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UPSC NDA I 2022 Mathematics Solved Paper

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Question : 47 of 120

Marks: +1, -0

What is the modulus of

\left(\frac{\sqrt{-3}}{2}-\frac{1}{2}\right)^{200}?

$\frac{1}{4}$
$\frac{1}{2}$
1
$2^{200}$

Solution:

Concept:

The modulus of a complex number raised to a power equals the modulus raised to that power:

|z^{n}| = |z|^{n}

.

Explanation:

Let

z = \frac{\sqrt{-3}}{2} - \frac{1}{2}

.

Since

\sqrt{-3} = i\sqrt{3}

, we write

z = \frac{i\sqrt{3}}{2} - \frac{1}{2}

.

The modulus of

z

is

|z| = \sqrt{x^2 + y^2}

, where

x = -\frac{1}{2}

and

y = \frac{\sqrt{3}}{2}

.

Compute

|z| = \sqrt{\left(-\frac{1}{2}\right)^2 + \left(\frac{\sqrt{3}}{2}\right)^2} = \sqrt{\frac{1}{4} + \frac{3}{4}} = \sqrt{1} = 1

.

Now, the required modulus is

|z^{200}| = |z|^{200} = 1^{200} = 1

.

Answer:

The modulus is

1

, which corresponds to option C.

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