2022 ACJC P1 Q5
Do not use a calculator in answering this question.
Two complex numbers are ${{z}_{1}}=2\left( \cos \frac{\pi }{18}-\mathbf{i}\sin \frac{\pi }{18} \right)$ and ${{z}_{2}}=2\mathbf{i}$.
(i)
Show that $\frac{{{z}_{1}}^{2}}{{{z}_{1}}^{*}}+{{z}_{2}}$ is $\sqrt{3}+\mathbf{i}$.
[3]
(ii)
A third complex number, ${{z}_{3}}$, is such that $\left( \frac{{{z}_{1}}^{2}}{{{z}_{1}}^{*}}+{{z}_{2}} \right){{z}_{3}}$ is real and $\left| \left( \frac{{{z}_{1}}^{2}}{{{z}_{1}}^{*}}+{{z}_{2}} \right){{z}_{3}} \right|=\frac{2}{3}$. Find the possible values of ${{z}_{3}}$ in the form of $r(\cos \theta +\mathbf{i}\sin \theta )$, where $r>0$ and $-\pi <\theta \le \pi $.
[4]
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