2021 YIJC P2 Q3
Do not use a calculator in answering this question.
(a)
The complex number $z$ is given by $z=\frac{{{\left( 1-\mathbf{i} \right)}^{3}}}{\sqrt{2}{{\left( a+\mathbf{i} \right)}^{2}}}$, where $a<0$.
(i) Given that $\left| z \right|=\frac{1}{2}$, show that $\arg z=-\frac{5\pi }{12}$.
[5]
(ii) Hence find the smallest positive integer $n$ for which ${{z}^{n}}$ has equal real and imaginary parts.
[2]
(b)
The complex number $q$ is given by $\frac{{{\text{e}}^{-\mathbf{i}\theta }}}{{{\text{e}}^{\mathbf{i}\theta }}-\mathbf{i}}$, where $0<\theta <\frac{\pi }{2}$. Show that
$\operatorname{Re}\left( q \right)=\frac{1}{2}\left( 1+2\sin \theta \right)$.
[4]
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