Complex Number Matching Solution

Modulus of \(\sqrt{3} + i\) Calculation

The modulus \( |z| \) of a complex number \( z = x + iy \) is calculated using the formula \( |z| = \sqrt{x^2 + y^2} \).

For the complex number \( \sqrt{3} + i \), we have \( x = \sqrt{3} \) and \( y = 1 \).

Therefore, the modulus is:

\( |\sqrt{3} + i| = \sqrt{(\sqrt{3})^2 + 1^2} = \sqrt{3 + 1} = \sqrt{4} = 2 \).

This matches element II from List - II.

Argument of \(\sqrt{3} + i\) Calculation

The principal argument \( \theta \) of a complex number \( z = x + iy \) is found using \( \tan(\theta) = \frac{y}{x} \), considering the quadrant.

For \( z = \sqrt{3} + i \), \( x = \sqrt{3} \) and \( y = 1 \). Since both \( x \) and \( y \) are positive, the number lies in the first quadrant.

\( \tan(\theta) = \frac{1}{\sqrt{3}} \).

The principal argument is \( \theta = \frac{\pi}{6} \).

This matches element IV from List - II.

Modulus of \(\frac{(1+i)^2}{1-i}\) Calculation

First, simplify the expression \( \frac{(1+i)^2}{1-i} \).

Calculate the numerator: \( (1+i)^2 = 1^2 + 2(1)(i) + i^2 = 1 + 2i - 1 = 2i \).

The expression becomes \( \frac{2i}{1-i} \).

To simplify further, multiply the numerator and denominator by the conjugate of the denominator (\( 1+i \)):

\( \frac{2i(1+i)}{(1-i)(1+i)} = \frac{2i + 2i^2}{1^2 - i^2} = \frac{2i - 2}{1 - (-1)} = \frac{-2 + 2i}{2} = -1 + i \).

Now, calculate the modulus of \( -1 + i \):

\( |-1 + i| = \sqrt{(-1)^2 + 1^2} = \sqrt{1 + 1} = \sqrt{2} \).

This matches element I from List - II.

Argument of \(\frac{(1+i)^2}{1-i}\) Calculation

From the previous calculation, the simplified complex number is \( -1 + i \).

Here, \( x = -1 \) and \( y = 1 \). Since \( x \) is negative and \( y \) is positive, the number lies in the second quadrant.

The reference angle \( \alpha \) is found using \( \tan(\alpha) = \left|\frac{y}{x}\right| = \left|\frac{1}{-1}\right| = 1 \), which gives \( \alpha = \frac{\pi}{4} \).

For the second quadrant, the principal argument \( \theta \) is \( \pi - \alpha \).

\( \theta = \pi - \frac{\pi}{4} = \frac{3\pi}{4} \).

This matches element III from List - II.

Final Matching Summary

The correct matches are:

• A matches with II (2)
• B matches with IV (\(\frac{\pi}{6}\))
• C matches with I (\(\sqrt{2}\))
• D matches with III (\(\frac{3\pi}{4}\))

Therefore, the correct option is A-II, B-IV, C-I, D-III.