Complex Numbers 5 Question 11
12. The value of $\sum _{k=1}^{6} \sin \frac{2 \pi k}{7}-i \cos \frac{2 \pi k}{7}$ is
$(1998,2 M)$
(a) -1
(b) 0
(c) $-i$
(d) $i$
Match the Columns
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Answer:
Correct Answer: 12. (d)
Solution:
- $\sum _{k=1}^{6} \sin \frac{2 k \pi}{7}-i \cos \frac{2 k \pi}{7}$
$$ \begin{aligned} & =\sum _{k=1}^{6}-i \cos \frac{2 k \pi}{7}+i \sin \frac{2 k \pi}{7} \\ & =-i \sum _{k=1}^{6} e^{\frac{i 2 k \pi}{7}}=-i{e^{i 2 \pi / 7}+e^{i 4 \pi / 7}+e^{i 6 \pi / 7}\right. \\ & \left.+e^{i 8 \pi / 7}+e^{i 10 \pi / 7}+e^{i 12 \pi / 7} } \\ & =-i e^{i 2 \pi / 7} \frac{\left(1-e^{i 12 \pi / 7}\right)}{1-e^{i 2 \pi / 7}} \\ & =-i \frac{e^{i 2 \pi / 7}-e^{i 14 \pi / 7}}{1-e^{i 2 \pi / 7}} \quad\left[\because e^{i 14 \pi / 7}=1\right] \\ & =-i \frac{e^{i 2 \pi / 7}-1}{1-e^{i 2 \pi / 7}=i} \end{aligned} $$
