2022 HCI P2 Q3

The complex number \(z\) is given by \(z=2(\cos \beta +\mathrm{i}\sin \beta )\) where \(0<\beta <\frac{\pi }{2}\).

1. Show that \(\frac{z}{4-{{z}^{2}}}=(k\operatorname{cosec}\beta )\mathrm{i}\), where \(k\) is a positive real constant to be determined.[3]
2. Given that the complex number \(w=-\sqrt{3}+\mathrm{i}\), find the three smallest positive integer values of \(n\) such that \(\left( \frac{z}{4-{{z}^{2}}} \right){{\left( {{w}^{*}} \right)}^{n}}\) is a real number.[4]