HSC Maths Extension 1 Syllabus: Every Topic for the 2027 HSC
HSC Mathematics Extension 1 is on a new syllabus. If you're starting Year 12 in Term 4 2026, you'll be in the first group to sit the HSC under the Mathematics Extension 1 11–12 Syllabus (2024), in 2027. This guide covers the course structure, each Year 12 topic, where marks usually go, and how to study.
What Extension 1 is
Mathematics Extension 1 is a 1-unit extension course taken alongside Mathematics Advanced. It goes deeper into functions, calculus, proof, vectors and statistics, and it suits students who enjoy maths and want a challenge, or who are heading into maths-heavy study.
Because it builds directly on Advanced, weaknesses in Advanced (especially algebra and calculus) show up quickly in Extension 1.
What's new for the 2027 HSC
The 2024 syllabus is taught to Year 12 from Term 4 2026 and first examined in 2027. The 2017 syllabus had its final exam in 2026. Much of the content will look familiar, but some details have changed. For example, the normal approximation to the binomial distribution isn't part of the new course.
So check older past paper questions against the new syllabus before relying on them. NESA's 2025 sample HSC exam materials for the new maths courses are the best guide to the new exam.
How the course is structured
Year 11 covers: Further work with functions; Polynomials; Further trigonometry; Permutations and combinations; and The binomial theorem.
Year 12 covers six topics, starting in Term 4 of Year 11.
Proof by mathematical induction
The structure of an inductive proof, induction for sums of series and for divisibility, and spotting flawed induction arguments.
Common mistakes: a weak inductive step that never clearly uses the assumption, and a missing or vague conclusion. Write the assumption, the target, and exactly where you use the assumption. Setting out is part of the marks.
Introduction to vectors
Vectors in two and three dimensions, magnitude and unit vectors, the scalar product, angles and projections, and using vectors to model motion, including projectile motion.
Common mistakes: mixing up the projection of one vector onto another with its perpendicular component, and sign errors when resolving components in projectile problems. Draw a diagram and define your positive directions first.
Inverse trigonometric functions
Why sine, cosine and tangent need restricted domains to have inverses, the domain and range of each inverse function, their graphs, identities, composite functions and transformations.
Common mistakes: forgetting the restricted range when evaluating, so an answer lands outside it. Learn the domain and range of each inverse function by sketching them, not memorising a table.
Further calculus skills
Parametric differentiation, derivatives of inverse functions and inverse trig functions, integrals that give inverse sine or inverse tangent forms, integration by substitution, and integrating squared sine and cosine.
Common mistakes: forgetting to change the limits in a definite integral after substitution, and missing the chain rule factor. When you substitute, either change the limits or convert back, never a mix of both.
Further applications of calculus
Sketching polynomials using multiplicity, related rates, rates proportional to a difference, areas between curves, volumes of solids of revolution, slope fields, and setting up and solving simple differential equations.
Common mistakes: setting up a related rates problem without clearly linking the rates with the chain rule, and rotating about the wrong axis in volume questions. State the formula you're using and which variable you're integrating with respect to.
The binomial distribution and sampling distribution of the mean
Bernoulli random variables, binomial probabilities, mean and variance, samples and sample means, the sampling distribution of the mean, and the central limit theorem.
Common mistakes: confusing a population with a sample, and using "at least" and "at most" the wrong way round. Write out which values of the variable you need before calculating.
How Extension 1 is assessed
Extension 1 results are reported in bands E1 to E4 based on a mark out of 50, with E4 covering 45 to 50. Your HSC mark is the average of your moderated school assessment mark and your HSC exam mark.
- School assessment tasks follow your school's program, typically topic tests and trial exams, sometimes with an investigation.
- The HSC exam is a written paper at the end of Year 12. Expect questions that combine ideas, and longer questions where clear reasoning matters as much as the final answer.
How to study Extension 1
- Secure your Advanced foundations. Algebra, logs, trig and calculus from Advanced are the tools Extension 1 uses constantly.
- Practise proofs as a routine. Induction and "show that" questions reward a consistent, careful layout. Do them until it's automatic.
- Do harder questions regularly. Aim for steady exposure to challenging problems rather than one big push before trials.
- Keep an error log. Most lost marks are algebra slips, missed restrictions and incomplete setting out.
- Mix topics. Exam questions often combine calculus with vectors, trig or statistics.
- Time yourself on the new sample materials and on older questions that still match the syllabus.
If a proof or method won't click, an AI study coach like otto can explain it another way and check each line of your working.
The bottom line
Extension 1 is challenging but rewarding with consistent practice. For the 2027 HSC it's a new syllabus, so plan from the new topic list, make sure your Advanced skills are strong, practise proof and setting out, and track the mistakes you keep making.
Extension 1 is where a second explanation helps most, and that's what otto gives you. It maps the new Extension 1 syllabus, builds lessons and harder practice sets, talks you through proofs and calculus problems 24/7, and checks your working line by line.
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Frequently Asked Questions
What topics are in HSC Maths Extension 1 Year 12?
Under the 2024 syllabus, Year 12 covers Proof by mathematical induction, Introduction to vectors, Inverse trigonometric functions, Further calculus skills, Further applications of calculus, and The binomial distribution and sampling distribution of the mean.
Is the Extension 1 syllabus new for 2027?
Yes. The Mathematics Extension 1 11–12 Syllabus (2024) is taught to Year 12 from Term 4 2026 and first examined in the 2027 HSC. The 2017 syllabus had its final exam in 2026.
How is Extension 1 graded?
Extension courses are reported in bands E1 to E4 on a mark out of 50, with E4 being 45 to 50. Like other courses, the HSC mark is the average of the moderated school assessment mark and the exam mark.
How should I study for Maths Extension 1?
Make sure your Mathematics Advanced foundations are solid, practise proofs until the structure is automatic, do challenging mixed questions regularly, and keep an error log. Extension 1 marks are often lost to setting out and small algebra slips rather than not knowing the idea.