<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">One of the roots of the equation \(w^4 - 2w^3 + 10w^2 + pw + q = 0\), where \(p\) and \(q\) are real, is \(2 + 3i\). Find the values of \(p\) and \(q\) and the other roots of the equation.</span><span class="inline-marks">[6]</span></div></li><li><div class="list-item-content"><span class="list-item-text">The complex numbers \(u\) and \(v\) are such that \(u = -\sqrt{2} + i\sqrt{2}\), and \(|v| = 3\) and \(\arg v = \theta\), where \(0 &lt; \theta &lt; \frac{\pi}{4}\). The points \(A\), \(B\) and \(C\) represents \(u\), \(v\) and \(u + v\) respectively on an Argand diagram.</span></div><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Find the modulus and argument of \(u\).</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Sketch the points \(A\), \(B\) and \(C\) on an Argand diagram.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">By finding the angle \(\angle OAC\) in terms of \(\theta\) or otherwise, show that \(|u + v|^2 = a + b\cos(\theta + K)\), where \(a\), \(b\) and \(K\) are constants to be determined.</span><span class="inline-marks">[3]</span></div></li></ol></li></ol>

One of the roots of the equation \(w^4 - 2w^3 + 10w^2 + pw + q = 0\), where \(p\) and \(q\) are real, is \(2 + 3i\). Find the values of \(p\) and \(q\) and the other roots of the equation.

The complex numbers \(u\) and \(v\) are such that \(u = -\sqrt{2} + i\sqrt{2}\), and \(|v| = 3\) and \(\arg v = \theta\), where \(0 < \theta < \frac{\pi}{4}\). The points \(A\), \(B\) and \(C\) represents \(u\), \(v\) and \(u + v\) respectively on an Argand diagram.

Sketch the points \(A\), \(B\) and \(C\) on an Argand diagram.

By finding the angle \(\angle OAC\) in terms of \(\theta\) or otherwise, show that \(|u + v|^2 = a + b\cos(\theta + K)\), where \(a\), \(b\) and \(K\) are constants to be determined.

Complex Numbers

International Baccalaureate Higher Level Mathematics

2025 ACJC P1 Q10

2025 ACJC P1 Q10