HSC Year 12 Mathematics:
Extension 1

Proof in HSC Mathematics Extension 1 introduces the formal methods of mathematical proof that are fundamental to university-level mathematics and are assessed in the HSC Extension 1 external examination — including proof by mathematical induction for results involving sums of series, divisibility, and inequalities, the structure of a valid inductive proof including the base case, inductive hypothesis, and inductive step, deductive proofs involving algebraic identities and geometric results, and proof by contradiction. Students are expected to write proofs that are logically complete, clearly structured, and expressed with precise mathematical language. The Proof topic is unique to the Extension 1 and Extension 2 courses and distinguishes these qualifications from Advanced Mathematics by developing the rigorous logical reasoning skills that are prerequisite for serious study of mathematics at university level.

Trigonometric Functions in HSC Mathematics Extension 1 extends the Advanced trigonometry content to the full Extension 1 level — including the sum and difference of angles identities for sin, cos, and tan, the double angle formulae, the products-to-sums and sums-to-products identities, the auxiliary angle method for expressing a sin x + b cos x in the form R sin(x + α) or R cos(x + α), the t-formulae expressing sin θ, cos θ, and tan θ in terms of t = tan(θ/2), solving harder trigonometric equations using these identities, and further applications of inverse trigonometric functions including their derivatives and integrals. The Trigonometric Functions content at Extension 1 level appears consistently in the HSC examination and demands that students can both identify and apply the most appropriate identity for a given situation under examination conditions.

Calculus in HSC Mathematics Extension 1 is the most extensively assessed and most demanding topic in the course — covering integration by substitution, integration of inverse trigonometric functions leading to results of the form sin⁻¹(x/a) + c and tan⁻¹(x/a) + c, volumes of solids of revolution formed by rotating a curve about the x-axis, first-order separable differential equations including forming and solving them from rates of change problems, exponential growth and decay modelled by dy/dx = ky and its solution y = Aeᴄᴕ, Newton’s law of cooling, logistic growth models, motion in a straight line using calculus including displacement, velocity, and acceleration as functions of time, simple harmonic motion and its characterisation through second-order differential equations, and applications of calculus to projectile motion. Extension 1 Calculus is the most heavily weighted topic in the HSC Extension 1 examination and requires confident fluency across all differentiation and integration techniques encountered in both the Advanced and Extension 1 courses.

Statistical Analysis in HSC Mathematics Extension 1 introduces the binomial distribution as the key discrete probability model at Extension 1 level — including the conditions for a binomial experiment, the binomial probability formula P(X = r) = ⁿCᵣ pʳ qⁿ₋ʳ, calculating probabilities for binomial random variables using the formula and technology, the mean E(X) = np and variance Var(X) = npq of a binomial distribution, and the normal approximation to the binomial distribution for large samples. Students are expected to identify whether a binomial model is appropriate for a given situation, apply the formula accurately, and interpret results in context. The Statistical Analysis topic at Extension 1 level connects the Year 11 combinatorics content directly to probability theory and appears in the HSC Extension 1 external examination as a distinct and assessable component.

Vectors in HSC Mathematics Extension 1 introduces the study of vectors in two dimensions, which is a topic unique to the Extension courses and directly prerequesite for the three-dimensional vector work in Extension 2 — including vector notation in component form and as column vectors, addition, subtraction, and scalar multiplication of vectors, the magnitude and direction of a vector, unit vectors and the standard basis vectors i and j, position vectors, displacement vectors, velocity vectors in two dimensions, the dot product of two vectors and its geometric interpretation including the angle between vectors and perpendicularity conditions, and applications of vectors to geometric proof and to the description of motion in two dimensions. Vectors appears regularly in the HSC Mathematics Extension 1 external examination and rewards students who can work fluently between geometric and algebraic representations of vector problems.