2021 HCI P1 Q5
Do not use a calculator for this question.
(a)
The three complex numbers ${{z}_{1}}$, ${{z}_{2}}$ and ${{z}_{3}}$ are given as ${{z}_{1}}=2\mathbf{i}$, ${{z}_{2}}=2{{\text{e}}^{\frac{\pi }{6}\,\mathbf{i}}}$ and ${{z}_{3}}=\frac{1}{7}\left( \cos \frac{2}{3}\pi +\mathbf{i}\sin \frac{2}{3}\pi \right)$. Find $\frac{1}{{{z}_{3}}}\left( {{z}_{2}}-{{z}_{1}} \right)$ in the form $r{{\text{e}}^{\mathbf{i}\,\theta }}$, where $r>0$ and $-\pi <\theta \le \pi $.
[3]
(b)
The complex number $z$ satisfies the equation $2{{z}^{3}}+a{{z}^{2}}+18z-9=0$, where $a$ is a real number. It is given that one root is of the form $z=k\,\mathbf{i}$, where $k$ is real and negative. Find the values of $a$ and $k$, and the other roots of the equation.
[6]
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