HSC Maths Extension 1: Complete Guide (2026)

Last updated: 20 July 2026

Extension 1 is a 1-unit course you take on top of Advanced Maths. It goes deeper into calculus, adds vectors and combinatorics, and introduces formal proof. It's also a prerequisite for Extension 2, which is the hardest maths course NSW offers.

This guide covers what's in the course, how the exam works, and how to manage the workload without falling behind.

What is Extension 1?

HSC Mathematics Extension 1 is worth 1 unit and must be taken alongside Advanced (2 units). So you end up with 3 units of maths total. The course extends the Advanced syllabus with harder content and additional topics.

Universities often list Extension 1 as a prerequisite or recommended study for engineering, actuarial science, and computer science. If you're aiming for those courses, Extension 1 gives you a buffer above the bare minimum.

Who should take it?

Consider Extension 1 if:

The main reason students drop Extension 1 isn't ability. It's time. You're adding another subject on top of an already full load. If you're already struggling to keep up with your other subjects, think carefully.

Syllabus topics

Extension 1 extends and builds on Advanced. Here's what's covered:

Functions

Extends the Advanced functions content. You'll work with harder function compositions, inverse functions, and more complex graphing problems. The emphasis shifts toward using functions as tools in problem-solving rather than studying them in isolation.

Trigonometric functions

Harder trig equations, more complex identities, and applications. You'll need to be comfortable manipulating trig expressions and solving equations that require multiple steps and identities.

Calculus: Differentiation

Goes beyond Advanced. Includes differentiation of inverse trig functions, harder chain rule problems, and more complex optimisation. The questions are longer and combine multiple concepts.

Calculus: Integration

This is where Extension 1 gets noticeably harder. You'll learn integration by substitution, which is the main new technique. Integration by parts is not in the syllabus (that's Extension 2), but substitution is used extensively. You'll also do harder area and volume problems.

Calculus: Differential equations

Solving first-order differential equations using separation of variables and other techniques. This is new content not covered in Advanced.

Exponential and logarithmic functions

Harder applications of exponentials and logs, including differential equations involving growth and decay models.

Statistical analysis

The binomial distribution is the main addition. You'll calculate binomial probabilities, expected value, and standard deviation for binomial experiments. Also covers harder normal distribution problems.

Combinatorics

New topic not in Advanced. Permutations and combinations, the binomial theorem, and Pascal's triangle. This is where a lot of students struggle because it requires a different style of thinking than calculus.

Vectors

Another new topic. Vector notation, operations (addition, subtraction, scalar multiplication), dot product, cross product, and geometric applications. Vectors show up in mechanics questions too.

Proof

Mathematical induction is the main proof technique. You'll prove statements about divisibility, sequences, and inequalities. This is unfamiliar territory for most students and requires precise, logical writing.

The exam

The Extension 1 exam is separate from the Advanced exam. It's a single paper:

The time pressure is significant. 70 marks in 90 minutes means about 1.3 minutes per mark. You need to be fast.

How marks are distributed

Calculus (differentiation, integration, and differential equations) makes up the largest chunk. Combinatorics and vectors each have a solid representation. Proof (induction) is usually one question worth 4-5 marks.

How to study for Extension 1

The biggest challenge isn't the difficulty of the content. It's the volume. You're learning Advanced content and Extension 1 content simultaneously, and the Extension 1 exam tests both.

Keep up with Advanced first

If your Advanced knowledge has gaps, Extension 1 will expose them brutally. The Extension 1 exam assumes you know all of Advanced cold. Make sure your Advanced foundations are solid before pouring energy into Extension 1 specific content.

Don't fall behind

Extension 1 content builds on itself fast. If you miss the concept of integration by substitution, every subsequent calculus topic becomes incomprehensible. If you don't understand something in class, fix it that same week. Not next week. Not before the exam. That week.

Practise combinatorics and vectors separately

These topics feel different from the calculus-heavy content you're used to. A lot of students neglect them because they're smaller sections, but they're worth easy marks if you put in the practice. Do 20 combinatorics questions and 20 vectors questions as a dedicated block.

Learn induction by doing

Mathematical induction is weird. You won't understand it from reading about it. You need to do 15-20 induction proofs before the structure clicks. Once it clicks, it's free marks. But it takes repetition.

Do trial papers from selective schools

Extension 1 trial papers from James Ruse, Sydney Boys, and North Sydney Boys are harder than the HSC. If you can handle those, the actual exam will feel manageable. Start with easier trial papers and work up.

Extension 1 vs Advanced: what's different?

The content overlaps significantly since you do both courses. The main differences:

Common mistakes

Neglecting Advanced content. The Extension 1 exam tests Advanced content too. Don't assume you can focus only on the Extension 1 specific topics.

Weak induction. A lot of students write sloppy induction proofs. The structure matters. State the base case, state the assumption, prove the inductive step, write the conclusion. Skip any of these and you lose marks.

Forgetting vector notation. Vectors have specific notation (bold, arrows, unit vectors). Using the wrong notation loses marks even if your answer is correct.

Time management. 90 minutes for 70 marks is tight. If you spend 15 minutes on a 3-mark question, you're in trouble. Skip and come back.

Not practising combinatorics enough. Students assume it's easy because it doesn't involve calculus. Then they bomb it because the counting logic is tricky and requires practice.

Resources

Want to practise?

PracticePapers.io covers Extension 1 with auto-marking, worked solutions, and weak-areas tracking. Free during beta.

Start practising free

Based on the current NESA HSC Mathematics Extension 1 syllabus. PracticePapers.io is not affiliated with NESA.