Arguments after rotation and reflection · source reference

Question

A complex number is given by $z=a+(a-3)i$, where $a$ is a positive real constant. It is given that $\arg z=\theta$, where $-\pi/2<\theta<0$. Find, leaving answers in terms of $\theta$:

(i) $\arg(-2z)$ [2];

(ii) $\arg(z-2a)$ [2].

[modified 2023/HCI/Prelim/II/3a]

Source working

Printed working

(i) $\arg(-2z_1)=\pi+\theta$.

(ii)

$$\begin{aligned}z_1-2a&=a+(a-3)i-2a=-a+(a-3)i\\\arg(z_1-2a)&=\arg[-a+(a-3)i]\\&=-\pi+|\theta|=-\pi+(-\theta)=-\pi-\theta.\end{aligned}$$

Source notation

The working uses $z_1$ for the question’s $z$; it is the same complex number. First diagram labels $z_1=(a,a-3)$, $-2z_1$, $a$, $a-3$, $-2a$, $-2(a-3)$, Re, Im, O, $\theta$ and $\pi+\theta$. Second diagram reflects $z_1$ horizontally to $z_1-2a=(-a,a-3)$, with labels $-a,a,a-3,|\theta|,-\pi,\theta$.

Printed answer key

(i) $\pi+\theta$; (ii) $-\pi-\theta$.

Native diagrams

Provenance

2025 TJC H2 Math (9758) JC2 MSM: COMPLEX NUMBERS. Question pages 1; solution pages 3. These are original document page numbers, retained as text metadata.