2019 TJC P1 Q8
Do not use a calculator in answering this question.
The complex numbers $z$ and $w$ are given by $z=\frac{{{\left( 1+\mathbf{i} \right)}^{4}}}{{{\left( 1-\mathbf{i} \right)}^{2}}}$ and $w=\frac{8}{{{\left( \sqrt{3}+\mathbf{i} \right)}^{2}}}$.
(i)
Express $z$ and $w$ in polar form $r\left( \cos \theta +\mathbf{i}\sin \theta \right)$, where $r>0$ and $-\pi <\theta \le \pi $. Give $r$ and $\theta $ in exact form.
[4]
(ii)
Given that ${{z}^{2}}$, $w$ and ${{w}^{*}}$ are the roots of the equation ${{x}^{3}}+b{{x}^{2}}+cx+d=0$ where $b$, $c$ and $d$ are real values, find the equation.
[3]
(iii)
Sketch on an Argand diagram with origin $O$, the points $P$, $Q$ and $R$ representing the complex numbers $z$, $w$ and $z+w$ respectively.
[2]
(iv)
By considering the quadrilateral $OPRQ$ and the argument of $z+w$, deduce that
$\tan \frac{5\pi }{12}=2+\sqrt{3}$.
[3]
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