JEE Advanced 2015 · Paper 2 · Mathematics Q41

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MathematicsInteger+4Question 41 of 60

For any integer $k$ , let $\alpha_k=\cos\left(\dfrac{k\pi}7\right)+i\sin\left(\dfrac{k\pi}7\right)$ , where $i=\sqrt{-1}$ . The value of the expression $\dfrac{\sum_{k=1}^{12}|\alpha_{k+1}-\alpha_k|}{\sum_{k=1}^{3}|\alpha_{4k-1}-\alpha_{4k-2}|}$ is

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Answer and worked solution

Official answer: 4

Points. The $\alpha_k$ lie on the unit circle at angles $\frac{k\pi}7$ . Consecutive points are always the same distance apart:
$|\alpha_{k+1}-\alpha_k|=2\sin\frac\pi{14}.$

Sums.

Numerator: 12 such terms.
Denominator: 3 terms, each again consecutive ( $4k-1$ and $4k-2$ ).

Ratio. $12/3=4$ .

Answer: 4

Topics:Geometry in the complex plane Roots of unity

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JEE Advanced 2015 · Paper 2, Mathematics question 41