Do not use calculator in answering this question. Three complex numbers are ${{z}_{1}}=1+\sqrt{3}\ma

Question

Do not use calculator in answering this question. Three complex numbers are ${{z}_{1}}=1+\sqrt{3}\mathrm{i}$, ${{z}_{2}}=1-\mathrm{i}$ and ${{z}_{3}}=2\left( \cos \frac{1}{6}\pi +\mathrm{i}\sin \frac{1}{6}\pi ight)$. Find $\frac{{{z}_{1}}}{{{z}_{2}}{{z}_{3}}}$ in the form $r\,(\cos 	heta +\mathbf{i}\sin 	heta )$, where $r>0$ and $-\pi [4] A fourth complex number, ${{z}_{4}}$, is such that $\frac{{{z}_{1}}{{z}_{4}}}{{{z}_{2}}{{z}_{3}}}$ is purely imaginary and $\left| \frac{{{z}_{1}}{{z}_{4}}}{{{z}_{2}}{{z}_{3}}} ight|=1$ . Find the possible values of ${{z}_{4}}$ in the form $r\,(\cos 	heta +\mathbf{i}\sin 	heta )$, where $r>0$ and $-\pi [3]

Accepted Answer

(i) $\frac{1}{\sqrt{2}}\left( \cos \frac{5}{12}\pi +\mathrm{i}\sin \frac{5}{12}\pi ight)$ (ii) ${{z}_{4}}=\sqrt{2}\left[ \cos \left( \frac{1}{12}\pi ight)+\mathrm{i}\sin \left( \frac{1}{12}\pi ight) ight]$ 	ext{ or } $\sqrt{2}\left[ \cos \left( -\frac{11}{12}\pi ight)+\mathrm{i}\sin \left( -\frac{11}{12}\pi ight) ight]$

N2020 P1 Q4

N2020 P1 Q4

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