Maths Extension 2 exam trends & analysis (2020–2025)
Across 2020–2025, Introduction to Complex Numbers (MEX-N1) is examined most (78 questions), ahead of Further Work with Vectors (MEX-V1) (58 questions) and Applications of Calculus to Mechanics (MEX-M1) (57 questions). By topic, Three-dimensional vectors, lines and spheres in HSC Mathematics Extension 2: component form, magnitude, the scalar product, vector equations of lines and the equation of a sphere, The nature of proof in HSC Mathematics Extension 2: direct proof, proof by contradiction, contrapositive and disproof by counterexample and De Moivre's theorem and roots of unity in HSC Mathematics Extension 2: powers in polar form, nth roots, the roots of unity and their geometry come up most, with Complex number arithmetic and the Argand plane in HSC Mathematics Extension 2: Cartesian and polar form, modulus, argument, conjugate and geometric interpretation and Resisted motion in HSC Mathematics Extension 2: rectilinear motion under gravity with resistance proportional to velocity or velocity squared, and terminal velocity also recurring.
Based on 288 questions across 6 official NESA exam papers, their marking guidelines and marking feedback.
Work in progress
These exam-trend insights are an early release. The frequencies, mark ranges and figures are still being verified against the official NESA past papers and may change. Treat them as a study guide, not a guarantee of what will be examined.
Most-examined dot points
By module
Every dot point, by exam frequency
Click any dot point for the full verbatim syllabus wording, worked answers and past questions.
Showing 18 of 18 dot points
| Dot point | Times | Marks | Years | Most common mistake |
|---|---|---|---|---|
| Three-dimensional vectors, lines and spheres in HSC Mathematics Extension 2: component form, magnitude, the scalar product, vector equations of lines and the equation of a spherevectors Careless algebra expanding brackets with negatives; not reading the question carefully | 33× | 1–4 | 2020, 2021, 2022, 2023, 2024, 2025 | Careless algebra expanding brackets with negatives; not reading the question carefully |
| The nature of proof in HSC Mathematics Extension 2: direct proof, proof by contradiction, contrapositive and disproof by counterexampleproof Reused same pronumeral for two variables; struggled with logs equal to fractions and non-e/10 bases | 29× | 1–4 | 2020, 2021, 2022, 2023, 2024, 2025 | Reused same pronumeral for two variables; struggled with logs equal to fractions and non-e/10 bases |
| De Moivre's theorem and roots of unity in HSC Mathematics Extension 2: powers in polar form, nth roots, the roots of unity and their geometrycomplex-numbers Showing the result by writing the complex number in modulus-argument form | 26× | 1–4 | 2020, 2021, 2022, 2023, 2024, 2025 | Showing the result by writing the complex number in modulus-argument form |
| Complex number arithmetic and the Argand plane in HSC Mathematics Extension 2: Cartesian and polar form, modulus, argument, conjugate and geometric interpretationcomplex-numbers Finding the modulus of a complex number | 25× | 1–4 | 2020, 2021, 2022, 2023, 2024, 2025 | Finding the modulus of a complex number |
| Resisted motion in HSC Mathematics Extension 2: rectilinear motion under gravity with resistance proportional to velocity or velocity squared, and terminal velocitymechanics Confused 50g as 50 grams when g=10; failed to write relevant calculations | 24× | 1–4 | 2020, 2021, 2022, 2023, 2024, 2025 | Confused 50g as 50 grams when g=10; failed to write relevant calculations |
| Vector geometric proofs in HSC Mathematics Extension 2: proving triangle, parallelogram and quadrilateral properties using position vectors and the scalar productvectors Poor vector notation (AC vector vs magnitude); weak use of given info and line equations | 20× | 1–3 | 2020, 2021, 2022, 2023, 2024, 2025 | Poor vector notation (AC vector vs magnitude); weak use of given info and line equations |
| Proving inequalities in HSC Mathematics Extension 2: the non-negativity of squares, the AM-GM inequality, and rigorous algebraic argumentproof Not knowing a standard method to prove this result | 19× | 1–3 | 2020, 2021, 2022, 2023, 2024, 2025 | Not knowing a standard method to prove this result |
| Integration by parts in HSC Mathematics Extension 2: the formula, choosing u and dv, repeated application, and the boomerang (recovery) techniqueintegration Wrong choice for u and dv/dx; careless applying the by-parts formula and substituting limits | 18× | 1–4 | 2020, 2021, 2022, 2023, 2024, 2025 | Wrong choice for u and dv/dx; careless applying the by-parts formula and substituting limits |
| Trigonometric and rationalising substitutions in HSC Mathematics Extension 2: substitutions for square-root integrands, completing the square, and the t = tan(x/2) substitutionintegration Did not clearly show substitutions and limits; failed to find a suitable trig substitution | 15× | 1–4 | 2020, 2021, 2022, 2023, 2024, 2025 | Did not clearly show substitutions and limits; failed to find a suitable trig substitution |
| Simple harmonic motion in HSC Mathematics Extension 2: the defining equation, displacement and velocity functions, amplitude, period and the velocity-displacement relationmechanics Did not know Reference Sheet; confused cos vs sine forms, phase, amplitude and finding n | 12× | 1–4 | 2020, 2021, 2022, 2023, 2024, 2025 | Did not know Reference Sheet; confused cos vs sine forms, phase, amplitude and finding n |
| Velocity and acceleration as functions in HSC Mathematics Extension 2: the calculus of rectilinear motion when acceleration depends on displacement or velocitymechanics Used wrong acceleration form; failed to simplify before integrating or to evaluate the constant | 11× | 1–3 | 2020, 2021, 2022, 2023, 2024, 2025 | Used wrong acceleration form; failed to simplify before integrating or to evaluate the constant |
| Modulus-argument form in HSC Mathematics Extension 2: polar form, the principal argument, and the geometry of multiplication and division of complex numberscomplex-numbers Did not grasp geometric meaning of polar form; unclear why moduli equal and angle is pi/3 | 10× | 1–3 | 2020, 2021, 2022, 2023, 2024, 2025 | Did not grasp geometric meaning of polar form; unclear why moduli equal and angle is pi/3 |
| Projectile motion with resistance in HSC Mathematics Extension 2: resolving motion into components and modelling air resistance proportional to velocity in two dimensionsmechanics Poor vector notation; quoting x-dot=u cos theta without justification; rote answers omitting detail | 10× | 1–5 | 2020, 2021, 2022, 2023, 2025 | Poor vector notation; quoting x-dot=u cos theta without justification; rote answers omitting detail |
| Polynomials over the complex field in HSC Mathematics Extension 2: solving quadratics with complex coefficients, the conjugate root theorem, and complete factorisationcomplex-numbers Did not know efficient method or how to find square root of a complex number; careless bracket/arithmetic errors | 9× | 1–4 | 2020, 2021, 2022, 2023, 2024 | Did not know efficient method or how to find square root of a complex number; careless bracket/arithmetic errors |
| Curves and regions in the complex plane in HSC Mathematics Extension 2: loci from modulus and argument conditions, including circles, perpendicular bisectors and rayscomplex-numbers Using Cartesian form when dealing with graphs involving complex numbers. | 8× | 1–3 | 2021, 2022, 2023, 2024, 2025 | Using Cartesian form when dealing with graphs involving complex numbers. |
| Further proof by induction in HSC Mathematics Extension 2: series, divisibility, inequalities and stronger induction structuresproof Struggled when base case isn't n=1; careless with algebraic fractions; confused when < vs = needed | 8× | 2–4 | 2020, 2021, 2022, 2023, 2024, 2025 | Struggled when base case isn't n=1; careless with algebraic fractions; confused when < vs = needed |
| Integration by partial fractions in HSC Mathematics Extension 2: decomposing rational functions over linear, repeated and irreducible quadratic factors and integratingintegration Solving simultaneous equations with three variables. | 6× | 1–4 | 2021, 2022, 2023, 2024, 2025 | Solving simultaneous equations with three variables. |
| Vector projection in HSC Mathematics Extension 2: the scalar projection, the vector projection, and resolving a vector into parallel and perpendicular componentsvectors Finding the dot product of a vector with itself | 5× | 1–4 | 2021, 2024, 2025 | Finding the dot product of a vector with itself |