CAIE A-Level Mathematics 3.9 Complex numbers Question Bank

Cambridge International AS & A Level Mathematics 9709 syllabus for exams in 2028, 2029 and 20302028–2030

Practise complex numbers in Cartesian, modulus-argument and Argand forms, including roots, conjugates, loci and transformations.

Exam points

convert between x+iy and re^{iθ}, choosing θ in the required interval
use conjugate-root facts or z=x+iy substitution to solve complex equations
shade Argand loci from modulus inequalities, including circles and perpendicular bisectors

Covered syllabus points

3.9.1—Complex numbers 3.9.2—Complex numbers 3.9.3—The result that 3.9.4—Complex numbers 3.9.5—Complex numbers 3.9.6—Complex numbers 3.9.7—Complex numbers 3.9.8—Complex numbers

[Maximum number: 4]

The complex numbers s and t are given by

s=5(\cos 0.25+\mathrm{i} \sin 0.25) \quad \text { and } \quad t=6 \mathrm{e}^{3 \mathrm{i}}

(a)

Express $\frac{s}{t}$ in the form $r \mathrm{e}^{\mathrm{i} \theta}$ , where $-\pi<\theta \leqslant \pi$ and r>0.

[ 2 ]

(b)

In an Argand diagram with origin O, the points A and B represent the complex numbers s and $\frac{s}{t}$ respectively.

By considering the line segments OA and OB, or otherwise, state the two geometric effects of dividing a complex number by $6e^{3i}$ .

[ 2 ]

[Maximum number: 4]

The complex number u is given by $u=-1-\mathrm{i} \sqrt{3}$ .

(a)

Express u in the form $r(\cos \theta+\mathrm{i} \sin \theta)$ , where r>0 and $-\pi<\theta \leqslant \pi$ . Give the exact values of r and $\theta$ .
The complex number v is given by $v=5\left(\cos \frac{1}{6} \pi+\mathrm{i} \sin \frac{1}{6} \pi\right)$ .

[ 2 ]

(b)

Express the complex number $\frac{v}{u}$ in the form $r \mathrm{e}^{\mathrm{i} \theta}$ where r>0 and $-\pi<\theta \leqslant \pi$ .

[ 2 ]

[Maximum number: 3]

$z=3 \mathrm{e}^{\frac{1}{4} \pi \mathrm{i}}$ is a root of the equation $z^{2}+b z+c=0$ , where b and c are real.

State the other root and hence find the values of b and c.

(a)

On a sketch of an Argand diagram with origin O, show points A and B representing the roots of the equation in part (a).

[ 1 ]

(b)

Find the exact modulus and argument of each root.

[ 3 ]

(a)

Given that z=1+y i and that y is a real number, express $\frac{1}{z}$ in the form a+b i, where a and b are functions of y.

[ 2 ]

(b)

On a single Argand diagram, sketch the loci given by the equations $\operatorname{Re}(z)=1$ and $\left|z-\frac{1}{2}\right|=\frac{1}{2}$ , where z is a complex number.

[ 3 ]

Exam points

Covered syllabus points

Question 3

Question 3(a)

Question 3(b)

Question 4

Question 4(a)

Question 4(b)

Question 4(b)

Question 5

Question 5(b)

Question 5(c)

Question 8

Question 8(a)

Question 8(c)