SACE Stage 2 Specialist Mathematics flashcards

32 cards. Read the term and the definition here. The deck at the top of this page is optional.

Mathematical induction

A proof technique: prove the statement for a base case, assume it for n = k, then prove it for n = k + 1; concludes the statement for all integers n ≥ base.

Strong induction

A variant where the inductive step assumes the statement for all integers from the base up to k, then proves it for k + 1; useful when the result depends on multiple previous cases.

Complex number modulus

For z = a + bi, the modulus |z| = √(a² + b²); it represents the distance of z from the origin in the complex plane.

Complex number argument

For z = a + bi, the argument arg(z) is the angle θ from the positive real axis to the vector (a, b), usually taken in (-π, π]; tan θ = b/a with quadrant adjustment.

Polar form

A complex number written as z = r(cos θ + i sin θ) = r cis θ, where r = |z| and θ = arg(z); convenient for multiplication, division, and powers.

De Moivre's theorem

For z = r cis θ and integer n, zⁿ = rⁿ cis(nθ); also gives roots: the n-th roots are r^(1/n) cis((θ + 2kπ)/n) for k = 0, 1, ..., n − 1.

Complex conjugate

For z = a + bi, the conjugate is z̄ = a − bi; properties include z z̄ = |z|² and z + z̄ = 2a, useful for division and real-coefficient polynomials.

Vector dot product

For vectors a and b, a · b = |a||b| cos θ = a₁b₁ + a₂b₂ + a₃b₃; the dot product is zero if and only if the vectors are perpendicular.

Vector cross product

For a = (a₁, a₂, a₃) and b = (b₁, b₂, b₃), a × b = (a₂b₃ − a₃b₂, a₃b₁ − a₁b₃, a₁b₂ − a₂b₁); it is perpendicular to both a and b, with |a × b| = |a||b| sin θ.

Scalar projection

The scalar projection of a onto b is (a · b)/|b|; the vector projection is ((a · b)/|b|²) b. It gives the component of a in the direction of b.

Vector equation of a line

A line can be written r = a + t d, where a is a position vector of a point on the line, d is a direction vector, and t is a real parameter.

Vector equation of a plane

A plane can be written r · n = d, where n is a normal vector and d is a constant; alternatively r = a + s u + t v using two direction vectors u and v.

Implicit differentiation

A method for finding dy/dx when y is defined implicitly by an equation F(x, y) = 0: differentiate both sides with respect to x and solve for dy/dx.

Related rates

Problems where quantities change with time and are related by an equation; differentiate the equation with respect to time using the chain rule to relate the rates.

Integration by parts

∫ u dv = uv − ∫ v du; choose u to simplify when differentiated and dv to be integrable, often for products of algebraic, exponential, or trigonometric functions.

Partial fractions

A technique to decompose a rational function into simpler fractions with linear or quadratic denominators, enabling term-by-term integration or other analysis.

Differential equation

An equation involving a function and its derivatives; solving it means finding functions that satisfy the equation, often with an initial condition to determine constants.

Separable differential equation

A first-order equation that can be written as dy/dx = f(x)g(y); solve by separating variables: ∫ (1/g(y)) dy = ∫ f(x) dx, then integrate.

Logistic differential equation

A model of limited growth: dP/dt = kP(1 − P/L), where L is the carrying capacity; solutions are logistic curves with an inflection point at P = L/2.

Simple harmonic motion

Motion described by x'' = −ω²x, with solution x = A cos(ωt + φ); the period is 2π/ω and the acceleration is proportional and opposite to displacement.

Confidence interval for mean

An interval estimate for a population mean: x̄ ± z* (σ/√n) when σ is known, or x̄ ± t* (s/√n) when σ is unknown and the sample is approximately normal.

Hypothesis test for mean

A procedure comparing a sample mean to a hypothesised population mean using a z- or t-statistic; the p-value is the probability of a result at least as extreme under the null hypothesis.

Type I and Type II errors

A Type I error rejects a true null hypothesis (probability α); a Type II error fails to reject a false null hypothesis (probability β). Power = 1 − β.

Mathematical investigation

A school-assessed task where students explore a problem, form conjectures, and justify conclusions; the report should show the process, reasoning, and verification, not only a final answer.

Skills and applications task

A school-assessed supervised task focusing on fluency and problem-solving across topics; conditions such as notes and calculator use are set by the school and documented in the assessment plan.

Performance standards

Descriptions of expected quality of learning at grades A–E; teachers use them to make judgements on assessment tasks, and they are used in moderation to ensure comparability.

Moderation

The process by which school-assessed evidence is reviewed externally to ensure judgements are consistent with the performance standards across schools.

External examination format

Typically a 3-hour written examination worth 30% of the subject grade, set and marked by the SACE Board; it assesses all topics and proficiency strands, with a formula sheet provided.

Hence or otherwise

In an examination, 'hence' requires using the previous part's result; 'hence or otherwise' allows either that method or an independent method for full marks.

Method marks

Marks awarded for correct reasoning and procedure even if the final answer is incorrect; clear working and justification are needed to earn them.

Exact values

Answers expressed in exact form, such as surds, fractions, or multiples of π, rather than decimal approximations; often required unless a question specifies otherwise.

Approved calculator

A calculator permitted in the external examination; students should be familiar with its functions for calculus, vectors, complex numbers, and statistics, and should still show working.