[Om Sharma]

The sum of all possible values of $\theta \in[-\pi, 2 \pi]$, for which $\frac{1+\mathrm{i} \cos \theta}{1-2 \mathrm{i} \cos \theta}$ is purely imaginary, is equal to [JEE (Main) 8 April 2024 S II]
(A) $4 \pi$
(B) $2 \pi$
(C) $5 \pi$
(D) $3 \pi$

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[Om Sharma]

Solution

$Z=\frac{(1+i \cos \theta)(1+2 i \cos \theta)}{(1-2 i \cos \theta)(1+2 i \cos \theta)} \Rightarrow Z=\frac{1-2 \cos ^{2} \theta+3 i \cos \theta}{1+4 \cos ^{2} \theta}$ is purely imaginary
So, $\operatorname{Re}(\mathrm{Z})=0 \Rightarrow \frac{1-2 \cos ^{2} \theta}{1+4 \cos ^{2} \theta}=0 \Rightarrow \cos ^{2} \theta=\frac{1}{2} \Rightarrow \theta=-\frac{\pi}{4},-\frac{3 \pi}{4}, \frac{\pi}{4}, \frac{3 \pi}{4}, \frac{5 \pi}{4}, \frac{7 \pi}{4}$ sum of all solutions $=3 \pi$

[Om Sharma]

Answer

D

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