Q61 / 155 · WBJEE 2018 · Mathematics

If $Z_{r}=\sin \frac{2 \pi}{11}-i \cos \frac{2 \pi}{11},$ then $\sum_{r=0}^{10} Z_{r}$ is equal to

  • A
    -1
  • Correct answer
    B
    0
  • C
    i
  • D
    $-i$

Solution

We have, $Z,=\sin \frac{2 \pi r}{11}-i \cos \frac{2 \pi}{11}$ $$ =-i\left(\cos \frac{2 \pi r}{11}+i \sin \frac{2 \pi r}{11}\right) $$ $\therefore$ $$ \begin{aligned} &=-i e^{\frac{i 2 \pi}{11}} \\ \sum_{r=0}^{10} Z_{r} &=-i \sum_{r=0}^{10} e^{\frac{i 2 \pi}{11}} \end{aligned} $$ $$ =-i \times 0=0 $$

Hence, the correct answer is option: 'B'.

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