2022 TMJC P1 Q9

The complex number \(z\) is given by \(z=x+y\mathbf{i}\), where \(x\) and \(y\) are non-zero real numbers. Given that \(\left| z \right|=1\), find the possible values of \(z\) for which \(\frac{{{\left( {{z}^{2}} \right)}^{*}}}{z}\) is real.[6]

Without the use of a calculator, find the roots of the equation \({{z}^{2}}=33+56\mathbf{i}\), expressing your answer in cartesian form \(x+\mathbf{i}y\) where \(x\) and \(y\) are real.[4]

Hence, find in cartesian form, the roots of the equation \({{w}^{2}}=-33+56\mathbf{i}\).[2]