$A$ value of $\theta$, for which $\frac{2+3 i \sin \theta}{1-2 i \sin \theta}, \mathrm{i}=\sqrt{-1}$ is…

$A$ value of $\theta$, for which $\frac{2+3 i \sin \theta}{1-2 i \sin \theta}, \mathrm{i}=\sqrt{-1}$ is purely imaginary, is
  1. $\frac{\pi}{6}$
  2. $\frac{\pi}{3}$
  3. $\sin ^{-1}\left(\frac{1}{\sqrt{3}}\right)$
  4. $\sin ^{-1}(\sqrt{3})$

Solution

$\mathrm{z}=\frac{2+3 i \sin \theta}{1-2 i \sin \theta} \times \frac{1+2 i \sin \theta}{1+2 i \sin \theta}=\frac{\left(2-6 \sin ^2 \theta\right)+\mathrm{i}(7 \sin \theta)}{1+4 \sin ^2 \theta}$ for $z$ to be purely imaginary $\operatorname{Re}(z)=0$ $\Rightarrow 6 \sin ^2 \theta=2$ $\Rightarrow \theta=\sin ^{-1}\left(\frac{1}{\sqrt{3}}\right)$

Asked in: MHT CET 2022 (11 Aug Shift 1)

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