CensaiHSC Mathematics Extension 1 · 2026 Forecastcensai.com.au
HSC Mathematics Extension 1 2026 Forecast
Every question from the six HSC Mathematics Extension 1 papers of 2020–2025 (175 questions) has been mapped to the syllabus dot point(s) it assesses. Multiple-choice questions 1–10 are kept whole; from Question 11 each letter (a), (b), … is a separate question, with roman (i), (ii) kept together. The 2026 paper is the final examination of the 2017 syllabus; a new syllabus is taught from 2026 and first examined in 2027. Each dot point is tagged with its fate in the 2024 syllabus (kept, reduced, or removed), because content that is about to disappear has only one more chance to be examined.
Based on the NESA HSC Mathematics Extension 1 examinations 2020–2025, the Mathematics Extension 1 Stage 6 Syllabus (2017) and the Mathematics Extension 1 11–12 Syllabus (2024). Counts are questions, not marks. A question is counted once for each dot point it touches, so column totals exceed 175. This forecast is a study guide to help you prioritise revision; it is not a guarantee of what will be examined.
175
Questions reviewed (6 papers)
72
Syllabus dot points tracked
244
Dot-point hits recorded
6
Examined all 6 years
7
Examined once
8
Never examined
7
Removed or reduced in 2024
21
Past hits on removed content
Questions per paper by primary topic
| Topic | 2020 | 2021 | 2022 | 2023 | 2024 | 2025 | Total |
|---|---|---|---|---|---|---|---|
| Functions | 5 | 4 | 8 | 6 | 4 | 4 | 31 |
| Trigonometric Functions | 3 | 4 | 2 | 3 | 5 | 6 | 23 |
| Calculus | 8 | 11 | 7 | 7 | 11 | 9 | 53 |
| Combinatorics | 3 | 3 | 3 | 3 | 2 | 3 | 17 |
| Proof | 1 | 1 | 1 | 1 | 1 | 1 | 6 |
| Vectors | 4 | 6 | 7 | 3 | 4 | 7 | 31 |
| Statistical Analysis | 1 | 2 | 3 | 3 | 3 | 2 | 14 |
| Total | 25 | 31 | 31 | 26 | 30 | 32 | 175 |
Calculus and Vectors together are about half of every paper. Proof is almost always a single induction. Statistics is a small number of questions but a large share of the last-chance marks.
Dot-point hits by subtopic, 2020–2025
| Subtopic | Hits | |
|---|---|---|
| F1 Further Work with Functions | 23 | |
| F2 Polynomials | 18 | |
| T1 Inverse Trigonometric Functions | 16 | |
| T2 Further Trigonometric Identities | 8 | |
| T3 Trigonometric Equations | 16 | |
| C1 Rates of Change | 9 | |
| A1 Working with Combinatorics | 21 | |
| P1 Proof by Mathematical Induction | 6 | |
| V1 Introduction to Vectors | 47 | |
| C2 Further Calculus Skills | 26 | |
| C3 Applications of Calculus | 29 | |
| S1 The Binomial Distribution | 25 |
Further calculus skills, DEs/volumes, and vectors dominate. Inverse trig and combinatorics are steady supporting strands. t-formulae do not appear on the chart because they have never been examined.
Syllabus dot points × years examined
Row shading: green = examined every year, amber = examined once, red = never examined. “2024 syllabus” column: Kept / Reduced / Removed.
| Code | Syllabus dot point | 2020 | 2021 | 2022 | 2023 | 2024 | 2025 | Total | Years | 2024 syllabus | Questions (year·number) |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Functions · ME-F1 Further Work with Functions | 23 | ||||||||||
| F1.1a | Sketch y=1/f(x) from y=f(x); asymptotes | 1 | · | · | · | 1 | · | 2 | 2/6 | Kept | 2020·11C, 2024·12E |
| F1.1b | Sketch y^2=f(x) and y=√f(x) from y=f(x) | · | · | · | · | · | · | 0 | 0/6 | Kept | |
| F1.1c | Sketch y=|f(x)| and y=f(|x|) from y=f(x) | · | · | 1 | 1 | · | · | 2 | 2/6 | Kept | 2022·02, 2023·08 |
| F1.1d | Sketch y=f(x)+g(x) and y=f(x)−g(x) | · | · | 1 | · | · | · | 1 | 1/6 | Kept | 2022·04 |
| F1.2a | Solve quadratic inequalities (algebra or graph) | 1 | · | · | · | 1 | · | 2 | 2/6 | Kept | 2020·01, 2024·11B |
| F1.2b | Solve rational inequalities (unknown in denominator) | · | · | 1 | · | 1 | · | 2 | 2/6 | Kept | 2022·11F, 2024·12E |
| F1.2c | Solve |ax+b| ≥ k / ≤ k / < k / > k | · | · | · | · | 1 | 1 | 2 | 2/6 | Kept | 2024·12E, 2025·01 |
| F1.3a | One-to-one, horizontal line test, restrict domain so an inverse exists | · | 1 | 2 | 2 | 1 | · | 6 | 4/6 | Kept | 2021·12D, 2022·09, 2022·13C, 2023·09, 2023·14A, 2024·14B |
| F1.3b | Find f^{−1}(x); domain and range of an inverse | 1 | 1 | · | · | · | 1 | 3 | 3/6 | Kept | 2020·02, 2021·12D, 2025·11A |
| F1.4a | Parametric ↔ Cartesian for lines, parabolas, circles; sketch | · | 1 | 1 | 1 | · | · | 3 | 3/6 | Kept | 2021·08, 2022·05, 2023·11A |
| Functions · ME-F2 Polynomials | 18 | ||||||||||
| F2.1a | Remainder theorem: remainder when P(x) is divided by x−α | 1 | 1 | · | 1 | · | · | 3 | 3/6 | Kept | 2020·11A, 2021·03, 2023·11C |
| F2.1b | Factor theorem; factorise a cubic or quartic | 2 | · | · | 2 | 1 | · | 5 | 3/6 | Kept | 2020·05, 2020·11A, 2023·11C, 2023·14B, 2024·01 |
| F2.1c | Polynomial division: P=AQ+R, degree of remainder | · | · | 1 | · | · | · | 1 | 1/6 | Kept | 2022·03 |
| F2.2a | Sums and products of roots of quadratic, cubic or quartic | 1 | 1 | 1 | 1 | 1 | 2 | 7 | 6/6 | Kept | 2020·14B, 2021·11H, 2022·13D, 2023·14B, 2024·01, 2025·11F, 2025·14E |
| F2.2b | Multiplicity of a root; if multiplicity > 1 then P'(α)=0 | 1 | · | 1 | · | · | · | 2 | 2/6 | Kept | 2020·05, 2022·13D |
| F2.2c | Graph a polynomial from factored form / root multiplicity | · | · | · | · | · | · | 0 | 0/6 | Kept | |
| Trigonometric Functions · ME-T1 Inverse Trigonometric Functions | 16 | ||||||||||
| T1a | Define sin^{−1}, cos^{−1}, tan^{−1}: domain, range, principal value | · | · | 2 | 1 | · | · | 3 | 2/6 | Kept | 2022·01, 2022·13C, 2023·07 |
| T1b | Sketch inverse-trig graphs and their transformations | · | 2 | · | · | 1 | 1 | 4 | 3/6 | Kept | 2021·09, 2021·12D, 2024·05, 2025·11D |
| T1c | sin(sin^{−1} x)=x and sin^{−1}(sin x)=x (and cos, tan) where valid | · | 1 | · | 1 | · | 1 | 3 | 3/6 | Kept | 2021·09, 2023·05, 2025·14D |
| T1d | Odd/even; sin^{−1}(−x); cos^{−1} x + sin^{−1} x = π/2 | · | · | · | 1 | 2 | 1 | 4 | 3/6 | Kept | 2023·07, 2024·04, 2024·14C, 2025·06 |
| T1e | Solve equations / problems involving inverse trig | 1 | · | · | · | 1 | · | 2 | 2/6 | Kept | 2020·13C, 2024·14C |
| Trigonometric Functions · ME-T2 Further Trigonometric Identities | 8 | ||||||||||
| T2a | Sum and difference expansions for sin, cos, tan | · | 1 | · | · | · | · | 1 | 1/6 | Kept | 2021·13D |
| T2b | Double-angle formulae | 2 | 1 | · | · | 1 | 1 | 5 | 4/6 | Kept | 2020·13A, 2020·14B, 2021·11G, 2024·13B, 2025·11B |
| T2c | t-formulae: sin, cos, tan in terms of t=tan(x/2) | · | · | · | · | · | · | 0 | 0/6 | Removed | |
| T2d | Product-to-sum and sum-to-product identities | 1 | · | · | · | · | 1 | 2 | 2/6 | Removed | 2020·12D, 2025·11C |
| Trigonometric Functions · ME-T3 Trigonometric Equations | 16 | ||||||||||
| T3a | R-method: a cos x + b sin x = R sin(x±α) or R cos(x±α) | 1 | 1 | 1 | 1 | 1 | 1 | 6 | 6/6 | Kept | 2020·11D, 2021·07, 2022·11E, 2023·11E, 2024·10, 2025·12E |
| T3b | Solve trig equations (factorise, compound or double angle) | 1 | 2 | · | 1 | 1 | 3 | 8 | 5/6 | Kept | 2020·11D, 2021·11G, 2021·13D, 2023·11E, 2024·06, 2025·05, 2025·11B, 2025·12E |
| T3c | Prove further identities (e.g. sin 3θ, cos 3θ) | 1 | · | · | · | · | 1 | 2 | 2/6 | Kept | 2020·14B, 2025·14E |
| Calculus · ME-C1 Rates of Change | 9 | ||||||||||
| C1.1a | Rates with respect to time; 1D particle language ẋ, ẍ | · | · | · | · | · | · | 0 | 0/6 | Reduced | |
| C1.2a | Exponential growth/decay Q=Ae^{kt} | · | · | · | · | · | · | 0 | 0/6 | Kept | |
| C1.2b | Newton's law of cooling / carrying capacity Q=P+Ae^{kt} | · | 2 | 1 | 1 | · | · | 4 | 3/6 | Kept | 2021·12B, 2021·14B, 2022·12D, 2023·01 |
| C1.3a | Related rates of change (chain rule in context) | 1 | 1 | · | 1 | 1 | 1 | 5 | 5/6 | Kept | 2020·10, 2021·11E, 2023·13A, 2024·11F, 2025·12A |
| Combinatorics · ME-A1 Working with Combinatorics | 21 | ||||||||||
| A1.1a | Multiplication principle; arrangements of distinct objects in a line | · | · | · | · | · | · | 0 | 0/6 | Kept | |
| A1.1b | Circular arrangements | · | · | · | 1 | · | 1 | 2 | 2/6 | Kept | 2023·10, 2025·13B |
| A1.1c | Permutations with restrictions or repeated objects | 1 | · | · | 1 | · | · | 2 | 2/6 | Kept | 2020·08, 2023·11B |
| A1.1d | Combinations nCr; committees and selections | 2 | 1 | 1 | · | 1 | · | 5 | 4/6 | Kept | 2020·08, 2020·12C, 2021·11D, 2022·07, 2024·09 |
| A1.1e | Pigeonhole principle (simple results and proofs) | · | 1 | 1 | · | 1 | 1 | 4 | 4/6 | Removed | 2021·10, 2022·12B, 2024·03, 2025·14A |
| A1.1f | Probability using permutations or combinations | 1 | · | · | · | 1 | · | 2 | 2/6 | Kept | 2020·12C, 2024·09 |
| A1.2a | Binomial expansion (a+b)^n; coefficient of a specified term | 1 | 1 | 1 | · | · | · | 3 | 3/6 | Kept | 2020·14A, 2021·11B, 2022·11C |
| A1.2b | Pascal identities; combinatorial proofs of binomial identities | 1 | · | · | 1 | · | 1 | 3 | 3/6 | Kept | 2020·14A, 2023·12D, 2025·13E |
| Proof · ME-P1 Proof by Mathematical Induction | 6 | ||||||||||
| P1a | Prove a sum identity by induction | 1 | 1 | · | 1 | · | 1 | 4 | 4/6 | Kept | 2020·12A, 2021·12C, 2023·12B, 2025·12C |
| P1b | Prove a divisibility result by induction | · | · | 1 | · | 1 | · | 2 | 2/6 | Kept | 2022·12F, 2024·12D |
| P1c | Identify errors in false inductions; when induction is not appropriate | · | · | · | · | · | · | 0 | 0/6 | Kept | |
| Vectors · ME-V1 Introduction to Vectors | 47 | ||||||||||
| V1.1a | i, j notation; addition, subtraction, scalar multiple | 1 | 2 | 1 | · | 1 | 1 | 6 | 5/6 | Kept | 2020·04, 2021·01, 2021·11A, 2022·11A, 2024·11A, 2025·08 |
| V1.1b | Geometric addition/subtraction (triangle or parallelogram law) | 1 | · | · | · | · | · | 1 | 1/6 | Kept | 2020·06 |
| V1.2a | Magnitude and unit vector | · | · | 1 | · | 1 | 1 | 3 | 3/6 | Kept | 2022·08, 2024·11A, 2025·09 |
| V1.2b | Direction of a vector; bearings and navigation | · | 1 | · | · | · | · | 1 | 1/6 | Kept | 2021·14A |
| V1.2c | Dot product, angle between vectors, perpendicular condition | 1 | 2 | 5 | 1 | 2 | 3 | 14 | 6/6 | Kept | 2020·11B, 2021·05, 2021·14C, 2022·06, 2022·08, 2022·11A, 2022·11D, 2022·13A, 2023·14C, 2024·12A, 2024·13C, 2025·09, 2025·13C, 2025·14C |
| V1.2d | Parallel vectors a=kb | · | · | 2 | · | · | 2 | 4 | 2/6 | Kept | 2022·06, 2022·14B, 2025·08, 2025·11E |
| V1.2e | Projection of one vector onto another | 1 | · | 1 | 1 | 1 | 1 | 5 | 5/6 | Kept | 2020·09, 2022·14B, 2023·06, 2024·13C, 2025·02 |
| V1.2f | Displacement, force or velocity problems in 2D | 1 | 1 | · | · | · | 2 | 4 | 3/6 | Kept | 2020·04, 2021·14A, 2025·13C, 2025·14C |
| V1.2g | Geometric proofs with vectors (parallelogram, midpoint, circle) | · | 1 | 1 | 1 | · | · | 3 | 3/6 | Reduced | 2021·14C, 2022·13A, 2023·14C |
| V1.3a | Projectile: derive/use x=Vt cos θ, y=Vt sin θ − ½gt²; range, height, time | · | 1 | · | 1 | 1 | · | 3 | 3/6 | Kept | 2021·13B, 2023·13B, 2024·14D |
| V1.3b | Projectile with calculus, unknown angle, or two particles | · | · | 1 | 1 | · | 1 | 3 | 3/6 | Kept | 2022·14C, 2023·13B, 2025·14B |
| Calculus · ME-C2 Further Calculus Skills | 26 | ||||||||||
| C2a | Integration by substitution (given substitution) | 2 | 1 | 1 | 1 | 2 | 2 | 9 | 6/6 | Kept | 2020·12D, 2020·13A, 2021·11C, 2022·11B, 2023·12A, 2024·11C, 2024·13D, 2025·03, 2025·11C |
| C2b | ∫sin² nx and ∫cos² nx using ½(1±cos 2nx) | · | 1 | · | · | 1 | 1 | 3 | 3/6 | Kept | 2021·02, 2024·13B, 2025·11G |
| C2c | Derivative of an inverse: (f^{−1})'(x)=1/f'(f^{−1}(x)) | 1 | 1 | 1 | 1 | · | 1 | 5 | 5/6 | Kept | 2020·13C, 2021·14E, 2022·09, 2023·14A, 2025·10 |
| C2d | Derivatives of sin^{−1}, cos^{−1}, tan^{−1} (and chain rule) | 1 | · | 1 | · | 2 | 1 | 5 | 4/6 | Kept | 2020·13C, 2022·12C, 2024·11E, 2024·14B, 2025·14D |
| C2e | Integrals 1/√(a²−x²) and 1/(a²+x²) | 1 | 1 | · | 1 | · | 1 | 4 | 4/6 | Kept | 2020·03, 2021·11F, 2023·11D, 2025·03 |
| Calculus · ME-C3 Applications of Calculus | 29 | ||||||||||
| C3.1a | Area of the region between curves | · | 1 | · | 1 | 2 | · | 4 | 3/6 | Kept | 2021·13C, 2023·04, 2024·02, 2024·11G |
| C3.1b | Volume of a solid of revolution about the x-axis | 1 | · | 1 | 1 | 1 | 1 | 5 | 5/6 | Kept | 2020·13B, 2022·13B, 2023·12E, 2024·12B, 2025·12B |
| C3.1c | Volume about the y-axis, or the region between two curves | · | 1 | · | · | · | · | 1 | 1/6 | Kept | 2021·13A |
| C3.2a | Direction (slope) fields: identify, sketch, particular solution | 1 | 2 | 2 | 1 | 1 | 1 | 8 | 6/6 | Kept | 2020·07, 2021·04, 2021·12A, 2022·10, 2022·12A, 2023·03, 2024·13A, 2025·07 |
| C3.2b | Solve dy/dx=f(x) | · | · | · | · | · | · | 0 | 0/6 | Kept | |
| C3.2c | Solve dy/dx=g(y) | 1 | · | 1 | · | · | 2 | 4 | 3/6 | Kept | 2020·11E, 2022·10, 2025·12D, 2025·13A |
| C3.2d | Separation of variables dy/dx=f(x)g(y); initial-value problems | 1 | · | 1 | · | 2 | 1 | 5 | 4/6 | Kept | 2020·12E, 2022·14A, 2024·11D, 2024·14A, 2025·12D |
| C3.2e | Logistic equation and modelling DEs | · | 1 | · | · | 1 | · | 2 | 2/6 | Kept | 2021·14B, 2024·13A |
| Statistical Analysis · ME-S1 The Binomial Distribution | 25 | ||||||||||
| S1.1a | Bernoulli random variable: E(X)=p, Var(X)=p(1−p) | · | · | · | · | · | 1 | 1 | 1/6 | Kept | 2025·04 |
| S1.1b | Binomial probabilities P(X=r)=nCr p^r (1−p)^{n−r} | · | 1 | 1 | 2 | 2 | · | 6 | 4/6 | Kept | 2021·06, 2022·12E, 2023·02, 2023·12C, 2024·07, 2024·12C |
| S1.1c | Binomial mean np and variance np(1−p) | 1 | · | 1 | 1 | · | · | 3 | 3/6 | Kept | 2020·12B, 2022·12E, 2023·12C |
| S1.2a | Sample proportion p̂ as a random variable | · | 1 | 2 | 2 | 1 | 1 | 7 | 5/6 | Removed | 2021·14D, 2022·13E, 2022·14D, 2023·02, 2023·11F, 2024·08, 2025·13D |
| S1.2b | Normal approximation to the distribution of p̂ | 1 | 1 | 2 | 1 | 2 | 1 | 8 | 6/6 | Removed | 2020·12B, 2021·14D, 2022·13E, 2022·14D, 2023·11F, 2024·08, 2024·12C, 2025·13D |
Most examined dot points by topic
Functions41 hits
- F1.3a inverse exists / restrict domain — 6 hits, 4 of 6 years
- F2.2a sums and products of roots — 7 hits, 6 of 6 years
- F1.3b find f^{−1} — 3; F1.2a–c inequalities rotate as the early algebra item
F1 carries most of the Functions load. Reciprocal and |f| graphs are rare; y^2=f(x) has never appeared. Polynomials show up as remainder/factor in Q11 and as sums of roots later.
Trigonometric Functions40 hits
- T3a R-method — 6 hits, every year
- T3b solve a trig equation — 8 hits, 5 of 6 years
- T1c/T1d inverse-trig identities — the usual inverse-trig MCQ
T2 identities are thin except as a tool for T3 or for ∫sin². t-formulae have never been used. Product-to-sum appeared only as an integrand (2020, 2025).
Calculus64 hits
- C3.2a direction fields — 8, every year
- C2a substitution — 9, every year
- C3.1b volume about the x-axis — 5, every year
C3 dominates the late paper (DE + volume/area). C1 is mostly related rates and cooling. Inverse-trig integrals (C2e) went quiet in 2024–2025.
Vectors47 hits
- V1.2c dot product / perpendicular — 14 hits, every year
- V1.1a add/subtract — 6
- V1.3 projectile — 6 hits across 5 of 6 years
Every paper has an early vector skill and a late vector or projectile item. Projection and geometric proof rotate as the harder vector part.
Combinatorics + Proof27 hits
- P1a/P1b induction — 6 hits, every year
- A1.1d combinations — 5; A1.1e pigeonhole — 4 (last chance)
- A1.2b Pascal identities — 3
One induction is fixed. Counting items rotate through committees, repeated letters, circles and pigeonhole. Straight (a+b)^n expansion is less common than Pascal.
Statistical Analysis25 hits
- S1.2b normal approximation to p̂ — 8 hits, every year
- S1.1b binomial probability — 6
- S1.1a Bernoulli mean/variance — 1 (2025 MCQ)
The z-table p̂ question is the most reliable last-chance item in the course. 2024 drops it for CLT / sampling mean, which cannot appear in 2026.
Recurring question archetypes (same concept, new context each year)
| Archetype | Instances | Frequency |
|---|---|---|
| Direction-field MCQ or 1-mark sketch | 2020 Q7, 2021 Q4 and Q12(a), 2022 Q10 and Q12(a), 2023 Q3, 2024 Q13(a), 2025 Q7 | Every year |
| Mathematical induction (sum or divisibility) | One induction in every paper 2020–2025 (sums four times, divisibility twice) | Every year |
| Integration by substitution, substitution given | Q11 in 2021, 2022, 2023, 2024; also 2020 Q13(a) and 2025 Q3 / Q12 | Every year |
| R-method a sin x + b cos x | 2020 Q11(d), 2021 Q7, 2022 Q11(e), 2023 Q11(e), 2024 Q10, 2025 Q12(e) | 6 of 6 years |
| Normal approximation to the sample proportion | At least one item every year 2020–2025, often a late 3–4 marker with a z-table | Every year — last chance |
| Projectile (standard or two-particle) | 2021 Q13(b), 2022 Q14(c), 2023 Q13(b), 2024 Q14(d), 2025 Q14(b) — absent 2020 | 5 of 6 years |
| Vector perpendicular condition a·b=0 | 2020 Q11(b), 2022 Q11(d), 2024 Q12(a), plus MCQ angle/dot items most years | Core skill |
| Remainder / factor theorem then factorise | 2020 Q11(a), 2021 Q3, 2023 Q11(c); multiplicity and sums of roots rotate around it | 5 of 6 years |
| Related rates (sphere, tank, sector) | 2020 Q10, 2021 Q11(e), 2023 Q13(a), 2024 Q11(f), 2025 Q12(a) | 5 of 6 years |
| Volume of a solid of revolution | 2020 Q13(b), 2021 Q13(a), 2022 Q13(b), 2023 Q12(e), 2024 Q12(b), 2025 Q12(b) | Every year |
| Pigeonhole principle | 2021 Q10, 2022 Q12(b), 2024 Q3, 2025 Q14(a) | 4 of 6 — last chance |
| Newton cooling / carrying capacity / logistic | 2021 Q12(b) and Q14(b), 2022 Q12(d), 2023 Q1, 2024 Q13(a) | Rotates with separable DEs |
2026 forecast
How to read the tiersTier 1 is the stable core — content examined in five or six of the last six papers. Treat it as near-certain and make it the backbone of your revision. Tier 2 lists dot points that have never, or only once, been examined: 2026 is their last chance under this syllabus, so they are plausible but unproven. Tier 3 is the recency signal — strong history, but skipped in 2025. Tier 4 is content that does not survive into the 2024 Extension 1 syllabus, so 2026 is the only remaining opportunity to examine it. The composite shortlist combines all four signals.
Tier 1 — Bankers: dot points examined 5–6 of 6 years
| Dot point | Predicted question | Likely form in 2026 | Basis |
|---|---|---|---|
| V1.2c | Perpendicular condition a·b=0, or an angle from the dot product | Solve for a parameter | Examined all 6 years (14 hits) |
| C2a | ∫ using a given substitution | Every year; usually Q11 | Examined all 6 years (9 hits) |
| C3.2a | Direction field: match the DE, or sketch the solution through a point | Every year, usually MCQ or a 1-mark sketch | Examined all 6 years (8 hits) |
| S1.2b | Normal approximation to p̂ with a z-table | Last chance — gone in 2024. Continuity correction sometimes | Examined all 6 years (8 hits); last chance (not in 2024 syllabus) |
| T3b | Solve a trig equation on [0,2π] by factorising or a double-angle identity | Often the ‘hence’ after R-method or an identity | Examined 5 of 6 years (8 hits) |
| F2.2a | Sums/products of roots of a cubic or quartic | 1/α + 1/β + … or α²+β²+γ² | Examined all 6 years (7 hits) |
| S1.2a | Sample proportion p̂ as a random variable | As in recent papers | Examined 5 of 6 years (7 hits); last chance (not in 2024 syllabus) |
| T3a | Express a sin x + b cos x as R sin(x±α) and solve | Almost every paper; sometimes graph only | Examined all 6 years (6 hits) |
| V1.1a | Add/subtract vectors in i, j or column form | Almost every paper, often 1 mark | Examined 5 of 6 years (6 hits) |
| C1.3a | Related rates: sphere, tank or sector | Given dV/dt or dA/dt, find dr/dt | Examined 5 of 6 years (5 hits) |
| C2c | (f^{−1})' from f' at the corresponding point | Sometimes a second-derivative version | Examined 5 of 6 years (5 hits) |
| C3.1b | Volume of revolution about the x-axis | π∫ y² dx | Examined 5 of 6 years (5 hits) |
| V1.2e | Projection of u onto v | Formula given or required; last-chance flavour if skipped | Examined 5 of 6 years (5 hits) |
Tier 2 — Due: never or once examined
| Dot point | Predicted question | Likely form in 2026 | Basis |
|---|---|---|---|
| A1.1a | Multiplication principle; arrangements of distinct objects in a line | As in recent papers | Never examined 2020–2025 |
| C1.1a | Rates with respect to time; 1D particle language ẋ, ẍ | As in recent papers | Never examined 2020–2025; reduced in 2024 |
| C1.2a | Exponential growth/decay Q=Ae^{kt} | As in recent papers | Never examined 2020–2025 |
| C3.2b | Solve dy/dx=f(x) | As in recent papers | Never examined 2020–2025 |
| F1.1b | Sketch y^2=f(x) and y=√f(x) from y=f(x) | As in recent papers | Never examined 2020–2025 |
| F2.2c | Graph a polynomial from factored form / root multiplicity | As in recent papers | Never examined 2020–2025 |
| P1c | Identify errors in false inductions; when induction is not appropriate | As in recent papers | Never examined 2020–2025 |
| T2c | t-formulae: sin, cos, tan in terms of t=tan(x/2) | As in recent papers | Never examined 2020–2025; last chance (not in 2024 syllabus) |
| C3.1c | Volume about the y-axis, or the region between two curves | As in recent papers | Once: 2021·13A |
| F1.1d | Sketch y=f(x)+g(x) and y=f(x)−g(x) | As in recent papers | Once: 2022·04 |
| F2.1c | Polynomial division: P=AQ+R, degree of remainder | As in recent papers | Once: 2022·03 |
| S1.1a | Bernoulli mean and variance | P(x) table with two outcomes | Once: 2025·04 |
| T2a | Sum and difference expansions for sin, cos, tan | As in recent papers | Once: 2021·13D |
| V1.1b | Geometric addition/subtraction (triangle or parallelogram law) | As in recent papers | Once: 2020·06 |
| V1.2b | Direction of a vector; bearings and navigation | As in recent papers | Once: 2021·14A |
Tier 3 — Strong history but absent in 2025
| Dot point | Predicted question | Likely form in 2026 | Basis |
|---|---|---|---|
| F1.3a | Show an inverse exists (HLT / one-to-one), or restrict the domain | Often chained with finding f^{−1} or (f^{−1})' | 4 of 6 years, 6 hits (2021·12D … 2024·14B) |
| S1.1b | Binomial probability in context | At least k successes; sometimes expected frequency | 4 of 6 years, 6 hits (2021·06 … 2024·12C) |
| A1.1d | Committee / selection using nCr | Sometimes then a probability | 4 of 6 years, 5 hits (2020·08 … 2024·09) |
| F2.1b | Factor a cubic given one root | Remainder/factor theorem pair in Q11 | 3 of 6 years, 5 hits (2020·05 … 2024·01) |
| C1.2b | Newton's law of cooling or a carrying-capacity model | Find k from two temperatures, then a later time | 3 of 6 years, 4 hits (2021·12B … 2023·01) |
| C3.1a | Area between curves | Signed pieces; exact value | 3 of 6 years, 4 hits (2021·13C … 2024·11G) |
| A1.2a | Expand (a+b)^n or find a specified coefficient | Short Q11 skill | 3 of 6 years, 3 hits (2020·14A … 2022·11C) |
| F1.4a | Parametric line, parabola or circle → Cartesian, or match a sketch | MCQ or 1–2 mark conversion | 3 of 6 years, 3 hits (2021·08 … 2023·11A) |
| F2.1a | Evaluate P(α) as a remainder | Show P(2)=0 then factor | 3 of 6 years, 3 hits (2020·11A … 2023·11C) |
| S1.1c | Binomial mean np and variance np(1−p) | As in recent papers | 3 of 6 years, 3 hits (2020·12B … 2023·12C) |
| V1.2g | Vector proof in a parallelogram, trapezium or circle | Show / hence geometric | 3 of 6 years, 3 hits (2021·14C … 2023·14C); reduced in 2024 |
| V1.3a | Standard projectile: time of flight, range or Cartesian path | Equations often given in recent papers | 3 of 6 years, 3 hits (2021·13B … 2024·14D) |
Tier 4 — Last chance: not kept in the 2024 Extension 1 syllabus
Cross-checked against the NESA Mathematics Extension 1 11–12 Syllabus (2024). “Removed” leaves Ext1 (and is not replaced by an Ext1 equivalent). “Reduced” stays in some form but is no longer a named 2017-style item. Ordered by how much examiners have relied on the content since 2020.
| Dot point | Predicted question | Likely form in 2026 | Basis |
|---|---|---|---|
| S1.2b | Normal approximation to p̂ with a z-table | Last chance — gone in 2024. Continuity correction sometimes | Examined all 6 years (8 hits); last chance (not in 2024 syllabus) |
| S1.2a | Sample proportion p̂ as a random variable | As in recent papers | Examined 5 of 6 years (7 hits); last chance (not in 2024 syllabus) |
| A1.1e | Pigeonhole: minimum n so that at least k are in one group | Last chance — dropped in 2024 | 4 of 6 years, 4 hits (2021·10 … 2025·14A); last chance (not in 2024 syllabus) |
| T2d | Product-to-sum and sum-to-product identities | As in recent papers | 2 of 6 years, 2 hits (2020·12D … 2025·11C); last chance (not in 2024 syllabus) |
| T2c | t-formulae: sin, cos, tan in terms of t=tan(x/2) | As in recent papers | Never examined 2020–2025; last chance (not in 2024 syllabus) |
| V1.2g | Vector proof in a parallelogram, trapezium or circle | Show / hence geometric | 3 of 6 years, 3 hits (2021·14C … 2023·14C); reduced in 2024 |
| C1.1a | Rates with respect to time; 1D particle language ẋ, ẍ | As in recent papers | Never examined 2020–2025; reduced in 2024 |
What the 2024 Ext1 syllabus drops or reduces
Removed: t-formulae (T2c); product-to-sum identities (T2d); pigeonhole principle (A1.1e); the normal approximation to the sample proportion p̂ (S1.2a–b) — 2024 Ext1 explicitly excludes the normal approximation to the binomial and replaces this strand with the sampling distribution of the mean and the central limit theorem.
Reduced: 1D particle language ẋ, ẍ as a standalone Ext1 skill (C1.1a); named vector theorems on parallelograms and midpoints (V1.2g) — 2024 keeps projection and the perpendicular condition, but those geometry proofs are no longer listed.
Reorganised, not dropped: inverse trig moves to Year 12; projectile stays, now grouped with 2D motion in vector form; binomial probabilities stay.
New in 2024 — cannot appear in 2026
- 3D vectors (i, j, k; magnitude; dot product in three dimensions)
- 3D trigonometry (moved from Advanced)
- Graphs of y=sec x, y=cosec x, y=cot x (moved from Advanced)
- Cubic inequalities in factored form
- Motion in vector form: r(t), v(t), a(t); Cartesian path of a particle; relative velocity / crosswind
- Perpendicular component of a vector a − proj_b a as a named formula
- Sampling distribution of the mean and the central limit theorem (replaces the normal approximation to p̂)
- Logistic DE solved using a given partial-fraction decomposition
If a practice paper includes 3D vectors, CLT, or sec/cosec/cot graphs as Ext1 content, it is written for the new syllabus and goes beyond what the 2026 exam can ask.
Composite shortlist — candidates with two or more signals
Signals: Core = examined 5–6 of 6 years · Absent 2025 = not in the most recent paper · Untested = 0–1 hits in six papers · Gone 2024 = removed in the 2024 syllabus. Green rows carry three or more signals, blue rows two.
| Predicted question | Dot points | Core | Absent 2025 | Untested | Gone 2024 | Verdict |
|---|---|---|---|---|---|---|
| Normal approximation to p̂ with a z-table | S1.2b | Yes | — | — | Yes | Strongest last-chance item: a z-table p̂ question every year since 2020. |
| Pigeonhole: minimum n so that at least k are in one group | A1.1e | — | — | — | Yes | Four of six years and leaving the syllabus; a 2–3 mark pigeonhole is very likely. |
| Product-to-sum and sum-to-product identities | T2d | — | — | — | Yes | Product-to-sum used to integrate (2020, 2025); last chance for the identity itself. |
| t-formulae: sin, cos, tan in terms of t=tan(x/2) | T2c | — | — | Yes | Yes | t-formulae have not appeared in six papers and are dropped in 2024 — plausible one-off. |
| Direction field: match the DE, or sketch the solution through a point | C3.2a | Yes | — | — | — | Direction field is a banker; 2025 was only the MCQ, so a sketch part is due. |
| Standard projectile: time of flight, range or Cartesian path | V1.3a | — | Yes | — | — | Projectile is a late-paper fixture 2021–2025; 2020 skipped it. |
| Induction on divisibility (usually by 7) | P1b | — | — | — | — | Divisibility induction in 2022 and 2024; 2025 was a sum, so divisibility is due. |
| Newton's law of cooling or a carrying-capacity model | C1.2b | — | Yes | — | — | Cooling/carrying-capacity absent in 2025 after four earlier appearances. |
| Sketch y=1/f(x) from a given graph | F1.1a | — | — | — | — | y=1/f(x) only in 2020 and as a graph in 2024 Q12(e); a sketch is overdue. |
| ∫ 1/√(a²−x²) or 1/(a²+x²) | C2e | — | — | — | — | Inverse-trig integral skipped in 2024–2025 after three earlier hits. |
| Expand (a+b)^n or find a specified coefficient | A1.2a | — | Yes | — | — | Straight binomial expansion last in 2022; 2025 used Pascal instead. |
| Projection of u onto v | V1.2e | Yes | — | — | — | Projection appeared in 2025 as a formula MCQ; a calculation part was last in 2024. |
| Logistic DE: carrying capacity, max growth rate, sketch on a slope field | C3.2e | — | — | — | — | Logistic on a slope field in 2024; not in 2025. |
| Identify errors in false inductions; when induction is not appropriate | P1c | — | — | Yes | — | False-induction / ‘when not appropriate’ has never been examined. |
| Sketch y^2=f(x) and y=√f(x) from y=f(x) | F1.1b | — | — | Yes | — | y^2=f(x) and y=√f(x) have never been examined. |
| Volume about the y-axis, or the region between two curves | C3.1c | — | — | Yes | — | Volume about the y-axis / between two curves only in 2021. |
| Sample proportion p̂ as a random variable | S1.2a | Yes | — | — | Yes | Sample proportion as a named random variable is leaving with the normal approximation. |
A plausible 2026 paper skeleton (10 MCQ + Q11–14, 70 marks)
- Q11 (a) Vector addition or 2a−b, then a magnitude
- Q11 (b) Quadratic or rational inequality
- Q11 (c) Integration by substitution (substitution given)
- Q11 (d) Remainder/factor theorem, or sums of roots of a cubic
- Q11 (e) R-method, or differentiate arcsin/arctan of a composite
- Q11 (f) Related rates (sphere or similar) — or inverse f^{−1}(x)
- Q12 (a) Direction field: sketch the solution through a given point
- Q12 (b) Combinations / circular arrangement / binomial coefficient
- Q12 (c) Induction: a sum, or divisibility by 7
- Q12 (d) Separable DE with an initial condition
- Q12 (e) Inverse-trig integral or ∫sin²/cos²
- Q13 (a) Volume of revolution, or area between curves
- Q13 (b) Projectile (possibly two particles or an unknown angle)
- Q13 (c) Vector projection, perpendicular, or a short geometric proof
- Q14 (a) Logistic DE on a slope field, or a hard separable DE (domain of the solution)
- Q14 (b) Inverse functions: exist / restrict / (f^{−1})'
- Q14 (c) Normal approximation to p̂ with a z-table — last chance
- Q14 (d) Pigeonhole or a Pascal-identity ‘hence’ — last chance; or a trig identity feeding sums of roots
Section I (10 MCQ) is predictable in type: an inequality or inverse, a vector diagram or projection, a direction field, an inverse-trig graph or identity, a binomial/Bernoulli item, a remainder or sums-of-roots item, a substitution or inverse-trig integral, a trig-equation count, a parametric or |f| graph, and one conceptual calculus item.
© Censai 2026. HSC examination papers and syllabuses are © NSW Education Standards Authority; this document references them and does not reproduce examination questions. The Tier 1 core is the backbone of Section II each year; use the lower tiers to round out your revision, not to replace it.