2026 RVHS P1 Q11

Do not use a calculator in answering this question.

1. The function \(\mathrm{f}\) is defined by \(\mathrm{f}(z)=(2-\mathrm{i})z^2+10z+20+10\mathrm{i}\).
   1. Find the roots of \(\mathrm{f}(z)=0\).[3]
      The function \(\mathrm{p}\) is defined by \(\mathrm{p}(z)=z^4+az^3+bz^2+cz+d\), where \(a\), \(b\), \(c\) and \(d\) are real numbers. It is given that \(\mathrm{f}(z)\) is a factor of \(\mathrm{p}(z)\).
   2. Without any further calculations, state all the roots of \(\mathrm{p}(z)=0\), explaining your answer clearly.[2]
2. Solve the simultaneous equations, giving \(z\) and \(w\) in the form \(a+b\mathrm{i}\), where \(a\) and \(b\) are real numbers. \[\begin{aligned}z^*w&=8-20\mathrm{i},\\w-2z&=10.\end{aligned}\][5]