Q79 / 150 · MHT CET 2022 (07 Aug Shift 1) · Mathematics

Argument of $\frac{1-i \sqrt{3}}{1+i \sqrt{3}}$ is

  • A
    $210^{\circ}$
  • B
    $120^{\circ}$
  • Correct answer
    C
    $240^{\circ}$
  • D
    $60^{\circ}$

Solution

$z=\frac{1-i \sqrt{3}}{1+i \sqrt{3}} \times \frac{1-i \sqrt{3}}{1-i \sqrt{3}}=\frac{-2-2 \sqrt{3} i}{4}=-\frac{1}{2}-\frac{1}{2} \sqrt{3} i$ $\begin{aligned} & \text { Now Arg }(Z)=\pi+\tan ^{-1}\left|\frac{\operatorname{Im}(Z)}{\operatorname{Re}(Z)}\right| \\ & =180^{\circ}+\tan ^{-1}(\sqrt{3})=180^{\circ}+60^{\circ}=240^{\circ}\end{aligned}$

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

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