HSC Maths Extension 1 rewards students who can set out clear reasoning, use the Reference Sheet well and adapt standard methods to unfamiliar problems. The 2025 exam included a related rates problem, an induction proof with factorials, a pigeonhole proof, a projectile problem with two particles and a clock-hands question solved with vectors. NESA's marking feedback shows exactly which skills separated the better responses.
This guide breaks down the 2025 HSC Maths Extension 1 exam question by question. Every question is classified by syllabus topic using NESA's own mapping grid, so you can see exactly where the marks were. It also covers the answers, what better responses did and the areas NESA's markers said students need to improve.
Key Takeaways
- Vectors was the biggest topic, worth 12 marks on its own and part of a further 3-mark question (Q14(c)). Calculus topics were worth 20 marks together.
- Year 12 content dominated: at least 47 of the 70 marks came from Year 12 topics, compared with 20 marks from Year 11 topics.
- Combinatorics was worth 7 marks, including a circular seating question, a Pascal's triangle proof and a pigeonhole principle proof.
- Markers repeatedly said students should use the Reference Sheet for trigonometric identities, integrals, derivatives and polynomial root relationships.
- Proofs needed complete, clearly set-out reasoning. Testing a few cases did not count as a proof for the general case.
HSC Mathematics Extension 1 Examination Structure
| Feature | Section I | Section II |
|---|---|---|
| Question type | 10 multiple-choice questions | Four multi-part extended-response questions (Q11–14) |
| Marks | 10 marks | 60 marks |
| Suggested time | About 15 minutes | About 1 hour 45 minutes |
| Skills tested | Concepts, quick calculations, interpreting graphs and diagrams | Calculations, proofs, graphing, modelling and multi-step problem solving |
The exam has 10 minutes of reading time and 2 hours of working time, and is worth 70 marks. NESA-approved calculators are allowed, and a Reference Sheet is provided. The HSC exam mark makes up 50% of the final HSC mark. The other 50% comes from school-based assessment.
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Marks by Topic in the 2025 Exam
Every question below is classified using the mapping grid in NESA's 2025 marking guidelines, which links each question to a syllabus topic. Two questions were mapped to two topics each, so they are shown separately.
| Topic | Year | Questions | Marks |
|---|---|---|---|
| ME-V1 Introduction to Vectors | 12 | MC 2, 8, 9; Q11(e), 13(c), 14(b) | 12 |
| ME-C3 Applications of Calculus | 12 | MC 7; Q12(b), 12(d), 13(a) | 9 |
| ME-C2 Further Calculus Skills | 12 | MC 3, 10; Q11(g), 14(d) | 8 |
| ME-T3 Trigonometric Equations | 12 | MC 5; Q11(b), 12(e) | 7 |
| ME-A1 Working with Combinatorics | 11 | Q13(b), 13(e), 14(a) | 7 |
| ME-S1 The Binomial Distribution | 12 | MC 4; Q13(d) | 5 |
| ME-F1 Further Work with Functions | 11 | MC 1; Q11(a) | 3 |
| ME-T1 Inverse Trigonometric Functions | 11 | MC 6; Q11(d) | 3 |
| ME-C1 Rates of Change | 11 | Q12(a) | 3 |
| ME-P1 Proof by Mathematical Induction | 12 | Q12(c) | 3 |
| ME-T2 Further Trigonometric Identities | 11 | Q11(c) | 2 |
| ME-F2 Polynomials | 11 | Q11(f) | 2 |
| Vectors and Trigonometric Equations | 12 | Q14(c) | 3 |
| Polynomials and Trigonometric Equations | 11 and 12 | Q14(e) | 3 |
Source: NESA 2025 HSC Mathematics Extension 1 marking guidelines (mapping grid).
Section I: Multiple-Choice Breakdown
| Question | Topic | What It Tested | Answer |
|---|---|---|---|
| 1 | Further Work with Functions | Solving the absolute value inequality ∣2x + 3∣ < 5 | A (−4 < x < 1) |
| 2 | Vectors | Projection of u = i + 2j onto v = 2i − 3j | B (−4/13 (2i − 3j)) |
| 3 | Further Calculus Skills | Applying the substitution x = 5 sin θ to a definite integral | B ((1/5)∫ sec θ dθ from −π/6 to π/6) |
| 4 | Binomial Distribution | Mean and variance of a Bernoulli random variable | C (E(X) = 2/3, Var(X) = 2/9) |
| 5 | Trigonometric Equations | Number of solutions of cos 5x + sin x = 0 for 0 ≤ x ≤ 2π | D (10) |
| 6 | Inverse Trigonometric Functions | Identifying the integral of an odd function over [−a, a] | D (x² tan⁻¹ x) |
| 7 | Applications of Calculus | Matching a slope field to its differential equation | A (dy/dx = x²) |
| 8 | Vectors | Order of collinear points when c = 3a − 2b | A (A lies between B and C) |
| 9 | Vectors | Angle between a and b when a + b + c = 0 | B (π/3) |
| 10 | Further Calculus Skills | Second derivative of an inverse function, g″(1) | C (−1/2) |
Multiple-Choice Questions to Watch
- Question 5 needed both families of solutions. Writing cos 5x = cos(x + π/2) gives one set of 4 solutions and another set of 6, for 10 in total.
- Question 6 tested odd and even functions. tan⁻¹ x is odd, so x² tan⁻¹ x is odd and its integral over [−a, a] is zero. cos⁻¹ x is neither odd nor even.
- Question 8 could be solved by rewriting c − a = 2(a − b), which shows that C lies on the line through B and A, beyond A.
- Question 10 used the result g″(x) = −f″(y) / [f′(y)]³, giving −4/2³ = −1/2.
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Section II: Question-by-Question Breakdown
| Question | Topic | What It Tested | Marks | Answer |
|---|---|---|---|---|
| 11(a) | Further Work with Functions | Inverse of f(x) = 1 − 1/(x − 2) | 2 | f⁻¹(x) = 2 − 1/(x − 1) |
| 11(b) | Trigonometric Equations | Solving sin 2θ − sin θ = 0 for 0 ≤ θ ≤ π | 3 | θ = 0, π/3, π |
| 11(c) | Further Trigonometric Identities | Integrating sin 3x cos x using products to sums | 2 | −(cos 4x)/8 − (cos 2x)/4 + C |
| 11(d) | Inverse Trigonometric Functions | Sketching y = (1/3)cos⁻¹(2x) | 2 | Domain [−1/2, 1/2], range [0, π/3], y-intercept π/6 |
| 11(e) | Vectors | Value of m that makes two vectors parallel | 1 | m = 3 |
| 11(f) | Polynomials | Sum of reciprocals of root products | 2 | −6 |
| 11(g) | Further Calculus Skills | Definite integral of cos²(3x) | 3 | π/12 |
| 12(a) | Rates of Change | Rate of change of a sector's radius when area is constant | 3 | −3.2 cm/s (decreasing) |
| 12(b) | Applications of Calculus | Volume of revolution about the y-axis for y = 1/x | 2 | π(1 − 1/a) |
| 12(c) | Proof by Mathematical Induction | Sum of k × k! equals (n + 1)! − 1 | 3 | Proof |
| 12(d) | Applications of Calculus | Solving dy/dx = (2 − y)(2 + y) with y = 1 when x = 0 | 3 | y = 2 sin(x + π/6) |
| 12(e)(i) | Trigonometric Equations | Writing √3 sin x − cos x as 2 sin(x − α) | 1 | 2 sin(x − π/6) |
| 12(e)(ii) | Trigonometric Equations | Solving √3 sin x = cos x + 1 for 0 ≤ x ≤ 2π | 2 | x = π/3, π |
| 13(a) | Applications of Calculus | Solving dy/dx = 5/y with y = −4 when x = 0 | 3 | y = −√(10x + 16) |
| 13(b) | Combinatorics | Round-table seating with two guests kept apart | 2 | 3600 |
| 13(c) | Vectors | Time when the angle between velocity and acceleration is π/4 | 4 | t = 9/2 |
| 13(d) | Binomial Distribution | Finding k using the normal approximation | 4 | k = 26 |
| 13(e)(i) | Combinatorics | Rearranging the Pascal's triangle relation | 1 | Proof |
| 13(e)(ii) | Combinatorics | Telescoping sum of binomial coefficients | 2 | Proof |
| 14(a) | Combinatorics | Pigeonhole proof: product of seven factors of 60 | 2 | Proof |
| 14(b) | Vectors | Two horizontally projected particles landing at the same point | 4 | V/U = √k |
| 14(c) | Vectors and Trigonometric Equations | When clock hands are perpendicular, using the scalar product | 3 | t ≈ 16.364 and 49.091 minutes |
| 14(d) | Further Calculus Skills | Derivative of f(x) = cos⁻¹(sin x) on (0, π) | 3 | −1 for 0 < x < π/2; 1 for π/2 < x < π |
| 14(e) | Polynomials and Trigonometric Equations | Smallest positive α + β + γ when tan α, tan β, tan γ are roots | 3 | 3π/4 |
Answers from NESA's 2025 marking guidelines.
What the Markers Said: Calculus (20 marks)
Question 11(g) – Integrating cos²(3x)
- Better responses used the Reference Sheet to rewrite cos²(3x) with a double angle identity and integrated cos(kx) correctly.
- Markers asked students to show enough working and to take care with brackets when taking out a common factor such as ½.
Question 12(a) – Related rates
- Better responses wrote down an appropriate chain rule, substituted correctly and interpreted the negative answer as a decreasing rate.
- Markers noted that some students treated θ or r as constants, or used an incorrect area formula instead of the one on the Reference Sheet.
Question 12(b) – Volume of revolution
- Better responses sketched the region first, wrote x in terms of y and set up the correct integral.
- Common errors were leaving out π and rotating about the x-axis instead of the y-axis.
Question 12(d) and Question 13(a) – Differential equations
- Better responses separated the variables correctly, integrated and used the given point to find the constant.
- In 12(d), markers wanted students to recognise the difference of two squares (4 − y²) and use the Reference Sheet to integrate. Some students made errors rearranging x = sin⁻¹(y/2) + C.
- In 13(a), better responses used the initial condition y = −4 to choose the negative square root and checked that their solution was reasonable.
Question 14(d) – Derivative of cos⁻¹(sin x)
- Better responses recognised that √(cos² x) = |cos x|, not cos x, so the derivative is −1 on one part of the domain and 1 on the other.
- They also identified that the derivative is undefined at x = π/2, where the denominator is zero.
- Markers asked students to use the mark value as a guide to how much working is needed.
What the Markers Said: Vectors (12 marks, plus Q14(c))
Question 11(e) – Parallel vectors
- Better responses recognised that one vector must be a scalar multiple of the other.
- Some students found a value for perpendicular vectors instead of parallel ones.
Question 13(c) – Angle between velocity and acceleration
- Better responses differentiated to find velocity and acceleration, applied the scalar product using components and solved the resulting equation.
- Markers flagged errors finding the length of a vector, manipulating fractions with square roots and not rejecting negative values of time.
Question 14(b) – Two projected particles
- Better responses wrote the position vectors of both particles, using a projection angle of zero and starting heights above the origin.
- They recognised that the two particles had different times of flight, equated the horizontal displacements and set each vertical displacement to zero.
- Markers said weaker responses used the same value of time for both particles and quoted standard projectile formulas without adapting them to the situation.
Question 14(c) – Perpendicular clock hands
- Better responses recognised that perpendicular hands mean OA · OB = 0, then turned the scalar product into a single compound angle expression.
- Markers asked students to practise general solutions of trigonometric equations and to use the Reference Sheet for compound angle formulas.
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What the Markers Said: Trigonometry (12 marks, plus Q14(c) and Q14(e))
Question 11(b) – Solving sin 2θ − sin θ = 0
- Better responses used the double angle formula, factorised and kept to the given domain.
- A common error was dividing by sin θ, which loses the solutions θ = 0 and θ = π. Markers also flagged inefficient methods such as the t-formula.
Question 11(c) – Products to sums
- Better responses recognised the product-to-sum identity and used the Reference Sheet to integrate.
- Markers said expanding sin 3x as sin(2x + x) was an inefficient method.
Question 11(d) – Sketching an inverse cosine graph
- Better responses knew the shape of y = cos⁻¹ x, applied the transformations correctly and labelled the y-intercept.
- Markers asked for larger, clearly labelled graphs.
Question 12(e) – Auxiliary angle method
- In 12(e)(i), better responses wrote the answer in the given form and gave α in radians.
- In 12(e)(ii), better responses used part (i) and kept to the domain. Markers flagged errors adding angles in radians.
Question 14(e) – Polynomial roots and tan(α + β + γ)
- Better responses wrote the relationships between roots and coefficients, used the tan(A + B) result twice to find tan(α + β + γ) = −1, and found the smallest positive angle, 3π/4.
- Markers asked students to take care with signs in the root relationships.
What the Markers Said: Combinatorics (7 marks)
Question 13(b) – Round-table seating
- Better responses used (n − 1)! for circular arrangements and counted the arrangements where the two guests sit together.
- Markers asked students to consider the complementary case: all arrangements minus those where the two guests sit together (7! − 2 × 6! = 3600).
Question 13(e) – Pascal's triangle
- In 13(e)(i), better responses substituted r = R + 1 and n = m + 1 into the given relation and justified each step.
- In 13(e)(ii), better responses applied part (i) to create a telescoping sum and recognised that the first terms cancel.
Question 14(a) – Pigeonhole principle
- Better responses grouped the factors of 60 into 6 pairs, each with a product of 60, and treated them as pigeonholes. Choosing 7 factors means at least one complete pair is chosen.
- Markers noted that testing particular sets of factors is not a proof of the general case.
What the Markers Said: Functions, Statistics and Proof (13 marks)
Question 11(a) – Inverse function
- Better responses swapped x and y and rearranged carefully.
- Markers flagged confusion between f⁻¹(x) and [f(x)]⁻¹, and errors with negative signs.
Question 11(f) – Polynomial roots
- Better responses wrote the expression over a common denominator and used the Reference Sheet for the sum and product of the roots.
- Markers flagged sign errors and incorrect values for the root relationships.
Question 12(c) – Proof by induction
- Better responses set out a clear, logical proof, unrolled (k + 1)! and factorised correctly.
- Markers asked students to test the initial case correctly, set out the n = k and n = k + 1 steps clearly and finish with a conclusion.
- Some students placed brackets incorrectly when replacing n with k + 1, for example writing (k + 1)! as k! + 1.
Question 13(d) – Normal approximation
- Better responses found p = 1/5 from E(X) = 20, calculated the variance (16) and used the z-score table to find k = 26.
- Markers flagged difficulty reading the z-score table and understanding what a z-score represents.
Common Mistakes Across the 2025 Exam
- Not using the Reference Sheet: Markers mentioned it for integrals, derivatives, trigonometric identities and polynomial roots.
- Losing solutions: Dividing by sin θ in Q11(b), and missing domain restrictions in trigonometric equations.
- Square roots and absolute values: √(cos² x) is |cos x|, and the sign of a square root must match the initial condition, as in Q13(a).
- Incomplete proofs: Missing conclusions in induction, unjustified steps and testing examples instead of proving the general case.
- Applying formulas without adapting them: Standard projectile formulas in Q14(b) and the wrong volume or area formula in Q12.
- Notation and sign errors: Confusing f⁻¹(x) with 1/f(x), misplaced brackets and dropped negative signs.
- Not enough working: Markers asked students to use the mark value as a guide to how much working to show.
How to Prepare for HSC Maths Extension 1
- Know the Reference Sheet thoroughly, including where each identity, integral and polynomial result appears.
- Practise vectors in context, including projectile motion with different start times and heights, and scalar product problems.
- Practise solving trigonometric equations with general solutions and domain restrictions, without dividing out factors.
- Write out full proofs for induction, combinatorics and vectors, with every step justified and a clear conclusion.
- Review inverse function results, such as the derivative of an inverse and the shapes of inverse trigonometric graphs.
- Complete full past papers under timed conditions to practise managing 2 hours of working time.
For more preparation:
- read our complete guide to HSC Maths Extension 1
- see our 2025 HSC Maths Advanced examiner report breakdown
- read how one of our tutors scored a 96 in HSC Maths Extension 1
- compare courses with our guide to HSC Maths: which course is right for you
- understand HSC performance bands
- see the HSC exam timetable 2026 for this year's exam date.
Need Help With HSC Maths Extension 1?
Our HSC Maths Extension 1 tutors have scored highly in the subject themselves and work one-on-one with students to master vectors, calculus and proofs, and to set out working the way markers expect.
Sources: NSW Education Standards Authority, 2025 HSC Mathematics Extension 1 examination, marking guidelines and HSC marking feedback 2025.
