2022 HCI P2 Q3
The complex number $z$ is given by $z=2(\cos \beta +\text{i}\sin \beta )$ where $0<\beta <\frac{\pi }{2}$.
(i)
Show that $\frac{z}{4-{{z}^{2}}}=(k\operatorname{cosec}\beta )\text{i}$, where $k$ is a positive real constant to be determined.
[3]
(ii)
Given that the complex number $w=-\sqrt{3}+\text{i}$, 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]
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