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 $a \sin(\omega t + \beta)$ .

[4]

Question 2

HLPaper 2

In a physics simulation, two forces are represented as complex numbers: Let $z = r \operatorname{cis} \theta$ and $w = 3 \operatorname{cis} 5$

Express $z/w$ in terms of $r$ and $\theta$ .

[2]

Express $w/z$ in terms of $r$ and $\theta$ .

[2]

Find $z/w^2$ in terms of $r$ and $\theta$ .

[2]

Question 3

HLPaper 2

In a telecommunications project, engineers use complex numbers to model signal processing.

Find the polar form for $z = 1 + i\sqrt{3}$

[3]

Find the polar form for $w = 2 - 2i$

[2]

Find the polar form of $z \cdot w$

[2]

Find the polar form of $\frac{w}{z}$

[2]

Question 4

HLPaper 1

Consider $w = 2\left(\cos\frac{\pi}{3} + i\sin\frac{\pi}{3}\right)$ . The points represented on an Argand diagram by $w^0, w^1, w^2$ and $w^3$ form the vertices of a quadrilateral, $Q$ .

Express $w^2$ and $w^3$ in modulus-argument form.

[3]

Sketch on an Argand diagram the points represented by $w^0$ , $w^1$ , $w^2$ and $w^3$ .

[2]

Show that the area of the quadrilateral $Q$ is $\frac{21\sqrt{3}}{2}$ .

[3]

Let $z = 2(\cos\frac{\pi}{n} + i\sin\frac{\pi}{n})$ , $n \in \mathbb{Z}^+$ . The points represented on an Argand diagram by $z^0$ , $z^1$ , $z^2$ , $\dots$ , $z^n$ form the vertices of a polygon $P_n$ .

Show that the area of the polygon $P_n$ can be expressed in the form $a(b^n - 1)\sin\frac{\pi}{n}$ , where $a, b \in \mathbb{R}$ .

[6]

Question 5

HLPaper 1

Consider the complex numbers $z_1 = 1 + \sqrt{3}\text{i}$ , $z_2 = 1 + \text{i}$ and $w = \frac{z_1}{z_2}$ .

By expressing $z_1$ and $z_2$ in modulus-argument form write down the modulus of $w$ .

[3]

By expressing $z_1$ and $z_2$ in modulus-argument form write down the argument of $w$ .

[1]

Find the smallest positive integer value of $n$ , such that $w^n$ is a real number.

[2]

Question 6

HLPaper 1

[Maximum Mark: 7]

Let $z_1 = a \left( \cos \frac{\pi}{4} + i \sin \frac{\pi}{4} \right)$ and $z_2 = b \left( \cos \frac{\pi}{3} + i \sin \frac{\pi}{3} \right)$ , where $a, b \in \mathbb{R}^+$ .

Express $\left( \frac{z_2}{z_1} \right)^2$ in Cartesian form.

[7]

Question 7

HLPaper 2

In a digital art project, artists use complex numbers to create intricate designs. Each design element is represented in Cartesian form, but they need to convert them to polar form for rendering.

Find the polar form $r \, \text{cis} \, \theta$ of $z_1 = 1 + \text{i}$ .

[2]

Find the polar form $r \, \text{cis} \, \theta$ of $z_2 = -1 + \text{i}$ .

[2]

Find the polar form $r \, \text{cis} \, \theta$ of $z_3 = -1 - \text{i}$ .

[2]

Find the polar form $r \, \text{cis} \, \theta$ of $z_4 = 1 - \text{i}$ .

[2]

Question 8

HLPaper 1

Let $A(t) = 10 \cos(3t + 4)$ and $B(t) = 20 \cos(3t + 5)$ . The functions $A(t)$ 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 9

HLPaper 1

Let $z = 1 - \cos 2\theta - \text{i}\sin 2\theta$ , $z \in \mathbb{C}$ , $0 \leqslant \theta \leqslant \pi$ .

Solve $2\sin (x + 60^\circ ) = \cos (x + 30^\circ )$ , $0^\circ \leqslant x \leqslant 180^\circ$ .

[5]

Show that $\sin 105^\circ + \cos 105^\circ = \frac{1}{\sqrt{2}}$ .

[3]

Find the modulus and argument of $z$ in terms of $\theta$ . Express each answer in its simplest form.

[9]

Hence find the cube roots of $z$ in modulus-argument form.

[5]

Question 10

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 $a+bi$ .

[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]