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

The 2020 Paper 1, averaging 60.1% across 469 marked attempts on AusGrader. The 2021 Paper 1 is the highest on 76.9%.

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

3.1.2.3 (3.2 marks at risk per 100), 4.1.2.5 (2.1 marks at risk per 100) and 3.1.2.5 (1.8 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?

20,380 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 20,380 marked attempts on AusGrader, QCE Mathematical Methods students average 60.8% on questions taken from board external papers. The lowest average of any command verb is evaluate on 55.0%, 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 20 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	67.4%	472
2025	Paper 2	63.7%	732
2024	Paper 1	66.3%	637
2024	Paper 2	60.5%	429
2023	Paper 1	64.2%	447
2023	Paper 2	65.2%	355
2022	Paper 1	70.2%	466
2022	Paper 2	68.3%	300
2021	Paper 1	76.9%	822
2021	Paper 2	73.3%	274
2020	Paper 1	60.1%	469
2020	Paper 2	65.9%	306

The lowest average belongs to the 2020 Paper 1 on 60.1% from 469 attempts, and the highest to the 2021 Paper 1 on 76.9% from 822 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	59.7%	16,081
Short answer	62.5%	4,299

Score by unit

Unit	Average	Attempts
Unit 3: Further calculus and introduction to statistics	60.1%	13,566
Unit 4: Further calculus, trigonometry and statistics	62.0%	7,858

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 20,380.

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%	61.8%	5,043	7.9
Further integration	12.2%	56.5%	2,961	5.3
Further applications of differentiation	10.9%	51.4%	3,312	5.3
Continuous random variables and the normal distribution	13.1%	62.6%	3,202	4.9
Discrete random variables	9.7%	59.3%	2,870	4.0
Differentiation of trigonometric functions and differentiation rules	6.9%	58.8%	2,372	2.8
Introduction to integration	9.6%	71.8%	2,114	2.7
Trigonometry	6.6%	63.8%	274	2.4
Interval estimates for proportions	6.5%	69.7%	941	2.0
Sampling and proportions	3.8%	68.6%	536	1.2

Differentiation of exponential and logarithmic functions tops the list on 7.9 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%	59.8%	3,491	7.1
The second derivative and applications of differentiation	10.9%	51.4%	3,312	5.3
Applications of integration	10.5%	50.6%	1,245	5.2
Binomial distributions	7.9%	60.8%	2,025	3.1
Normal distribution	7.2%	58.5%	1,672	3.0
Anti-differentiation	9.6%	71.8%	2,114	2.7
Cosine and sine rules	6.6%	63.8%	274	2.4
General continuous random variables	5.9%	66.1%	1,803	2.0
Confidence intervals for proportions	6.5%	69.7%	941	2.0
Calculus of trigonometric functions	3.7%	55.4%	884	1.6
Differentiation rules	3.2%	60.9%	1,696	1.3
Sample proportions	3.1%	65.5%	424	1.1
Calculus of exponential functions	2.9%	66.9%	1,817	1.0
Fundamental theorem of calculus and definite integrals	1.6%	62.1%	1,968	0.6
Bernoulli distributions	1.3%	78.4%	96	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%	64.4%	1,104	3.2
4.1.2.5	Model and solve problems that involve definite integrals, including motion problems, with and without technology	3.6%	42.1%	539	2.1
3.1.2.5	Model and solve problems that involve derivatives of exponential and logarithmic functions, with and without technology	4.1%	55.0%	801	1.8
4.3.2.4	Calculate probabilities and quantiles associated with a given normal distribution, using technology	4.2%	56.4%	886	1.8
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%	59.7%	132	1.4
3.3.1.3	Understand the concepts of concavity and points of inflection and their relationship with the second derivative	2.9%	52.4%	470	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%	43.9%	342	1.3
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%	63.5%	1,039	1.2
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%	47.6%	1,580	1.1
4.3.2.5	Model and solve problems that involve normal distributions, with and without technology (distribution tables are not required)	2.4%	55.8%	1,257	1.1
3.2.1.3	Model and solve problems that involve derivatives of trigonometric functions, with and without technology	2.1%	49.7%	637	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%	59.7%	1,408	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%	47.1%	1,132	1.0
4.1.2.4	Calculate total change by integrating instantaneous or marginal rates of change, with and without technology	1.9%	48.3%	57	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%	55.3%	177	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%	66.5%	868	0.8
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%	54.4%	829	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%	71.8%	1,357	0.7
4.5.1.6	Model and solve problems that involve interval estimates for proportions, with and without technology	2.3%	70.9%	562	0.7
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%	53.8%	269	0.7

Which command verbs cost the most marks

Verb	Share of marks	Average	Attempts	Marks at risk
determine	57.5%	60.7%	1,991	22.6
evaluate	7.0%	55.0%	132	3.1
state	1.9%	71.1%	116	0.6
justify	1.2%	55.9%	123	0.5
calculate	2.1%	79.2%	233	0.4
explain	0.7%	59.4%	62	0.3
estimate	0.1%	57.9%	52	0.0
show	0.1%	78.9%	92	0.0

Evaluate has the lowest average on the page at 55.0%, and it is not where the marks go. Determine averages 60.7% but carries 57.5% of the paper against 7.0%, so it puts 22.6 marks per 100 at risk against 3.1. 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.