Mathematics Specialist ATAR Course Year 12 Syllabus (for teaching from 2025)
Units 3 and 4 are assessed through school-based tasks (Response 40%, Investigation 20%, Examination 40% of the school mark) and one ATAR course examination that is combined with the moderated school mark. The examination has two sections sat as separate papers: Section One: Calculator-free (5 minutes reading, 50 minutes working, 35% of the examination; 8 questions and 48 marks on the 2026 cover, 5-10 questions allowed by the design brief) and Section Two: Calculator-assumed (10 minutes reading, 100 minutes working, 65%; 10 questions and 86 marks on the 2026 cover, 8-13 allowed), with a changeover of up to 15 minutes between them. All questions are compulsory written response. Section One allows standard items only; Section Two adds drawing instruments, templates, notes on two unfolded A4 sheets and up to three calculators (CAS assumed). A formula sheet is supplied for both sections. For any question or part worth more than two marks, valid working or justification is required for full marks.
Past papers on this subject span more than one syllabus. Papers written under an older one still work as practice, but the units and content areas they test have changed — the index labels every paper with the syllabus it was set under.
Year 12 syllabus for teaching from 2025 (current, 2013/28123 v15) · 2025–presentPrevious Year 12 syllabus version (before the 2025 minor changes) · 2021–2024
The units and content areas, one by one
Each area below lists the concepts named in the syllabus, what the SCSA exam asks of them, and the mistake that most often costs marks.
- Unit 3 — Topic 3.1: Complex numbers
- Unit 3 — Topic 3.2: Functions and sketching graphs
- Unit 3 — Topic 3.3: Vectors in three dimensions
- Unit 4 — Topic 4.1: Integration and applications of integration
- Unit 4 — Topic 4.2: Rates of change and differential equations
- Unit 4 — Topic 4.3: Statistical inference
Area 1 of 6
Unit 3 — Topic 3.1: Complex numbers
Moves from Cartesian to polar form: modulus and argument identities, multiplication and division as rotation and dilation, de Moivre's theorem for integral powers, loci and regions in the Argand plane, nth roots of unity and of complex numbers, and the factor, remainder and conjugate root theorems for solving real polynomial equations.
What the syllabus lists under this area · 4 points
- Modulus, argument, polar form and de Moivre's theorem for integral powers
- Addition as vector addition, multiplication as rotation and dilation, and complex-number geometry
- Loci and regions in the complex plane: circles, rays, perpendicular bisectors and |z - a| = k|z - b|
- nth roots of unity and of complex numbers; factor and remainder theorems, conjugate roots and polynomial equations
What the exam asks
Every recent paper has two or three complex-number questions across both sections: cube and fourth roots from a diagram or in cis form (2023, 2025), solving quartics and quintics with given complex roots (2023, 2024), loci sketches and descriptions without x and y (2024, 2025), and geometric questions treating complex numbers as vectors - a regular hexagon in 2025 and repeated multiplication by w in 2023 and 2025.
Where marks go missing
Treating complex numbers only algebraically. The 2025 report singled out weak vector interpretation (Arg(z5 - z3) is the direction from z3 to z5), and the 2023 and 2024 reports flagged expansion errors in polynomial work. Use principal arguments in (-pi, pi].
13 real SCSA questions indexed on this area →
Area 2 of 6
Unit 3 — Topic 3.2: Functions and sketching graphs
Composition of functions and when it is defined, one-to-one functions and inverses with the reflection property, absolute value, the graphs of 1/f(x), |f(x)| and f(|x|), and sketching rational functions of low degree with vertical, horizontal and oblique asymptotes.
What the syllabus lists under this area · 3 points
- Composition of functions: when f(g(x)) is defined, its domain and range
- One-to-one functions, inverse functions and the reflection property
- Absolute value, y = 1/f(x), y = |f(x)|, y = f(|x|) and rational functions with vertical, horizontal and oblique asymptotes
What the exam asks
A calculator-free question built on printed graphs appears every year: inverse and composite sketches, the domain of a composite, and transformations such as f(-|x|). Section Two adds defining a rational function from its graph (2025 Q9 and 2023 Q18) and non-standard absolute-value families (2024 W-graphs).
Where marks go missing
The domain of a composite function was the most reported weakness in 2023, 2024 and 2025: f(g(x)) needs the range of g inside the domain of f, and the answer is a set of x values, not the range.
9 real SCSA questions indexed on this area →
Area 3 of 6
Unit 3 — Topic 3.3: Vectors in three dimensions
3D vector algebra and proof, spheres, vector equations of lines, segments, curves and planes (normals from the cross product), systems of three linear equations with their three solution cases and geometric meaning, and vector calculus for motion including projectile and circular motion.
What the syllabus lists under this area · 4 points
- 3D vector algebra, scalar product and vector proofs of geometric results
- Lines, line segments, spheres and parametric curves; do the paths cross or do the particles meet?
- Cross product, planes and systems of three linear equations with their geometric interpretation
- Vector functions of time: velocity, acceleration, distance travelled, projectile and circular motion
What the exam asks
The largest share of marks: a system of equations with a parameter (calculator-free, 2024 and 2025), line-plane-sphere work, a 3D proof (2025 pyramid), and a vector-motion question asking for a Cartesian path, path length, speed or closest approach (2023-2025).
Where marks go missing
Confusing distance travelled (the integral of speed |r'(t)|) with change in displacement (the integral of r'(t)) - the 2025 report's main vector-motion weakness. For systems, describe the geometry precisely: three planes meeting in a line, versus no common point.
11 real SCSA questions indexed on this area →
Area 4 of 6
Unit 4 — Topic 4.1: Integration and applications of integration
Integration using the sin^2, cos^2 and tan^2 identities, substitution u = g(x), the integral of 1/x and f'(x)/f(x), and partial fractions; areas between curves in x or in y, volumes of solids of revolution about either axis, and numerical integration with technology.
What the syllabus lists under this area · 3 points
- Integration with trigonometric identities and the substitution u = g(x), including exact definite integrals
- Integral of 1/x, f'(x)/f(x) and partial fractions
- Areas between curves in x and in y, volumes of solids of revolution about either axis, and numerical integration
What the exam asks
Calculator-free papers always include an exact definite integral by a given substitution (well answered every year) and a partial-fractions integral; Section Two usually has a volume-of-revolution context such as a vase, torus or glass, often linked to related rates.
Where marks go missing
Setting up the volume integral is where marks go: the 2024 report flagged the torus derivation. Square the radius in terms of the right variable, and change limits when you substitute.
10 real SCSA questions indexed on this area →
Area 5 of 6
Unit 4 — Topic 4.2: Rates of change and differential equations
Implicit differentiation, related rates and the increments formula; separable first-order differential equations and slope fields; formulating growth models including, but not limited to, the logistic equation; and rectilinear motion with variable acceleration, v dv/dx and simple harmonic motion.
What the syllabus lists under this area · 4 points
- Implicit differentiation, related rates and the increments formula
- Separable differential equations and slope fields
- Formulating differential equations: exponential, logistic and other growth-rate models
- Rectilinear motion with variable acceleration, v dv/dx, and simple harmonic motion
What the exam asks
Expect a growth or DE question (logistic brumbies in 2024, a non-logistic fish growth rate in 2025), a slope field, an SHM question and a related-rates question in most papers. Show that ... by separation of variables is a regular 4-mark part.
Where marks go missing
Assuming every growth model is logistic: the 2025 report noted many candidates drew the logistic shape for a different rate equation. Read the given dP/dt, find its equilibria and the sign of the rate before sketching.
9 real SCSA questions indexed on this area →
Area 6 of 6
Unit 4 — Topic 4.3: Statistical inference
The sample mean as a random variable with mean mu and standard deviation sigma/sqrt(n), its approximate normality for large n, approximate confidence intervals x-bar +/- z s/sqrt(n) for a population mean, interval width and sample size, and what intervals do and do not tell you.
What the syllabus lists under this area · 2 points
- The sample mean as a random variable: distribution of X-bar and the central limit theorem
- Confidence intervals for a population mean: construction, sample size, width and interpretation
What the exam asks
One substantial Section Two question every year (11-15 marks): distribution and probabilities for X-bar, constructing an interval, recovering n or s from a reported interval, and two or more 'comment on this claim' parts about intervals and repeated sampling.
Where marks go missing
Paraphrasing a remembered answer instead of responding to the claim. The 2025 report asked candidates to answer the specific question; say what a confidence level means for repeated samples, not that mu is 95% likely to lie in one fixed interval.
4 real SCSA questions indexed on this area →
Common questions
Which Mathematics Specialist syllabus applies to the 2026 exam?
The Year 12 syllabus for teaching from 2025 (2013/28123 version 15, effective 1 January 2025). It made only minor wording changes from the earlier version, so the 2021-2024 papers remain good practice.
How is the exam structured?
Two sections sat as separate papers. Section One: Calculator-free has 5 minutes reading and 50 minutes working, 8 questions and 48 marks (35%). Section Two: Calculator-assumed has 10 minutes reading and 100 minutes working, 10 questions and 86 marks (65%). That is the 2026 front cover; there is a changeover of up to 15 minutes between the sections.
Why do the raw marks not match 35% and 65%?
SCSA fixes the percentage weights, and raw marks change every year: 49/92 in 2021, 48/86 in 2022, 48/89 in 2023, 47/85 in 2024, 45/93 in 2025 and 48/86 on the 2026 cover. Each section's raw score is scaled to its percentage.
What can I take into each section?
Section One: standard items only (pens, pencils, sharpener, correction fluid or tape, eraser, ruler, highlighters). Section Two: also drawing instruments, templates, notes on two unfolded A4 sheets and up to three calculators, which may include CAS. The supervisor supplies a formula sheet that you keep for both sections.
Are there multiple-choice questions?
No. Both sections are written response. The multiple-choice drills in this hub are an extra revision format, not an exam format.
How much working do I need to show?
For any question or part worth more than two marks, valid working or justification is needed for full marks, and a wrong answer without reasoning scores nothing. Examiner reports repeatedly ask for clear, logically ordered working that ends in a stated conclusion.
When is the 2026 exam?
Monday 9 November 2026, 9.20 am session, according to the SCSA 2026 Year 12 ATAR course written examinations timetable. Arrive 30 minutes before the start.