Showing your working, find the complex numbers $v$ and $w$ which satisfy the following simultane

Question

Showing your working, find the complex numbers $v$ and $w$ which satisfy the following simultaneous equations. $ \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! v + 2w = 5$ $3v - w^* = 1 + 5\mathbf{i}$[4] Points $O$, $V$ and $W$ on an Argand diagram represent the complex numbers $0$, $v$ and $w$ respectively. Plot these points on an Argand diagram and state the transformation that maps the point $W$ onto $V$.[2] The roots of the equation $z^4 - 2z^3 - 3z^2 + az + b = 0$, where $a$ and $b$ are real numbers, are $z_1$, $z_2$, $z_3$ and $z_4$. It is given that $z_1^2 + z_2^2 + z_3^2 + z_4^2 < 0$. Explain why at most two of $z_1$, $z_2$, $z_3$ and $z_4$ are real.[2]

Accepted Answer

(a)(i) $w=2-\mathbf{i}$; $v=1+2\mathbf{i}$(ii) Transformation: $W$ is rotated anti-clockwise through $\frac{\pi }{2}$ about $O$ to obtain $V$; or $W$ is rotated clockwise through $\frac{3\pi }{2}$ about $O$ to obtain $V$. (b) Since $z_{1}^{2}+z_{2}^{2}+z_{3}^{2}+z_{4}^{2}<0$, the equation has at least one non-real root. In addition, since the coefficients of the polynomial equation are all real, by conjugate root theorem, there is at least $1$ pair of complex conjugate roots. Hence, at most two of ${{z}_{1}}$, ${{z}_{2}}$, ${{z}_{3}}$ and ${{z}_{4}}$ are real.

2024 VJC Promo Q7

2024 VJC Promo Q7