2025 VJC P2 Q4

Do not use a calculator in answering this question.

The complex number \(z\) has modulus \(1\) and argument \(\theta\), where \(\frac{\pi}{2} < \theta < \pi\), and the complex number \(w\) is given by \(w = \mathrm{i}\sqrt{3}z\). The point \(P\) on the Argand diagram represents \(z\).

1. On the copy of the Argand diagram with origin \(O\) in the Printed Answer Booklet, plot the points \(Q\) and \(R\) to represent \(w\) and \(z - w\) respectively. Show clearly the geometrical relationship between the points \(P\), \(Q\) and \(R\).[3]
2. Find the area of the quadrilateral \(ORPQ\).[1]

It is given that \(\theta = \frac{3\pi}{4}\).

3. Find \(z\) in the form \(x + y\mathrm{i}\), where \(x\) and \(y\) are real numbers.[2]
4. Show that \(z - w = k\left[ (\sqrt{3} - 1) + (\sqrt{3} + 1)\mathrm{i} \right]\), where \(k\) is a constant to be determined.[2]
5. Hence show that \(\tan \frac{5\pi}{12} = \frac{\sqrt{3} + 1}{\sqrt{3} - 1}\).[1]