Which topic carries the most marks in the QCE Mathematical Methods external exam?

Differentiation of exponential and logarithmic functions carries 20.6% of the marks across the 12 QCAA papers from 2020 to 2025, ahead of Continuous random variables and the normal distribution on 13.1%.

How are marks split between question types in the QCE Mathematical Methods exam?

Short answer carries 82.1% of the marks and multiple choice carries 17.9% of the marks, measured across 12 papers and 669 marks from 2020 to 2025.

Has the topic balance changed in recent QCE Mathematical Methods papers?

Yes. Differentiation of exponential and logarithmic functions moved up 3.4 percentage points, Continuous random variables and the normal distribution moved down 6.3 percentage points, Further integration moved down 3.2 percentage points, Discrete random variables moved up 4.0 percentage points, Introduction to integration moved up 4.2 percentage points, Trigonometry moved down 4.2 percentage points and Sampling and proportions moved up 3.1 percentage points between the 2020 to 2022 and 2023 to 2025 papers.

How many past papers is this QCE Mathematical Methods analysis based on?

12 QCAA external papers from 2020 to 2025, covering 232 questions and 669 marks. 3.1% of those marks carry no dot-point mapping and sit outside the percentages.

QCE Mathematical Methods most tested topics

Across the 12 QCAA Mathematical Methods external papers from 2020 to 2025, Differentiation of exponential and logarithmic functions carries 20.6% of the marks against 3.8% for Sampling and proportions, so the paper rewards Differentiation of exponential and logarithmic functions more than any other topic. At dot-point level 3.1.2.3 carries 9.1% of paper marks, and 15 of the 88 dot points that the board assesses externally have not appeared in one of these papers. Every percentage below covers the 96.9% of marks that map to a syllabus dot point.

Computed from 12 QCAA Mathematical Methods external papers, 2020 to 2025: 232 questions, 316 question parts and 669 marks.

Unit, topic and dot-point shares below are of the 648 marks that map to a dot point, so each of those tables adds to 100%. 140 of the 316 parts are assessed against more than one dot point, and their 373 marks are divided evenly between the dot points they cover.

What these papers are

Year	Paper	Questions	Marks
2025	Paper 1	19	55
2025	Paper 2	19	55
2024	Paper 1	19	55
2024	Paper 2	19	55
2023	Paper 1	19	55
2023	Paper 2	19	55
2022	Paper 1	19	54
2022	Paper 2	19	55
2021	Paper 1	20	55
2021	Paper 2	20	55
2020	Paper 1	20	60
2020	Paper 2	20	60

Marks by unit

Unit	Share of marks
Unit 3: Further calculus and introduction to statistics	57.7%
Unit 4: Further calculus, trigonometry and statistics	42.3%

Marks by topic

Topic	Share of marks	Papers it appears in	Years
Differentiation of exponential and logarithmic functions	20.6%	12 of 12	6 of 6
Continuous random variables and the normal distribution	13.1%	10 of 12	6 of 6
Further integration	12.2%	12 of 12	6 of 6
Further applications of differentiation	10.9%	12 of 12	6 of 6
Discrete random variables	9.7%	12 of 12	6 of 6
Introduction to integration	9.6%	12 of 12	6 of 6
Differentiation of trigonometric functions and differentiation rules	6.9%	11 of 12	6 of 6
Trigonometry	6.6%	10 of 12	6 of 6
Interval estimates for proportions	6.5%	11 of 12	6 of 6
Sampling and proportions	3.8%	8 of 12	5 of 6

Marks by subtopic

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

Subtopic	Share of marks	Papers
Calculus of logarithmic functions	17.7%	12 of 12
Calculus of exponential functions	2.9%	11 of 12
Normal distribution	7.2%	9 of 12
General continuous random variables	5.9%	10 of 12
Applications of integration	10.5%	10 of 12
Fundamental theorem of calculus and definite integrals	1.6%	8 of 12
The second derivative and applications of differentiation	10.9%	12 of 12
Binomial distributions	7.9%	12 of 12
Bernoulli distributions	1.3%	4 of 12
General discrete random variables	0.5%	3 of 12
Anti-differentiation	9.6%	12 of 12
Calculus of trigonometric functions	3.7%	10 of 12
Differentiation rules	3.2%	8 of 12
Cosine and sine rules	6.6%	10 of 12
Confidence intervals for proportions	6.5%	11 of 12
Sample proportions	3.1%	7 of 12
Random sampling	0.7%	3 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
3.1.2.3	Solve equations involving exponential and logarithmic functions with base $e$ , with and without technology	Differentiation of exponential and logarithmic functions	9.1%	12 of 12
4.3.2.4	Calculate probabilities and quantiles associated with a given normal distribution, using technology	Continuous random variables and the normal distribution	4.2%	7 of 12
3.1.2.5	Model and solve problems that involve derivatives of exponential and logarithmic functions, with and without technology	Differentiation of exponential and logarithmic functions	4.1%	7 of 12
4.1.2.5	Model and solve problems that involve definite integrals, including motion problems, with and without technology	Further integration	3.6%	8 of 12
4.2.1.4	Model and solve problems that involve the sine rule, cosine rule and the area formula in two- and three-dimensional contexts (including bearings, directions and angles of elevation and depression), with and without technology	Trigonometry	3.6%	6 of 12
4.3.1.2	Understand the concepts of a probability density function, cumulative distribution function, and probabilities associated with a continuous random variable given by integrals; examine simple types of continuous random variables and use them in appropriate contexts	Continuous random variables and the normal distribution	3.4%	7 of 12
3.3.1.3	Understand the concepts of concavity and points of inflection and their relationship with the second derivative	Further applications of differentiation	2.9%	9 of 12
3.5.3.3	Determine and use the probabilities $P(X = r) = \binom{n}{r} p^r (1 - p)^{n-r}$ associated with the binomial distribution with parameters $n$ and $p$	Discrete random variables	2.6%	8 of 12
4.3.2.5	Model and solve problems that involve normal distributions, with and without technology (distribution tables are not required)	Continuous random variables and the normal distribution	2.4%	7 of 12
3.1.1.4	Use the rules $\frac{d}{dx}e^x = e^x$ and $\frac{d}{dx}e^{f(x)} = f'(x)e^{f(x)}$	Differentiation of exponential and logarithmic functions	2.4%	10 of 12
3.1.2.1	Recognise and determine the qualitative features of the graph of $y = \ln(x) = \log_e(x)$ , including asymptote and intercept	Differentiation of exponential and logarithmic functions	2.3%	8 of 12
4.5.1.6	Model and solve problems that involve interval estimates for proportions, with and without technology	Interval estimates for proportions	2.3%	6 of 12
4.1.2.1	Calculate the area enclosed by a curve and the $x$ -axis over a given domain, with and without technology	Further integration	2.3%	8 of 12
3.2.1.3	Model and solve problems that involve derivatives of trigonometric functions, with and without technology	Differentiation of trigonometric functions and differentiation rules	2.1%	6 of 12
3.3.1.6	Model and solve optimisation problems from a wide variety of fields using first and second derivatives, where the function to be optimised is either given or to be developed	Further applications of differentiation	2.1%	6 of 12
4.5.1.2	Understand and use the approximate confidence interval, $\left(\hat{p} - z\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}, \hat{p} + z\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\right)$ , as an interval estimate for $p$ , the population proportion, where $z$ is the appropriate quantile for the standard normal distribution	Interval estimates for proportions	2.0%	6 of 12
4.4.2.1	Understand the concept of the sample proportion $\hat{p}$ as a random variable whose value varies between samples, and the formulas for the mean $p$ and standard deviation $\sqrt{p(1 - p)/n}$ of the sample proportion $\hat{p}$ , where $n$ is the sample size	Sampling and proportions	2.0%	7 of 12
3.1.2.4	Use the rules $\frac{d}{dx}\ln(x) = \frac{1}{x}$ and $\frac{d}{dx}\ln(f(x)) = \frac{f'(x)}{f(x)}$	Differentiation of exponential and logarithmic functions	1.9%	6 of 12
4.1.2.4	Calculate total change by integrating instantaneous or marginal rates of change, with and without technology	Further integration	1.9%	7 of 12
4.2.1.3	Use the formula $area = \frac{1}{2}bc \sin(A)$ to calculate the area of a triangle	Trigonometry	1.9%	5 of 12

These 20 dot points carry 59.1% of the paper marks between them. Another 53 assessed dot points share the rest, and 15 of the 88 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, seven topics have moved by more than 3.0 percentage points.

Topic	2020 to 2022 share	2023 to 2025 share	Movement
Differentiation of exponential and logarithmic functions	18.9%	22.3%	up 3.4 points
Continuous random variables and the normal distribution	16.2%	9.9%	down 6.3 points
Further integration	13.7%	10.5%	down 3.2 points
Discrete random variables	7.8%	11.8%	up 4.0 points
Introduction to integration	7.6%	11.8%	up 4.2 points
Trigonometry	8.7%	4.5%	down 4.2 points
Sampling and proportions	2.3%	5.4%	up 3.1 points

How the papers are built

Short answer carries 82.1% of the marks across these 12 papers.

Question type	Share of marks	Marks	Parts
Short answer	82.1%	549	196
Multiple choice	17.9%	120	120

Verb	Share of marks	Marks	Parts
determine	57.5%	385	137
evaluate	7.0%	47	10
calculate	2.1%	14	6
verify	2.1%	14	3
state	1.9%	13	6
make	1.3%	9	2
justify	1.2%	8	2
explain	0.7%	5	3

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: Differentiation of exponential and logarithmic functions, Continuous random variables and the normal distribution, Further integration, Further applications of differentiation, Discrete random variables, Introduction to integration, Differentiation of trigonometric functions and differentiation rules, Trigonometry, Interval estimates for proportions and Sampling and proportions. For how students actually score on this content, read QCE Mathematical Methods hardest topics, and for the full question bank start at QCE Mathematical Methods.

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.