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NDA 2017 Maths · Mathematics · Complex Numbers and Quadratic Equations · hard

The value of Σk=1^6(sin (2kπ)/(7)-icos (2kπ)/(7)) is

Single correct option

The value of ∑k=16(sin⁡2kπ7−icos⁡2kπ7)\displaystyle\sum_{k=1}^{6}\left(\sin\dfrac{2k\pi}{7}-i\cos\dfrac{2k\pi}{7}\right) is
  1. A. −i-i
  2. B. 00
  3. C. ii
  4. D. −1-1
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Answer: (C) ii

Write sin⁡θ−icos⁡θ=−i(cos⁡θ+isin⁡θ)=−ieiθ\sin\theta-i\cos\theta=-i(\cos\theta+i\sin\theta)=-ie^{i\theta}. The sum of e2kπi/7e^{2k\pi i/7} for k=1,…,6k=1,\dots,6 is −1-1 (all 7th roots of unity sum to 0, minus the term k=0k=0 or 7). So the total is −i⋅(−1)=i-i\cdot(-1)=i.
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Original question written in the NDA 2017 Maths pattern and checked twice for the correct answer. Not an official question paper. How we write and check questions · Corrections policy · Report a mistake