Mathematics Extension 1 — past papers by topic
Every official NESA exam and examiner report we index, organised by topic and subtopic and labelled with the study design it was written under. Open the official paper, work the question in your own workspace, or bring a tutor into it live.
Papers and examiner reports open on the official NESA website — they stay official and up to date. Question descriptions below are our own; we never republish exam content. Prefer to be walked through it? Book a tutor →
Official papers & examiner reports, year by year
All URLs fetched from the official nsw.gov.au NESA host and copied exactly as returned. Two full exam PDFs (2025 and 2016) were opened and read directly to seed the questions array, so those descriptions are grounded in real content. Syllabus-era correction found during verification: the true boundary between syllabus eras is 2019 to 2020, not 2018 to 2019 as first suspected. The 2020 exam page carries no previous-syllabus warning while 2019, 2018, 2017, 2016 and 2015 pages explicitly do, and NESA states the 2017 syllabus rolled out Year 11 in 2019 then Year 12 in 2020. This is corroborated by content: the 2016 paper covers projectile motion and circle/parabola geometry with no vectors or statistics, while the 2025 paper has vectors and binomial-distribution statistics, topics that only exist in the 2017-syllabus Year 12 course. The topics array reflects only the current 2017 syllabus structure; old-syllabus topic labels in 2016-2019 questions are mapped as closely as reasonable but some (circle-geometry proof, projectile motion via parametrics, parabola-locus geometry) have no true current-topic equivalent and are flagged in their subtopic field. The next syllabus change (2024 syllabus) only takes effect for HSC exams from 2027 and touches none of the years covered here; it is included in sdEras purely for forward badging completeness. Coverage note: only the 2025 and 2016 PDFs were actually opened and read line by line to extract questions, in line with the instruction not to guess; all 10-11 years still carry real, verified paper/marking-guidelines/marking-feedback URLs. Marking feedback documents are titled with a Section II suffix across all years, meaning NESA's public feedback covers only the extended-response section, not the multiple-choice section, a limit of the source itself. Every reportUrl is the official marking-guidelines PDF, NESA's closest equivalent to an examination report for this subject.
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Functions · question map rolling out
Further work with functions
Question-level mapping for this subtopic is being verified — use the year table above; every official paper is one click away.
Polynomials
Question-level mapping for this subtopic is being verified — use the year table above; every official paper is one click away.
Trigonometric Functions · 5 mapped questions
Inverse trigonometric functions
- 2025SD 2020-2026 (Mathematics Extension 1 Stage 6 Syllabus, 2017)Exam Q14(d)3 marksDifferentiate a composite of inverse and ordinary trigonometric functions on a restricted domain.Official paper ↗Examiner report ↗Work through it with a tutor →
Further trigonometric identities
Question-level mapping for this subtopic is being verified — use the year table above; every official paper is one click away.
Trigonometric equations
- 2025SD 2020-2026 (Mathematics Extension 1 Stage 6 Syllabus, 2017)Exam Q11(b)3 marksSolve a trigonometric equation involving a double-angle identity over a restricted domain.Official paper ↗Examiner report ↗Work through it with a tutor →
- 2016Pre-2017 syllabus (old 3-Unit-era Extension 1 course)Exam Q61 markMultiple-choice: find the general solution of a quadratic-in-sine trigonometric equation.Official paper ↗Examiner report ↗Work through it with a tutor →
Applications of trigonometric functions
- 2025SD 2020-2026 (Mathematics Extension 1 Stage 6 Syllabus, 2017)Exam Q14(c)3 marksModel clock-hand positions with trigonometric parametrisation and find the times the hands are first and second perpendicular after midnight.Official paper ↗Examiner report ↗Work through it with a tutor →
Applications of trigonometric functions - old syllabus SHM
- 2016Pre-2017 syllabus (old 3-Unit-era Extension 1 course)Exam Q13(a)4 marksModel a tide using simple harmonic motion and find the earliest time the tide rises at its fastest rate. Old syllabus era; SHM sits outside the current topics array.Official paper ↗Examiner report ↗Work through it with a tutor →
Calculus · 8 mapped questions
Rates of change
Question-level mapping for this subtopic is being verified — use the year table above; every official paper is one click away.
Further calculus skills
- 2025SD 2020-2026 (Mathematics Extension 1 Stage 6 Syllabus, 2017)Exam Q101 markMultiple-choice: apply inverse-function differentiation rules twice to find the second derivative of an inverse function at a point.Official paper ↗Examiner report ↗Work through it with a tutor →
Further applications of calculus
- 2025SD 2020-2026 (Mathematics Extension 1 Stage 6 Syllabus, 2017)Exam Q12(a)3 marksRelated-rates problem: find the rate of change of a sector's radius given a constant area and a known rate of change of angle.Official paper ↗Examiner report ↗Work through it with a tutor →
- 2025SD 2020-2026 (Mathematics Extension 1 Stage 6 Syllabus, 2017)Exam Q12(d)3 marksSolve a separable first-order differential equation with a logistic-type right-hand side, given an initial condition.Official paper ↗Examiner report ↗Work through it with a tutor →
- 2016Pre-2017 syllabus (old 3-Unit-era Extension 1 course)Exam Q12(a)6 marksRelated-rates problem for a leaking conical container: relate radius to height, derive dV/dh, then find dh/dt and the rate of volume change at a given height.Official paper ↗Examiner report ↗Work through it with a tutor →
- 2016Pre-2017 syllabus (old 3-Unit-era Extension 1 course)Exam Q12(b)5 marksSet up and solve a first-order differential equation modelling a chemical reaction where mass transfer rate is proportional to the remaining reactant, given an initial rate.Official paper ↗Examiner report ↗Work through it with a tutor →
Further applications of calculus (projectile motion)
- 2025SD 2020-2026 (Mathematics Extension 1 Stage 6 Syllabus, 2017)Exam Q14(b)4 marksTwo-particle projectile scenario: show that a velocity ratio between two horizontally-launched particles landing at the same point depends only on a height ratio.Official paper ↗Examiner report ↗Work through it with a tutor →
Applications of calculus - old syllabus simple harmonic motion
- 2016Pre-2017 syllabus (old 3-Unit-era Extension 1 course)Exam Q71 markMultiple-choice: find the maximum velocity of a particle whose displacement is a sum of sine and cosine terms. Old syllabus era; SHM sits outside the current topics array.Official paper ↗Examiner report ↗Work through it with a tutor →
Further applications of calculus - old syllabus projectile motion via parametrics
- 2016Pre-2017 syllabus (old 3-Unit-era Extension 1 course)Exam Q13(b)7 marksMulti-part projectile-motion problem: prove a maximum-height formula, find the height at which a projectile strikes a wall, then analyse its rebound trajectory to the ground. Old syllabus era; projectile motion via parametrics has no direct current-topic equivalent.Official paper ↗Examiner report ↗Work through it with a tutor →
Combinatorics · 4 mapped questions
Permutations and combinations
- 2025SD 2020-2026 (Mathematics Extension 1 Stage 6 Syllabus, 2017)Exam Q13(b)2 marksCount circular seating arrangements for a group of guests subject to a must-not-sit-together restriction.Official paper ↗Examiner report ↗Work through it with a tutor →
- 2016Pre-2017 syllabus (old 3-Unit-era Extension 1 course)Exam Q81 markMultiple-choice: count team selections from a group subject to an at-least-one composition constraint.Official paper ↗Examiner report ↗Work through it with a tutor →
The binomial theorem
- 2025SD 2020-2026 (Mathematics Extension 1 Stage 6 Syllabus, 2017)Exam Q13(e)3 marksProve a binomial-coefficient telescoping identity using the Pascal's triangle recurrence.Official paper ↗Examiner report ↗Work through it with a tutor →
- 2016Pre-2017 syllabus (old 3-Unit-era Extension 1 course)Exam Q14(b)3 marksDerive two standard binomial-coefficient identities from the binomial expansion, then combine them to prove a weighted-sum identity.Official paper ↗Examiner report ↗Work through it with a tutor →
Proof · 8 mapped questions
Proof by mathematical induction
- 2025SD 2020-2026 (Mathematics Extension 1 Stage 6 Syllabus, 2017)Exam Q12(c)3 marksProve a summation identity involving factorials by mathematical induction.Official paper ↗Examiner report ↗Work through it with a tutor →
- 2016Pre-2017 syllabus (old 3-Unit-era Extension 1 course)Exam Q14(a)4 marksFactorise a cubic expression, then use the result to prove a summation identity by mathematical induction.Official paper ↗Examiner report ↗Work through it with a tutor →
Sum and product of roots (polynomials)
- 2025SD 2020-2026 (Mathematics Extension 1 Stage 6 Syllabus, 2017)Exam Q11(f)2 marksUse root-coefficient relationships for a cubic to evaluate a symmetric expression in its roots.Official paper ↗Examiner report ↗Work through it with a tutor →
- 2025SD 2020-2026 (Mathematics Extension 1 Stage 6 Syllabus, 2017)Exam Q14(e)3 marksUse a cubic's roots expressed as tangents of three angles to find the smallest positive value of the angle sum.Official paper ↗Examiner report ↗Work through it with a tutor →
Proof techniques (divisibility)
- 2025SD 2020-2026 (Mathematics Extension 1 Stage 6 Syllabus, 2017)Exam Q14(a)2 marksProve a divisibility claim about the product of a chosen set of distinct factors of a fixed integer.Official paper ↗Examiner report ↗Work through it with a tutor →
Sum and product of roots (polynomials) - old syllabus series notation
- 2016Pre-2017 syllabus (old 3-Unit-era Extension 1 course)Exam Q11 markMultiple-choice: identify which expanded series matches a given sigma-notation summation. Old syllabus era; no direct current-topic equivalent.Official paper ↗Examiner report ↗Work through it with a tutor →
Geometric proof - old syllabus circle geometry
- 2016Pre-2017 syllabus (old 3-Unit-era Extension 1 course)Exam Q13(c)4 marksCircle-geometry proof: show a quadrilateral formed by two chords and their intersection points is cyclic, then use this to prove a perpendicularity result. Old syllabus era; circle-geometry proof has no direct current-topic equivalent.Official paper ↗Examiner report ↗Work through it with a tutor →
Locus and parametric proof - old syllabus parabola geometry
- 2016Pre-2017 syllabus (old 3-Unit-era Extension 1 course)Exam Q14(c)7 marksParabola locus problem: find coordinates of a point on the directrix, show a related point traces a second parabola, state its focal length, then minimise the distance between two points on the two parabolas. Old syllabus era; parabola-locus geometry has no direct current-topic equivalent.Official paper ↗Examiner report ↗Work through it with a tutor →
Vectors · 5 mapped questions
Introduction to vectors
- 2025SD 2020-2026 (Mathematics Extension 1 Stage 6 Syllabus, 2017)Exam Q21 markMultiple-choice: compute the vector projection of one 2D vector onto another.Official paper ↗Examiner report ↗Work through it with a tutor →
- 2025SD 2020-2026 (Mathematics Extension 1 Stage 6 Syllabus, 2017)Exam Q81 markMultiple-choice: determine what must be true about the relative position of three collinear points defined by position vectors.Official paper ↗Examiner report ↗Work through it with a tutor →
- 2025SD 2020-2026 (Mathematics Extension 1 Stage 6 Syllabus, 2017)Exam Q91 markMultiple-choice: use a vector-sum-to-zero condition and given magnitudes to find the angle between two of three vectors.Official paper ↗Examiner report ↗Work through it with a tutor →
- 2025SD 2020-2026 (Mathematics Extension 1 Stage 6 Syllabus, 2017)Exam Q11(e)1 markFind the unknown component that makes one given vector parallel to another.Official paper ↗Examiner report ↗Work through it with a tutor →
Further work with vectors
Question-level mapping for this subtopic is being verified — use the year table above; every official paper is one click away.
Further work with vectors (calculus of vectors)
- 2025SD 2020-2026 (Mathematics Extension 1 Stage 6 Syllabus, 2017)Exam Q13(c)4 marksGiven a vector-valued position function, find the time at which the angle between the velocity and acceleration vectors equals a specified value.Official paper ↗Examiner report ↗Work through it with a tutor →
Statistical Analysis · 2 mapped questions
The binomial distribution
- 2025SD 2020-2026 (Mathematics Extension 1 Stage 6 Syllabus, 2017)Exam Q41 markMultiple-choice: find the mean and variance of a Bernoulli random variable from a given probability function.Official paper ↗Examiner report ↗Work through it with a tutor →
The sampling distribution of the mean
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The sampling distribution of the mean / normal approximation
- 2025SD 2020-2026 (Mathematics Extension 1 Stage 6 Syllabus, 2017)Exam Q13(d)4 marksUse a normal approximation to a binomial setting, given the mean and a tail probability, to find an unknown threshold value.Official paper ↗Examiner report ↗Work through it with a tutor →
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