A level · OCR

Further Mathematics H245

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

  • 3 papers
  • 180 sub-topics
  • Y12Y13When each is usually taught

Official specification (PDF)

Pure Core

Compulsory

Papers Y540 and Y541. Either paper can assess any Pure Core content.

4.01ProofMathematical induction.Y121 sub-topic
  1. 4.01a-bMathematical inductionY12
4.02Complex numbersThe language of complex numbers, basic operations, solution of equations, Argand diagrams, Euler's formula, loci, De Moivre's theorem and more.9 sub-topics
  1. 4.02a-dThe language of complex numbersY12
  2. 4.02e-fBasic operationsY12
  3. 4.02g-jSolution of equationsY12
  4. 4.02k-mArgand diagramsY12
  5. 4.02nEuler's formulaY13
  6. 4.02o-pLociY12
  7. 4.02qDe Moivre's theoremY13
  8. 4.02rnth rootsY13
  9. 4.02sRoots of unityY13
4.03MatricesThe language of matrices, matrix addition and multiplication, linear transformations, 3-D transformations, invariance, determinants, inverses and more.Y129 sub-topics
  1. 4.03aThe language of matricesY12
  2. 4.03b-cMatrix addition and multiplicationY12
  3. 4.03d-eLinear transformationsY12
  4. 4.03f3-D transformationsY12
  5. 4.03gInvarianceY12
  6. 4.03h-mDeterminantsY12
  7. 4.03n-qInversesY12
  8. 4.03r-sSolution of simultaneous equationsY12
  9. 4.03tIntersection of planesY12
4.04Further vectorsEquation of a straight line, equation of a plane, scalar product, intersections, vector product, shortest distances.6 sub-topics
  1. 4.04aEquation of a straight lineY12
  2. 4.04bEquation of a planeY12
  3. 4.04c-dScalar productY12
  4. 4.04e-fIntersectionsY12
  5. 4.04gVector productY12
  6. 4.04h-jShortest distancesY13
4.05Further algebraRoots of equations, transformation of equations, partial fractions.3 sub-topics
  1. 4.05aRoots of equationsY12
  2. 4.05bTransformation of equationsY12
  3. 4.05cPartial fractionsY13
4.06SeriesSummation of series, method of differences.2 sub-topics
  1. 4.06aSummation of seriesY12
  2. 4.06bMethod of differencesY13
4.07Hyperbolic functionsDefinitions, differentiation and integration, inverse hyperbolic functions.Y133 sub-topics
  1. 4.07a-cDefinitionsY13
  2. 4.07dDifferentiation and integrationY13
  3. 4.07e-fInverse hyperbolic functionsY13
4.08Further calculusMaclaurin series, improper integrals, volumes of solids of revolution, mean values, partial fractions, inverse trigonometric and hyperbolic functions, further integration.7 sub-topics
  1. 4.08a-bMaclaurin seriesY13
  2. 4.08cImproper integralsY13
  3. 4.08dVolumes of solids of revolutionY12
  4. 4.08eMean valuesY13
  5. 4.08fPartial fractionsY13
  6. 4.08gInverse trigonometric and hyperbolic functionsY13
  7. 4.08hFurther integrationY13
4.09Polar coordinatesPolar coordinates, sketching curves, area.Y133 sub-topics
  1. 4.09aPolar coordinatesY13
  2. 4.09bSketching curvesY13
  3. 4.09cAreaY13
4.10Differential equationsGeneral and particular solutions, modelling, integrating factor method, second order homogeneous differential equations, second order non-homogeneous differential equations, simple harmonic motion, damped oscillations and more.Y138 sub-topics
  1. 4.10aGeneral and particular solutionsY13
  2. 4.10bModellingY13
  3. 4.10cIntegrating factor methodY13
  4. 4.10dSecond order homogeneous differential equationsY13
  5. 4.10eSecond order non-homogeneous differential equationsY13
  6. 4.10fSimple harmonic motionY13
  7. 4.10gDamped oscillationsY13
  8. 4.10hLinear systemsY13

Statistics

Optional

Optional paper Y542.

5.01ProbabilityProbability.Y13Option S · 1 sub-topic
  1. 5.01a-bProbabilityY13
5.02Discrete random variablesProbability distributions for general discrete random variables, the binomial distribution, the discrete uniform distribution, the geometric distribution, the Poisson distribution.Option S · 5 sub-topics
  1. 5.02a-cProbability distributions for general discrete random variablesY12
  2. 5.02dThe binomial distributionY12
  3. 5.02eThe discrete uniform distributionY12
  4. 5.02f-hThe geometric distributionY13
  5. 5.02i-nThe Poisson distributionY12
5.03Continuous random variablesContinuous random variables, probability density functions, cumulative distribution functions.Y13Option S · 3 sub-topics
  1. 5.03aContinuous random variablesY13
  2. 5.03b-dProbability density functionsY13
  3. 5.03e-gCumulative distribution functionsY13
5.04Linear combinations of random variablesLinear combinations of any random variables, linear combinations of normal random variables.Option S · 2 sub-topics
  1. 5.04aLinear combinations of any random variablesY13
  2. 5.04bLinear combinations of normal random variablesY12
5.05Hypothesis tests and confidence intervalsThe distribution of the sample mean and the central limit theorem, unbiased estimates of population mean and variance, using the normal distribution in hypothesis tests, confidence intervals.Y13Option S · 4 sub-topics
  1. 5.05aThe distribution of the sample mean and the central limit theoremY13
  2. 5.05bUnbiased estimates of population mean and varianceY13
  3. 5.05cUsing the normal distribution in hypothesis testsY13
  4. 5.05dConfidence intervalsY13
5.06Chi-squared testsContingency tables, fitting a theoretical distribution, goodness of fit test.Option S · 3 sub-topics
  1. 5.06aContingency tablesY12
  2. 5.06b-cFitting a theoretical distributionY12
  3. 5.06dGoodness of fit testY13
5.07Non-parametric testsNon-parametric tests, the basis of non-parametric tests, single-sample hypothesis tests, paired-sample and two-sample hypothesis tests, normal approximations.Option S · 5 sub-topics
  1. 5.07aNon-parametric testsY13
  2. 5.07bThe basis of non-parametric testsY13
  3. 5.07cSingle-sample hypothesis testsY13
  4. 5.07d-ePaired-sample and two-sample hypothesis testsY13
  5. 5.07fNormal approximationsY13
5.08CorrelationPearson's product-moment correlation coefficient, hypothesis tests using Pearson's coefficient, Spearman's rank correlation coefficient, hypothesis tests using Spearman's coefficient, comparison of coefficients.Option S · 5 sub-topics
  1. 5.08a-cPearson's product-moment correlation coefficientY13
  2. 5.08dHypothesis tests using Pearson's coefficientY13
  3. 5.08eSpearman's rank correlation coefficientY13
  4. 5.08fHypothesis tests using Spearman's coefficientY13
  5. 5.08gComparison of coefficientsY13
5.09Linear regressionDependent and independent variables, calculation of the equation of the regression line, use of the regression line.Option S · 3 sub-topics
  1. 5.09aDependent and independent variablesY13
  2. 5.09b-dCalculation of the equation of the regression lineY13
  3. 5.09eUse of the regression lineY13

Mechanics

Optional

Optional paper Y543.

6.01Dimensional analysisDimensional analysis.Y12Option M · 1 sub-topic
  1. 6.01a-eDimensional analysisY12
6.02Work, energy and powerWork, energy, Hooke's law, conservation of energy, power.Option M · 5 sub-topics
  1. 6.02a-cWorkY12
  2. 6.02d-fEnergyY12
  3. 6.02g-hHooke's lawY13
  4. 6.02i-jConservation of energyY13
  5. 6.02k-mPowerY13
6.03Impulse and momentumLinear momentum, impulse, restitution.Option M · 3 sub-topics
  1. 6.03a-dLinear momentumY12
  2. 6.03e-hImpulseY13
  3. 6.03i-lRestitutionY12
6.04Centre of massCentre of mass, rigid bodies.Y13Option M · 2 sub-topics
  1. 6.04a-dCentre of massY13
  2. 6.04eRigid bodiesY13
6.05Motion in a circleUniform motion in a circle, motion in a vertical circle.Y13Option M · 2 sub-topics
  1. 6.05a-cUniform motion in a circleY13
  2. 6.05d-fMotion in a vertical circleY13
6.06Further dynamics and kinematicsLinear motion under a variable force.Y13Option M · 1 sub-topic
  1. 6.06aLinear motion under a variable forceY13

Discrete Mathematics

Optional

Optional paper Y544.

7.01Mathematical preliminariesTypes of problem, set notation, the pigeonhole principle, arrangement and selection problems, the inclusion-exclusion principle.Option D · 5 sub-topics
  1. 7.01aTypes of problemY13
  2. 7.01bSet notationY13
  3. 7.01cThe pigeonhole principleY13
  4. 7.01d-jArrangement and selection problemsY13
  5. 7.01k-mThe inclusion-exclusion principleY13
7.02Graphs and networksTerminology and notation, complete graphs, bipartite graphs, Eulerian graphs, hamiltonian graphs, isomorphism, digraphs and more.Option D · 9 sub-topics
  1. 7.02a-cTerminology and notationY12
  2. 7.02dComplete graphsY13
  3. 7.02e-fBipartite graphsY12
  4. 7.02gEulerian graphsY12
  5. 7.02h-iHamiltonian graphsY13
  6. 7.02jIsomorphismY13
  7. 7.02kDigraphsY13
  8. 7.02l-oPlanar graphsY12
  9. 7.02p-rUsing graphs and networksY12
7.03AlgorithmsDefinition of an algorithm, awareness of the uses and practical limitations of algorithms, working with algorithms, the order of an algorithm, efficiency and complexity, strategies for sorting, strategies for packing.Option D · 7 sub-topics
  1. 7.03aDefinition of an algorithmY12
  2. 7.03bAwareness of the uses and practical limitations of algorithmsY13
  3. 7.03cWorking with algorithmsY13
  4. 7.03dThe order of an algorithmY12
  5. 7.03e-iEfficiency and complexityY13
  6. 7.03j-kStrategies for sortingY12
  7. 7.03l-mStrategies for packingY12
7.04Network algorithmsLeast weight path between two vertices, least weight set of arcs connecting all vertices, least weight cycle through all vertices, least weight route that traverses every arc at least once, network problems.Option D · 5 sub-topics
  1. 7.04aLeast weight path between two verticesY12
  2. 7.04bLeast weight set of arcs connecting all verticesY12
  3. 7.04c-dLeast weight cycle through all verticesY13
  4. 7.04eLeast weight route that traverses every arc at least onceY12
  5. 7.04fNetwork problemsY12
7.05Decision making in project managementCritical path analysis.Y12Option D · 1 sub-topic
  1. 7.05a-eCritical path analysisY12
7.06Graphical linear programmingFormulating LP problems, working with constraints, graphical solutions.Y12Option D · 3 sub-topics
  1. 7.06a-bFormulating LP problemsY12
  2. 7.06cWorking with constraintsY12
  3. 7.06d-fGraphical solutionsY12
7.07The simplex algorithmUse a simplex tableau, terminology, graphical and algebraic interpretation of iterations.Y13Option D · 3 sub-topics
  1. 7.07a-cUse a simplex tableauY13
  2. 7.07dTerminologyY13
  3. 7.07e-fGraphical and algebraic interpretation of iterationsY13
7.08Game theoryPay-off matrix, pure strategies, mixed strategies.Y13Option D · 3 sub-topics
  1. 7.08a-bPay-off matrixY13
  2. 7.08c-dPure strategiesY13
  3. 7.08e-fMixed strategiesY13

Additional Pure Mathematics

Optional

Optional paper Y545.

8.01Sequences and seriesRecurrence relations, properties of sequences, Fibonacci and related numbers, solving recurrence systems, modelling.Option A · 5 sub-topics
  1. 8.01a-bRecurrence relationsY13
  2. 8.01c-dProperties of sequencesY13
  3. 8.01eFibonacci and related numbersY13
  4. 8.01f-gSolving recurrence systemsY13
  5. 8.01h-iModellingY13
8.02Number theoryNumber bases, divisibility tests, the division algorithm, finite (modular) arithmetics, prime numbers, Euclid's lemma, Fermat's little theorem and more.Option A · 9 sub-topics
  1. 8.02aNumber basesY13
  2. 8.02b-cDivisibility testsY13
  3. 8.02dThe division algorithmY13
  4. 8.02e-hFinite (modular) arithmeticsY13
  5. 8.02i-jPrime numbersY13
  6. 8.02kEuclid's lemmaY13
  7. 8.02lFermat's little theoremY13
  8. 8.02m-nThe order of a modulo pY13
  9. 8.02oBinomial theoremY13
8.03GroupsBinary operations, definition of a group, orders of elements and groups, subgroups, cyclic groups, generators, properties of groups and more.Option A · 10 sub-topics
  1. 8.03a-bBinary operationsY12
  2. 8.03c-dDefinition of a groupY13
  3. 8.03eOrders of elements and groupsY13
  4. 8.03fSubgroupsY13
  5. 8.03gCyclic groupsY13
  6. 8.03hGeneratorsY13
  7. 8.03i-jProperties of groupsY13
  8. 8.03kLagrange's theoremY13
  9. 8.03lIsomorphismY13
  10. 8.03mAbstract groupsY12
8.04Further vectorsVector product.Y12Option A · 1 sub-topic
  1. 8.04a-eVector productY12
8.05Surfaces and partial differentiation3-D surfaces, sections and contours, partial differentiation, stationary points, tangent planes.Option A · 5 sub-topics
  1. 8.05a-b3-D surfacesY13
  2. 8.05cSections and contoursY13
  3. 8.05dPartial differentiationY13
  4. 8.05e-fStationary pointsY13
  5. 8.05gTangent planesY13
8.06Further calculusReduction formulae, arc lengths and surface areas.Y13Option A · 2 sub-topics
  1. 8.06aReduction formulaeY13
  2. 8.06bArc lengths and surface areasY13

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
AKAssumed knowledgeFurther Maths builds on A level Maths, so any of that content can appear in your papers.1 sub-topic
  1. AKGCSE and A level Mathematics contentY13

The exam papers

Pure Core 11 h 30 min · 75 marks · 25%Y540
Pure Core 21 h 30 min · 75 marks · 25%Y541
Two optional papers: Statistics, Mechanics, Discrete Mathematics or Additional Pure Mathematics1 h 30 min each · 75 marks · 25% eachY542 to Y545