For any integer k, let α_k=cos⁡(kπ)/7+isin⁡(kπ)/7, where i= √(-1). The value of the expression (∑k=1)/(12α_(k+1)-α_k∑k=13α_(4k-1)-α_(4k-2)) is

Correct answer: 4. S= (∑k=112α_(k+1)-α_k)/(∑k=13α_(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

Q24 / 60 · JEE Advanced 2015 (Paper 2) · Mathematics

For any integer $k,$ let $α_{k} = \cos (\frac{k π}{7}) + i \sin (\frac{k π}{7}), w h e r e i = \sqrt{- 1} .$ The value of the expression $\frac{\sum_{k = 1}^{12} |α_{k + 1} - α_{k}|}{\sum_{k = 1}^{3} |α_{4 k - 1} - α_{4 k - 2}|}$ is

Correct answer: 4

Solution

$S = \frac{\sum_{k = 1}^{12} |α_{k + 1} - α_{k}|}{\sum_{k = 1}^{3} |α_{4 k - 1} - α_{4 k - 2}|}$
$| α_{r + 1} - α_{r} |$ = side length of the polygon having 14 sides = a

$S = \frac{(α_{2} - α_{1}) + (α_{3} - α_{2}) + (α_{4} - α_{3}) + . . . . . . . . . . + (α_{13} - α_{12})}{(α_{3} - α_{2}) + (α_{7} - α_{6}) + (α_{11} - α_{10})}$
So, $S = \frac{12 a}{3 a} = 4$

Hence, the correct answer is 4.

For any integer k, let αk=cos⁡kπ7+isin⁡kπ7, where i= -1. The value of the expression ∑k=112αk+1-αk∑k=13α4k-1-α4k-2 is

Solution

More Complex Number Questions

For any integer $k,$ let $α_{k} = \cos (\frac{k π}{7}) + i \sin (\frac{k π}{7}), w h e r e i = \sqrt{- 1} .$ The value of the expression $\frac{\sum_{k = 1}^{12} |α_{k + 1} - α_{k}|}{\sum_{k = 1}^{3} |α_{4 k - 1} - α_{4 k - 2}|}$ is