2021 ASRJC P2 Q1
The complex numbers $z$ and $w$ have moduli $k$ and $3{{k}^{2}}$ respectively and arguments $\alpha $ and $4\alpha $ respectively, where $k$ is a positive constant and $-\frac{\pi }{7}<\alpha \le \frac{\pi }{7}$.
(i)
Express $\frac{{{z}^{3}}}{{{w}^{*}}}$ in the form $r{{\text{e}}^{\mathbf{i}\theta }}$ where $r>0$ and $-\pi <\theta \le \pi $.
[2]
(ii)
It is given that $\alpha =\frac{\pi }{21}$. Find the integer values of $n$ such that ${{\left( \frac{{{z}^{3}}}{{{w}^{*}}} \right)}^{n}}$ is real.
[2]
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