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NDA 2022
Exam
NDA
Year
2022
Subject
Mathematics
Chapter
Complex Number

NDA 2022 · Mathematics · Complex Number

What is the modulus of ( -3 2 -(1)/(2))^ 200 ?

What is the modulus of (−32−12)200\left(\frac{\sqrt{-3}}{2}-\frac{1}{2}\right)^{200} ?

Options

  1. A14\frac{1}{4}
  2. B12\frac{1}{2}
  3. C1
  4. D22002^{200}

Correct answer

C. 1

Step-by-step solution

∣(−32−12)200∣=∣(−12+32i)200∣=∣−12+32i∣200[∵∣Zn∣=∣Zn]=∣14+34∣200=1\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}
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