2025 ASRJC P2 Q6

The points \(P\) and \(Q\) are represented by the complex numbers \(p\) and \(q\) respectively, where \(p = \sqrt{2} + \sqrt{2}i, \quad \arg(q) = \frac{2}{3}\pi \quad \text{and} \quad |q| = 2.\)

1. Find \(|p|\) and \(\arg(p)\) in exact form.[2]
2. Sketch the points \(P\) and \(Q\) on an Argand diagram.[2]
3. Use the Argand diagram to deduce \(\mathrm{Re}(q)\) and \(\mathrm{Im}(q)\), giving your answers in exact form.[2]
4. The point \(R\) is represented by the complex number \(p+q\). What can you deduce about the shape of quadrilateral \(OPRQ\), where \(O\) is the origin?[1]
5. By considering \(\arg(p+q)\) or otherwise, show that \(\tan\left(\frac{11}{24}\pi\right) = \sqrt{6} + \sqrt{3} + \sqrt{2} + 2\).[3]