What is in QCE Specialist Maths Units 3 and 4

Ten topics in two units. Unit 3 is further complex numbers, proof, vectors and matrices; Unit 4 is further calculus and statistical inference. Both papers of the external examination can test all of Units 3 and 4.

Weeks before the external examination

Hours a week for Specialist Maths

Complex numbers and proof

Vectors and matrices

Calculus

Statistical inference

26 weeks, 104 hours in all

Start with Calculus

38.4 of your hours. It is 4 of the 10 units in the course, 40%.

Hours per area
Area	Your hours	Share of the course
Calculus	38.4	40%
Vectors and matrices	28.8	30%
Complex numbers and proof	19.2	20%
Statistical inference	9.6	10%

Week by week

Week 1Calculus
Week 2Calculus
Week 3Calculus
Week 4Calculus
Week 5Calculus
Week 6Calculus
Week 7Calculus
Week 8Calculus
Week 9Calculus
Week 10Calculus, Vectors and matrices
Week 11Vectors and matrices
Week 12Vectors and matrices
Week 13Vectors and matrices
Week 14Vectors and matrices
Week 15Vectors and matrices
Week 16Vectors and matrices
Week 17Vectors and matrices, Complex numbers and proof
Week 18Complex numbers and proof
Week 19Complex numbers and proof
Week 20Complex numbers and proof
Week 21Complex numbers and proof
Week 22Complex numbers and proof, Statistical inference
Week 23Statistical inference
Week 24Statistical inference
Week 25Timed paper 1 (Paper 1 style, technology-free), then review every lost mark
Week 26Timed paper 2 (Paper 2 style, technology-active), then review every lost mark

Check your units with the free test

External assessment	One examination worth 50%, set and marked by the QCAA
Paper 1	Technology-free, 25%: short response and multiple choice
Paper 2	Technology-active, 25%: short response and multiple choice
Time	5 minutes perusal and 90 minutes working time for each paper; the formula book is provided
Internal assessment	Set by your school: a modelling task 20%, two short response examinations 15% each

Unit 3: Further complex numbers, proof, vectors and matrices

Further complex numbers: polar form, De Moivre's theorem, roots of complex numbers and factorising polynomials over the complex numbers.
Mathematical induction for sums, divisibility and De Moivre's theorem, and trigonometric identities proved with De Moivre's theorem.
Vectors in three dimensions: the scalar and vector products, projections, and equations of lines, planes and spheres; then vector calculus for motion in a plane.
Further matrices: determinants and inverses, systems of linear equations by Gaussian elimination, and Dominance and Leslie matrices.

Unit 4: Further calculus and statistical inference

Integration techniques: trigonometric identities, substitution, inverse trigonometric functions, partial fractions and integration by parts.
Applications of integration: areas between curves, volumes of solids of revolution, Simpson's rule and the exponential random variable.
Implicit differentiation, related rates and first-order differential equations, including separation of variables and slope fields; then modelling motion with forces and simple harmonic motion.
Statistical inference: the distribution of the sample mean and approximate confidence intervals for a population mean.

How this course maps the syllabus

The course gives each of the ten topics its own unit, plus a unit on the external examination. Each has a lesson, multiple choice questions and short response questions in the style of both papers. Internal assessment is set by your school, so the course describes it but does not practise it. Set the planner above with the weeks you have and how sure you are of each area, and it splits your hours.

Try 3 questions

Paper 1 · Multiple choice · 1 mark · Further complex numbers
In polar form with the principal argument, $(\sqrt{3} + i)^{4}$ is
Answer it, every step marked
Paper 1 · Short response · 7 marks · Further complex numbers
Let $z = a + b i$ , where $a$ and $b$ are real and not both zero.
Answer it, every step marked
Paper 2 · Multiple choice · 1 mark · Further complex numbers
The fifth roots of unity are the vertices of a regular pentagon in the complex plane. Given that $\sin\left(\frac{2 \pi}{5}\right) = 0.9511$ , the area of the pentagon, correct to two decimal places, is
Answer it, every step marked

Next step

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Questions people ask

What is the external examination worth?

50%. It has two papers, Paper 1 (technology-free) and Paper 2 (technology-active), each worth 25%. Internal assessment set by your school makes up the other 50%.

Which syllabus does the course follow?

The QCAA Specialist Mathematics 2025 syllabus (version 1.4), Units 3 and 4.

Does this course cover Mathematical Methods?

No. It is built for Specialist Mathematics only. The syllabus assumes knowledge from Mathematical Methods, which the course does not teach again.