Q48 / 160 · AP EAMCET 2024 (20 May Shift 2) · Mathematics

If $\frac{3-2 i \sin \theta}{1+2 i \sin \theta}$ is purely imaginary number, then $\theta=$

  • A
    $2 n \pi \pm \frac{\pi}{4}$
  • B
    $2 n \pi \pm \frac{\pi}{2}$
  • Correct answer
    C
    $n \pi \pm \frac{\pi}{3}$
  • D
    $n \pi \pm \frac{\pi}{6}$

Solution

$\frac{3-2 i \sin \theta}{1+2 i \sin \theta}=\frac{(3-2 i \sin \theta)(1-2 i \sin \theta)}{1+4 \sin ^2 \theta}=\frac{3-4 \sin ^2 \theta}{1+4 \sin ^2 \theta}$ From the given information real part $=0$ $\Rightarrow \sin ^2 \theta=\frac{3}{4} \Rightarrow \theta=n \pi \pm \frac{\pi}{3}$

Hence, the correct answer is option: 'C'.

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