2024 SJI P2 Q9

The complex numbers \(z\) and \(w\) are represented by Z and W respectively on an Argand diagram. It is given that \(z=-\sqrt3-\mathrm i\) and \(w=1-\mathrm i\).

1. Plot point Z and point W on an Argand diagram.[2]
2. Find the exact value of the modulus and argument of \(z\) and of \(w\).[3]
3. Hence find the area of triangle OZW, where O is the origin.[2]
4. Given that \(v=\dfrac w{z^*}\), find \(\arg v\) where \(-\pi<\arg v\leq\pi\).[3]
5. Find the smallest positive integer \(n\) such that \(v^n\) is purely imaginary.[3]