2023 HCI P2 Q3

A complex number \({{z}_{1}}\) is given \({{z}_{1}}=a+\left( a-3 \right)\,\,\mathbf{i}\), where \(a\) is a positive real constant. It is given that \(\arg {{z}_{1}}=\theta \), where \(-\frac{\pi }{2}<\theta <0\).

1. Find, leaving your answers in terms of \(\theta \),
   1. \(\arg \left( -2{{z}_{1}} \right)\)[2]
   2. \(\arg \left( {{z}_{1}}-2a \right)\)[2]

Another complex number \({{z}_{2}}\) is given by \({{z}_{2}}=1+3\,\mathbf{i}\).

2. Without using a calculator, find the range of values of \(a\) such that
   \(\frac{{{\left| {{z}_{1}}{{z}_{2}} \right|}^{2}}}{\operatorname{Im}\left( {{z}_{1}}{{z}_{2}} \right)}\le 10\).[5]