SATHEE: Complex Numbers 4 Question 19

Complex Numbers 4 Question 19

19.

For any integer $k$ , let $α_{k} = \cos \frac{k π}{7} + i \sin \frac{k π}{7}$ , where $i = \sqrt{- 1}$ . The value of the expression

$\frac{\sum_{k = 1}^{12} | α_{k + 1} - α_{k} |}{\sum_{k = 1}^{3} | α_{4 k - 1} - α_{4 k - 2} |} is$

(2016 Adv.)

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Answer:

Correct Answer: 19. (4)

Solution:

Given, $α_{k} = \cos \frac{k π}{7} + i \sin \frac{k π}{7}$

$= \cos \frac{2 k π}{14} + i \sin \frac{2 k π}{14}$

$∴ α_{k}$ are vertices of regular polygon having 14 sides.

Let the side length of regular polygon be $a$ .

$∴ | α_{k + 1} - α_{k} | =$ length of a side of the regular polygon

$= a$

and $| α_{4 k - 1} - α_{4 k - 2} | =$ length of a side of the regular polygon

$∴ \frac{\sum_{k = 1}^{12} | α_{k + 1} - α_{k} |}{\sum_{k = 1}^{3} | α_{4 k - 1} - α_{4 k - 2} |} = \frac{12 (a)}{3 (a)} = 4$

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Complex Numbers 4 Question 19

19.

Answer:

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