Edexcel A-Level Mathematics

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Pure Mathematics (Papers 1 and 2)

2 Algebra and functions

Pure 2.1 - Laws of indices for rational exponentsIndices let you work exactly with roots, reciprocals and powers. You will explain what rational indices mean, choose the right index law and simplify numerical and algebraic expressions without losing their conditions.
Pure 2.2 - Surds and rationalising denominatorsSurds keep answers exact while you change their form. You will simplify and multiply square-root expressions, then rationalise single-term and two-term denominators without changing the value of a fraction.
Pure 2.3 - Quadratic functions and graphsA quadratic's graph, roots and minimum or maximum are different views of the same expression. You will complete the square, interpret the discriminant, solve quadratic equations and recognise quadratics hidden inside powers, exponentials, trigonometric functions and logarithms.
Pure 2.4 - Simultaneous equations by elimination and substitutionSimultaneous equations ask for values that satisfy both equations together. You will choose elimination or substitution, solve linear and quadratic systems, and return every matching coordinate pair with checks in the original equations.
Pure 2.5 - Linear and quadratic inequalitiesAn inequality describes every value in a solution set, rather than just the roots of an equation. You will solve linear, quadratic and fractional inequalities, express their intervals clearly, and represent solutions on number lines and coordinate graphs.
Pure 2.6 - Algebraic manipulation of polynomialsThe same polynomial can reveal different information in expanded, factorised or quotient form. You will choose a useful form, divide by linear expressions, use the factor theorem to factorise cubics, and simplify rational expressions while preserving excluded inputs.
Pure 2.7 - Graphs of functions and the modulus graphA useful sketch records what a function does without calculating every point. Learn to build polynomial, modulus, reciprocal and proportion graphs from their key features, then read equations and inequalities as intersections and relative heights.
Pure 2.8 - Composite and inverse functionsFunctions describe processes with one output for each allowed input. Learn to combine those processes, undo them and use their domains and graphs to decide which answers make sense. Keeping track of the direction of each mapping is the central skill.
Pure 2.9 - Transformations of functionsA graph can move, stretch, reflect or fold without losing its connection to the original function. Learn to track key points and asymptotes through those changes, then sketch combinations for polynomial, reciprocal, trigonometric and exponential functions.
Pure 2.10 - Partial fractionsOne fraction can hide several simpler ones. Learn how to uncover those pieces, handle repeated factors and check that your answer is an identity. This prepares rational functions for later integration, differentiation and series work.
Pure 2.11 - Functions in modellingA function model links a real input to a predicted output. Learn to interpret its constants, choose a suitable family, fit it to data and decide where its predictions can be trusted. A useful refinement changes a particular weakness in the model.

3 Coordinate geometry in the (x, y) plane

Pure 3.1 - Equation of a straight lineA straight-line equation describes every point on a line with just a few constants. You will connect those constants to gradients, intercepts and perpendicular directions, then build and interpret straight-line models.
Pure 3.2 - Coordinate geometry of the circleA circle's equation records one fixed distance from one centre. You will uncover that centre by completing the square, use chord properties to find circumcircles, and find tangents from perpendicular radii.
Pure 3.3 - Parametric equations and Cartesian formParametric equations generate both coordinates from one shared input. You will read those coordinate pairs, convert familiar curves between parametric and Cartesian forms, and retain the domain that decides how much of a curve is actually traced.
Pure 3.4 - Parametric equations in modellingParametric models keep time, position and the order of a journey together. You will build constant-velocity models from observations, use their valid intervals, and model shapes with an angle while explaining what each simplification assumes.

4 Sequences and series

Pure 4.1 - Binomial expansionA bracket raised to a power can hide a long polynomial, a square root or a reciprocal. You will use binomial coefficients to uncover its terms, extend the method to rational powers, and decide when a truncated expansion gives a sensible approximation. You will also combine expansions after partia...
Pure 4.2 - SequencesA sequence is an ordered list, but the rule behind the list tells you much more than its first few values. You will generate terms from position formulae and recurrence relations, establish increasing or decreasing behaviour, and identify the shortest repeating cycle. You will also see how changi...
Pure 4.3 - Sigma notationSigma notation writes a whole sum by specifying its terms and its endpoints. You will read, expand and construct finite sums, count their terms, and use the notation to express totals accurately. The aim is to see the addition hiding behind the symbol.
Pure 4.4 - Arithmetic sequences and seriesAdding the same amount repeatedly creates an arithmetic sequence; adding its terms creates an arithmetic series. You will find terms and totals, prove the sum formula, and solve equations involving an unknown number of terms. Contexts with decreasing payments will show why a mathematically valid...
Pure 4.5 - Geometric sequences and seriesRepeated multiplication creates geometric sequences and their sums. You will find unknown terms and ratios, prove the finite sum formula, decide when a sum to infinity exists, and work backwards from sums. Logarithms will then let you find how many terms are needed to pass a target.
Pure 4.6 - Sequences and series in modellingRepeated changes become useful models when you define exactly what each term measures. You will choose arithmetic, geometric and relation-defined models, calculate individual quantities and meaningful totals, and interpret their limitations. Timing, units and the difference between a payment and...

5 Trigonometry

Pure 5.1 - Unit-circle trigonometry, triangle rules and radiansUse a point on a circle to understand sine, cosine and tangent for any angle, then choose triangle rules and radian formulae to solve geometry problems. You will also learn why the sine rule can describe two different triangles, and how to separate a circular segment from its sector.
Pure 5.2 - Small angle approximationsReplace trigonometric expressions near zero by simple algebra, while keeping track of radians and the size of the whole angle. You will simplify quotients, estimate a small root and decide whether an approximation makes sense.
Pure 5.3 - Sine, cosine and tangent functionsBuild exact trigonometric values from two triangles, then use signs, symmetry and periodicity to reach any angle. Read the key features of the three graphs and see how a shift or a change of scale affects them.
Pure 5.4 - Reciprocal and inverse trigonometric functionsDistinguish a reciprocal value from an inverse-function angle, then use domains, ranges and graphs to choose valid answers. You will calculate exact values of secant, cosecant and cotangent, interpret principal inverse functions and use their ranges to decide signs.
Pure 5.5 - Trigonometric identitiesUse the unit-circle identity and its reciprocal forms to rewrite expressions, prove identities and solve equations. You will keep track of domains and learn how to avoid losing a solution when a factor could be zero.
Pure 5.6 - Compound and double angles, half-angles and R-formUse and understand compound-angle formulae, derive double-angle and half-angle results, and combine sine and cosine into one shifted wave. You will use geometry to justify the formulae and solve equations on a stated interval.
Pure 5.7 - Simple trigonometric equationsFind every solution of a trigonometric equation in a stated interval. You will handle shifted and multiplied angles in degrees or radians, solve quadratics in trig functions, and explain why careless cancellation or an incomplete interval can lose solutions.
Pure 5.8 - Proofs with trigonometric identitiesProve trigonometric identities by recognising compound-angle patterns, choosing useful algebra and handling reciprocal fractions. You will keep a valid chain of reasoning, explain necessary domain restrictions, and distinguish a proof from a numerical check.
Pure 5.9 - Trigonometric functions in contextRead and build sinusoidal models for repeating motion and changing quantities, solve their equations in meaningful time intervals, and resolve vectors, velocities and forces. You will connect each parameter to its units and judge what a model can reasonably predict.

6 Exponentials and logarithms

Pure 6.1 - Exponential functions and graphsRead exponential graphs from their equations: recognise growth and decay, use the natural base e, and find the intercept, range and asymptote after a transformation. A few exact points will tell you more than a screenful of rounded calculator values.
Pure 6.2 - Gradients of exponential functionsFind the gradient of e^(kx) and connect that gradient to the current height of the curve. Use the connection to recognise and interpret exponential growth and decay models, including their coefficients, signs and units.
Pure 6.3 - Logarithms and natural logarithmsA logarithm finds a missing exponent. Learn its definition and restrictions, read the graph of the natural logarithm, and use inverse operations to solve exponential and logarithmic equations exactly.
Pure 6.4 - Laws of logarithmsUse the three logarithm laws in both directions, and understand why they follow from the laws of indices. Expand expressions, handle reciprocals and roots, and condense logarithms to solve equations while keeping their original domain restrictions.
Pure 6.5 - Solving exponential equationsFind an unknown exponent using logarithms, and choose a common-base shortcut when one is available. You will solve equations with linear expressions in the exponent, keep exact answers exact and check that a real solution is possible.
Pure 6.6 - Logarithmic graphs for estimating parametersTurn curved positive data into a straight line so you can estimate a model's constants. You will choose the correct logarithmic axes for power and exponential relationships, recover parameters from a gradient and intercept, and judge how far a fitted model can be trusted.
Pure 6.7 - Exponential growth and decay in modellingBuild exponential models from observations, interpret their constants and use logarithms to find when a threshold is reached. You will distinguish continuous rates from percentage changes over a whole year, then check whether the model's long-term predictions fit the context.

7 Differentiation

Pure 7.1 - Derivatives, tangents and second derivativesA curve keeps changing its steepness, so how can it have an exact gradient at one point? Build the derivative from nearby chord gradients, prove the first rules, and use first and second derivatives to read a curve's movement and shape.
Pure 7.2 - Derivatives of standard functionsBuild a small, reliable toolkit for differentiating powers, exponentials, logarithms and trigonometric functions. Learn to rewrite an expression before you differentiate, keep its real domain in view, and recognise which formulae you need to remember.
Pure 7.3 - Gradients, tangents, normals, maxima and minimaUse derivatives to make decisions about a curve: find its tangent and normal, locate its turning points, and identify where it rises, falls or changes concavity. Then turn a practical maximum or minimum problem into a function, with a sensible domain and a justified answer.
Pure 7.4 - Product, quotient and chain rulesDifferentiate expressions built from products, quotients and nested functions by reading their structure first. Use the same chain-rule idea to handle reciprocal trigonometric functions, inverse relations and connected rates of change.
Pure 7.5 - Implicit and parametric differentiationFind a curve's gradient even when its equation does not give y directly in terms of x. Differentiate implicit relations and parametric equations, then use the resulting first derivative to find tangents and normals, including horizontal and vertical cases.
Pure 7.6 - Constructing simple differential equationsTurn a statement about a changing quantity into a differential equation. Choose the correct derivative, translate proportionality, find a constant from a measured rate, and check that the resulting equation matches the words, signs and units.

8 Integration

Pure 8.1 - Fundamental Theorem of CalculusCan a gradient function tell you exactly where a curve is? The Fundamental Theorem of Calculus links differentiation to accumulation, and integration recovers a function from its derivative. You will see why a constant is needed, use a point to find one particular curve, and evaluate definite int...
Pure 8.2 - Integrating standard functionsHow can you recover a curve when its derivative contains powers, fractions and trigonometric functions together? You will build a small set of integration rules, rewrite unfamiliar-looking expressions into those forms, and use a point to identify the required curve. Choosing the form before calcu...
Pure 8.3 - Definite integrals and area under a curveCan a region have positive area when its definite integral is zero? You will distinguish signed accumulation from geometric area, choose the boundaries of a region, and integrate the height of a vertical strip. The same idea works below the axis, between two curves and with parametric equations.
Pure 8.4 - Integration as the limit of a sumHow can rectangles with flat tops give the exact area beneath a curved graph? You will connect a sum of thin strips to a definite integral, read Pearson's limiting-sum notation and evaluate the result exactly. The crucial distinction is between a finite approximation and the value it approaches a...
Pure 8.5 - Integration by substitution and by partsSome difficult integrals become familiar when you change the variable; others become easier when you undo the product rule. You will choose and carry out substitution and integration by parts, keep limits and logarithm domains consistent, and check an answer by differentiating it.
Pure 8.6 - Integration using partial fractionsOne rational expression can hide several easy integrals. You will split linear denominator factors into partial fractions, integrate logarithmic and repeated-factor terms, and evaluate exact definite integrals while keeping coefficients, signs and domains under control.
Pure 8.7 - Separable first-order differential equationsA differential equation describes a function through its rate of change. You will separate variables, integrate to find a family of solutions, use a condition to choose one member, and check its branch, domain and sketch.
Pure 8.8 - Interpreting solutions of differential equationsSolving a differential equation gives a formula; interpreting it tells you what the formula predicts. You will connect initial values, rates, limits and valid domains to a model's quantities and units, including velocity and acceleration, and explain when long-term predictions become unreliable.

9 Numerical methods

Pure 9.1 - Locating roots by sign changesTrap an awkward root inside an interval without finding its exact value. You will use signs and continuity to justify roots and rounded answers, then recognise the two traps: missed crossings and breaks in a graph.
Pure 9.2 - Iteration, cobweb and staircase diagramsBuild successive approximations to a root, and see the same calculation as a path between a curve and a diagonal. You will rearrange equations, calculate accurately, draw iteration paths and judge whether they approach a fixed point.
Pure 9.3 - Newton-Raphson and recurrence relationsUse recurrence relations to approach roots, and derive Newton-Raphson by following a tangent to the axis. You will calculate successive approximations, simplify a requested recurrence and recognise when a fast method becomes unreliable.
Pure 9.4 - Numerical integration and the trapezium ruleEstimate an integral using straight-topped strips when a formula or a table supplies the function values. You will organise the calculation, use a sketch to judge the error, and turn suitable estimates into bounds for the true area.
Pure 9.5 - Numerical methods in contextTurn a model into a numerical target, then interpret the result with its units and limits. You will find a model's maximum by iteration, estimate totals from rate readings and separate calculation accuracy from confidence in the model.

10 Vectors

Pure 10.1 - Vectors in two and three dimensionsA vector records a movement, including its direction. You will read and translate column vectors and unit-vector notation in two and three dimensions, keep zero components in their proper places, and use equality of vectors to find unknown components. A reliable order for the components makes all...
Pure 10.2 - Magnitude, direction and component formComponents tell you how to move along the axes; magnitude and direction tell you the length and orientation of the same movement. You will convert between these descriptions, choose the correct quadrant for an angle, and use a unit vector to build a vector of a required length. The methods work i...
Pure 10.3 - Vector addition and scalar multiplicationVector addition combines directed movements, while scalar multiplication changes a vector's length and sometimes its direction. You will connect triangle and parallelogram diagrams to component calculations, combine scaled vectors, and establish when two vectors are parallel. The diagram chooses...
Pure 10.4 - Position vectors and distance between pointsA position vector locates a point relative to a chosen origin. You will subtract positions to find a directed displacement, calculate distances in two and three dimensions, and use a distance condition to find an unknown coordinate. Keeping vector and length notation distinct also explains why ch...
Pure 10.5 - Vectors in pure mathematics and contextVectors turn geometric statements and directed quantities into equations. You will find missing vertices, divide segments, establish collinearity, combine displacements and forces, and solve problems that need more than one vector route. The central choice is which directed journeys the wording m...

Statistics (Paper 3, Section A)

2 Data presentation and interpretation

Statistics 2.1 - Single-variable data diagramsRead histograms, frequency polygons, cumulative frequency diagrams and box plots by identifying what their scales mean. Use them to compare distributions and turn estimated frequencies into probabilities, while keeping track of what grouped data cannot tell you exactly.
Statistics 2.2 - Scatter diagrams and regression linesRead paired data, describe its relationship and interpret a supplied regression line in context. Decide when a prediction is defensible, distinguish association from cause, and use logarithms to recover a curved model from a transformed straight line.
Statistics 2.3 - Measures of central tendency and variationChoose and interpret averages and measures of spread, calculate exact summaries, estimate grouped percentiles, and recover original summaries after coding. Use the supplied standard-deviation formulas with clear units, careful rounding and a reason for the comparison you make.
Statistics 2.4 - Outliers in data setsRecognise possible outliers, apply a supplied numerical rule and interpret unusual points on diagrams. Clean missing values, coding issues and genuine errors without automatically deleting valid extremes, and explain how those decisions affect the conclusions.

3 Probability

Statistics 3.1 - Mutually exclusive and independent eventsDoes “one event tells you nothing about another” mean that they cannot happen together? You will distinguish independence from mutual exclusivity, calculate probabilities using regions and paths, and connect the same event language to discrete and continuous distributions.
Statistics 3.2 - Conditional probability with diagrams and tables“Given that” changes which outcomes you consider. You will calculate conditional probabilities using event totals, Venn diagrams, trees and two-way tables, then choose a representation that makes the denominator clear.
Statistics 3.3 - Probability modelling and assumptionsA probability calculation is only as useful as its assumptions. You will build and compare models, use observed evidence carefully, and explain how changing fairness, independence or constant-probability assumptions changes a prediction.

4 Statistical distributions

Statistics 4.1 - Discrete and binomial probability distributionsUse a probability distribution to describe a whole set of possible outcomes. Build Binomial models, calculate individual and cumulative probabilities, and decide whether the assumptions fit the situation.
Statistics 4.2 - Normal distribution as a modelUse the Normal distribution to turn measurements into probabilities and probabilities back into unknown values. Connect its shape to data, recover its parameters, and approximate suitable Binomial counts with a continuity correction.
Statistics 4.3 - Choosing an appropriate probability distributionChoose a probability model from what the random variable counts or measures. Justify its assumptions in context, recognise unsuitable models, and link a Normal measurement to a Binomial count when the question changes variable.

5 Statistical hypothesis testing

Statistics 5.1 - Language of statistical hypothesis testingHow unusual must a sample be before you question a population claim? You will learn to set up hypotheses, recognise extreme results and make careful decisions using critical regions or p-values. The same language will then let you test a claim about linear correlation.
Statistics 5.2 - Hypothesis test for a binomial proportionA high sample proportion can be evidence of a population change, but how high is high enough? You will carry out exact binomial tests, choose inclusive tails, find critical regions and explain their actual significance. You will also check whether the sampling and binomial assumptions support the...
Statistics 5.3 - Hypothesis test for a normal meanA sample mean below a packet's advertised mass might be ordinary sampling variation, or evidence that the filling machine has shifted. You will test a Normal population mean when its variance is known, given or assumed, using the distribution of the sample mean. Critical values and p-values will...

Mechanics (Paper 3, Section B)

7 Kinematics

Mechanics 7.1 - The language of kinematicsDescribe a journey precisely by separating position, displacement, distance, velocity, speed and acceleration. You will calculate average rates and use signs to explain motion, ready for the graphs, formulae and calculus that follow.
Mechanics 7.2 - Kinematics graphs in straight-line motionRead a motion graph as information about gradients, areas and direction. You will use displacement-time and velocity-time graphs to solve straight-line journeys, including reversals and journeys with several stages.
Mechanics 7.3 - Constant acceleration formulaeDerive the constant-acceleration formulae from a velocity-time graph, then use them in straight-line and two-dimensional motion. You will choose equations, preserve vector components and distinguish displacement from position and speed.
Mechanics 7.4 - Calculus in kinematicsUse differentiation and integration to connect position, velocity and acceleration when acceleration may vary. You will apply initial conditions in one and two dimensions, calculate speed and direction at a specified time, and account for reversals when finding distance.
Mechanics 7.5 - Motion under gravity using vectorsModel a projectile with constant downward acceleration and a shared time for its horizontal and vertical motion. You will derive flight and path results, solve point-passing problems and recognise when launch and landing heights change the equations you need.

8 Forces and Newton's laws

Mechanics 8.1 - Force and Newton's first lawLearn to name the forces on one body, add them with their directions and use Newton's first law to decide whether its velocity changes. You will build force diagrams and use zero-resultant equations for objects at rest and objects already moving.
Mechanics 8.2 - Newton's second law and resolving forcesConnect a force diagram to acceleration using Newton's second law. You will form signed equations along a straight line, work with perpendicular components and simple two-dimensional vectors, and choose useful axes for particles on smooth inclined planes.
Mechanics 8.3 - Weight and motion under gravityUse weight and gravitational acceleration in straight-line motion. You will distinguish kilograms from newtons, choose signs for vertical motion, find greatest heights and impact speeds, and judge the accuracy and limits of a constant-gravity model.
Mechanics 8.4 - Newton's third law and equilibriumLearn to decide which body a force acts on, then build equilibrium or motion equations from its free-body diagram. You will connect Newton's third law to contact forces, lift reactions, ideal pulleys and particles on inclined planes. A careful choice of body does much of the work before the algeb...
Mechanics 8.5 - Addition and resultant forces in dynamicsCombine forces by adding their vectors, then use the resultant to find acceleration in a plane. You will turn magnitude-and-direction descriptions into components, recover directions without losing the quadrant, and use a given acceleration direction to determine unknown forces. The same two comp...
Mechanics 8.6 - Friction, limiting friction and static frictionDecide how much friction a rough surface actually supplies, and when that amount reaches its limiting value. You will model rest and motion on horizontal and inclined surfaces, check what happens after stopping, and account for an angled force changing the normal reaction. The state of motion com...