2025 RI Complex Numbers Assignment Q3

Do not use a calculator in answering this question

It is given that \({{z}_{1}}\), \({{z}_{2}}\) and \({{z}_{3}}\) are the roots of the equation

\(2{{z}^{3}}+p{{z}^{2}}+qz-4=0\)

such that \(\arg {{z}_{1}}<\arg {{z}_{2}}<\arg {{z}_{3}}\) and \({{z}_{1}}=1-\mathbf{i}\sqrt{3}\). Find the values of the real numbers \(p\) and \(q\) and the remaining roots of the equation.[6]

Hence solve the equation \(2iz^3+pz^2-qiz+4=0\), leaving your answers in exact Cartesian form.[2]