Do not use a calculator in answering this question. One of the roots of the equation \(w^4 - 2w^3 +

Question

Do not use a calculator in answering this question. 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.[6] 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. Find the modulus and argument of $u$.[2] Sketch the points $A$, $B$ and $C$ on an Argand diagram.[2] 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.[3]

Accepted Answer

(a) $p=6,q=65$; other roots $2-3i,-1\pm2i$. (b)(i) $|u|=2,\arg u=\frac{3\pi}4$. (ii) Diagram shown. (iii) $|u+v|^2=13-12\cos(\theta+\frac\pi4)$, so $a=13,b=-12,K=\frac\pi4$.

2025 ACJC P1 Q10

2025 ACJC P1 Q10

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