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.

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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
Find the modulus of the complex number $z = 8 + 6i$ . (2 marks)
Level 1 · 2 marksTry it
2
Find the modulus and argument of $z = 1 + \sqrt{3}\,i$ , giving the argument as an exact multiple of $\pi$ . (3 marks)
Level 2 · 3 marksTry it
3
The complex number $z = 2 + 5i$ is shown on an Argand diagram.
(a) Describe the geometric relationship between the points representing $z$ and $z^{*}$ . (1 mark)
(b) Describe the geometric relationship between the points representing $z$ and $-z$ . (1 mark)
(c) Find $|z|$ , giving your answer as an exact surd. (2 marks)
Level 2 · 4 marksTry it
4
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
5
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

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