N2008 P1 Q8

A graphic calculator is not to be used in answering this question.

1. It is given that \({{z}_{1}}=1+\sqrt{3}\mathbf{i}\). Find the value of \({{z}_{1}}^{3}\), showing clearly how you obtain your answer. [3]
2. Given that \(1+\sqrt{3}\mathbf{i}\) is a root of the equation \(2{{z}^{3}}+a{{z}^{2}}+bz+4=0\), find the values of the real numbers \(a\) and \(b\). [4]

3. For these values of \(a\) and \(b\), solve the equation in part (ii), and show all the roots on an Argand diagram.

   [4]