Math Methods Q&A by dot point

Define arithmetic and geometric sequences, find the $n$th term and the sum of the first $n$ terms, and apply to real-world contexts

Counting techniques (multiplication principle, permutations and combinations), simple probability, conditional probability and the addition and multiplication rules

Apply simple interest, compound interest and depreciation models to financial calculations, including future value, present value and effective annual rate

Functions and graphs introduced in Year 11, including linear, quadratic, cubic, polynomial, exponential and logarithmic functions; their key features, intercepts and transformations

Sketch and analyse linear and quadratic functions, finding gradient, intercepts, vertex and discriminant, and solving linear and quadratic equations and inequalities

Index and logarithm laws, factorisation techniques, solving polynomial equations, and the relationship between exponential and logarithmic functions

Sketch and analyse polynomial functions of degree 3 and 4, using factored form to read roots and multiplicities, and applying the factor and remainder theorems

Apply the rules of probability (addition, multiplication, conditional), permutations and combinations to calculate probabilities of compound events

Arithmetic and geometric sequences and series, including the general term formulas, sum formulas, and applications to growth and decay problems

Solve systems of simultaneous linear equations in two and three variables, including by substitution, elimination, and matrix methods, and interpret the results graphically

Simplify expressions involving surds and apply the laws of indices to rational and negative exponents

Apply translations, dilations and reflections to the graph of a function, including the form $y = a f(b(x - h)) + k$ and the effect of each parameter

Define the derivative of a function as a limit and use first principles to find the derivative of a polynomial function

Exponential, logarithmic and trigonometric functions (including their graphs and transformations), and applications to growth and decay and periodic phenomena

Graph and analyse exponential functions of the form $y = a \cdot b^x + c$, identifying key features (intercepts, asymptote, domain, range) and applying transformations

Model exponential growth and decay using $y = A \cdot r^t$ or $y = A e^{kt}$, including problems involving population growth, radioactive decay, depreciation and continuous compound interest

Recall and apply the laws of indices to simplify expressions and solve equations involving rational and negative exponents

Introduction to differential calculus, including the gradient at a point, the derivative as a function, and the power rule for derivatives of polynomial functions

Define logarithms as the inverse of exponentials, apply the laws of logarithms, and solve exponential equations using logarithms

Apply the power rule, the sum rule, and the constant-multiple rule to differentiate polynomial functions, and use the derivative to find tangent and normal line equations

Discrete probability distributions, including the uniform discrete distribution and an introduction to the Bernoulli distribution, with calculations of expected value and variance

Define radian measure of angle and relate to arc length; evaluate exact values of sine, cosine and tangent of common angles using the unit circle

Use the derivative to find stationary points of a polynomial function and classify them, and apply differentiation to simple optimisation problems

Sketch and analyse graphs of $y = a \sin(b(x - h)) + k$ and $y = a \cos(b(x - h)) + k$, identifying amplitude, period, phase shift and vertical translation

State and apply the Pythagorean identity $\sin^2 \theta + \cos^2 \theta = 1$, and use it together with related identities to simplify expressions and solve equations

Find antiderivatives of standard functions including polynomial, exponential and trigonometric forms, evaluate definite integrals using the Fundamental Theorem of Calculus, and recognise the definite integral as the limit of a Riemann sum

Apply the definite integral to find the area under a curve, the area between two curves, the average value of a function, and to solve kinematics problems involving displacement, velocity and acceleration

Differentiate exponential and logarithmic functions, including compositions of the form $e^{f(x)}$ and $\ln(f(x))$, and apply the derivatives to model and analyse rates of change

Differentiate trigonometric functions, including compositions of the form $\sin(f(x))$, $\cos(f(x))$ and $\tan(f(x))$, working in radians

Define a discrete random variable and its probability distribution, calculate the expected value $E(X)$ and the variance $\mathrm{Var}(X)$ and standard deviation, and recognise the Bernoulli distribution as the single-trial case

Use the first and second derivative to analyse the behaviour of a function (intervals of increase and decrease, stationary points and their nature, concavity and inflection), and apply the derivative to solve optimisation and rates of change problems in context

Apply the product, quotient and chain rules, including in combination, to differentiate functions built from polynomial, exponential, logarithmic and trigonometric components

Recognise the binomial distribution $X \sim \mathrm{Bin}(n, p)$ as the count of successes in $n$ independent Bernoulli trials, apply the binomial probability formula and use CAS, and use the formulas $E(X) = np$ and $\mathrm{Var}(X) = np(1 - p)$

Apply the definite integral to compute the area between curves (including curves that change relative order), the average value of a function, and kinematics quantities (displacement, distance, position) from velocity and acceleration

Define a continuous random variable, its probability density function (pdf), cumulative distribution function (cdf), and compute probabilities, expected value (mean), variance and standard deviation as definite integrals

Apply the product, quotient and chain rules to differentiate composite functions involving exponential, logarithmic, polynomial and trigonometric pieces, including logarithmic differentiation and the differentiation of inverse functions

Apply implicit differentiation to find $\frac{dy}{dx}$ from equations relating $x$ and $y$ that cannot be expressed in the form $y = f(x)$, and apply differentiation to related rates problems

Integrate trigonometric functions including $\sin(kx)$, $\cos(kx)$ and $\sec^2(kx)$, and apply the linear reverse-chain rule for integrals of the form $f(ax+b)$

Apply the normal distribution $N(\mu, \sigma^2)$ and the standardisation $Z = (X - \mu)/\sigma$ to compute normal probabilities and inverse probabilities, including the empirical 68-95-99.7 rule

Apply the sampling distribution of the sample proportion $\hat{p}$ (mean $p$, standard deviation $\sqrt{p(1-p)/n}$) and construct approximate confidence intervals $\hat{p} \pm z^* \sqrt{\hat{p}(1-\hat{p})/n}$ for a population proportion