9758/2020/P1/Q04

Do not use a calculator in answering this question.

Three complex numbers are $z_{1}=1+\sqrt{3} i, z_{2}=1-i$ and $z_{3}=2\left(\cos \frac{1}{6} \pi+i \sin \frac{1}{6} \pi\right)$.

(i) Find $\frac{z_{1}}{z_{2} z_{3}}$ in the form $r(\cos \theta+i \sin \theta)$, where $r>0$ and $-\pi<\theta \leqslant \pi$.

[4]

A fourth complex number, $z_{4}$, is such that $\frac{z_{1} z_{4}}{z_{2} z_{3}}$ is purely imaginary and $\left|\frac{z_{1} z_{4}}{z_{2} z_{3}}\right|=1$.

(ii) Find the possible values of $z_{4}$ in the form $r(\cos \theta+\mathrm{i} \sin \theta)$, where $r>0$ and $-\pi<\theta \leqslant \pi$.

[3]