Let A =left{θ ∈(0,2 π): frac{1+2 i sin θ}{1- i sin θ}right. is purely imaginary } . Then sum o

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4-1.Complex numbers

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Let $A =\left\{\theta \in(0,2 \pi): \frac{1+2 i \sin \theta}{1- i \sin \theta}\right.$ is purely imaginary $\}$. Then the sum of the elements in $A$ is

A

$\pi$

B

$2 \pi$

C

$4 \pi$

D

$3 \pi$

(JEE MAIN-2023)

Solution

$z=\frac{1+2 i \sin \theta}{1-i \sin \theta} \times \frac{1+i \sin \theta}{1+i \sin \theta}$

$z=\frac{1-2 \sin ^2 \theta+i(3 \sin \theta)}{1+\sin ^2 \theta}$

$\operatorname{Re}(z)=0$

$\frac{1-2 \sin ^2 \theta}{1+\sin ^2 \theta}=0$

$\sin \theta=\frac{ \pm 1}{\sqrt{2}}$

$A=\left\{\frac{\pi}{4}, \frac{3 \pi}{4}, \frac{5 \pi}{4}, \frac{7 \pi}{4}\right\}$

$\text { sum }=4 \pi(\text { Option } 3)$

Standard 11

Mathematics

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