QCE Specialist Mathematics: complete 2026 guide to Units 3 and 4 (General subject)

QCE General Specialist Mathematics Units 3 and 4 is the Year 12 sequence assessed across three internal assessments and one External Assessment. It is designed to be studied alongside Mathematical Methods, extending its calculus and statistics. Unit 3 is the priority for IA2; Unit 4 is the priority for IA3; the EA tests Units 3 and 4 cumulatively at the end of the year.

This page is the index. Below you will find the structure of the course, what each instrument assesses, and links to every dot-point answer we have written for QCE Specialist Mathematics.

The four instruments in 2026

IA1: Problem-solving and modelling task

A school-based response developing and evaluating a mathematical model for a given scenario, submitted as a report with assumptions, solution and justified decisions. It assesses modelling judgement and communication rather than timed technique. Weighting 20 percent.

IA2: Examination (Unit 3)

An internal examination drawn from Unit 3 subject matter (induction, vectors, matrices and complex numbers), with short-response and extended items. Weighting 15 percent.

IA3: Examination (Unit 4)

An internal examination drawn from Unit 4 subject matter (integration, differential equations and statistical inference). Weighting 15 percent.

EA: External Assessment

A centrally set external examination at the end of Unit 4, marked by QCAA, assessing Units 3 and 4 cumulatively. Weighting 50 percent. Three of the four instruments (IA2, IA3 and the EA) are examinations.

Unit 3: Mathematical induction, and further vectors, matrices and complex numbers

Unit 3 develops rigorous proof and the algebra of vectors, matrices and complex numbers. It is the basis of IA2 and contributes to the EA.

Proof by mathematical induction

The base step and inductive step structure applied to sums, divisibility and inequalities.

Vectors in two and three dimensions

Dot and cross products, angles, scalar and vector projections, and the equations of lines.

Further matrices

Determinants, inverses and the interpretation of $2\times 2$ matrices as linear transformations of the plane.

Further complex numbers

Polar and exponential form, de Moivre's theorem, and the roots of a complex number on the Argand plane.

Unit 4: Further calculus, and statistical inference

Unit 4 extends integration and introduces statistical inference. It is the basis of IA3 and contributes to the EA.

Integration and applications

Substitution, partial fractions, trigonometric integrals and volumes of solids of revolution.

Rates of change and differential equations

First-order equations solved by separation of variables, including growth, decay, cooling and logistic models.

Statistical inference

The sampling distribution of the mean, the central limit theorem, and confidence intervals for a population mean.

Our 2026 QCE Specialist Mathematics dot-point answers

Every link below is a focused answer to one QCAA subject-matter dot point. Each page identifies the dot point, gives a rigorous worked answer with correct notation, and flags the mistakes QCAA markers penalise.

Unit 3: Induction, vectors, matrices and complex numbers

• Mathematical induction
• Trigonometric proofs and methods of proof
• Vectors in three dimensions, dot and cross products
• Vector and Cartesian equations of lines
• Vector and Cartesian equations of planes
• Matrices and linear transformations of the plane
• Systems of linear equations and larger matrices
• Complex numbers in polar form and de Moivre's theorem
• Factorising polynomials over the complex field
• Regions and curves in the complex plane

Unit 4: Further calculus and statistical inference

• Integration techniques: substitution and partial fractions
• Integration by parts and trigonometric integrals
• Areas between curves and volumes of revolution
• Simpson's rule for numerical integration
• First-order differential equations and separation of variables
• Slope fields of first-order differential equations
• Implicit differentiation and related rates
• Growth, decay, cooling and logistic models
• Distribution of the sample mean and the central limit theorem
• Confidence intervals for a population mean