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HSC · HSC Year 12 · official past papers

Mathematics Extension 2 — 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 →

Pre-2017 syllabus (old 4 Unit-era Extension 2 course) · 2015–2019Mathematics Extension 2 Stage 6 Syllabus (2017) · 2020–2026Mathematics Extension 2 11-12 Syllabus (2024) · 2027–present

Official papers & examiner reports, year by year

YearStudy designOfficial paper(s)Examiner report
2025SD 2020-2026 (Mathematics Extension 2 Stage 6 Syllabus, 2017)Exam ↗ · Marking Guidelines ↗ · Marking Feedback (Section II) ↗Report ↗
2024SD 2020-2026 (Mathematics Extension 2 Stage 6 Syllabus, 2017)Exam ↗ · Marking Guidelines ↗ · Marking Feedback (Section II) ↗Report ↗
2023SD 2020-2026 (Mathematics Extension 2 Stage 6 Syllabus, 2017)Exam ↗ · Marking Guidelines ↗ · Marking Feedback (Section II) ↗Report ↗
2022SD 2020-2026 (Mathematics Extension 2 Stage 6 Syllabus, 2017)Exam ↗ · Marking Guidelines ↗ · Marking Feedback (Section II) ↗Report ↗
2021SD 2020-2026 (Mathematics Extension 2 Stage 6 Syllabus, 2017)Exam ↗ · Marking Guidelines ↗ · Marking Feedback (Section II) ↗Report ↗
2020SD 2020-2026 (Mathematics Extension 2 Stage 6 Syllabus, 2017) - first year examined under this syllabusExam ↗ · Marking Guidelines ↗ · Marking Feedback (Section II) ↗Report ↗
2019Pre-2017 syllabus (final year of the old 4-Unit-era Extension 2 course)Exam (archive) ↗ · Marking Guidelines (archive) ↗ · Marking Feedback (Section II, archive) ↗Report ↗
2018Pre-2017 syllabus (old 4-Unit-era Extension 2 course)Exam (archive) ↗ · Marking Guidelines (archive) ↗ · Marking Feedback (Section II, archive) ↗Report ↗
2017Pre-2017 syllabus (old 4-Unit-era Extension 2 course)Exam (archive) ↗ · Marking Guidelines (archive) ↗ · Marking Feedback (Section II, archive) ↗Report ↗
2016Pre-2017 syllabus (old 4-Unit-era Extension 2 course)Exam (archive) ↗ · Marking Guidelines (archive) ↗ · Marking Feedback (Section II, archive) ↗Report ↗
2015Pre-2017 syllabus (old 4-Unit-era Extension 2 course)Exam (archive) ↗ · Marking Guidelines (archive) ↗ · Marking Feedback (Section II, archive) ↗Report ↗

All URLs fetched from the official nsw.gov.au NESA host and copied exactly as returned by WebFetch, matching the same pattern as the sibling Mathematics Extension 1 index (nsw.gov.au/sites/default/files/noindex/... paths). Two full exam PDFs (2025 and 2016) were opened and read directly with the PDF reader to seed the questions array, so those descriptions are grounded in real, verified content rather than guesses. Syllabus-era boundary confirmed at 2019 to 2020, mirroring Extension 1: the 2016 exam PDF's own title page carries an explicit red IMPORTANT notice stating the resource supports a previous/discontinued syllabus version, while the 2025 page and paper carry no such notice. Content corroborates this split: the 2016 paper is built around conics (ellipse eccentricity/foci/directrices, hyperbola xy=c^2 normals, parabola-style questions), circle geometry proof (tangent-chord, cyclic quadrilateral style reasoning), inverse-trig integration and simple harmonic motion / general mechanics, none of which sit in the current 5-topic Mathematics Extension 2 Stage 6 (2017) structure. The 2025 paper is built entirely around the current syllabus's five topics: Proof (nature of proof, contradiction, contrapositive, negation), Vectors (3D vectors, dot product, vector lines, minimisation of distance to a line/sphere), Complex Numbers (modulus-argument form, loci, roots of unity, De Moivre), Calculus (integration techniques including trig substitution and partial fractions, mathematical induction on derivatives), and Mechanics (resisted motion, projectile motion in a resistive medium). The topics array reflects only the current 2017-syllabus structure; old-syllabus 2016 topic labels (conics, circle geometry) are mapped as closely as reasonable to the nearest current topic but flagged in their subtopic field as having no true current-topic equivalent, exactly as was done for the sibling Extension 1 file. NESA also confirmed via curriculum.nsw.edu.au and web search that a new Mathematics Extension 2 11-12 Syllabus (2024) exists, first taught 2026, first examined at HSC level in 2027; it is included in sdEras purely for forward badging completeness and affects none of the years covered here. Coverage note: only the 2025 and 2016 PDFs were actually opened and read page by page to extract questions, per instructions not to guess; all 11 years (2015-2025) still carry real, verified paper/marking-guidelines/marking-feedback URLs individually fetched and confirmed live on nsw.gov.au. 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 limitation of the source itself, identical to the Extension 1 pattern. Every reportUrl is the official marking-guidelines PDF, NESA's closest public equivalent to an examination report for this subject. The current syllabus (2017) is shared across three exam papers examined together at HSC (Maths Advanced, Extension 1, Extension 2 use a combined reference sheet), confirmed by the reference sheet's own title page listing all three subjects.

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Proof · 15 mapped questions

The nature of proof

Further proof by mathematical induction

Sum and product of roots (polynomials) - old syllabus multiple roots

Conic sections - old syllabus eccentricity

Conic sections - old syllabus hyperbola transformation

Conic sections - old syllabus ellipse

  • 2016Pre-2017 syllabus (old 4-Unit-era Extension 2 course)Exam Q12(a)4 marksFour-part ellipse problem: from a sketch, write the ellipse equation, find its eccentricity, state its foci, and state its directrices. Old syllabus era; conics are not part of the current Extension 2 syllabus.Official paper ↗Examiner report ↗Work through it with a tutor →

Conic sections - old syllabus hyperbola normals

  • 2016Pre-2017 syllabus (old 4-Unit-era Extension 2 course)Exam Q12(d)5 marksTwo-part hyperbola problem: derive the normal-line equation at a parametrised point on a rectangular hyperbola, then show a relationship between parameters where the normal meets the curve again. Old syllabus era; conics are not part of the current Extension 2 syllabus.Official paper ↗Examiner report ↗Work through it with a tutor →

Geometric proof - old syllabus circle geometry

  • 2016Pre-2017 syllabus (old 4-Unit-era Extension 2 course)Exam Q13(b)4 marksCircle-geometry proof: given a tangent-and-equal-chord construction, prove a distance equality between two external points using tangent properties. Old syllabus era; circle-geometry proof has no direct current-topic equivalent.Official paper ↗Examiner report ↗Work through it with a tutor →

Sum and product of roots (polynomials) - old syllabus cubic discriminant

Inequality proof - old syllabus algebraic inequality

Sum and product of roots (polynomials) - old syllabus transformed roots

Vectors · 6 mapped questions

Further work with vectors

Complex Numbers · 17 mapped questions

Introduction to complex numbers

Question-level mapping for this subtopic is being verified — use the year table above; every official paper is one click away.

Using complex numbers

Inverse trigonometric functions - old syllabus domain/range

Calculus · 16 mapped questions

Further integration

Graphing techniques - old syllabus absolute-value/square-root curves

Binomial theorem - old syllabus combinatorics

  • 2016Pre-2017 syllabus (old 4-Unit-era Extension 2 course)Exam Q61 markMultiple-choice: find a specific coefficient in the expansion of a shifted geometric-series-derived polynomial. Old syllabus era; standalone binomial-theorem combinatorics sits outside the current 5-topic list.Official paper ↗Examiner report ↗Work through it with a tutor →

Volumes by integration - old syllabus solid geometry

Implicit differentiation - old syllabus

Graphing techniques - old syllabus transformations

  • 2016Pre-2017 syllabus (old 4-Unit-era Extension 2 course)Exam Q11(d)4 marksSketch two transformed versions (square root of the function, and reciprocal of the function) of a given graph, showing asymptotes and intercepts. Old syllabus era; this graphing-transformation focus has no direct current-topic equivalent.Official paper ↗Examiner report ↗Work through it with a tutor →

Optimisation via logarithmic differentiation - old syllabus

Combinatorics - old syllabus derangements

  • 2016Pre-2017 syllabus (old 4-Unit-era Extension 2 course)Exam Q16(c)7 marksFive-part derangement-counting problem building a recursive formula step by step, culminating in a proof by induction of a closed-form summation formula. Old syllabus era; standalone derangement combinatorics sits outside the current 5-topic list.Official paper ↗Examiner report ↗Work through it with a tutor →
Mechanics · 9 mapped questions

Applications of calculus to mechanics

Circular motion - old syllabus conical surface

  • 2016Pre-2017 syllabus (old 4-Unit-era Extension 2 course)Exam Q81 markMultiple-choice: identify the correct pair of force-balance equations for an object on a rotating conical surface. Old syllabus era; this specific circular-motion-on-a-cone setup has no direct current-topic equivalent.Official paper ↗Examiner report ↗Work through it with a tutor →

Circular motion - old syllabus conical pendulum

Applications of calculus to mechanics - old syllabus inverse-square motion

  • 2016Pre-2017 syllabus (old 4-Unit-era Extension 2 course)Exam Q15(b)6 marksThree-part particle-motion problem under an inverse-square-law acceleration: derive a velocity-distance relationship, use a trigonometric substitution to find a time integral, then find the limiting time to reach the origin.Official paper ↗Examiner report ↗Work through it with a tutor →

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