Do not use a calculator in answering this question. The complex numbers $z$ and $w$ satisfy the

Question

Do not use a calculator in answering this question. The complex numbers $z$ and $w$ satisfy the following equations. $\mathrm{i}w + z = 3\mathrm{i}$ $2w^* - 3z = 12 - \mathrm{i}$ Find $z$ and $w$, giving your answers in the form $x + \mathrm{i}y$ where $x$ and $y$ are real numbers.[4] It is given that $\mathrm{f}(x) = ax^4 + x^3 + bx^2 - 9x + 35$, where $a$ and $b$ are real numbers, and $\mathrm{f}(x) = 0$ has no repeated roots. The graph of $y = \mathrm{f}(x)$ intersects the $x$-axis exactly once. Explain why $a = 0$.[2] One of the roots of $\mathrm{f}(x) = 0$ is $1 + 2\mathrm{i}$. Find the other roots and the value of $b$.[3]

Accepted Answer

(a) $w=-4i$ and $z=-4+3i$ (b)(ii) other roots are $x=1-2i$ and $x=-7$ , $b =5$

2025 YIJC P1 Q8

2025 YIJC P1 Q8