2019 ACJC P1 Q8

Do not use a calculator in answering this question.

1. The complex numbers \(z\) and \(w\) satisfy the following equations
   \(w-2z=9\),
   \(3w-wz^*=17-30\mathrm{i}\).
   Find \(w\) and \(z\) in the form \(a+b\mathrm{i}\), where \(a\) and \(b\) are real and \(\operatorname{Re}(z)<0\).[4]
2. 1. Given that \(-\mathrm{i}\) is a root of the equation
      \({{z}^{3}}+k{{z}^{2}}+\left( 8+2\sqrt{2}\mathrm{i} \right)z+8\mathrm{i}=0\),
      where \(k\) is a constant to be determined, find the other roots, leaving your answers in exact cartesian form \(x+y\mathrm{i}\), showing your working.[3]
   2. Hence solve the equation \(\mathrm{i}{{z}^{3}}+k{{z}^{2}}+\left( 2\sqrt{2}-8\mathrm{i} \right)z-8\mathrm{i}=0\), leaving your answers in exact cartesian form.[2]
   3. Let \({{z}_{0}}\) be the root in (i) such that \(\arg ({{z}_{0}})>0\). Find the smallest positive integer value of \(n\) such that\({{(\mathrm{i}{{z}_{0}})}^{n}}\) is a purely imaginary number.[2]