2024 RI Promo Q7

Do not use a calculator in answering this question.

1. Given that \(x = 1 + 3\mathrm{i}\) 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.
   \(\begin{aligned} w^* + z &= 4 - 6\mathrm{i} \\ w - 2z &= 1 + 10\mathrm{i} \end{aligned}\)
   Find \(z\) and \(w\), giving your answers in the form \(c + \mathrm{i}d\), where \(c\) and \(d\) are real numbers.[5]