Year 11: QCE Mathematical Methods

Listed Mathematics syllabus sub-topics at Units 1 and 2, 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.

18syllabus sub-topics
92skills mapped
100%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 1: Surds, algebra, functions and probability

Topic 1 · SurdsUnderstand 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, e.g. √7 √3 = √7 √3 × √3 √3 = √7×√3 √3×√3 = √21 Use the four operations to simplify surds, e.g. √5 − 2√5 + 4√5 = 3√5 and
- Level 1: Rational indices
- Level 2: Rational indices
Topic 1 · Quadratic functionsRecognise and determine features of the graphs of 𝑦 = 𝑥^2, 𝑦 = 𝑎𝑥^2 + 𝑏𝑥 + 𝑐, 𝑦 = 𝑎(𝑥 − ℎ)^2 + 𝑘 and 𝑦 = 𝑎(𝑥 − 𝑥_1)(𝑥 − 𝑥_2), 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 expansionUnderstand 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 = 𝑥^𝑛 + (𝑛 choose 1) 𝑥^(𝑛−1) 𝑦 + ... + (𝑛 choose 𝑟) 𝑥^(𝑛−𝑟) 𝑦^𝑟 + ... + 𝑦^𝑛 to expand expressions, e.g. (2𝑥 − 1)^3
Topic 2 · Cubic functionsIdentify 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 · Introduction to functions and relationsUnderstand 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.
Topic 3 · Graphs of relationsRecognise 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.
Topic 3 · Reciprocal functionsRecognise features of the graphs of 𝑦 = 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 · Circular measure and radian measureDefine and use radian measure and understand its relationship with degree measure. Calculate lengths of arcs and areas of sectors in circles.
- Conversions between radians and degrees
- Trigonometric relations involving circles
Topic 4 · Introduction to trigonometric functionsUnderstand 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 and π . 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 · Language of events and setsUse the concepts and language of outcomes, sample spaces and events as sets of outcomes. 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 everyday occurrences to illustrate set descriptions and representations of events, and set operations, including the use of Venn diagrams.
Topic 5 · Conditional probability and independenceUse 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 · Indices and index lawsUse indices (including negative and fractional indices) and the index laws. Convert radicals to and from fractional indices. Understand and use scientific notation.
Topic 1 · Introduction to exponential functionsRecognise and determine the qualitative features of the graph of 𝑦 = 𝑟^𝑥 (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 lawsDefine logarithms as indices, where 𝑎^𝑥 = 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(𝑥^𝑛) = 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.
- Evaluation of logarithms
- Simplification of logarithmic expressions
Topic 2 · Logarithmic functionsRecognise and determine the qualitative features of the graph of 𝑦 = log𝑎(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 = log𝑎(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.
- Level 1: Using logarithms to solve equations involving exponents
- Level 2: Using logarithms to solve equations involving exponents
Topic 3 · Rates of change and the concept of derivativesDetermine 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 (𝑥^𝑛) = 𝑛𝑥^(𝑛−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 · Graphical applications of derivativesDetermine 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 · Differentiation rulesUse 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)/𝑣^2, 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.

Other Year 11 skills

Year 11 skills not currently linked to an entry on this page. These may include revision, extension practice, or skills still awaiting a curriculum mapping.