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HSC · HSC Year 12 · official past papers

HSC Mathematics Extension 2 past exams 2015–2025, by year and topic

The official NESA exams, marking guides and examiner reports 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.

Index last synced 19 August 2026.

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ATARMAxxing indexes 22 official NESA Mathematics Extension 2 papers from 2015 to 2025, with 63 questions mapped to 5 topics. Every paper opens on the NESA website.

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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

HSC Mathematics Extension 2 exams by year: official papers & marking guidance

Past exams indexed: 2025, 2024, 2023, 2022, 2021, 2020, 2019, 2018, 2017, 2016, 2015. Open the official NESA papers, with marking guidance where available.

Year	Study design	Official paper(s)	Marking guidance
2025 HSC Mathematics Extension 2 exam	SD 2020-2026 (Mathematics Extension 2 Stage 6 Syllabus, 2017)	Exam ↗ · Marking Guidelines ↗	Marking guidance ↗
2024 HSC Mathematics Extension 2 exam	SD 2020-2026 (Mathematics Extension 2 Stage 6 Syllabus, 2017)	Exam ↗ · Marking Guidelines ↗	Marking guidance ↗
2023 HSC Mathematics Extension 2 exam	SD 2020-2026 (Mathematics Extension 2 Stage 6 Syllabus, 2017)	Exam ↗ · Marking Guidelines ↗	Marking guidance ↗
2022 HSC Mathematics Extension 2 exam	SD 2020-2026 (Mathematics Extension 2 Stage 6 Syllabus, 2017)	Exam ↗ · Marking Guidelines ↗	Marking guidance ↗
2021 HSC Mathematics Extension 2 exam	SD 2020-2026 (Mathematics Extension 2 Stage 6 Syllabus, 2017)	Exam ↗ · Marking Guidelines ↗	Marking guidance ↗
2020 HSC Mathematics Extension 2 exam	SD 2020-2026 (Mathematics Extension 2 Stage 6 Syllabus, 2017) - first year examined under this syllabus	Exam ↗ · Marking Guidelines ↗	Marking guidance ↗
2019 HSC Mathematics Extension 2 exam	Pre-2017 syllabus (final year of the old 4-Unit-era Extension 2 course)	Exam (archive) ↗ · Marking Guidelines (archive) ↗	Marking guidance ↗
2018 HSC Mathematics Extension 2 exam	Pre-2017 syllabus (old 4-Unit-era Extension 2 course)	Exam (archive) ↗ · Marking Guidelines (archive) ↗	Marking guidance ↗
2017 HSC Mathematics Extension 2 exam	Pre-2017 syllabus (old 4-Unit-era Extension 2 course)	Exam (archive) ↗ · Marking Guidelines (archive) ↗	Marking guidance ↗
2016 HSC Mathematics Extension 2 exam	Pre-2017 syllabus (old 4-Unit-era Extension 2 course)	Exam (archive) ↗ · Marking Guidelines (archive) ↗	Marking guidance ↗
2015 HSC Mathematics Extension 2 exam	Pre-2017 syllabus (old 4-Unit-era Extension 2 course)	Exam (archive) ↗ · Marking Guidelines (archive) ↗	Marking guidance ↗

22 official papers across 11 years (2015–2025), 11 with marking guidance, published by NESA. These span more than one study design; the most recent is Mathematics Extension 2 11-12 Syllabus (2024). Earlier papers may cover material that has since changed.

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Proof · 15 mapped questions

The nature of proof

2025SD 2020-2026 (Mathematics Extension 2 Stage 6 Syllabus, 2017)Exam Q21 markMultiple-choice: state the correct logical negation of an existential quantifier statement about integers.Official paper ↗Marking guidance ↗Check tutors for this question →
2025SD 2020-2026 (Mathematics Extension 2 Stage 6 Syllabus, 2017)Exam Q51 markMultiple-choice: identify the correct contrapositive of a given conditional statement.Official paper ↗Marking guidance ↗Check tutors for this question →
2025SD 2020-2026 (Mathematics Extension 2 Stage 6 Syllabus, 2017)Exam Q11(e)2 marksProve an inequality involving sums of square roots by contradiction.Official paper ↗Marking guidance ↗Check tutors for this question →
2025SD 2020-2026 (Mathematics Extension 2 Stage 6 Syllabus, 2017)Exam Q13(c)4 marksTwo-part inequality proof: establish the AM-GM inequality for two positive reals, then apply it to prove a related fraction inequality.Official paper ↗Marking guidance ↗Check tutors for this question →
2025SD 2020-2026 (Mathematics Extension 2 Stage 6 Syllabus, 2017)Exam Q14(d)3 marksProve an inequality about four positive reals in arithmetic-sequence-of-reciprocals relationship.Official paper ↗Marking guidance ↗Check tutors for this question →

Further proof by mathematical induction

2025SD 2020-2026 (Mathematics Extension 2 Stage 6 Syllabus, 2017)Exam Q12(b)3 marksUse mathematical induction to prove a formula for the nth derivative of a function of the form x times an exponential.Official paper ↗Marking guidance ↗Check tutors for this question →

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

2016Pre-2017 syllabus (old 4-Unit-era Extension 2 course)Exam Q21 markMultiple-choice: identify which polynomial has a repeated root at a given value. Old syllabus era; polynomial multiple-root testing has no direct current-topic equivalent within the current 5-topic list.Official paper ↗Marking guidance ↗Check tutors for this question →

Conic sections - old syllabus eccentricity

2016Pre-2017 syllabus (old 4-Unit-era Extension 2 course)Exam Q31 markMultiple-choice: given a sum of eccentricities, determine which pair of conic types is consistent. Old syllabus era; conics are not part of the current Extension 2 syllabus.Official paper ↗Marking guidance ↗Check tutors for this question →

Conic sections - old syllabus hyperbola transformation

2016Pre-2017 syllabus (old 4-Unit-era Extension 2 course)Exam Q71 markMultiple-choice: find the constant in a rotated-axes hyperbola equation equivalent to a given rectangular hyperbola. Old syllabus era; conics are not part of the current Extension 2 syllabus.Official paper ↗Marking guidance ↗Check tutors for this question →

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 ↗Marking guidance ↗Check tutors for this question →

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 ↗Marking guidance ↗Check tutors for this question →

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 ↗Marking guidance ↗Check tutors for this question →

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

2016Pre-2017 syllabus (old 4-Unit-era Extension 2 course)Exam Q13(d)4 marksTwo-part cubic-polynomial problem: deduce a single-real-root condition from a discriminant-like inequality, then determine root multiplicity in a boundary case.Official paper ↗Marking guidance ↗Check tutors for this question →

Inequality proof - old syllabus algebraic inequality

2016Pre-2017 syllabus (old 4-Unit-era Extension 2 course)Exam Q14(c)3 marksProve an algebraic inequality involving square roots of a variable for all non-negative values.Official paper ↗Marking guidance ↗Check tutors for this question →

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

2016Pre-2017 syllabus (old 4-Unit-era Extension 2 course)Exam Q15(a)2 marksFind a cubic equation with integer coefficients whose roots are the squares of the roots of a given cubic.Official paper ↗Marking guidance ↗Check tutors for this question →

Vectors · 6 mapped questions

Further work with vectors

2025SD 2020-2026 (Mathematics Extension 2 Stage 6 Syllabus, 2017)Exam Q11 markMultiple-choice: identify the correct vector equation of a line through two given points.Official paper ↗Marking guidance ↗Check tutors for this question →
2025SD 2020-2026 (Mathematics Extension 2 Stage 6 Syllabus, 2017)Exam Q101 markMultiple-choice: identify which vector-valued parametrisation traces the same curve as a given one.Official paper ↗Marking guidance ↗Check tutors for this question →
2025SD 2020-2026 (Mathematics Extension 2 Stage 6 Syllabus, 2017)Exam Q11(d)3 marksThree-part force problem: verify a force vector from magnitude and direction, find a resultant of two forces, then compute a dot product of the resultant with a displacement vector.Official paper ↗Marking guidance ↗Check tutors for this question →
2025SD 2020-2026 (Mathematics Extension 2 Stage 6 Syllabus, 2017)Exam Q13(b)4 marksFind all unit vectors perpendicular to two given 3D vectors using the cross-product condition.Official paper ↗Marking guidance ↗Check tutors for this question →
2025SD 2020-2026 (Mathematics Extension 2 Stage 6 Syllabus, 2017)Exam Q15(a)4 marksUse the cross product to find the area of a parallelogram defined by two given 3D vectors.Official paper ↗Marking guidance ↗Check tutors for this question →
2025SD 2020-2026 (Mathematics Extension 2 Stage 6 Syllabus, 2017)Exam Q16(c)6 marksThree-part optimisation problem: minimise the distance from a point to a parametrised line, prove a perpendicularity result, then find the shortest distance from the line to a sphere.Official paper ↗Marking guidance ↗Check tutors for this question →

Complex Numbers · 17 mapped questions

Using complex numbers

2025SD 2020-2026 (Mathematics Extension 2 Stage 6 Syllabus, 2017)Exam Q31 markMultiple-choice: find the square roots of a given complex number.Official paper ↗Marking guidance ↗Check tutors for this question →
2025SD 2020-2026 (Mathematics Extension 2 Stage 6 Syllabus, 2017)Exam Q61 markMultiple-choice: given two unit-modulus complex numbers and the modulus of their sum, find the modulus of their difference.Official paper ↗Marking guidance ↗Check tutors for this question →
2025SD 2020-2026 (Mathematics Extension 2 Stage 6 Syllabus, 2017)Exam Q71 markMultiple-choice: read off the range of possible argument values for a translated point on the unit circle from a diagram.Official paper ↗Marking guidance ↗Check tutors for this question →
2025SD 2020-2026 (Mathematics Extension 2 Stage 6 Syllabus, 2017)Exam Q91 markMultiple-choice: given vector-style conditions on four complex numbers, classify the resulting quadrilateral.Official paper ↗Marking guidance ↗Check tutors for this question →
2025SD 2020-2026 (Mathematics Extension 2 Stage 6 Syllabus, 2017)Exam Q11(a)2 marksGiven a plotted complex number, mark the locations of its conjugate and a rotated conjugate on an Argand diagram.Official paper ↗Marking guidance ↗Check tutors for this question →
2025SD 2020-2026 (Mathematics Extension 2 Stage 6 Syllabus, 2017)Exam Q12(c)3 marksSketch the region of the complex plane satisfying a strict inequality between two distances from fixed points.Official paper ↗Marking guidance ↗Check tutors for this question →
2025SD 2020-2026 (Mathematics Extension 2 Stage 6 Syllabus, 2017)Exam Q14(c)4 marksRoots-of-unity problem: show a complex number is a 7th root of unity, then use a related quadratic to find another root and a coefficient value.Official paper ↗Marking guidance ↗Check tutors for this question →
2025SD 2020-2026 (Mathematics Extension 2 Stage 6 Syllabus, 2017)Exam Q15(c)7 marksThree-part complex-numbers problem: prove a cotangent identity, use De Moivre's theorem to find sixth roots of negative one, then apply this to solve a related equation.Official paper ↗Marking guidance ↗Check tutors for this question →
2025SD 2020-2026 (Mathematics Extension 2 Stage 6 Syllabus, 2017)Exam Q16(a)4 marksUse proof by contradiction and the triangle inequality to show all solutions of a trigonometric-complex equation lie outside a given circle.Official paper ↗Marking guidance ↗Check tutors for this question →
2016Pre-2017 syllabus (old 4-Unit-era Extension 2 course)Exam Q41 markMultiple-choice: given an Argand diagram showing two complex numbers in different quadrants, identify which transformed expression could lie in the third quadrant.Official paper ↗Marking guidance ↗Check tutors for this question →
2016Pre-2017 syllabus (old 4-Unit-era Extension 2 course)Exam Q51 markMultiple-choice: determine the rotation produced by multiplying by a given complex fraction, using its modulus-argument form.Official paper ↗Marking guidance ↗Check tutors for this question →
2016Pre-2017 syllabus (old 4-Unit-era Extension 2 course)Exam Q101 markMultiple-choice: given a reciprocal-sum condition on a variable, evaluate a related high-power reciprocal-sum expression using complex roots of unity reasoning.Official paper ↗Marking guidance ↗Check tutors for this question →
2016Pre-2017 syllabus (old 4-Unit-era Extension 2 course)Exam Q11(a)4 marksThree-part complex-numbers problem: convert a complex number to modulus-argument form, show a power of it is real, then find a power that is purely imaginary.Official paper ↗Marking guidance ↗Check tutors for this question →
2016Pre-2017 syllabus (old 4-Unit-era Extension 2 course)Exam Q12(c)3 marksUse the real part of a fourth power of a complex exponential to derive a multiple-angle cosine identity, then rewrite it purely in terms of cosine.Official paper ↗Marking guidance ↗Check tutors for this question →
2016Pre-2017 syllabus (old 4-Unit-era Extension 2 course)Exam Q16(a)5 marksTwo-part complex-numbers geometry problem: show three points satisfying a sum-to-zero condition form an equilateral triangle on the Argand diagram, then extend the result to a differently-scaled case.Official paper ↗Marking guidance ↗Check tutors for this question →
2016Pre-2017 syllabus (old 4-Unit-era Extension 2 course)Exam Q16(b)3 marksTwo-part complex-numbers problem: prove an algebraic identity for an equilateral triangle with a vertex at the origin, then give a concrete example of two complex numbers satisfying it.Official paper ↗Marking guidance ↗Check tutors for this question →

Inverse trigonometric functions - old syllabus domain/range

2016Pre-2017 syllabus (old 4-Unit-era Extension 2 course)Exam Q11(e)2 marksState the domain and range of a function built from x multiplied by an inverse sine expression.Official paper ↗Marking guidance ↗Check tutors for this question →

Also in this topic: Introduction to complex numbers — question-level mapping in progress; the year table above links every official paper.

Calculus · 16 mapped questions

Further integration

2025SD 2020-2026 (Mathematics Extension 2 Stage 6 Syllabus, 2017)Exam Q11(f)2 marksIntegrate a rational function involving a square root of a quadratic by completing the square.Official paper ↗Marking guidance ↗Check tutors for this question →
2025SD 2020-2026 (Mathematics Extension 2 Stage 6 Syllabus, 2017)Exam Q12(d)4 marksIntegrate a rational function using partial fractions with a mixed linear and irreducible-quadratic denominator.Official paper ↗Marking guidance ↗Check tutors for this question →
2025SD 2020-2026 (Mathematics Extension 2 Stage 6 Syllabus, 2017)Exam Q13(d)4 marksEvaluate a definite trigonometric integral using a given substitution that exploits symmetry of the interval.Official paper ↗Marking guidance ↗Check tutors for this question →
2025SD 2020-2026 (Mathematics Extension 2 Stage 6 Syllabus, 2017)Exam Q14(a)5 marksTwo-part reduction-formula problem: derive a recursive relationship for a cotangent-power integral, then use it to evaluate a specific case.Official paper ↗Marking guidance ↗Check tutors for this question →
2016Pre-2017 syllabus (old 4-Unit-era Extension 2 course)Exam Q11(b)3 marksIntegrate a product of x and an exponential function using integration by parts.Official paper ↗Marking guidance ↗Check tutors for this question →
2016Pre-2017 syllabus (old 4-Unit-era Extension 2 course)Exam Q12(b)3 marksUse a differentiation identity involving an integral to derive the standard antiderivative of inverse tangent.Official paper ↗Marking guidance ↗Check tutors for this question →
2016Pre-2017 syllabus (old 4-Unit-era Extension 2 course)Exam Q14(a)5 marksThree-part integration problem: verify an antiderivative for sine cubed, explain a symmetry-based zero-integral result graphically, then use both to find a volume of revolution by cylindrical shells.Official paper ↗Marking guidance ↗Check tutors for this question →
2016Pre-2017 syllabus (old 4-Unit-era Extension 2 course)Exam Q14(b)7 marksThree-part reduction-formula problem on a family of rational integrals: evaluate a base case by substitution, prove a sum relation between two members, then find a specific higher-order member.Official paper ↗Marking guidance ↗Check tutors for this question →
2016Pre-2017 syllabus (old 4-Unit-era Extension 2 course)Exam Q15(c)7 marksThree-part partial-fractions problem: verify a four-term partial-fraction decomposition, generalise the pattern to n terms with binomial-coefficient numerators, then use the result to find a limiting sum.Official paper ↗Marking guidance ↗Check tutors for this question →

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

2016Pre-2017 syllabus (old 4-Unit-era Extension 2 course)Exam Q11 markMultiple-choice: match a given cusp-shaped graph to its equation involving absolute value and square root. Old syllabus era; this graphing-technique focus has no direct current-topic equivalent.Official paper ↗Marking guidance ↗Check tutors for this question →

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 ↗Marking guidance ↗Check tutors for this question →

Volumes by integration - old syllabus solid geometry

2016Pre-2017 syllabus (old 4-Unit-era Extension 2 course)Exam Q91 markMultiple-choice: choose the correct integral expression for the volume of an irregular polyhedron from cross-sectional dimensions.Official paper ↗Marking guidance ↗Check tutors for this question →

Implicit differentiation - old syllabus

2016Pre-2017 syllabus (old 4-Unit-era Extension 2 course)Exam Q11(c)2 marksFind dy/dx for a curve defined implicitly by a cubic relation between x and y.Official paper ↗Marking guidance ↗Check tutors for this question →

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 ↗Marking guidance ↗Check tutors for this question →

Optimisation via logarithmic differentiation - old syllabus

2016Pre-2017 syllabus (old 4-Unit-era Extension 2 course)Exam Q13(a)3 marksFind the minimum point of a power-tower-style function by differentiating its logarithm.Official paper ↗Marking guidance ↗Check tutors for this question →

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 ↗Marking guidance ↗Check tutors for this question →

Mechanics · 9 mapped questions

Applications of calculus to mechanics

2025SD 2020-2026 (Mathematics Extension 2 Stage 6 Syllabus, 2017)Exam Q41 markMultiple-choice: find the amplitude of simple-harmonic-motion given a velocity-squared-versus-displacement expression.Official paper ↗Marking guidance ↗Check tutors for this question →
2025SD 2020-2026 (Mathematics Extension 2 Stage 6 Syllabus, 2017)Exam Q81 markMultiple-choice: given a velocity-displacement graph, choose the matching acceleration-displacement graph.Official paper ↗Marking guidance ↗Check tutors for this question →
2025SD 2020-2026 (Mathematics Extension 2 Stage 6 Syllabus, 2017)Exam Q12(e)3 marksResisted-motion problem: derive the stopping distance of a particle under constant and velocity-squared resistive forces.Official paper ↗Marking guidance ↗Check tutors for this question →
2025SD 2020-2026 (Mathematics Extension 2 Stage 6 Syllabus, 2017)Exam Q14(b)4 marksGiven an acceleration-displacement function for a particle, derive its velocity function and find the time to travel a given distance.Official paper ↗Marking guidance ↗Check tutors for this question →
2025SD 2020-2026 (Mathematics Extension 2 Stage 6 Syllabus, 2017)Exam Q15(b)4 marksSimple-harmonic-motion problem: given period and a speed-at-displacement condition, find the maximum positive acceleration.Official paper ↗Marking guidance ↗Check tutors for this question →
2025SD 2020-2026 (Mathematics Extension 2 Stage 6 Syllabus, 2017)Exam Q16(b)5 marksResisted projectile-motion problem: derive the Cartesian equation of motion for a particle launched into a medium with velocity-proportional resistance.Official paper ↗Marking guidance ↗Check tutors for this question →

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 ↗Marking guidance ↗Check tutors for this question →

Circular motion - old syllabus conical pendulum

2016Pre-2017 syllabus (old 4-Unit-era Extension 2 course)Exam Q13(c)4 marksTwo-part conical-pendulum-style problem: show a tension formula in terms of angular velocity, then find the angular-velocity range for one tension to exceed another.Official paper ↗Marking guidance ↗Check tutors for this question →

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 ↗Marking guidance ↗Check tutors for this question →

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