It is given that $z=1+2\mathrm{i}$. Without using a calculator, find the values of ${{z}^{2}}$ and $

Question

It is given that $z=1+2\mathrm{i}$. Without using a calculator, find the values of ${{z}^{2}}$ and $\frac{1}{{{z}^{3}}}$ in cartesian form $x+\mathrm{i}y$, showing your working. [4] The real numbers $p$ and $q$ are such that $p{{z}^{2}}+\frac{q}{{{z}^{3}}}$ is real. Find, in terms of $p$, the value of $q$ and the value of $p{{z}^{2}}+\frac{q}{{{z}^{3}}}$. [3]

Accepted Answer

$19p$

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