2025 YIJC P1 Q8

Do not use a calculator in answering this question.

1. The complex numbers \(z\) and \(w\) satisfy the following equations.
   \(\mathrm{i}w + z = 3\mathrm{i}\)
   \(2w^* - 3z = 12 - \mathrm{i}\)
   Find \(z\) and \(w\), giving your answers in the form \(x + \mathrm{i}y\) where \(x\) and \(y\) are real numbers.[4]
2. It is given that \(\mathrm{f}(x) = ax^4 + x^3 + bx^2 - 9x + 35\), where \(a\) and \(b\) are real numbers, and \(\mathrm{f}(x) = 0\) has no repeated roots. The graph of \(y = \mathrm{f}(x)\) intersects the \(x\)-axis exactly once.
   1. Explain why \(a = 0\).[2]

   One of the roots of \(\mathrm{f}(x) = 0\) is \(1 + 2\mathrm{i}\).

   2. Find the other roots and the value of \(b\).

      [3]