2014 SAJC P2 Q4 [Modified]
(a)
Find the complex number $z$ in the form $x+y\,\mathbf{i}$ where $x$, $y\in \mathbb{R}$ such that
$\frac{\mathbf{i}z}{z-2{{z}^{*}}-2}=-1$.
[3]
(b)
Show that that for any complex number $z=r{{\text{e}}^{\mathbf{i}\theta }}$, where $r>0$ and $-\pi <\theta \le \pi $,
$\frac{z}{z-r}=\frac{1}{2}-\frac{1}{2}\mathbf{i}\cot \left( \frac{\theta }{2} \right)$.
[2]
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