The complex number $z$ has modulus $2$ and argument $\frac{\pi }{8}$ . It is also given that $w=1+\m

Question

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

Accepted Answer

(i) $\frac{z^n}{w^*} = 2^{n-\frac{1}{2}}e^{i(\frac{n\pi}{8} + \frac{\pi}{4})}$ (ii) $n = 6, 22$

2020 CJC P1 Q2

2020 CJC P1 Q2

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