2023 VJC Promo Q7

Do not use a calculator in answering this question.

1. One of the roots of the equation \(3{{x}^{3}}+p{{x}^{2}}+qx+3=0\), where \(p\) and \(q\) are real, is \(1+\sqrt{2}\,\mathbf{i}\).
   Find the other roots and the values of  \(p\) and \(q\).[5]
2. The complex numbers \(w\) and \(z\), with \(\left| w \right|<\left| z \right|\), satisfy the simultaneous equations
   \(w{{z}^{*}}=3-\mathbf{i}\) and \(w+{{z}^{*}}=2-\mathbf{i}\).
   Find \(w\) and \(z\).[5]