A-level · Complex numbers

Complex numbers in Cartesian formExam-style questions

The imaginary unit i with i² = −1, real and imaginary parts, equality of complex numbers, and the four operations in the form a + bi — addition, subtraction, multiplication, and division by realising the denominator with the conjugate.

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17 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 Level 5

1
The complex numbers $7 + \mathrm{i}$ and $3 + 4\mathrm{i}$ are denoted by $u$ and $v$ respectively.
(a) Find $\dfrac{u}{v}$ in the form $x + \mathrm{i}y$ , where $x$ and $y$ are real.

(3 marks)
(b) State the argument of $\dfrac{u}{v}$ .

(1 mark)

In an Argand diagram, with origin $O$ , the points $A$ , $B$ and $C$ represent the complex numbers $u$ , $v$ and $u + v$ respectively.
(c) State fully the geometrical relationship between $AC$ and $OB$ .

(2 marks)
(d) Show that angle $OAC = \tfrac{3}{4}\pi$ radians.

(2 marks)
Level 4 · 8 marksTry it
2
Solve the equation $(2 + i)z = 3 - i$ , giving your answer for $z$ in the form $a + bi$ . (3 marks)
Level 4 · 3 marksTry it
3
Find the two square roots of $3 - 4i$ , giving your answers in the form $a + bi$ where $a$ and $b$ are real. (5 marks)
Level 4 · 5 marksTry it
4
The equation $z^{2} + bz + c = 0$ , where $b$ and $c$ are real, has a root $2 - 5i$ .

Find the values of $b$ and $c$ . (4 marks)
Level 4 · 4 marksTry it
5
The complex number $z = 2 + i$ .
(a) Express $\dfrac{1}{z}$ in the form $a + bi$ . (2 marks)
(b) Hence find $z + \dfrac{1}{z}$ in the form $a + bi$ . (2 marks)
Level 4 · 4 marksTry it

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