HSC Year 12 Mathematics:
Advanced

Functions in HSC Mathematics Advanced builds on the Year 11 Preliminary content to the full depth required for the HSC external examination — including further graphical transformations and curve sketching, graphical analysis of polynomial, rational, and exponential functions, logarithmic functions and their graphs including the natural logarithm ln x, the relationship between exponential and logarithmic functions as inverses, solving equations involving exponential and logarithmic expressions, and applying functions to model and interpret real-world situations involving growth, decay, and other phenomena. Functions in HSC Mathematics Advanced connects directly to the Calculus topic through the differentiation and integration of exponential and logarithmic functions, and students are expected to demonstrate fluency with graphical and algebraic representations throughout the HSC examination.

Trigonometric Functions in HSC Mathematics Advanced extends the Year 11 content to the full HSC level — including further work with the unit circle, the Pythagorean identity and related identities, solving trigonometric equations over specified domains, graphing transformations of trigonometric functions including changes to amplitude, period, and phase, differentiating trigonometric functions sin x, cos x, and tan x and their composites using the chain rule, integrating trigonometric functions including sin x and cos x, and applying trigonometric functions and their calculus to model and interpret periodic real-world phenomena. Trigonometric Functions is a consistently assessed area of the HSC Mathematics Advanced external examination and requires students to move fluently between graphical, algebraic, and applied representations of trigonometric relationships.

Calculus in HSC Mathematics Advanced is the most extensively assessed topic in the entire course — covering differentiation of polynomials, exponential functions, logarithmic functions, and trigonometric functions using the product rule, quotient rule, and chain rule, second derivatives and their role in identifying concavity and points of inflection, curve sketching using first and second derivative analysis, optimisation problems in applied contexts, integration as the reverse of differentiation, the definite integral and the calculation of areas under and between curves, integration of polynomial, exponential, trigonometric, and selected rational functions, and the trapezoidal rule as a numerical integration technique. Calculus accounts for a significant proportion of the HSC Mathematics Advanced external examination and demands both procedural fluency and the ability to apply differentiation and integration to unfamiliar, multi-step problems in context.

Financial Mathematics in HSC Mathematics Advanced covers the financial content that connects algebraic and geometric sequences to practical financial applications — including arithmetic sequences and series and their application to simple interest problems, geometric sequences and series and their application to compound interest problems, the future value and present value of lump sum investments and loans, superannuation and annuities as geometric series with regular contributions, the limiting sum of a geometric series in the context of perpetuities, and using recursive formulae and spreadsheets to model loan repayments and investment growth. Financial Mathematics in HSC Mathematics Advanced is examined at a greater level of algebraic rigour than the Standard courses and requires students to derive and apply general formulae, reason about convergence, and communicate financial conclusions precisely in the context of the HSC external examination.

Statistical Analysis in HSC Mathematics Advanced covers the probability and statistics content at Advanced level — including discrete probability distributions including the construction and use of probability distribution tables, expected value E(X) and its applications, variance and standard deviation of a discrete random variable, continuous random variables and their probability density functions, the normal distribution as the most important continuous distribution, using the empirical rule and z-scores to find probabilities and compare data values, applications of the normal distribution to real-world contexts, and sample means and their distributions including the central limit theorem concept. Statistical Analysis is assessed in the HSC Mathematics Advanced external examination with a greater level of theoretical and algebraic depth than the Standard courses, requiring students to work with probability density functions and distribution concepts as well as applying the normal distribution in extended problem-solving contexts.