A-level · Complex numbers

The Argand diagram, modulus and argumentExam-style questions

Picturing a complex number z = a + bi as a point (or vector) on the Argand diagram, its modulus |z| = √(a²+b²) as the distance from the origin, and its argument arg z as the angle from the positive real axis — taken as the principal value in (−π, π], which means getting the quadrant right, not just arctan(b/a). Geometric effects: the conjugate reflects in the real axis, addition is vector addition, and |z₁ − z₂| is the distance between the two points.

Start practising Read the lesson

15 questions · A Level level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 1 Level 2 Level 3 Level 4

1
The complex number $z = -2 + 2i$ . Find $|z|$ and $\arg z$ , giving the argument as an exact multiple of $\pi$ . (3 marks)
Level 3 · 3 marksTry it
2
Find the modulus and argument of $z = -\sqrt3 - i$ , giving the argument as an exact multiple of $\pi$ in the range $-\pi < \arg z \le \pi$ . (4 marks)
Level 3 · 4 marksTry it
3
On an Argand diagram, points $A$ and $B$ represent the complex numbers $z_1 = 6 + 2i$ and $z_2 = 2 - i$ . Find the distance $AB$ . (3 marks)
Level 3 · 3 marksTry it
4
The complex number $z = -3 - 4i$ .
(a) Find $|z|$ . (1 mark)
(b) Find $\arg z$ in radians to $2$ decimal places. (3 marks)
Level 3 · 4 marksTry it
5
Given $z_{1} = 3 + 4i$ and $z_{2} = 5 - 12i$ :
(a) Find $|z_{1}z_{2}|$ . (2 marks)
(b) Write down $\left|\dfrac{z_{1}}{z_{2}}\right|$ . (1 mark)
(c) Find $\arg z_{1}$ in radians to $3$ decimal places. (2 marks)
Level 3 · 5 marksTry it

3 more questions on the argand diagram, modulus and argument

Free with an account — just an email, no card. The whole bank, every mark scheme and worked solution.

Start free