Mathematical Methods · QCAA
Full Mathematical Methods syllabus
Drill from units to topics, subtopics and individual learning objectives. Every LO is wired up to AI-marked practice questions.
▶1 — Surds, algebra, functions and probability40 LOs
▶Surds and quadratic functions9 LOs
▶Surds4 LOs
●Rationalise the denominator of fractional expressions involving square roots, e.g. √7 √3 = √7 √3 × √3 √3 = √7×√3 √3×√3 = √21 3
●Simplify square roots of natural numbers which contain perfect square factors, e.g. √45 = √9 × 5 = √9√5 = 3√5
●Understand the concept of a surd as an irrational number represented using a square root or a radical sign.
●Use the four operations to simplify surds, e.g. √5 − 2√5 + 4√5 = 3√5 and 2√3 × 5√11 = 10√33
▶Quadratic functions5 LOs
●Determine turning points and zeros of quadratic functions, with and without technology.
●Model and solve problems that involve quadratic functions, with and without technology.
●Sketch the graphs of quadratic functions, with or without technology.
●Solve quadratic equations algebraically using factorisation, the quadratic formula (both exact and approximate solutions), completing the square and using technology.
●Use the discriminant to determine the number of solutions to a quadratic equation.
▶Binomial expansion and cubic functions7 LOs
▶Binomial expansion2 LOs
●Understand the notion of a combination as an unordered set of 𝑟 objects taken from a set of 𝑛 distinct objects.
●Use the binomial theorem (𝑥 + 𝑦)𝑛 = 𝑥𝑛 + (𝑛 1) 𝑥𝑛−1𝑦+... + (𝑛 𝑟) 𝑥𝑛−𝑟 𝑦𝑟 +... + 𝑦𝑛 to expand expressions, e.g.(2𝑥 − 1)3 Mathematical Methods 2025 v1.3
▶Cubic functions5 LOs
●Expand quadratic and cubic polynomials from factors.
●Identify the coefficients and the degree of a polynomial.
●Model and solve problems that involve cubic functions, with and without technology.
●Sketch the graphs of cubic functions, with and without technology.
●Solve cubic equations using technology, and algebraically in cases where the equation is factorised.
▶Functions and relations6 LOs
▶Introduction to functions and relations2 LOs
●Model and solve problems that involve piece-wise functions with and without technology.
●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.
▶Graphs of relations2 LOs
●Model and solve problems that involve relations, with and without technology.
●Sketch the graphs of relations, with and without technology.
▶Reciprocal functions2 LOs
●Model and solve problems that involve reciprocal functions, with and without technology.
●Sketch the graphs of reciprocal functions, with and without technology. Mathematical Methods 2025 v1.3
▶Trigonometric functions8 LOs
▶Circular measure and radian measure2 LOs
●Calculate lengths of arcs and areas of sectors in circles.
●Define and use radian measure and understand its relationship with degree measure.
▶Introduction to trigonometric functions6 LOs
●Model and solve problems that involve trigonometric functions, 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.
●Sketch the graphs of 𝑦 = sin(𝑥), 𝑦 = cos(𝑥) and 𝑦 = tan(𝑥) on extended domains.
●Solve trigonometric equations, with and without technology, including the use of the Pythagorean identity sin2 (𝐴) + cos2(𝐴) = 1.
●Understand and use the exact values of cos(𝜃), sin(𝜃) and tan(𝜃) at integer multiples of π 6 and π 4.
●Understand the unit circle definition of cos(𝜃), sin (𝜃) and tan (𝜃) and periodicity using radians.
▶Probability10 LOs
▶Language of events and sets3 LOs
●Use everyday occurrences to illustrate set descriptions and representations of events, and set operations, including the use of Venn diagrams.
●Use set language and notation for events, including 𝐴 or 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 the concepts and language of outcomes, sample spaces and events as sets of outcomes.
▶Conditional probability and independence7 LOs
●Model and solve problems that involve probability, with and without technology.
●Understand and use the notion of independence of an event 𝐴 from an event 𝐵, as defined by 𝑃(𝐴|𝐵) = 𝑃(𝐴).
●Understand the notion of a conditional probability and recognise and use language that indicates conditionality.
●Use relative frequencies obtained from data as point estimates of conditional probabilities and as indications of possible independence of events.
●Use the formula 𝑃(𝐴 ∩ 𝐵) = 𝑃(𝐴)𝑃(𝐵) for independent events 𝐴 and 𝐵.
●Use the notation 𝑃(𝐴|𝐵) and the formula 𝑃(𝐴 ∩ 𝐵) = 𝑃(𝐴|𝐵)𝑃(𝐵) to solve problems.
●Use the rules 𝑃(𝐴) = 1 − 𝑃(𝐴) and 𝑃(𝐴 ∪ 𝐵) = 𝑃(𝐴) + 𝑃(𝐵) − 𝑃(𝐴 ∩ 𝐵).
▶2 — Calculus and further functions28 LOs
▶Exponential functions6 LOs
▶Indices and index laws3 LOs
●Convert radicals to and from fractional indices.
●Understand and use scientific notation.
●Use indices (including negative and fractional indices) and the index laws.
▶Introduction to exponential functions3 LOs
●Model and solve problems that involve exponential functions, with and without technology. Mathematical Methods 2025 v1.3
●Sketch the graphs of exponential functions, with and without technology.
●Solve equations involving exponential functions, with and without technology.
▶Logarithms and logarithmic functions6 LOs
▶Logarithms and logarithmic laws3 LOs
●Define logarithms as indices, where 𝑎𝑥 = 𝑏 is equivalent to 𝑥 = log𝑎 (𝑏), and convert between both forms.
●Solve equations involving indices using logarithms, with and without technology.
●Use logarithmic laws and definitions log𝑎 (𝑥) + log𝑎(𝑦) = log𝑎 (𝑥𝑦) log𝑎 (𝑥) − log𝑎(𝑦) = log𝑎 (𝑥 𝑦) log𝑎 (𝑥𝑛) = 𝑛 log𝑎 (𝑥) log𝑎 (𝑥) = log𝑏(𝑥) log𝑏(𝑎) log𝑎 (𝑎) = 1 log𝑎 (1) = 0
▶Logarithmic functions3 LOs
●Model and solve problems that involve logarithmic functions, e.g. decibels in acoustics and the Richter scale for earthquake magnitude, with and without technology.
●Sketch graphs of logarithmic functions, with and without technology.
●Solve equations involving logarithmic functions with and without technology.
▶Introduction to differential calculus7 LOs
▶Rates of change and the concept of derivatives7 LOs
●Calculate derivatives of power and polynomial functions. Mathematical Methods 2025 v1.3
●Determine average rate of change in a variety of practical contexts.
●Interpret the derivative as the gradient of a tangent line of the graph of 𝑦 = 𝑓(𝑥).
●Interpret the derivative as the instantaneous rate of change.
●Understand the concept of the derivative as a function.
●Use the rule 𝑑 𝑑𝑥 𝑥𝑛 = 𝑛𝑥𝑛−1 for positive integers.
●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.
▶Applications of differential calculus5 LOs
▶Graphical applications of derivatives5 LOs
●Construct and interpret displacement-time graphs, with velocity as the slope of the tangent.
●Determine instantaneous rates of change.
●Determine the equation of a tangent and a normal of the graph of 𝑦 = 𝑓(𝑥).
●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 → − ∞.
●Use the first derivative of a function to determine and identify the nature of stationary points.
▶Further differentiation4 LOs
▶Differentiation rules4 LOs
●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.
●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 = 𝑣𝑑𝑢 𝑑𝑥−𝑢𝑑𝑣 𝑑𝑥 𝑣2, to determine the derivative of quotients of functions involving power and polynomial functions.
▶3 — Further calculus and introduction to statistics43 LOs
▶Differentiation of exponential and logarithmic functions5 LOs
▶Calculus of exponential functions2 LOs
●Estimate the limit of 𝑎ℎ−1 ℎ as ℎ → 0, using technology, for various values of 𝑎 > 0.
●Use the rules 𝑑 𝑑𝑥 𝑒𝑥 = 𝑒𝑥 and 𝑑 𝑑𝑥 𝑒𝑓(𝑥) = 𝑓′(𝑥) 𝑒𝑓(𝑥).
▶Calculus of logarithmic functions3 LOs
●Model and solve problems that involve derivatives of exponential and logarithmic functions, with and without technology.
●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).
▶Differentiation of trigonometric functions and differentiation rules7 LOs
▶Calculus of trigonometric functions3 LOs
●Model and solve problems that involve derivatives of trigonometric functions, with and without technology.
●Use the rules 𝑑 𝑑𝑥 cos(𝑥) = − sin(𝑥) and 𝑑 𝑑𝑥 cos(𝑓(𝑥)) = −𝑓′(𝑥) sin(𝑓(𝑥)).
●Use the rules 𝑑 𝑑𝑥 sin (𝑥) = cos(𝑥) and 𝑑 𝑑𝑥 sin (𝑓(𝑥)) = 𝑓′(𝑥) cos(𝑓(𝑥)).
▶Differentiation rules4 LOs
●Solve problems that involve combinations of the chain rule, product rule and quotient rule to differentiate exponential, logarithmic and trigonometric functions. Mathematical Methods 2025 v1.3
●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.
▶Further applications of differentiation5 LOs
▶The second derivative and applications of differentiation5 LOs
●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.
●Sketch the graph of a function using first and second derivatives to locate stationary points and points of inflection.
●Understand and use the second derivative test for finding local maxima and minima.
●Understand the concept of the second derivative as the rate of change of the first derivative function.
●Understand the concepts of concavity and points of inflection and their relationship with the second derivative.
▶Introduction to integration11 LOs
▶Anti-differentiation11 LOs
●Determine displacement given acceleration and initial values of displacement and vel ocity.
●Determine displacement given velocity and the initial value of displacement.
●Determine 𝑓(𝑥) given 𝑓′(𝑥) and an initial condition 𝑓(𝑎) = 𝑏.
●Determine indefinite integrals of the form ∫ 𝑓(𝑎𝑥 + 𝑏)𝑑𝑥.
●Model and solve problems that involve indefinite integrals, with and without technology. Mathematical Methods 2025 v1.3
●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.
●Use the formula ∫ 1 𝑥 𝑑𝑥 = ln(𝑥) + 𝑐, for 𝑥 > 0.
●Use the formula ∫ 𝑒𝑥 𝑑𝑥 = 𝑒𝑥 + 𝑐.
●Use the formula ∫ 𝑥𝑛 𝑑𝑥 = 𝑥𝑛+1 𝑛+1 + 𝑐 for 𝑛 ≠ − 1.
●Use the formulas ∫ sin (𝑥) 𝑑𝑥 = − cos(𝑥) + 𝑐 and ∫ cos(𝑥) 𝑑𝑥 = sin (𝑥) + 𝑐.
●Use the notation ∫ 𝑓(𝑥) 𝑑𝑥 for anti-derivatives or indefinite integrals.
▶Discrete random variables15 LOs
▶General discrete random variables6 LOs
●Determine and use the mean (expected value) of a discrete random variable as a measurement of centre, 𝐸(𝑋) = 𝜇 = ∑ 𝑝𝑖 𝑥𝑖 where 𝑝𝑖 is the probability of outcome 𝑥𝑖 occurring.
●Determine and use the standard deviation of a discrete random variable, √𝑉𝑎𝑟(𝑋), as a measure of spread.
●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.
●Model and solve problems that involve discrete random variables and associated probabilities, with and without technology.
●Understand the concepts of a discrete random variable and its associat ed 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.
▶Bernoulli distributions3 LOs
●Identify contexts suitable for modelling by Bernoulli random variables.
●Model and solve problems that involve Bernoulli random variables and associated probabilities, with and without technology.
●Use a Bernoulli random variable as a model for two-outcome situations.
▶Binomial distributions6 LOs
●Calculate the mean 𝑛𝑝 and variance 𝑛𝑝(1 − 𝑝) of a binomial distribution using technology and algebraic methods.
●Determine and use the probabilities 𝑃(𝑋 = 𝑟) = (𝑛 𝑟) 𝑝𝑟 (1 − 𝑝)𝑛−𝑟 associated with the binomial distribution with parameters 𝑛 and 𝑝.
●Identify contexts suitable for modelling by binomial random variables.
●Model and solve problems that involve binomial distributions and associated probabilities with and without technology.
●Understand the concepts of Bernoulli trials and the concept of a binomial rand om variable as the number of ‘successes’, 𝑟, in 𝑛 independent Bernoulli trials, with the same probability of success 𝑝 in each trial.
●Use the language of probability, including at most, at least, no more than, no less than, inclusive and between.
▶4 — Further calculus, trigonometry and statistics31 LOs
▶Further integration8 LOs
▶Fundamental theorem of calculus and definite integrals3 LOs
●Understand the fundamental theorem of calculus, ∫ 𝑓(𝑥) 𝑏 𝑎 𝑑𝑥 = 𝐹(𝑏) − 𝐹(𝑎), and use it to calculate definite integrals.
●Use sums of the form ∑ 𝑓(𝑥𝑖) 𝛿𝑥𝑖𝑖 to estimate the area under the curve 𝑦 = 𝑓(𝑥).
●Use the definite integral ∫ 𝑓(𝑥) 𝑏 𝑎 𝑑𝑥 to determine the area under the curve 𝑦 = 𝑓(𝑥) between 𝑥 = 𝑎 and 𝑥 = 𝑏 if 𝑓(𝑥) > 0 over this interval.
▶Applications of integration5 LOs
●Calculate the area between curves, with and without technology.
●Calculate the area enclosed by a curve and the 𝑥-axis over a given domain, 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.
●Use the trapezoidal rule, ∫ 𝑓(𝑥) 𝑏 𝑎 𝑑𝑥 ≈ 𝑤 2 [𝑓(𝑥0) + 2(𝑓(𝑥1) + 𝑓(𝑥2) + 𝑓(𝑥3)+... 𝑓(𝑥𝑛−1)) + 𝑓(𝑥𝑛)], where 𝑤 = 𝑏−𝑎 𝑛, to approximate an area and the value of a definite integral, with and without technology.
▶Trigonometry4 LOs
▶Cosine and sine rules4 LOs
●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. Mathematical Methods 2025 v1.3
●Use the cosine rule, 𝑐2 = 𝑎2 + 𝑏2 − 2𝑎𝑏 cos(𝐶).
●Use the formula area = 1 2 𝑏 𝑐 sin (𝐴) to calculate the area of a triangle.
●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.
▶Continuous random variables and the normal distribution8 LOs
▶General continuous random variables5 LOs
●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.
●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.
●Use relative frequencies and histograms obtained from data to estimate probabilities associated with a continuous random variable.
▶Normal distributions3 LOs
●Calculate probabilities and quantiles associated with a given normal distribution, using technology.
●Identify contexts, e.g. naturally occurring variations, that are suitable for modelling by normal random variables.
●Model and solve problems that involve normal distributions, with and without technology (distribution tables are not required). Mathematical Methods 2025 v1.3
▶Sampling and proportions5 LOs
▶Random sampling3 LOs
●Identify and use procedures to ensure randomness.
●Understand sources of bias in samples, and procedures to ensure randomness.
●Understand the concept of a random sample.
▶Sample proportions2 LOs
●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.
●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.
▶Interval estimates for proportions6 LOs
▶Confidence intervals for proportions6 LOs
●Model and solve problems that involve interval estimates for proportions, with and without technology.
●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.
●Understand the concept of an interval estimate for a parameter associated with a random variable.
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