Past exam question navigator — topic × difficulty (2018–2025)

SACE Stage 2 Mathematical Methods • every question of the last eight external exams • 93 questions • difficulty from the official cohort mark data • response patterns from the official marker advice

What this is

Every question from the 2018–2025 external examinations, tagged by topic and by how the whole state cohort actually performed that year. The difficulty data comes from the SACE Board’s official Subject Assessment Advice, which reports the percentages of students reaching each mark in every question. Use it to aim your revision at the right level of question — not too easy, not (yet) too hard.

In the Cohort performance column, a icon shows that the examiners described what more successful and less successful responses commonly did for that question in the official advice. Click the icon to read their findings. If a question has no icon, the advice for that year gave no point-by-point response commentary.

The last column of each table has two icons: a 👁 view icon that opens just that question in a new window, and a ↓ download icon that saves it as its own PDF (e.g. 2025-Q1.pdf) from the past-exams-by-questions folder. These PDFs use the absolute page ranges you see printed on each exam page.

Accessible

Most of the cohort earned most of the marks (e.g. “85% scored 6–7 of 7”, “average 7 of 9”). These are the marks everyone can bank.

Moderate

The cohort lands mid-range: a routine start, then a harder finish (e.g. “mean 56%”, “37% full marks but 27% scored 3 or fewer”). The biggest share of exam marks lives here.

Demanding

The cohort mean was below about 50%, or very few students finished it (e.g. “8% full marks, 22% scored zero”). Usually the last question of a booklet, a conjecture-and-prove finish, or an unfamiliar twist. Early parts are still worth attempting — see below.

How to use it in your revision

Start from your diagnostic quiz results: pick your Amber/Red topic from the buttons below.
Work Accessible rows in that topic first, then Moderate, then Demanding. The numbered Year / Question columns tell you exactly which past paper to open.
Even in Demanding questions the marks are front-loaded: SACE advice repeatedly notes that most students earn at least half the marks on the routine openings of the big questions. So never skip a demanding row entirely — bank its early parts.
Check the Marks column before you start. Questions worth 12–16 marks are expected to take 15–20 minutes; a 4–5 mark question should take about 5.
Rows from 2018–2019 come from the older, longer exam format (142–146 marks). Treat them as a topic question bank when you are short of time; the 2020–2025 rows match the current 100-mark format exactly.

A guide by goal:

Securing a pass: make every Accessible row automatic across all six topics, plus the routine openings of Moderate questions.
Reaching a B: all of the above, plus the full Moderate set — and know how to start every question in the paper.
Reaching an A: add the Demanding rows — the conjecture-and-prove finishes, exact-value algebra and interpretation parts are what separate the top band.

Topic:

Difficulty:

Topic 1 — Further differentiation and applications

What the data says: the derivative warm-up (Q1) is the most reliable bankable mark in the exam, appearing every year in the Accessible band. Reading graphs of f′ and the conjecture-and-prove finishes are the persistent low-scoring spots.

Year	Q	Marks	What it tests	Difficulty	Cohort performance (that year)
2018	1	8	Derivative warm-up: product, quotient, chain rules with ln	Accessible	47% full marks; ~95% earned at least half
2018	3	8	Stormwater tank V(t): differentiate, interpret flow both ways, sketch V′	Moderate	~75% scored over half; 20% full marks
2018	5	5	First principles for a 1 − 5/x²-type function	Moderate	over half full marks; 30% scored 3 or fewer
2018	9	7	Sketch f given the graph of f′	Moderate	60% scored ≥4/7; 36% scored ≥6
2018	13	11	Tangent to a trig curve with exact radian values	Demanding	38% scored ≤2/11; 14% full marks
2018	16	12	y = xe^−x: tangent family, maximise the y-intercept	Demanding	~20% scored 0–1; 46% scored over half; 17% scored ≥10/12
2019	1	8	Derivative + definite integral warm-up	Accessible	44% full marks; over 90% earned at least half
2019	3	9	Exponential graph: f′, f″, inflections “using algebra”	Moderate	~⅓ full marks; ~⅔ scored over half
2019	6	5	Identify features from f, f′, f″; sketch f′	Moderate	over ¼ full marks; ~1 in 10 scored zero
2019	8	12	Average vs instantaneous rate; first principles (5-mark part)	Moderate	~⅔ scored over half
2019	9	11	Logistic model: rates, fastest growth, tipping-point targets	Demanding	statistically the hardest of 2019; mean below half
2020	1	9	Derivative + integral warm-up	Accessible	74% scored 8–9/9
2020	4	8	First principles on a rational function + accuracy discussion	Moderate	general success; large variation in marks
2020	7	9	f, g stationary/inflection tick boxes; sketch g′	Demanding	only 5% full marks
2020	8	10	x² ln x: tangent family table, conjecture, prove	Moderate	~¼ full marks; largest variation of the paper
2020	10	13	Traffic C(t): rates, total cars, best start time (also uses T3)	Demanding	lowest average of 2020; only 4% full marks
2021	1	7	Derivative warm-up	Accessible	85% scored 6–7/7
2021	2	8	Rational function: stationary points, inflection conditions	Moderate	only 20% scored over 6/8
2021	6	12	Caffeine surge model: sketch, rates, maximum safe dose	Moderate	~70% scored at least half
2021	7	8	Chord slopes → exact tangent; exponential form	Moderate	~70% scored at least half
2022	1	11	First principles + derivatives + definite integral warm-up	Accessible	~60% scored 10–11/11
2022	3	7	Sign diagrams for f′, f″ from a graph; sketch f′	Moderate	mean 4.15/7 (≈59%)
2022	5	8	Trig stationary points; exact value of a; fastest decrease	Accessible	mean 71.6% — third highest of 2022
2022	7	9	x ln x family: stationary point, table, conjecture (primary topic: T4)	Moderate	mean 56.1%; 21.5% full marks; high zero rate
2022	9	11	Normal-line geometry: x-intercept optimisation	Demanding	mean 49.6%; about half scored over half the marks
2023	1	7	Derivative warm-up	Accessible	~80% scored 6–7/7
2023	3	7	kx² + 1: first principles, tangent, find k	Accessible	42% full marks — second-highest BK1 average
2023	5	12	Rational family: stationary points table, conjecture, prove	Demanding	23% full marks; near-uniform spread 0–9
2023	6	8	f′ graph from a description: sign diagram, ticks, sketch	Moderate	lowest BK1 mean (56%)
2023	8	10	Damped-trig lean: rates, exact stationary times	Moderate	mean ≈56%
2024	5	5	First principles on x/(x − 2)	Moderate	55% full marks; 27% scored ≤2/5
2024	7	10	Tangent family: x-intercept table, conjecture, prove	Moderate	mean 6.06/10 (≈61%)
2024	9	12	Exponential family: stationary x- and y-coordinate formulas	Demanding	mean ≈41%
2024	10	13	e^x cos x: f′, f″, exact area difference (also uses T3)	Demanding	~half scored ≥4/12
2025	1	8	Derivative + definite integral warm-up	Accessible	average 6/8; 40% full marks
2025	5	7	Read the f′ graph: sign diagram, sketch f″ and f	Moderate	only ~15% full marks (graph-drawing)
2025	6	8	Valley tangent: minimum slope, exact tangent through O†	Moderate	lowest BK1 (no figure quoted); strong start, hard finish
2025	10	12	100 ln(1 + sin²x): stationary points, conjecture, d_k	Demanding	only 9% scored ≥10/12
2025	11	13	Hyperbolic functions: sketch, prove derivatives, exact inflection	Demanding	over 30% scored over half — better than expected
2018	7	8	x(ln x)² derivative; integral table conjecture (primary topic: T4)	Moderate	~30% scored ≤2; 43% scored ≥5; ~30% full marks
2018	10	13	Product-rule derivative feeding a k sin x pdf (primary topic: T5)	Moderate	⅔ scored over half; 20% full marks
2019	11	16	D(x) = 20 ln x − f(x): exact maximum by algebra (primary topic: T4)	Demanding	toughest in BK2; mean below half
2020	2	7	xe^−2x² pdf: mode via f′, exact and algebraic (primary topic: T5)	Moderate	37% full marks; 27% scored ≤3/7
2022	6	14	Rate model m(t): show m′, maximum, queue size (primary topic: T3)	Demanding	BK1 lowest mean (49.8%); 3.34% full marks
2022	7	9	(x ln x)/k family: stationary point, conjecture (primary topic: T4)	Moderate	mean 56.1%; 21.5% full marks; high zero rate
2024	3	6	ln(x − 3): derivative, parameter b, transformation (primary topic: T4)	Moderate	over ¼ full marks, but flagged as challenging

Topic 2 — Discrete random variables

What the data says: routine distribution-table and binomial parts are consistently among the stronger results in the paper, but the non-routine twists — three-draw sums via combinations, moment-generating functions, geometric-type games — split the cohort sharply.

Year	Q	Marks	What it tests	Difficulty	Cohort performance (that year)
2018	14	12	Zener cards ESP: binomial, threshold k by trial-and-error, claim	Demanding	82% scored ≥4/14, but only 35% scored at least half
2018	15	14	Duck game: E(X), sd, three-duck sums via combinations, profit	Moderate	~¾ scored ≥6/14; only 16% scored ≥12
2019	5	11	Nestlings: distribution table, binomial, interpretation	Moderate	⅔ scored over half; large spread
2019	13	11	Ceiling fans: profit DRV, V(X), fixed-price decision	Moderate	~⅔ scored over half; large spread
2020	3	7	Robots: binomial conditions, E(T), model claim	Accessible	majority scored close to full marks
2021	5	9	Hotel ratings: distribution, binomial, group size for 90%	Moderate	~60% scored ≥5/9; 26% full marks
2022	2	7	Benford’s law DRV + CI for a mean (also uses T6)	Accessible	mean 74.1% — second highest of 2022
2022	8	16	Choc-pricots: binomial, model p(x), 90% CI, rate bounds (also uses T6)	Moderate	highest BK2 mean (60.7%); only 4.35% full marks; mode 12/16
2023	4	10	Astro Cats: binomial, disprove a “100%” claim, 90% CI (also uses T6)	Moderate	strong performance, but full marks rare†
2023	9	10	Poisson-type pmf via M(t): table, show μ, σ, Pr within one sd	Demanding	only 7% full marks
2025	4	9	Cat Eyes: binomial, prize threshold, p̂, 95% CI (also uses T6)	Accessible	~¾ scored ≥6/9
2025	8	11	Dice game: geometric-type pmf, quadratic condition for n	Moderate	~60% scored ≥6/11; mode 10
2024	4	12	Binomial count of CI misses across 50 simulations (primary topic: T6)	Moderate	~50% earned at least 75% of the marks

Topic 3 — Integral calculus

What the data says: rectangle estimates and straightforward definite integrals land in the Moderate band with decent full-marks rates; kinematics is reliably Moderate; rate-in/rate-out models and exact-area chains with logarithms are among the hardest questions SACE has set.

Year	Q	Marks	What it tests	Difficulty	Cohort performance (that year)
2018	6	7	Braking: velocity → distance integral in terms of k, solve	Moderate	20% scored 0–1; ⅓ full marks
2018	11	10	Exponential area estimates: rectangles, convexity, f″ = 0	Moderate	70% scored at least half; ~¼ scored 9–10/10
2019	4	4	Areas between curves from labelled regions (integral vs area)	Moderate	~⅓ full marks
2019	14	12	Cos family: areas for k = 1, 2, conjecture, prove	Moderate	⅓ full marks
2019	15	14	x ln x rectangles + factorial inequality (n/e)&supn; < n!	Demanding	over half scored under half marks
2020	5	6	Upper/lower estimates; show (U + L)/2 = 24	Moderate	almost half scored 7–8 marks (near-full)
2020	9	16	Monte Carlo area: p̂, proportion CI, minimum m, exact area	Demanding	mean ≈8/16
2021	10	9	x^x rectangles: underestimate, convexity, which is closer	Moderate	over 60% scored at least half
2021	11	15	Progress bars: CI + integral models for task times (also uses T6)	Demanding	mean ≈50%; only 7% scored ≥14/15
2022	6	14	Sample-queue rate model: show m′, estimates, queue size (also uses T1)	Demanding	BK1 lowest mean (49.8%); 3.34% full marks — lowest of both booklets
2023	10	14	ln((x + 1)²): estimates, ∫2 ln x dx, exact area (also uses T4)	Demanding	mean ≈40%; under 4% full marks
2024	2	9	Two cars: velocity → displacement, compare distances	Moderate	~60% scored ≥6/9
2024	8	12	Parking-garage rates + EV weight sums via CLT (also uses T6)	Moderate	highest BK2 mean; 30% scored ≥10/12
2025	3	10	Parabola: U₂ rectangles, f″, exact area	Moderate	~¼ full marks
2025	9	8	Areas with sin x: integral statements, maximise ∫, sketch	Moderate	~63% scored ≥4/8
2019	1	8	Warm-up includes one definite integral (primary topic: T1)	Accessible	44% full marks; over 90% earned at least half
2020	10	13	Total cars over a time interval by integration (primary topic: T1)	Demanding	lowest average of 2020; only 4% full marks
2023	2	8	Exponential pdf integrals, indefinite integral (primary topic: T5)	Moderate	~50% scored 7–8/8; sharp drop-off at 5–6
2024	6	12	Mean of a k/x pdf by integration (primary topic: T5)	Moderate	lowest BK1 mean; 30% scored ≥10/12
2024	10	13	Exact area difference between f and f′ (primary topic: T1)	Demanding	~half scored ≥4/12
2025	1	8	Warm-up includes one definite integral (primary topic: T1)	Accessible	average 6/8; 40% full marks

Topic 4 — Logarithmic functions

What the data says: standalone log-law and log-transformation questions have produced some of the worst cohort results in eight years (2018’s Q12 was the single hardest question of that exam) — yet short ln derivatives inside the warm-ups are reliably banked. The gap is law fluency and transformation language, not calculus.

Year	Q	Marks	What it tests	Difficulty	Cohort performance (that year)
2018	7	8	x(ln x)² derivative; integral table conjecture (also uses T1)	Moderate	~30% scored ≤2; 43% scored ≥5; ~30% full marks
2018	12	6	Log-law simplification to a + b ln x; graph relationship	Demanding	hardest of 2018: over ⅓ scored 0–1; only 16% full marks
2019	11	16	D(x) = 20 ln x − f(x): sketch, exact maximum (also uses T1)	Demanding	toughest in BK2; mean below half
2021	4	7	ln x vs ln 3x; ln(2x + 6) as a ln(x + b)	Moderate	only 18% full marks (transformation language)
2022	7	9	(x ln x)/k family: stationary point, table, conjecture (also uses T1)	Moderate	mean 56.1%; 21.5% full marks; high zero rate
2024	3	6	ln(x − 3): derivative, parameter b, transformation (also uses T1)	Moderate	over ¼ full marks, but flagged as challenging
2022	1	11	Warm-up includes ln(x + 5) and ∫3/(x − 4) dx (primary topic: T1)	Accessible	~60% scored 10–11/11
2023	10	14	ln((x + 1)²) area chain with log laws (primary topic: T3)	Demanding	mean ≈40%; under 4% full marks
2024	9	12	Exponential family: logs to extract stationary coordinates (primary topic: T1)	Demanding	mean ≈41%

Topic 5 — Continuous random variables (incl. normal distributions)

What the data says: normal-distribution computation questions are regularly among the best answered in the paper, but pdf calculus (normalising constants, exact means by integration, modes and medians via calculus) is reliably Moderate-to-Demanding and polarises the cohort.

Year	Q	Marks	What it tests	Difficulty	Cohort performance (that year)
2018	2	5	Normal curves: match areas to Pr statements (tick boxes)	Accessible	over 80% full marks — most accessible of 2018
2018	10	13	k sin x pdf: show k, Pr, show E(X) = π/2, sd (also uses T1)	Moderate	⅔ scored over half; 20% full marks
2019	2	7	Egg/bread shelf life: two normals, sketch, z-score comparison	Demanding	only 15% full marks; ~30% scored under half
2019	10	9	kx(1 − x)ⁿ pdf family: k, area, conjecture, prove	Moderate	nearly 30% full marks, but the same share scored under ¼ of the marks
2020	2	7	xe^−2x² pdf: validity, Pr, exact algebraic mode (also uses T1)	Moderate	37% full marks; 27% scored ≤3/7
2020	6	15	Oysters: normal, S₁₂, CI, claim chain (also uses T6)	Moderate	~¾ scored at least half the marks
2021	8	10	x/8 pdf: Var, Pr, conjecture, prove, max m	Moderate	11% full marks but 11% scored zero; mean ≈60%
2022	4	9	Oat milk: normal, S₆ with √n, largest σ (also uses T6)	Moderate	~¼ full marks
2022	10	8	Service-life pdf (Weibull-type): Pr, median, exact k	Demanding	8% full marks; 21.8% scored zero
2023	2	8	Exponential wait-time pdf: integrals, exact 80th percentile (also uses T3)	Moderate	~50% scored 7–8/8; sharp drop-off at 5–6 marks
2023	7	14	Milk: normal, S₉₀, CI, 2000 L claim (also uses T6)	Accessible	mean 75% — very high, but low full-marks rate
2024	1	9	Normal + X̄₄ distribution, sketch its pdf (also uses T6)	Accessible	average 7/9
2024	6	12	Sand bags: k/x pdf, mean by integral, 99% CI (also uses T6, T3)	Moderate	lowest BK1 mean; 30% scored ≥10/12
2025	2	8	Tofu: normal, recover σ via Z, box of 25 sums (also uses T6)	Moderate	~50% scored ≥6/8
2025	7	6	TDEE pdf: integrate, quartile claim, CI twist (also uses T6)	Demanding	only ~12% full marks; final two parts
2018	4	8	Sampling distributions of means and sums (primary topic: T6)	Moderate	under ¼ scored half or less; only ⅓ scored 7–8/8

Topic 6 — Sampling and confidence intervals

What the data says: computing a confidence interval is one of the best-banked skills in the exam (means of 74–75% appear repeatedly) — what caps marks is the interpretation finish: claims, interval positions, reverse-engineering sample sizes, and the lower-bound-zero edge case.

Year	Q	Marks	What it tests	Difficulty	Cohort performance (that year)
2018	4	8	Nut packets: sample means vs sums, CLT description, label claim (also uses T5)	Moderate	under ¼ scored half or less; only ⅓ scored 7–8/8
2018	8	8	Persian cats: CI for a mean, press-release claim, endpoints as estimates	Moderate	80% scored at least half; only 19% scored 7–8/8
2019	7	5	Electricity: CI for a mean + interpretation choice	Accessible	second-highest mean score of the exam
2019	12	12	Goggles: CI for a proportion chain, claim logic, minimum n	Demanding	25% scored ≥10/12; fairly uniform spread below
2021	3	7	Donations: why not normal; S₉₀₀; CLT; Pr	Accessible	mean 73% — second highest of 2021
2021	9	8	Marbles: 99% CI, p̂ = 2.58²/(n + 2.58²) edge case	Demanding	lowest mean of 2021 (43%)
2024	4	12	50 simulated CIs: fill boxes, contradict claim, binomial count of misses (also uses T2)	Moderate	~50% earned at least 75% of the marks
2020	6	15	CI for a mean inside the oysters chain (primary topic: T5)	Moderate	~¾ scored at least half the marks
2020	9	16	Proportion CI on a Monte Carlo estimate; why n = 3 fails (primary topic: T3)	Demanding	mean ≈8/16
2021	11	15	CIs for task-time means inside an integral model (primary topic: T3)	Demanding	mean ≈50%; only 7% scored ≥14/15
2022	2	7	CI for a mean on Benford digit averages (primary topic: T2)	Accessible	mean 74.1% — second highest of 2022
2022	4	9	CLT justification for six-pack sums (primary topic: T5)	Moderate	~¼ full marks
2022	8	16	Proportion CI + rate-bounds claim (primary topic: T2)	Moderate	highest BK2 mean (60.7%); only 4.35% full marks
2023	4	10	90% proportion CI and website claim (primary topic: T2)	Moderate	strong performance, but full marks rare†
2023	7	14	CI for a mean + 2000 L claim (primary topic: T5)	Accessible	mean 75% — very high, but low full-marks rate
2024	1	9	X̄₄ distribution: mean, sd, sketch (primary topic: T5)	Accessible	average 7/9
2024	6	12	99% CI for a mean and “mean reduced” claim (primary topic: T5)	Moderate	lowest BK1 mean; 30% scored ≥10/12
2024	8	12	CLT weight-sum probability for vehicles (primary topic: T3)	Moderate	highest BK2 mean; 30% scored ≥10/12
2025	2	8	Box-of-25 sums: show mean and sd, Pr (primary topic: T5)	Moderate	~50% scored ≥6/8
2025	4	9	p̂, 95% CI and tick-box claim evaluation (primary topic: T2)	Accessible	~¾ scored ≥6/9
2025	7	6	Unfamiliar CI question: width given, recover the sample mean (primary topic: T5)	Demanding	only ~12% full marks; final two parts

Reading the table — three things worth knowing

Question position predicts difficulty. Q1 of Booklet 1 was Accessible in every single year (44–85% full marks). The last question of each booklet is consistently Demanding — but its early parts are usually routine: SACE advice notes ~80% of students earn at least half marks on the routine openings of the big questions. Bank the openings, then push into the finish.
Some rows appear under two topics. Multi-topic questions are listed under each topic they substantially draw on, with their home topic noted. Filtering by a topic shows all of its rows.
Demanding ≠ “not for you”. The band describes how the whole statewide cohort went. In a topic you are strong at, a Demanding row may feel like Moderate — that is exactly the signal to attempt it.