AQA A-Level Mathematics

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Pure Mathematics

Algebra

B1, B2 - Indices and surdsUse rational powers confidently, keeping the real-number conditions that make the index laws valid. Then simplify and combine surds and rationalise denominators, so exact answers stay exact even when individual numbers are irrational.
B3 - Quadratic graphs and discriminantsSketch quadratics from their intercepts and turning points, and use completed-square form to explain their maximum or minimum values. Then use the discriminant to count real roots and find parameter conditions, including values where an equation stops being quadratic.
B3 - Quadratic equations in unfamiliar formsChoose a quadratic-solving method, recognise quadratic structure inside a less familiar equation and return to the original variable. Keep exact answers and use domain restrictions to decide which candidate roots really solve the original problem.
B4 - Simultaneous equationsSolve equations together using elimination and substitution, then interpret each solution as a shared point of two graphs. You will handle linear and quadratic relations, decide how many intersections exist and keep the answers that fit a stated context.
B5 - Linear and quadratic inequalitiesLearn to solve inequalities with brackets and fractions, use a quadratic graph to choose whole intervals, and describe single or combined solution sets precisely. You will see why finding boundary values is only the beginning of solving an inequality.
B5 - Regions defined by inequalitiesLearn to represent linear and quadratic inequalities as regions of the coordinate plane, including strict boundaries and overlaps. You will also recover inequalities from a shaded diagram and check which boundary points actually belong to a common region.
B6 - Polynomials and the factor theoremLearn to expand and factorise polynomials, divide by a linear expression and use the factor theorem to connect factors with roots. You will also use factorisation to explain a pattern that holds for every integer.
B6 - Rational expressions and algebraic divisionLearn to simplify and combine algebraic fractions while retaining every excluded input. You will also divide by a linear expression to separate an improper fraction into a polynomial and a proper fractional remainder.

Functions and graphs

B7 - Function graphs and intersectionsAn equation tells you much more than a collection of plotted points. You will sketch polynomial, modulus and reciprocal graphs, use their intersections to solve equations and inequalities, and recognise proportional relationships on suitable axes.
B8 - Composite and inverse functionsA function has a rule and a set of permitted inputs. You will describe its outputs, combine functions in a specified order, and reverse a function when each output identifies just one original input. Domains and ranges keep each operation meaningful.
B9 - Transforming function graphsUse a known graph to build a new one by moving, stretching or reflecting its points. You will distinguish changes to outputs from changes to inputs, track domains and ranges, and describe combinations in an order that produces the required equation.
B10 - Partial fractionsRewrite a rational expression as simpler fractions by choosing a complete denominator pattern and matching a polynomial identity. You will prepare expressions first, handle distinct and repeated linear factors, and check the result by recombining its terms.
B11 - Functions as modelsTranslate a situation into a function, fit its parameters and use its outputs with sensible inputs. You will interpret what a model assumes, compare predictions with observations, and refine a model when evidence shows that an assumption needs to change.

Coordinate geometry

C1 - Straight lines and linear modelsFind line equations, use coordinates to justify lengths and angles, and build straight-line models with meaningful rates and valid input ranges. You will combine these ideas to prove the shape of a quadrilateral without trusting the appearance of a sketch.
C2 - Circles, chords and tangentsUse a circle's centre and radius to write its equation, recover them by completing squares, and solve coordinate problems involving chords, right angles and tangents. You will see how one distance condition explains all these geometric relationships.
C3 - Parametric and Cartesian equationsDescribe a curve using a shared parameter, trace it in the correct direction, and convert between parametric and Cartesian forms. You will retain the restrictions that distinguish the actual curve from an equation containing extra points.
C4 - Parametric modelsConstruct and use models with a shared parameter, interpret their outputs and valid intervals, and check when observations require a model to change. You will distinguish a geometric path crossing from two objects occupying the same place at the same time.

Sequences and series

D1 - Binomial expansions for integer powersLearn how binomial coefficients count choices in an expansion. You will expand positive integer powers, select individual coefficients and explain why the same counting appears in binomial probabilities.
D1 - General binomial expansions and approximationsUse rational powers to build useful approximations near a convenient value. You will rewrite brackets into the right form, state a strict validity range and check whether a truncated expansion is accurate enough for its purpose.
D2, D3 - Sequences, recurrence and sigma notationGenerate and analyse sequences from direct formulae and recurrence rules (D2). Read and write sigma sums (D3), keeping a term's value separate from its index and a sequence separate from its sum.
D4 - Arithmetic sequences and seriesUse a constant difference to locate terms, recover unknown parameters and derive arithmetic sums. You will solve equations and inequalities for term counts, checking their integer domain and any extra conditions.
D5 - Geometric sequences and seriesUse a common ratio to locate geometric terms and derive finite sums. Decide when an infinite series converges, then use its remaining tail to control accuracy while retaining the signs and the conditions of every formula.
D6 - Sequence and series modelsTranslate repeated changes into arithmetic, geometric or recurrence models. Define the index and update order, distinguish an individual amount from a cumulative total, and interpret predictions through their assumptions and physical limits.

Trigonometry

E1 - Trigonometric ratios and triangle rulesGive trigonometric ratios a meaning for every rotation, choose a rule from the information in a triangle, and calculate its area. You will also discover why two sides and one angle can describe two different triangles.
E1 - Radians, arcs and sectorsMeasure rotations in radians and use them to find arc lengths, sector areas and circular boundaries. You will see why an angle measured as an arc-to-radius ratio makes circle formulae unusually simple.
E2 - Small-angle approximationsReplace trigonometric functions by simpler expressions when their arguments are small in radians. You will track transformed arguments, avoid losing terms through cancellation, and decide whether an approximation is accurate enough for a particular purpose.
E3 - Trigonometric graphs and exact valuesDerive exact trigonometric values, extend them through symmetry and repeated turns, and describe sine, cosine and tangent graphs. You will use exact landmarks to understand transformations instead of relying on a calculator picture.
E4 - Reciprocal and inverse trigonometric functionsDistinguish reciprocating a trigonometric ratio from undoing a function. You will use secant, cosecant and cotangent, select principal inverse angles, and describe the graphs, domains and ranges of all six functions.
E5, E8 - Trigonometric identities and proofsUse the quotient and Pythagorean identities in E5 to find exact values and simplify expressions. Then build the general arguments required in E8, checking domains and explaining why each algebraic step is valid.
E6 - Compound and double-angle formulaeDerive compound-angle formulae from geometric projections and use them to find exact values. Obtain the double-angle formulae from those results, choose a form that fits the information, and connect the formulae in proofs.
E6 - Combining sine and cosineReplace a sum of sine and cosine by one shifted wave. Match coefficients to choose its amplitude and phase, then use that form to find extreme values and solve equations over a complete interval.
E7 - Solving trigonometric equationsFind every solution in a stated interval, using symmetry, periodicity and algebra. Transform changed arguments, solve quadratics in trigonometric functions, and check that admissibility restrictions or cancelled factors have not altered the solution set.
E9 - Trigonometric models and applicationsTranslate rotation, geometric fit, displacements and forces into trigonometric relations. Choose the appropriate method, interpret every solution in context, and identify the assumptions that make each model useful.

Exponentials and logarithms

F1, F3 - Exponential and logarithmic functionsExponential functions turn an input into a power; logarithmic functions recover that input. Use F1 to recognise exponential graphs and F3 to find their inverses, including the domains, ranges and transformations of the natural functions.
F4, F5 - Logarithm laws and equationsUse F4 logarithm laws to combine expressions and check their domains, then use F5 to solve equations with an unknown exponent. Exact logarithmic answers and checks in the original equation keep the algebra reliable.
F6 - Linearising exponential and power dataChoose logarithmic axes that turn a power or exponential model into a straight line. Use its gradient and intercept to estimate the original constants, then check the model against data and the range of any prediction.
F2, F7 - Exponential growth and decay modelsUse F2 to connect an exponential's gradient to its current value. Apply that proportional-rate idea in F7 models of continuous interest, population growth and decay, then test assumptions and refine models with a nonzero baseline.

Differentiation

G1 - Derivatives from first principlesLearn how a chord gradient approaches a tangent gradient, and use that idea to derive derivatives of polynomial powers. Connect the resulting gradient function with instantaneous rates and with the shape of a curve.
G1 - First principles for sine and cosineDerive the sine and cosine derivatives from first principles, explaining why radians matter. Use second derivatives to connect changing gradients with convexity, concavity and inflection.
G2 - Standard derivativesDifferentiate rational powers, exponentials, logarithms and trigonometric functions with linear arguments. Combine the rules in sums, keep the real domain in view and use numerical gradients as checks on your algebra.
G3 - Tangents, normals and monotonicityFind tangent and normal equations from the point and derivative, including horizontal and vertical cases. Use derivative signs and gradient graphs to justify where a function increases or decreases, with care over isolated zero slopes and gaps in the domain.
G3 - Stationary points and optimisationClassify stationary points and inflections using evidence from derivatives. Distinguish local from global extrema, compare boundaries and construct a constrained volume model whose optimum is physically possible.
G4 - Product and quotient rulesA product or quotient has two changing pieces. Learn how their changes combine, choose a convenient derivative rule and keep the original domain when an expression simplifies.
G4 - Chain rule and composite derivativesIdentify the inner and outer operations of a composite function, track every layer of change and combine the chain rule with product and quotient rules. Keep the function's domain alongside the algebra.
G4 - Connected rates and inverse derivativesConnect rates through a derivative, keep their signs and units meaningful, and find inverse derivatives at corresponding inputs. Check that the inverse and a finite reciprocal slope are valid before reversing a rate.
G5 - Implicit and parametric differentiationFind first derivatives when a curve is given by a relation between its coordinates or by a parameter. Use them for tangents, normals and horizontal tangent points, checking the original relation and allowed parameters before interpreting a gradient.
G6 - Constructing differential equationsTranslate rate and proportionality language into differential equations, calibrate constants from measurements and state initial conditions. Check signs, units and assumptions, and verify a proposed function by differentiation.

Integration and differential equations

H1, H2 - Antiderivatives and the fundamental theoremRecover a function from its derivative, integrate rational powers and use an antiderivative to measure accumulated change. H2 supplies the integration methods; H1 explains why they also evaluate a definite integral.
H2 - Standard exponential and trigonometric integralsIntegrate exponentials, reciprocals, sine and cosine by reversing their derivatives. Learn how the coefficient inside an argument changes the answer, why logarithms need a valid domain and how to combine standard forms.
H3 - Definite integrals and areasEvaluate definite integrals exactly and use them to measure regions under and between curves. Distinguish signed accumulation from geometric area, locate crossings before integrating and keep the correct horizontal direction for a parametric curve.
H4 - Integration as a limit of sumsBuild area estimates from rectangle widths and heights, then see why their limiting value is a definite integral. Recognise an integral hidden in a sum and connect a finite numerical estimate with an exact antiderivative calculation.
H5 - Integration by substitutionTurn a complicated-looking integral into a simpler one by choosing one new variable. You will reverse the chain rule, rewrite the whole integrand, handle square-root domains and change definite-integral limits consistently.
H5 - Integration by partsIntegrate products by reversing the product rule. You will choose which factor to differentiate, integrate logarithms, apply the method twice when needed and keep both boundary terms in definite integrals.
H6 - Integration using partial fractionsSplit rational expressions into fractions you can integrate using logarithms and powers. You will handle distinct and repeated linear factors, prepare an improper fraction and keep the original domain when evaluating definite integrals.
H7, H8 - Separable differential equations and modelsSolve simple first-order differential equations by separating variables and using initial conditions (H7). Then interpret cooling and motion models, checking solution domains and the assumptions that limit their predictions (H8).

Numerical methods

I1 - Locating roots and sign-change limitsLocate solutions by finding intervals that contain roots, justify a requested accuracy, and recognise when endpoint signs cannot support a conclusion. A calculator can suggest a decimal; a short argument can establish which digits you may trust.
I2 - Fixed-point iterationRearrange equations into iterative rules, generate accurate calculator sequences, and construct staircase and cobweb diagrams. You will see why equivalent equations can produce very different sequences, and how to distinguish numerical evidence from a guaranteed root accuracy.
I2 - Newton-Raphson iterationDerive Newton–Raphson from a tangent, use its recurrence to approximate roots, and justify the digits you report. You will also recognise starting values that cause undefined steps, cycles, divergence or convergence to the wrong root.
I3, I4 - Trapezium rule and numerical modelsEstimate integrals and bound areas using numerical integration (I3), then use numerical methods to find totals and threshold times in context (I4). You will distinguish a careful calculation from the reliability of the measurements and model behind it.

Vectors

J1, J2, J3 - Vector components and operationsDescribe directed quantities in two and three dimensions, combine their effects, and move between components and magnitude with a clearly stated direction. J1 supplies the notation, J2 connects an arrow to its length and angles, and J3 explains what addition and multiplication by a scalar do geom...
J4 - Position vectors and geometryUse position vectors to locate points, subtract them to find a displacement, and measure the distance between fixed or moving points. You will also locate midpoints and ratio points, while keeping the difference between a whole line and a segment clear.
J5 - Vector proofs and contextual problemsUse directed-vector relationships to prove geometric claims and locate constrained intersections. Apply the same component reasoning to forces and planar motion, stating what the model assumes and checking that the answer describes the physical event asked for.

Mechanics

Motion and modelling

P1, Q1 - Mechanical quantities and modellingUse SI units consistently and describe a journey using position, displacement, distance, velocity, speed and acceleration. You will also choose modelling assumptions that keep the information needed for a particular question.
Q2 - Displacement and velocity graphsRead velocity and acceleration from gradients, and displacement and distance from areas. You will connect two graphs of the same journey and construct a graph from a description of motion.
Q3 - Constant-acceleration equationsDerive and use the equations for motion with constant acceleration along a straight line. You will choose signs consistently, interpret the roots of vertical-motion equations and join stages whose accelerations differ.
Q3 - Vector constant-acceleration motionExtend constant-acceleration equations to two dimensions, calculate speed and direction from velocity components, and solve conditions that must hold at the same time. You will distinguish where a particle is from the direction in which it is moving, then recover a motion model from observations.
Q4 - Calculus for straight-line motionUse differentiation and integration when acceleration varies, determine constants from conditions at stated times, and calculate total distance by finding reversals. You will also check whether a mathematical motion model supports the physical conclusions drawn from it.
Q4 - Calculus for vector motionDifferentiate and integrate two-dimensional motion component by component, use vector conditions to determine constants, and interpret changing speeds and directions. You will distinguish acceleration from the rate of change of speed, and use a common time when solving a direction condition.
Q5 - Projectile motionModel a projectile in a vertical plane using horizontal and vertical components, derive its trajectory, and interpret heights, ranges and impact times. You will use the assumptions of the model explicitly and choose the physical solution when an equation gives several possible times.

Forces and Newton's laws

R1, R3 - Forces, weight and Newton's first lawIdentify the interactions acting on one body and turn them into a clear force diagram (R1). You will distinguish mass from gravitational weight (R3), then use Newton's first law to find unknown forces when a body rests or moves with constant velocity.
R2, R5 - Resultant forces and Newton's second lawAdd forces as vectors and use their resultant to determine acceleration (R2, R5). You will connect constant forces to familiar motion equations, then use calculus and differential equations when a constant-mass particle experiences a force that varies with time or speed.
R2, R4 - Resolving forces and equilibriumReplace angled forces by signed perpendicular components (R2, R4), then use those components in equilibrium and motion equations. You will choose axes to match a surface and account for an angled pull in both the reaction and the acceleration.
R3 - Vertical dynamicsModel vertical motion with signed force equations, explain what a lift scale measures, and use gravity-only motion in stages. You will also distinguish mass from weight, use the local value of gravitational acceleration and decide when a simple model needs changing.
R4 - Newton's third law and connected particlesIdentify which bodies receive a third-law pair, connect the assumptions about a string and pulley to their mathematical consequences, and write separate equations for connected bodies. You will also rebuild the model when a string breaks or becomes slack.
R6 - Friction and limiting equilibriumFind the friction actually needed for rest, compare it with the available capacity and recognise limiting equilibrium. You will resolve angled forces, check contact, and find complete ranges of coefficients or tensions on rough horizontal and inclined surfaces.
R2, R4, R6 - Rough slopes and connected dynamicsResolve forces on rough slopes, choose friction from the actual motion or tendency to move, and solve connected-particle models with separate equations. You will check rest before predicting release, and rebuild the model at a stop, reversal, slackening or loss of contact.

Statistics

Sampling and the AQA large data set

K1, 3.21.1 - The AQA car data set and populationsIdentify the group behind a statistical claim and read the real AQA car workbook with confidence. You will distinguish measurements from category codes, interpret its selection rules and decide what a blank or zero can mean before calculating.
K1 - Sampling techniques and inferenceChoose and describe a sampling method that fits a real investigation. You will make selections reproducible, compare practical advantages with possible bias and explain why two well-chosen samples can still give different results.
K1, L4, 3.21 - Using technology to investigate real dataUse the actual AQA car workbook to turn a real question into a reproducible investigation. You will preserve records when filtering and sampling, justify field-specific cleaning and use calculator or spreadsheet summaries to reach a careful contextual conclusion.

Presenting and interpreting data

L1 - Data displays and histogramsRead a display by asking what its positions, heights and areas represent. You will interpret histograms, boxplots and cumulative frequency graphs, compare distributions and turn recorded frequencies into proportions.
L3 - Location, spread and standard deviationCalculate measures of centre and variation from raw lists, frequency tables and summary statistics. You will check calculator outputs, recognise grouped estimates and choose summaries that answer a question about the data.
L4 - Outliers, missing values and data choicesInvestigate possible outliers, distinguish missing values from genuine zeros and choose displays that support a fair conclusion. Use the actual AQA car data to make field-specific cleaning decisions and keep an honest account of what remains uncertain.
L2 - Scatter diagrams, regression and correlationInterpret paired observations, supplied correlation coefficients and the coefficients of a given regression line. You will recognise distinct groups, keep the units and pairings intact, and distinguish an association from evidence of a causal effect.

Probability

M1 - Events, independence and probabilityUse sets to describe chance events, combine their probabilities and test independence. You will distinguish “cannot happen together” from “does not affect the probability”, then see why an exact value behaves differently in discrete and continuous models.
M2 - Conditional probabilityCalculate probabilities after information restricts the possible outcomes. Use two-way tables, Venn diagrams and probability trees to identify the right conditioning population, including when a question reverses the information you were given.
M3 - Probability models and assumptionsBuild probability models from clearly defined events and assumptions, then check what their predictions mean. Refine dependence, population weights and physical constraints, and explain how each change affects the answer and what uncertainty remains.

Statistical distributions

N1, N3 - Discrete and binomial distributionsDescribe separate random outcomes with a probability distribution, then recognise when a count has a binomial model. N1 develops exact and cumulative probabilities; N3 helps you judge the assumptions and interpret binomial mean and spread.
N2 - Normal distributions and probabilitiesUse a Normal curve to model continuous measurements, calculate probabilities and find cutoffs. You will connect the curve's shape to its parameters, recover unknown parameters from probability information, and use one measurement's probability in a binomial count.
N2, N3 - Choosing and checking distribution modelsChoose a distribution from the variable and the process that generates it, then check its predictions against data and physical limits. N2 connects Normal shape to histograms and binomial counts; N3 develops reasons for using, questioning and refining a model.

Hypothesis testing

O1, O2 - Binomial hypothesis testsUse a sample to investigate a claim about a population proportion. O1 supplies the language of hypotheses, tails and significance; O2 connects exact binomial probabilities to a cautious conclusion about the population.
O1 - Correlation hypothesis testsInterpret a supplied correlation coefficient and test for population linear correlation using a supplied critical value or p-value. Learn why strength, significance and causation answer different questions, and state what the evidence can reasonably support.
O3 - Normal mean hypothesis testsTest a population mean using a Normal model with known, given or assumed population variance. Find the distribution of a sample mean, use the correct tail or critical values, and interpret a complete test without confusing the sample result with the population claim.