N2014 P1 Q5

It is given that \(z=1+2\mathrm{i}\).

1. 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]
2. 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]