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 \(w=\frac{8}{{{\left( \sqrt{3}+\mathbf{i} \right)}^{2}}}\).

1. 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]
2. 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]
3. 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]
4. By considering the quadrilateral \(OPRQ\) and the argument of \(z+w\), deduce that
   \(\tan \frac{5\pi }{12}=2+\sqrt{3}\).[3]