2021 RI P1 Q6

Do not use a calculator in answering this question.

1. Show that \(z=2\mathbf{i}\) is a root of the equation \({{z}^{3}}+2z+4\mathbf{i}=0\).[2]
   Hence find the other roots.[3]
2. Let \({{w}_{1}}=-\frac{\sqrt{6}}{2}+\frac{\sqrt{2}}{2}\mathbf{i}\) and \({{w}_{2}}=1+\mathbf{i}\).
   Find the smallest positive integer \(n\) such that \(\arg {{\left( \frac{{{w}_{2}}}{{{w}_{1}}} \right)}^{n}}=-\frac{\pi }{2}\).[4]