Do not use a calculator in answering this question.
(a) Show that $z=2 \mathrm{i}$ is a root of the equation $z^3+2 z+4 \mathrm{i}=0$.

[2]

Hence find the other roots.

[3]

(b) Let $w_1=-\frac{\sqrt{6}}{2}+\frac{\sqrt{2}}{2} \mathrm{i}$ and $w_2=1+\mathrm{i}$.
Find the smallest positive integer $n$ such that $\arg \left(\frac{w_2}{w_1}\right)^n=-\frac{\pi}{2}$.

[4]