Which QCE Mathematical Methods past paper do students score lowest on?

The 2023 Paper 1, averaging 56.0% across 233 marked attempts on AusGrader. The 2021 Paper 1 is the highest on 78.7%.

Which QCE Mathematical Methods dot points give the best return on revision time?

3.1.2.3 (3.3 marks at risk per 100), 4.1.2.5 (2.1 marks at risk per 100) and 4.3.2.4 (2.1 marks at risk per 100). Marks at risk combines a dot point's share of paper marks with the marks students drop on it, so it ranks by recoverable marks and not by score alone.

How many attempts is each QCE Mathematical Methods figure based on?

17,488 marked attempts overall, with the per-row count shown in every table. Any cut below 50 attempts is withheld instead of published.

Do these averages show how the QCAA cohort performed?

No. They are AusGrader users' marked attempts, a self-selected group practising when they chose to and often without exam timing. They corroborate what QCAA publishes about this subject and do not replace it.

QCE Mathematical Methods hardest topics

Across 17,488 marked attempts on AusGrader, QCE Mathematical Methods students average 58.9% on questions taken from board external papers. The lowest average of any command verb is evaluate on 51.0%, and the verb that costs the most marks is determine, which carries 57.5% of the paper against 7.0% for evaluate. These are self-selected users practising when they chose to, not the QCAA cohort under exam conditions.

How to read these numbers

The figures are AusGrader users' marked attempts, not QCAA results. They corroborate what the board publishes about this subject and do not stand in for it.
Any cut with fewer than 50 attempts is withheld, which on this page is 23 dot points and 18 verbs. Every figure shown carries its attempt count.
Scores cover questions mapped to the QCE Mathematical Methods syllabus from any board's external papers, which is why the sample is larger than the 12 QCAA papers alone. The paper table below is the exception and uses QCAA papers only.
Internal assessment and school-uploaded exams are excluded throughout, so these averages differ from the ones on QCE Mathematical Methods performance stats, which count every attempt.

Score by past paper

Year	Paper	Average	Attempts
2025	Paper 1	65.0%	178
2025	Paper 2	60.7%	454
2024	Paper 1	65.3%	370
2024	Paper 2	56.5%	320
2023	Paper 1	56.0%	233
2023	Paper 2	60.5%	211
2022	Paper 1	71.6%	269
2022	Paper 2	60.6%	170
2021	Paper 1	78.7%	660
2021	Paper 2	75.0%	166
2020	Paper 1	56.8%	285
2020	Paper 2	60.9%	191

The lowest average belongs to the 2023 Paper 1 on 56.0% from 233 attempts, and the highest to the 2021 Paper 1 on 78.7% from 660 attempts. A paper's average reflects both how hard it was and who chose to sit it, so treat the spread as a guide to which papers make demanding practice.

Score by question type

Question type	Average	Attempts
Multiple choice	57.7%	14,481
Short answer	61.3%	3,007

Score by unit

Unit	Average	Attempts
Unit 3: Further calculus and introduction to statistics	58.3%	11,768
Unit 4: Further calculus, trigonometry and statistics	59.9%	6,596

An attempt counts once per unit, so a question assessed across two units appears in both rows and the column adds to slightly more than 17,488.

The priority list: heavy topics with low scores

Topic	Share of marks	Average	Attempts	Marks at risk
Differentiation of exponential and logarithmic functions	20.6%	60.5%	4,220	8.1
Further integration	12.2%	54.7%	2,614	5.5
Further applications of differentiation	10.9%	50.7%	3,108	5.4
Continuous random variables and the normal distribution	13.1%	60.9%	2,710	5.1
Discrete random variables	9.7%	56.7%	2,513	4.2
Differentiation of trigonometric functions and differentiation rules	6.9%	55.7%	2,021	3.1
Introduction to integration	9.6%	71.3%	1,768	2.8
Trigonometry	6.6%	61.9%	207	2.5
Interval estimates for proportions	6.5%	69.1%	723	2.0
Sampling and proportions	3.8%	65.7%	365	1.3

Differentiation of exponential and logarithmic functions tops the list on 8.1 marks at risk per 100 paper marks, 2.6 ahead of Further integration.

The same cut at subtopic level

Subtopic	Share of marks	Average	Attempts	Marks at risk
Calculus of logarithmic functions	17.7%	58.9%	3,134	7.3
Applications of integration	10.5%	47.9%	1,094	5.5
The second derivative and applications of differentiation	10.9%	50.7%	3,108	5.4
Binomial distributions	7.9%	58.4%	1,771	3.3
Normal distribution	7.2%	55.3%	1,439	3.2
Anti-differentiation	9.6%	71.3%	1,768	2.8
Cosine and sine rules	6.6%	61.9%	207	2.5
General continuous random variables	5.9%	65.9%	1,513	2.0
Confidence intervals for proportions	6.5%	69.1%	723	2.0
Calculus of trigonometric functions	3.7%	52.0%	745	1.8
Differentiation rules	3.2%	57.8%	1,419	1.4
Sample proportions	3.1%	63.8%	313	1.1
Calculus of exponential functions	2.9%	65.9%	1,247	1.0
Fundamental theorem of calculus and definite integrals	1.6%	60.6%	1,762	0.6
Bernoulli distributions	1.3%	74.6%	53	0.3

The same cut at dot-point level

Dot point	Content	Share of marks	Average	Attempts	Marks at risk
3.1.2.3	Solve equations involving exponential and logarithmic functions with base $e$ , with and without technology	9.1%	63.5%	983	3.3
4.1.2.5	Model and solve problems that involve definite integrals, including motion problems, with and without technology	3.6%	40.2%	491	2.1
4.3.2.4	Calculate probabilities and quantiles associated with a given normal distribution, using technology	4.2%	50.8%	733	2.1
3.1.2.5	Model and solve problems that involve derivatives of exponential and logarithmic functions, with and without technology	4.1%	53.9%	640	1.9
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	3.6%	59.0%	110	1.5
3.3.1.3	Understand the concepts of concavity and points of inflection and their relationship with the second derivative	2.9%	52.3%	407	1.4
4.1.2.1	Calculate the area enclosed by a curve and the $x$ -axis over a given domain, with and without technology	2.3%	40.5%	292	1.4
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	3.4%	62.9%	866	1.3
4.3.2.5	Model and solve problems that involve normal distributions, with and without technology (distribution tables are not required)	2.4%	53.7%	1,124	1.1
3.2.1.3	Model and solve problems that involve derivatives of trigonometric functions, with and without technology	2.1%	47.0%	572	1.1
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 $p$	2.6%	57.5%	1,254	1.1
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	2.1%	47.5%	1,564	1.1
3.1.2.4	Use the rules $\frac{d}{dx}\ln(x) = \frac{1}{x}$ and $\frac{d}{dx}\ln(f(x)) = \frac{f'(x)}{f(x)}$	1.9%	45.7%	1,074	1.0
3.3.1.1	Understand the concept of the second derivative as the rate of change of the first derivative function	1.8%	51.4%	136	0.9
3.2.2.2	Use the product rule to determine the derivative of exponential, logarithmic and trigonometric functions, expressing derivatives in simplest and factorised form	1.6%	50.5%	705	0.8
3.1.2.1	Recognise and determine the qualitative features of the graph of $y = \ln(x) = \log_e(x)$ , including asymptote and intercept	2.3%	66.3%	814	0.8
4.1.2.3	Use the trapezoidal rule, $\int_a^b f(x) dx \approx \frac{w}{2}[f(x_0) + 2(f(x_1) + f(x_2) + f(x_3) + \dots + f(x_{n-1})) + f(x_n)]$ , where $w = \frac{b-a}{n}$ , to approximate an area and the value of a definite integral, with and without technology	1.4%	50.2%	239	0.7
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	2.0%	65.0%	310	0.7
3.1.1.4	Use the rules $\frac{d}{dx}e^x = e^x$ and $\frac{d}{dx}e^{f(x)} = f'(x)e^{f(x)}$	2.4%	71.1%	929	0.7
4.5.1.6	Model and solve problems that involve interval estimates for proportions, with and without technology	2.3%	70.6%	477	0.7

Which command verbs cost the most marks

Verb	Share of marks	Average	Attempts	Marks at risk
determine	57.5%	58.9%	1,373	23.6
evaluate	7.0%	51.0%	97	3.4
justify	1.2%	52.7%	85	0.6
state	1.9%	71.3%	86	0.6
calculate	2.1%	82.4%	175	0.4
show	0.1%	81.8%	73	0.0

Evaluate has the lowest average on the page at 51.0%, and it is not where the marks go. Determine averages 58.9% but carries 57.5% of the paper against 7.0%, so it puts 23.6 marks per 100 at risk against 3.4. For the plain ranking of verbs by score, and what QCAA has said about each, read the QCE Mathematical Methods study guide.

What this does not measure

Where to practise

Work the priority list from the top: Differentiation of exponential and logarithmic functions and Further integration first, then the dot points above. Each topic page holds real QCAA questions with marking criteria attached. For what the papers actually cover, read QCE Mathematical Methods most tested topics.

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.