<p><strong>Do not use a calculator for this question.</strong></p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">The three complex numbers ${{z}_{1}}$, ${{z}_{2}}$ and ${{z}_{3}}$ are given as ${{z}_{1}}=2\mathbf{i}$, ${{z}_{2}}=2{{\mathrm{e}}^{\frac{\pi }{6}\,\mathbf{i}}}$ 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 $r{{\mathrm{e}}^{\mathbf{i}\,\theta }}$, where $r>0$ and $-\pi <\theta \le \pi $.</span></div><p style="text-align: right;">[3]</p></li><li><div class="list-item-content"><span class="list-item-text">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.</span></div><p style="text-align: right;">[6]</p></li></ol>

**Do not use a calculator for this question.**

The three complex numbers ${{z}_{1}}$, ${{z}_{2}}$ and ${{z}_{3}}$ are given as ${{z}_{1}}=2\mathbf{i}$, ${{z}_{2}}=2{{\mathrm{e}}^{\frac{\pi }{6}\,\mathbf{i}}}$ 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 $r{{\mathrm{e}}^{\mathbf{i}\,\theta }}$, where $r>0$ and $-\pi <\theta \le \pi $.

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.

2021 HCI P1 Q5

2021 HCI P1 Q5

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