2022 NJC P1 Q8
Two complex numbers are $z=2\left( \cos \frac{\pi }{4}-\text{i}\sin \frac{\pi }{4} \right)$ and $w=\left( -\text{i}\sqrt{3} \right)z$.
(i)
Show that $z+{{w}^{*}}=r{{\text{e}}^{\text{i}\left( \frac{3\pi }{4} \right)}}$ for some positive constant $r$ to be determined exactly.
[3]
(ii)
Hence find the values of $n$ such that ${{\left( z+{{w}^{*}} \right)}^{n}}$ is purely imaginary.
[2]
It is given that $v=\frac{z+{{w}^{*}}}{{{z}^{*}}w}$.
(iii)
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]
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