2025 MI P2 Q3

The complex numbers \(z\) and \(w\) satisfy the simultaneous equations\[z + 2z^* + w = 6 + \mathrm{i} \quad \text{and} \quad \mathrm{i}z - w = 3\]Find \(z\) and \(w\), leaving your answers in the form \(a + b\mathrm{i}\), where \(a\) and \(b\) are real numbers.[4]

The diagram shows the curve with equation \(y = \mathrm{f}(x)\), where \(\mathrm{f}(x)\) is given by \(x^3 + mx^2 + nx + 5\), and \(m\) and \(n\) are real constants. The curve crosses the \(x\)-axis at the point \(P\).

One of the roots of the equation \(\mathrm{f}(x) = 0\) is \(2 + \mathrm{i}\). Determine the coordinates of the point \(P\).[4]