Mathematics Extension 1 · 2021–2025 · 10 min read
Five years of Extension 1:
what keeps coming back?
We analysed every atomic question in the five most recent HSC papers to find the content, skills and question structures students repeatedly need.
The short version
Calculus, vectors and connected reasoning dominate.
Vectors and differential equations carried the largest primary-category mark totals. Projectile/vector motion, binomial probability, advanced trigonometric equations, polynomial roots and inverse functions also appeared in every paper.
The deeper pattern is about form as much as topic: 76.9% of all marks were attached to worked solutions or show/prove questions. Students need fluent techniques, but they also need to connect results, justify steps and communicate a chain of reasoning.
Common question types
The highest-volume categories
These totals use each atomic question’s primary concept category. Every mark is allocated once, so category percentages add to 100% across the full analysis.
Recurring every year
The reliable practice list
Vectors, Projections & Geometry
9.2 marks per paper on average · 25 atomic questions
Separable Differential Equations
8.4 marks per paper on average · 19 atomic questions
Vector Motion & Projectiles
5.6 marks per paper on average · 9 atomic questions
Binomial Distribution & Approximation
5.2 marks per paper on average · 14 atomic questions
Extension Trigonometric Equations
4.6 marks per paper on average · 12 atomic questions
Polynomial Roots & Coefficient Relations
4.0 marks per paper on average · 10 atomic questions
Inverse Functions
3.8 marks per paper on average · 11 atomic questions
Inverse Trigonometric Functions
3.6 marks per paper on average · 13 atomic questions
Combinatorics & Counting
3.4 marks per paper on average · 11 atomic questions
Proof by Mathematical Induction
3.0 marks per paper on average · 5 atomic questions
Volumes of Revolution
3.0 marks per paper on average · 5 atomic questions
Rates of Change & Motion
2.4 marks per paper on average · 7 atomic questions
Paper-by-paper pattern
Primary-category marks by year
| Question category | 2021 | 2022 | 2023 | 2024 | 2025 | Average |
|---|---|---|---|---|---|---|
| Vectors, Projections & Geometry | 10 | 12 | 8 | 9 | 7 | 9.2 |
| Separable Differential Equations | 9 | 10 | 6 | 10 | 7 | 8.4 |
| Vector Motion & Projectiles | 4 | 4 | 8 | 4 | 8 | 5.6 |
| Binomial Distribution & Approximation | 1 | 9 | 6 | 5 | 5 | 5.2 |
| Extension Trigonometric Equations | 6 | 3 | 3 | 4 | 7 | 4.6 |
| Polynomial Roots & Coefficient Relations | 3 | 4 | 7 | 1 | 5 | 4.0 |
| Inverse Functions | 6 | 3 | 4 | 3 | 3 | 3.8 |
| Inverse Trigonometric Functions | 3 | 1 | 4 | 4 | 6 | 3.6 |
| Combinatorics & Counting | 3 | 3 | 5 | 1 | 5 | 3.4 |
| Anti-differentiation Techniques | 3 | 3 | 3 | 6 | — | 3.0 |
| Proof by Mathematical Induction | 3 | 3 | 3 | 3 | 3 | 3.0 |
| Volumes of Revolution | 3 | 3 | 4 | 3 | 2 | 3.0 |
| Definite Integrals & Areas Under Curves | 3 | — | 1 | 7 | 3 | 2.8 |
| Rates of Change & Motion | 3 | 1 | 3 | 2 | 3 | 2.4 |
Syllabus outcomes
Average marks associated with each outcome
An atomic question can assess more than one outcome. In that case its full marks are associated with each listed outcome, matching the official mapping approach. These figures overlap and must not be added together.
How marks are earned
The examination rewards written mathematical reasoning.
Worked solutions
191 marks across five papers
Show/prove questions
78 marks across five papers
Multiple choice
50 marks across five papers
Short answers
13 marks across five papers
Sketching
10 marks across five papers
Written explanations
8 marks across five papers
Multiple choice is a fixed 10 marks per paper. The larger opportunity—and larger risk—is Section II, where method, justification and linked reasoning matter.
A hidden examination skill
Practise question chains as chains.
Our dependency audit found 20 linked groups containing 45 atomic parts. A later “hence” part may rely on the same diagram, parameters or a result established earlier.
Do not always extract subparts in isolation. Practise carrying notation and results forward, and learn how to continue after an earlier error by using a stated result.
A practical study plan
Turn the pattern into preparation
- 01
Build the high-volume core
Prioritise vectors, differential equations, vector/projectile motion, binomial probability and advanced trigonometric equations.
- 02
Keep the annual staples active
Include polynomial roots, inverse functions, inverse trigonometric functions, combinatorics, induction, volumes and rates every week.
- 03
Write complete solutions
More than three quarters of the marks were in worked or show/prove formats. State substitutions, restrictions, reasons and conclusions.
- 04
Practise linked parts together
Vary the shared stem once, then solve the entire chain. This protects dependencies that isolated worksheets often lose.
- 05
Finish with full timed papers
Category drills build fluency; complete papers train selection, pacing, recovery and communication under examination conditions.
Methodology and sources
What we counted
We used the official 2021–2025 NSW HSC Mathematics Extension 1 papers and marking materials. Each paper contributes 70 marks. Multi-part questions were separated into their smallest meaningful assessed units while their shared contexts and dependencies were retained.
- 2021: exam paper · marking guidelines · marking feedback
- 2022: exam paper · marking guidelines · marking feedback
- 2023: exam paper · marking guidelines · marking feedback
- 2024: exam paper · marking guidelines · marking feedback
- 2025: exam paper · marking guidelines
Category totals use one primary concept per atomic unit. Outcome totals use every outcome listed in the official mapping metadata, so multi-outcome questions count towards each relevant outcome. No E1–E4 difficulty band has been assigned to individual questions.