2022 DHS P2 Q2

Do not use a calculator in answering this question.

The complex number \(w\) is given by \(2-3\mathbf{i}\).

1. Find \({{w}^{3}}\) in the form \(x+\mathbf{i}y\), showing your working.[2]

\(w\) is a root of the equation \({{z}^{3}}-5{{z}^{2}}+az+b=0\) where \(a\) and \(b\) are real constants. This equation also has a real root \(\alpha \).

2. Find the values of \(a\) and \(b\).[2]
3. Find a cubic polynomial that has roots \(\mathbf{i}w\), \(\mathbf{i}{{w}^{*}}\) and \(\mathbf{i}\alpha \) in the form \({{z}^{3}}+p{{z}^{2}}+qz+r=0\), where \(p\), \(q\) and \(r\) are constants.[2]