Conjugation and component equations · source reference

Question

(a) Given $\arg(a+ib)=\theta$, where $a,b$ are real and positive, find in terms of $\theta$ and/or $\pi$:

(i) $\arg(a-ib)$;

(ii) $\arg(b+ia)$. [3]

(b) Solve $iz+2w=0$ and $z-w^*=3$, giving $z,w$ in the form $a+ib$, where $a,b$ are real. [5]

[2015/JJC/Prelim/II/1]

Source working

Printed working

(a)(i) $\arg(a-ib)=-\theta$.

(a)(ii) $\arg(b+ia)=\pi/2-\theta$.

(b)

$$\begin{aligned}iz+2w&=0\quad(1)\\z-w^*&=3\Longrightarrow z=3+w^*\quad(2)\\i(3+w^*)+2w&=0\\3i+iw^*+2w&=0.\end{aligned}$$

Let $w=x+iy$:

$$\begin{aligned}3i+i(x-iy)+2(x+iy)&=0\\(y+2x)+i(3+x+2y)&=0.\end{aligned}$$

Comparing real and imaginary parts:

$$y+2x=0\quad(3),\qquad3+x+2y=0\quad(4).$$

Substitute (3) into (4): $3+x-4x=0$, hence $x=1,y=-2$. Thus $w=1-2i$ and $z=3+1+2i=4+2i$.

Diagram labels and geometry

The first diagram places $u(a,b)$ above the real axis at angle $\theta$, and $u^*(a,-b)$ below at $-\theta$. The second places $(b,a)$ at angle $\pi/2-\theta$; the positive imaginary axis is $\pi/2$ from the real axis. Both diagrams label Re, Im and origin $0$.

Printed answer key

(a)(i) $-\theta$; (ii) $\pi/2-\theta$. (b) $z=4+2i,w=1-2i$.

Native diagrams

Provenance

2025 TJC H2 Math (9758) JC2 MSM: COMPLEX NUMBERS. Question pages 1; solution pages 2, 3. These are original document page numbers, retained as text metadata.