The QCE Mathematical Methods external assessment is worth 50% of your final subject result, and it is the one assessment every Queensland student sits under the same conditions on the same day. The 2025 exam tested inflection points, logarithmic equations, motion problems, confidence intervals and conditional probability, and QCAA's subject report shows exactly where students gained and lost marks.
This guide breaks down the external assessment section of QCAA's 2025 Mathematical Methods subject report. It covers how every multiple-choice question was answered across the state, the short-response questions QCAA singled out, what effective responses did and the advice QCAA gave for future students. For the school-based side of the subject, see our companion breakdown of the 2025 QCE Maths Methods internal assessments.
Key Takeaways
- 12,616 students completed Units 3 and 4 in 2025, a 9.45% increase on 2024, and 96.27% received a C or higher.
- An A required a final mark of 86 or more out of 100. 28.46% of students achieved an A.
- Students averaged 70.4% on the Paper 1 multiple-choice questions and 68.7% on Paper 2.
- The hardest multiple-choice question was Paper 1, Question 8: only 28.72% chose the correct answer, while 58.88% chose the same wrong option.
- QCAA's biggest concern was Objective 4: evaluating the reasonableness of solutions. Students lost marks for not proving a change in concavity, not checking solutions and not using a confidence interval to judge a claim.
The 2025 Cohort at a Glance
| Statistic | 2025 |
|---|---|
| Schools offering Mathematical Methods | 431 |
| Students who completed Unit 1 | 16,348 |
| Students who completed Unit 2 | 14,464 |
| Students who completed Units 3 and 4 | 12,616 |
| Change in enrolment since 2024 | +9.45% |
| Students who received a C or higher | 96.27% |
Source: QCAA Mathematical Methods subject report, 2025 cohort.
About 77% of the students who completed Unit 1 went on to complete Units 3 and 4. Units 1 and 2 were also harder than many students expect: 1,850 students (12.8%) received an Unsatisfactory result for Unit 2.
QCE Mathematical Methods External Exam Structure
| Feature | Paper 1 | Paper 2 |
|---|---|---|
| Technology | Technology-free | Technology-active (graphics calculator allowed) |
| Section 1 | 10 multiple-choice questions (10 marks) | 10 multiple-choice questions (10 marks) |
| Section 2 | Short-response questions (45 marks) | Short-response questions (45 marks) |
| Total | 55 marks | 55 marks |
| Time | 5 minutes perusal + 90 minutes working | 5 minutes perusal + 90 minutes working |
The two papers are combined into a mark out of 50, which makes up 50% of your final result. The other 50% comes from your three internal assessments: IA1 (20%), IA2 (15%) and IA3 (15%).
The most common external assessment results in 2025 fell between the high 30s and mid 40s out of 50. Very few students scored 49 or 50, which shows that full marks on the external exam are rare even for strong students.
💡 Get ready for the exam with our step-by-step guide on how to prepare for QCE external exams.
2025 Grade Boundaries and Results
| Grade | Final Mark (out of 100) | Number of Students | Percentage of Students |
|---|---|---|---|
| A | 86–100 | 3,590 | 28.46% |
| B | 69–85 | 4,842 | 38.38% |
| C | 45–68 | 3,713 | 29.43% |
| D | 23–44 | 464 | 3.68% |
| E | 0–22 | 7 | 0.06% |
Source: QCAA Mathematical Methods subject report, 2025 cohort.
Your final mark is your internal assessment total (out of 50) plus your external assessment mark (out of 50). Because so many students do well on their internal assessments, the external exam is often what separates a B from an A.
Paper 1 (Technology-Free): Multiple-Choice Results
| Question | Correct Answer | % Correct | Most Common Wrong Answer |
|---|---|---|---|
| 1 | B | 84.88% | A (7.81%) |
| 2 | A | 65.77% | D (16.28%) |
| 3 | D | 85.59% | C (9.03%) |
| 4 | B | 88.25% | C (10.45%) |
| 5 | D | 62.12% | B (22.51%) |
| 6 | D | 59.88% | C (23.15%) |
| 7 | C | 89.73% | B (4.93%) |
| 8 | C | 28.72% | B (58.88%) |
| 9 | A | 67.49% | D (15.40%) |
| 10 | C | 71.86% | A (16.15%) |
Source: QCAA Mathematical Methods subject report, 2025 cohort. Some students did not answer every question.
What the Paper 1 Results Show
- Question 8 was by far the hardest question on the paper. Twice as many students chose option B as chose the correct answer, C. When most of the state picks the same wrong answer, it usually points to a common misconception rather than a careless slip.
- Questions 5 and 6 each had one wrong option chosen by more than 20% of students, which again suggests an attractive trap.
- Questions 1, 3, 4 and 7 were answered correctly by more than 84% of students. These are marks you cannot afford to drop.
- Across the paper, students averaged 70.4%, so the typical student lost about 3 of the 10 multiple-choice marks.
Paper 2 (Technology-Active): Multiple-Choice Results
| Question | Correct Answer | % Correct | Most Common Wrong Answer |
|---|---|---|---|
| 1 | A | 96.70% | D (1.30%) |
| 2 | C | 89.37% | B (6.24%) |
| 3 | A | 85.58% | B (6.77%) |
| 4 | A | 56.75% | D (23.27%) |
| 5 | D | 44.09% | C (27.15%) |
| 6 | D | 71.39% | A (10.85%) |
| 7 | C | 64.08% | B (18.93%) |
| 8 | C | 73.54% | B (14.86%) |
| 9 | B | 30.60% | D (36.98%) |
| 10 | B | 74.73% | A (9.17%) |
Source: QCAA Mathematical Methods subject report, 2025 cohort. Some students did not answer every question.
What the Paper 2 Results Show
- Question 9 was the hardest on Paper 2. More students chose D (36.98%) than the correct answer, B (30.60%), and responses were spread across all four options.
- Question 5 was answered correctly by fewer than half of students. Options A and C each drew around a quarter of responses.
- Question 1 was answered correctly by 96.70% of students, the easiest question across both papers.
- Paper 2's average (68.7%) was slightly lower than Paper 1's, even though a calculator was allowed. Having technology available does not make a question easier if you misread what it is asking.
💡 Check out why past papers are the best way to study for exams.
Short-Response Questions QCAA Highlighted
QCAA's report gave detailed feedback and sample responses for five short-response questions. Here is what each one asked and what separated effective responses.
Paper 1, Question 13: Proving a Point of Inflection (4 marks)
Students were given the second derivative of a function (f″(x) = 2x − 6 in the sample responses) and asked to show that the function has a point of inflection at a specified value (x = 3).
Effective responses:
- showed that f″(3) = 0, which is the necessary condition
- tested the sign of f″(x) on either side of x = 3, for example f″(2) = −2 < 0 and f″(4) = 2 > 0
- concluded that the function changes from concave down to concave up, which is the sufficient condition, so x = 3 is a point of inflection.
Where students lost marks: Many students correctly showed that f″(3) = 0 and stopped there. QCAA said a significant number could not fully justify the inflection point by showing the change in concavity. A second derivative of zero on its own does not prove a point of inflection.
QCAA's sample responses used three different ways to show the change in concavity: testing values either side, sketching the graph of f″(x), and a sign table. All three earned full marks.
Paper 1, Question 15: Solving a Logarithmic Equation
Students solved an equation of the form (log₂ x)² = log₂(x²) + 8.
Effective responses:
- applied the log law of powers to rewrite log₂(x²) as 2 log₂ x
- recognised the resulting quadratic by letting a = log₂ x, giving a² = 2a + 8
- solved the quadratic to find both solutions, a = 4 and a = −2, so x = 16 or x = ¼
- checked the reasonableness of a solution by substituting it back into the original equation.
Where students lost marks: QCAA recommended more practice with logarithmic equations that need several log laws before they become a quadratic. Full marks needed both solutions and a check of reasonableness, so stopping at one value or skipping the check cost marks.
Paper 1, Question 18: Motion of Two Objects
Students analysed two moving objects: one given by a displacement function with an unknown constant k, the other by a constant acceleration of 4 m/s². They had to find when the objects next had equal velocities and the difference in how far they travelled over that time.
Effective responses:
- differentiated the first object's displacement to find its velocity
- integrated the second object's acceleration and used the initial condition to find its velocity
- used the condition that the velocities were equal at t = 1 to relate the two unknown constants, then solved a quadratic to find the next time (t = 4)
- calculated each object's change in displacement between t = 1 and t = 4. The k terms cancelled, showing that the second object travelled 4.5 m further.
Where students lost marks: QCAA listed this as a question where students needed more practice building their own model. The sample response QCAA chose shows an effective way to handle the two unknown constants (k and the constant of integration): write one in terms of the other, rather than trying to find both.
Paper 2, Question 18: Using a Confidence Interval to Evaluate a Claim
Students were given an incomplete 95% confidence interval (the lower bound, 0.648, from a sample of 500) and had to decide whether a claim that 75% of employees were satisfied was reasonable.
Effective responses:
- set up p̂ − 1.96 × √(p̂(1 − p̂)/500) = 0.648 and solved it with technology to find p̂ ≈ 0.6886
- used p̂ to calculate the upper bound of the interval (about 0.729)
- stated that 0.75 lies outside the confidence interval, so the claim is not supported.
Where students lost marks: QCAA said many students did not fully use the confidence interval to justify a claim. Calculating the interval was not enough. To earn the final marks, students had to explain clearly what the interval meant for the claim.
Paper 2, Question 19: Conditional Probability With Two Beetle Species
Students modelled the horn lengths of two beetle species. 70% of beetles were species A, whose horn lengths were normally distributed with a mean of 20 mm and a standard deviation of 2 mm. 30% were species B, and 14.6% of species B had horns shorter than 18 mm. Students had to find the probability that a beetle with a horn shorter than 18 mm was species A.
Effective responses:
- used the normal distribution to find P(A horn < 18 mm) ≈ 0.1587
- found the proportion of all beetles with horns under 18 mm: 0.7 × 0.1587 + 0.3 × 0.146 ≈ 0.1549
- applied the conditional probability rule: 0.7 × 0.1587 ÷ 0.1549 ≈ 0.717 (about 71.7%)
- communicated clearly, with defined notation, labelled diagrams and a concluding statement.
Where students lost marks: QCAA listed this as a complex question where students needed more practice developing a model. The key step was recognising that the question needed conditional probability, not just a normal distribution calculation.
What Students Did Well
QCAA said the 2025 cohort responded well to:
- solving exponential equations and modelling with exponential functions, in both simple familiar and complex familiar questions
- differentiating and integrating simple logarithmic, exponential and trigonometric expressions
- discrete random variables and their probabilities
- using technology to graph functions and find key features, solve equations, evaluate derivatives and rates of change, and do statistical calculations.
Areas QCAA Wants Students to Improve
1. Proving Inflection Points
Know the difference between the necessary condition (f″(x) = 0) and the sufficient condition (f″(x) changes sign). Practise with a range of functions, including ones where f″(x) = 0 but the concavity does not change.
2. Multi-Step Logarithmic Equations
Practise equations that need several log laws before they can be solved, especially ones that become a quadratic in log x. Always check that each solution is valid in the original equation.
3. Confidence Intervals
Practise confidence intervals for a population proportion in both technology-free situations (QCAA singled out Paper 1, Question 12) and technology-active ones. QCAA also wants students to use the interval to make a judgement about a claim, not just calculate it.
4. Complex Modelling Questions
QCAA wants more practice on unfamiliar questions where students must build the model themselves. It singled out three 2025 examples:
- Paper 1, Question 18: differentiation and anti-differentiation to compare the velocities of two objects
- Paper 1, Question 19: trigonometry in an optimisation problem
- Paper 2, Question 19: conditional probability in a statistical model.
5. Evaluating the Reasonableness of Solutions (Objective 4)
QCAA gave three examples where students were expected to evaluate their answer:
- Paper 1, Question 15: substituting a solution back into a logarithmic equation
- Paper 2, Question 14: using the second derivative to justify the nature of a stationary point
- Paper 2, Question 18: using a confidence interval to judge a claim.
If a question asks you to justify, verify or evaluate, the final marks depend on a clear written conclusion, not just a correct calculation.
💡 Not sure what "evaluate", "justify" or "determine" are asking for? Read our complete guide to QCE cognitive verbs.
Exam-Day Advice From QCAA
- Units 1 and 2 are assumed knowledge. Content from Year 11 can be used in external exam questions, so revise it as well.
- Read each question carefully and check that you have answered what was asked.
- Use the marks as a guide. A 4-mark question needs more working and justification than a 1-mark question.
- Use the right pages for extra working. Extra responses go on the pages labelled "Additional pages for student responses" at the back of the book, not on pages set aside for other questions.
- Fill in the multiple-choice bubbles. Work in the multiple-choice question book is not marked. Use a 2B pencil, fill each A, B, C or D bubble completely and make sure every question has an answer.
How to Prepare for the QCE Maths Methods External Exam
- Practise both papers separately. Build technology-free fluency with algebra, calculus and logs, and practise using your calculator efficiently for Paper 2.
- Learn the full justifications. Inflection points, the nature of stationary points and claims about confidence intervals all need a written conclusion.
- Revise Year 11 content alongside Units 3 and 4.
- Practise complex unfamiliar questions, especially motion, optimisation and conditional probability problems where you must set up the model yourself.
- Check your answers. Substitute solutions back in and ask whether the answer makes sense in context.
- Sit full past papers under timed conditions to practise managing 90 minutes per paper.
For more preparation:
- read our guide to getting an A in QCE Mathematical Methods
- see our breakdown of the 2025 QCE Maths Methods internal assessments
- check the QCE exam timetable 2026 for this year's exam dates
- learn how QCE subject scaling works
- understand how the QCE ATAR is calculated.
Need Help With QCE Maths Methods?
Our QCE Mathematical Methods tutors work one-on-one with students to build technology-free fluency, master calculus and statistics, and write the clear justifications QCAA markers look for.
Source: Queensland Curriculum & Assessment Authority, Mathematical Methods subject report, 2025 cohort (January 2026).
