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SACE · SACE Stage 2 · official past papers

SACE Specialist Mathematics past exams 2023–2025, by year and topic

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ATARMAxxing indexes 3 official SACE Board Specialist Mathematics papers from 2023 to 2025, with 19 questions mapped to 6 topics. Every paper opens on the SACE Board website.

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Subject Outline accredited from 2017 (reissues 2017–2024) · 2017–2024Subject Outline reissued for 2025 (current) · 2025–present

SACE Specialist Mathematics exams by year: official papers & marking guidance

Past exams indexed: 2025, 2024, 2023. Open the official SACE Board papers, with marking guidance where available.

YearStudy designOfficial paper(s)Marking guidance
2025 SACE Specialist Mathematics examSubject Outline reissued for 2025 (current)Examination paper ↗Marking guidance ↗
2024 SACE Specialist Mathematics examSubject Outline accredited from 2017 (reissues 2017–2024)Examination paper ↗Marking guidance ↗
2023 SACE Specialist Mathematics examSubject Outline accredited from 2017 (reissues 2017–2024)Examination paper ↗Marking guidance ↗

3 official papers across 3 years (2023–2025), 3 with marking guidance, published by SACE Board. These span more than one study design; the most recent is Subject Outline reissued for 2025 (current). Earlier papers may cover material that has since changed.

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Topic 1: Mathematical induction · 1 mapped question

Subtopic 1.1 nth-derivative patterns proved by induction

Also in this topic: Subtopic 1.1 Proof by mathematical induction: the initial statement and the inductive step · Divisibility results proved by induction · Closed forms for sums of series proved by induction · Product formulae and nth-derivative patterns proved by induction · De Moivre's theorem proved by induction (see Subtopic 2.1) · Out of scope: proofs using inequalities or recursion formulae — question-level mapping in progress; the year table above links every official paper.

Topic 2: Complex numbers · 4 mapped questions

Subtopic 2.3 Solving z^n = c with de Moivre's theorem

  • 2025Subject Outline reissued for 2025 (current)Examination paper Q3(a)–(b)5 marksVerifies one polar solution of a fifth-degree complex equation, then uses de Moivre's theorem to establish the full set of five solutions, factors the quintic into a linear and a quartic factor, and states the quartic's solutions in exact rcis form.Official paper ↗Marking guidance ↗Check tutors for this question →

Subtopic 2.3 The nth roots of unity and their spacing

Subtopic 2.2 Loci: circles, lines, rays and regions

Subtopic 2.2 Distance between points and equations in terms of z

  • 2025Subject Outline reissued for 2025 (current)Examination paper Q6(c)–(e)6 marksWrites the equation of a general member of a doubling family of circles in terms of z, shows by a distance argument that a given complex number lies inside one of them, and finds with reasoning the smallest positive integer index for which a second complex number lies inside.Official paper ↗Marking guidance ↗Check tutors for this question →

Also in this topic: Subtopic 2.1 Cartesian form, real and imaginary parts, and Cartesian arithmetic · Modulus, argument and polar (rcis) form; conversion both ways · Multiplication and division in polar form as dilation and rotation · De Moivre's theorem, including negative and rational indices · Subtopic 2.2 The complex (Argand) plane: addition as vector addition, multiplication as a linear transformation, multiplication by i as a right-angle rotation · Distance between points, the triangle inequality, and loci: circles, lines, rays and regions · Subtopic 2.3 Roots of complex numbers: solving z^n = c and the nth roots of unity · Subtopic 2.4 Polynomial division, the factor and remainder theorems, conjugate roots and real quadratic factors · Solving real cubic and quartic equations and the fundamental theorem of algebra — question-level mapping in progress; the year table above links every official paper.

Topic 3: Functions and sketching graphs · 1 mapped question

Subtopic 3.2 Inverse functions and symmetry about y = x

  • 2025Subject Outline reissued for 2025 (current)Examination paper Q7(b)–(d)6 marksSketches the inverse of a restricted trigonometric-polynomial function on supplied axes, identifies from four options how a second supplied graph relates to the original function, and finds the exact shaded area of a symmetric region formed by the function, its inverse and their reflections inside a square.Official paper ↗Marking guidance ↗Check tutors for this question →

Also in this topic: Subtopic 3.1 Composition of functions and when a composition is defined · Subtopic 3.2 One-to-one functions, the horizontal line test, inverse functions and symmetry about y = x · Subtopic 3.3 The absolute value function and its properties · Compositions involving absolute values and reciprocals of linear, quadratic and trigonometric functions · Graphs of rational functions (numerator up to degree 3, denominator up to degree 2) and their asymptotes — question-level mapping in progress; the year table above links every official paper.

Topic 4: Vectors in three dimensions · 4 mapped questions

Subtopic 4.2 Vector (cross) product and areas

  • 2025Subject Outline reissued for 2025 (current)Examination paper Q5(a)–(b)6 marksEstablishes a cross-product identity for a combination of two vectors, then uses a labelled figure to express one displacement in terms of the two base vectors and to show exact expressions for the area of a triangle and of a quadrilateral in terms of the magnitude of their cross product.Official paper ↗Marking guidance ↗Check tutors for this question →

Subtopic 4.1 Component form and magnitude in three dimensions

Subtopic 4.2 Planes, distances and normals

  • 2025Subject Outline reissued for 2025 (current)Examination paper Q8(a)–(b)9 marksShows a stated point lies on a general plane and derives the standard formula for the distance between two parallel planes, then applies it to two specific parallel planes, finds the normal line through a given point and shows where that normal meets the second plane.Official paper ↗Marking guidance ↗Check tutors for this question →

Subtopic 4.2 Planes and vector methods of proof

  • 2025Subject Outline reissued for 2025 (current)Examination paper Q8(c)–(d)6 marksFinds a third parallel plane from a stated ratio of distances, explains by the triangle inequality why a sum of two displacement magnitudes is at least the magnitude of a third, and finds the coordinates of the point where equality holds.Official paper ↗Marking guidance ↗Check tutors for this question →

Also in this topic: Subtopic 4.1 Vectors in space, component form and the unit vectors i, j and k · Subtopic 4.2 Cartesian coordinates, spheres and simple planes · Vector, parametric and Cartesian equations of a line; parallel, perpendicular and skew lines; closest point and angle between lines · Two particles as vector functions of time: crossing paths versus meeting · Scalar (dot) product and vector (cross) product, areas, and the cross product by determinant · The Cartesian equation of a plane, line–plane intersection, angles, and point-to-plane distance · Vector methods of proof in two and three dimensions · Subtopic 4.3 Systems of linear equations, row operations on an augmented matrix, and the geometry of intersecting planes — question-level mapping in progress; the year table above links every official paper.

Topic 5: Integration techniques and applications · 3 mapped questions

Subtopic 5.1 Integration by parts

Subtopic 5.2 Volumes of solids of revolution about the x-axis

Subtopic 5.1 Trigonometric identities and substitution

Also in this topic: Subtopic 5.1 Trigonometric identities for squared functions and substitution for composite integrands · Inverse trigonometric functions, their principal domains and their derivatives · Integrals producing arcsine and arctangent forms · Partial fractions for rational integrands in simple cases · Integration by parts · Subtopic 5.2 Areas between curves determined by functions · Volumes of solids of revolution about the x-axis and about the y-axis — question-level mapping in progress; the year table above links every official paper.

Topic 6: Rates of change and differential equations · 6 mapped questions

Subtopic 6.2 Related rates linked by the chain rule

  • 2025Subject Outline reissued for 2025 (current)Examination paper Q1(a)–(c)6 marksA flask fills at a constant volumetric rate with the volume given as a cubic-type function of the liquid height; the parts ask for the height at a stated volume, a 'show that' rearrangement giving dh/dt in terms of h and dV/dt, and then the rate of change of height at that instant to three significant figures.Official paper ↗Marking guidance ↗Check tutors for this question →

Subtopic 6.4 Parametric curves and their sketches

  • 2025Subject Outline reissued for 2025 (current)Examination paper Q9(a)–(b)5 marksShows an antiderivative for a square-rooted expression in one-minus-cosine using the double-angle formula, then sketches on supplied axes the parametric path of a pendulum wrapping around a fixed structure over a stated parameter interval.Official paper ↗Marking guidance ↗Check tutors for this question →

Subtopic 6.4 Velocity components and arc length of a parametric curve

  • 2025Subject Outline reissued for 2025 (current)Examination paper Q9(c)–(d)9 marksDifferentiates the general pendulum parameterisation, shows the arc-length integral reduces to a stated form, uses the earlier antiderivative to find the integer constant relating arc length to string length, and states the arc length for a particular string length.Official paper ↗Marking guidance ↗Check tutors for this question →

Subtopic 6.2 Formulating and solving differential equations

  • 2025Subject Outline reissued for 2025 (current)Examination paper Q10(a)–(c)6 marksSets up a chemical-reaction model: integrates a simple proportional rate to relate one unreacted mass to the mass of product, substitutes both unreacted masses into a product rate law to show a stated quadratic differential equation, and verifies a partial-fraction identity for its right-hand side.Official paper ↗Marking guidance ↗Check tutors for this question →

Subtopic 6.2 Separation of variables and limiting values

  • 2025Subject Outline reissued for 2025 (current)Examination paper Q10(d)8 marksSeparates the variables, integrates using the verified partial fractions, applies the stated initial condition to show a given closed-form solution, rearranges it to a simpler equivalent form and states the limiting value of the product mass as time increases.Official paper ↗Marking guidance ↗Check tutors for this question →

Subtopic 6.2 Slope fields and solution curves

Also in this topic: Subtopic 6.1 Implicit differentiation, including justifying the derivative of the logarithm function · Differential equations of the form dy/dx = f(x) and dy/dx = g(y), and separation of variables · Slope fields and reconstructing a solution curve from an initial value · Modelling: exponential change, Newton's law of cooling, spread of rumours and the logistic differential equation · Subtopic 6.3 Polynomial parameterisations: uniform motion, the vector form of a line, and objects in free flight · Subtopic 6.4 Cartesian path by eliminating the parameter; velocity vector, tangency, speed and arc length in two and three dimensions · Subtopic 6.5 Trigonometric parameterisations, circular motion, centripetal acceleration and non-circular trigonometric paths — question-level mapping in progress; the year table above links every official paper.

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