Stage 2 Specialist Mathematics Subject Outline
Stage 2 Specialist Mathematics is 70% school assessment and 30% external assessment. School assessment is Assessment Type 1: Skills and Applications Tasks (50%, five or six supervised tasks) and Assessment Type 2: Mathematical Investigation (20%, one report of at most 12 single-sided A4 pages). Assessment Type 3 is the examination (30%): a 130-minute written paper based on the key questions and key concepts in all six topics. It is printed as two question booklets rather than lettered sections — Question booklet 1 is Questions 1 to 7 worth 55 marks and Question booklet 2 is Questions 8 to 10 worth 45 marks, 100 marks in total, each booklet headed 'Allow approximately 65 minutes'. Every question is compulsory; there is no choice and no multiple choice. The 2023, 2024 and 2025 papers all use exactly this shape, with booklet 1 questions worth 5 to 10 marks and booklet 2 questions worth 14 to 16 marks. A formula sheet is supplied as the final page of the paper, you may take two unfolded A4 sheets (four sides) of handwritten notes into the room, and approved calculators may be used (no computer algebra systems). The paper prints a single 'Total time: 130 minutes' with no separate reading period, and asks for answers to three significant figures unless a question says otherwise. School and external components are combined into one A+ to E- grade.
Past papers on this subject span more than one subject outline. Papers written under an older one still work as practice, but the topics they test have changed — the index labels every paper with the subject outline it was set under.
Subject Outline accredited from 2017 (reissues 2017–2024) · 2017–2024Subject Outline reissued for 2025 (current) · 2025–present
The topics, one by one
Each area below lists the concepts named in the subject outline, what the SACE Board exam asks of them, and the mistake that most often costs marks.
- Topic 1: Mathematical induction
- Topic 2: Complex numbers
- Topic 3: Functions and sketching graphs
- Topic 4: Vectors in three dimensions
- Topic 5: Integration techniques and applications
- Topic 6: Rates of change and differential equations
Area 1 of 6
Topic 1: Mathematical induction
One subtopic, but it sets the standard of proof for the whole subject. You state the proposition P(n), verify the initial statement, assume P(k), derive P(k+1) from that assumption, and close with the conclusion for all positive integers n. The outline's examples are divisibility results, closed forms for sums of series, product formulae and nth-derivative patterns, and induction reappears inside other topics — proving de Moivre's theorem in 2.1 and extending the triangle inequality in 2.2.
What the subject outline lists under this area · 6 points
- 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
What the exam asks
Usually one five-to-seven-mark part in booklet 1, often after an easier derivative or algebra part whose result the induction then uses. The 2025 paper differentiated a linear-times-exponential function and then asked for an induction proof of its nth derivative; the 2024 paper set an induction proof on a complex-number identity.
Where marks go missing
Losing marks on setting out rather than on mathematics. The proposition must be stated before P(1), P(k) and P(k+1) are used; 'P(1) is true' needs the working that justifies it; the (k+1)th derivative must be written as the derivative of the kth; and the final statement earns its mark only if the proposition was stated and the inductive step genuinely used the assumption. Induction proofs using inequalities or recursion formulae are not part of this subject.
1 real SACE Board question indexed on this area →
Area 2 of 6
Topic 2: Complex numbers
Four subtopics. Subtopic 2.1 moves from Cartesian arithmetic to modulus, argument and rcis form, and to de Moivre's theorem including negative and rational indices. Subtopic 2.2 reads the plane geometrically: addition as vector addition, multiplication as dilation and rotation, multiplication by i as a right-angle turn, the modulus of a difference as a distance, the triangle inequality, and loci — circles, lines, rays and the regions they bound — converted to and from Cartesian form. Subtopic 2.3 solves z^n = c and places the roots of unity symmetrically on a circle whose sum is zero. Subtopic 2.4 returns to real polynomials: division, the factor and remainder theorems, conjugate pairs giving a real quadratic factor, and the fundamental theorem of algebra.
What the subject outline lists under this area · 9 points
- 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
What the exam asks
Typically two booklet-1 questions. The 2025 paper set the five solutions of a quintic in exact rcis form and used their symmetry to find exact polygon areas, and separately a circle-and-ray question that asked for a family of circles written in terms of z and for reasoning about whether given complex numbers lay inside them. The 2024 paper asked for a real quadratic factor from a supplied zero, and for roots forming a regular hexagon.
Where marks go missing
Complex loci. The 2025 Subject Assessment Advice names the circle-and-ray question as the hardest in the paper: a ray starts at its point and excludes it, and a question asking for an equation 'in terms of z' is not answered with a Cartesian equation. Elsewhere, omit brackets around the argument in rcis form and the working collapses, and a conjugate pair must be multiplied out to a real quadratic before division.
4 real SACE Board questions indexed on this area →
Area 3 of 6
Topic 3: Functions and sketching graphs
Three subtopics. Composition is defined exactly where the inner output lands in the outer domain, so finding the domain is an inequality to solve, not an afterthought. One-to-one functions, tested by the horizontal line test, are the only ones with inverses; a restriction may be needed first, and the inverse is the reflection in y = x with domain and range swapped. Sketching covers the absolute value function and the compositions built from a given f: |f(x)|, f(|x|), 1/f(x), 1/|f(x)| and |1/f(x)|, plus rational functions with numerator up to degree 3 and denominator up to degree 2, including oblique asymptotes.
What the subject outline lists under this area · 5 points
- 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
What the exam asks
Usually one booklet-1 question, often fused with integration: the 2025 paper asked for the inverse of a restricted function sketched on supplied axes, then a multiple-choice-style 'circle the correct option' identification of a transformed graph, then an exact shaded area exploiting the symmetry of the function and its inverse. The 2024 paper asked for a curve to be drawn, an inverse to be shown, and its exact domain stated.
Where marks go missing
Sketches that ignore the restriction. Adding arrows to a restricted inverse, drawing a broken line, or failing to mark the corner where an absolute value folds all cost marks, as does stating an inverse's domain without the exact endpoints. For 1/f(x), the zeros of f become the vertical asymptotes and the points where f equals one are fixed — sketch those anchors before the curve.
1 real SACE Board question indexed on this area →
Area 4 of 6
Topic 4: Vectors in three dimensions
Three subtopics and the biggest single block of marks. Subtopic 4.1 extends component form, magnitude and unit vectors to space. Subtopic 4.2 carries lines in vector, parametric and Cartesian form with parallel, perpendicular and skew cases, closest points and angles; particles as vector functions of time, where crossing paths and meeting are different questions; the dot and cross products with their perpendicularity and area meanings and the determinant evaluation; the Cartesian equation of a plane from a normal and a point, line–plane intersection, angles, the point-to-plane distance formula; and vector proof of geometric results. Subtopic 4.3 solves systems of linear equations by row operations on an augmented matrix and reads the three cases as configurations of planes.
What the subject outline lists under this area · 8 points
- 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
What the exam asks
Almost always the long opening question of booklet 2, plus a shorter cross-product question in booklet 1. The 2025 booklet-2 vectors question derived the distance between parallel planes, applied it, found a normal line and where it met a second plane, built a third plane from a distance ratio, and closed with a triangle-inequality argument and the coordinates where equality holds. The 2024 paper used row operations to show three planes meet along a line, and a vector proof using perpendicular medians.
Where marks go missing
Notation and logic. A vector needs a tilde or an arrow, the cross product needs its own symbol, and a 'show that' on a plane must substitute the point into one side rather than setting both sides equal from the start. When a later part gives you the answer, validate the parameter in both the line and the plane rather than working backwards from the printed point.
4 real SACE Board questions indexed on this area →
Area 5 of 6
Topic 5: Integration techniques and applications
Two subtopics. Subtopic 5.1 widens the class of integrable functions: identities for squared trigonometric functions, substitution for composite integrands, the inverse trigonometric functions with their principal domains and derivatives obtained implicitly, the standard arcsine and arctangent integrals, partial fractions in simple cases, and integration by parts set out in the form printed on the formula sheet. Subtopic 5.2 applies them to the area between two curves and to volumes of solids of revolution about either axis, the y-axis case requiring a one-to-one positive function.
What the subject outline lists under this area · 7 points
- 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
What the exam asks
Spread through both booklets and frequently fused with another topic. The 2025 paper used a supplied antiderivative and integration by parts to establish a result, then rotated a graph about the x-axis for an exact volume; later it turned a cubed sine into an integrable form and used that exact value again in a symmetry-based area. The 2024 paper set an exact area bounded by two curves after a 'show that' step.
Where marks go missing
Notation and exactness. Omitting the dx or dt is not logically correct and is flagged every year; integration by parts must follow the formula-sheet structure rather than an ad hoc split; limits must be in the right order after a substitution; and an exact answer stays as a surd or a multiple of pi. When an earlier part gives an antiderivative, the later part is meant to use it.
3 real SACE Board questions indexed on this area →
Area 6 of 6
Topic 6: Rates of change and differential equations
Five subtopics and the subject's payoff. Implicit differentiation (6.1) finds gradients without an explicit rule and justifies the derivative of the logarithm. Subtopic 6.2 covers related rates through the chain rule, differential equations of the forms dy/dx = f(x) and dy/dx = g(y), separation of variables, slope fields and solution curves, and the standard models — exponential change, Newton's law of cooling, spread of rumours, and the logistic equation with its limiting value. Subtopics 6.3 to 6.5 treat a moving point (x(t), y(t)): polynomial components giving lines and free flight, the velocity vector as tangent with speed as its magnitude, arc length in two and three dimensions, and trigonometric parameterisations including circular motion with centripetal acceleration.
What the subject outline lists under this area · 8 points
- Subtopic 6.1 Implicit differentiation, including justifying the derivative of the logarithm function
- Subtopic 6.2 Related rates linked by the chain rule
- 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
What the exam asks
Both a short booklet-1 related-rates opener and the long final question of booklet 2. The 2025 paper opened with a flask filling at a constant volumetric rate, and finished with a chemical-reaction model: a simple integration, a quadratic differential equation set up from a product rate law, a verified partial-fraction identity, separation of variables to a printed closed form, the limiting value, and a solution curve drawn on a supplied slope field. The 2025 middle question of booklet 2 set a pendulum's parametric path, its sketch, and its arc-length integral.
Where marks go missing
Not seeing that the parts are a chain. Later parts say 'hence' or 'using part (a)' because the earlier antiderivative or identity is the intended tool; students who restart from scratch usually run out of time or algebra. Also: answer in the variable the question uses, keep the constant of integration until the initial condition fixes it, and when drawing on a slope field follow the printed direction segments rather than sketching a remembered curve shape.
6 real SACE Board questions indexed on this area →
Common questions
Are the practice papers official past exams?
No. They are original ATARMAxxing questions with fully worked responses and mark-by-mark guides. The official 2023, 2024 and 2025 examination papers and the Subject Assessment Advice are linked separately, and the platform is not affiliated with the SACE Board or SATAC.
What does the real paper look like?
Two question booklets, no sections and no multiple choice. Question booklet 1 is Questions 1 to 7 worth 55 marks; Question booklet 2 is Questions 8 to 10 worth 45 marks. Total 100 marks in 130 minutes, with about 65 minutes suggested per booklet, and every question compulsory. Booklet 1 questions run 5 to 10 marks and booklet 2 questions 14 to 16 marks in each of the last three papers.
What may I take into the examination?
A formula sheet is supplied as the final page of the paper. The subject outline allows two unfolded A4 sheets (four sides) of handwritten notes. Approved calculators may be used — approved graphics calculators or a scientific calculator without external memory, with computer algebra systems not permitted — and the paper prints a box for up to two. Black or blue pen, and a sharp dark pencil for diagrams.
How should I write a 'show that' answer?
Start from one side of the statement and work towards the other. Do not write the two sides as an equation and manipulate both together — the markers treat that as not logically valid. Because the answer is printed in the question, the marks are entirely for the working steps that reach it, so every line of logic has to be there, with brackets and with dx or dt written in any integral.
Is there a separate reading time?
No. The papers print a single 'Total time: 130 minutes', and the SACE Board's information sheet on examination lengths confirms that the published length already absorbs the ten minutes formerly called reading time. The 2026 timetable places Specialist Mathematics [2MSC20] on Tuesday 10 November 2026 in the 9 am South Australian session for 130 minutes.
Are all six topics examinable?
Yes. The subject outline states that the examination is based on the key questions and key concepts in the six topics, and that the considerations for developing teaching and learning strategies may supply contexts for questions. Recent papers draw on every topic, and single questions routinely interleave two or three of them.