QCE Mathematical Methods Syllabus Guide (Mathematics)

QCE Mathematical Methods runs across four units, with Units 3 and 4 counting towards your ATAR. This guide lists every unit, topic and subject-matter point from the QCAA syllabus.

The external exam is worth 50% of your final result - one of the highest-scaling QCE subjects - so it rewards a complete grasp of the syllabus.

Unit 1: Surds, algebra, functions and probability

Topic 1: Surds and quadratic functions

  • Understand the concept of a surd as an irrational number represented using a square root or a radical sign.
  • Simplify square roots of natural numbers which contain perfect square factors, e.g. √45 = √9 × 5 = √9√5 = 3√5.
  • Rationalise the denominator of fractional expressions involving square roots.
  • Use the four operations to simplify surds, e.g. √5 − 2√5 + 4√5 = 3√5 and 2√3 × 5√11 = 10√33.
  • Recognise and determine features of the graphs of y = x², y = ax² + bx + c, y = a(x − h)² + k and y = a(x − x₁)(x − x₂), including their parabolic nature, turning points, axes of symmetry and intercepts.
  • Solve quadratic equations algebraically using factorisation, the quadratic formula (both exact and approximate solutions), completing the square and using technology.
  • Sketch the graphs of quadratic functions, with or without technology.
  • Use the discriminant to determine the number of solutions to a quadratic equation.
  • Determine turning points and zeros of quadratic functions, with and without technology.
  • Model and solve problems that involve quadratic functions, with and without technology.

Topic 2: Binomial expansion and cubic functions

  • Understand the notion of a combination as an unordered set of r objects taken from a set of n distinct objects.
  • Recognise and use the link between Pascal's triangle and the notation (n choose r).
  • Use the binomial theorem (x + y)ⁿ = xⁿ + (n choose 1)xⁿ⁻¹y + ... + (n choose r)xⁿ⁻ʳyʳ + ... + yⁿ to expand expressions, e.g. (2x − 1)³.
  • Identify the coefficients and the degree of a polynomial.
  • Expand quadratic and cubic polynomials from factors.
  • Recognise and determine features of the graphs of y = x³, y = a(x − h)³ + k and y = a(x − x₁)(x − x₂)(x − x₃), including shape, intercepts, and behaviour as x → ∞ and x → −∞.
  • Solve cubic equations using technology, and algebraically in cases where the equation is factorised.
  • Sketch the graphs of cubic functions, with and without technology.
  • Model and solve problems that involve cubic functions, with and without technology.

Topic 3: Functions and relations

  • Understand the concept of a relation as a mapping between sets, a graph and as a rule or a formula that defines one variable quantity in terms of another.
  • Recognise the distinction between functions and relations and use the vertical line test to determine whether a relation is a function.
  • Recognise and use function notation, domain and range, and independent and dependent variables.
  • Recognise and use piece-wise functions as a combination of multiple sub-functions with restricted domains.
  • Model and solve problems that involve piece-wise functions with and without technology.
  • Recognise and determine features of the graphs of x² + y² = r² and (x − h)² + (y − k)² = r², including their circular shapes, centres and radii.
  • Recognise and determine features of the graph of y² = x, including its parabolic shape and axis of symmetry.
  • Recognise and determine features of the graphs of y = a√(x − h) + k, including their shape, intercepts, and behaviour as x → ∞ and x → −∞.
  • Sketch the graphs of relations, with and without technology.
  • Model and solve problems that involve relations, with and without technology.
  • Recognise features of the graphs of y = 1/x and y = a/(x − h) + k, including their hyperbolic shape, intercepts, asymptotes, and behaviour as x → ∞ and x → −∞.
  • Model and solve problems that involve reciprocal functions, with and without technology.
  • Sketch the graphs of reciprocal functions, with and without technology.

Topic 4: Trigonometric functions

  • Define and use radian measure and understand its relationship with degree measure.
  • Calculate lengths of arcs and areas of sectors in circles.
  • Understand the unit circle definition of cos(θ), sin(θ) and tan(θ) and periodicity using radians.
  • Understand and use the exact values of cos(θ), sin(θ) and tan(θ) at integer multiples of π/6 and π/4.
  • Sketch the graphs of y = sin(x), y = cos(x) and y = tan(x) on extended domains.
  • Recognise and determine the effect of the parameters a, b, h and k on the graphs of y = a sin(b(x − h)) + k, y = a cos(b(x − h)) + k, with and without technology.
  • Sketch the graphs of y = a sin(b(x − h)) + k, y = a cos(b(x − h)) + k, with and without technology.
  • Solve trigonometric equations, with and without technology, including the use of the Pythagorean identity sin²(A) + cos²(A) = 1.
  • Model and solve problems that involve trigonometric functions, with and without technology.

Topic 5: Probability

  • Use the concepts and language of outcomes, sample spaces and events as sets of outcomes.
  • Use set language and notation for events, including A′ for the complement of an event A, A ∩ B for the intersection of event A and event B, and A ∪ B for the union of event A and event B, and recognise mutually exclusive events.
  • Use everyday occurrences to illustrate set descriptions and representations of events, and set operations, including the use of Venn diagrams.
  • Use the rules P(A′) = 1 − P(A) and P(A ∪ B) = P(A) + P(B) − P(A ∩ B).
  • Understand the notion of a conditional probability and recognise and use language that indicates conditionality.
  • Use the notation P(A|B) and the formula P(A ∩ B) = P(A|B)P(B) to solve problems.
  • Understand and use the notion of independence of an event A from an event B, as defined by P(A|B) = P(A).
  • Use the formula P(A ∩ B) = P(A)P(B) for independent events A and B.
  • Use relative frequencies obtained from data as point estimates of conditional probabilities and as indications of possible independence of events.
  • Model and solve problems that involve probability, with and without technology.

Unit 2: Calculus and further functions

Topic 1: Exponential functions

  • Use indices (including negative and fractional indices) and the index laws.
  • Convert radicals to and from fractional indices.
  • Understand and use scientific notation.
  • Recognise and determine the qualitative features of the graph of y = rˣ (where r > 0), including asymptote and intercept.
  • Recognise and determine the effect of the parameters h, k and r on the graph of y = r^(x−h) + k (where r > 0), with and without technology.
  • Sketch the graphs of exponential functions, with and without technology.
  • Solve equations involving exponential functions, with and without technology.
  • Model and solve problems that involve exponential functions, with and without technology.

Topic 2: Logarithms and logarithmic functions

  • Define logarithms as indices, where aˣ = b is equivalent to x = log_a(b), and convert between both forms.
  • Use logarithmic laws and definitions: log_a(x) + log_a(y) = log_a(xy); log_a(x) − log_a(y) = log_a(x/y); log_a(xⁿ) = n log_a(x); log_a(x) = log_b(x)/log_b(a); log_a(a) = 1; log_a(1) = 0.
  • Solve equations involving indices using logarithms, with and without technology.
  • Recognise and determine the qualitative features of the graph of y = log_a(x) (where a > 1), including asymptote and intercept.
  • Recognise and determine the effect of the parameters a, h and k on the graph of y = log_a(x − h) + k (where a > 1), with and without technology.
  • Sketch graphs of logarithmic functions, with and without technology.
  • Solve equations involving logarithmic functions with and without technology.
  • Model and solve problems that involve logarithmic functions, e.g. decibels in acoustics and the Richter scale for earthquake magnitude, with and without technology.

Topic 3: Introduction to differential calculus

  • Determine average rate of change in a variety of practical contexts.
  • Use the rule f′(x) = lim(h→0) [f(x+h) − f(x)]/h to determine the derivative of simple power functions and polynomial functions from first principles.
  • Interpret the derivative as the instantaneous rate of change.
  • Interpret the derivative as the gradient of a tangent line of the graph of y = f(x).
  • Use the rule d/dx(xⁿ) = nxⁿ⁻¹ for positive integers.
  • Understand the concept of the derivative as a function.
  • Recognise and use properties of the derivative: d/dx(f(x) + g(x)) = d/dx f(x) + d/dx g(x).
  • Calculate derivatives of power and polynomial functions.

Topic 4: Applications of differential calculus

  • Determine instantaneous rates of change.
  • Determine the equation of a tangent and a normal of the graph of y = f(x).
  • Construct and interpret displacement-time graphs, with velocity as the slope of the tangent.
  • Recognise that velocity is the instantaneous rate of change of displacement with respect to time.
  • Use the first derivative of a function to determine and identify the nature of stationary points.
  • Sketch curves associated with power functions and polynomials up to degree 4; find stationary points and local and global maxima and minima with and without technology; and examine behaviour as x → ∞ and x → −∞.

Topic 5: Further differentiation

  • Use the chain rule, if y = f(u) and u = g(x) then dy/dx = dy/du × du/dx, to determine the derivative of composite functions involving power and polynomial functions.
  • Use the product rule, d(uv)/dx = u dv/dx + v du/dx, to determine the derivative of products of functions involving power and polynomial functions.
  • Use the quotient rule, d(u/v)/dx = (v du/dx − u dv/dx)/v², to determine the derivative of quotients of functions involving power and polynomial functions.
  • Solve problems that involve combinations of the chain rule, product rule and quotient rule to differentiate functions involving power and polynomial functions, expressing derivatives in simplest and factorised form.

Unit 3: Further calculus and introduction to statistics

Topic 1: Differentiation of exponential and logarithmic functions

  • Estimate 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 = eˣ, including asymptote and intercept.
  • Use the rules d/dx(eˣ) = eˣ and d/dx(e^f(x)) = f′(x)e^f(x).
  • Recognise 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 = eˣ 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.

Topic 2: Differentiation of trigonometric functions and differentiation rules

  • Use 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.
  • Use 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: Further applications of differentiation

  • Understand 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: Introduction to integration

  • Recognise anti-differentiation as the reverse of differentiation.
  • Use the notation ∫f(x)dx for anti-derivatives or indefinite integrals.
  • Use the formula ∫xⁿdx = x^(n+1)/(n+1) + c for n ≠ −1.
  • Use the formula ∫eˣdx = eˣ + 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 ∫kf(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: Discrete random variables

  • Understand 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 = ∑pᵢxᵢ where pᵢ is the probability of outcome xᵢ occurring.
  • Determine and use the variance of a discrete random variable as a measure of spread, Var(X) = ∑pᵢ(xᵢ − μ)² where pᵢ is the probability of outcome xᵢ 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.
  • Use 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.
  • Understand 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.

Unit 4: Further calculus, trigonometry and statistics

Topic 1: Further integration

  • Use sums of the form ∑f(xᵢ)δxᵢ to estimate the area under the curve y = f(x).
  • Recognise the definite integral ∫ from a to b f(x)dx as a limit of sums of the form ∑f(xᵢ)δxᵢ.
  • Understand the fundamental theorem of calculus, ∫ from a to b f(x)dx = F(b) − F(a), and use it to calculate definite integrals.
  • Use the definite integral ∫ from a to 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.
  • Calculate 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 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: Trigonometry

  • Use 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, c² = a² + b² − 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.

Topic 3: Continuous random variables and the normal distribution

  • Use 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 = ∫ from −∞ to ∞ xp(x)dx, of a continuous random variable where p(x) is the probability density function.
  • Calculate the variance, Var(X) = σ² = ∫ from −∞ to ∞ (x − μ)²p(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.
  • Identify 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, σ²).
  • 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).

Topic 4: Sampling and proportions

  • Understand 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.
  • Understand 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.

Topic 5: Interval estimates for proportions

  • Understand 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.

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Syllabus structure from the QCAA Mathematical Methods 2025senior syllabus, © State of Queensland (Queensland Curriculum & Assessment Authority), licensed under CC BY 4.0. For the authoritative version see the official QCAA syllabus. Polarbear is not affiliated with or endorsed by the QCAA. Snapshot generated 2026-07-20.

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