Maths Extension 1 exam trends & analysis (2020–2025)
Across 2020–2025, Calculus (ME-C1, C2, C3) is examined most (64 questions), ahead of Trigonometric Functions (ME-T1, T2, T3) (34 questions) and Functions (ME-F1, ME-F2) (33 questions). By topic, Separable differential equations: separating variables, integrating both sides, and initial conditions, Inverse trigonometric functions: definitions, principal branches, domains, ranges and graphs and The scalar (dot) product: component formula, geometric formula, angle between vectors and orthogonality come up most, with Polynomial and rational inequalities: sign analysis, critical points and excluded values and Combinations: counting unordered selections with $\\binom{n}{r}$ also recurring.
Based on 209 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 36 of 36 dot points
| Dot point | Times | Marks | Years | Most common mistake |
|---|---|---|---|---|
| Separable differential equations: separating variables, integrating both sides, and initial conditionscalculus Following tangent lines to produce the sketch | 18× | 1–4 | 2020, 2021, 2022, 2023, 2024, 2025 | Following tangent lines to produce the sketch |
| Inverse trigonometric functions: definitions, principal branches, domains, ranges and graphstrigonometric-functions Reading the question; clear logical explanations | 14× | 1–4 | 2020, 2021, 2022, 2023, 2024, 2025 | Reading the question; clear logical explanations |
| The scalar (dot) product: component formula, geometric formula, angle between vectors and orthogonalityvectors Used specific values instead of a general proof | 12× | 1–4 | 2020, 2021, 2022, 2023, 2024, 2025 | Used specific values instead of a general proof |
| Polynomial and rational inequalities: sign analysis, critical points and excluded valuesfunctions Must multiply by denominator squared; exclude x making it zero | 11× | 1–3 | 2020, 2021, 2022, 2023, 2024, 2025 | Must multiply by denominator squared; exclude x making it zero |
| Combinations: counting unordered selections with $\\binom{n}{r}$combinatorics Added outcomes instead of multiplying them | 10× | 1–2 | 2020, 2021, 2022, 2023, 2024 | Added outcomes instead of multiplying them |
| Derivatives and integrals of inverse trigonometric functionscalculus Confused tangent/normal and function/inverse gradients | 10× | 1–3 | 2021, 2022, 2023, 2024, 2025 | Confused tangent/normal and function/inverse gradients |
| Vector arithmetic: addition, scalar multiplication, magnitude and unit vectorsvectors Errors adding and subtracting like vectors | 10× | 1–3 | 2020, 2021, 2022, 2024, 2025 | Errors adding and subtracting like vectors |
| Volumes of revolution: discs about the x-axis and y-axiscalculus Identifying outer vs inner volume; choosing limits | 10× | 1–4 | 2020, 2021, 2022, 2023, 2024, 2025 | Identifying outer vs inner volume; choosing limits |
| General solutions of trigonometric equations: $\\sin$, $\\cos$ and $\\tan$trigonometric-functions Dividing out a factor lost solutions; missed plus/minus square root | 8× | 1–3 | 2020, 2021, 2024, 2025 | Dividing out a factor lost solutions; missed plus/minus square root |
| Normal approximation of the binomial distribution: continuity, validity and z-scoresstatistical-analysis Confused variance and standard deviation; using empirical rule | 8× | 1–4 | 2020, 2021, 2022, 2023, 2024, 2025 | Confused variance and standard deviation; using empirical rule |
| Roots and coefficients of polynomials: Vieta's formulas for cubics and quarticsfunctions Recalling quartic root-coefficient formulae; combining algebraic fractions | 8× | 1–3 | 2020, 2021, 2022, 2023, 2024, 2025 | Recalling quartic root-coefficient formulae; combining algebraic fractions |
| Graphing polynomials: leading-term behaviour, intercepts and root multiplicityfunctions Recognising restricted-domain parabola; showing intercepts | 7× | 1–3 | 2020, 2021, 2022, 2023 | Recognising restricted-domain parabola; showing intercepts |
| Integration by substitution in HSC Maths Extension 1: choosing $u$, transforming the integral and changing limitscalculus Integrating fractional indices; forgot to re-substitute and add constant | 7× | 1–4 | 2020, 2021, 2022, 2023, 2024, 2025 | Integrating fractional indices; forgot to re-substitute and add constant |
| The binomial distribution: definition, probability mass function, mean and variancestatistical-analysis Identifying the binomial distribution formula on Reference Sheet | 7× | 1–2 | 2020, 2021, 2022, 2023, 2024 | Identifying the binomial distribution formula on Reference Sheet |
| The binomial theorem and Pascal's triangle: expansion of $(a + b)^n$ and the general termcombinatorics Mishandled the negative sign in the expansion | 6× | 1–2 | 2020, 2021, 2022, 2023, 2025 | Mishandled the negative sign in the expansion |
| Integrals giving inverse trig functions: $\\arcsin$, $\\arctan$ and the patterns to recognisecalculus Evaluating limits to an answer in radians | 5× | 1–3 | 2021, 2023, 2024 | Evaluating limits to an answer in radians |
| Polynomial division and the remainder and factor theoremsfunctions Applying remainder and factor theorems correctly | 5× | 1–3 | 2020, 2022, 2023, 2024 | Applying remainder and factor theorems correctly |
| Projectile motion: parametric equations, range, maximum height and time of flightcalculus Used wrong time with displacement; range not double time to max | 5× | 2–4 | 2021, 2022, 2023, 2024, 2025 | Used wrong time with displacement; range not double time to max |
| Related rates of change: linking changing quantities via implicit differentiationcalculus Recalling sphere volume; building and using the chain rule | 5× | 1–3 | 2020, 2021, 2023, 2024, 2025 | Recalling sphere volume; building and using the chain rule |
| Exponential growth and decay: $\\frac{dN}{dt} = k N$, $N = N_0 e^{kt}$, doubling and half-lifecalculus Log-to-exponential conversion; calculator with logarithms | 4× | 1–3 | 2021, 2022 | Log-to-exponential conversion; calculator with logarithms |
| Geometric proofs with vectors: parallel, perpendicular, midpoint and ratio propertiesvectors Using the part (i) result; distinguishing vectors from magnitudes | 4× | 3–4 | 2021, 2022, 2023 | Using the part (i) result; distinguishing vectors from magnitudes |
| Mathematical induction for series identitiesproof Simplifying algebraic fractions; manipulating both sides at once | 4× | 3 | 2020, 2021, 2023, 2025 | Simplifying algebraic fractions; manipulating both sides at once |
| Product-to-sum and sum-to-product identities for trigonometric expressionstrigonometric-functions Substituting part (i); antiderivative of cos4x; evaluating limits | 4× | 2–3 | 2020, 2024, 2025 | Substituting part (i); antiderivative of cos4x; evaluating limits |
| Sum and difference identities for sin, cos and tan: expansions, simplifications and exact valuestrigonometric-functions Taking identities from Reference Sheet; careful algebra | 4× | 2 | 2020, 2021, 2024 | Taking identities from Reference Sheet; careful algebra |
| Vector projection: scalar projection $\\frac{\\mathbf{a} \\cdot \\mathbf{b}}{|\\mathbf{b}|}$ and vector projection $\\frac{\\mathbf{a} \\cdot \\mathbf{b}}{|\\mathbf{b}|^2} \\mathbf{b}$vectors Drawing labelled diagram; where the projection lies | 4× | 1–4 | 2020, 2022, 2024, 2025 | Drawing labelled diagram; where the projection lies |
| Auxiliary angle: writing $a \\sin x + b \\cos x$ as $R \\sin(x + \\alpha)$trigonometric-functions Equating coefficients for R and alpha; checking the quadrant | 3× | 1–4 | 2020, 2022, 2025 | Equating coefficients for R and alpha; checking the quadrant |
| Permutations: counting ordered arrangements with the multiplication principlecombinatorics Identifying and dividing out repeated letters | 3× | 1–2 | 2020, 2023, 2025 | Identifying and dividing out repeated letters |
| The pigeonhole principle: guaranteed coincidences in counting problemscombinatorics Indicating remainder to justify an extra hole (ceiling) | 3× | 2 | 2020, 2022, 2025 | Indicating remainder to justify an extra hole (ceiling) |
| Binomial probability calculations: exact values, cumulative probabilities and complementsstatistical-analysis Applying binomial probability correctly | 2× | 1–2 | 2023 | Applying binomial probability correctly |
| Mathematical induction for divisibility: standard technique and algebraic restructuringproof Showing base case; setting out all induction steps | 2× | 3 | 2022, 2024 | Showing base case; setting out all induction steps |
| Parametric equations: parameter elimination, sketches, and standard curvesfunctions Solving simultaneous equations | 2× | 1–2 | 2023 | Solving simultaneous equations |
| Parametric vector equations of lines: point and direction form, parameter eliminationvectors Recognising horizontal displacements equal at collision | 2× | 1–2 | 2023 | Recognising horizontal displacements equal at collision |
| Bernoulli trials: definition, parameters, mean and variancestatistical-analysis | 1× | 1 | 2025 | — |
| The t-formula: rational expressions for $\\sin \\theta$, $\\cos \\theta$ and $\\tan \\theta$ via $t = \\tan(\\theta/2)$trigonometric-functions Solving quadratic from t-substitution; solving over the domain | 1× | 3 | 2023 | Solving quadratic from t-substitution; solving over the domain |
| Mathematical induction for general statements: recurrence relations and propertiesproof | — | — | not yet | — |
| Mathematical induction for inequalities: the technique and the algebraic careproof | — | — | not yet | — |