Q197 / 270 · NDA 2022 (Phase 1) · Mathematics

What is the modulus of $\left(\frac{\sqrt{-3}}{2}-\frac{1}{2}\right)^{200}$ ?

  • A
    $\frac{1}{4}$
  • B
    $\frac{1}{2}$
  • Correct answer
    C
    1
  • D
    $2^{200}$

Solution

$\begin{aligned} & \left|\left(\frac{\sqrt{-3}}{2}-\frac{1}{2}\right)^{200}\right|=\left|\left(-\frac{1}{2}+\frac{\sqrt{3}}{2} \mathrm{i}\right)^{200}\right| \\ & =\left|-\frac{1}{2}+\frac{\sqrt{3}}{2} \mathrm{i}\right|^{200} \quad\left[\because\left|\mathrm{Z}^n\right|=\mid \mathrm{Z}^n\right] \\ & =\left|\sqrt{\frac{1}{4}+\frac{3}{4}}\right|^{200}=1\end{aligned}$

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

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