If real part of complex number z=frac{3+2 i cos θ}{1-3 i cos θ}, θ ∈left(0, frac{π}{2}right) i

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4-1.Complex numbers

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If the real part of the complex number $z=\frac{3+2 i \cos \theta}{1-3 i \cos \theta}, \theta \in\left(0, \frac{\pi}{2}\right)$ is zero, then the value of $\sin ^{2} 3 \theta+\cos ^{2} \theta$ is equal to $.....$

A

$1$

B

$3$

C

$2$

D

$4$

(JEE MAIN-2021)

Solution

$\operatorname{Re}(z)=\frac{3-6 \cos ^{2} \theta}{1+9 \cos ^{2} \theta}=0$

$\Rightarrow \theta=\frac{\pi}{4}$

Hence, $\sin ^{2} 3 \theta+\cos ^{2} \theta=1$

Standard 11

Mathematics

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