Year 12: QCE Mathematical Methods

Listed Mathematics syllabus sub-topics at Units 3 and 4, in the words of the Queensland Curriculum and Assessment Authority (QCAA), with the practice on this site that covers it. Checked against the source on 4 September 2026.

2025 syllabus, version 1.3. Queensland numbers its units and topics but gives the content itself no codes, so each entry here is a sub-topic and its text is the sub-topic's own bullet list. Mathematical notation is flattened out of the syllabus PDF.

17syllabus sub-topics
54skills mapped
94%have practice

Coverage is the percentage of listed syllabus sub-topics with at least one linked practice skill; it does not measure how fully every part of an entry is practised. A skill can practise more than one syllabus sub-topic, so some appear more than once below.

Unit 3: Further calculus and introduction to statistics

Topic 1 · Calculus of exponential functionsEstimate the limit of (𝑎^ℎ − 1)/ℎ as ℎ → 0, using technology, for various values of 𝑎 > 0. Recognise that 𝑒 is the unique number 𝑎 for which the above limit is 1. Recognise and determine the qualitative features of the graph of 𝑦 = 𝑒^𝑥, including asymptote and intercept. Use the rules 𝑑/𝑑𝑥 (𝑒^𝑥) = 𝑒^𝑥 and 𝑑/𝑑𝑥 (𝑒^𝑓(𝑥)) = 𝑓′(𝑥)𝑒^𝑓(𝑥).
- Level 1: Differentiation of mixed functions
- Level 2: Differentiation of mixed functions
Topic 1 · Calculus of logarithmic functionsRecognise and determine the qualitative features of the graph of 𝑦 = ln(𝑥) = log_𝑒(𝑥), including asymptote and intercept. Recognise and use the inverse relationship of the functions 𝑦 = 𝑒^𝑥 and 𝑦 = ln(𝑥). Solve equations involving exponential and logarithmic functions with base 𝑒, with and without technology. Use the rules 𝑑/𝑑𝑥 ln(𝑥) = 1/𝑥 and 𝑑/𝑑𝑥 ln(𝑓(𝑥)) = 𝑓′(𝑥)/𝑓(𝑥). Model and solve problems that involve derivatives of exponential and logarithmic functions, with and without technology.
- Level 1: Differentiation of logarithmic functions
- Level 2: Differentiation of logarithmic functions
Topic 2 · Calculus of trigonometric functionsUse the rules 𝑑/𝑑𝑥 sin(𝑥) = cos(𝑥) and 𝑑/𝑑𝑥 sin(𝑓(𝑥)) = 𝑓′(𝑥) cos(𝑓(𝑥)). Use the rules 𝑑/𝑑𝑥 cos(𝑥) = − sin(𝑥) and 𝑑/𝑑𝑥 cos(𝑓(𝑥)) = −𝑓′(𝑥) sin(𝑓(𝑥)). Model and solve problems that involve derivatives of trigonometric functions, with and without technology.
Topic 2 · Differentiation rulesUse the chain rule to determine the derivative of composite functions involving exponential, logarithmic and trigonometric functions, expressing derivatives in simplest and factorised form. Use the product rule to determine the derivative of exponential, logarithmic and trigonometric functions, expressing derivatives in simplest and factorised form. Use the quotient rule to determine the derivative of exponential, logarithmic and trigonometric functions, expressing derivatives in simplest and factorised form. Solve problems that involve combinations of the chain rule, product rule and quotient rule to differentiate exponential, logarithmic and trigonometric functions.
Topic 3 · The second derivative and applications of differentiationUnderstand the concept of the second derivative as the rate of change of the first derivative function. Recognise acceleration as the second derivative of displacement position with respect to time. Understand the concepts of concavity and points of inflection and their relationship with the second derivative. Understand and use the second derivative test for finding local maxima and minima. Sketch the graph of a function using first and second derivatives to locate stationary points and points of inflection. Model and solve optimisation problems from a wide variety of fields using first and second derivatives, where the function to be optimised is either given or to be developed.
Topic 4 · Anti-differentiationRecognise anti-differentiation as the reverse of differentiation. Use the notation ∫𝑓(𝑥) 𝑑𝑥 for anti-derivatives or indefinite integrals. Use the formula ∫ 𝑥^𝑛 𝑑𝑥 = 𝑥^(𝑛+1)/(𝑛+1) + 𝑐 for 𝑛 ≠ −1. Use the formula ∫ 𝑒^𝑥 𝑑𝑥 = 𝑒^𝑥 + 𝑐. Use the formula ∫ (1/𝑥) 𝑑𝑥 = ln(𝑥) + 𝑐, for 𝑥 > 0. Use the formulas ∫sin(𝑥) 𝑑𝑥 = − cos(𝑥) + 𝑐 and ∫cos(𝑥) 𝑑𝑥 = sin(𝑥) + 𝑐. Understand and use the formulas ∫(𝑓(𝑥) + 𝑔(𝑥))𝑑𝑥 = ∫𝑓(𝑥)𝑑𝑥 + ∫ 𝑔(𝑥)𝑑𝑥 and ∫𝑘 𝑓(𝑥)𝑑𝑥 = 𝑘 ∫ 𝑓(𝑥)𝑑𝑥. Determine indefinite integrals of the form ∫𝑓(𝑎𝑥 + 𝑏)𝑑𝑥. Determine 𝑓(𝑥) given 𝑓′(𝑥) and an initial condition 𝑓(𝑎) = 𝑏. Determine displacement given velocity and the initial value of displacement. Determine displacement given acceleration and initial values of displacement and velocity. Model and solve problems that involve indefinite integrals, with and without technology.
Topic 5 · General discrete random variablesUnderstand the concepts of a discrete random variable and its associated probability function, and its use in modelling data. Use relative frequencies obtained from data to determine point estimates of probabilities associated with a discrete random variable. Recognise uniform discrete random variables and use them to model random phenomena with equally likely outcomes. Recognise non-uniform discrete random variables and use them to model random phenomena. Determine and use the mean (expected value) of a discrete random variable as a measurement of centre, 𝐸(𝑋) = 𝜇 = ∑ 𝑝_𝑖 𝑥_𝑖 where 𝑝_𝑖 is the probability of outcome 𝑥_𝑖 occurring. Determine and use the variance of a discrete random variable as a measure of spread, 𝑉𝑎𝑟(𝑋) = ∑ 𝑝_𝑖 (𝑥_𝑖 − 𝜇)^2 where 𝑝_𝑖 is the probability of outcome 𝑥_𝑖 occurring, 𝜇 is the mean. Determine and use the standard deviation of a discrete random variable, √𝑉𝑎𝑟(𝑋), as a measure of spread. Model and solve problems that involve discrete random variables and associated probabilities, with and without technology.
Topic 5 · Bernoulli distributionsUse a Bernoulli random variable as a model for two-outcome situations. Identify contexts suitable for modelling by Bernoulli random variables. Recognise and determine the mean 𝑝 and variance 𝑝(1 − 𝑝) of the Bernoulli distribution with parameter 𝑝. Model and solve problems that involve Bernoulli random variables and associated probabilities, with and without technology.
- Bernoulli distribution: mean and variance
- Bernoulli distribution: modelling practical problems
Topic 5 · Binomial distributionsUnderstand the concepts of Bernoulli trials and the concept of a binomial random variable as the number of ‘successes’, 𝑟, in 𝑛 independent Bernoulli trials, with the same probability of success 𝑝 in each trial. Identify contexts suitable for modelling by binomial random variables. Determine and use the probabilities 𝑃(𝑋 = 𝑟) = (𝑛 choose 𝑟) 𝑝^𝑟 (1 − 𝑝)^(𝑛−𝑟) associated with the binomial distribution with parameters 𝑛 and 𝑝. Calculate the mean 𝑛𝑝 and variance 𝑛𝑝(1 − 𝑝) of a binomial distribution using technology and algebraic methods. Use the language of probability, including at most, at least, no more than, no less than, inclusive and between. Model and solve problems that involve binomial distributions and associated probabilities with and without technology.
- Level 1: Multi-stage probability experiments
- Level 2: Multi-stage probability experiments

Unit 4: Further calculus, trigonometry and statistics

Topic 1 · Fundamental theorem of calculus and definite integralsUse sums of the form ∑_𝑖 𝑓(𝑥_𝑖) 𝛿𝑥_𝑖 to estimate the area under the curve 𝑦 = 𝑓(𝑥). Recognise the definite integral ∫_𝑎^𝑏 𝑓(𝑥) 𝑑𝑥 as a limit of sums of the form ∑_𝑖 𝑓(𝑥_𝑖) 𝛿𝑥_𝑖. Understand the fundamental theorem of calculus, ∫_𝑎^𝑏 𝑓(𝑥) 𝑑𝑥 = 𝐹(𝑏) − 𝐹(𝑎), and use it to calculate definite integrals. Use the definite integral ∫_𝑎^𝑏 𝑓(𝑥) 𝑑𝑥 to determine the area under the curve 𝑦 = 𝑓(𝑥) between 𝑥 = 𝑎 and 𝑥 = 𝑏 if 𝑓(𝑥) > 0 over this interval.
Topic 1 · Applications of integrationCalculate the area enclosed by a curve and the 𝑥-axis over a given domain, with and without technology. Calculate the area between curves, with and without technology. Use the trapezoidal rule, ∫_𝑎^𝑏 𝑓(𝑥) 𝑑𝑥 ≈ (𝑤/2)[𝑓(𝑥_0) + 2(𝑓(𝑥_1) + 𝑓(𝑥_2) + 𝑓(𝑥_3) + ... 𝑓(𝑥_(𝑛−1))) + 𝑓(𝑥_𝑛)], where 𝑤 = (𝑏−𝑎)/𝑛, to approximate an area and the value of a definite integral, with and without technology. Calculate total change by integrating instantaneous or marginal rates of change, with and without technology. Model and solve problems that involve definite integrals, including motion problems, with and without technology.
Topic 2 · Cosine and sine rulesUse the sine rule (ambiguous case is required), 𝑎/sin(𝐴) = 𝑏/sin(𝐵) = 𝑐/sin(𝐶), where 𝑎, 𝑏 and 𝑐 are the side lengths of the triangle and 𝐴, 𝐵 and 𝐶 are the corresponding opposite angles. Use the cosine rule, 𝑐^2 = 𝑎^2 + 𝑏^2 − 2𝑎𝑏 cos(𝐶). Use the formula area = (1/2)𝑏𝑐 sin(𝐴) to calculate the area of a triangle. Model and solve problems that involve the sine rule, cosine rule and the area formula in two- and three-dimensional contexts (including bearings, directions and angles of elevation and depression), with and without technology.
- Trigonometric relations involving complex geometric shapes
- Trigonometric relations involving circles
Topic 3 · General continuous random variablesUse relative frequencies and histograms obtained from data to estimate probabilities associated with a continuous random variable. Understand the concepts of a probability density function, cumulative distribution function, and probabilities associated with a continuous random variable given by integrals; examine simple types of continuous random variables and use them in appropriate contexts. Calculate the expected value, 𝐸(𝑋) = 𝜇 = ∫_(−∞)^∞ 𝑥𝑝(𝑥)𝑑𝑥, of a continuous random variable where 𝑝(𝑥) is the probability density function. Calculate the variance, 𝑉𝑎𝑟(𝑋) = 𝜎^2 = ∫_(−∞)^∞ (𝑥 − 𝜇)^2 𝑝(𝑥)𝑑𝑥, and standard deviation 𝜎, of a continuous random variable. Understand standardised normal variables (𝑧-values, 𝑧-scores) and use these to compare samples.
Topic 3 · Normal distributionsIdentify contexts, e.g. naturally occurring variations, that are suitable for modelling by normal random variables. Recognise features of the graph of the probability density function of the normal distribution with mean 𝜇 and standard deviation 𝜎 and the use of the standard normal distribution. Recognise and use the link between the normal distribution and the notation 𝑋 ∼ 𝑁(𝜇, 𝜎 2). Calculate probabilities and quantiles associated with a given normal distribution, using technology. Model and solve problems that involve normal distributions, with and without technology (distribution tables are not required).
- Normal distribution: qualitative graph features
- Normal distribution: probabilities and quantiles
Topic 4 · Random samplingUnderstand the concept of a random sample. Understand sources of bias in samples, and procedures to ensure randomness. Identify and use procedures to ensure randomness. Recognise and use graphical displays of real and simulated data of random samples from various types of distributions, including uniform, Bernoulli, binomial and normal.
No practice for this one yet.
Topic 4 · Sample proportionsUnderstand the concept of the sample proportion 𝑝̂ as a random variable whose value varies between samples, and the formulas for the mean 𝑝 and standard deviation √(𝑝(1 − 𝑝)/𝑛) of the sample proportion 𝑝̂, where 𝑛 is the sample size. Recognise and use the approximate normality of the distribution of 𝑝̂ for large samples. Use repeated random sampling data, for a variety of values of 𝑝 and a range of sample sizes, to examine the distribution of 𝑝̂ and the approximate standard normality of (𝑝̂ − 𝑝)/√(𝑝̂(1−𝑝̂)/𝑛), where the closeness of the approximation depends on both 𝑛 and 𝑝.
- Sample proportion: mean and standard deviation
Topic 5 · Confidence intervals for proportionsUnderstand the concept of an interval estimate for a parameter associated with a random variable. Understand and use the approximate confidence interval, (𝑝̂ − 𝑧√(𝑝̂(1−𝑝̂)/𝑛), 𝑝̂ + 𝑧√(𝑝̂(1−𝑝̂)/𝑛)), as an interval estimate for 𝑝, the population proportion, where 𝑧 is the appropriate quantile for the standard normal distribution. Understand and use the approximate margin of error, 𝑧√(𝑝̂(1−𝑝̂)/𝑛). Understand and use the relationship between margin of error, level of confidence and sample size. Understand that there are variations in confidence intervals between samples and that most, but not all, confidence intervals contain 𝑝. Model and solve problems that involve interval estimates for proportions, with and without technology.
- Confidence interval and margin of error for a proportion

Other Year 12 skills

Year 12 skills not currently linked to an entry on this page. These may include revision, extension practice, or skills still awaiting a curriculum mapping.