NSW · NESASyllabus
Maths Advanced syllabus, dot point by dot point
Every dot point in the NSW Maths Advanced syllabus, with a focused answer for each one. Click any dot point for a worked explainer, past exam questions, and links to related dot points. Generated by Claude Opus and reviewed by Better Tuition Academy tutors.
Year 12: Calculus
Module overview →- How can the derivative be used to analyse curves and solve optimisation and rate problems?Use the first and second derivatives to find stationary points, points of inflection, and to solve optimisation and related rates problems10 min answer →
- How do we use definite integrals to compute areas between curves and volumes of solids of revolution?Calculate the area under a curve, the area between two curves, and the volume of a solid of revolution about the $x$ or $y$ axis9 min answer →
- How do we differentiate functions built from products, quotients and compositions of standard functions?Apply the product, quotient and chain rules, and differentiate exponential, logarithmic and trigonometric functions8 min answer →
- How do we model continuous growth, decay and cooling using differential equations?Establish and solve differential equations of the form $\frac{dN}{dt} = k N$ and $\frac{dT}{dt} = k(T - T_a)$ and apply them to growth, decay and Newton's law of cooling9 min answer →
- How do we find antiderivatives and use the Fundamental Theorem of Calculus to evaluate definite integrals?Find antiderivatives of standard functions, apply integration by substitution and evaluate definite integrals using the Fundamental Theorem of Calculus10 min answer →
- How do we differentiate and integrate exponential and logarithmic functions, and how do they appear in modelling?Find derivatives and integrals of $e^x$ and $\ln x$, including composed forms, and apply them to modelling problems8 min answer →
- How do we use calculus to analyse the motion of a particle moving in a straight line?Apply calculus to motion in a straight line, with displacement, velocity and acceleration as derivatives and integrals with respect to time8 min answer →
- How do we differentiate and integrate trigonometric functions and use them to model periodic phenomena?Find derivatives and integrals of $\sin$, $\cos$ and $\tan$ (with linear inside arguments) and apply them to model simple harmonic and periodic motion8 min answer →
Year 12: Financial Mathematics
Module overview →- How do equal regular contributions to an investment grow over time, and what is the future value formula for an annuity?Derive and use the future value formula for an annuity to find the accumulated value of a series of equal regular contributions9 min answer →
- How do geometric sequences and series model repeated payments and recurring growth, and when does an infinite series converge?Use the formulas for the nth term and the sum of n terms of a geometric sequence, and the limiting sum, in financial contexts9 min answer →
- How are reducing-balance loan repayments calculated, and how much of each payment goes to interest versus principal?Use recurrence relations and the present value of an annuity to find loan repayments, outstanding balances and total interest paid10 min answer →
- How do simple and compound interest accumulate value over time, and how do we move money between present and future?Use simple and compound interest formulas to find future values, present values, interest rates and time periods9 min answer →
Year 12: Functions
Module overview →- How do we sketch graphs built from sums, differences, products, quotients and reciprocals of standard functions?Sketch graphs of sums, differences, products, quotients, squares and reciprocals of two known functions9 min answer →
- What are the key features of exponential and logarithmic graphs, and how do transformations and the inverse relationship link them?Sketch and interpret graphs of exponential and logarithmic functions, including transformations, and use the inverse relationship between them8 min answer →
- When does a function have an inverse, and how do we form, evaluate and graph composite and inverse functions?Form composite functions, determine when a function has an inverse, find and graph the inverse, and use restriction of domain to invert non-one-to-one functions9 min answer →
- How do translations, reflections and dilations transform the graph of a function in a predictable way?Apply translations, reflections and dilations to the graph of a function and identify the resulting equation9 min answer →
Year 12: Statistical Analysis
Module overview →- How do we describe and model the relationship between two numerical variables?Construct scatter plots, calculate and interpret Pearson's correlation coefficient, and fit and use the least-squares regression line9 min answer →
- How do probability density functions describe continuous random variables, and how do we extract probabilities and summary statistics from them?Use probability density functions and cumulative distribution functions to find probabilities, medians, modes, means and variances of continuous random variables10 min answer →
- How do we describe a discrete random variable and summarise its distribution with mean and variance?Define a discrete random variable by its probability distribution, and calculate the expected value, variance and standard deviation9 min answer →
- How do we use the normal distribution and z-scores to compute probabilities and compare observations?Use the normal distribution, z-scores, the empirical rule and the standard normal table to find probabilities and percentiles9 min answer →
Year 12: Trigonometric Functions
Module overview →- How do amplitude, period, phase shift and vertical shift transform the graphs of sine, cosine and tangent?Sketch and interpret graphs of $y = a \sin(b x + c) + d$, $y = a \cos(b x + c) + d$ and $y = a \tan(b x + c) + d$, identifying amplitude, period, phase shift and vertical shift9 min answer →
- How are radians defined, and how do we use them to find arc length and sector area?Use radian measure to find arc length, the area of a sector, and the area of a segment of a circle8 min answer →
- How do we find all solutions of a trigonometric equation in a given interval, including equations involving multiple angles and identities?Solve trigonometric equations over a given interval using exact values, the unit circle, and identities to reduce to a single trig function9 min answer →
- Which trigonometric identities are essential for simplifying expressions and proving equivalences in HSC Maths Advanced?Use Pythagorean, ratio, double angle and complementary identities to simplify expressions and prove equalities9 min answer →