2021 HCI P1 Q5

Do not use a calculator for this question.

1. 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 \(-\pi <\theta \le \pi \).[3]
2. 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]