2022 RI P1 Q5
Do not use a calculator in answering this question.
The complex numbers $z$ and $w$ are given by
$z=\sin \left( \frac{\pi }{6} \right)+\text{i}\cos \left( \frac{\pi }{6} \right)$ and $w=\sqrt{2}\left[ \sin \left( \frac{\pi }{6} \right)+\text{i}\cos \left( \frac{\pi }{3} \right) \right]$.
(i)
Find $\left| z \right|$ and $\arg \left( z \right)$. Hence find the value of ${{z}^{3}}$.
[3]
(ii)
By considering some suitable form of $w$ or otherwise, find ${{w}^{4}}$.
[3]
(iii)
Hence find the value of ${{z}^{2022}}-{{w}^{2020}}$.
[2]
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