SACE Stage 2 Mathematical Methods cheat sheet

School assessment contributes 70% of the subject grade: Skills and Applications Tasks (SATs) plus the Mathematical Investigation; the external examination contributes the remaining 30%.

A SAT is an in-school, supervised test marked against the performance standards; it may include routine procedures, problem-solving and interpretation, and reasoning questions drawn from the topics taught.

The Mathematical Investigation is a folio-style task: formulate a question, use mathematical models, collect or generate data, and communicate findings with justification, not just a final number.

Performance standards describe five levels of achievement (A–E); grades are awarded on the quality of understanding, fluency, problem-solving and reasoning demonstrated, not on percentage of items correct alone.

Reasoning marks are earned by showing why a result follows - stating the rule or theorem used, showing key algebraic steps, and interpreting the answer in context.

Fluency marks are earned by accurate, efficient execution: correct derivative, integral, or probability value with appropriate units and sensible rounding.

Solutions to equations must be checked against the domain of the model; for example, logarithmic equations require arguments to be positive, and trigonometric equations have infinitely many solutions unless a domain is stated.

When a question gives a real-world context, define variables and units explicitly before calculating; final answers without interpretation typically lose communication marks.

The external examination is a single 3-hour paper worth 30% of the subject grade; a scientific calculator and the provided formula sheet are permitted, and questions cover the whole syllabus.

Examination questions are structured in three styles: routine procedures, problem-solving tasks involving unfamiliar contexts, and reasoning tasks requiring proof or justification.

For any function f, the derivative from first principles is f'(x) = lim(h→0)[f(x+h) − f(x)]/h; the limit definition underpins all differentiation rules.

Chain rule: d/dx f(g(x)) = f'(g(x))·g'(x). Product rule: (uv)' = u'v + uv'. Quotient rule: (u/v)' = (u'v − uv')/v².

Integration by recognition: ∫ f'(x)·g(f(x)) dx = G(f(x)) + C, where G is an antiderivative of g; substitution reverses the chain rule.

Definite integrals: ∫ₐᵇ f(x) dx = F(b) − F(a); the area between curves is ∫ₐᵇ |f(x) − g(x)| dx, with intersection points found first.

Normal distribution: standardise with z = (x − μ)/σ, then use the standard normal table or calculator; the inverse normal finds x given a probability.

Binomial distribution: P(X = k) = C(n,k)·pᵏ·(1−p)^(n−k), with mean np and variance np(1−p); use it for a fixed number of independent trials with constant success probability.

In a reasoning question, write a logical chain: given information, the theorem or identity applied, the algebraic manipulation, and the conclusion with a statement of what has been proven.