Question

If \(Z_{r}=\sin \frac{2 \pi r}{11}-i \cos \frac{2 \pi r}{11}\) then \(\sum_{r=0}^{10} Z_{r}=\)
(A) \(-1\)
(B) 0
(C) \(\mathrm{i}\)
(D) \(-\mathrm{i}\)

Answer 1

Ans: (B)
Hint \(: Z_{r}=-i\left(\cos \frac{2 \pi r}{11}+i \sin \frac{2 \pi r}{11}\right)=-i e^{i 2 \pi}\)
$$
\Rightarrow \sum_{r=0}^{10} Z_{r}=-i \sum_{r=0}^{10} e^{i \frac{2 \pi r}{11}}=-i \times 0=0
$$

Answer 2

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