2020 CJC P1 Q2
The complex number $z$ has modulus $2$ and argument $\frac{\pi }{8}$ . It is also given that $w=1+\text{i}$.
(i)
Given that $n$ is an integer, find $\frac{{{z}^{n}}}{w*}$ in terms of $n$, giving your answer in the form $r{{e}^{i\theta }}$ .
[3]
(ii)
Hence, find the smallest two positive integers $n$ such that $\frac{{{z}^{n}}}{w*}$ is real and negative.
[3]
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