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AHL 1.13—Polar and Euler form - IB Questionbank | RevisionDojo

Practice AHL 1.13—Polar and Euler form with authentic IB Mathematics Analysis and Approaches (AA) exam questions for both SL and HL students. This question bank mirrors Paper 1, 2, 3 structure, covering key topics like functions and equations, calculus, complex numbers, sequences and series, and probability and statistics. Get instant solutions, detailed explanations, and build exam confidence with questions in the style of IB examiners.

Paper

Level

Question 1

HLPaper 1

The complex number $z$ is defined by $z = \frac{1+i}{1-i\sqrt{3}}$ . Find, showing all your working,

an expression for $z$ in the form $re^{i\theta}$ , where $r>0$ and $-\pi < \theta \le \pi$ .

[5]

the two square roots of $z$ , giving your answers in the form $re^{i\theta}$ , where $r>0$ and $-\pi < \theta \le \pi$ .

[5]

Question 2

HLPaper 1

The complex number $z$ is defined by $z = \frac{4-4i}{1+i}$ .

Find an expression for $z$ in the form $re^{i\theta}$ , where $r>0$ and $-\pi < \theta \le \pi$ .

[5]

Find the two square roots of $z$ , giving your answers in the form $re^{i\theta}$ , where $r>0$ and $-\pi < \theta \le \pi$ .

[3]

Question 3

HLPaper 1

Find the modulus and argument of each root of: $z^2 - 2iz - 4 = 0$ .

[3]

Given that $w = 2\left(\cos\frac{\pi}{8} + i\sin\frac{\pi}{8}\right)$ , write $w^4$ in the form $re^{i\theta}$ , where $r>0$ and $-\pi < \theta \le \pi$ .

[3]

Question 4

HLPaper 1

The complex number $z$ is given by $z = 1+i$ .

Find the modulus of $z$ .

[1]

Find the argument of $z$ , giving your answer in radians.

[2]

Express $z$ in the form $re^{i\theta}$ .

[1]

Question 5

HLPaper 1

Given the complex numbers $z_1 = 4\left(\cos\frac{\pi}{3} + i\sin\frac{\pi}{3}\right)$ and $z_2 = \frac{1+i\sqrt{3}}{1-i}$ .

Write $z_1$ in exponential form.

[1]

Write $z_2$ in exponential form.

[3]

Find and simplify an expression for $\frac{z_1}{z_2}$ in exponential form.

[3]

Question 6

HLPaper 1

The complex number $z$ is given in Euler form as $z = 4e^{i\frac{\pi}{3}}$ . Express $z$ in the form $x+iy$ , where $x$ and $y$ are real.

Express $z$ in the form $x+iy$ , where $x$ and $y$ are real.

[3]

Question 7

HLPaper 1

The complex number $z$ is given by $z = \sqrt{3} + i$ .

Find the modulus of $z$ .

[1]

Find the argument of $z$ , giving your answer in radians in terms of $\pi$ .

[2]

Express $z$ in polar form $r(\cos \theta + i \sin \theta)$ .

[1]

Express $z$ in Euler form $re^{i\theta}$ .

[1]

Question 8

HLPaper 1

The complex number $w$ is defined by $w = \frac{3\sqrt{3} + 3i}{1+i\sqrt{3}}$ .

Express $w$ in the form $re^{i\theta}$ , where $r>0$ and $0 \le \theta < 2\pi$ .

[5]

Find the square roots of $w$ , giving your answers in the form $re^{i\phi}$ , where $r>0$ and $0 \le \phi < 2\pi$ .

[4]

Without using a calculator, show that $w^6$ is a real number.

[3]

Question 9

HLPaper 1

Consider the equation $w^4 = -4$ , where $w \in \mathbb{C}$ .

Solve the equation, giving the solutions in the form $c + \text{i}d$ , where $c, d \in \mathbb{R}$ .

[5]

The solutions form the vertices of a polygon in the complex plane. Find the area of the polygon.

[2]

Question 10

HLPaper 1

A circuit designer is analyzing an AC circuit where the voltage is represented by the complex number $z = 4 e^{i \frac{\pi}{6}}$ . The designer needs to compute the fifth power of this voltage to understand its behavior over time.

Calculate $z^5$ using De Moivre's Theorem and express the result in polar form.

[4]

Convert your answer from part (a) into rectangular form.

[3]

AHL 1.13—Polar and Euler form Questionbank