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§-Math Methods syllabus
QLD · QCAA← Math Methods

Math Methods syllabus, dot point by dot point

Every dot point in the QLD Math Methods syllabus, in syllabus order, with a focused answer for each. Open any dot point for the answer, exam-style questions and links to related points.

Unit 1: Surds, algebra, functions and probability

Unit overview →

Topic 1: Surds and quadratic functions

Sketch and analyse linear and quadratic functions, finding gradient, intercepts, vertex and discriminant, and solving linear and quadratic equations and inequalities
8 min read →
Simplify expressions involving surds and apply the laws of indices to rational and negative exponents
7 min read →

Topic 2: Binomial expansion and cubic 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
8 min read →
Expand expressions of the form (a+b)n(a + b)^n for a positive integer nn using Pascal's triangle and binomial coefficients, and find specific terms and coefficients
7 min read →

Topic 3: Functions and relations

Sketch reciprocal functions of the form y=ax−h+ky = \dfrac{a}{x - h} + k, identify asymptotes, domain and range, and find the inverse of a one-to-one function
7 min read →
Functions and graphs introduced in Year 11, including linear, quadratic, cubic, polynomial, exponential and logarithmic functions; their key features, intercepts and transformations
9 min read →
Apply translations, dilations and reflections to the graph of a function, including the form y=af(b(x−h))+ky = a f(b(x - h)) + k and the effect of each parameter
7 min read →

Topic 5: Probability

Counting techniques (multiplication principle, permutations and combinations), simple probability, conditional probability and the addition and multiplication rules
9 min read →
Apply the rules of probability (addition, multiplication, conditional), permutations and combinations to calculate probabilities of compound events
8 min read →

How are arithmetic and geometric sequences analysed?

Define arithmetic and geometric sequences, find the nnth term and the sum of the first nn terms, and apply to real-world contexts
8 min read →

How are simultaneous equations solved?

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

How are arithmetic and geometric sequences and series defined and computed in QCE Math Methods Unit 1?

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

How are financial calculations done?

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

What additional algebraic skills does QCE Math Methods Unit 1 introduce?

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

Unit 2: Calculus and further functions

Unit overview →

Topic 1: Exponential functions

Recall and apply the laws of indices to simplify expressions and solve equations involving rational and negative exponents
7 min read →
Model exponential growth and decay using y=A⋅rty = A \cdot r^t or y=Aekty = A e^{kt}, including problems involving population growth, radioactive decay, depreciation and continuous compound interest
7 min read →
Graph and analyse exponential functions of the form y=a⋅bx+cy = a \cdot b^x + c, identifying key features (intercepts, asymptote, domain, range) and applying transformations
7 min read →

Topic 2: Logarithms and logarithmic functions

Define logarithms as the inverse of exponentials, apply the laws of logarithms, and solve exponential equations using logarithms
7 min read →
Sketch and analyse logarithmic functions of the form y=log⁡b(x−h)+ky = \log_b(x - h) + k, state their domain, range and asymptote, relate them to exponential functions as inverses, and solve logarithmic equations
7 min read →

Topic 3: Introduction to differential calculus

Define the derivative of a function as a limit and use first principles to find the derivative of a polynomial function
7 min read →
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
7 min read →

Topic 4: Trigonometric functions (Unit 1 in the 2025 syllabus)

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
7 min read →
State and apply the Pythagorean identity sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1, and use it together with related identities to simplify expressions and solve equations
7 min read →
Sketch and analyse graphs of y=asin⁡(b(x−h))+ky = a \sin(b(x - h)) + k and y=acos⁡(b(x−h))+ky = a \cos(b(x - h)) + k, identifying amplitude, period, phase shift and vertical translation
7 min read →

Topic 4: Applications of differential calculus

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

Topic 5: Further differentiation

Apply the chain rule, the product rule and the quotient rule to differentiate functions built from polynomial and power functions, and use the results to find gradients and tangents
8 min read →

How are exponential, logarithmic and trigonometric functions extended in QCE Math Methods Unit 2?

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

How is differential calculus introduced in QCE Math Methods Unit 2?

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

What discrete probability distributions does QCE Math Methods Unit 2 introduce?

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

Unit 3: Further calculus and introduction to statistics

Unit overview →

Topic 1: Differentiation of exponential and logarithmic functions

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

Topic 2: Differentiation of trigonometric functions and differentiation rules

Differentiate trigonometric functions, including compositions of the form sin⁡(f(x))\sin(f(x)), cos⁡(f(x))\cos(f(x)) and tan⁡(f(x))\tan(f(x)), working in radians
7 min read →
Apply the product, quotient and chain rules, including in combination, to differentiate functions built from polynomial, exponential, logarithmic and trigonometric components
8 min read →

Topic 3: Further applications of differentiation

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
9 min read →

Topic 4: Introduction to integration

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
9 min read →
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
9 min read →

Topic 5: Discrete random variables

Define a discrete random variable and its probability distribution, calculate the expected value E(X)E(X) and the variance Var(X)\mathrm{Var}(X) and standard deviation, and recognise the Bernoulli distribution as the single-trial case
8 min read →
Recognise the binomial distribution X∼Bin(n,p)X \sim \mathrm{Bin}(n, p) as the count of successes in nn independent Bernoulli trials, apply the binomial probability formula and use a graphics calculator, and use the formulas E(X)=npE(X) = np and Var(X)=np(1−p)\mathrm{Var}(X) = np(1 - p)
9 min read →

Unit 4: Further calculus, trigonometry and statistics

Unit overview →

Topic 1: Further integration

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

Topic 3: Continuous random variables and the normal distribution

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
9 min read →
Apply the normal distribution N(μ,σ2)N(\mu, \sigma^2) and the standardisation Z=(X−μ)/σZ = (X - \mu)/\sigma to compute normal probabilities and inverse probabilities, including the empirical 68-95-99.7 rule
9 min read →

Specialist Mathematics content (not a Unit 4 Methods topic)

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

Topics 4 and 5: Sampling and proportions; Interval estimates for proportions

Apply the sampling distribution of the sample proportion p^\hat{p} (mean pp, standard deviation p(1−p)/n\sqrt{p(1-p)/n}) and construct approximate confidence intervals p^±z∗p^(1−p^)/n\hat{p} \pm z^* \sqrt{\hat{p}(1-\hat{p})/n} for a population proportion
9 min read →

Further differentiation (beyond the Unit 4 topic list)

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
9 min read →

Based on the official QCAA syllabus. Not individually reviewed by a person; always check QCAA and your school's materials.

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