FMPRFurther Maths, Maths & Physics Revision
AQA A-level · 7367

Further Mathematics

Every topic AQA can examine. Tap a topic to see its specification points; Y12 and Y13 show which year each one is taught.

Official specification (PDF)

Compulsory content

Sections A to J, assessed on Papers 1 and 2.

AProofProof 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.1 point
  1. A1Proof by mathematical induction: sums of series, divisibility, powers of matricesY12
Practise this topic 10 questions
BComplex 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 points
  1. B1Solve quadratics with real coefficients; cubics and quartics given a root or quadratic factorY12
  2. B2Add, subtract, multiply and divide complex numbers in the form x + iy; real and imaginary partsY12
  3. B3Complex conjugates; non-real roots of real polynomials occur in conjugate pairsY12
  4. B4Use and interpret Argand diagramsY12
  5. B5Convert between Cartesian and modulus–argument formY12
  6. B6Multiply and divide complex numbers in modulus–argument formY12
  7. B7Construct and interpret simple loci in the Argand diagram, such as |z − a| > r and arg(z − a) = θY12
  8. B8De Moivre's theorem: multiple angle formulae and sums of seriesY13
  9. B9Exponential form: e^(iθ) = cos θ + i sin θ and z = re^(iθ)Y13
  10. B10The n distinct nth roots of re^(iθ); vertices of a regular n-gonY13
  11. B11Complex roots of unity to solve geometric problemsY13
Practise this topic 10 questions
CMatricesMatrices as transformations, determinants, inverses, solving equations in three unknowns, and eigenvalues, eigenvectors and diagonalisation.11 points
  1. C1Add, subtract and multiply conformable matrices; multiply a matrix by a scalarY12
  2. C2Zero and identity matricesY12
  3. C3Matrices as linear transformations in 2D; single reflections and rotations in 3D; successive transformationsY12
  4. C4Invariant points and invariant linesY12
  5. C5Determinants as scale factors and effect on orientation (2 × 2 in AS; 3 × 3 at A level)Y12–13
  6. C6Singular and non-singular matrices; inverses (2 × 2 in AS; 3 × 3 at A level)Y12–13
  7. C7Solve three linear simultaneous equations using the inverse matrixY13
  8. C8Interpret geometrically the solution, or failure of solution, of three simultaneous equationsY13
  9. C9Factorise determinants using row and column operationsY13
  10. C10Eigenvalues and eigenvectors of 2 × 2 and 3 × 3 matrices; characteristic equationY13
  11. C11Diagonalise matrices, M = UDU⁻¹; powers of MY13
Practise this topic 10 questions
DFurther algebra and functionsRoots of polynomials, summing series, Maclaurin series, limits, inequalities, the modulus function, rational functions and conic sections.16 points
  1. D1Relationships between roots and coefficients of polynomials up to quarticsY12
  2. D2Form a polynomial whose roots are a linear transformation of the roots of anotherY12
  3. D3Standard results for sums of integers, squares and cubesY12
  4. D4Method of differences, including with partial fractionsY12
  5. D5Find the Maclaurin series of a function, including the general termY13
  6. D6Maclaurin series of eˣ, ln(1 + x), sin x, cos x and (1 + x)ⁿ with ranges of validityY12
  7. D7Evaluate limits using Maclaurin series or l'Hôpital's ruleY13
  8. D8Solve cubic and quartic polynomial inequalitiesY12
  9. D9Solve inequalities of the form (ax + b)/(cx + d) < (ex + f)/(gx + h)Y12
  10. D10Modulus functions and related inequalitiesY13
  11. D11Sketch y = 1/f(x) given the graph of y = f(x)Y13
  12. D12Graphs of rational functions (ax + b)/(cx + d): asymptotes, intercepts, inequalitiesY12
  13. D13Graphs of rational functions with quadratic numerator or denominator, including oblique asymptotesY12–13
  14. D14Use quadratic theory (not calculus) to find ranges and stationary points of rational functionsY12
  15. D15Sketch the parabola, ellipse, hyperbola and rectangular hyperbolaY12
  16. D16Transformations of curves: translations, stretches, reflections, rotations and enlargementsY12–13
Practise this topic 10 questions
EFurther calculusImproper integrals, volumes of revolution, mean values, inverse trigonometric functions, arc length, surface area and reduction formulae.9 points
  1. E1Improper integralsY13
  2. E2Derive and use formulae for volumes of revolutionY12
  3. E3Mean value of a functionY12
  4. E4Integrate using partial fractions, including quadratic factorsY13
  5. E5Differentiate inverse trigonometric functionsY13
  6. E6Integrate 1/√(a² − x²) and 1/(a² + x²); choose trigonometric substitutionsY13
  7. E7Arc length and area of surface of revolution, Cartesian and parametricY13
  8. E8Derive and use reduction formulaeY13
  9. E9Limits of xᵏeˣ and xᵏ ln x for improper integralsY13
Practise this topic 10 questions
FFurther vectorsLines and planes in three dimensions, the scalar and vector products, angles, intersections and shortest distances.6 points
  1. F1Vector and Cartesian forms of the equation of a straight line in 3DY12
  2. F2Vector and Cartesian forms of the equation of a planeY13
  3. F3Scalar product: angles between lines, planes, and a line and a planeY12–13
  4. F4Check whether vectors are perpendicular using the scalar productY12
  5. F5Vector product; (r − a) × b = 0; area of a triangleY13
  6. F6Intersections of lines and planes; perpendicular distancesY12–13
Practise this topic 10 questions
GPolar 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.3 points
  1. G1Polar coordinates; convert between polar and Cartesian formsY12
  2. G2Sketch curves with r given as a function of θY12
  3. G3Areas enclosed by polar curvesY13
Practise this topic 10 questions
HHyperbolic 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.7 points
  1. H1Definitions, domains, ranges and graphs of sinh, cosh, tanh, sech, cosech and cothY12
  2. H2Differentiate and integrate hyperbolic functionsY13
  3. H3Inverse hyperbolic functions: definitions, domains and rangesY12
  4. H4Derive and use logarithmic forms of the inverse hyperbolic functionsY12
  5. H5Integrate 1/√(x² + a²) and 1/√(x² − a²); choose substitutionsY13
  6. H6Hyperbolic identities, including tanh x = sinh x / cosh xY12
  7. H7Construct proofs involving hyperbolic functions and identitiesY13
Practise this topic 10 questions
IDifferential equationsIntegrating factors, second-order equations, simple harmonic motion, damping and coupled equations such as predator–prey models.11 points
  1. I1Integrating factor for dy/dx + P(x)y = Q(x)Y13
  2. I2General and particular solutionsY13
  3. I3Differential equations in kinematics and other modelling contextsY13
  4. I4Second order homogeneous equations using the auxiliary equationY13
  5. I5Second order non-homogeneous equations: complementary function and particular integralY13
  6. I6Sign of the discriminant and the form of the solutionY13
  7. I7Simple harmonic motion, ẍ = −ω²xY13
  8. I8Damped oscillations: light, critical and heavy dampingY13
  9. I9Coupled first order simultaneous differential equationsY13
  10. I10Hooke's law to form the equation of simple harmonic motionY13
  11. I11Damped motion with damping force proportional to velocityY13
Practise this topic 10 questions
JNumerical methodsApproximating integrals with the mid-ordinate and Simpson's rules, and solving differential equations step by step with Euler's methods.3 points
  1. J1Mid-ordinate rule and Simpson's ruleY13
  2. J2Euler's step by step method for first order differential equationsY13
  3. J3Improved Euler methodY13

Optional application 1: mechanics

Paper 3, component 3M.

MADimensional analysisChecking that formulae make sense using dimensions (mass, length, time), and predicting formulae from them.2 points
  1. MA1Dimensions of quantities; dimensional consistencyY12
  2. MA2Predict formulae; find powers in a proposed formulaY12
MBMomentum and collisionsMomentum, impulse and collisions, including the coefficient of restitution.4 points
  1. MB1Conservation of momentum, including velocities as vectorsY12–13
  2. MB2Coefficient of restitution and Newton's experimental law, including fixed smooth surfacesY12–13
  3. MB3Impulse and momentum, Ft = mv − muY12–13
  4. MB4Impulse of a variable force, I = ∫F dtY12
MCWork, energy and powerWork, energy and power, including elastic strings and springs.7 points
  1. MC1Work done by a force, WD = Fd cos θY12
  2. MC2Gravitational potential energyY12
  3. MC3Kinetic energyY12
  4. MC4Hooke's law and modulus of elasticityY12
  5. MC5Work done by a variable force, WD = ∫F dxY12
  6. MC6Elastic potential energyY12
  7. MC7Power, P = FvY12–13
MDCircular motionMotion in horizontal and vertical circles, the conical pendulum and angular speed.6 points
  1. MD1Motion of a particle in a circle at constant speedY12
  2. MD2Angular speedY12
  3. MD3Relationships between speed, angular speed, radius and accelerationY12
  4. MD4Position, velocity and acceleration vectors in circular motionY13
  5. MD5Conical pendulum, with one or two stringsY13
  6. MD6Motion in a vertical circleY13
MECentres of mass and momentsCentres of mass of particles, laminas and solids, and when bodies slide or topple.6 points
  1. ME1Centre of mass of a system of particlesY13
  2. ME2Centre of mass of a composite bodyY13
  3. ME3Centre of mass of a lamina by integrationY13
  4. ME4Centres of mass of solids of revolutionY13
  5. ME5Sliding and toppling, including suspended bodies and inclined planesY13
  6. ME6Rigid bodies in equilibrium: moments and couplesY13

Optional application 2: statistics

Paper 3, component 3S.

SADiscrete random variables (DRVs) and expectationDiscrete random variables: probabilities, expectation, variance and the discrete uniform distribution.7 points
  1. SA1Discrete random variables as tables or functionsY12
  2. SA2Probabilities for a DRVY12
  3. SA3Mean, variance, standard deviation, mode and median of a DRVY12
  4. SA4E(X), E(X²) and Var(X)Y12
  5. SA5Expectation and variance of functions of a DRVY12
  6. SA6Discrete uniform distributionY12
  7. SA7Proofs of the mean and variance of the discrete uniform distributionY12
SBPoisson distributionThe Poisson distribution for counts of random events, and tests about its mean.5 points
  1. SB1Conditions for a Poisson model; X ~ Po(λ)Y12
  2. SB2Poisson probabilitiesY12
  3. SB3Mean, variance and standard deviationY12
  4. SB4Sums of independent Poisson variablesY12
  5. SB5Hypothesis test for the mean of a Poisson distributionY12
SCType I and Type II errorsType I and Type II errors and the power of a test.2 points
  1. SC1Type I and Type II errors; probability of a Type I errorY12–13
  2. SC2Power of a test; probability of a Type II errorY13
SDContinuous random variables (CRVs)Continuous random variables: probability density functions, cumulative distribution functions, means, variances and the rectangular distribution.8 points
  1. SD1Probability density functionsY12
  2. SD2Probability over an intervalY12
  3. SD3Median and quartilesY12
  4. SD4Mean, variance and standard deviationY12
  5. SD5Expectation and variance of functions of a CRVY12
  6. SD6Cumulative distribution functionsY13
  7. SD7Rectangular distribution; proofs of its mean and varianceY13
  8. SD8Expectation and variance of sums of independent random variablesY12
SEChi-squared tests for associationχ² tests for association using contingency tables.5 points
  1. SE1n × m contingency tablesY12
  2. SE2The χ² statistic and degrees of freedomY12
  3. SE3Convention that expected frequencies exceed 5Y12
  4. SE4Sources of association in contextY12
  5. SE5Yates' correctionY13
SFExponential distributionThe exponential distribution for waiting times between random events.4 points
  1. SF1Conditions for an exponential model; pdf and cdfY13
  2. SF2Probabilities from the exponential distributionY13
  3. SF3Proofs of the mean, variance and standard deviationY13
  4. SF4Time between Poisson events is exponentially distributedY13
SGInference – one sample t-distributionThe t-test for a mean when the variance is unknown.1 point
  1. SG1t-test for the mean of a normal distribution with unknown varianceY13
SHConfidence intervalsConfidence intervals for a mean, with known variance, large samples and the t-distribution.4 points
  1. SH1Confidence interval for a normal mean with known varianceY12
  2. SH2Large-sample confidence interval with unknown varianceY12
  3. SH3Inferences from confidence intervalsY12
  4. SH4Confidence intervals using the t-distributionY13

Optional application 3: discrete mathematics

Paper 3, component 3D.

DAGraphsGraph theory: vertices, edges, degrees, Eulerian and Hamiltonian graphs, planarity and isomorphism.7 points
  1. DA1Language of graphs: vertex, edge, trail, cycle, degree, subgraphY12
  2. DA2Eulerian, semi-Eulerian and Hamiltonian graphsY12
  3. DA3Euler's formula for connected planar graphsY12
  4. DA4Kuratowski's theoremY13
  5. DA5Complete and bipartite graphs; adjacency matrices; complementsY12
  6. DA6Simple graphs, connected graphs and treesY12
  7. DA7Isomorphism between graphsY13
DBNetworksNetworks: minimum spanning trees, route inspection and the travelling salesperson problem.5 points
  1. DB1Language of networks: node, arc, weightY12
  2. DB2Minimum spanning treesY12
  3. DB3Route inspectionY12
  4. DB4Travelling salesperson: upper and lower boundsY12
  5. DB5Evaluate and refine network modelsY12
DCNetwork flowsFlows in networks: cuts, the maximum flow–minimum cut theorem and the labelling procedure.7 points
  1. DC1Flows in directed networksY12
  2. DC2Cuts and their valuesY12
  3. DC3Maximum flow–minimum cut theoremY12
  4. DC4Supersources and supersinksY12
  5. DC5Augment flows to find the maximum flowY13
  6. DC6Arcs with upper and lower capacitiesY13
  7. DC7Nodes of restricted capacityY13
DDLinear programmingWriting real problems as linear programs and solving them graphically and with the Simplex algorithm.4 points
  1. DD1Formulate constrained optimisation problemsY12
  2. DD2Graphical solutionY12
  3. DD3Simplex algorithm with slack variablesY13
  4. DD4Interpret a Simplex tableauY13
DECritical path analysisPlanning projects with activity networks: early and late times, critical paths, float, Gantt charts and scheduling.6 points
  1. DE1Activity-on-node precedence networksY12
  2. DE2Earliest and latest start and finish timesY12
  3. DE3Critical activities, critical paths and floatY12
  4. DE4Refine models and assess changesY12
  5. DE5Gantt (cascade) charts and resource histogramsY13
  6. DE6Resource levellingY13
DFGame theory for zero-sum gamesZero-sum games between two players: play-safe strategies, dominance and mixed strategies.6 points
  1. DF1Pay-off matricesY12
  2. DF2Play-safe strategies and the value of a gameY12
  3. DF3Stable solutionsY12
  4. DF4Dominated strategiesY12
  5. DF5Optimal mixed strategies, including graphical methodsY12
  6. DF6Convert larger games to linear programming problemsY13
DGBinary operationsBinary operations and groups: Cayley tables, identities, inverses, subgroups, Lagrange's theorem and isomorphisms.12 points
  1. DG1Binary operations, including modular arithmetic and matrix multiplicationY12
  2. DG2CommutativityY12
  3. DG3AssociativityY12
  4. DG4Cayley tablesY12
  5. DG5Identity elementsY12
  6. DG6InversesY12
  7. DG7Language of groups: order, period, subgroupY13
  8. DG8Group axiomsY13
  9. DG9Finite and infinite groups; cyclic and abelian groups; symmetry groupsY13
  10. DG10Lagrange's theoremY13
  11. DG11GeneratorsY13
  12. DG12Isomorphism between finite groupsY13

Across the whole course

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 points
  1. OT1.1Construct and present mathematical argumentsY12–13
  2. OT1.2Mathematical language and syntaxY12–13
  3. OT1.3Set notationY12–13
  4. OT1.4FunctionsY12–13
  5. OT1.5Critique arguments and proofsY12–13
OT2Mathematical problem solvingSolving unfamiliar, unstructured problems and explaining what the answer means. Applies to every paper.5 points
  1. OT2.1Recognise the underlying structure of a problemY12–13
  2. OT2.2Extended arguments for unstructured problemsY12–13
  3. OT2.3Interpret solutions in contextY12–13
  4. OT2.6The problem-solving cycleY12–13
  5. OT2.7Diagrams in problem solvingY12–13
OT3Mathematical modellingTurning a real situation into maths, using the model, then judging and improving it. Applies to every paper.5 points
  1. OT3.1Build a modelY12–13
  2. OT3.2Use a modelY12–13
  3. OT3.3Interpret a model's outputsY12–13
  4. OT3.4Refine and evaluate a modelY12–13
  5. OT3.5Modelling assumptionsY12–13
AKAssumed knowledgeFurther Maths builds on A level Maths, so any of that content can appear in your papers.1 point
  1. AKGCSE and A level Mathematics contentY12–13

The exam papers

Paper 1: compulsory content, sections A to J2 hours · 100 marks · 33⅓%7367/1
Paper 2: compulsory content, sections A to J2 hours · 100 marks · 33⅓%7367/2
Paper 3: two optional applications2 hours · 100 marks · 33⅓%7367/3