2024 RI Promo Q7

Do not use a calculator in answering this question.

1. Given that \(x=1+3\mathbf{i}\) is a root of the equation \(x^3 + ax^2 + 18x + b = 0\), find the values of the real numbers \(a\) and \(b\), and the other roots.[5]
2. The complex numbers \(z\) and \(w\) satisfy the following equations.

\(w^* + z = 4 - 6\mathbf{i}\)
\(w - 2z = 1 + 10\mathbf{i}\)

Find \(z\) and \(w\), giving your answers in the form \(c + \mathbf{i}d\), where \(c\) and \(d\) are real numbers.[5]