2022 DHS P2 Q2
Do not use a calculator in answering this question.
The complex number $w$is given by $2-3\mathbf{i}$.
(a)
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 $.
(b)
Find the values of $a$ and $b$.
[2]
(c)
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]
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