A level · Pearson Edexcel

Further Mathematics 9FM0

Every topic Pearson Edexcel can examine you on. Open a topic to see its sub-topics, lessons and practice questions.

  • 4 papers
  • 216 sub-topics
  • Y12Y13When each is usually taught

Official specification (PDF)

Core Pure Mathematics

Compulsory

Papers 1 and 2. Either paper can assess any Core Pure content.

1ProofProof by induction: show a statement is true for n = 1, assume it for n = k, then prove it for n = k + 1. Used for sums of series, divisibility and powers of matrices.Y121 sub-topic
  1. 1.1Proof by mathematical inductionY12
2Complex numbersNumbers of the form x + iy, where i² = −1. You add, multiply and divide them, find complex roots of polynomials, draw them on an Argand diagram, use modulus–argument and exponential form, de Moivre's theorem, nth roots and loci.11 sub-topics
  1. 2.1Solving polynomial equations with complex rootsY12
  2. 2.2Arithmetic in Cartesian formY12
  3. 2.3Complex conjugatesY12
  4. 2.4Argand diagramsY12
  5. 2.5Modulus–argument formY12
  6. 2.6Multiplying and dividing in modulus–argument formY12
  7. 2.7Loci and regions in the Argand diagramY12
  8. 2.8De Moivre's theoremY13
  9. 2.9Exponential (Euler) formY13
  10. 2.10nth roots of a complex numberY13
  11. 2.11Roots of unity and geometryY13
3MatricesGrids of numbers that act on vectors. You add and multiply matrices, find determinants and inverses, use them as transformations (rotations, reflections, enlargements), find invariant points and lines, and solve three equations in three unknowns.Y128 sub-topics
  1. 3.1Matrix operationsY12
  2. 3.2Zero and identity matricesY12
  3. 3.3Linear transformationsY12
  4. 3.4Invariant points and invariant linesY12
  5. 3.5DeterminantsY12
  6. 3.6Inverse matricesY12
  7. 3.7Solving three linear equationsY12
  8. 3.8Geometry of three planesY12
4Further algebra and functionsLinks between the roots and coefficients of polynomials, summing series with standard formulae and the method of differences, and Maclaurin series that approximate functions as polynomials.6 sub-topics
  1. 4.1Roots and coefficients of polynomialsY12
  2. 4.2Transforming rootsY12
  3. 4.3Standard results for sumsY12
  4. 4.4Method of differencesY13
  5. 4.5Maclaurin seriesY13
  6. 4.6Standard Maclaurin seriesY13
5Further calculusVolumes made by rotating a curve about an axis, improper integrals (where a limit is infinite), the mean value of a function, and integrals that lead to inverse trigonometric functions.7 sub-topics
  1. 5.1Volumes of revolution (Cartesian)Y12
  2. 5.1Volumes of revolution (parametric and modelling)Y13
  3. 5.2Improper integralsY13
  4. 5.3Mean value of a functionY13
  5. 5.4Integration using partial fractionsY13
  6. 5.5Differentiating inverse trigonometric functionsY13
  7. 5.6Integrals leading to inverse trig functionsY13
6Further vectorsLines and planes in three dimensions. You write their equations, find angles and intersections, and use the scalar product to find distances between points, lines and planes.Y125 sub-topics
  1. 6.1Equations of lines in 3DY12
  2. 6.2Equations of planesY12
  3. 6.3Scalar product, planes and anglesY12
  4. 6.4Perpendicular vectorsY12
  5. 6.5Intersections and perpendicular distancesY12
7Polar coordinatesDescribing points by a distance r and an angle θ instead of x and y. You sketch curves like r = 1 + cos θ, convert to and from Cartesian form, and find areas inside curves and tangents.Y133 sub-topics
  1. 7.1Polar and Cartesian conversionY13
  2. 7.2Sketching polar curvesY13
  3. 7.3Areas and tangentsY13
8Hyperbolic functionssinh, cosh and tanh: functions built from eˣ and e⁻ˣ that behave like the trigonometric functions. You graph them, use their identities, inverses and logarithmic forms, and differentiate and integrate them.Y135 sub-topics
  1. 8.1Definitions and graphsY13
  2. 8.2Differentiating and integrating hyperbolic functionsY13
  3. 8.3Inverse hyperbolic functionsY13
  4. 8.4Logarithmic formsY13
  5. 8.5Integrals leading to inverse hyperbolic functionsY13
9Differential equationsEquations that involve derivatives. You solve first-order equations with an integrating factor and second-order equations with constant coefficients, then use them to model motion, oscillations and populations.Y139 sub-topics
  1. 9.1Integrating factor methodY13
  2. 9.2General and particular solutionsY13
  3. 9.3Modelling with first-order equationsY13
  4. 9.4Second-order homogeneous equationsY13
  5. 9.5Second-order non-homogeneous equationsY13
  6. 9.6Auxiliary equation casesY13
  7. 9.7Simple harmonic motionY13
  8. 9.8Damped and forced oscillationsY13
  9. 9.9Coupled first-order systemsY13

Further Pure Mathematics 1 (FP1)

Optional

Paper 3, option 3A.

1Further trigonometryThe t-formulae, where t = tan(θ/2), turn trigonometric expressions into algebra, which helps you solve equations and prove identities.Y12Option FP1 · 3 sub-topics
  1. 1.1The t-formulaeY12
  2. 1.2Proving identities with t-formulaeY12
  3. 1.3Solving equations with t-formulaeY12
2Further calculusTaylor series, using series to find limits, Leibnitz's theorem for repeated differentiation, l'Hôpital's rule and the Weierstrass substitution.Y13Option FP1 · 5 sub-topics
  1. 2.1Taylor seriesY13
  2. 2.2Series expansions to evaluate limitsY13
  3. 2.3Leibnitz's theoremY13
  4. 2.4L'Hôpital's ruleY13
  5. 2.5Weierstrass substitutionY13
3Further differential equationsUsing Taylor series to find approximate solutions of differential equations, and substitutions that turn harder equations into ones you can solve.Y13Option FP1 · 2 sub-topics
  1. 3.1Taylor series solutions of differential equationsY13
  2. 3.2Reducible differential equationsY13
4Coordinate systemsConic sections: the parabola and rectangular hyperbola in Year 12, then the ellipse and hyperbola in Year 13. You use their Cartesian and parametric equations, tangents, normals, foci and directrices.Option FP1 · 6 sub-topics
  1. 4.1Conics: parabola and rectangular hyperbolaY12
  2. 4.1Conics: ellipse and hyperbolaY13
  3. 4.2Focus–directrix propertiesY12
  4. 4.2Eccentricity, foci and directrices of ellipse and hyperbolaY13
  5. 4.3Tangents and normals to conicsY12
  6. 4.4Loci problemsY12
5Further vectorsThe vector product, the scalar triple product for volumes, and writing lines and planes using the vector product.Option FP1 · 3 sub-topics
  1. 5.1Vector productY12
  2. 5.2Scalar triple productY12
  3. 5.3Vector geometry with the vector productY13
6Further numerical methodsNumerical methods for differential equations, such as Euler's method and its improved versions, and Simpson's rule for integrals.Option FP1 · 2 sub-topics
  1. 6.1Numerical solution of differential equationsY12
  2. 6.2Simpson's ruleY13
7InequalitiesSolving inequalities that involve fractions and the modulus function, by algebra and with sketches.Y12Option FP1 · 1 sub-topic
  1. 7.1InequalitiesY12

Further Statistics 1 (FS1)

Optional

Paper 3, option 3B.

1Discrete probability distributionsRandom variables that take separate values. You find their mean (expected value) and variance, including for functions of the variable.Y12Option FS1 · 1 sub-topic
  1. 1.1Discrete probability distributionsY12
2Poisson and binomial distributionsThe Poisson distribution for counts of random events, and how it combines with and approximates the binomial distribution.Y12Option FS1 · 3 sub-topics
  1. 2.1The Poisson distributionY12
  2. 2.2Mean and variance of binomial and PoissonY12
  3. 2.3Poisson approximation to the binomialY12
3Geometric and negative binomial distributionsThe geometric and negative binomial distributions: how many trials until the first (or rth) success.Y13Option FS1 · 3 sub-topics
  1. 3.1Geometric and negative binomial modelsY13
  2. 3.2Mean and variance of the geometric distributionY13
  3. 3.3Mean and variance of the negative binomialY13
4Hypothesis testingHypothesis tests for the mean of a Poisson distribution and the parameter of a geometric distribution, including critical regions.Option FS1 · 2 sub-topics
  1. 4.1Hypothesis tests for the mean of a Poisson distributionY12
  2. 4.2Hypothesis tests for the parameter of a geometric distributionY13
5The central limit theoremThe Central Limit Theorem: averages of large samples are approximately normally distributed, whatever the original distribution.Y13Option FS1 · 1 sub-topic
  1. 5.1The central limit theoremY13
6Chi-squared testsχ² (chi-squared) tests: checking whether data fits a model (goodness of fit) and whether two factors are linked (contingency tables).Option FS1 · 2 sub-topics
  1. 6.1Chi-squared tests (discrete uniform, binomial, Poisson)Y12
  2. 6.1Chi-squared tests for geometric distributionsY13
7Probability generating functionsProbability generating functions, a compact way to store a distribution that makes means, variances and sums of variables quick to find.Y13Option FS1 · 3 sub-topics
  1. 7.1Probability generating functionsY13
  2. 7.2Mean and variance from PGFsY13
  3. 7.3PGF of a sum of independent variablesY13
8Quality of testsHow good a test is: Type I and Type II errors, the size of a test and its power.Y13Option FS1 · 1 sub-topic
  1. 8.1Quality of testsY13

Further Mechanics 1 (FM1)

Optional

Paper 3, option 3C.

1Momentum and impulseMomentum and impulse, in one and two dimensions, and how momentum is conserved in collisions.Option FM1 · 2 sub-topics
  1. 1.1Momentum and impulseY12
  2. 1.2Momentum as a vectorY13
2Work, energy and powerWork done by forces, kinetic and potential energy, power, and solving problems with conservation of energy.Y12Option FM1 · 1 sub-topic
  1. 2.1Work, energy and powerY12
3Elastic strings and springs and elastic energyHooke's law for strings and springs, the energy they store, and problems that combine elastic energy with other energies.Y13Option FM1 · 2 sub-topics
  1. 3.1Hooke's lawY13
  2. 3.2Elastic potential energyY13
4Elastic collisions in one dimensionCollisions in a straight line, using Newton's law of restitution and conservation of momentum, including loss of kinetic energy.Y12Option FM1 · 2 sub-topics
  1. 4.1Direct impact and Newton's law of restitutionY12
  2. 4.2Successive direct impactsY12
5Elastic collisions in two dimensionsOblique collisions with a wall and between smooth spheres, splitting velocities into components along and across the line of centres.Y13Option FM1 · 2 sub-topics
  1. 5.1Oblique impact of smooth elastic spheresY13
  2. 5.2Successive oblique impactsY13

Decision Mathematics 1 (D1)

Optional

Paper 3, option 3D.

1Algorithms and graph theoryWhat an algorithm is, sorting and bin-packing algorithms, and the language of graphs: vertices, edges, degrees, paths and cycles.Option D1 · 4 sub-topics
  1. 1.1Algorithms and orderY12
  2. 1.2Bin packing and sortingY12
  3. 1.3Graph theoryY12
  4. 1.4Planarity algorithm and Hamiltonian cyclesY13
2Algorithms on graphsMinimum spanning trees with Kruskal's and Prim's algorithms, and shortest paths with Dijkstra's and Floyd's algorithms.Option D1 · 3 sub-topics
  1. 2.1Minimum spanning treesY12
  2. 2.2Dijkstra's algorithmY12
  3. 2.2Floyd's algorithmY13
3Algorithms on graphs IIRoute inspection (the Chinese postman problem) and the travelling salesperson problem, with upper and lower bounds.Option D1 · 2 sub-topics
  1. 3.1Route inspection (Chinese postman)Y12
  2. 3.2Travelling salesman problemY13
4Critical path analysisPlanning projects with activity networks: early and late times, critical paths, float, Gantt charts and scheduling.Option D1 · 6 sub-topics
  1. 4.1Activity networksY12
  2. 4.2Precedence tablesY12
  3. 4.3Critical path analysisY12
  4. 4.4Float and Gantt chartsY12
  5. 4.5Resource histograms and levellingY13
  6. 4.6SchedulingY13
5Linear programmingWriting real problems as linear programs and solving them graphically and with the Simplex algorithm.Option D1 · 5 sub-topics
  1. 5.1Formulating linear programming problemsY12
  2. 5.1Slack, surplus and artificial variablesY13
  3. 5.2Graphical solution of LP problemsY12
  4. 5.3The simplex algorithmY13
  5. 5.4Two-stage simplex and big-MY13

Further Pure Mathematics 2 (FP2)

Optional

Paper 4, option 4A.

1GroupsGroups: sets with an operation that follow four rules. You use Cayley tables, orders, subgroups, Lagrange's theorem and isomorphisms.Y13Option FP2 · 5 sub-topics
  1. 1.1Group axiomsY13
  2. 1.2Examples of groups and Cayley tablesY13
  3. 1.3Order and subgroupsY13
  4. 1.4Lagrange's theoremY13
  5. 1.5IsomorphismY13
2Further calculusReduction formulae for integrals, arc lengths and areas of surfaces of revolution.Y13Option FP2 · 2 sub-topics
  1. 2.1Reduction formulaeY13
  2. 2.2Arc length and area of surface of revolutionY13
3Further matrix algebraEigenvalues and eigenvectors, diagonalising matrices, and the Cayley–Hamilton theorem.Y13Option FP2 · 3 sub-topics
  1. 3.1Eigenvalues and eigenvectorsY13
  2. 3.2DiagonalisationY13
  3. 3.3Cayley–Hamilton theoremY13
4Further complex numbersLoci and regions in the Argand diagram, and transformations of the complex plane.Y13Option FP2 · 2 sub-topics
  1. 4.1Further loci and regionsY13
  2. 4.2Transformations of the complex planeY13
5Number theoryDivisibility, the Euclidean algorithm, modular arithmetic, Fermat's little theorem and counting problems.Y13Option FP2 · 7 sub-topics
  1. 5.1Division algorithm and Euclidean algorithmY13
  2. 5.2Bézout's identityY13
  3. 5.3Modular arithmeticY13
  4. 5.4Fermat's little theoremY13
  5. 5.5Divisibility testsY13
  6. 5.6Solving congruence equationsY13
  7. 5.7CombinatoricsY13
6Further sequences and seriesFirst- and second-order recurrence relations, solved in closed form and used in modelling.Y13Option FP2 · 3 sub-topics
  1. 6.1Recurrence relationsY13
  2. 6.2Solving recurrence relationsY13
  3. 6.3Proving closed forms by inductionY13

Further Statistics 2 (FS2)

Optional

Paper 4, option 4B.

1Linear regressionFitting a least squares regression line and checking it with residuals.Y13Option FS2 · 2 sub-topics
  1. 1.1Least squares linear regressionY13
  2. 1.2ResidualsY13
2Continuous probability distributionsContinuous random variables: probability density and cumulative distribution functions, means, variances, medians and the continuous uniform distribution.Y13Option FS2 · 4 sub-topics
  1. 2.1Continuous random variablesY13
  2. 2.2Relationship between pdf and cdfY13
  3. 2.3Summary measuresY13
  4. 2.4Continuous uniform distributionY13
3CorrelationMeasuring correlation with the product moment and Spearman's rank coefficients, and testing whether it is significant.Y13Option FS2 · 3 sub-topics
  1. 3.1Product moment correlation coefficientY13
  2. 3.2Spearman's rank correlationY13
  3. 3.3Testing for zero correlationY13
4Combinations of random variablesMeans and variances of sums and differences of independent normal random variables.Y13Option FS2 · 1 sub-topic
  1. 4.1Combinations of random variablesY13
5Estimation, confidence intervals and tests using a normal distributionEstimators and bias, confidence intervals for a normal mean, tests for the difference between two means, and large-sample methods.Y13Option FS2 · 5 sub-topics
  1. 5.1Estimators and biasY13
  2. 5.2Confidence intervalsY13
  3. 5.3Confidence limits for a normal mean (variance known)Y13
  4. 5.4Test for the difference of two normal means (variances known)Y13
  5. 5.5Large-sample methodsY13
6Other hypothesis tests and confidence intervalsTests and confidence intervals for a normal variance, and the F-test for comparing two variances.Y13Option FS2 · 2 sub-topics
  1. 6.1Tests and intervals for a normal varianceY13
  2. 6.2F-test for equality of variancesY13
7Confidence intervals and tests using the t-distributionThe t-distribution for small samples: confidence intervals and tests for means, including paired data.Y13Option FS2 · 3 sub-topics
  1. 7.1t-tests and intervals for a normal mean (variance unknown)Y13
  2. 7.2Paired t-testY13
  3. 7.3Two-sample t-test (equal unknown variances)Y13

Further Mechanics 2 (FM2)

Optional

Paper 4, option 4C.

1Motion in a circleCircular motion at constant and varying speed, including horizontal circles and motion in a vertical circle.Y13Option FM2 · 2 sub-topics
  1. 1.1Horizontal circular motionY13
  2. 1.2Motion in a vertical circleY13
2Centres of mass of plane figuresFinding the centre of mass of laminas, using standard results and integration.Y13Option FM2 · 2 sub-topics
  1. 2.1Moments and discrete centres of massY13
  2. 2.2Centres of mass of plane figures and frameworksY13
3Further centres of massCentres of mass of solids, and when objects hang in equilibrium, slide or topple.Y13Option FM2 · 3 sub-topics
  1. 3.1Centres of mass by integrationY13
  2. 3.2Equilibrium of rigid bodiesY13
  3. 3.3Toppling and slidingY13
4Further dynamicsMotion under a force that varies, using differential equations, and simple harmonic motion.Y13Option FM2 · 2 sub-topics
  1. 4.1Variable force in one dimensionY13
  2. 4.2Simple harmonic motionY13
5Further kinematicsKinematics with variable acceleration, using calculus with displacement, velocity and acceleration.Y13Option FM2 · 1 sub-topic
  1. 5.1Further kinematicsY13

Decision Mathematics 2 (D2)

Optional

Paper 4, option 4D.

1Transportation problemsTransportation problems: moving goods from sources to destinations at minimum cost.Y13Option D2 · 3 sub-topics
  1. 1.1Transportation: initial solutionY13
  2. 1.2Transportation: improving the solutionY13
  3. 1.3Transportation as a linear programY13
2Allocation (assignment) problemsAllocation problems: matching workers to tasks with the Hungarian algorithm.Y13Option D2 · 2 sub-topics
  1. 2.1Allocation and the Hungarian algorithmY13
  2. 2.2Allocation as a linear programY13
3Flows in networksFlows in networks: cuts, the maximum flow–minimum cut theorem and the labelling procedure.Y13Option D2 · 5 sub-topics
  1. 3.1CutsY13
  2. 3.2Labelling procedureY13
  3. 3.3Max-flow min-cut theoremY13
  4. 3.4Supersources, supersinks and vertex capacitiesY13
  5. 3.5Upper and lower capacitiesY13
4Dynamic programmingDynamic programming: solving multi-stage problems by working backwards through stages and states.Y13Option D2 · 1 sub-topic
  1. 4.1Dynamic programmingY13
5Game theoryZero-sum games between two players: play-safe strategies, dominance and mixed strategies.Y13Option D2 · 5 sub-topics
  1. 5.1Two-person zero-sum gamesY13
  2. 5.2Play-safe strategies and stable solutionsY13
  3. 5.3DominanceY13
  4. 5.4Mixed strategies: graphical methodY13
  5. 5.5Mixed strategies via linear programmingY13
6Recurrence relationsRecurrence relations and their closed-form solutions.Y13Option D2 · 2 sub-topics
  1. 6.1Recurrence relations: modellingY13
  2. 6.2Solving recurrence relationsY13
7Decision analysisDecision trees, expected monetary value and utility.Y13Option D2 · 2 sub-topics
  1. 7.1Decision treesY13
  2. 7.2Expected monetary value and utilityY13

Across the whole course

Compulsory

Not a topic of its own: the specification applies these to every paper.

OT1Mathematical argument, language and proofWriting clear mathematical arguments: correct notation, logic words such as 'implies', set notation and functions. Applies to every paper.5 sub-topics
  1. OT1.1Construct and present mathematical argumentsY13
  2. OT1.2Mathematical language and syntaxY13
  3. OT1.3Set notationY13
  4. OT1.4FunctionsY13
  5. OT1.5Critique arguments and proofsY13
OT2Mathematical problem solvingSolving unfamiliar, unstructured problems and explaining what the answer means. Applies to every paper.5 sub-topics
  1. OT2.1Recognise the underlying structure of a problemY13
  2. OT2.2Extended arguments for unstructured problemsY13
  3. OT2.3Interpret solutions in contextY13
  4. OT2.6The problem-solving cycleY13
  5. OT2.7Diagrams in problem solvingY13
OT3Mathematical modellingTurning a real situation into maths, using the model, then judging and improving it. Applies to every paper.5 sub-topics
  1. OT3.1Build a modelY13
  2. OT3.2Use a modelY13
  3. OT3.3Interpret a model's outputsY13
  4. OT3.4Refine and evaluate a modelY13
  5. OT3.5Modelling assumptionsY13
A1Formulae you must recall (not in the formulae booklet)Formulae that are not in the exam formulae booklet, so you must learn them by heart.6 sub-topics
  1. PPure formulae from A level MathematicsY13
  2. CPFurther pure formulaeY13
  3. CCalculusY13
  4. VVectorsY13
  5. SStatisticsY13
  6. MMechanicsY13
AKAssumed knowledgeFurther Maths builds on A level Maths, so any of that content can appear in your papers.1 sub-topic
  1. AKA level Mathematics contentY13

The exam papers

Paper 1: Core Pure Mathematics 11 h 30 min · 75 marks · 25%9FM0/01
Paper 2: Core Pure Mathematics 21 h 30 min · 75 marks · 25%9FM0/02
Paper 3: Further Mathematics Option 11 h 30 min · 75 marks · 25%9FM0/3A to 3D
Paper 4: Further Mathematics Option 21 h 30 min · 75 marks · 25%9FM0/4A to 4D