N2025 P1 Q3

Do not use a caulculator in answering this question.

A complex number \(z\) is given by \(z = \frac{\sqrt{3} + \mathrm{i}}{\sqrt{3} - \mathrm{i}}\).

1. Find \(\vert{}z\vert{}\) and \(\arg(z)\).[4]

The point \(A\) represents the complex number \(z\) on an Argand diagram. The point \(A\) is rotated anticlockwise through \(\frac{1}{2}\pi\) about the origin \(O\) to the point \(B\).

2. Plot the points \(A\) and \(B\) on an Argand diagram. State in terms of \(z\) the complex number that is represented by \(B\), explaining your answer.[3]