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
Express $\dfrac{5 + i}{2 - 3i}$ in the form $a + bi$ , where $a$ and $b$ are real. (3 marks)
Level 3 · 3 marksTry it
2
Find the real numbers $x$ and $y$ such that

$(x + yi)(2 - i) = 5 + 5i$ . (4 marks)
Level 3 · 4 marksTry it
3
The complex number $z = 3 - 4i$ .
(a) Find $z^2$ , giving your answer in the form $a + bi$ . (2 marks)
(b) Express $\dfrac{1}{z}$ in the form $a + bi$ . (2 marks)
Level 3 · 4 marksTry it
4
Express $\dfrac{7 - 4i}{1 + 2i}$ in the form $a + bi$ , where $a$ and $b$ are real. (3 marks)
Level 3 · 3 marksTry it
5
Find the complex number $z$ satisfying

$z + 2z^{*} = 9 - 2i$ (4 marks)
Level 3 · 4 marksTry it

1 more question on complex numbers in cartesian form

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