Mathematics 9709 · syllabus 3.9
Cambridge A Level Mathematics 9709 — Complex numbers topical past papers
Complex numbers in Cartesian and polar form, conjugate roots of polynomials, square roots, and loci and regions drawn on an Argand diagram.
- Questions
- 136
- Years
- 2009–2026
- With mark scheme
- 100%
- Questions 2022–2026
- 56
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What Complex numbers covers
A complex number z = x + iy, where i² = −1, has a real part, an imaginary part, a modulus |z| and an argument arg z. Paper 3 (section 3.9) asks you to add, subtract, multiply and divide in Cartesian form, dividing by multiplying the top and bottom by the conjugate, to equate real and imaginary parts, and to find the two square roots of a complex number. For a polynomial with real coefficients, non-real roots come in conjugate pairs, which lets you finish a cubic or quartic once one complex root is known.
On an Argand diagram, z is the point (x, y). In polar form, z = r(cos θ + i sin θ) = re^(iθ); multiplying multiplies the moduli and adds the arguments, and dividing divides the moduli and subtracts the arguments. Loci such as a circle |z − a| = k, a perpendicular bisector |z − a| = |z − b| and a half-line arg(z − a) = α are sketched, and inequalities shade the regions between them; questions then ask for the greatest or least value of |z| or arg z in a region.
Complex numbers appear in Paper 3 only. The topic uses trigonometry for arguments and exact values, coordinate geometry for circles and perpendicular bisectors, and the factor theorem for polynomial roots. Arguments are usually given in the range −π < θ ≤ π unless a question says otherwise. Multiplication, division and square roots are expected with the algebra written out, so practise them by hand rather than relying on a calculator's complex mode.
In the syllabus
The headings this topic has in the Cambridge International 2026–2027 syllabus, as the board prints them.
- 3.9 Complex numbers
What you should be able to do
- Add, subtract, multiply and divide complex numbers in Cartesian form
- Find the modulus and argument of a complex number and convert between Cartesian and polar forms
- Find the square roots of a complex number by equating real and imaginary parts
- Use conjugate pairs to solve polynomial equations with real coefficients
- Multiply and divide in polar form using moduli and arguments
- Sketch loci and regions on an Argand diagram and find the greatest or least values of |z| or arg z
Key ideas
- i² = −1; z = x + iy has conjugate z* = x − iy, and zz* = x² + y² = |z|²
- |z| = √(x² + y²); arg z is the angle θ with tan θ = y/x in the correct quadrant, usually −π < θ ≤ π
- z = r(cos θ + i sin θ) = re^(iθ); |z₁z₂| = |z₁||z₂| and arg(z₁z₂) = arg z₁ + arg z₂
- Non-real roots of a polynomial equation with real coefficients occur in conjugate pairs
- |z − a| = k is a circle with centre a and radius k; |z − a| = |z − b| is the perpendicular bisector of a and b; arg(z − a) = α is a half-line from a
How Complex numbers has been examined
From 2022 to 2026, Complex numbers was on Paper 3 · Pure Mathematics 3 in 11 of 11 exam series, worth about 14% of that paper’s marks.
| Paper | Questions | Share of marks | Series |
|---|---|---|---|
| Paper 3 · Pure Mathematics 3 | 56 | 14% | 11 of 11 |
Momentum’s own count, over the questions in its bank for these years: a series is one exam session, every zone included, and a question filed under two chapters counts in both. Median marks per question: 5.
Filed in Momentum under: Complex Numbers.
A worked example
Find the two square roots of −7 + 24i, giving your answers in the form x + iy, where x and y are real.
- Let (x + iy)² = −7 + 24i. Expanding gives x² − y² + 2xyi.
- Equate real and imaginary parts: x² − y² = −7 and 2xy = 24, so y = 12/x.
- Substitute: x² − 144/x² = −7, so x⁴ + 7x² − 144 = 0, which factorises as (x² + 16)(x² − 9) = 0.
- x is real, so x² = 9: x = 3 gives y = 4, and x = −3 gives y = −4.
- Check: (3 + 4i)² = 9 + 24i + 16i² = 9 + 24i − 16 = −7 + 24i.
Answer: 3 + 4i and −3 − 4i
Written by Momentum for this page, with its own numbers; it is not taken from a past paper.
Mistakes that cost marks
- Taking arg z straight from tan⁻¹(y/x): for −1 − i the calculator gives π/4, but the point is in the third quadrant, so arg z = −3π/4.
- Drawing arg(z − a) = α as a full line through a: it is a half-line starting at a, and the point a itself is not included.
- Reading the centre of |z + 2 − i| = 3 as (2, −1): the locus is |z − (−2 + i)| = 3, so the centre is (−2, 1).
- Giving only one square root: every non-zero complex number has two, and they are negatives of each other.
- Writing down a quotient from the calculator without the working: multiplication and division need the conjugate step shown in full.
Exam technique
- Show the working for every multiplication and division, including the conjugate you multiply by.
- Plot the point before giving an argument, so the quadrant is clear, and give arguments in radians unless told otherwise.
- For the greatest or least |z| on a circle, use the distance from the origin to the centre plus or minus the radius.
Practising in Momentum
- Every question in a chapter as one list, from every paper and year in the bank, newest first or shuffled.
- Each question turns over to its own part of the mark scheme, where the board published one.
- Narrow by paper, year, session, zone and marks, and star the questions worth keeping.
- Pick chapters, a number of questions and a time, and the exam builder draws a mock to sit and mark.
Past-paper questions and mark schemes are the copyright of Cambridge International. Momentum shows them to signed-in learners for revision and does not publish them on these pages.