AHL 1.13—Complex numbers continued Questionbank

Practice AHL 1.13—Complex numbers continued with authentic IB Mathematics Applications & Interpretation (AI) exam questions for both SL and HL students. This question bank mirrors Paper 1, 2, 3 structure, covering key topics like core principles, advanced applications, and practical problem-solving. Get instant solutions, detailed explanations, and build exam confidence with questions in the style of IB examiners.

Paper

Level

Question 1

HLPaper 1

Let $A(t)=5.1 \sin (\omega t)$ and $B(t)=3.2 \sin (\omega t+5)$

Find $A(t)+B(t)$ in the form $a \sin (\omega t+B)$ .

[4]

Question 2

HLPaper 1

[Maximum Mark : 7] Let $z_{1}=a\left(\cos \frac{\pi}{4}+i \sin \frac{\pi}{4}\right)$ and $z_{2}=b\left(\cos \frac{\pi}{3}+i \sin \frac{\pi}{3}\right)$ Express $\left(\frac{z_{2}}{z_{1}}\right)^{2}$ in Cartesian form.

[3]

Question 3

HLPaper 1

Let $A(t)=10 \cos (3 t+4)$ and $B(t)=20 \cos (3 t+5)$ $A(t)$ and $B(t)$ are the real parts of the complex numbers $z_{A}$ and $z_{B}$ respectively.

Express $z_{A}$ and $z_{B}$ in the form $r \operatorname{cis}(\theta)$ and $r e^{i \theta}$ , where $\theta$ is in terms of $t$ .

[3]

Find $z_{A}+z_{B}$ in the form $r e^{i \theta}$ , where $\theta$ is in terms of $t$ .

[3]

Question 4

HLPaper 1

Let $z_{1}=\frac{\sqrt{6}-i \sqrt{2}}{2}$ and $z_{2}=1-i$

Find the value of $\frac{z_{1}}{z_{2}}$ in the form of $a+b i$

[2]

Write $z_{1}$ and $z_{2}$ in the form $r \operatorname{cis}(\theta)$ where $r>0$

[2]

Show that $\frac{z_{1}}{z_{2}}=\cos \frac{\pi}{12}+i \sin \frac{\pi}{12}$

[4]

Find the exact values of $\cos \frac{\pi}{12}$ and $\sin \frac{\pi}{12}$ . Justify your answer.

[2]

Question 5

HLPaper 1

Let $z=\sqrt{3}+i$

Find the polar form $r \operatorname{cis}(\theta)$ of the complex numbers $z$ and $-z$ .

[2]

Find the polar form $r \operatorname{cis}(\theta)$ of the complex numbers $\bar{z}$ and $-\bar{z} .[6]$

And represent the four complex numbers above on the Complex plane.

[2]

Question 6

HLPaper 1

A complex number $z=x+i y$ is such that $|z|=|z-3 i|$

Show that the imaginary part of $z$ is $\frac{3}{2}$

[2]

Let $z_{1}$ and $z_{2}$ be the two possible values of $z$ , such that $|z|=3$ .

Sketch a diagram to show the points which represent $z_{1}$ and $z_{2}$ in the complex plane, where $z_{1}$ is in the first quadrant.

[2]

Show that $\arg z_{1}=\frac{\pi}{6}$

[3]

Find $\arg z_{3}$

[2]

Write down the values of $z_{1}$ and $z_{2}$ in polar form.

[2]

Question 7

HLPaper 1

Let $z=2+3 i$ $

Find $z i, z i^{2}, z i^{3}$ and represent them on the diagram above.

[6]

Find $r$

[2]

Express $z i, z i^{2}, z i^{3}$ in Euler form in terms of $\theta$ and describe geometrically the result.

[3]

Question 8

HLPaper 1

Find polar form of the complex numbers

$z=1+i$ and $w=\sqrt{3}-i$

[2]

Find $z w$ by using the Cartesian forms.

[2]

Find $z w$ by using the polar forms.

[3]

Deduce the exact values of $\cos 15^{\circ}, \sin 15^{\circ}$ and $\tan 15^{\circ}$

[4]

Question 9

HLPaper 1

Consider the complex number $z$ and $w$ , where $z=\sqrt{3}-i$ , imaginary part of $\frac{z^{2}}{w}$ is 0 and $\left|\frac{z^{2}}{w}\right|=\frac{1}{2}|z|$ .

Use geometrical reasoning to find the two possibilities for $w$ , giving your answers in exponential form.

[6]

Question 10

HLPaper 1

Given that $z=(b+i)^{2}$ , where $b$ is real and positive,

find the exact value of $b$ when $\arg z=60^{\circ}$ .

[3]