2023 MI P1 Q7

Do not use a calculator in answering this question.

The roots of the equation \({{z}^{2}}=18\,\mathbf{i}\) are \({{z}_{1}}\) and \({{z}_{2}}\).

1. Find \({{z}_{1}}\) and \({{z}_{2}}\) in cartesian form \(x+y\mathbf{i}\), showing your working clearly.[4]

The complex number \(w\) is given by \(-1-\mathbf{i}\sqrt{3}\).

2. Find the complex conjugate of \(\frac{w}{{{z}^{2}}}\) in the form \(r{{\mathrm{e}}^{\mathbf{i}\,\theta }}\), where \(r>0\) and \(-\pi <\theta \le \pi \).[4]