Q108 / 180 · KCET 2008 · Mathematics
The value of $\sum_{k=1}^{6}\left(\sin \frac{2 k \pi}{7}-i \cos \frac{2 k \pi}{7}\right)$ is
- Correct answerA$\mathrm{i}$
- B0
- C$-\mathrm{i}$
- D$-1$
Solution
$\sum_{k=1}^{6}\left(\sin \frac{2 k \pi}{7}-i \cos \frac{2 k \pi}{7}\right)$
$$
=-i \sum_{k=1}^{6}\left(\cos \frac{2 k \pi}{7}+i \sin \frac{2 k \pi}{7}\right)
$$
$$
\begin{aligned}
&=-\mathrm{i} \sum_{\mathrm{k}=1}^{6} \alpha^{\mathrm{k}}\left(\text { where } \alpha=\cos \frac{2 \mathrm{k} \pi}{7}+\mathrm{i} \sin \frac{2 \mathrm{k} \pi}{7}\right) \\
&=-\mathrm{i}\left(\frac{\alpha\left(1-\alpha^{6}\right)}{1-\alpha}\right)=-\mathrm{i}\left(\frac{\alpha-\alpha^{7}}{1-\alpha}\right) \\
&=-\mathrm{i}\left(\frac{\alpha-1}{1-\alpha}\right)=\mathrm{i}
\end{aligned}
$$
Hence, the correct answer is option: 'A'.