Q166 / 270 · NDA 2022 (Phase 2) · Mathematics

Paragraph: Let $z=\frac{1+i \sin \theta}{1-i \sin \theta}$ where $i=\sqrt{-1}$
Question: What is the modulus of $z$ ?

  • Correct answer
    A
    $1$
  • B
    $\sqrt{2}$
  • C
    $1+\sin ^2 \theta$
  • D
    $\frac{1+\sin ^2 \theta}{1-\sin ^2 \theta}$

Solution

$\begin{aligned} & \because z=\frac{1+i \sin \theta}{1-i \sin \theta} \\ & \therefore|z|=\frac{|1+i \sin \theta|}{|1-i \sin \theta|}=\frac{\sqrt{1+\sin ^2 \theta}}{\sqrt{1+\sin ^2 \theta}}=1\end{aligned}$

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

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