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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

Given that $z = r(\cos\theta + i\sin\theta)$ , where $r$ is a positive real number and $\theta$ is an angle.

Find and simplify an expression for $z^n$ , where $n$ is a positive integer.

[2]

Given a real constant $k$ , find and simplify an expression for $\frac{k}{z^n}$ , where $n$ is a positive integer.

[3]

Question 10

HLPaper 1

The complex number $w$ is defined by $w = 2e^{i\frac{\pi}{4}}$ . In an Argand diagram, the points $A$ , $B$ and $C$ represent the complex numbers $w$ , $w^*$ and $w^2$ respectively (where $w^*$ denotes the complex conjugate of $w$ ).

Draw the Argand diagram showing the points $A$ , B and C, and calculate the area of triangle $ABC$ .

[5]

AHL 1.13—Polar and Euler form Questionbank