<ol class="abc-list"><li><div class="list-item-paragraph">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&nbsp;\(a\) and \(b\) are real numbers.<span class="inline-marks">[4]</span></div></li></ol>
<ol class="abc-list" start="2"><li><div class="list-item-paragraph">The diagram shows the curve with equation \(y = \mathrm{f}(x)\), where&nbsp;\(\mathrm{f}(x)\) 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 \(P\).</div><div class="content-block list-item-image" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/public/question-images/pasted/1768725032197-dg94i83v9nb.png" alt="List item image" class="aligncenter" style="max-width: 100%; height: auto;" /></div><div class="list-item-paragraph">One of the roots of the equation \(\mathrm{f}(x) = 0\) is \(2 + \mathrm{i}\). Determine the coordinates of the point \(P\).<span class="inline-marks">[4]</span></div></li></ol>

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

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.

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 \(P\).

2025 MI P2 Q3

2025 MI P2 Q3