For any integer k, let α _ k = ((kπ)/(7)) + isin ((kπ)/(7)), wherei = - 1 . The value of the expression ∑ _ k = 1 ^ 12 | α _ k + 1 - α _ k | ∑ _ k = 1 ^ 3 | α _ 4k - 1 - α _ 4k - 2 | is

Correct answer: 0 S = ∑ _ k = 1 ^ 12 | α _ k + 1 - α _ k | ∑ _ k = 1 ^ 3 | α _ 4k - 1 - α _ 4k - 2 | | α _ r + 1 - α _ r | = side length of the polygon having 14 sides = a S = ( α _ 2 - α _ 1 ) + ( α _ 3 - α _ 2 ) + ( α _ 4 - α _ 3 ) + . . . . . . . . . . + ( α _ 13 - α _ 12 ) ( α _ 3 - α _ 2 ) + ( α _ 7 - α _ 6 ) + ( α _ 11 - α _ 10 ) So, S = (12a)/(3a) = 4

JEE Advanced 2015Paper 2

Exam: JEE Advanced
Year: 2015
Paper: Paper 2
Subject: Mathematics
Chapter: Complex Number

JEE Advanced 2015 · Paper 2 · Mathematics · Complex Number

For any integer k, let α _ k = ((kπ)/(7)) + isin ((kπ)/(7)), wherei...

For any integer

k,

let

{\alpha}_{k} = \cos \left(\dfrac{k\pi}{7}\right) + isin \left(\dfrac{k\pi}{7}\right), wherei = \sqrt{- 1} .

The value of the expression

\dfrac{\mathop{∑}_{k = 1}^{12}\left|{\alpha}_{k + 1} - {\alpha}_{k}\right|}{\mathop{∑}_{k = 1}^{3}\left|{\alpha}_{4k - 1} - {\alpha}_{4k - 2}\right|}

Correct answer

Step-by-step solution

$S = \dfrac{\mathop{∑}_{k = 1}^{12}\left|{\alpha}_{k + 1} - {\alpha}_{k}\right|}{\mathop{∑}_{k = 1}^{3}\left|{\alpha}_{4k - 1} - {\alpha}_{4k - 2}\right|}$
$| {\alpha}_{r + 1} - {\alpha}_{r} |$ = side length of the polygon having 14 sides = a

$S = \dfrac{\left({\alpha}_{2} - {\alpha}_{1}\right) + \left({\alpha}_{3} - {\alpha}_{2}\right) + \left({\alpha}_{4} - {\alpha}_{3}\right) + . . . . . . . . . . + \left({\alpha}_{13} - {\alpha}_{12}\right)}{\left({\alpha}_{3} - {\alpha}_{2}\right) + \left({\alpha}_{7} - {\alpha}_{6}\right) + \left({\alpha}_{11} - {\alpha}_{10}\right)}$
So, $S = \dfrac{12a}{3a} = 4$

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