2021 MI P1 Q9

Do not use a calculator in answering this question.

1. Two complex numbers are \(w=1-\sqrt{3}\mathrm{i}\) and \(z=\sqrt{2}\left( \cos \frac{3}{4}\pi -\mathrm{i}\sin \frac{3}{4}\pi \right)\).
   1. Find \({{w}^{2}}{{z}^{*}}\) in the form \(r\left( \cos \theta +\mathrm{i}\sin \theta \right)\), where \(r>0\) and \(-\pi <\theta \le \pi \).[4]
   2. Show that there is no integer value of \(n\) for which the real part of \({{w}^{n}}\) is zero.[2]
2. 1. One of the roots of the equation \(3{{z}^{3}}+13{{z}^{2}}+20z+14=0\) is \(-1+\mathrm{i}\). Find the other roots of the equation in cartesian form, \(p+\mathrm{i}q\), showing your working.[4]
   2. Hence find the roots of the equation \({{w}^{3}}+13{{w}^{2}}+60w+126=0\).[2]