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CBSE Class 12-science Answered

<div> <p class="MsoNormal"><span style="font-size: 20.0pt; mso-bidi-font-size: 44.0pt; line-height: 115%; font-family: &quot;Calibri&quot;,&quot;sans-serif&quot;; mso-ascii-theme-font: minor-latin; mso-fareast-font-family: Calibri; mso-fareast-theme-font: minor-latin; mso-hansi-theme-font: minor-latin; mso-bidi-font-family: &quot;Times New Roman&quot;; mso-bidi-theme-font: minor-bidi; mso-ansi-language: EN-US; mso-fareast-language: EN-US; mso-bidi-language: AR-SA;"><strong>z<sub>1 </sub>, z<sub>2 </sub>, z<sub>3 , </sub>, z<sub>4<span style="mso-spacerun: yes;">&nbsp; </span></sub>,<sub> </sub><span style="mso-spacerun: yes;">&nbsp;</span>. . . . . . , z</strong><sub><strong>n-1 are the non real</strong>&nbsp;<span style="font-size: 27px;"><strong style="mso-bidi-font-weight: normal;">n<sup>th &nbsp;<span style="font-family: Verdana, Arial, Helvetica, sans-serif; font-weight: normal; line-height: normal; font-size: 10px;"><strong><span style="font-size: 16.0pt; font-family: &quot;Times New Roman&quot;,&quot;serif&quot;; mso-fareast-font-family: &quot;Times New Roman&quot;; color: black;">roots of unity, then find the value of &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;<span style="font-family: Verdana, Arial, Helvetica, sans-serif; font-weight: normal; font-size: 10px;"><strong><span style="font-size: 12.0pt; font-family: &quot;Verdana&quot;,&quot;sans-serif&quot;; mso-fareast-font-family: &quot;Times New Roman&quot;; mso-bidi-font-family: &quot;Times New Roman&quot;; color: black;">(3-z<sub>1</sub>)<sup>-1</sup>&nbsp;+ (3-z<sub>2</sub>)<sup>-1</sup>&nbsp;+ (3-z<sub>3</sub>)<sup>-1</sup>&nbsp;+ . . . . . . + (3-z<sub>n-1</sub>)<sup>-1</sup></span></strong></span></span></strong></span></sup></strong></span></sub></span></p> <p class="MsoNormal" style="mso-margin-top-alt: auto; mso-margin-bottom-alt: auto; text-indent: .5in; line-height: normal;"><strong><span style="font-size: 12.0pt; font-family: &quot;Times New Roman&quot;,&quot;serif&quot;; mso-fareast-font-family: &quot;Times New Roman&quot;; color: black;"></span></strong></p> <p class="MsoNormal"><span style="font-size: 20.0pt; mso-bidi-font-size: 44.0pt; line-height: 115%; font-family: &quot;Calibri&quot;,&quot;sans-serif&quot;; mso-ascii-theme-font: minor-latin; mso-fareast-font-family: Calibri; mso-fareast-theme-font: minor-latin; mso-hansi-theme-font: minor-latin; mso-bidi-font-family: &quot;Times New Roman&quot;; mso-bidi-theme-font: minor-bidi; mso-ansi-language: EN-US; mso-fareast-language: EN-US; mso-bidi-language: AR-SA;"><sub><span style="font-size: 27px;"><strong style="mso-bidi-font-weight: normal;"><sup><span style="font-family: Verdana, Arial, Helvetica, sans-serif; font-weight: normal; line-height: normal; font-size: 10px;"><strong><span style="font-size: 16.0pt; font-family: &quot;Times New Roman&quot;,&quot;serif&quot;; mso-fareast-font-family: &quot;Times New Roman&quot;; color: black;"><span style="font-family: Verdana, Arial, Helvetica, sans-serif; font-weight: normal; font-size: 10px;"><strong style="mso-bidi-font-weight: normal;"><sup><span style="font-size: 14.0pt; line-height: 115%;"><span style="mso-spacerun: yes;">&nbsp;</span></span></sup></strong></span>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;</span></strong></span></sup></strong></span></sub></span></p> <p class="MsoNormal" style="mso-margin-top-alt: auto; mso-margin-bottom-alt: auto; text-indent: .5in; line-height: normal;"><strong><span style="font-size: 16.0pt; font-family: &quot;Times New Roman&quot;,&quot;serif&quot;; mso-fareast-font-family: &quot;Times New Roman&quot;; color: black;"></span></strong></p> <p class="MsoNormal"><strong style="mso-bidi-font-weight: normal;"><sup></sup></strong></p> <p class="MsoNormal"><strong style="mso-bidi-font-weight: normal;"><sup></sup></strong></p> <p class="MsoNormal"><strong style="mso-bidi-font-weight: normal;"><sup></sup></strong></p> <p class="MsoNormal" style="text-indent: .5in;"><strong style="mso-bidi-font-weight: normal;"><sup> </sup></strong></p> <p class="MsoNormal" style="text-indent: .5in;"><strong style="mso-bidi-font-weight: normal;"><sup> </sup></strong></p> <strong style="mso-bidi-font-weight: normal;"><strong style="mso-bidi-font-weight: normal;"> <p class="MsoNormal" style="text-indent: 0.5in; display: inline !important;"><strong style="mso-bidi-font-weight: normal;">&nbsp;</strong></p> <strong style="mso-bidi-font-weight: normal;"> <p class="MsoNormal" style="text-indent: .5in;">&nbsp;</p> </strong> <p>&nbsp;</p> </strong></strong><strong style="mso-bidi-font-weight: normal;"><strong style="mso-bidi-font-weight: normal;"> <p class="MsoNormal" style="text-indent: 0.5in; display: inline !important;"><strong style="mso-bidi-font-weight: normal;">&nbsp;</strong></p> <strong style="mso-bidi-font-weight: normal;"> <p class="MsoNormal" style="text-indent: .5in;">&nbsp;</p> </strong> <p>&nbsp;</p> </strong></strong><strong style="mso-bidi-font-weight: normal;"><strong style="mso-bidi-font-weight: normal;"> <p class="MsoNormal" style="text-indent: 0.5in; display: inline !important;"><strong style="mso-bidi-font-weight: normal;">&nbsp;</strong></p> <strong style="mso-bidi-font-weight: normal;"> <p class="MsoNormal" style="text-indent: .5in;">&nbsp;</p> </strong> <p>&nbsp;</p> </strong></strong><strong style="mso-bidi-font-weight: normal;"><strong style="mso-bidi-font-weight: normal;"> <p class="MsoNormal" style="text-indent: 0.5in; display: inline !important;"><strong style="mso-bidi-font-weight: normal;"><br /></strong></p> </strong></strong> <p>&nbsp;</p> <p class="MsoNormal">&nbsp;&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;</p> </div>
Asked by Sourav | 08 May, 2015, 12:12: PM
Expert Answer
I f space Z subscript 1 comma space Z subscript 2 comma space Z subscript 3 comma space..... Z subscript n minus 1 end subscript space a r e space t h e space n o n space r e a l space n t h space r o o t s space o f space u n i t y comma space w e space c a n space w r i t e x to the power of n minus 1 equals open parentheses x minus 1 close parentheses open parentheses x minus Z subscript 1 close parentheses open parentheses x minus Z subscript 2 close parentheses open parentheses x minus Z subscript 3 close parentheses..... open parentheses x minus Z subscript n minus 1 end subscript close parentheses T a k i n g space log space o n space b o t h space s i d e s ln open parentheses x to the power of n minus 1 close parentheses equals ln open square brackets open parentheses x minus 1 close parentheses open parentheses x minus Z subscript 1 close parentheses open parentheses x minus Z subscript 2 close parentheses open parentheses x minus Z subscript 3 close parentheses..... open parentheses x minus Z subscript n minus 1 end subscript close parentheses close square brackets D i f f e r e n t i a t i n g space b o t h space s i d e s space w e space g e t fraction numerator n x to the power of n minus 1 end exponent over denominator x to the power of n minus 1 end fraction equals fraction numerator 1 over denominator x minus 1 end fraction plus fraction numerator 1 over denominator x minus Z subscript 1 end fraction plus fraction numerator 1 over denominator x minus Z subscript 2 end fraction plus..... plus fraction numerator 1 over denominator x minus Z subscript n minus 1 end subscript end fraction P u t t i n g space x equals 3 comma space w e space g e t fraction numerator n open parentheses 3 close parentheses to the power of n minus 1 end exponent over denominator 3 to the power of n minus 1 end fraction equals 1 half plus fraction numerator 1 over denominator 3 minus Z subscript 1 end fraction plus fraction numerator 1 over denominator 3 minus Z subscript 2 end fraction plus...... plus fraction numerator 1 over denominator 3 minus Z subscript n minus 1 end subscript end fraction  rightwards double arrow fraction numerator 1 over denominator 3 minus Z subscript 1 end fraction plus fraction numerator 1 over denominator 3 minus Z subscript 2 end fraction plus...... plus fraction numerator 1 over denominator 3 minus Z subscript n minus 1 end subscript end fraction equals fraction numerator n open parentheses 3 close parentheses to the power of n minus 1 end exponent over denominator 3 to the power of n minus 1 end fraction minus 1 half
Answered by satyajit samal | 11 May, 2015, 06:25: AM
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