys half this amount plus half a kg of rice. The second customer
buys half the remaining amount plus half a kg of rice. Then the third
customer also buys half the remaining amount plus half a kg of rice.
Thereafter, no rice is left in the shop. Which of the following best
describes the value of <I>x</I>?</SPAN></P>
<P><SPAN class=Normal><B>[CAT 2008]</B></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>(1) 2 <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/less_equal.png">
<I>x <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/less_equal.png"></I>
6 <SPAN style="MARGIN-LEFT: 35px"></SPAN>(2) 5 <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/less_equal.png">
<I>x <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/less_equal.png"></I>
8 </SPAN></P>
<P><SPAN class=Normal>(3) 9 <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/less_equal.png">
<I>x <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/less_equal.png"></I>
12 <SPAN style="MARGIN-LEFT: 25px"></SPAN>(4) 11 <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/less_equal.png">
<I>x <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/less_equal.png"></I>
14 </SPAN></P>
<P><SPAN class=Normal>(5) 13 <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/less_equal.png">
<I>x <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/less_equal.png"></I>
18 </SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>The store has <I>x</I> kg of rice, initially.
(Henceforth, <I>x</I> is assumed to be measured in kg.)</SPAN></P>
<P><SPAN class=Normal>The first customer buys half the total rice in the
store, and another half kg.</SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/016_024.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/016_025.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/016_026.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>The second customer buys half of this, and another
half kg.</SPAN></P>
<P><SPAN class=Normal>Amount purchased by second customer</SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/016_027.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/016_028.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>The third customer also buys half the remaining
rice, and another half kg.</SPAN></P>
<P><SPAN class=Normal>Amount purchased by third customer</SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/016_029.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Since we know that after this purchase, there was no
more rice left in the store, we conclude that:</SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/016_030.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/016_031.png"></SPAN></P>
<P><SPAN class=Normal>Hence, <B>option
2.</B></SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel3><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel3">
<DIV id=016_solvedex_21>
<TABLE style="WIDTH: 100%" id=016_solvedex_21_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_21>Example
22:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>In second year, students at a business school can
opt for Systems, Operations or HR electives only. The number of girls
opting for Operations and the number of boys opting for Systems elective
is 37. Twenty two students opt for operations elective. Twenty girls opt
for Systems and Operations electives. The number of students opting for
Systems elective and the number of boys opting for Operations electives is
37. Twenty-five students opt for HR electives.</SPAN></P>
<P><SPAN class=Normal><B>[XAT 2008]</B></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><B>Question1:</B></SPAN></P>
<P><SPAN class=Normal>The number of students in the second year
is?</SPAN></P>
<P><SPAN class=Normal>(1) 73<SPAN style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN>(2) 74</SPAN></P>
<P><SPAN class=Normal>(3) 75<SPAN style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN>(4) 76</SPAN></P>
<P><SPAN class=Normal>(5) 77</SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<DIV class=tblCenter>
<TABLE style="BORDER-COLLAPSE: collapse" cellSpacing=1 align=center>
<TBODY>
<TR>
<TD class=visible_border_table></TD>
<TD class=visible_border_table><B>Systems</B></TD>
<TD class=visible_border_table><B>Operations</B></TD>
<TD class=visible_border_table><B>HR</B></TD></TR>
<TR>
<TD class=visible_border_table><B>Boys</B></TD>
<TD class=visible_border_table><I>a</I></TD>
<TD class=visible_border_table><I>c</I></TD>
<TD class=visible_border_table></TD></TR>
<TR>
<TD class=visible_border_table><B>Girls</B></TD>
<TD class=visible_border_table><I>b</I></TD>
<TD class=visible_border_table>20 – <I>b</I></TD>
<TD class=visible_border_table></TD></TR>
<TR>
<TD class=visible_border_table><B>Students</B></TD>
<TD class=visible_border_table><I>d</I></TD>
<TD class=visible_border_table>22</TD>
<TD class=visible_border_table>25</TD></TR></TBODY></TABLE></DIV>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>From the given data we have,</SPAN></P>
<P><SPAN class=Normal>20 – <I>b</I> + <I>a</I> = 37<SPAN
style="MARGIN-LEFT: 13px"></SPAN>… (i)</SPAN></P>
<P><SPAN class=Normal><I>c</I> + <I>d</I> = 37<SPAN
style="MARGIN-LEFT: 47px"></SPAN>… (ii)</SPAN></P>
<P><SPAN class=Normal><I>a</I> + <I>b</I> = <I>d</I><SPAN
style="MARGIN-LEFT: 55px"></SPAN>… (iii)</SPAN></P>
<P><SPAN class=Normal><I>c</I> – <I>b</I> = 2<SPAN
style="MARGIN-LEFT: 55px"></SPAN>… (iv)</SPAN></P>
<P><SPAN class=Normal>On solving these equations we get,</SPAN></P>
<P><SPAN class=Normal><I>a </I>= 23, <I>b </I>= 6, <I>c </I>= 8 and
<I>d</I> = 29</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/therefore.png">
Total number of students in second year</SPAN></P>
<P><SPAN class=Normal>= 29 + 25 + 22</SPAN></P>
<P><SPAN class=Normal>=76</SPAN></P>
<P><SPAN class=Normal>Hence, <B>option 4.</B></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><B>Question 2:</B></SPAN></P>
<P><SPAN class=Normal>If 20% of the girls opt for HR electives, then the
total number of boys in the second year is</SPAN></P>
<P><SPAN class=Normal>(1) 54<SPAN style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN>(2) 53</SPAN></P>
<P><SPAN class=Normal>(3) 52<SPAN style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN>(4) 51</SPAN></P>
<P><SPAN class=Normal>(5) 50</SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<DIV class=tblCenter>
<TABLE style="BORDER-COLLAPSE: collapse" cellSpacing=1 align=center>
<TBODY>
<TR>
<TD class=visible_border_table></TD>
<TD class=visible_border_table><B>Systems</B></TD>
<TD class=visible_border_table><B>Operations</B></TD>
<TD class=visible_border_table><B>HR</B></TD></TR>
<TR>
<TD class=visible_border_table><B>Boys</B></TD>
<TD class=visible_border_table>23</TD>
<TD class=visible_border_table>8</TD>
<TD class=visible_border_table><I>x</I></TD></TR>
<TR>
<TD class=visible_border_table><B>Girls</B></TD>
<TD class=visible_border_table>6</TD>
<TD class=visible_border_table>14</TD>
<TD class=visible_border_table><I>y</I></TD></TR>
<TR>
<TD class=visible_border_table><B>Students</B></TD>
<TD class=visible_border_table>29</TD>
<TD class=visible_border_table>22</TD>
<TD class=visible_border_table>25</TD></TR></TBODY></TABLE></DIV>
<P><SPAN class=Normal>Now, <I>y </I>= 20% of total number of
girls.</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/therefore.png">
16 + 4 = 20 = 80% of total number of girls.</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/therefore.png">
Total number of girls = 25</SPAN></P>
<P><SPAN class=Normal>As total number of students = 76 and</SPAN></P>
<P><SPAN class=Normal>total number of girls = 25</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/c0223cf0-f84c-4a39-89db-d7d3951aff66/016_images/therefore.png">
Total number of boys = 76 – 25 = 51</SPAN></P>
<P><SPAN class=Normal>Hence, <B>option
4</B>.</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV></DIV></BODY></HTML>
guruhttp://www.blogger.com/profile/01591375209227117455noreply@blogger.comtag:blogger.com,1999:blog-7850540640045081515.post-30332204856636382392015-03-03T03:49:00.001-08:002019-09-14T18:55:03.848-07:00ineq<div dir="ltr" style="text-align: left;" trbidi="on">
<br /></div>
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
<HTML xmlns="http://www.w3.org/1999/xhtml"><HEAD><TITLE>Inequalities</TITLE>
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<P class=oTOCTitle><SPAN class=oTOCTitleFont><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018.htm#Title">Contents</A></SPAN></P>
<OL class=oTOCLevel1>
<LI class=oTocLiClass><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018.htm#I">INTRODUCTION</A>
</LI>
<LI class=oTocLiClass><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018.htm#II">PROPERTIES
OF INEQUALITIES</A> </LI>
<LI class=oTocLiClass><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018.htm#III">SOME
USEFUL INEQUATIONS</A> </LI>
<LI class=oTocLiClass><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018.htm#IV">SOLVING
INEQUALITIES</A> </LI>
<LI class=oTocLiClass><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018.htm#V">INEQUALITIES
AND ABSOLUTE VALUES</A> </LI>
<LI class=oTocLiClass><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018.htm#VI">PROPERTIES
OF MODULUS</A> </LI>
<LI class=oTocLiClass><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018.htm#VII">QUADRATIC
INEQUALITIES</A> </LI></OL>
<P id=Title class=pTitle><SPAN class=Title>Inequalities</SPAN></P>
<HR>
<P class=pLevel2><SPAN class=Normal></SPAN>
<DIV id=018_I>
<TABLE id=018_I_bookmark class=tableh1 border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD
style="PADDING-BOTTOM: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 100%; PADDING-RIGHT: 0px; PADDING-TOP: 0px">
<OL class="oLevel1 Heading1">
<LI value=1><A id=I>INTRODUCTION</A></LI></OL></TD></TR></TBODY></TABLE>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>An <B>inequality</B> as the name suggests,
is a relationship that states that two quantities are not equal. It means that
one expression is greater than the other in value.</SPAN></P>
<P class=pLevel2><SPAN class=Normal>Inequality can be expressed by using the
following symbols:</SPAN></P>
<UL class=oLevel3>
<LI value=0><I>x</I> ≠ <I>y</I> means <I>x</I> is not equal to
<I>y</I></LI></UL>
<UL class=oLevel3>
<LI value=1><I>x</I> > <I>y</I> means <I>x</I> is greater than
<I>y</I></LI></UL>
<UL class=oLevel3>
<LI value=2><I>x</I> < <I>y</I> means <I>x</I> is less than
<I>y</I></LI></UL>
<UL class=oLevel3>
<LI value=3><I>x</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/greater_equal.png">
<I>y</I> means <I>x</I> is greater than or equal to <I>y</I>, or <I>x</I> is
not less than <I>y</I></LI></UL>
<UL class=oLevel3>
<LI value=4><I>x</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/less_equal.png">
<I>y</I> means <I>x</I> is less than or equal to <I>y</I>, or <I>x</I> is not
greater than <I>y</I></LI></UL>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>Inequalities of the last two forms are
called <B>inequations</B>.</SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>For example, consider the statement “the
total number of people (<I>P</I>) at a given time inside the “City Plaza” mall
should not exceed 1500”. This statement can be expressed mathematically as
<I>P</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/less_equal.png">
1500</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>Equalities involving variables often give us
exact values of the variables but inequalities involving variables often give a
range of values for the variables.</SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>For example, If 3<I>x</I> > 9, then
<I>x</I> > 3, which means all real numbers greater than 3 satisfy the
condition.</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P></DIV>
<DIV id=018_II>
<TABLE id=018_II_bookmark class=tableh1 border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD
style="PADDING-BOTTOM: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 100%; PADDING-RIGHT: 0px; PADDING-TOP: 0px">
<OL class="oLevel1 Heading1">
<LI value=2><A id=II>PROPERTIES OF
INEQUALITIES</A></LI></OL></TD></TR></TBODY></TABLE>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>The properties of inequalities are better
understood if numbers are substituted in place of variables. It is important
that inequalities be tested for positive as well as negative numbers.</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal><B>Transitivity</B></SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>For real numbers <I>a</I>, <I>b</I> and
<I>c</I></SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<UL class=oLevel3>
<LI value=5>If <I>a</I> > <I>b</I> and <I>b</I> > <I>c</I>, then
<I>a</I> > <I>b</I> > <I>c</I> and <I>a</I> > <I>c</I></LI></UL>
<UL class=oLevel3>
<LI value=6>If <I>a</I> < <I>b</I> and <I>b</I> < <I>c</I>, then
<I>a</I> < <I>b</I> < <I>c</I> and <I>a</I> < <I>c</I></LI></UL>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal><B>Sum and Difference</B></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<UL class="oLevel3 ">
<LI value=7>If <I>a</I> > <I>b</I>, then <I>a</I> + <I>c</I> > <I>b</I>
+ <I>c</I> and <I>a</I> – <I>c</I> > <I>b</I> – <I>c</I></LI></UL>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal><SPAN style="MARGIN-LEFT: 17px"></SPAN>For
example,</SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal><SPAN style="MARGIN-LEFT: 17px"></SPAN>If
<I>a</I> = 12, <I>b</I> = 10 and <I>c</I> = 5</SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal><SPAN
style="MARGIN-LEFT: 17px"></SPAN><I>a</I> + <I>c</I> = 17 and <I>b</I> +
<I>c</I> = 15</SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal><SPAN style="MARGIN-LEFT: 17px"></SPAN><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
<I>a</I> + <I>c</I> > <I>b</I> + <I>c</I></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal><SPAN style="MARGIN-LEFT: 17px"></SPAN><I>a
</I>– <I>c </I>= 7<I> </I>and<I> b </I>– <I>c </I>= 5</SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal><SPAN style="MARGIN-LEFT: 17px"></SPAN><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png"></I>
<I>a </I>–<I> c ></I><I> b </I>–<I> c</I></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<UL class=oLevel3>
<LI value=8>If <I>a</I> > <I>b</I> and <I>c</I> > <I>d</I>, then
<I>a</I> + <I>c</I> > <I>b</I> + <I>d</I></LI></UL>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal><SPAN style="MARGIN-LEFT: 17px"></SPAN>For
example,</SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal><SPAN style="MARGIN-LEFT: 17px"></SPAN>If<I>
a </I>= 12<I>, b </I>= 10<I>, c </I>= 2<I> </I>and<I> d </I>= 1</SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal><SPAN style="MARGIN-LEFT: 17px"></SPAN><I>a
</I>+<I> c </I>= 14<I>, b </I>+<I> d </I>= 11</SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal><SPAN style="MARGIN-LEFT: 17px"></SPAN><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png"><I>
a</I> + <I>c</I> > <I>b</I> + <I>d</I></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal><SPAN style="MARGIN-LEFT: 17px"></SPAN>If
<I>a</I> = 12, <I>b</I> = 10, <I>c</I> = –5, <I>d</I> = –6</SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal><SPAN
style="MARGIN-LEFT: 17px"></SPAN><I>a</I> + <I>c</I> = 7, <I>b</I> + <I>d</I> =
4</SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal><SPAN style="MARGIN-LEFT: 17px"></SPAN><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png"><I>
a</I> + <I>c</I> > <I>b</I> + <I>d</I></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal><SPAN style="MARGIN-LEFT: 17px"></SPAN>This
rule can be generalised as follows.</SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal><SPAN style="MARGIN-LEFT: 17px"></SPAN>If
<I>a</I> > <I>b</I>, <I>c</I> > <I>d</I>, <I>e</I> > <I>f</I>, and so
on, then</SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal><SPAN
style="MARGIN-LEFT: 17px"></SPAN><I>a</I> + <I>c</I> + <I>e</I> + … >
<I>b</I> + <I>d</I> + <I>f</I> + …</SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<UL class=oLevel3>
<LI value=1>If <I>a</I> > <I>b</I> and <I>c</I> > <I>d</I>, then
<I>a</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">
<I>c</I> > <I>b</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">
<I>d</I> is not necessarily true.</LI></UL>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal><SPAN
style="MARGIN-LEFT: 17px"></SPAN>However, <I>a</I> – <I>d</I> > <I>b</I> –
<I>c</I></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal><SPAN style="MARGIN-LEFT: 17px"></SPAN>For
example,</SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal><SPAN style="MARGIN-LEFT: 17px"></SPAN>If
<I>a</I> = 12, <I>b</I> = 10, <I>c</I> = 7 and <I>d</I> = 6</SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal><SPAN
style="MARGIN-LEFT: 17px"></SPAN><I>a</I> – <I>d</I> = 6, <I>b</I> – <I>c</I> =
3</SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal><SPAN style="MARGIN-LEFT: 17px"></SPAN><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png"><I>
a</I> – <I>d</I> > <I>b</I> – <I>c</I></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal><B>Multiplication and
Division</B></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<UL class=oLevel3>
<LI value=9>If <I>a</I> > <I>b</I> and <I>c</I> > 0 then <I>a</I> <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/multiplication.png">
<I>c</I> > <I>b</I> <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/multiplication.png">
<I>c</I> and <I>a</I>/<I>c</I> > <I>b</I>/<I>c</I></LI></UL>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal><SPAN style="MARGIN-LEFT: 17px"></SPAN>For
example,</SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal><SPAN style="MARGIN-LEFT: 17px"></SPAN>If
<I>a</I> = 12, <I>b</I> = 10, <I>c</I> = 5 and <I>d</I> = 4</SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal><SPAN
style="MARGIN-LEFT: 17px"></SPAN><I>ac</I> = 60 and <I>bc</I> = 50</SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal><SPAN style="MARGIN-LEFT: 17px"></SPAN><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
<I>ac</I> > <I>bc</I></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal><SPAN
style="MARGIN-LEFT: 17px"></SPAN><I>a</I>/<I>d</I> = 3 and <I>b</I>/<I>d</I> =
2.5</SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal><SPAN style="MARGIN-LEFT: 17px"></SPAN><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
<I>a</I>/<I>d</I> > <I>b</I>/<I>d</I></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<UL class=oLevel3>
<LI value=10>If <I>a</I> > <I>b</I> and <I>c</I> < 0 then <I>a</I> <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/multiplication.png">
<I>c</I> < <I>b</I> <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/multiplication.png">
<I>c</I> and <I>a</I>/<I>c</I> < <I>b</I>/<I>c</I></LI></UL>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal><SPAN style="MARGIN-LEFT: 17px"></SPAN>For
example,</SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal><SPAN style="MARGIN-LEFT: 17px"></SPAN>If
<I>a</I> = 12, <I>b</I> = 10, <I>c</I> = –5 and <I>d</I> = –4</SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal><SPAN
style="MARGIN-LEFT: 17px"></SPAN><I>ac</I> = –60, <I>bc</I> = –50</SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal><SPAN style="MARGIN-LEFT: 17px"></SPAN><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
<I>ac</I> < <I>bc</I></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal><SPAN
style="MARGIN-LEFT: 17px"></SPAN><I>a</I>/<I>d</I> = –3 and <I>b</I>/<I>d</I> =
–2.5</SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal><SPAN style="MARGIN-LEFT: 17px"></SPAN><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
<I>a</I>/<I>d</I> < <I>b</I>/<I>d</I></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal><SPAN style="MARGIN-LEFT: 17px"></SPAN>It
follows that if <I>a</I> > <I>b</I>, then –<I>a</I> < –<I>b</I></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<UL class=oLevel3>
<LI value=11>If <I>a</I> > <I>b</I> and <I>c</I> > <I>d</I>, and
<I>a</I>, <I>b</I>, <I>c</I>, <I>d</I> > 0, then <I>ac</I> >
<I>bd</I></LI></UL>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal><SPAN style="MARGIN-LEFT: 17px"></SPAN>Let
<I>a</I> = 12, <I>b</I> = 10, <I>c</I> = 5 and <I>d</I> = 4</SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal><SPAN
style="MARGIN-LEFT: 17px"></SPAN><I>ac</I> = 60 and <I>bd</I> = 40</SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal><SPAN style="MARGIN-LEFT: 17px"></SPAN><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
<I>ac</I> ><I> bd</I></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal><SPAN style="MARGIN-LEFT: 17px"></SPAN>This
rule can be generalised as follows.</SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal><SPAN style="MARGIN-LEFT: 17px"></SPAN>If
<I>a</I>, <I>b</I>, <I>c</I>, <I>d</I>, <I>e</I>, <I>f</I>, … are all positive
and if <I>a</I> > <I>b</I>, <I>c</I> > <I>d</I>, <I>e</I> > <I>f</I>,
and so on, </SPAN></P>
<P class=pLevel4><SPAN class=Normal><SPAN style="MARGIN-LEFT: 17px"></SPAN>then
<I>ace</I>… > <I>bdf</I>…</SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<UL class=oLevel3>
<LI value=2>If <I>a</I> > <I>b</I> and <I>a</I>, <I>b</I>, <I>n</I> > 0,
then <I>a<SUP>n</I></SUP> > <I>b<SUP>n</I></SUP></LI></UL>
<UL class=oLevel3>
<LI value=3>If <I>a</I> > <I>b</I> and <I>a</I>, <I>b</I>, <I>n</I> > 0,
then <I>a</I><SUP>1/<I>n ></I></SUP>; <I>b</I><SUP>1/<I>n</I></SUP></LI></UL>
<UL class=oLevel3>
<LI value=4>If <I>a</I> > <I>b</I> and <I>a</I>, <I>b</I>, <I>n</I> > 0,
then 1/<I>a<SUP>n</I></SUP> < 1/<I>b<SUP>n</I></SUP> or
<I>a<SUP>–n</I></SUP> < <I>b<SUP>–n</I></SUP></LI></UL>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal><SPAN style="MARGIN-LEFT: 17px"></SPAN>It
follows that if <I>a</I> > <I>b</I>, then 1/<I>a</I> <
1/<I>b</I></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorNote divLevel2">
<P><SPAN class=Note>REMEMBER</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<UL type=disc>
<LI value=1>Working with inequalities is similar to working with equalities,
except for the fact that when both the sides of the inequality are multiplied
or divided by a negative number, the inequality gets reversed.</LI></UL></DIV>
<P class=pLevel4><SPAN class=Normal></SPAN></P></DIV>
<DIV id=018_III>
<TABLE id=018_III_bookmark class=tableh1 border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD
style="PADDING-BOTTOM: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 100%; PADDING-RIGHT: 0px; PADDING-TOP: 0px">
<OL class="oLevel1 Heading1">
<LI value=3><A id=III>SOME USEFUL
INEQUATIONS</A></LI></OL></TD></TR></TBODY></TABLE>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>We know that the square of any positive
number is always positive. Thus,</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>(<I>x</I> – <I>y</I>)<SUP>2</SUP> <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/greater_equal.png">
0</SPAN></P>
<P class=pLevel2><SPAN class=Normal><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
<I>x</I><SUP>2</SUP> + <I>y</I><SUP>2</SUP> – 2<I>xy</I> <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/greater_equal.png">
0</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
<I>x</I><SUP>2</SUP> + <I>y</I><SUP>2</SUP> <IMG style="VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/greater_equal.png">
2<I>xy<B> </I></B>(The two sides of the expression are equal only if <I>x</I> =
<I>y</I>)</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>We can state a few important results based
on this inequation.</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<UL class="oLevel3 ">
<LI value=12><I>x</I><SUP>2</SUP> + <I>y</I><SUP>2</SUP> <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/greater_equal.png">
2<I>xy </I></LI></UL>
<P class=pLevel5><SPAN class=Normal>Let <I>x</I><SUP>2</SUP> = <I>a</I> and
<I>y</I><SUP>2</SUP> = <I>b</I></SPAN></P>
<P class=pLevel5><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/018_001.png"></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel5><SPAN class=Normal></SPAN><IMG style="VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/018_070.png"></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel5><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/018_002.png">
<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/018_003.png"></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel5><SPAN class=Normal><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
The arithmetic mean of two numbers is greater than the geometric
mean.</SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<UL class=oLevel3>
<LI value=13><I>a</I><SUP>2</SUP> + <I>b</I><SUP>2</SUP> <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/greater_equal.png">
2<I>ab</I></LI></UL>
<P class=pLevel5><SPAN class=Normal>If <I>a</I> and <I>b</I> both are positive
or negative, <I>ab</I> > 0</SPAN></P>
<P class=pLevel5><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/018_004.png"></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel5><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/018_005.png"></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel5><SPAN class=Normal><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
The sum of a positive number and its reciprocal is always greater than
2.</SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<UL class=oLevel3>
<LI value=14><I>a</I><SUP>2</SUP> + <I>b</I><SUP>2</SUP> <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/greater_equal.png">
2<I>ab</I></LI></UL>
<P class=pLevel5><SPAN class=Normal>Let <I>a</I> + <I>b</I> > 0</SPAN></P>
<P class=pLevel5><SPAN class=Normal><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
(<I>a</I><SUP>2</SUP> + <I>b</I><SUP>2</SUP>)(<I>a</I> + <I>b</I>) <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/greater_equal.png">
2<I>ab</I>(<I>a</I> + <I>b</I>)</SPAN></P>
<P class=pLevel5><SPAN class=Normal><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
<I>a</I><SUP>3</SUP> + <I>ab</I><SUP>2</SUP> + <I>a</I><SUP>2</SUP><I>b</I> +
<I>b</I><SUP>3</SUP> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/greater_equal.png">
2<I>a</I><SUP>2</SUP><I>b</I> + 2<I>ab</I><SUP>2</SUP></SPAN></P>
<P class=pLevel5><SPAN class=Normal><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
<I>a</I><SUP>3</SUP> + <I>b</I><SUP>3</SUP> <IMG style="VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/greater_equal.png">
<I>ab</I>(<I>a</I> + <I>b</I>) (The two sides of the expression are equal only
if <I>x</I> = <I>y</I>)</SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<UL class=oLevel3>
<LI value=15><I>x</I><SUP>3</SUP> + <I>y</I><SUP>3</SUP> <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/greater_equal.png">
<I>xy</I>(<I>x</I> + <I>y</I>)</LI></UL>
<P class=pLevel5><SPAN class=Normal>Let <I>x</I> = <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/018_006.png">
and <I>y</I> = <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/018_007.png"></SPAN>
<SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN><SPAN class=Normal></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel5><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/018_008.png"></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel5><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/018_009.png"></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<UL class=oLevel3>
<LI value=16>Let <I>a</I> ≠ <I>b</I> ≠ <I>c</I> Then,</LI></UL>
<P class=pLevel5><SPAN class=Normal><I>a</I><SUP>2</SUP> + <I>b</I><SUP>2</SUP>
> 2<I>ab</I></SPAN></P>
<P class=pLevel5><SPAN class=Normal><I>b</I><SUP>2</SUP> + <I>c</I><SUP>2</SUP>
> 2<I>bc</I></SPAN></P>
<P class=pLevel5><SPAN class=Normal><I>c</I><SUP>2</SUP> + <I>a</I><SUP>2</SUP>
> 2<I>ac</I></SPAN></P>
<P class=pLevel5><SPAN class=Normal>Adding,</SPAN></P>
<P class=pLevel5><SPAN class=Normal>2(<I>a</I><SUP>2</SUP> +
<I>b</I><SUP>2</SUP> + <I>c</I><SUP>2</SUP>) > 2(<I>ab</I> + <I>bc</I> +
<I>ac</I>)</SPAN></P>
<P class=pLevel5><SPAN class=Normal><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
<I>a</I><SUP>2</SUP> + <I>b</I><SUP>2</SUP> + <I>c</I><SUP>2</SUP> >
(<I>ab</I> + <I>bc</I> + <I>ac</I>)</SPAN></P>
<UL class=oLevel3>
<LI value=17>(<I>a</I> + <I>b</I>)(<I>b</I> + <I>c</I>)(<I>a</I> + <I>c</I>) =
2<I>abc</I> + <I>b</I>(<I>a</I><SUP>2</SUP> + <I>c</I><SUP>2</SUP>) +
<I>a</I>(<I>b</I><SUP>2</SUP> + <I>c</I><SUP>2</SUP>) +
<I>c</I>(<I>b</I><SUP>2</SUP> + <I>a</I><SUP>2</SUP>)</LI></UL>
<P class=pLevel5><SPAN class=Normal>Also, <I>a</I><SUP>2</SUP> +
<I>c</I><SUP>2</SUP> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/greater_equal.png">
2<I>ac</I>,</SPAN></P>
<P class=pLevel5><SPAN class=Normal><I>a</I><SUP>2</SUP> + <I>b</I><SUP>2</SUP>
<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/greater_equal.png">
2<I>ab</I> and</SPAN></P>
<P class=pLevel5><SPAN class=Normal><I>b</I><SUP>2</SUP> + <I>c</I><SUP>2</SUP>
<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/greater_equal.png">
2<I>bc</I></SPAN></P>
<P class=pLevel5><SPAN class=Normal><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
If <I>a</I>, <I>b</I> and <I>c</I> are positive,</SPAN></P>
<P class=pLevel5><SPAN class=Normal><I>b</I>(<I>a</I><SUP>2</SUP> +
<I>c</I><SUP>2</SUP>) <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/greater_equal.png">
2<I>abc</I>,</SPAN></P>
<P class=pLevel5><SPAN class=Normal><I>c</I>(<I>a</I><SUP>2</SUP> +
<I>b</I><SUP>2</SUP>) <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/greater_equal.png">
2<I>abc</I> and</SPAN></P>
<P class=pLevel5><SPAN class=Normal><I>a</I>(<I>b</I><SUP>2</SUP> +
<I>c</I><SUP>2</SUP>) <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/greater_equal.png">
2<I>abc</I></SPAN></P>
<P class=pLevel5><SPAN class=Normal><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">2<I>abc</I>
+ <I>b</I>(<I>a</I><SUP>2</SUP> + <I>c</I><SUP>2</SUP>) +
<I>a</I>(<I>b</I><SUP>2</SUP> + <I>c</I><SUP>2</SUP>) +
<I>c</I>(<I>b</I><SUP>2</SUP> + <I>a</I><SUP>2</SUP>)</SPAN></P>
<P class=pLevel5><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/greater_equal.png">
2<I>abc</I> + 2<I>abc</I> + 2<I>abc</I> + 2<I>abc</I></SPAN></P>
<P class=pLevel5><SPAN class=Normal><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
(<I>a</I> + <I>b</I>)(<I>b</I> + <I>c</I>)(<I>a</I> + <I>c</I>) <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/greater_equal.png">
8<I>abc</I>, if <I>a</I>, <I>b</I> and <I>c</I> are positive.</SPAN></P>
<UL class=oLevel3>
<LI value=1>The arithmetic mean of 1, 2, 3, …, <I>n</I> is <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/018_010.png"></LI></UL>
<P class=pLevel5><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/018_011.png"></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel5><SPAN class=Normal>Since Arithmetic Mean <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/greater_equal.png">
Geometric Mean,</SPAN></P>
<P class=pLevel5><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/018_012.png"></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel5><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/018_013.png"></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel5><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/018_014.png"></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel5><SPAN class=Normal>(<I>n </I>+ 1)<I><SUP>n</SUP> </I><IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/greater_equal.png"><I>
</I>2<I><SUP>n</SUP> </I>(<I>n</I>!)</SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<UL class=oLevel3>
<LI value=2>The arithmetic mean of 2, 4, 6, … , 2<I>n</I> is <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/018_015.png"></LI></UL>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel5><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/018_016.png"></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel5><SPAN class=Normal>Since Arithmetic Mean <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/greater_equal.png">
Geometric Mean,</SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel5><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/018_017.png"></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel5><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/018_018.png">
<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/018_019.png"></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel5><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/018_020.png"></SPAN></P>
<P class=pLevel5><SPAN class=Normal></SPAN></P>
<P class=pLevel5><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/018_021.png"></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel5><SPAN class=Normal>2 <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/multiplication.png">
4 <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/multiplication.png">
6 <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/multiplication.png">
… <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/multiplication.png">
2<I>n</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/less_equal.png">
(<I>n</I> + 1)<I><SUP>n</I></SUP></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<UL class=oLevel3>
<LI value=3>Using the binomial theorem, <IMG style="VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/018_022.png"></LI></UL>
<UL class="oLevel3 ">
<LI value=4>If <I>x </I>and<I> y</I> are positive and<I> x</I> > <I>y</I>,
then log <I>x</I> > log <I>y</I></LI></UL>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel2">
<DIV id=018_solvedex_1>
<TABLE style="WIDTH: 100%" id=018_solvedex_1_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_1>Example
1:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>A = <I>x</I><SUP>2</SUP>(2<I>x</I> + 1) +
<I>y</I><SUP>2</SUP>(2<I>y</I> + 1)</SPAN></P>
<P><SPAN class=Normal><I>B</I> = 2<I>xy</I>(<I>x</I> + <I>y</I> + 1) where
<I>x</I> and <I>y</I> are distinct positive numbers.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Is <I>A</I> greater than <I>B</I>?</SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal><I>A</I> = 2<I>x</I><SUP>3</SUP> +
<I>x</I><SUP>2</SUP> + 2<I>y</I><SUP>3</SUP> +<I>
y</I><SUP>2</SUP></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
<I>A</I> = 2(<I>x</I><SUP>3</SUP> + <I>y</I><SUP>3</SUP>) +
(<I>x</I><SUP>2</SUP> + <I>y</I><SUP>2</SUP>)</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
<I>A</I>/2 = <I>x</I><SUP>3</SUP> + <I>y</I><SUP>3</SUP> +
(<I>x</I><SUP>2</SUP> + <I>y</I><SUP>2</SUP>)/2</SPAN></P>
<P><SPAN class=Normal><I>x</I><SUP>3</SUP> + <I>y</I><SUP>3</SUP> >
<I>xy</I>(<I>x</I> + <I>y</I>) and (<I>x</I><SUP>2</SUP> +
<I>y</I><SUP>2</SUP>)/2 > <I>xy</I></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
<I>A</I>/2 > <I>xy</I>(<I>x</I> + <I>y</I>) + <I>xy</I></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
<I>A</I> > 2<I>xy</I>(<I>x </I>+ <I>y</I> + 1)</SPAN></P>
<P><SPAN class=Normal>Hence, <I>A</I> is greater than
<I>B</I>.</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel2">
<DIV id=018_solvedex_2>
<TABLE style="WIDTH: 100%" id=018_solvedex_2_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_2>Example
2:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>The sum of the squares of two numbers <I>m</I> and
<I>n</I> is 5 and the sum of the squares of other two numbers <I>p</I> and
<I>q</I> is 1. What can be said about the possible range of values of the
sum (<I>mp</I> + <I>nq</I>) if all the four numbers are
unequal?</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal><I>m</I><SUP>2</SUP> + <I>n</I><SUP>2</SUP> = 5 and
<I>p</I><SUP>2</SUP> + <I>q</I><SUP>2</SUP> = 1</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
<I>m</I><SUP>2</SUP> + <I>n</I><SUP>2</SUP> + <I>p</I><SUP>2</SUP> +
<I>q</I><SUP>2</SUP> = 6</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
<I>m</I><SUP>2</SUP> + <I>p</I><SUP>2</SUP> + <I>n</I><SUP>2</SUP> +
<I>q</I><SUP>2</SUP> = 6</SPAN></P>
<P><SPAN class=Normal>As arithmetic mean > geometric mean for unequal
numbers,</SPAN></P>
<P><SPAN class=Normal><I>m</I><SUP>2</SUP> + <I>p</I><SUP>2</SUP> >
2<I>mp</I> and <I>n</I><SUP>2</SUP> + <I>q</I><SUP>2</SUP> >
2<I>nq</I></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Thus, <I>m</I><SUP>2</SUP> + <I>p</I><SUP>2</SUP> +
<I>n</I><SUP>2</SUP> + <I>q</I><SUP>2</SUP> > 2<I>mp</I> +
2<I>nq</I></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
2 (<I>mp</I> + <I>nq</I>) < 6</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
<I>mp</I> + <I>nq</I> < 3</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel2">
<DIV id=018_solvedex_3>
<TABLE style="WIDTH: 100%" id=018_solvedex_3_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_3>Example
3:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/018_023.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P><SPAN class=Normal><IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/018_024.png"></SPAN>
<P></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/018_025.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Since Arithmetic Mean <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/greater_equal.png">
Geometric Mean,</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/018_026.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/018_027.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
<I>x</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/greater_equal.png">
2</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal><B>Maximizing the product of <I>n</I>
numbers when the sum is known.</B></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>Consider any two numbers <I>a</I> and
<I>b</I>.</SPAN></P>
<P class=pLevel2><SPAN class=Normal>Let <I>S</I> = <I>a</I> + <I>b</I> and
<I>P</I> = <I>ab</I></SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>(<I>a</I> + <I>b</I>)<SUP>2</SUP> –
(<I>a</I> – <I>b</I>)<SUP>2</SUP> = 4<I>ab</I></SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
<I>S</I><SUP>2</SUP> – (<I>a</I> – <I>b</I>)<SUP>2</SUP> = 4<I>P</I></SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
<I>P</I> will be maximum when (<I>a</I> – <I>b</I>)<SUP>2</SUP> = 0</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
The product of two numbers, when their sum is known, is maximum when they are
equal.</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>Also, <I>S</I><SUP>2 </SUP>= 4<I>P</I> +
(<I>a</I> – <I>b</I>)<SUP>2</SUP></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
<I>S</I> is minimum when (<I>a</I> – <I>b</I>)<SUP>2</SUP> = 0</SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
The sum of two numbers, when their product is known, is minimum when they are
equal.</SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>Now consider <I>n</I> numbers <I>a</I>,
<I>b</I>, <I>c</I>, <I>d</I>, …, <I>k</I>, such that <I>a</I> + <I>b</I> +
<I>c</I> + <I>d</I> + … + <I>k</I> = <I>S</I></SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>As stated earlier, the product of two
numbers with a fixed sum is maximum when the two numbers are made
equal.</SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>Consider <I>a</I> and <I>b</I>. The product
<I>ab</I> will be maximised, while keeping their sum constant, when <I>a</I> and
<I>b</I> are both made equal to (<I>a</I> + <I>b</I>)/2.</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>Thus, the product of <I>n</I> numbers
<I>a</I>, <I>b</I>, <I>c</I>, <I>d</I>, …, <I>k</I> is maximum when each of the
<I>n</I> numbers is made equal to <I>S</I>/<I>n</I>.</SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/018_028.png"></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/018_029.png"></SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>This result also states that the arithmetic
mean of <I>n</I> quantities is greater than their geometric mean when the
<I>n</I> quantities are unequal. The arithmetic mean of <I>n</I> quantities is
equal to their geometric mean when the <I>n</I> quantities are equal.</SPAN></P>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel2">
<DIV id=018_solvedex_4>
<TABLE style="WIDTH: 100%" id=018_solvedex_4_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_4>Example
4:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>The sum of three numbers is 42. Find the maximum
value that the product of these three numbers can have.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal><I>x</I> + <I>y</I> + <I>z</I> = 42</SPAN></P>
<P><SPAN class=Normal>The product of the three numbers is maximum when
<I>x</I> = <I>y</I> = <I>z </I>= 42/3 = 14</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>The maximum value of the product = 14 <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/multiplication.png">
14 <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/multiplication.png">
14 = 2744</SPAN></P>
<P><SPAN class=Normal></SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel2">
<DIV id=018_solvedex_5>
<TABLE style="WIDTH: 100%" id=018_solvedex_5_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_5>Example
5:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>If the positive real numbers <I>a</I>, <I>b</I> and
<I>c</I> are in Arithmetic Progression, such that <I>abc</I> = 4, then
minimum possible value of <I>b</I> is:</SPAN></P>
<P><SPAN class=Normal><B>[IIFT 2008]</B></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/018_030.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/018_031.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>The arithmetic mean of <I>n</I> numbers is greater
than or equal to their geometric mean.</SPAN></P>
<P><SPAN class=Normal>As <I>a</I>, <I>b</I> and <I>c</I> are in Arithmetic
Progression, their arithmetic mean = <I>b</I></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/018_032.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/018_033.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/018_034.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/018_035.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Hence, <B>option
2.</B></SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel4><SPAN class=Normal></SPAN></P></DIV>
<DIV id=018_IV>
<TABLE id=018_IV_bookmark class=tableh1 border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD
style="PADDING-BOTTOM: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 100%; PADDING-RIGHT: 0px; PADDING-TOP: 0px">
<OL class="oLevel1 Heading1">
<LI value=4><A id=IV>SOLVING
INEQUALITIES</A></LI></OL></TD></TR></TBODY></TABLE>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>Solution of an inequality is the set of
values of the variable(s) that satisfy the inequality. The solution involves
reducing one side of the inequality to one single variable and algebraically
solving to find the required range of values.Solutions of inequalities are
represented using two kinds of notations.</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>The first one uses <, >, <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/less_equal.png">
and <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/greater_equal.png">
signs to denote the range of values that a variable can take. For example,
<I>a</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/less_equal.png">
<I>x</I> < <I>y</I>.</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>The second one uses brackets to denote the
range. A round bracket on either or both the sides denotes that the number on
that side is not included in the range. A square bracket on either or both the
sides indicates that the number on that side is included in the range. Thus,
<I>a</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/less_equal.png">
<I>x</I> < <I>y </I>is same as [<I>a</I>, <I>y</I>).</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>For example,</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>If <I>x</I> lies between 5 and 8, the
solution can be written as 5 < <I>x</I> < 8 or (5, 8).</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>5 <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/less_equal.png">
<I>x</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/less_equal.png">
8 can also be written as [5, 8].</SPAN></P>
<P class=pLevel2><SPAN class=Normal>5 <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/less_equal.png">
<I>x</I> < 8 can be written as [5, 8).</SPAN></P>
<P class=pLevel2><SPAN class=Normal>5 < <I>x</I> <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/less_equal.png">
8 can be written as (5, 8].</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel2">
<DIV id=018_solvedex_6>
<TABLE style="WIDTH: 100%" id=018_solvedex_6_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_6>Example
6:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>Find the range of solutions for <I>x</I> for the
inequation 7<I>x</I> + 6 < 18.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>7<I>x</I> + 6 < 18</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
7<I>x</I> < 12</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
<I>x</I> < 12/7</SPAN></P>
<P><SPAN class=Normal>All values of <I>x</I> less than 12/7 will satisfy
the given inequality.</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel2">
<DIV id=018_solvedex_7>
<TABLE style="WIDTH: 100%" id=018_solvedex_7_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_7>Example
7:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>For which values of <I>x</I> is 7<I>x</I> + 9 <
8<I>x</I> – 19?</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>7<I>x</I> + 9 < 8<I>x</I> <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">19</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png"><I>x</I>
< <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">28</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
<I>x</I> > 28</SPAN></P>
<P><SPAN class=Normal>(Since we are dividing both sides of the inequation
by a negative quantity, the sense of the inequality will
change.)</SPAN></P>
<P><SPAN class=Normal>Here all values of <I>x</I> greater than 28 will
satisfy the given inequality.</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel2">
<DIV id=018_solvedex_8>
<TABLE style="WIDTH: 100%" id=018_solvedex_8_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_8>Example
8:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>If <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">7
< <I>x</I> < 18 and 9 < <I>y</I> < 20, then find the range of
<I>x</I> + <I>y</I>.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>We can add two inequalities of the same type. Hence,
to find the range of <I>x</I> + <I>y</I>, we add the extreme values of
both the inequalities.</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
The range of <I>x</I> + <I>y</I> is given by, 2 < <I>x</I> + <I>y</I>
< 38</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel2">
<DIV id=018_solvedex_9>
<TABLE style="WIDTH: 100%" id=018_solvedex_9_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_9>Example
9:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>If <I>x</I> > 2 and <I>y</I> > – 1, Then which
of the following statements is necessarily true?</SPAN></P>
<P><SPAN class=Normal><B>[CAT 2000]</B></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>(1) <I>xy</I> > –2 <SPAN
style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN>(2) –<I>x</I> < 2<I>y</I></SPAN></P>
<P><SPAN class=Normal>(3) <I>xy</I> < –2 <SPAN
style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN>(4) –<I>x</I> > 2<I>y</I></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal><I>y</I> > –1</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
–2<I>y</I> < 2 < <I>x </I></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
–<I>x</I> < 2<I>y</I></SPAN></P>
<P><SPAN class=Normal>Hence, <B>option
2.</B></SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel2">
<DIV id=018_solvedex_10>
<TABLE style="WIDTH: 100%" id=018_solvedex_10_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_10>Example
10:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>If 32 < <I>x</I> < 54 and <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">21
< <I>y</I> < 78, then find the range of <I>x</I> <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">
<I>y</I>.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>Since we cannot subtract one inequality from the
other, we find the range of <I>x</I> + (<IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png"><I>y</I>).</SPAN></P>
<P><SPAN class=Normal>To find the range of <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png"><I>y</I>,
multiply the inequality by <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">1
and reverse the inequality.</SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">78
< <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png"><I>y</I>
< 21</SPAN></P>
<P><SPAN class=Normal>So to find the range of <I>x</I> <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">
<I>y</I>, add the two inequalities, 32 < <I>x</I> < 54 and <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">78
< <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png"><I>y</I>
< 21</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">46
< <I>x</I> + (<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png"><I>y</I>)
< 75 or <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">46
< <I>x</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">
<I>y</I> < 75</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal><B>Solving a System of
Inequalities</SPAN></P>
<P class=pLevel2><SPAN class=Normal></B></SPAN></P>
<P class=pLevel2><SPAN class=Normal>A number of inequalities in one unknown,
which can be solved to find a common set of values that satisfy all
inequalities, are called a <B>System of Inequalities</B>.</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel2">
<DIV id=018_solvedex_11>
<TABLE style="WIDTH: 100%" id=018_solvedex_11_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_11>Example
11:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>Find the range of values of <I>x</I> that satisfy
the following in equations.</SPAN></P>
<P><SPAN class=Normal>–17 <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/less_equal.png">
3<I>x</I> + 10 <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/less_equal.png">
–2</SPAN></P>
<P><SPAN class=Normal>–22 <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/less_equal.png">
5<I>x</I> + 13 <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/less_equal.png">
3</SPAN></P>
<P><SPAN class=Normal>–19 <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/less_equal.png">
2<I>x</I> – 9 <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/less_equal.png">
–3</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>–17 <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/less_equal.png">
3<I>x</I> + 10 <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/less_equal.png">
–2</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
–27 <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/less_equal.png">
3<I>x</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/less_equal.png">
–12</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
–9 <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/less_equal.png">
<I>x</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/less_equal.png">
–4<SPAN style="MARGIN-LEFT: 25px"></SPAN>…(i)</SPAN></P>
<P><SPAN class=Normal>–22 <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/less_equal.png">
5<I>x</I> + 13 <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/less_equal.png">
3</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
–35 <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/less_equal.png">
5<I>x</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/less_equal.png">
–10</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
–7 <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/less_equal.png">
<I>x</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/less_equal.png">
–2<SPAN style="MARGIN-LEFT: 25px"></SPAN>…(ii)</SPAN></P>
<P><SPAN class=Normal>–19 <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/less_equal.png">
2<I>x</I> – 9 <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/less_equal.png">
–3</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
–10 <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/less_equal.png">
2<I>x</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/less_equal.png">
6</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
–5 <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/less_equal.png">
<I>x</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/less_equal.png">
3<SPAN style="MARGIN-LEFT: 30px"></SPAN>…(iii)</SPAN></P>
<P><SPAN class=Normal>The common range of values that satisfies the three
equations is [–5, –4]</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">5
<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/less_equal.png">
<I>x</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/less_equal.png">
<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">4</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel2">
<DIV id=018_solvedex_12>
<TABLE style="WIDTH: 100%" id=018_solvedex_12_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_12>Example
12:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal><I>x</I> and <I>y</I> are real numbers satisfying
the conditions 2 < <I>x</I> < 3 and –8 < <I>y</I> < –7. Which
of the following expressions will have the least value?</SPAN></P>
<P><SPAN class=Normal><B>[CAT 2001]</B></SPAN></P>
<P><SPAN class=Normal>(1) <I>x</I><SUP>2</SUP><I>y<SPAN
style="MARGIN-LEFT: 25px"></SPAN><SPAN style="MARGIN-LEFT: 25px"></SPAN>
</I>(2) <I>xy</I><SUP>2</SUP></SPAN></P>
<P><SPAN class=Normal>(3) 5<I>xy</I> <SPAN
style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 20px"></SPAN>(4) None of these</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>Option 2 can easily be eliminated as it will give
positive value, while both option 1 and 3 will be negative.</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/symb23.png">
<I>x</I><SUP>2</SUP> will fall in the range 4 < <I>x</I><SUP>2</SUP>
< 9.</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/symb23.png">
5<I>x</I> will fall between 10 < 5<I>x</I> < 15.</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
5<I>x</I> > <I>x</I><SUP>2</SUP></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/symb23.png">
Both <I>x</I><SUP>2</SUP><I>y </I>and 5<I>xy</I> are negative and
5<I>x</I> ><I> x</I><SUP>2</SUP></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
5<I>xy</I> represents the least value.</SPAN></P>
<P><SPAN class=Normal>Hence, <B>option
3.</B></SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel2">
<DIV id=018_solvedex_13>
<TABLE style="WIDTH: 100%" id=018_solvedex_13_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_13>Example
13:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>Given that <IMG style="VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">1
<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/less_equal.png">
<I>v</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/less_equal.png">
1, <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">2
<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/less_equal.png">
<I>u</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/less_equal.png">
<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">0.5
and <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">2
<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/less_equal.png">
<I>z</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/less_equal.png">
<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">0.5
and <I>w</I> = <I>vz</I>/<I>u</I>, then which of the following is
necessarily true?</SPAN></P>
<P><SPAN class=Normal><B>[CAT 2003 Leaked Test]</B></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>(1) <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">0.5
<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/less_equal.png">
<I>w</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/less_equal.png">
2<SPAN style="MARGIN-LEFT: 25px"></SPAN>(2) <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">4
<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/less_equal.png">
<I>w</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/less_equal.png">
4</SPAN></P>
<P><SPAN class=Normal>(3) <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">4
<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/less_equal.png">
<I>w</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/less_equal.png">
2<SPAN style="MARGIN-LEFT: 33px"></SPAN> (4) <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">2
<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/less_equal.png">
<I>w</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/less_equal.png">
<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">0.5</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal><I>w</I> = <I>vz</I>/<I>u</I></SPAN></P>
<P><SPAN class=Normal>From the given range of values for <I>u</I>,
<I>v</I> and <I>z</I>, we have,</SPAN></P>
<P><SPAN class=Normal>Maximum possible value of <I>w</I> is 4 when
<I>v</I> is 1, <I>z</I> is <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">2
and <I>u</I> is <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">0.5.</SPAN></P>
<P><SPAN class=Normal>Also minimum value of <I>w</I> is <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">4
when <I>v</I> is <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">1,
<I>z</I> is <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">2
and <I>u</I> is <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">0.5.</SPAN></P>
<P><SPAN class=Normal>Hence, <B>option
2.</B></SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel2><SPAN class=Normal></SPAN></P></DIV>
<DIV id=018_V>
<TABLE id=018_V_bookmark class=tableh1 border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD
style="PADDING-BOTTOM: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 100%; PADDING-RIGHT: 0px; PADDING-TOP: 0px">
<OL class="oLevel1 Heading1">
<LI value=5><A id=V>INEQUALITIES AND ABSOLUTE
VALUES</A></LI></OL></TD></TR></TBODY></TABLE>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>The <B>absolute value</B> or <B>modulus</B>
of a number <I>x</I> denoted by |<I>x</I>| is defined as its distance from zero
on the number line. Since distance cannot be negative, the absolute value of any
number is always positive.</SPAN></P>
<P class=pLevel2><SPAN class=Normal>Mathematically, the absolute value of
<I>x</I> is defined as,</SPAN></P>
<P class=pLevel2><SPAN class=Normal>|<I>x</I>| = <I>x</I>, if <I>x</I> <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/greater_equal.png"><U></U>
0</SPAN></P>
<P class=pLevel2><SPAN class=Normal>|<I>x</I>| = <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png"><I>x</I>,
if <I>x</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/less_equal.png">
0</SPAN></P>
<UL type=disc>
<LI class=pLevel2 value=5>If |<I>x</I>| < <I>p</I> then this means <I>x</I>
< <I>p</I> and <I>x</I> > <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png"><I>p</I>,
which can be written as <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png"><I>p</I>
< <I>x</I> < <I>p</I>, and can be stated as ‘<I>x</I> lies between
negative <I>p</I> and positive <I>p</I>’.</LI></SPAN></UL>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<UL type=disc>
<LI class=pLevel2>If |<I>x</I>| > <I>p</I>, then this can be interpreted as
<I>x</I> > <I>p</I> or <I>x</I> < <IMG style="VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png"><I>p</I>
which can also be stated as ‘<I>x</I> is either greater than positive <I>p</I>
or <I>x</I> is less than negative <I>p</I>’.</LI></UL>
<P class=pLevel4><SPAN class=ListParagraph></SPAN></P></DIV>
<DIV id=018_VI>
<TABLE id=018_VI_bookmark class=tableh1 border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD
style="PADDING-BOTTOM: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 100%; PADDING-RIGHT: 0px; PADDING-TOP: 0px">
<OL class="oLevel1 Heading1">
<LI value=6><A id=VI>PROPERTIES OF
MODULUS</A></LI></OL></TD></TR></TBODY></TABLE>
<UL type=disc>
<LI class=pLevel2 value=5>For a real number <I>x</I>, |<I>x</I>| = max
{<I>x</I>, <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png"><I>x</I>}</LI></UL>
<UL type=disc>
<LI class=pLevel2 value=6>For all real numbers <I>x</I>,
|<I>x</I>|<SUP>2</SUP> = <I>x</I><SUP>2</SUP> = |<IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png"><I>x</I>|<SUP>2</SUP></LI></UL>
<P class=pLevel4><SPAN class=Normal><SPAN
style="MARGIN-LEFT: 10px"></SPAN><B>Explanation</B></SPAN></P>
<P class=pLevel4><SPAN class=Normal><SPAN style="MARGIN-LEFT: 10px"></SPAN>By
definition,</SPAN></P>
<P class=pLevel4><SPAN class=Normal><SPAN
style="MARGIN-LEFT: 10px"></SPAN>|<I>x</I>| = <I>x</I>, if <I>x</I> <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/greater_equal.png">
0</SPAN></P>
<P class=pLevel4><SPAN class=Normal><SPAN style="MARGIN-LEFT: 10px"></SPAN><SPAN
style="MARGIN-LEFT: 23px"></SPAN>= <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png"><I>x</I>,
if <I>x</I> < 0</SPAN></P>
<P class=pLevel4><SPAN class=Normal><SPAN style="MARGIN-LEFT: 10px"></SPAN>In
either case,</SPAN></P>
<P class=pLevel4><SPAN class=Normal><SPAN
style="MARGIN-LEFT: 10px"></SPAN>|<I>x</I>|<SUP>2</SUP> =
<I>x</I><SUP>2</SUP></SPAN></P>
<P class=pLevel4><SPAN class=Normal><SPAN style="MARGIN-LEFT: 10px"></SPAN><IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
|<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png"><I>x</I>|<SUP>2</SUP>
= (<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png"><I>x</I>)<SUP>2</SUP>
= <I>x</I><SUP>2</SUP></SPAN></P>
<P class=pLevel4><SPAN class=Normal><SPAN
style="MARGIN-LEFT: 10px"></SPAN>Hence, |<I>x</I>|<SUP>2</SUP> =
<I>x</I><SUP>2</SUP> = |<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png"><I>x</I>|<SUP>2</SUP></SPAN></P>
<UL type="disc ">
<LI class=pLevel2 value=7>For any real number <I>x</I>, |<I>x</I>| =|<IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png"><I>x</I>|</LI></UL>
<P class=pLevel4><SPAN class=Normal><SPAN
style="MARGIN-LEFT: 10px"></SPAN><B>Explanation:</B></SPAN></P>
<P class=pLevel4><SPAN class=Normal><SPAN style="MARGIN-LEFT: 10px"></SPAN>|<IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png"><I>x</I>|
= max{<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png"><I>x</I>,
<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">
(<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png"><I>x</I>)}</SPAN></P>
<P class=pLevel4><SPAN class=Normal><SPAN style="MARGIN-LEFT: 10px"></SPAN><SPAN
style="MARGIN-LEFT: 33px"></SPAN>= max {<IMG style="VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png"><I>x</I>,
<I>x</I>}</SPAN></P>
<P class=pLevel4><SPAN class=Normal><SPAN style="MARGIN-LEFT: 10px"></SPAN><SPAN
style="MARGIN-LEFT: 33px"></SPAN>= max {<I>x</I>, <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png"><I>x</I>}</SPAN></P>
<P class=pLevel4><SPAN class=Normal><SPAN style="MARGIN-LEFT: 10px"></SPAN><SPAN
style="MARGIN-LEFT: 33px"></SPAN>= |<I>x</I>|</SPAN></P>
<UL type=disc>
<LI class=pLevel2 value=8>For real numbers, <I>x</I> and <I>y</I>, |<I>xy</I>|
= |<I>x</I>| |<I>y</I>|</LI></UL>
<P class=pLevel4><SPAN class=Normal><SPAN
style="MARGIN-LEFT: 10px"></SPAN><B>Explanation:</B></SPAN></P>
<P class=pLevel4><SPAN class=Normal><SPAN
style="MARGIN-LEFT: 10px"></SPAN>|<I>xy</I>|<SUP>2</SUP> =
(<I>xy</I>)<SUP>2</SUP></SPAN></P>
<P class=pLevel4><SPAN class=Normal><SPAN style="MARGIN-LEFT: 10px"></SPAN><SPAN
style="MARGIN-LEFT: 33px"></SPAN>= <I>x</I><SUP>2</SUP>
<I>y</I><SUP>2</SUP></SPAN></P>
<P class=pLevel4><SPAN class=Normal><SPAN style="MARGIN-LEFT: 10px"></SPAN><SPAN
style="MARGIN-LEFT: 33px"></SPAN>= (|<I>x</I>|)<SUP>2</SUP>
(|<I>y</I>|)<SUP>2</SUP></SPAN></P>
<P class=pLevel4><SPAN class=Normal><SPAN style="MARGIN-LEFT: 10px"></SPAN><SPAN
style="MARGIN-LEFT: 33px"></SPAN>= (|<I>x</I>|
|<I>y</I>|)<SUP>2</SUP></SPAN></P>
<P class=pLevel4><SPAN class=Normal><SPAN style="MARGIN-LEFT: 10px"></SPAN>As,
|<I>xy</I>| and |<I>x</I>||<I>y</I>| are both non negative, therefore, taking
the positive square root on both sides we have, |<I>xy</I>| = |<I>x</I>|
|<I>y</I>|</SPAN></P>
<UL type=disc>
<LI class=pLevel2 value=9><B>Triangle Inequality</B></LI></UL>
<P class=pLevel4><SPAN class=Normal><SPAN style="MARGIN-LEFT: 10px"></SPAN>For
all real numbers <I>x</I> and <I>y</I>,</SPAN></P>
<P class=pLevel4><SPAN class=Normal><SPAN
style="MARGIN-LEFT: 10px"></SPAN>|<I>x</I> + <I>y</I>| <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/less_equal.png">
|<I>x</I>| + |<I>y</I>|</SPAN></P>
<UL type=disc>
<LI class=pLevel2 value=10>For all real numbers <I>x</I> and <I>y</I>,</LI></UL>
<P class=pLevel4><SPAN class=Normal><SPAN
style="MARGIN-LEFT: 10px"></SPAN>|<I>x</I> <IMG style="VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">
<I>y</I>| <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/less_equal.png">
||<I>x</I>| <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">
|<I>y</I>||</SPAN></P>
<P class=pLevel5><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel2">
<DIV id=018_solvedex_14>
<TABLE style="WIDTH: 100%" id=018_solvedex_14_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_14>Example
14:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>Find the value of <I>x</I>, if |<I>x</I> <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">
7| < 18.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>|<I>x</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">
7| < 18</SPAN></P>
<P><SPAN class=Normal>This can be written as <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">18
< <I>x</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">
7 < 18</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">11
< <I>x</I> < 25</SPAN></P>
<P><SPAN class=Normal>This means all values of <I>x</I> between <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">11
and 25 satisfy the given
inequality.</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel2">
<DIV id=018_solvedex_15>
<TABLE style="WIDTH: 100%" id=018_solvedex_15_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_15>Example
15:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>Find the value of <I>x</I>, if |6<I>x</I> <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">
14| > 12.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>|6<I>x</I> <IMG style="VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">
14| > 12</SPAN></P>
<P><SPAN class=Normal>This can be written as (6<I>x</I> <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">
14) > 12 or (6<I>x</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">
14) < <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">12</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
6<I>x</I> > 26 or 6<I>x</I> < 2</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
<I>x</I> > 13/3 or <I>x</I> < 1/3</SPAN></P>
<P><SPAN class=Normal>This means that all values of <I>x</I>, except
values between and including 13/3 and 1/3, will satisfy the
inequality.</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel2">
<DIV id=018_solvedex_16>
<TABLE style="WIDTH: 100%" id=018_solvedex_16_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_16>Example
16:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>If |<I>b</I>| <IMG style="VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/greater_equal.png">
1 and <I>x</I> = –|<I>a</I>|<I>b</I>, then which one of the following is
necessarily true?</SPAN></P>
<P><SPAN class=Normal><B>[CAT 2003 Re-Test]</B></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>(1)<I> a</I> – <I>xb</I> < 0<SPAN
style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN>(2) <I>a</I> – <I>xb</I> <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/greater_equal.png">
0</SPAN></P>
<P><SPAN class=Normal>(3)<I> a</I> – <I>xb</I> > 0<SPAN
style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN>(4)<I> a</I> – <I>xb</I> <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/less_equal.png">
0</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>|<I>b</I>| <IMG style="VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/greater_equal.png">
1 and <I>x</I> = –|<I>a</I>|<I>b</I></SPAN></P>
<P><SPAN class=Normal>Consider the case when <I>b</I> is
negative:</SPAN></P>
<P><SPAN class=Normal>i.e. say <I>b</I> = –<I>k</I>, where <I>k</I> <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/greater_equal.png">
1</SPAN></P>
<P><SPAN class=Normal>Then, <I>x</I> = –|<I>a</I>|<I>b</I> = –|<I>a</I>|
<IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/multiplication.png">
(–<I>k</I>) = |<I>a</I>|<I>k</I></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
<I>xb</I> = –|<I>a</I>|<I>k</I><SUP>2</SUP></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
<I>a</I> – <I>xb</I> = <I>a</I> +
|<I>a</I>|<I>k</I><SUP>2</SUP></SPAN></P>
<P><SPAN class=Normal>Now,</SPAN></P>
<P><SPAN class=Normal>If <I>a</I> > 0, then <I>a</I> – <I>xb</I> =
<I>a</I> + |<I>a</I>|<I>k</I><SUP>2</SUP> > 0 since all the terms will
be positive</SPAN></P>
<P><SPAN class=Normal>If <I>a</I> < 0 (say <I>a</I> = –2), then
<I>a</I> – <I>xb</I> = –2 + 2<I>k</I><SUP>2</SUP> <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/greater_equal.png">
0, since 2<I>k</I><SUP>2</SUP> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/greater_equal.png">
2 as <I>k</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/greater_equal.png">
1</SPAN></P>
<P><SPAN class=Normal>However, if <I>a</I> = 0, then <I>a</I> – <I>xb</I>
= 0 + 0 = 0</SPAN></P>
<P><SPAN class=Normal>Hence, when <I>b</I> is negative, <I>a</I> –
<I>xb</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/greater_equal.png">
0</SPAN></P>
<P><SPAN class=Normal>Now, consider the case when <I>b</I> is
positive:</SPAN></P>
<P><SPAN class=Normal>i.e. say <I>b</I> = +<I>k</I>, where <I>k</I> <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/greater_equal.png">
1</SPAN></P>
<P><SPAN class=Normal>Then, <I>x</I> = –|<I>a</I>|<I>b</I> = –|<I>a</I>|
<IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/multiplication.png">
(<I>k</I>) = –|<I>a</I>|<I>k</I></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
<I>xb</I> = –|<I>a</I>|<I>k</I><SUP>2</SUP></SPAN></P>
<P><SPAN class=Normal>This is the same value of <I>xb</I> as we got in the
previous case. Hence, the same conclusions will hold.</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
For all cases, <I>a</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">
<I>xb </I><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/greater_equal.png">
0</SPAN></P>
<P><SPAN class=Normal>Hence, <B>option
2.</B></SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel4><SPAN class=Normal></SPAN></P></DIV>
<DIV id=018_VII>
<TABLE id=018_VII_bookmark class=tableh1 border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD
style="PADDING-BOTTOM: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 100%; PADDING-RIGHT: 0px; PADDING-TOP: 0px">
<OL class="oLevel1 Heading1">
<LI value=7><A id=VII>QUADRATIC
INEQUALITIES</A></LI></OL></TD></TR></TBODY></TABLE>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>A <B>quadratic inequality</B> is an
inequality of the form <I>ax</I><SUP>2</SUP> + <I>bx</I> + <I>c</I> < 0 or
<I>ax</I><SUP>2</SUP> + <I>bx</I> + <I>c</I> > 0 or <I>ax</I><SUP>2</SUP> +
<I>bx</I> + <I>c</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/greater_equal.png">
0 or <I>ax</I><SUP>2</SUP> + <I>bx</I> + <I>c</I> <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/less_equal.png">
0.</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>If <I>y</I> = <I>ax</I><SUP>2</SUP> +
<I>bx</I> + <I>c</I> where <I>a</I>, <I>b</I> and <I>c</I> are real and <I>a</I>
≠ 0, then <I>y</I> represents a parabola whose axis is parallel to the
<I>y</I>–axis. Let <I><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/alpha.png"></I>
and <I><IMG style="PADDING-BOTTOM: 2px; VERTICAL-ALIGN: bottom" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/beta.png"></I>
be the two roots of the equation where <I><IMG style="VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/alpha.png"></I>
< <I><IMG style="PADDING-BOTTOM: 2px; VERTICAL-ALIGN: bottom" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/beta.png"></I>.
<I>y</I> may be positive, negative or zero depending on the value of <I>x</I>.
If <I>a</I> > 0, then the parabola opens upwards and if <I>a</I> < 0, the
parabola opens downwards. The parabola intersects the <I>x</I>-axis at <I><IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/alpha.png"></I>
and <I><IMG style="PADDING-BOTTOM: 2px; VERTICAL-ALIGN: bottom" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/beta.png"></I>,
if <I><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/alpha.png"></I>
and <I><IMG style="PADDING-BOTTOM: 2px; VERTICAL-ALIGN: bottom" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/beta.png"></I>
are real. If <I><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/alpha.png"></I>
and <I><IMG style="PADDING-BOTTOM: 2px; VERTICAL-ALIGN: bottom" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/beta.png">
</I>are imaginary, the parabola does not intersect the <I>x</I>-axis.</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>For a quadratic expression of the
form</SPAN></P>
<P class=pLevel2><SPAN class=Normal><I>ax</I><SUP>2</SUP> + <I>bx</I> + <I>c</I>
= 0, the discriminant is given by</SPAN></P>
<P class=pLevel2><SPAN class=Normal>∆ = <I>b</I><SUP>2</SUP> <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">
4<I>ac</I>.</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>The nature of the roots of the quadratic
equation depends on the value of the discriminant.</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>Consider the following cases.</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal><B>Case 1:</B> ∆ < 0</SPAN></P>
<P class=pLevel2><SPAN class=Normal>Roots of <I>ax</I><SUP>2</SUP> + <I>bx</I> +
<I>c</I> = 0 are imaginary, when ∆ < 0.</SPAN></P>
<P class=pImage><SPAN><IMG alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/018_036.png"></SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>Thus the value of <I>y</I> =
<I>ax</I><SUP>2</SUP> + <I>bx</I> + <I>c</I> has the same sign as <I>a</I> when
∆ < 0.</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal><B>Case 2:</B> ∆ = 0</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>Roots of <I>ax</I><SUP>2</SUP> + <I>bx</I> +
<I>c</I> = 0 are real and equal when ∆ = 0.</SPAN></P>
<P class=pImage><SPAN><IMG alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/018_037.png"></SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>Thus when ∆ = 0, the value of <I>y</I> =
<I>ax</I><SUP>2</SUP> + <I>bx</I> + <I>c</I> has the same sign as <I>a</I>,
except when <I>x</I> = <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png"><I>b</I>/2<I>a</I>.</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal><B>Case 3:</B> ∆ > 0</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>Roots of <I>ax</I><SUP>2</SUP> + <I>bx</I> +
<I>c</I> = 0 are real and distinct when ∆ > 0.</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pImage><SPAN><IMG alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/018_038.png"></SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>Thus, if <I><IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/alpha.png"></I>
and <I><IMG style="PADDING-BOTTOM: 2px; VERTICAL-ALIGN: bottom" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/beta.png"></I>
are the two roots of the equation <I>ax</I><SUP>2</SUP> + <I>bx</I> + <I>c</I> =
0 then,</SPAN></P>
<P class=pLevel2><SPAN class=Normal>if <I>a</I> > 0, <I>ax</I><SUP>2</SUP> +
<I>bx</I> + <I>c</I> is negative in the interval (<I><IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/alpha.png"></I>,
<I><IMG style="PADDING-BOTTOM: 2px; VERTICAL-ALIGN: bottom" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/beta.png"></I>),
zero at <I>x</I> = <I><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/alpha.png"></I>
and <I>x </I>= <I><IMG style="PADDING-BOTTOM: 2px; VERTICAL-ALIGN: bottom"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/beta.png"></I>,
and positive otherwise;</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>if <I>a</I> < 0, <I>ax</I><SUP>2</SUP> +
<I>bx</I> + <I>c </I>is positive in the interval (<I><IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/alpha.png"></I>,
<I><IMG style="PADDING-BOTTOM: 2px; VERTICAL-ALIGN: bottom" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/beta.png"></I>),
zero at <I>x</I> = <I><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/alpha.png"></I>
and <I>x </I>= <I><IMG style="PADDING-BOTTOM: 2px; VERTICAL-ALIGN: bottom"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/beta.png"></I>,
and negative otherwise.</SPAN></P>
<DIV class="divTextBox divColorNote divLevel2">
<P><SPAN class=Note>REMEMBER</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<UL type=disc>
<LI value=2>If a quadratic equation <I>ax</I><SUP>2</SUP> + <I>bx</I> +
<I>c</I> = 0; <I>a</I> > 0 has real roots <I><IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/alpha.png"></I>
and <I><IMG style="PADDING-BOTTOM: 2px; VERTICAL-ALIGN: bottom" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/beta.png"></I>
such that <I><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/alpha.png"></I>
< <I><IMG style="PADDING-BOTTOM: 2px; VERTICAL-ALIGN: bottom" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/beta.png"></I>.</LI></UL>
<P><SPAN class=Normal></SPAN></P>
<UL type=disc>
<LI value=3>If <I>ax</I><SUP>2</SUP> + <I>bx</I> + <I>c</I> < 0, then
<I><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/alpha.png"></I>
< <I>x</I> < <I><IMG style="PADDING-BOTTOM: 2px; VERTICAL-ALIGN: bottom"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/beta.png"></I>,
i.e. <I>x</I> will lie between the roots.</LI></UL>
<P><SPAN class=Normal></SPAN></P>
<UL type=disc>
<LI value=4>If <I>ax</I><SUP>2</SUP> + <I>bx</I> + <I>c</I> > 0, then
<I>x</I> < <I><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/alpha.png"></I>
or <I>x</I> > <I><IMG style="PADDING-BOTTOM: 2px; VERTICAL-ALIGN: bottom"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/beta.png"></I>,
i.e. <I>x</I> will not lie between the roots.</LI></UL></DIV>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel2">
<DIV id=018_solvedex_17>
<TABLE style="WIDTH: 100%" id=018_solvedex_17_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_17>Example
17:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>Solve <I>x</I><SUP>2</SUP> + 8<I>x</I> + 15 <
0.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal><I>x</I><SUP>2</SUP> + 8<I>x</I> +15 <
0</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
<I>x</I><SUP>2</SUP> + 5<I>x</I> + 3<I>x</I> +15 < 0</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
(<I>x</I> + 5)(<I>x</I> + 3) < 0</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">5
< <I>x</I> < <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">3</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel2">
<DIV id=018_solvedex_18>
<TABLE style="WIDTH: 100%" id=018_solvedex_18_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_18>Example
18:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>Solve <I>x</I><SUP>2</SUP> + 10<I>x</I> <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">
119 > 0.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal><I>x</I><SUP>2</SUP> + 10<I>x</I> <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">119
> 0</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
<I>x</I><SUP>2</SUP> + 17<I>x</I> <IMG style="VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">
7<I>x</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">
119 > 0</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
(<I>x</I> + 17)(<I>x</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">
7) > 0</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
<I>x</I> < <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">17
or <I>x</I> > 7</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel2">
<DIV id=018_solvedex_19>
<TABLE style="WIDTH: 100%" id=018_solvedex_19_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_19>Example
19:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>Solve <I>x</I><SUP>2</SUP> – 7<I>x</I> + 15 >
0.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal><I>x</I><SUP>2</SUP> – 7<I>x</I> + 15 >
0</SPAN></P>
<P><SPAN class=Normal>Δ = 49 <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">
4(15)</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
Δ < 0</SPAN></P>
<P><SPAN class=Normal>Also, the coefficient of <I>x</I><SUP>2</SUP> is
positive.</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
The expression <I>x</I><SUP>2</SUP> – 7<I>x</I> + 15 is positive for all
values of <I>x</I>.</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel2">
<DIV id=018_solvedex_20>
<TABLE style="WIDTH: 100%" id=018_solvedex_20_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_20>Example
20:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>Solve <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">2<I>x</I><SUP>2</SUP>
+ 5<I>x</I> – 6 < 0.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">2<I>x</I><SUP>2</SUP>
+ 5<I>x</I> – 6 < 0</SPAN></P>
<P><SPAN class=Normal>Δ = 25 – 4(<IMG style="VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">6)(<IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">2)</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
Δ = 25 – 48 < 0</SPAN></P>
<P><SPAN class=Normal>Also, the coefficient of <I>x</I><SUP>2</SUP> is
negative.</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
The expression <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">2<I>x</I><SUP>2</SUP>
+ 5<I>x</I> – 6 is negative for all values of
<I>x</I>.</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel2">
<DIV id=018_solvedex_21>
<TABLE style="WIDTH: 100%" id=018_solvedex_21_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_21>Example
21:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal><I>x</I><SUP>2</SUP> – 12<I>x</I> + 72 <
0</SPAN></P>
<P><SPAN class=Normal>For what values of <I>x </I>does the inequality
hold?</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal><I>x</I><SUP>2</SUP> – 12<I>x</I> + 72 <
0</SPAN></P>
<P><SPAN class=Normal>Δ = 144 – 4(72) < 0</SPAN></P>
<P><SPAN class=Normal>Coefficient of <I>x</I><SUP>2</SUP> is
positive.</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
The expression <I>x</I><SUP>2</SUP> – 12<I>x</I> + 72 will be positive for
all values of <I>x</I>.</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
The inequality does not hold for any value of
<I>x.</I></SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel2">
<DIV id=018_solvedex_22>
<TABLE style="WIDTH: 100%" id=018_solvedex_22_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_22>Example
22:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/018_039.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/018_040.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>In this case we cannot cross multiply since the sign
of variable <I>m</I> is not known.</SPAN></P>
<P><SPAN class=Normal>So we rewrite it as,</SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/018_041.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/018_042.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/018_043.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Since the numerator is negative and the inequation
is positive, the denominator has to be negative.</SPAN></P>
<P><SPAN class=Normal><I>m</I> + 1 < 0</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
<I>m</I> < <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">1</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel2">
<DIV id=018_solvedex_23>
<TABLE style="WIDTH: 100%" id=018_solvedex_23_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_23>Example
23:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>Solve</SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/018_044.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/018_045.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/018_046.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/018_047.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Since LHS is negative, we have 2 cases.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><B>Case 1:</B></SPAN></P>
<P><SPAN class=Normal>When the numerator is negative and denominator is
positive,</SPAN></P>
<P><SPAN class=Normal>9<I>x</I> + 3 < 0 and (<I>x</I> + 1)(<I>x <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png"></I>
2) > 0</SPAN></P>
<P><SPAN class=Normal><I>x</I> < <IMG style="VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">1/3
and <I>x</I> > 2 or <I>x</I> < <IMG style="VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">1</SPAN></P>
<P><SPAN class=Normal>So <I>x</I> < <IMG style="VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">1</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><B>Case 2:</B></SPAN></P>
<P><SPAN class=Normal>When the numerator is positive and denominator is
negative,</SPAN></P>
<P><SPAN class=Normal>9<I>x</I> + 3 > 0 and (<I>x</I> + 1)(<I>x</I>
<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">
2) < 0</SPAN></P>
<P><SPAN class=Normal><I>x</I> > <IMG style="VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">1/3
and <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">1
< <I>x</I> < 2</SPAN></P>
<P><SPAN class=Normal>So, <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">1/3
< <I>x</I> < 2</SPAN></P>
<P><SPAN class=Normal>Hence, <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">1/3
< <I>x</I> < 2 or <I>x</I> < <IMG style="VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">1</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel2">
<DIV id=018_solvedex_24>
<TABLE style="WIDTH: 100%" id=018_solvedex_24_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_24>Example
24:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/018_048.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/018_049.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>The above expression can be negative in two
cases.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><B>Case 1:</B></SPAN></P>
<P><SPAN class=Normal>The numerator is negative and the denominator is
positive.</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
(<I>x</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">1)<SUP>2</SUP>(<I>x</I>
+ 2) < 0</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
(<I>x</I> + 2) < 0 (<IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/symb23.png">
(<I>x</I> – 1)<SUP>2</SUP> > 0)</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
<I>x</I> < <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">2<SPAN
style="MARGIN-LEFT: 40px"></SPAN>… (i)</SPAN></P>
<P><SPAN class=Normal>(<I>x</I> – 5) > 0</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
<I>x</I> > 5<SPAN style="MARGIN-LEFT: 55px"></SPAN>…(ii)</SPAN></P>
<P><SPAN class=Normal>(i) and (ii) are not possible
simultaneously.</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
Case 1 is not true.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><B>Case 2:</B></SPAN></P>
<P><SPAN class=Normal>The numerator is positive and the denominator is
negative.</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
(<I>x</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">1)<SUP>2</SUP>(<I>x</I>
+ 2) > 0</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
(<I>x</I> + 2) > 0 </SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
<I>x</I> > <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">2<SPAN
style="MARGIN-LEFT: 37px"></SPAN>… (iii)</SPAN></P>
<P><SPAN class=Normal>(<I>x</I> – 5) < 0</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
<I>x</I> < 5<SPAN style="MARGIN-LEFT: 53px"></SPAN>…(iv)</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">2
< <I>x</I> < 5</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel2">
<DIV id=018_solvedex_25>
<TABLE style="WIDTH: 100%" id=018_solvedex_25_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_25>Example
25:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>Solve the inequality (<I>p</I> –
4)<I>x</I><SUP>2</SUP> – 3<I>x</I> + 1 > 0.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>Δ = 9 – 4(1)(<I>p</I> – 4)</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
Δ = 25 – 4<I>p</I></SPAN></P>
<P><SPAN class=Normal>As the value of <I>p</I> is not known, we cannot
find the exact value of Δ.</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
We consider three cases.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><B>Case 1:</SPAN></P>
<P><SPAN class=Normal></B>Δ < 0</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
25 – 4<I>p</I> < 0</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
4<I>p</I> > 25</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
<I>p</I> > 25/4</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
<I>p</I> > 6.25</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
(<I>p</I> – 4) > 0</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
The inequality holds for all values of <I>x</I>.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><B>Case 2:</SPAN></P>
<P><SPAN class=Normal></B>Δ = 0</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
<I>p</I> = 6.25</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
(<I>p</I> – 4) > 0</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
The inequality holds for all values of <I>x</I>, except when <I>x</I> is
the root of the equation</SPAN></P>
<P><SPAN class=Normal>i.e. at <I>x </I>= 3/2(<I>p</I> – 4)</SPAN></P>
<P><SPAN class=Normal>i.e. at <I>x</I> = 2/3</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><B>Case 3:</SPAN></P>
<P><SPAN class=Normal></B>Δ > 0</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
<I>p</I> < 25/4</SPAN></P>
<P><SPAN class=Normal>We find the roots of the equation at <I>p</I> <
25/4 and <I>p</I> ≠ 4.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/018_050.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>If (<I>p</I> – 4) > 0, the inequality is true
when <I>x</I> < <I><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/alpha.png"></I>
and <I>x > <IMG style="PADDING-BOTTOM: 2px; VERTICAL-ALIGN: bottom"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/beta.png"></I>.</SPAN></P>
<P><SPAN class=Normal>If (<I>p</I> – 4) < 0, the inequality is true
when <I><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/alpha.png"></I>
< <I>x</I> < <I><IMG
style="PADDING-BOTTOM: 2px; VERTICAL-ALIGN: bottom" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/beta.png"></I>.</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel4><SPAN class=ExampleHeading></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel2">
<DIV id=018_solvedex_26>
<TABLE style="WIDTH: 100%" id=018_solvedex_26_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_26>Example
26:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>Solve the inequality <I>x</I><SUP>2</SUP> + <I>x</I>
– 4 > |3<I>x</I> + 4|.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>|3<I>x</I> + 4| < <I>x</I><SUP>2</SUP> + <I>x</I>
– 4</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
3<I>x</I> + 4 < <I>x</I><SUP>2</SUP> + <I>x</I> – 4 or 3<I>x </I>+ 4
> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png"><I>x</I><SUP>2</SUP>
<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">
<I>x</I> + 4</SPAN></P>
<P><SPAN class=Normal>If 3<I>x</I> + 4 < <I>x</I><SUP>2</SUP> +
<I>x</I> – 4,</SPAN></P>
<P><SPAN class=Normal><I>x</I><SUP>2</SUP> <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">
2<I>x </I>– 8 > 0</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
(<I>x </I>– 4)(<I>x </I>+ 2) > 0</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
<I>x</I> < <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">2
or <I>x</I> > 4<SPAN style="MARGIN-LEFT: 37px"></SPAN>… (i)</SPAN></P>
<P><SPAN class=Normal>If <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png"><I>x</I><SUP>2</SUP>
<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">
<I>x</I> + 4 < 3<I>x </I>+ 4</SPAN></P>
<P><SPAN class=Normal><I>x</I><SUP>2</SUP> + 4<I>x</I> > 0</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
<I>x</I>(<I>x</I> + 4) > 0</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
<I>x</I> < <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">4
or <I>x</I> > 0<SPAN style="MARGIN-LEFT: 37px"></SPAN>… (ii)</SPAN></P>
<P><SPAN class=Normal>From (i) and (ii),</SPAN></P>
<P><SPAN class=Normal><I>x</I> < <IMG style="VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">4
or <I>x</I> > 4</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel2">
<DIV id=018_solvedex_27>
<TABLE style="WIDTH: 100%" id=018_solvedex_27_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_27>Example
27:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>Find the range of values of <I>x</I> for which the
following inequality is true.</SPAN></P>
<P><SPAN class=Normal>(|<I>x</I> – 2| <IMG style="VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">
5)(|<I>x</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">
3| <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">
7) < 0</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>(|<I>x</I> – 2| <IMG style="VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">
5)(|<I>x</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">
3| <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">
7) < 0</SPAN></P>
<P><SPAN class=Normal>The LHS is negative in two cases.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><B>Case 1:</B></SPAN></P>
<P><SPAN class=Normal>(|<I>x</I> – 2| <IMG style="VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">
5) < 0 and (|<I>x</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">
3| <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">
7) > 0</SPAN></P>
<P><SPAN class=Normal>If |<I>x</I> – 2| < 5</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">3
< <I>x</I> < 7<SPAN style="MARGIN-LEFT: 60px"></SPAN>…
(i)</SPAN></P>
<P><SPAN class=Normal>If |<I>x</I> <IMG style="VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">
3| > 7</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
<I>x</I> <<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">4
or <I>x</I> > 10<SPAN style="MARGIN-LEFT: 15px"></SPAN>…
(ii)</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>(i) and (ii) cannot be true
simultaneously.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><B>Case 2:</B></SPAN></P>
<P><SPAN class=Normal>(|<I>x</I> – 2| <IMG style="VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">
5) > 0 and (|<I>x</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">
3| <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">
7) < 0</SPAN></P>
<P><SPAN class=Normal>If |<I>x</I> – 2| > 5</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
<I>x</I> > 7 or <I>x</I> < <IMG style="VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">3<SPAN
style="MARGIN-LEFT: 25px"></SPAN>…(iii)</SPAN></P>
<P><SPAN class=Normal>If |<I>x</I> <IMG style="VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">
3| < 7</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">4
< <I>x</I> < 10<SPAN style="MARGIN-LEFT: 50px"></SPAN>…
(iv)</SPAN></P>
<P><SPAN class=Normal>From (iii) and (iv),</SPAN></P>
<P><SPAN class=Normal>7 < <I>x</I> < 10 and <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">4
< <I>x</I> < <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">3</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel2">
<DIV id=018_solvedex_28>
<TABLE style="WIDTH: 100%" id=018_solvedex_28_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_28>Example
28:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/018_051.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/018_052.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>If <I>x</I> + 1 <IMG style="VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/less_equal.png">
0, the above inequality is true.</SPAN></P>
<P><SPAN class=Normal>But, <I>x</I> + 3 <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/greater_equal.png">
0</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
<I>x</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/greater_equal.png">
<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">3</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">3
<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/less_equal.png">
<I>x</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/less_equal.png">
<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">1<SPAN
style="MARGIN-LEFT: 25px"></SPAN>…(i)</SPAN></P>
<P><SPAN class=Normal>If <I>x</I> + 1 > 0,</SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/018_053.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
<I>x</I> + 3 > <I>x</I><SUP>2</SUP> + 2<I>x</I> + 1</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
<I>x</I><SUP>2</SUP> + <I>x</I> – 2 < 0</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
(<I>x</I> + 2)(<I>x </I>– 1) < 0</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">2
< <I>x</I> < 1 </SPAN></P>
<P><SPAN class=Normal>But <I>x</I> > <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">1</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">1
< <I>x</I> < 1<SPAN
style="MARGIN-LEFT: 33px"></SPAN>…(ii)</SPAN></P>
<P><SPAN class=Normal>From (i) and (ii),</SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">3
<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/less_equal.png">
<I>x</I> < 1</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel2">
<DIV id=018_solvedex_29>
<TABLE style="WIDTH: 100%" id=018_solvedex_29_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_29>Example
29:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal><I>x</I><SUP>2</SUP> – 17|<I>x</I>| + 72 <
0</SPAN></P>
<P><SPAN class=Normal>Find the range of values of <I>x</I> that satisfy
the inequality.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal><I>x</I><SUP>2</SUP> – 17|<I>x</I>| + 72 <
0</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
|<I>x</I>|<SUP>2</SUP> – 17|<I>x</I>| + 72 < 0</SPAN></P>
<P><SPAN class=Normal>(|<I>x</I>| <IMG style="VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">
9)(|<I>x</I>| <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">
8) < 0</SPAN></P>
<P><SPAN class=Normal>This can be true in two cases.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><B>Case 1:</B></SPAN></P>
<P><SPAN class=Normal>(|<I>x</I>| <IMG style="VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">
9) < 0 and (|<I>x</I>| <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">
8) > 0</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">9
< <I>x</I> < 9 and <I>x</I> > 8 or <I>x</I> < <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">8</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">9
< <I>x</I> < <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">8
and 8 < <I>x</I> < 9</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><B>Case 2:</B></SPAN></P>
<P><SPAN class=Normal>(|<I>x</I>| <IMG style="VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">
9) > 0 and (|<I>x</I>| <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">
8) < 0</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
<I>x</I> > 9 or <I>x</I> < <IMG style="VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">9
and <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">8
< <I>x</I> < 8</SPAN></P>
<P><SPAN class=Normal>This is not possible.</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
The inequality is true for</SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">9
< <I>x</I> < <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">8
and 8 < <I>x</I> < 9.</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel2">
<DIV id=018_solvedex_30>
<TABLE style="WIDTH: 100%" id=018_solvedex_30_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_30>Example
30:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/018_054.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/018_055.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/018_056.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><I>x</I><SUP>2</SUP> – <I>x</I> + 1 > 0 for all
values of <I>x</I> as the coefficient of <I>x</I><SUP>2</SUP> is positive
and Δ for the equation is negative.</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
For the inequation to hold,</SPAN></P>
<P><SPAN class=Normal>(<I>x </I>– 5)(<I>x </I>+ 4) < 0</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">4
< <I>x</I> < 5</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel2">
<DIV id=018_solvedex_31>
<TABLE style="WIDTH: 100%" id=018_solvedex_31_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_31>Example
31:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/018_057.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/018_058.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>The numerator of the LHS has only even powers of
<I>y</I>.</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
Numerator > 0</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
For the inequation to hold,</SPAN></P>
<P><SPAN class=Normal><I>y</I><SUP>2</SUP> – <I>y </I>– 72 <
0</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
(<I>y</I> – 9)(<I>y</I> + 8) < 0</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">8
< <I>y</I> < 9</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel2">
<DIV id=018_solvedex_32>
<TABLE style="WIDTH: 100%" id=018_solvedex_32_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_32>Example
32:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/018_059.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/018_060.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/018_061.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/018_062.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/018_063.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/018_064.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/018_065.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>This is possible in two cases.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><B>Case 1:</B></SPAN></P>
<P><SPAN class=Normal>Both the numerator and the denominator are
negative.</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">7
< <I>x</I> < 9 and <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">8
< <I>x</I> < 7</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">7
< <I>x</I> < 7</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><B>Case 2:</B></SPAN></P>
<P><SPAN class=Normal>Both the numerator and the denominator are
positive.</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
(<I>x</I> < <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">7,
<I>x</I> > 9) and (<I>x</I> < <IMG style="VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">8,
<I>x</I> > 7)</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
<I>x</I> < <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">8
and <I>x</I> > 9</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
The integer values that do not satisfy the inequality are <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">8,
<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/minus.png">7,
7, 8 and 9.</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/therefore.png">
There are five integer values that do not satisfy the
inequality.</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorNote divLevel2">
<P><SPAN class=Note>IMPORTANT</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<UL type=disc>
<LI value=1>During the exam, multiple choice questions on inequalities can
often be solved by evaluating options. However, having an in-depth
understanding of inequalities is important.</LI></UL></DIV>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorNote divLevel2">
<P><SPAN class=Note>REMEMBER</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<UL class=Normal>
<LI value=7>If <I>a</I> > <I>b</I> and <I>b</I> > <I>c</I>, then
<I>a</I> > <I>b</I> > <I>c</I> and <I>a</I> > <I>c</I></LI></UL>
<UL class=Normal>
<LI value=8>If <I>a</I> < <I>b</I> and <I>b</I> < <I>c</I>, then
<I>a</I> < <I>b</I> < <I>c</I> and <I>a</I> < <I>c</I></LI></UL>
<UL class=Normal>
<LI value=9>If <I>a</I> > <I>b</I>, then <I>a</I> + <I>c</I> > <I>b</I>
+ <I>c</I> and <I>a</I> – <I>c</I> > <I>b</I> – <I>c</I></LI></UL>
<UL class=Normal>
<LI value=10>If <I>a</I> > <I>b</I> and <I>c</I> > <I>d</I>, then
<I>a</I> + <I>c</I> > <I>b</I> + <I>d</I></LI></UL>
<UL class=Normal>
<LI value=11>If <I>a</I> > <I>b</I> and <I>c</I> > <I>d</I>, then
<I>a</I> – <I>d</I> > <I>b</I> – <I>c</I></LI></UL>
<UL class=Normal>
<LI value=12>If <I>a</I> > <I>b</I> and <I>c</I> > 0 then <I>a</I> <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/multiplication.png">
<I>c</I> > <I>b</I> <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/multiplication.png">
<I>c</I> and <I>a</I>/<I>c</I> > <I>b</I>/<I>c</I></LI></UL>
<UL class=Normal>
<LI value=13>If <I>a</I> > <I>b</I> and <I>c</I> < 0 then <I>a</I> <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/multiplication.png">
<I>c</I> < <I>b</I> <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/multiplication.png">
<I>c</I> and <I>a</I>/<I>c</I> < <I>b</I>/<I>c</I></LI></UL>
<UL class=Normal>
<LI value=14>If <I>a</I> > <I>b</I> and <I>c</I> > <I>d</I>, and
<I>a</I>, <I>b</I>, <I>c</I>, <I>d</I> > 0, then <I>ac</I> >
<I>bd</I></LI></UL>
<UL class=Normal>
<LI value=15>If <I>a</I> > <I>b</I> and <I>a</I>, <I>b</I>, <I>n</I> >
0, then <I>a<SUP>n</I></SUP> > <I>b<SUP>n</SUP></I></LI></UL>
<UL class=Normal>
<LI value=16>If <I>a</I> > <I>b</I> and <I>a</I>, <I>b</I>, <I>n</I> >
0, then <I>a<SUP>1/n</SUP></I> ><I>b<SUP>1/n</SUP></I></LI></UL>
<UL class=Normal>
<LI value=17>If <I>a</I> > <I>b</I> and <I>a</I>, <I>b</I>, <I>n</I> >
0, then 1/<I>a<SUP>n</I></SUP> < 1/<I>b<SUP>n</SUP></I> or
<I>a<SUP>–n</SUP></I> < <I>b<SUP>–n</SUP></I></LI></UL>
<UL class=Normal>
<LI value=18>If <I>a</I> > <I>b</I>, then 1/<I>a</I> <
1/<I>b</I></LI></UL>
<UL class=Normal>
<LI value=19><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/018_071.png"></LI></UL>
<P><SPAN class=Normal></SPAN></P>
<UL class=Normal>
<LI value=20><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/018_072.png"></LI></UL>
<P><SPAN class=Normal></SPAN></P>
<UL class=Normal>
<LI value=21><I>a</I><SUP>3</SUP> + <I>b</I><SUP>3</SUP> <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/greater_equal.png">
<I>ab</I>(<I>a</I> + <I>b</I>), if <I>a</I> and <I>b</I> are positive.</LI></UL>
<P><SPAN class=Normal></SPAN></P>
<UL class=Normal>
<LI value=22><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/018_073.png"></LI></UL>
<P><SPAN class=Normal></SPAN></P>
<UL class=Normal>
<LI value=23><I>a</I><SUP>2</SUP> + <I>b</I><SUP>2</SUP> +
<I>c</I><SUP>2</SUP> > (<I>ab</I> + <I>bc</I> + <I>ac</I>)</LI></UL>
<UL class=Normal>
<LI value=24>(<I>a</I> + <I>b</I>)(<I>b</I> + <I>c</I>)(<I>c</I> + <I>a</I>)
<IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/greater_equal.png">
8<I>abc</I> if <I>a</I>, <I>b</I> and <I>c</I> are all positive.</LI></UL>
<UL class=Normal>
<LI value=22>(<I>n </I>+ 1)<I><SUP>n</I></SUP> <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/greater_equal.png">
2<I><SUP>n</I></SUP> (<I>n</I>!)</LI></UL>
<UL class=Normal>
<LI value=25>2 <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/multiplication.png">
4 <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/multiplication.png">
6 <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/multiplication.png">
… <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/multiplication.png">
2<I>n</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/less_equal.png">
(<I>n</I> + 1)<I><SUP>n</SUP></I></LI></UL>
<UL class=Normal>
<LI value=26><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/b78fb690-6a5b-4134-8584-176651950975/018_images/018_074.png"></LI></UL>
<P><SPAN class=Normal></SPAN></P>
<UL class=Normal>
<LI value=27>Given the sum of two positive quantities, their product is the
maximum when they are equal.</LI></UL>
<UL class=Normal>
<LI value=28>Given the product of two positive quantities, their sum is the
least when they are equal.</LI></UL>
<UL class=Normal>
<LI value=29>If <I>x </I>and<I> y</I> are positive and<I> x</I> > <I>y</I>,
then log <I>x</I> > log <I>y</I>.</SPAN>
<P></P></LI></UL></DIV></DIV></BODY></HTML>
guruhttp://www.blogger.com/profile/01591375209227117455noreply@blogger.comtag:blogger.com,1999:blog-7850540640045081515.post-57056662430636242962015-03-03T03:46:00.004-08:002019-09-14T18:55:04.242-07:00algerbaic sys<div dir="ltr" style="text-align: left;" trbidi="on">
<br /></div>
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
<HTML
xmlns="http://www.w3.org/1999/xhtml"><HEAD><TITLE>Algebraic Symbols</TITLE>
<META content="text/html; charset=unicode" http-equiv=Content-Type>
<SCRIPT type=text/javascript
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<SCRIPT type=text/javascript
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<LINK rel=stylesheet type=text/css
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<META name=GENERATOR content="MSHTML 8.00.6001.19190"></HEAD>
<BODY onload="lesson_Function_1(); lesson_Function_2()">
<P class=oTOCTitle><SPAN class=oTOCTitleFont><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/8b368873-0de3-4bf6-ade2-d215b5baee16/012.htm#Title">Contents</A></SPAN></P>
<OL class=oTOCLevel1>
<LI class=oTocLiClass><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/8b368873-0de3-4bf6-ade2-d215b5baee16/012.htm#I">INTRODUCTION</A>
</LI>
<LI class=oTocLiClass><A class=header
href="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/8b368873-0de3-4bf6-ade2-d215b5baee16/012.htm#II">SOLVED
EXAMPLES</A> </LI></OL>
<P id=Title class=pTitle><SPAN class=Title>Algebraic Symbols</SPAN></P>
<HR>
<P class=pLevel2><SPAN class=Normal></SPAN>
<DIV id=012_I>
<TABLE id=012_I_bookmark class=tableh1 border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD
style="PADDING-BOTTOM: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 100%; PADDING-RIGHT: 0px; PADDING-TOP: 0px">
<OL class="oLevel1 Heading1">
<LI value=1><A id=I>INTRODUCTION</A></LI></OL></TD></TR></TBODY></TABLE>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>In mathematics, the four basic operations,
addition, subtraction, multiplication and division are known as fundamental
operations and are represented by standard symbols +, <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/8b368873-0de3-4bf6-ade2-d215b5baee16/012_images/minus.png">,
<IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/8b368873-0de3-4bf6-ade2-d215b5baee16/012_images/multiplication.png">
and <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/8b368873-0de3-4bf6-ade2-d215b5baee16/012_images/division.png">
respectively. But many times some special symbols are used to represent a
mathematical operation.</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>For example, if we represent the average of
two numbers <I>a </I>and <I>b </I>by<I> </I>(<I>a </I>$<I> b</I>) then the
result of (3 $ 6) is equal to (3 + 6)/2 or 4.5.</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>We will refer to these types of symbols as
<B>algebraic symbols</B>.</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>To solve problems related to symbols we will
use the concepts of<I> </I>binary operations (a calculation involving two
operands). A binary operation, for example (<I>a </I>$<I> b</I>), depends on the
ordered pair (<I>a</I>,<I> b</I>). So to find ((<I>a </I>$<I> b</I>) $
<I>c</I>), first operate on the ordered pair (<I>a</I>,<I> b</I>) and then
operate on the result of that using the ordered pair ((<I>a</I>,<I> b</I>),
<I>c</I>).</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<P class=pLevel2><SPAN class=Normal>For example, if we represent the LCM of two
numbers <I>m</I> and <I>n </I>by<I> </I>(<I>m</I> # <I>n</I>), then the result
of</SPAN></P>
<P class=pLevel2><SPAN class=Normal>((2 # 3) # 8) will be ((6) # 8) =
24.</SPAN></P>
<P class=pLevel2><SPAN class=Normal></SPAN></P></DIV>
<DIV id=012_II>
<TABLE id=012_II_bookmark class=tableh1 border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD
style="PADDING-BOTTOM: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 100%; PADDING-RIGHT: 0px; PADDING-TOP: 0px">
<OL class="oLevel1 Heading1">
<LI value=2><A id=II>SOLVED EXAMPLES</A></LI></OL></TD></TR></TBODY></TABLE>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel2">
<DIV id=012_solvedex_1>
<TABLE style="WIDTH: 100%" id=012_solvedex_1_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_1>Example
1:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>If <I>a </I>$ <I>b </I>is the average of <I>a</I>
and<I> b</I>, and <I>m</I> # <I>n</I> is the LCM of <I>m </I>and <I>n</I>,
then find the value of (3 $ (4 # 5)).</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>Solving the inner bracket,</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>(4 # 5) = LCM of 4 and 5 = 20</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>((3 $ (4 # 5)) = (3 $ (20))</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>(3 $ 20) = Average of 3 and 20</SPAN></P>
<P><SPAN class=Normal><SPAN style="MARGIN-LEFT: 59px"></SPAN>= (3 + 20)/2
= 23/2 = 11.5</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel2">
<DIV id=012_solvedex_2>
<TABLE style="WIDTH: 100%" id=012_solvedex_2_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_2>Example
2:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>If & (<I>a</I>,<I> b</I>) = the remainder when
<I>a</I> is divided by<I> b</SPAN></P>
<P><SPAN class=Normal>#</I> (<I>a</I>,<I> b</I>) = number of ways to
select <I>a </I>items from <I>b </I>distinct items</SPAN></P>
<P><SPAN class=Normal>$(<I>a</I>,<I> b</I>) = <I>a </I>if <I>a ></I>
<I>b </I>or <I>b </I>if <I>b ></I><I> a</I></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>then, find the value of $(#(&(7, 4), 5),
9).</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>$(# (& (7, 4), 5), 9)</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>= $(# (3, 5), 9) <SPAN
style="MARGIN-LEFT: 25px"></SPAN>… [<IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/8b368873-0de3-4bf6-ade2-d215b5baee16/012_images/symb23.png">
& (7, 4) = 3]</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>= $(10, 9) <SPAN style="MARGIN-LEFT: 58px"></SPAN>…
[<IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/8b368873-0de3-4bf6-ade2-d215b5baee16/012_images/symb23.png">
# (3, 5) = <SUP>5</SUP>C<SUB>3</SUB> = 10]</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>= 10<SPAN style="MARGIN-LEFT: 70px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN> … [<IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/8b368873-0de3-4bf6-ade2-d215b5baee16/012_images/symb23.png">
$(10, 9) = 10]</SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel2">
<DIV id=012_solvedex_3>
<TABLE style="WIDTH: 100%" id=012_solvedex_3_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_3>Example
3:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>A, B, C are three numbers. Let</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>@ (A, B) = Average of A and B,</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>/ (A, B) = Product of A and B, and</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>X (A, B) = The result of dividing A by B</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><B>[CAT 2000]</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Question 1:</B></SPAN></P>
<P><SPAN class=Normal>The sum of A and B is given by</SPAN></P>
<P><SPAN class=Normal>(1) /(@( A, B), 2)</SPAN></P>
<P><SPAN class=Normal>(2) X(@(A, B), 2)</SPAN></P>
<P><SPAN class=Normal>(3) @(/(A, B), 2) </SPAN></P>
<P><SPAN class=Normal>(4) @(X(A, B), 2)</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>Sum of A and B = (Average of A and B) <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/8b368873-0de3-4bf6-ade2-d215b5baee16/012_images/multiplication.png">
2</SPAN></P>
<P><SPAN class=Normal><SPAN style="MARGIN-LEFT: 110px"></SPAN>= /(@(A, B),
2)</SPAN></P>
<P><SPAN class=Normal><IMG style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/8b368873-0de3-4bf6-ade2-d215b5baee16/012_images/therefore.png">
Hence, <B>option 1</B>.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><B>Question 2:</B></SPAN></P>
<P><SPAN class=Normal>Average of A, B and C is given by</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>(1) @(/(@(/(B, A), 2), C), 3)</SPAN></P>
<P><SPAN class=Normal>(2) X(@(/(@(B, A), 3), C), 2)</SPAN></P>
<P><SPAN class=Normal>(3) /(@(X(@(B, A), 2), C), 3)</SPAN></P>
<P><SPAN class=Normal>(4) /(X(@(/(@(B, A), 2), C), 3), 2)</SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>We need to find average of A, B and C i.e. (A + B +
C)/3</SPAN></P>
<P><SPAN class=Normal>The only operator that can give 3 in the denominator
is X.</SPAN></P>
<P><SPAN class=Normal><B>Option 1</B> cannot be the answer, as there is no
X.</SPAN></P>
<P><SPAN class=Normal><B>Option 2</B> also cannot be the answer as
X(@(/(@(/(B, A), 2), C), 3), 2) gives 2 in the denominator and not
3.</SPAN></P>
<P><SPAN class=Normal>Similarly <B>option 3</B> is not the
answer.</SPAN></P>
<P><SPAN class=Normal>Hence, option 4 must be the answer.</SPAN></P>
<P><SPAN class=Normal><B>Option 4:</B> </SPAN></P>
<P><SPAN class=Normal>/( X (@ (/ (@ (B, A), 2), C), 3), 2)</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/8b368873-0de3-4bf6-ade2-d215b5baee16/012_images/012_001.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>=/(X(@((A + B), C), 3), 2)</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/8b368873-0de3-4bf6-ade2-d215b5baee16/012_images/012_019.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/8b368873-0de3-4bf6-ade2-d215b5baee16/012_images/012_002.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Hence, <B>option
4.</B></SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel2">
<DIV id=012_solvedex_4>
<TABLE style="WIDTH: 100%" id=012_solvedex_4_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_4>Example
4:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>Let <I>x</I> and <I>y</I> denote two numbers.
Then,</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>$(<I>x</I>, <I>y</I>) = Average of <I>x</I> and
<I>y</I>,</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>&(<I>x</I>, <I>y</I>) = Product of <I>x</I> and
<I>y</I>, and</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Q(<I>x</I>, <I>y</I>) = The result of dividing
<I>x</I> by <I>y</I></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>If <I>z</I> denotes a third number, then the average
of <I>x</I>, <I>y</I> and <I>z</I> is given by:</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>(1) & ($(Q(&(<I>x</I>, <I>y</I>), 2),
<I>z</I>), 3)</SPAN></P>
<P><SPAN class=Normal>(2) Q (& (Q(& ($(<I>x</I>, <I>y</I>), 3),
<I>z</I>), 3), 2)</SPAN></P>
<P><SPAN class=Normal>(3) & (Q($(& ($(<I>x</I>, <I>y</I>), 2),
<I>z</I>), 3), 2)</SPAN></P>
<P><SPAN class=Normal>(4) Q($(& ($(&(<I>x</I>, <I>y</I>), 2),
<I>z</I>), 3), 2)</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>The best way to solve this is by working through the
options:</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><B>Option 1:</B> & ($(Q(&(<I>x</I>,
<I>y</I>), 2), <I>z</I>), 3)</SPAN></P>
<P><SPAN class=Normal>&(<I>x</I>, <I>y</I>) = <I>xy</SPAN></P>
<P><SPAN class=Normal>Q</I>(<I>xy</I>, 2) = <I>xy</I>/2</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/8b368873-0de3-4bf6-ade2-d215b5baee16/012_images/012_003.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/8b368873-0de3-4bf6-ade2-d215b5baee16/012_images/012_004.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Hence, option 1 is eliminated.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><B>Option 2: </B>Q(&(Q(&($(<I>x</I>,
<I>y</I>),3), <I>z</I>),3),2)</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/8b368873-0de3-4bf6-ade2-d215b5baee16/012_images/012_005.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/8b368873-0de3-4bf6-ade2-d215b5baee16/012_images/012_006.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/8b368873-0de3-4bf6-ade2-d215b5baee16/012_images/012_007.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>(It should be apparent by now that this option will
not lead to the required answer.)</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/8b368873-0de3-4bf6-ade2-d215b5baee16/012_images/012_008.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/8b368873-0de3-4bf6-ade2-d215b5baee16/012_images/012_009.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Hence, option 2 is also eliminated.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><B>Option 3:</B> &(Q($(&($(<I>x</I>,
<I>y</I>), 2), <I>z</I>), 3), 2)</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/8b368873-0de3-4bf6-ade2-d215b5baee16/012_images/012_010.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/8b368873-0de3-4bf6-ade2-d215b5baee16/012_images/012_011.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/8b368873-0de3-4bf6-ade2-d215b5baee16/012_images/012_012.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/8b368873-0de3-4bf6-ade2-d215b5baee16/012_images/012_013.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/8b368873-0de3-4bf6-ade2-d215b5baee16/012_images/012_014.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>which is the average of x, y and z.</SPAN></P>
<P><SPAN class=Normal>Hence, <B>option
3.</B></SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel2">
<DIV id=012_solvedex_5>
<TABLE style="WIDTH: 100%" id=012_solvedex_5_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_5>Example
5:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>Two binary operations <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/8b368873-0de3-4bf6-ade2-d215b5baee16/012_images/ex-or.png">
and * are defined over the set {a, e, f, g, h} as per the following
tables</SPAN></P>
<DIV class=tblCenter>
<TABLE style="BORDER-COLLAPSE: collapse" cellSpacing=1 align=center>
<TBODY>
<TR>
<TD class=visible_border_table><B><IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/8b368873-0de3-4bf6-ade2-d215b5baee16/012_images/ex-or.png"></B></TD>
<TD class=visible_border_table><B>a</B></TD>
<TD class=visible_border_table><B>e</B></TD>
<TD class=visible_border_table><B>f</B></TD>
<TD class=visible_border_table><B>g</B></TD>
<TD class=visible_border_table><B>h</B></TD></TR>
<TR>
<TD class=visible_border_table><B>a</B></TD>
<TD class=visible_border_table>a</TD>
<TD class=visible_border_table>e</TD>
<TD class=visible_border_table>f</TD>
<TD class=visible_border_table>g</TD>
<TD class=visible_border_table>h</TD></TR>
<TR>
<TD class=visible_border_table><B>e</B></TD>
<TD class=visible_border_table>e</TD>
<TD class=visible_border_table>f</TD>
<TD class=visible_border_table>g</TD>
<TD class=visible_border_table>h</TD>
<TD class=visible_border_table>a</TD></TR>
<TR>
<TD class=visible_border_table><B>f</B></TD>
<TD class=visible_border_table>f</TD>
<TD class=visible_border_table>g</TD>
<TD class=visible_border_table>h</TD>
<TD class=visible_border_table>a</TD>
<TD class=visible_border_table>e</TD></TR>
<TR>
<TD class=visible_border_table><B>g</B></TD>
<TD class=visible_border_table>g</TD>
<TD class=visible_border_table>h</TD>
<TD class=visible_border_table>a</TD>
<TD class=visible_border_table>e</TD>
<TD class=visible_border_table>f</TD></TR>
<TR>
<TD class=visible_border_table><B>h</B></TD>
<TD class=visible_border_table>h</TD>
<TD class=visible_border_table>a</TD>
<TD class=visible_border_table>e</TD>
<TD class=visible_border_table>f</TD>
<TD class=visible_border_table>g</TD></TR></TBODY></TABLE></DIV>
<P><SPAN class=Normal></SPAN></P>
<DIV class=tblCenter>
<TABLE style="BORDER-COLLAPSE: collapse" cellSpacing=1 align=center>
<TBODY>
<TR>
<TD class=visible_border_table><B>*</B></TD>
<TD class=visible_border_table><B>a</B></TD>
<TD class=visible_border_table><B>e</B></TD>
<TD class=visible_border_table><B>f</B></TD>
<TD class=visible_border_table><B>g</B></TD>
<TD class=visible_border_table><B>h</B></TD></TR>
<TR>
<TD class=visible_border_table><B>a</B></TD>
<TD class=visible_border_table>a</TD>
<TD class=visible_border_table>a</TD>
<TD class=visible_border_table>a</TD>
<TD class=visible_border_table>a</TD>
<TD class=visible_border_table>a</TD></TR>
<TR>
<TD class=visible_border_table><B>e</B></TD>
<TD class=visible_border_table>a</TD>
<TD class=visible_border_table>e</TD>
<TD class=visible_border_table>f</TD>
<TD class=visible_border_table>g</TD>
<TD class=visible_border_table>h</TD></TR>
<TR>
<TD class=visible_border_table><B>f</B></TD>
<TD class=visible_border_table>a</TD>
<TD class=visible_border_table>f</TD>
<TD class=visible_border_table>h</TD>
<TD class=visible_border_table>e</TD>
<TD class=visible_border_table>g</TD></TR>
<TR>
<TD class=visible_border_table><B>g</B></TD>
<TD class=visible_border_table>a</TD>
<TD class=visible_border_table>g</TD>
<TD class=visible_border_table>e</TD>
<TD class=visible_border_table>h</TD>
<TD class=visible_border_table>f</TD></TR>
<TR>
<TD class=visible_border_table><B>h</B></TD>
<TD class=visible_border_table>a</TD>
<TD class=visible_border_table>h</TD>
<TD class=visible_border_table>g</TD>
<TD class=visible_border_table>f</TD>
<TD class=visible_border_table>e</TD></TR></TBODY></TABLE></DIV>
<P><SPAN class=Normal>Thus, according to the first table f <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/8b368873-0de3-4bf6-ade2-d215b5baee16/012_images/ex-or.png">
g = a, while according to the second table g * h = f, and so
on.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Also, let f<SUP>2</SUP> = f * f, g<SUP>3</SUP> = g *
g * g, and so on.</SPAN></P>
<P><SPAN class=Normal><B>[CAT 2003 – Re-test]</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Question 1:</B></SPAN></P>
<P><SPAN class=Normal>What is the smallest positive integer <I>n</I> such
that g<I><SUP>n</I></SUP> = e?</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>(1) 4 <SPAN style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN>(2) 5 <SPAN
style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN>(3) 2<SPAN
style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN>(4) 3</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>g<SUP>2</SUP> = g * g = h</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>g<SUP>3</SUP> = g * g * g = h * g = f</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>g<SUP>4</SUP> = g * g * g * g = f * g = e</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Hence, <B>option 1</B>.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><B>Question 2:</B></SPAN></P>
<P><SPAN class=Normal>Upon simplification, f <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/8b368873-0de3-4bf6-ade2-d215b5baee16/012_images/ex-or.png">
[f *{f <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/8b368873-0de3-4bf6-ade2-d215b5baee16/012_images/ex-or.png">
(f * f)}] equals</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>(1) e<SPAN style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN> (2) f<SPAN
style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN>(3) g<SPAN
style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN>(4) h</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>f * f = h</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>f <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/8b368873-0de3-4bf6-ade2-d215b5baee16/012_images/ex-or.png">
h = e</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>f * e = f</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>f <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/8b368873-0de3-4bf6-ade2-d215b5baee16/012_images/ex-or.png">
f = h</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Hence, <B>option 4</B>.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><B>Question 3:</B></SPAN></P>
<P><SPAN class=Normal>Upon simplification, {a<SUP>10</SUP> *
(f<SUP>10</SUP> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/8b368873-0de3-4bf6-ade2-d215b5baee16/012_images/ex-or.png">
g<SUP>9</SUP>)} <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/8b368873-0de3-4bf6-ade2-d215b5baee16/012_images/ex-or.png">
e<SUP>8</SUP> equals</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>(1) e<SPAN style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN>(2) f<SPAN
style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN>(3) g <SPAN
style="MARGIN-LEFT: 25px"></SPAN><SPAN
style="MARGIN-LEFT: 25px"></SPAN>(4) h</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>f<SUP>10</SUP> = (f<SUP>2</SUP>)<SUP>5</SUP> =
h<SUP>5</SUP> = h * (h<SUP>2</SUP>)<SUP>2</SUP> = h * e<SUP>2</SUP> = h *
e = h</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>g<SUP>9</SUP> = g * (g<SUP>2</SUP>)<SUP>4</SUP> = g
* h<SUP>4</SUP> = g * e = g</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Also, e<SUP>8 </SUP>= e (<IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/8b368873-0de3-4bf6-ade2-d215b5baee16/012_images/symb23.png">
e * e = e) and a<SUP>10</SUP> = a (<IMG
style="PADDING-BOTTOM: 1px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/8b368873-0de3-4bf6-ade2-d215b5baee16/012_images/symb23.png">
a * a = a)</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Now,</SPAN></P>
<P><SPAN class=Normal>f<SUP>10</SUP> <IMG style="VERTICAL-ALIGN: middle"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/8b368873-0de3-4bf6-ade2-d215b5baee16/012_images/ex-or.png">
g<SUP>9 </SUP>= h <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/8b368873-0de3-4bf6-ade2-d215b5baee16/012_images/ex-or.png">
g = f</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>a<SUP>10</SUP> * f = a * f = a</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>a <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/8b368873-0de3-4bf6-ade2-d215b5baee16/012_images/ex-or.png">
e<SUP>8</SUP> = a <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/8b368873-0de3-4bf6-ade2-d215b5baee16/012_images/ex-or.png">
e = e</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Hence, <B>option
1.</B></SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel2">
<DIV id=012_solvedex_6>
<TABLE style="WIDTH: 100%" id=012_solvedex_6_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_6>Example
6:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>A florist hires some workers who can stack flowers
in her store within 10 days. Which of the following gives the fraction of
work the men do in a day?</SPAN></P>
<P><SPAN class=Normal>Symbols used:</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><I>a</I> @ <I>b</I> is the average of <I>a</I> and
<I>b.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>a</I> & <I>b</I> is the product of <I>a</I> and
<I>b.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>a</I> # <I>b</I> is the sum of the reciprocals of
<I>a</I> and <I>b.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>a</I> $ <I>b</I> is the difference between the
reciprocals of <I>a</I> and <I>b.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>a </I>! <I>b</I> is the result when <I>a</I> is
divided by <I>b</I>.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>(1) (((2 @ 3) & (2 ! (3 # 3))) $ ((2 & 2)
& 2))</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>(2) ((((2 @ 3) ! (3 $ 2)) # 2) & (3 &
3))</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>(3) (((2 @ 3) ! (3 ! 2)) $ 2)</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>(4) ((2 & (3 @ (2 # (2 ! (3 &
3))))))</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>If a certain number of men do some work in 10 days,
then the same number of men will do 1/10<SUP>th</SUP> of the work per
day.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Going through each option, we find that only option
3 gives the answer as 1/10:</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>(((2 @ 3) ! (3 ! 2)) $ 2)</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/8b368873-0de3-4bf6-ade2-d215b5baee16/012_images/012_015.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/8b368873-0de3-4bf6-ade2-d215b5baee16/012_images/012_016.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/8b368873-0de3-4bf6-ade2-d215b5baee16/012_images/012_017.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/8b368873-0de3-4bf6-ade2-d215b5baee16/012_images/012_018.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Hence, <B>option
3.</B></SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV>
<P class=pLevel2><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorNote divLevel2">
<P><SPAN class=Note>REMEMBER:</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<UL class=ListParagraph>
<LI class=ListStyleUL5>It is not necessary<I> </I>that the questions on
algebraic symbols are always based on binary operations (or operations on two
operands). There may be questions with operations on three or more
operands.</LI></UL>
<P><SPAN class=ListParagraph></SPAN></P>
<UL><SPAN class=ListParagraph>For example, if we represent the geometric mean
of three numbers <I>a</I>, <I>b</I> and <I>c</I> by @ (<I>a</I>, <I>b</I>,
<I>c</I>), then,</SPAN></UL>
<P><SPAN class=ListParagraph></SPAN>
<UL></UL>
<UL><SPAN class=ListParagraph>@ (2, 4, 27) = (2 <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/8b368873-0de3-4bf6-ade2-d215b5baee16/012_images/multiplication.png">
4 <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/8b368873-0de3-4bf6-ade2-d215b5baee16/012_images/multiplication.png">
27)<SUP>1/3</SUP> = (8<SUP>1/3</SUP> <IMG
style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/8b368873-0de3-4bf6-ade2-d215b5baee16/012_images/multiplication.png">
27<SUP>1/3</SUP>) or (2 <IMG style="PADDING-BOTTOM: 0px; MARGIN-BOTTOM: 0px"
alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/8b368873-0de3-4bf6-ade2-d215b5baee16/012_images/multiplication.png">
3) = 6</SPAN></UL></DIV>
<P class=pLevel4><SPAN class=Normal></SPAN></P>
<DIV class="divTextBox divColorExampleHeading divLevel2">
<DIV id=012_solvedex_7>
<TABLE style="WIDTH: 100%" id=012_solvedex_7_bookmark border=0 cellSpacing=0
cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=ExampleHeading><A id=ex_7>Example
7:</A></SPAN></P></TD></TR></TBODY></TABLE>
<TABLE border=0 cellSpacing=0 cellPadding=0>
<TBODY>
<TR>
<TD style="BACKGROUND-COLOR: transparent; WIDTH: 100%">
<P><SPAN class=Normal>Aaron is Babitti’s husband. Together, they have two
daughters; Corinne and Deborah. Deborah is married to Xerxis and they have
a beautiful baby girl named Ginger. It so happens that Xerxis’s parents,
Yates (father) and Zara (mother), are very close friends of Aaron and
Babitti. Also, their (i.e. Yates and Zara’s) daughter, Winnie, is married
to Pierre and they have a son named Eomer.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><B>Notations:</B></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><I>a</I> $ <I>b</I>: <I>a</I> is <I>b</I>’s
sister</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><I>a</I> # <I>b</I>: <I>a</I> is <I>b</I>’s
brother</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><I>a</I> & <I>b</I>: <I>a</I> is <I>b</I>’s
wife</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><I>a</I> <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/8b368873-0de3-4bf6-ade2-d215b5baee16/012_images/rightarrow.png">
<I>b</I>: <I>a</I> is a parent of <I>b</I></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><I>a</I> ← <I>b</I>: <I>a</I> is a child of
<I>b</I></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><I>a</I> ^ <I>b</I>: <I>a</I> is <I>b</I>’s
sister-in-law</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><I>a</I> % <I>b</I>: <I>a</I> is <I>b</I>’s
brother-in-law</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Using the above notations, for example, <I>a</I> $
<I>b</I> & <I>c</I> would mean that <I>a</I> is <I>b</I>’s sister, who
in turn is <I>c</I>’s wife.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Which of the following statements <B>cannot</B> be
derived from the above information?</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>(1) Corinne $ Deborah & Xerxis ← Zara &
Yates <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/8b368873-0de3-4bf6-ade2-d215b5baee16/012_images/rightarrow.png">
Winnie <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/8b368873-0de3-4bf6-ade2-d215b5baee16/012_images/rightarrow.png">
Eomer</SPAN></P>
<P><SPAN class=Normal>(2) Ginger ← Xerxis % Corinne $ Deborah ^ Winnie
& Pierre <IMG style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/8b368873-0de3-4bf6-ade2-d215b5baee16/012_images/rightarrow.png">
Eomer</SPAN></P>
<P><SPAN class=Normal>(3) Zara & Yates <IMG
style="VERTICAL-ALIGN: middle" alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/8b368873-0de3-4bf6-ade2-d215b5baee16/012_images/rightarrow.png">
Xerxis # Winnie % Deborah $ Corinne ← Babitti & Aaron</SPAN></P>
<P><SPAN class=Normal>(4) All the above statements are true and can be
derived.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=SolutionHeading>Solution:</SPAN></P>
<P><SPAN class=Normal>We will use only the first letter of each person to
denote them. (M) and (F) denote Male and Female respectively. According to
the information given:</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=pImage><IMG alt=""
src="file:///C:/Documents%20and%20Settings/All%20Users/Application%20Data/TestFunda/website/Content/lessons/8b368873-0de3-4bf6-ade2-d215b5baee16/012_images/012_020.png"></SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Now, let’s work through the options:</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><B>Option 1:</B> C is D’s sister; D is X’s wife; X
is a child of Z; Z is Y’s wife; Y is a parent of W; W is a parent of
E.</SPAN></P>
<P><SPAN class=Normal>All of these are from the information given in the
passage; hence, option 1 can be derived.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><B>Option 2: </B>G is a child of X; X is C’s
brother-in-law; C is D’s sister; D is W’s sister-in-law; W is P’s wife; P
is a parent of E.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>All of these are from the information given in the
passage; hence, option 2 can be derived.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal><B>Option 3: </B>Z is Y’s wife; Y is the parent of
X; X is W’s brother; W is D’s brother-in-law.</SPAN></P>
<P><SPAN class=Normal></SPAN></P>
<P><SPAN class=Normal>Winnie is Deborah’s sister-in-law, not
brother-in-law. This statement is false.</SPAN></P>
<P><SPAN class=Normal>Hence, <B>option
3.</B></SPAN></P></TD></TR></TBODY></TABLE></DIV></DIV></DIV></BODY></HTML>
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