2022 NJC P1 Q8

Two complex numbers are \(z=2\left( \cos \frac{\pi }{4}-\mathrm{i}\sin \frac{\pi }{4} \right)\) and \(w=\left( -\mathrm{i}\sqrt{3} \right)z\).

1. Show that \(z+{{w}^{*}}=r{{\mathrm{e}}^{\mathrm{i}\left( \frac{3\pi }{4} \right)}}\) for some positive constant \(r\) to be determined exactly.[3]
2. Hence find the values of \(n\) such that \({{\left( z+{{w}^{*}} \right)}^{n}}\) is purely imaginary.[2]

It is given that \(v=\frac{z+{{w}^{*}}}{{{z}^{*}}w}\).

3. By finding \(\arg \left( v \right)\) or otherwise, find an equation relating \(\operatorname{Re}\left( v \right)\) and \(\operatorname{Im}\left( v \right)\). Also, find \(\left| v \right|\) exactly.[4]