Which topic carries the most marks in the QCE Specialist Mathematics external exam?

Statistical inference carries 14.8% of the marks across the 12 QCAA papers from 2020 to 2025, ahead of Vectors in two and three dimensions on 13.8%.

How are marks split between question types in the QCE Specialist Mathematics exam?

Short answer carries 83.8% of the marks and multiple choice carries 16.2% of the marks, measured across 12 papers and 740 marks from 2020 to 2025.

Has the topic balance changed in recent QCE Specialist Mathematics papers?

Yes. Vector calculus moved up 5.2 percentage points and Modelling motion moved down 3.0 percentage points between the 2020 to 2022 and 2023 to 2025 papers.

How many past papers is this QCE Specialist Mathematics analysis based on?

12 QCAA external papers from 2020 to 2025, covering 227 questions and 740 marks. 0.7% of those marks carry no dot-point mapping and sit outside the percentages.

QCE Specialist Mathematics most tested topics

Across the 12 QCAA Specialist Mathematics external papers from 2020 to 2025, Statistical inference carries 14.8% of the marks against 5.5% for Modelling motion, so the paper rewards Statistical inference more than any other topic. At dot-point level 4.5.1.5 carries 3.5% of paper marks, and four of the 83 dot points that the board assesses externally have not appeared in one of these papers. Every percentage below covers the 99.3% of marks that map to a syllabus dot point.

Computed from 12 QCAA Specialist Mathematics external papers, 2020 to 2025: 227 questions, 340 question parts and 740 marks.

Unit, topic and dot-point shares below are of the 735 marks that map to a dot point, so each of those tables adds to 100%. 100 of the 340 parts are assessed against more than one dot point, and their 319 marks are divided evenly between the dot points they cover.

What these papers are

Year	Paper	Questions	Marks
2025	Paper 1	19	60
2025	Paper 2	18	60
2024	Paper 1	19	60
2024	Paper 2	19	60
2023	Paper 1	19	60
2023	Paper 2	19	60
2022	Paper 1	19	60
2022	Paper 2	19	60
2021	Paper 1	19	65
2021	Paper 2	19	65
2020	Paper 1	19	65
2020	Paper 2	19	65

Marks by unit

Unit	Share of marks
Unit 3: Further complex numbers, proof, vectors and matrices	51.0%
Unit 4: Further calculus and statistical inference	49.0%

Marks by topic

Topic	Share of marks	Papers it appears in	Years
Statistical inference	14.8%	12 of 12	6 of 6
Vectors in two and three dimensions	13.8%	12 of 12	6 of 6
Further complex numbers	13.1%	12 of 12	6 of 6
Integration techniques	10.7%	12 of 12	6 of 6
Rates of change and differential equations	10.4%	12 of 12	6 of 6
Vector calculus	10.2%	10 of 12	6 of 6
Further matrices	7.7%	10 of 12	6 of 6
Applications of integral calculus	7.6%	11 of 12	6 of 6
Mathematical induction and trigonometric proofs	6.2%	10 of 12	6 of 6
Modelling motion	5.5%	10 of 12	6 of 6

Marks by subtopic

Each topic above breaks into the subtopics below, in the same order.

Subtopic	Share of marks	Papers
Sample means	7.6%	11 of 12
Confidence intervals for means	7.3%	12 of 12
Vector and Cartesian equations	7.0%	12 of 12
Algebra of vectors in three dimensions	4.3%	11 of 12
Vectors in three dimensions	2.5%	7 of 12
Complex arithmetic using polar form	5.6%	11 of 12
Factorisation of polynomials	3.9%	10 of 12
Roots of complex numbers	3.6%	10 of 12
Integration techniques	10.7%	12 of 12
Differential equations	5.9%	8 of 12
Rates of change	4.4%	9 of 12
Vector calculus	10.2%	10 of 12
Matrix algebra and systems of equations	4.3%	9 of 12
Applications of matrices	3.4%	7 of 12
Applications of integral calculus	7.6%	11 of 12
Mathematical induction	5.3%	8 of 12
Trigonometric proofs using De Moivre’s theorem	1.0%	2 of 12
Modelling motion	5.5%	10 of 12

Marks by dot point

Dot-point numbers and wording are QCAA's own. Where a dot point is written as a list, its items run together here separated by semicolons.

Dot point	Content	Topic	Share of marks	Papers
4.5.1.5	Model and solve problems that involve sample means, with and without technology	Statistical inference	3.5%	6 of 12
3.5.2.1	Model and solve problems that involve real-life situations using matrices, including Dominance and Leslie matrices	Further matrices	3.4%	7 of 12
4.5.2.7	Model and solve problems that involve interval estimates for sample means, with and without technology	Statistical inference	3.2%	7 of 12
4.3.1.2	Model and solve related rates problems as instances of the chain rule including situations that involve surface area and volume of cones, pyramids and spheres, with and without technology	Rates of change and differential equations	3.0%	4 of 12
4.3.2.1	Determine general and particular solutions of first-order differential equations of the form $\frac{dy}{dx} = f(x)$ , differential equations of the form $\frac{dy}{dx} = g(y)$ and differential equations of the form $\frac{dy}{dx} = f(x)g(y)$ using separation of variables	Rates of change and differential equations	2.9%	6 of 12
3.1.1.1	Prove complex number identities involving modulus and argument, e.g. $z\bar{z} = \\|z\\|^2$ , $\\|z_1\\|\\|z_2\\| = \\|z_1 z_2\\|$ and $\text{arg}(z_1 z_2) = \text{arg}(z_1) + \text{arg}(z_2)$	Further complex numbers	2.9%	5 of 12
3.3.3.5	Use vector methods in applications, including areas of shapes and determining vector and Cartesian equations of a plane and of regions in a plane. vector equation of plane: $\boldsymbol{r} \cdot \boldsymbol{n} = \boldsymbol{a} \cdot \boldsymbol{n}$ ; Cartesian equation of plane: $ax + by + cz + d = 0$	Vectors in two and three dimensions	2.8%	11 of 12
3.1.1.2	Use De Moivre’s theorem for integral powers. $z^n = r^n \text{cis}(n\theta)$	Further complex numbers	2.8%	7 of 12
3.2.1.1	Understand the nature of inductive proof including the use of initial statement, assumption statement, inductive step and conclusion	Mathematical induction and trigonometric proofs	2.5%	8 of 12
4.2.1.3	Use Simpson’s rule to approximate an area and the value of a definite integral, with and without technology. $\int_a^b f(x) dx \approx \frac{w}{3} [f(x_0) + 4[f(x_1) + f(x_3) + \dots] + 2[f(x_2) + f(x_4) + \dots] + f(x_n)]$ ; where $w = \frac{b - a}{n}$	Applications of integral calculus	2.4%	4 of 12
3.1.2.1	Determine and examine the $n$ th roots of unity and their location on the unit circle	Further complex numbers	2.4%	7 of 12
3.4.1.4	Differentiate and integrate a vector function with respect to time	Vector calculus	2.4%	7 of 12
4.1.1.1	Integrate using the trigonometric identities $\sin^2(x) = \frac{1}{2}(1 - \cos(2x))$ , $\cos^2(x) = \frac{1}{2}(1 + \cos(2x))$ , $1 + \tan^2(x) = \sec^2(x)$ and $\cot^2(x) + 1 = \text{cosec}^2(x)$	Integration techniques	2.3%	5 of 12
3.4.1.3	Understand and use the position of two particles, each described as a vector function of time, and determine if their paths cross or if the particles meet	Vector calculus	2.3%	5 of 12
4.4.1.4	Model and solve problems that involve motion in a straight line with both constant and non-constant acceleration, including simple harmonic motion, vertical motion under gravity with and without air resistance, and motion of a body in non-equilibrium situations on a smooth inclined plane (excluding situations with pulleys and connected bodies). If $\frac{d^2x}{dt^2} = -\omega^2 x$ then $x = A \sin(\omega t + \alpha)$ or $x = A \cos(\omega t + \beta)$ ; $v^2 = \omega^2(A^2 - x^2)$ ; $T = \frac{2\pi}{\omega}$ ; $f = \frac{1}{T}$	Modelling motion	2.3%	6 of 12
3.3.3.3	Determine vector, parametric and Cartesian equations of straight lines and straight-line segments given the position of two points, or equivalent information, in both two and three dimensions. vector equation of line: $\boldsymbol{r} = \boldsymbol{a} + t\boldsymbol{d}$ ; parametric equations of line: $x = a_1 + t d_1, y = a_2 + t d_2, z = a_3 + t d_3$ ; Cartesian equation of line: $\frac{x - a_1}{d_1} = \frac{y - a_2}{d_2} = \frac{z - a_3}{d_3}$	Vectors in two and three dimensions	2.2%	6 of 12
3.5.1.3	Use matrix algebra to solve matrix equations that involve matrices of beyond dimension $2 \times 2$ , including those of the form $AX = B$ , $XA = B$ and $AX + BX = C$ , with technology	Further matrices	2.2%	5 of 12
4.5.2.2	Understand and use the approximate confidence interval $(\bar{x} - z \frac{s}{\sqrt{n}}, \bar{x} + z \frac{s}{\sqrt{n}})$ , as an interval estimate for $\mu$ , the population mean, where $z$ is the appropriate quantile for the standard normal distribution	Statistical inference	2.1%	8 of 12
3.4.1.6	Apply vector calculus to model and solve problems that involve motion in a plane, including projectile and circular motion, with and without technology	Vector calculus	2.0%	4 of 12
4.2.1.2	Determine volumes of solids of revolution about either axis, with and without technology. about the $x$ -axis: $V = \pi \int_a^b [f(x)]^2 dx$ ; about the $y$ -axis: $V = \pi \int_a^b [f(y)]^2 dy$	Applications of integral calculus	2.0%	6 of 12

These 20 dot points carry 51.6% of the paper marks between them. Another 59 assessed dot points share the rest, and 4 of the 83 externally assessable dot points have not been assessed in any of these 12 papers. Content that has not appeared is still examinable.

What has moved between papers

Comparing the 2020 to 2022 papers with the 2023 to 2025 papers, two topics have moved by more than 3.0 percentage points.

Topic	2020 to 2022 share	2023 to 2025 share	Movement
Vector calculus	7.6%	12.8%	up 5.2 points
Modelling motion	7.0%	4.0%	down 3.0 points

How the papers are built

Short answer carries 83.8% of the marks across these 12 papers.

Question type	Share of marks	Marks	Parts
Short answer	83.8%	620	220
Multiple choice	16.2%	120	120

Verb	Share of marks	Marks	Parts
determine	55.1%	408	137
prove	7.0%	52	9
show	5.8%	43	15
evaluate	5.3%	39	11
estimate	2.2%	16	3
state	1.6%	12	11
justify	1.5%	11	4
verify	1.1%	8	5

What these percentages do not tell you

Where to practise

Every topic above links to its own page of real QCAA questions with marking criteria and average scores attached: Statistical inference, Vectors in two and three dimensions, Further complex numbers, Integration techniques, Rates of change and differential equations, Vector calculus, Further matrices, Applications of integral calculus, Mathematical induction and trigonometric proofs and Modelling motion. For how students actually score on this content, read QCE Specialist Mathematics hardest topics, and for the full question bank start at QCE Specialist Mathematics.

Syllabus and assessment material referenced in this guide is © State of Queensland (Queensland Curriculum and Assessment Authority), licensed under CC BY 4.0. See our QCAA licensing notice. AusGrader is an independent study tool and is not affiliated with, endorsed by, or operated by the QCAA.