In the Argand diagram, the ${{P}_{1}}$ and ${{P}_{2}}$ represent the complex numbers $z$ and ${{z}^{

Question

In the Argand diagram, the ${{P}_{1}}$ and ${{P}_{2}}$ represent the complex numbers $z$ and ${{z}^{2}}$ respectively, where $z=\sqrt{3}+\mathrm{i}\sqrt{3}$. Find the exact modulus and argument of $z$. [2] Mark the points ${{P}_{\mathrm{1}}}$ and ${{P}_{2}}$ on an Argand diagram and find the area of the triangle $O{{P}_{1}}{{P}_{2}}$, where $O$ represents the complex number $0$. [3] Let $w=2{{\mathrm{e}}^{\mathrm{i}\left( -\frac{\pi }{3} \right)}}$. Find the set of integer values $n$ such that $\arg \left( {{w}^{n}}{{z}^{3}} \right)=-\frac{\pi }{4}$. [4]

Accepted Answer

(i) $|z|=\sqrt{6}$, $\arg z = \frac{\pi}{4}$ (ii) Area $= 3\sqrt{3}$ units² (iii) $\{n \in \mathbb{Z} : n = 3(1-2k), k \in \mathbb{Z}\}$

2019 EJC P1 Q7

2019 EJC P1 Q7

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