Edexcel A-Level International Mathematics

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Pure Mathematics 1

Quadratics and inequalities

P1 1.3, P1 1.5 - Quadratic graphs and completed squaresRead the key features of a quadratic graph and rewrite its equation to reveal its turning point. P1 1.3 connects the algebra to the graph; P1 1.5 supplies completing the square, which can reveal a minimum or maximum without plotting a table of values.
P1 1.5 - Solving quadratic equationsSolve quadratic equations by factorisation, completing the square, the quadratic formula and a calculator. Choose a method that suits the equation, retain every solution and distinguish an exact answer from a rounded approximation.
P1 1.4 - Counting real rootsUse the discriminant to decide how many real roots a quadratic has before solving it. Then turn requirements such as “two distinct real roots” into conditions on an unknown coefficient.
P1 1.6 - Simultaneous equationsSolve two equations by replacing one unknown with an equivalent expression. You will find exact intersection coordinates, then use a quadratic's discriminant to explain why a line and a curve can meet twice, once or not at all.
P1 1.7, P1 1.9 - Solving inequalitiesFind complete ranges of values that satisfy linear and quadratic inequalities. P1 1.9 supplies the algebraic solutions; P1 1.7 connects them to graph positions, including a curve below a line and fractions with excluded values.
P1 1.8 - Shading inequalitiesRepresent linear and quadratic inequalities as regions of the coordinate plane. You will choose the correct boundary style, test which side satisfies a condition, and locate the overlap when several inequalities must hold together.

Graphs and straight lines

P1 2.1, P1 2.2 - Straight-line equationsUse coordinates and direction conditions to determine a straight line exactly. P1 2.1 develops point-gradient and general equations; P1 2.2 supplies the parallel and perpendicular conditions that let you choose the required direction.
P1 1.11 - Cubic and reciprocal graphsSketch simple cubic and reciprocal curves using roots, signs, intercepts and asymptotes. You will also interpret an equation as a search for shared points on two graphs, connecting an exact algebraic solution with its picture.
P1 1.12 - Moving and stretching graphsMove, stretch and reflect a graph by following its coordinates. Use the same four transformations on familiar curves and on a graph whose equation you do not know, keeping track of intercepts, turning points and asymptotes.

Trigonometry

P1 3.1 - Solving trianglesFind missing sides, angles and areas in triangles that do not contain a right angle. Choose the sine rule or cosine rule from the information available, then connect the result to the area formula.
P1 3.1 - The ambiguous sine ruleDecide when two sides and a non-included angle determine no triangle, one triangle or two triangles. Find both sine-rule candidates, test them geometrically and finish each valid triangle without losing a solution.
P1 3.2 - Radians and sectorsMeasure angles using arc length rather than degrees. Convert between the two measures, calculate arc lengths and sector areas, and build composite areas and perimeters from the correct curved and straight boundaries.
P1 3.3, P1 1.11, P1 1.12 - Trigonometric graphsUnderstand the shapes, symmetries and periods of sine, cosine and tangent in radians. Transform their key features and read intersections to solve equations graphically on a stated interval.

Pure Mathematics 2

Proof and polynomial algebra

P2 1.1–P2 1.3 - Proof and counterexamplesLearn how assumptions support a logical conclusion, how exhaustion covers every possible case, and how one counterexample can overturn a universal claim. P2 1.1 supplies the structure of an argument, P2 1.2 gives a proof method, and P2 1.3 explains what is enough to disprove a statement.
P2 2.1 - Dividing polynomialsDivide a polynomial by a linear expression, identify its quotient and remainder, and verify the result by multiplication. The division identity makes these answers precise and prepares you for the factor and remainder theorems.
P2 2.1 - Factor and remainder theoremsFind a remainder by substitution, recognise linear factors, factorise cubics and determine unknown coefficients. These theorems come from the division identity, so the value you substitute and the equation you form both have a reason.

Exponentials and logarithms

P2 5.1 - Exponential functionsUnderstand how a variable exponent produces growth or decay, and use exact values to describe exponential graphs. You will identify their intercepts and asymptotes, including after a vertical translation.
P2 5.2 - Logarithms and their lawsRead a logarithm as a missing exponent, then use logarithm laws to simplify expressions and solve equations. Positive arguments matter throughout: you will learn why algebra can produce candidates that the original logarithms do not allow.
P2 5.3 - Solving exponential equationsSolve equations in which the unknown occurs in an exponent, using a common base or logarithms. You will derive change of base, calculate approximations without losing accuracy, and keep exact answers when finding graph intersections.

Sequences and series

P2 4.1, P2 4.3 - Describing sequencesUse a formula or a recurrence rule to generate a sequence, as in P2 4.1, then describe increasing, decreasing and periodic behaviour, as in P2 4.3. You will distinguish a term's position from its value and use the rule to justify behaviour beyond a few listed terms.
P2 4.2 - Arithmetic sequences and sumsFind terms and finite sums of arithmetic sequences, and understand the proof behind the sum formula. You will use sigma notation, sum the first natural numbers and recover unknown sequence parameters from given information.
P2 4.4 - Geometric sequences and finite sumsFind terms and totals when each new term is obtained by multiplying by the same number. You will derive the finite-sum formula, then use logarithms to work backwards from a total to the number of terms.
P2 4.4 - Infinite geometric sumsUnderstand how infinitely many terms can have a finite total, and recognise exactly when a geometric series has such a sum. Then combine finite and infinite sums to recover an unknown ratio and first term.
P2 4.5 - Binomial expansionExpand a positive-integer power without multiplying every bracket separately. You will use binomial coefficients to find particular terms, then track the contributions when an expansion is multiplied by another bracket.

Calculus applications

P2 7.1 - Stationary points and curve shapeUse the sign of a derivative to decide where a curve rises or falls. Find stationary points, distinguish maxima and minima from flat points that do not turn, and combine this information with intercepts to sketch a curve.
P2 7.1 - Optimisation in contextTurn a practical constraint into a function of one variable, then use differentiation to find and justify an optimum. Keep the physical domain, allowed endpoints and requested quantity in view so that a stationary input becomes a useful answer.
P2 8.1, P2 8.2 - Definite integrals and areaEvaluate a definite integral by substituting its limits into an antiderivative. Distinguish signed integrals from positive area, then calculate regions bounded by curves and straight lines, including areas between two curves that cross.
P2 8.3 - The trapezium ruleApproximate an integral using equally spaced values from a function or a table. Understand the trapezium weights, use curve shape to identify overestimates and underestimates, and judge how refinement affects accuracy and what an error estimate can establish.

Pure Mathematics 3

Functions and graphs

P3 1.1 - Simplifying rational expressionsSimplify algebraic fractions by factorising, combining them and dividing polynomials. Keep track of excluded inputs so that a shorter expression still describes the original fraction accurately.
P3 1.2 - Functions domain and rangeDescribe a function using its rule, its allowed inputs and its possible outputs. Use domains, graphs and algebra to distinguish one-one from many-one mappings and to find complete ranges.
P3 1.2 - Composite and inverse functionsCombine functions in a specified order and reverse a one-one function with an inverse. Domains decide which combinations exist, which inverse branch to choose and which portion of a reflected graph to draw.
P3 1.3 - Modulus graphs and equationsInterpret modulus as distance, sketch graphs involving absolute values and solve equations and inequalities from their intersections. Distinguish changing a function's output from changing its input.
P3 1.4 - Combining graph transformationsCombine familiar graph transformations by tracking inputs, outputs and key points. Use their order to distinguish similar equations, then build transformed trigonometric sketches with exact coordinates and periods.

Trigonometric functions

P3 2.1 - Reciprocal trigonometric functionsA sine value can be small while its reciprocal is large. In P3 2.1 you will connect secant, cosecant and cotangent to familiar trigonometric functions, explain their graph branches and asymptotes, and use the relationships to solve equations in degrees and radians.
P3 2.1, P3 2.2 - Inverse functions and identitiesSine takes an angle to a number, but many angles give the same number. P3 2.1 uses restricted domains to make arcsine, arccosine and arctangent genuine inverse functions and explain their graphs. P3 2.2 then extends the Pythagorean identity to secant, cosecant and cotangent.
P3 2.3 - Compound and double anglesKnowing the sine of two angles does not mean you can add their sine values to find the sine of their sum. P3 2.3 gives formulae for sums, differences and double angles; you will use them for exact values, half angles and algebraic proofs of identities.
P3 2.3 - Combining sine and cosineTwo waves of the same frequency can combine into one shifted wave. In P3 2.3 you will rewrite a sine–cosine combination with a positive amplitude, choose the phase angle from both coefficient signs, and use the result to solve equations and find greatest and least values.

Differentiation

P3 4.1 - Differentiating standard functionsExtend differentiation beyond powers to exponential, logarithmic and trigonometric functions. Learn how their gradients change, handle linear expressions inside functions and combine the results to find tangents.
P3 4.2 - Product and chain rulesDifferentiate functions built by putting one function inside another or multiplying two functions together. Use their structure to choose the chain rule, the product rule or both, and check that no changing factor has been overlooked.
P3 4.2, P3 4.3 - Quotient and inverse derivativesUse the quotient rule from P3 4.2 to differentiate fractions of functions and obtain the derivatives of secant, cosecant and cotangent. Then use P3 4.3 to find a gradient when the curve is given as x in terms of y, taking care over reciprocal rates and inverse branches.
P3 4.4 - Modelling growth and decayUse exponential functions to model quantities that grow or decay, and differentiate them to find instantaneous rates. Interpret initial values and long-term predictions, then use observations and physical limits to test and improve a model.

Pure Mathematics 4

Proof and algebra

P4 1.1 - Proof by contradictionProve a statement by showing that its negation leads to an impossibility. Build precise assumptions, prove that √(2) is irrational and that there are infinitely many primes, then adapt the same reasoning to unfamiliar statements.
P4 2.1 - Partial fractionsA single algebraic fraction can hide a much simpler sum. Learn how to recover that sum, including repeated factors and fractions needing division first, then check that the result is an identity on the original domain.
P4 4.1 - Binomial series with rational powersA square root or reciprocal can be approximated by a polynomial, even though its full expansion never ends. Learn to build binomial series, state where they are valid and combine them with partial fractions to approximate rational functions.

Parametric curves and rates

P4 3.1 - Parametric equationsDescribe a curve by giving both coordinates in terms of a third variable. Learn to move between parametric and Cartesian descriptions while keeping the allowed points, excluded values and direction of travel.
P4 5.1 - Implicit and parametric differentiationFind gradients without first writing a curve as y=f(x). Extend the chain rule to implicit and parametric descriptions, use the gradients for tangents and normals, and investigate points where a derivative formula cannot be used directly.
P4 5.2 - Connected rates and differential equationsConnect changing quantities with the chain rule, then translate a physical rate statement into a differential equation. Learn to choose signs, constants, units and initial conditions so that the equation describes the stated model.

Integration methods

P4 6.2 - Integration by substitutionUse a new variable to turn a complicated integral into a familiar one. You will connect substitution to the chain rule, change every part of the integrand consistently, and evaluate definite integrals with appropriate limits and domains.
P4 6.2 - Integration by partsReverse the product rule to integrate products whose factors cannot be handled by substitution. Learn how to choose a useful factor to differentiate, integrate logarithms, and continue when the method must be applied more than once.
P4 6.3 - Integrating rational functionsUse partial fractions to turn rational functions into logarithms and powers that you can integrate. Distinguish simple from repeated factors, divide improper fractions first, and check the domain before evaluating a definite integral.
P4 6.4 - Solving separable differential equationsRecover a function from a differential equation by separating its variables and integrating. Distinguish general from particular solutions, keep solutions that division might lose, and use initial conditions and domain restrictions to interpret a model.
P4 6.5 - Areas from parametric equationsCalculate area without eliminating a curve's parameter. Convert vertical-strip widths using the derivative of its x-coordinate, match parameter limits to the required boundaries, and keep the direction and sign of each contribution consistent.
P4 6.1 - Volumes of revolutionFind volumes by rotating a region about the x-axis, using either Cartesian or parametric equations. Understand why the radius is squared, how the limits follow the region, and how to keep a physical volume positive and correctly scaled.

Vectors

P4 7.1–P4 7.3 - Vectors in two and three dimensionsDescribe vectors using two or three components, find their magnitudes and unit directions, and combine them algebraically. Connect component calculations to arrows, journeys and parallel directions so that the algebra has a geometric meaning.
P4 7.4–P4 7.6 - Position vectors and linesUse position vectors to find displacements and distances, then describe a straight line with a vector equation. Decide whether lines coincide, are parallel, intersect or are skew by checking their directions and all three coordinates.
P4 7.7 - The scalar productUse the scalar product to connect vector components with angles and perpendicular directions. Choose the correct arrows for an angle, distinguish vectors from lines, and solve geometric problems by finding a perpendicular foot.

Mechanics 1

Forces and motion

M1 4.1, M1 5.1 - Newton's laws and force componentsConnect force diagrams to acceleration using Newton's laws in M1 4.1. Resolve forces into perpendicular components using M1 5.1, then build scalar and vector equations for the chosen object.
M1 4.4, M1 4.2 - Motion with frictionFind the contact forces on a sliding particle, then use them to predict its motion on a horizontal surface or a slope. You will also split a journey into stages when an applied force changes. Take g=9.8 m s⁻² and give non-exact final numerical answers to three significant figures.
M1 4.2 - Connected particlesUse the string model to connect two particles' motions, then write one force equation for each particle. You will find acceleration and tension, including what changes when one particle hits the ground. Take g=9.8 m s⁻² and round non-exact final numerical answers to three signif...
M1 4.3 - Momentum and impulseTrack direction as well as speed when a force acts briefly or two particles collide. You will calculate momentum and impulse, conserve total momentum when external impulse is negligible, and interpret a negative velocity after impact.

Mechanics 2

Impacts and rigid bodies

M2 4.1, M2 4.2 - Momentum and restitutionUse momentum and impulse as vectors, then combine conservation of momentum with Newton's law of restitution for direct impacts. Interpret the coefficient of restitution and calculate the kinetic energy lost in a collision.
M2 4.3 - Successive collisionsFollow velocities through successive direct impacts, including three-particle systems and normal rebounds from a fixed smooth plane. Use positions and relative velocities to decide whether the assumed next collision occurs.
M2 5.1, M2 5.2 - Rigid-body equilibriumBalance forces and moments for rods and ladders in equilibrium. Resolve non-parallel forces, choose perpendicular moment arms and distinguish ordinary friction from limiting friction at smooth or rough contacts. Use g=9.8 m s⁻² when a numerical value is needed.

Statistics 1

Data and modelling

S1 1.1 - Modelling with dataUse a statistical model to turn a practical question into a prediction, then compare that prediction with observations. Learn why assumptions make a model usable and why checking a model can lead you to change it.
S1 2.1 - Reading statistical diagramsRead histograms, stem-and-leaf diagrams and box plots, including displays with unequal intervals and two groups. Use their centre and spread to make comparisons in context, and understand the information each display keeps or loses.
S1 2.2, S1 2.3 - Location and spreadUse measures of location in S1 2.2 to describe a typical value, and measures of dispersion in S1 2.3 to describe variability. Calculate and interpret summaries for raw and grouped data, estimate percentiles by interpolation, and recover summaries after coding.
S1 2.4 - Skewness and outliersDescribe skewness using evidence from a distribution, apply a stated rule for outliers, and explain what an unusual observation does and does not tell you. Choose suitable measures when comparing groups whose extreme values may affect their summaries.

Probability

S1 3.1, S1 3.2 - Events and sample spacesUse elementary probabilities in S1 3.1 and sample spaces, complements, exclusive events, unions and intersections in S1 3.2. Translate “not”, “and” and “or” into precise events, then count outcomes or combine supplied probabilities without counting an overlap twice.
S1 3.2–S1 3.4 - Conditional probability and independenceUse conditional probability in S1 3.2, independence in S1 3.3, and sum and product laws with Venn diagrams in S1 3.4. Calculate how new information changes a probability and recognise when it leaves the chance unchanged.
S1 3.4 - Tree diagrams and repeated samplingBuild probability trees, multiply conditional probabilities along a path and add disjoint paths. Model repeated random sampling with and without replacement, including probabilities given a specified result.

Statistics 2

Continuous random variables

S2 2.1, S2 2.2 - Probability density functionsFor a continuous measurement, probability belongs to an interval rather than to an individual value. Learn how density represents probability as area, check whether a proposed density is valid, and calculate probabilities across polynomial and piecewise densities.
S2 2.2, S2 2.3, S2 2.5 - Cumulative functions and quantilesA cumulative distribution function collects probability from left to right. Build one from a density, recover the density by differentiation, and use the two functions to locate medians, quartiles and modes.
S2 2.4 - Continuous expectation and varianceCalculate a continuous distribution's mean and variance by weighting values with density. Use these moments alongside total probability to determine unknown parameters, and check that the resulting density remains valid.

Sampling and hypothesis tests

S2 4.1, S2 4.2 - Samples and sampling distributionsA sample gives one set of observations; another sample could give a different result. S2 4.1 explains who or what is being sampled and the tradeoffs between a census and a sample survey. S2 4.2 turns a numerical sample summary into a random variable with its own sampling distribution.
S2 4.3–S2 4.5 - Hypotheses and critical regionsAn unusual sample can challenge a probability model without proving that it is false. S2 4.3 introduces hypotheses and conclusions, S2 4.4 uses a test statistic and critical region to make a decision, and S2 4.5 distinguishes one-tailed and two-tailed tests. You will connect each decision to the...
S2 4.6 - Testing a binomial probabilityA sample proportion can differ from a claimed probability simply through chance. You will use exact binomial tails to construct critical regions and test a population probability, handle two-sided alternatives, and use a Normal approximation when a suitable large-sample test calls for it.
S2 4.6 - Testing a Poisson meanA rate claim becomes testable only after it is matched to the time or area actually observed. You will scale the null Poisson mean, find exact one-tailed and two-tailed critical regions, calculate actual significance levels and draw conclusions about the rate in context.

Decision Mathematics 1

Algorithms

D1 1.1 - Following an algorithmFollow instructions written as a flowchart or a numbered procedure, recording how each value changes. You will also learn the middle-item convention used throughout searching and sorting, where choosing the wrong middle item changes the whole trace.
D1 1.2 - Sorting listsSort numerical lists using bubble sort and quicksort, showing the intermediate states that distinguish the two algorithms. You will track each pivot and sublist and recognise when the specified algorithm has finished.
D1 1.2 - Searching and bin packingUse binary search to locate an item in an ordered list, then allocate whole items to bins using three packing methods. You will distinguish a valid allocation from a proven minimum, using capacity and item sizes to justify your conclusion.

Networks and routes

D1 2.1 - Graphs and network matricesDescribe connections using graph language, distinguish a spanning tree from a network containing cycles, and translate between a weighted network and its matrix. These representations let you record exactly which connections exist before choosing an algorithm to use them.
D1 2.1 - Minimum spanning treesConnect every vertex of a network with the least possible total edge weight, using Kruskal's and Prim's algorithms. You will apply Prim's method directly to a matrix, handle equal weights, and check that a proposed connector really is a spanning tree.
D1 2.2 - Dijkstra's algorithmFind a shortest route through a weighted network by keeping tentative distances and fixing them in a safe order. You will show Dijkstra's working values, recover the route from the final labels, and interpret changes when a road closes.
D1 3.1 - Route inspectionFind a shortest inspection route that travels along every edge and returns to its start. You will identify odd vertices, compare every possible pairing, specify the repeated edges, and adapt the parity reasoning when different start and finish vertices are prescribed.
D1 3.2–D1 3.4 - Travelling salesman boundsVisit every vertex and return to the start while keeping the total journey short. D1 3.2 distinguishes practical and classical tours; D1 3.3 builds complete shortest-distance networks and spanning-tree bounds; D1 3.4 supplies the nearest neighbour algorithm. Bounds can sometimes prove an optimum...

Project planning

D1 4.1, D1 4.2 - Activity networksD1 4.1 turns a project's immediate-predecessor table into an activity-on-arc network, including dummy activities when needed. D1 4.2 reads the network backwards to recover its table. Both directions let you check that the model imposes exactly the intended dependencies.
D1 4.3 - Critical paths and event timesFind a project's minimum completion time using a forward pass, then work backwards to find how late each event may occur without delaying completion. You will identify all critical paths and calculate earliest and latest start and finish times for individual activities.
D1 4.4 - Float and schedulingA project's minimum completion time also has to fit the workers available. Calculate total float, represent activity windows on a Gantt chart, and construct schedules that respect both precedence and resource limits.