HSC Maths Extension 1: the 70 mark exam, the Year 12 topics and how E1 to E4 are earned

The HSC Maths Extension 1 exam

Checked against the NESA syllabus pages on 11 November 2026. The new Mathematics Extension 1 11 to 12 Syllabus (2024) started teaching in Year 12 in Term 4 2026, so 2027 is the first HSC exam for it.

The exam is based on the Year 12 course and focuses on Year 12 outcomes. Mathematics Advanced is assumed knowledge, and the Extension 1 Year 11 course may be examined. The 2027 HSC written exams run from Wednesday 13 October to no later than Friday 5 November 2027. NESA publishes the timetable on Friday 30 April 2027, so the date of your Extension 1 paper is not known yet.

What Year 12 Extension 1 covers

Year 11 adds further work with functions, polynomials, further trigonometry, permutations and combinations, and the binomial theorem. Both years are 60 hours.

How the HSC mark and school assessment work

Your Extension 1 mark is a 50:50 combination of the HSC exam mark and your school-based assessment mark. In Year 12 the school assessment is weighted 50% knowledge and understanding of course content and 50% skills in working mathematically. Your school sets the number and weighting of tasks, each normally 10% to 40%, with at most one take-home task worth up to 15%. Extension courses report bands E1 to E4, with E4 the highest.

Worked examples

Proof by induction

Prove that 7ⁿ − 1 is divisible by 6 for every positive integer n. For n = 1 it is 6, which is true. Assume 7ᵏ − 1 = 6m for some integer m. Then:

This is a multiple of 6, so the statement holds for k + 1. So the result is true for every positive integer n, by the principle of mathematical induction. Always write the conclusion in words: marks are given for the base case, the assumption and the final sentence.

Vectors and inverse trigonometric functions

The projection of a = (3, 4) onto b = (1, 0) is the scalar (a · b) ÷ |b| = 3 ÷ 1 = 3, so the vector projection is (3, 0). For inverse functions, note that the derivative of sin⁻¹(3x) is 3 ÷ √(1 − 9x²), and that:

Differentiate to check, and see the Derivative Calculator for other examples.

Binomial and sampling distributions

If X is the number of successes in 10 trials with probability 0.3 each, then P(X = 3) = C(10, 3) × 0.3³ × 0.7⁷ ≈ 0.267, with mean 10 × 0.3 = 3 and standard deviation √(10 × 0.3 × 0.7) ≈ 1.45. For the sampling distribution of the mean, a population with mean 50 and standard deviation 12 gives sample means of size 36 with mean 50 and standard deviation 12 ÷ √36 = 2.

Using the daily papers as a trial

Each daily paper has 8 original questions and 25 to 35 marks, so it is a short timed trial of 35 to 45 minutes, not a full Extension 1 paper. For your school's own trial exams and their dates, see our HSC trials guide.

• On the Further Maths paper, sit the proof by induction questions. Most of the rest is Extension 2 content, so skip complex numbers, hyperbolic functions and volumes of revolution unless your teacher has set them.
• On the Year 12 calculus paper, sit the rates of change and chain, product and quotient rule questions. Substitution questions fit if your school has covered further calculus skills; skip parametric differentiation.
• Pair this with the HSC Maths guide for Advanced, which you also sit.

Exam answers

Answers and a worked mark scheme for the daily paper come with today's paper once it's marked. They are ours: NESA releases its own marking guidelines after each HSC.

Common mistakes in HSC Maths Extension 1

• An induction with no conclusion. Write the base case, the assumption, the step, and the final sentence.
• Forgetting the domain of inverse trig functions. sin⁻¹ x has a range of −π/2 to π/2, so check that your angle lies in it.
• Mixing up the sample and the sampling distribution. The mean of the sample means is the population mean, but the spread is σ ÷ √n.
• Ignoring the "show that" structure. Work toward the given result and write each step.

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