2025 ACJC P1 Q10

Do not use a calculator in answering this question.

1. One of the roots of the equation \(\omega^4 - 2\omega^3 + 10\omega^2 + p\omega + q = 0\), where \(p\) and \(q\) are real, is \(2 + 3\mathrm{i}\). Find the values of \(p\) and \(q\) and the other roots of the equation.[6]
2. The complex numbers \(u\) and \(v\) are such that \(u = -\sqrt{2} + \mathrm{i}\sqrt{2}\), and \(\left| v \right| = 3\) 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.
   1. Find the modulus and argument of \(u\).[2]
   2. Sketch the points \(A\), \(B\) and \(C\) on an Argand diagram.[2]

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

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