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UPSC NDA I and II 2020 Math Solved Paper

Question : 15 of 120

Marks: +1, -0

\frac{\cos\theta + i\sin\theta}{\cos\theta - i\sin\theta}

i=\sqrt{-1}

Solution:

Concept:

The modulus of a quotient of two complex numbers equals the quotient of their moduli:

\left| \frac{z_1}{z_2} \right| = \frac{|z_1|}{|z_2|}

For any real

\theta

|\cos\theta + i\sin\theta| = \sqrt{\cos^2\theta + \sin^2\theta} = 1

Explanation:

We need

\left| \frac{\cos\theta + i\sin\theta}{\cos\theta - i\sin\theta} \right|

Using the property of modulus for division:

\left| \frac{\cos\theta + i\sin\theta}{\cos\theta - i\sin\theta} \right| = \frac{|\cos\theta + i\sin\theta|}{|\cos\theta - i\sin\theta|}

Compute the numerator modulus:

|\cos\theta + i\sin\theta| = \sqrt{\cos^2\theta + \sin^2\theta} = \sqrt{1} = 1

Compute the denominator modulus:

|\cos\theta - i\sin\theta| = \sqrt{\cos^2\theta + (-\sin\theta)^2} = \sqrt{\cos^2\theta + \sin^2\theta} = 1

Therefore the required modulus is

\frac{1}{1} = 1

Answer:

1 (Option B)

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