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WACE ATAR Mathematics Methods was sat by 4,187 Western Australian students in 2025, more than in any of the previous three years. The SCSA's examination report and marking keys show which questions caused the most trouble, and why: logarithms, rectilinear motion, trigonometric functions and interpreting confidence intervals.
This guide breaks down the 2025 ATAR Mathematics Methods examination. It covers how students performed in each section, every question on both papers with the answers from the official marking keys, the questions the examiners singled out and the SCSA's advice for future candidates.
Key Takeaways
- 4,187 students sat the 2025 exam, up 5.3% on 2024. The average was 65.03%, and at least one student scored 100%.
- The Calculator-free and Calculator-assumed sections had almost identical averages: 64.70% and 65.22%.
- The hardest areas were logarithms (Questions 7 and 17), rectilinear motion (Question 6) and trigonometric functions (Question 14).
- Students calculated confidence intervals well but struggled to interpret them against a claim. A confidence interval never proves a claim is wrong.
- Marks were lost for unclear written explanations, disorganised working, graphs that didn't follow instructions and "show" questions without enough steps.
2025 Exam Results at a Glance
| Year | Students Who Sat the Exam | Absent |
|---|---|---|
| 2025 | 4,187 | 46 |
| 2024 | 3,975 | 37 |
| 2023 | 3,611 | 38 |
| 2022 | 3,590 | 65 |
Source: SCSA 2025 ATAR course examination report: Mathematics Methods.
| Section | Raw Marks | Weighting | Average (%) |
|---|---|---|---|
| Section One: Calculator-free | 47 | 35% | 64.70% |
| Section Two: Calculator-assumed | 97 | 65% | 65.22% |
| Whole exam | – | 100% | 65.03% |
Source: SCSA 2025 ATAR course examination report: Mathematics Methods.
Methods enrolments have grown by about 17% since 2022. Exam scores were spread widely, with most students scoring between about 45% and 95%, so there is plenty of room to move up with better technique.
WACE Mathematics Methods Exam Structure
| Section | Questions | Reading Time | Working Time | Marks | Weighting |
|---|---|---|---|---|---|
| Section One: Calculator-free | 7 | 5 minutes | 50 minutes | 47 | 35% |
| Section Two: Calculator-assumed | 10 | 10 minutes | 100 minutes | 97 | 65% |
All questions are compulsory, and a formula sheet is provided. For any question worth more than two marks, valid working or justification is required for full marks.
💡 For more on how WACE exams work, read how WACE exams are structured.
Section One: Calculator-Free Questions and Answers
| Question | Topic | What It Tested | Marks | Answer |
|---|---|---|---|---|
| 1(a) | Continuous random variables | Mean of X with pdf p(x) = 1/(x ln 5) for 1 ≤ x ≤ 5 | 3 | 4/ln 5 |
| 1(b) | Continuous random variables | Median of X | 3 | √5 |
| 2(a) | Fundamental theorem of calculus | F′(x) where F(x) = ∫₀ˣ t sin(t) dt | 1 | x sin(x) |
| 2(b) | Fundamental theorem of calculus | Sign of F(0) and F(2π), justified using the graph | 4 | F(0) = 0; F(2π) is negative |
| 3(a) | Chain rule | Derivative of the water level h(t) = √(8t) | 2 | dh/dt = 4/√(8t) |
| 3(b) | Rates of change | Rate of change of water level at t = 8 | 2 | ½ cm/s |
| 3(c) | Increments formula | Approximate change in water level from t = 8 to 8.1 | 2 | 1/20 cm |
| 4(a) | Uniform distribution | Show k = ⅓ for a pdf on 5 ≤ t ≤ 8 | 2 | Proof |
| 4(b) | Conditional probability | P(T > 5.5 given T < 6) | 2 | ½ |
| 4(c) | Binomial distribution | Ride takes 7 to 8 minutes on exactly two of three days | 3 | 2/9 |
| 5(a) | Area between curves | Integral expression for a shaded region | 4 | Integral expression |
| 5(b) | Area between curves | Evaluate the area | 4 | 6 square units |
| 6(a)–(c) | Rectilinear motion | Using a sign table of x, v and a for a soccer player | 6 | Returns to start between t = 3 and 4; stops between t = 1 and 2 and between t = 4 and 5; slowing down at t = 1, 3 and 4 |
| 6(d) | Rectilinear motion | Sketch a possible displacement graph | 3 | Graph |
| 7(a) | Logarithms | Solve 2 ln(x) = ln(9) + 4 | 3 | x = 3e² |
| 7(b) | Logarithms | Express log₃(10) in terms of m = log₃(6) and n = log₆(5) | 3 | mn + m − 1 |
Answers from the SCSA 2025 Mathematics Methods Calculator-free marking key.
What the Examiners Said About Section One
Students did well on area, probability and rates of change and increments. They struggled with:
- functions defined by an integral (Question 2)
- logarithm laws (Question 7)
- rectilinear motion relationships (Question 6)
- arithmetic with fractions
- plotting graphs accurately, including turning points and intercepts.
Question 6: Reading a Sign Table (9 marks)
A soccer player ran back and forth, and students were given only the signs of displacement, velocity and acceleration at each second. The key ideas:
- The player returns to the start when displacement changes sign (between t = 3 and t = 4).
- The player stops when velocity changes sign (between t = 1 and 2, and between t = 4 and 5).
- The player is slowing down when velocity and acceleration have opposite signs (t = 1, 3 and 4).
- A sketch of displacement needed a local maximum between t = 1 and 2, a local minimum between t = 4 and 5, and inflection points where acceleration changes sign.
Question 7: Logarithms (6 marks)
- In 7(a), write 4 as ln(e⁴) and combine the logs to get x² = 9e⁴, so x = 3e². The marking key required students to give only the positive solution, since ln(x) needs x > 0.
- In 7(b), use the change of base formula to get log₃(5) = mn, and the product rule to get log₃(2) = m − 1, so log₃(10) = mn + m − 1.
Section Two: Calculator-Assumed Questions and Answers
| Question | Topic | What It Tested | Marks | Key Answers |
|---|---|---|---|---|
| 8 | Discrete random variables | Two balls drawn without replacement from 2 red and 3 blue | 8 | P(X = 1) = 3/5, P(X = 2) = 1/10; E(X) = 0.8; Var(X) = 0.36; charge at least $4; replace the first ball for a binomial |
| 9 | Exponential functions and calculus | Growth model for the mass of an apple | 11 | 180.8 g at maturity; 100 g after 107.8 days; fastest growth at t ≈ 111.3 days |
| 10 | Normal distribution and sample proportions | Cognitive test scores, X ~ N(50, 10²) | 12 | 8.08% score 64+; top 1% needs 73.27; P(average) = 0.5161; P(p̂ ≤ 0.46) = 0.2137; Y has mean 100, SD 15; a = 1.5, b = 25 |
| 11 | Logarithmic functions | Stopping distance model s = 40.6 ln(0.15D) | 4 | 40 km/h; ds/dD = 40.6/D; roads with higher targets are affected less |
| 12 | Differentiation and tangents | A dog chasing a rolling ball | 9 | C(4, −⅔); minimum at (3, −4⁄3); ball at (4, −√3 − ⅔) |
| 13 | Continuous random variables | Butterfly life expectancy pdf | 8 | Not normal (not symmetric); P(T < 0.5) = 0.7776 |
| 14 | Trigonometric functions | Electric toothbrush rotation | 7 | 150 cycles per second; oscillates through 45°; max velocity 45π²/2 ≈ 222 cm/s |
| 15 | Binomial distribution and confidence intervals | A robot moving forward or backward | 13 | 0.48; 0 after 3 moves; X ~ Bin(10, 0.6); 0.3823; 50 forward moves; 90% CI (0.4178, 0.5822); the CI does not prove the claim wrong |
| 16 | Sampling and confidence intervals | Exercise-induced asthma in athletes | 12 | 95% CI (0.2410, 0.4632); margin of error 0.1111; minimum sample size 601; proportion differs from the US 20% |
| 17 | Logarithms | Sound intensity in decibels | 13 | 1000 times louder; 0 dB at the reference intensity; I₀ = 10⁻¹² W/m²; L = 10 log₁₀(I) + 120; 160 dB |
Answers from the SCSA 2025 Mathematics Methods Calculator-assumed marking key.
What the Examiners Said About Section Two
Students generally performed well, but Questions 13, 14, 15 and 17 were the most challenging. The issues were:
- not enough steps in "show" questions (such as Question 13(c), showing the value of a)
- applying the properties of trigonometric functions
- forgetting to use radian mode when differentiating or integrating trig functions on the CAS
- interpreting contextual scenarios
- fluency with logarithms.
Question 14: The Electric Toothbrush (7 marks)
Students read the period from a graph (three cycles in 0.02 s, so 150 cycles per second) and found the head oscillates through 45° (amplitude 22.5°). In part (c), the tangential velocity was v(t) = (π/300) dθ/dt, so students had to differentiate the sine model for θ and find the amplitude of the result: a maximum of 45π²/2 ≈ 222 cm/s. The SCSA specifically reminded students to stay in radian mode when differentiating trig functions on the calculator.
Questions 15 and 16: Interpreting Confidence Intervals
These two questions show the difference the SCSA highlighted:
- Question 15(f): The robot's 90% CI (0.4178, 0.5822) does not contain the claimed 0.6, but it does not prove the claim is wrong. About 10% of 90% confidence intervals won't contain the true proportion, and we can't know whether this one is among them.
- Question 16(f): The Australian 95% CI (0.2410, 0.4632) does not contain the American 20%, so we can conclude the proportions are different at the 95% level of confidence.
The key is the wording: a confidence interval can provide evidence at a level of confidence, but it is never proof.
Question 17: Sound Intensity (13 marks)
This question tested log laws throughout. Students needed to show that 90 dB is 1000 times the intensity of 60 dB, find I₀ = 10⁻¹² W/m², and rewrite the formula as L = 10 log₁₀(I) + 120. In part (f), students had to read the value from their graph (160 dB), not just calculate it. The SCSA said students should show clear use of the graph when asked.
💡 Not sure what "show", "justify" or "determine" are asking for? Read our guide to WACE key words in exam questions.
The SCSA's Advice for Candidates
- Write clearly and legibly, using pens that don't smudge.
- Use the marks as a guide to how much detail to give, and keep answers concise and mathematically accurate.
- Use correct terms and notation from the syllabus, not calculator syntax.
- Know when to use the CAS and when to show working. "Show" questions need enough steps to justify the result.
- Check that your answer is reasonable, for example a probability between 0 and 1, or an integer when one is needed.
- Practise fraction and arithmetic skills for the Calculator-free section.
- Include units in applied questions.
- When asked to use a graph, show how you used it, rather than relying on the calculator.
Advice to Teachers That Students Should Know
- Strengthen understanding of logarithms.
- Interpret confidence intervals in terms of statistical significance, not proof.
- Understand the relationships between displacement, velocity and acceleration.
- Stay in radian mode for trig calculus.
- Practise clear, precise written explanations.
How to Prepare for the WACE ATAR Mathematics Methods Exam
- Drill log laws by hand, including change of base and solving log equations, for both sections.
- Practise sign-table and graph-based motion questions until "stops", "returns to start" and "slowing down" are automatic.
- Set your CAS to radians and check it before every trig calculus question.
- Write a sentence of interpretation for every confidence interval question, using "evidence" rather than "proof".
- Set out "show" questions in full, with every intermediate step.
- Practise fractions and exact values without a calculator.
- Sit full past papers under timed conditions: 50 minutes for Calculator-free and 100 minutes for Calculator-assumed.
For more preparation:
- read our guide to acing WACE Mathematics Methods
- see our 2025 WACE English examination report breakdown
- check the WACE exam timetable 2026 for this year's exam dates
- learn how WACE scaling works.
Need Help With WACE Maths Methods?
Our WACE Mathematics Methods tutors work one-on-one with students to master calculus, logarithms and statistics, and to set out working the way SCSA markers expect.
Sources: School Curriculum and Standards Authority, 2025 ATAR course examination: Mathematics Methods (Calculator-free and Calculator-assumed), marking keys and summary examination report.
