<p><strong>Do not use a calculator in answering this question.</strong></p>
<ol class="abc-list"><li><div class="list-item-paragraph">Given that&nbsp;\(x = 1 + 3\mathrm{i}\) is a root of the equation</div><div class="list-item-paragraph" style="text-align: center">\(x^3 + ax^2 + 18x + b = 0,\)</div><div class="list-item-paragraph">find the values of the real numbers \(a\) and \(b\), and the other roots.<span class="inline-marks">[5]</span></div></li><li><div class="list-item-paragraph">The complex numbers \(z\) and \(w\) satisfy the following equations.</div><div class="list-item-paragraph" style="text-align: center">\(\begin{aligned} w^* + z &= 4 - 6\mathrm{i} \\ w - 2z &= 1 + 10\mathrm{i} \end{aligned}\) </div><div class="list-item-paragraph">Find \(z\) and \(w\), giving your answers in the form \(c + \mathrm{i}d\), where&nbsp;\(c\) and \(d\) are real numbers.<span class="inline-marks">[5]</span></div></li></ol>

**Do not use a calculator in answering this question.**

Given that \(x = 1 + 3\mathrm{i}\) is a root of the equation

find the values of the real numbers \(a\) and \(b\), and the other roots.

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.

2024 RI Promo Q7

2024 RI Promo Q7