N2016 P1 Q7

Do not use a calculator in answering this question.

1. Verify that \(-1+5\mathbf{i}\) is a root of the equation \({{w}^{2}}+\left( -1-8\mathbf{i} \right)w+\left( -17+7\mathbf{i} \right)=0\). Hence, or otherwise, find the second root of the equation in Cartesian form, \(p+\mathbf{i}q\), showing your working.[5]
2. The equation \(z^3-5z^2+16z+k=0\), where \(k\) is a real constant, has a root \(z=1+a\mathrm i\), where \(a\) is a positive real constant. Find the values of \(a\) and \(k\), showing your working.[5]