The complex numbers $w$ and $z$ are such that $w=-1-\mathrm{i}\sqrt{3}$ and ${{z}^{*}}+z
e 0$. The

Question

The complex numbers $w$ and $z$ are such that $w=-1-\mathrm{i}\sqrt{3}$ and ${{z}^{*}}+z
e 0$. The points $P$ and $Q$ in the Argand diagram represent the complex numbers $w$ and ${{z}^{*}}+z$ respectively. Find $\arg (w)$, leaving your answer in exact form. Hence, state the possible values of angle $POQ$, where $O$ is the origin. [3] Given that ${{z}_{1}}$ and ${{z}_{2}}$ are complex numbers such that $\frac{3{{z}_{1}}-{{z}_{2}}}{3{{z}_{1}}+{{z}_{2}}}$ is purely imaginary, show that $\left| {{z}_{2}} ight|=3\left| {{z}_{1}} ight|$. [3]

Accepted Answer

(a) $\arg(w)=-\frac{2\pi}{3}$, possible angles $\frac{2\pi}{3}$ or $\frac\pi3$; (b) $|z_2|=3|z_1|$

2020 VJC P2 Q3

2020 VJC P2 Q3

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