Paper 1 · Pure Mathematics 1
Specimen 2020 · 9709/01
Cambridge (CIE) · A Level
Mathematics 9709
Develop algebra, functions, trigonometry and calculus, then explore mechanics or statistics through the papers in your chosen route.
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What you'll cover
- Pure Mathematics 1 · Paper 1
- Pure Mathematics 2 · Paper 2 (AS only)
- Pure Mathematics 3 · Paper 3 (A Level)
- Mechanics · Paper 4
- Probability & Statistics 1 · Paper 5
- Probability & Statistics 2 · Paper 6 (A Level)
Published syllabus exam years: 2023, 2024, 2025, 2026, 2027, 2028, 2029, 2030. Confirm exam-series and centre availability with your school and Cambridge.
Your route through AS & A Level Mathematics
Version 4
38 of 38 sections have teaching and practice
Start the learning pathwayStudy options and assessment
312 sequenced lessons provide 936 guided, independent and application problems covering the 160 objectives in sections 1.1–1.8, 2.1–2.6, 3.1–3.9, 4.1–4.5, 5.1–5.5 and 6.1–6.5. The map lists all 38 sections across six alternative components; all eight Pure Mathematics 1 sections and all six Pure Mathematics 2 sections, from Paper 2 Algebra through Numerical solution of equations, currently have full teaching sequences. Paper 3 Algebra, section 3.1, links the six shared foundation lessons plus six lessons on partial fractions and rational binomial expansions; Paper 3 logarithms, trigonometry and numerical equations use the shared lessons, with the Paper 1 graph foundation linked into trigonometry. Paper 3 differentiation combines the shared methods with its inverse-tangent derivative lesson. Paper 3 Integration links four shared standard and identity lessons plus seven advanced lessons, covering all six section 3.5 requirements. Eight Vectors lessons cover all six section 3.7 requirements, with 2D and 3D line and solid applications. Nine Differential equations lessons cover all four section 3.8 requirements, including Paper 3 integration techniques, initial conditions, equilibrium solutions and conditional model interpretations. Eleven Complex numbers lessons cover all eight section 3.9 requirements, with seven Argand diagrams and explicit argument and boundary checks. All nine Pure Mathematics 3 sections now have teaching linked to every mapped requirement. Nine Forces and equilibrium lessons cover all seven section 4.1 requirements, with six force diagrams, signed incline-friction intervals, reaction-dependent pull and push limits and separate-body third-law checks. Nine Kinematics lessons cover the four official section 4.2 requirements, with the graph requirement mapped to three relationships. Six motion diagrams support directed quantities, graph slopes and signed areas, Paper 1 time-calculus methods, reversal distances, constant-acceleration stages and shared-clock particle meetings. Six Momentum lessons cover both section 4.3 requirements, including signed system totals, conservation assumptions, outgoing velocities, coalescence, unknown masses and successive impacts. Four diagrams show system momentum through the stipulated events, and supplied kinetic-energy checks distinguish conservation of momentum from changes in kinetic energy. Twelve Newton’s laws lessons cover all four section 4.4 requirements, with vertical support and release stages, rough horizontal and incline motion, inferred parameters, hanging and surface pulley systems, rigid tow-bar thrust and slack-string transitions. Seven force diagrams support the separate-particle balances. Nine Energy, work and power lessons cover all five section 4.5 requirements, with angle-dependent work, kinetic and reference potential energy, signed net-work balances, smooth curved slides, rough and driven slopes, ideal connected-system energy and instantaneous power on hills. Five diagrams distinguish the energy inputs, losses and valid positive-speed force model. All five Mechanics sections now have teaching and staged practice linked to their mapped requirements. Thirteen Representation of data lessons cover all five section 5.1 requirements, including choice and limitations of displays, ordered and back-to-back stems, raw summaries and box plots, unequal-width density histograms, cumulative-frequency interpolation, descriptive moment calculations, grouped estimates, coding, combining two datasets and correcting totals. Six accessible diagrams retain repeated observations and distinguish numerical spread, frequency area and estimated readings. Ten Permutations and combinations lessons cover both official section 5.2 requirements, with selection and ordered roles, repeated objects, adjacency blocks, non-adjacency gaps, leading-zero cases, labelled multi-row seating and mixed restrictions. Six accessible diagrams show outcome trees, ordered versus unordered choices, repeated copies, blocks, gaps and distinct row seats. Eleven Probability lessons cover all four section 5.3 requirements, with equiprobable elementary outcomes, selection-based probabilities, complements and disjoint regions, independence tests, restricted sample spaces, two-way tables, replacement and removal trees, reversed conditions and unknown parameters. Seven accessible diagrams show dice outcomes, random subsets, event regions, a conditioned grid, a count table and two probability trees. Twelve Discrete random variables lessons cover all three official section 5.4 requirements, with complete probability tables, moment and parameter checks, binomial assumptions, exact probabilities and moments, geometric first-success probabilities, tails, expectation and integer reliability thresholds. Six accessible diagrams distinguish outcome grouping, weighted moments, fixed trials, discrete probability bars, waiting conventions and integer tail deadlines. Twelve Normal distribution lessons cover all three official section 5.5 requirements, with continuous-area interpretation, MF19 cumulative tables, full standardisation, tail and interval regions, conditional probabilities, percentiles, one- and two-parameter inference, binomial suitability, continuity corrections and approximate integer thresholds. Seven accessible diagrams show model areas, table reading, aligned scales, intervals, quantiles, matched binomial and normal shapes and whole integer cells. All five Statistics 1 sections now have teaching and staged practice linked to their mapped requirements. Fourteen Poisson distribution lessons cover all five official section 6.1 requirements, with homogeneous event-count assumptions, exposure scaling, exact masses and tails, conditional count events, moments, parameter inference, probability duration targets, disjoint and overlapping windows, rare-count binomial approximations, matched normal approximations and continuity corrections. Nine accessible diagrams show event exposures, unit conversions, discrete bars, complete tail complements, population moments, shared windows, approximate count models and integer cells. Exact checks distinguish approximate and exact close-target decisions. Fourteen Linear combinations lessons cover the six listed results in official section 6.2, with affine moments, signed scales and constant cases, expectation without independence, independent squared variance contributions, fixed and random fees, independent copies versus reused values, affine normal events, independent normal sums and differences, weighted and conditional areas, coefficient inference, exact independent Poisson sums and overlapping exposure limits. Nine accessible diagrams distinguish scale changes, variance contributions, component costs, copy mechanisms, aligned normal scales, summed areas, Poisson count pairs and unsupported distribution claims. Fourteen Continuous random variables lessons cover both official section 6.3 requirements, with density validity and units, support-aware intervals, complements and conditional regions, first and second moments, percentiles by direct density areas, symmetry checks, parameter inference, infinite-support power and exponential integrals, divergent-moment limits and combined applications. Nine accessible diagrams distinguish density height from probability, normalisation, negative-density failures, central and conditional areas, mean versus median, percentile locations and partial infinite tails. Explicit cumulative distribution function knowledge is not required. Eighteen Sampling and estimation lessons cover all eight official section 6.4 requirements, with population and frame definitions, random-number selection, sampling-quality limits, independent sample-mean moments, normal means, CLT conditions, SE scaling, raw, summarised and coded unbiased estimates, known-variance and large-sample mean intervals, repeated-sampling interpretation, precision planning, proportion intervals, sparse-outcome limits and contextual applications. Ten accessible diagrams distinguish selection frames, number rejection, repeated sample means, population versus mean density, CLT models, variance divisors, interval targets, illustrative coverage, integer size checks and proportion margins. Eighteen Hypothesis tests lessons cover all five official section 6.5 requirements, with pre-specified alternatives, inclusive count tails, explicit equal-tail conventions, integer critical regions and actual significance, exposure-aware Poisson tests, continuity-corrected count approximations, normal and large-sample mean tests, confidence-interval connections, error classifications, fixed-region binomial, Poisson and normal error probabilities and sample-size power comparisons. Nine accessible diagrams show the testing procedure, exact count regions, unequal Poisson tails, discrete significance jumps, corrected count boundaries, matching mean intervals and alternative error distributions. All shared lessons are counted once in the course total. All 38 sections now link teaching and practice. Eight earlier overview lessons remain available, but do not count as full section coverage. This is a content-availability measure, not full programme completion or exam readiness. Written practice is self-checked, not officially marked. Cumulative assessment and human subject-teacher review remain pending. Check the 2028–2030 syllabus separately if sitting an examination from 2028; this teaching map uses 2026–2027 requirements.
Choose your paper route
- AS · Pure Mathematics onlyPapers 1 + 2
An AS-only route. Its result cannot be carried forward towards A Level Mathematics.
- AS · Pure Mathematics and MechanicsPapers 1 + 4
For staged A Level, follow with Papers 3 + 5.
- AS · Pure Mathematics and StatisticsPapers 1 + 5
For staged A Level, follow with Papers 3 + 4 or Papers 3 + 6.
- A Level · Pure Mathematics, Mechanics and StatisticsPapers 1 + 3 + 4 + 5
Take all four in one series, or use a permitted staged AS route above.
- A Level · Pure Mathematics and StatisticsPapers 1 + 3 + 5 + 6
Take all four in one series, or carry forward Papers 1 + 5 and add Papers 3 + 6.
Paper 1 is the foundation for every other component. Papers 2 and 3 build on it, and Papers 4 and 5 assume its content. Paper 6 builds on Paper 5. Papers 2 and 3 cannot be taken in the same series; Papers 4 and 6 do not form a valid A Level combination. Confirm examination entry and carry-forward rules with your school.
View syllabus coverage
Choose the components for your route with your school. The six papers are alternatives within defined combinations, not six compulsory papers. All eight Pure Mathematics 1 sections, 1.1–1.8, and all six Pure Mathematics 2 sections, 2.1–2.6, now have detailed teaching. Paper 2 belongs to the AS Pure Mathematics route, which cannot be carried forward to A Level. Paper 3 Algebra, section 3.1, also has teaching linked to all five objectives, using shared foundation lessons and its separate partial-fraction and rational-binomial lessons. Paper 3 sections 3.2, 3.3 and 3.6 use the corresponding shared teaching; section 3.4 adds the inverse-tangent derivative to the shared differentiation lessons. Paper 3 Integration, section 3.5, now combines four shared standard and identity lessons with seven lessons on scaled inverse-tangent primitives, linear and quadratic partial fractions, logarithmic patterns, single and repeated parts, and given substitutions. Paper 3 Vectors, section 3.7, now has eight lessons covering notation, arithmetic and midpoints, magnitudes and unit directions, line equations, line relationships, scalar-product angles, perpendicular feet and 3D solids. Paper 3 Differential equations, section 3.8, now has nine lessons on formulating rate laws, separation and equilibrium solutions, initial-value branches, advanced integration methods, growth and decay, limiting values, physical boundaries and saturation with independent model checks. Paper 3 Complex numbers, section 3.9, now has eleven lessons on Cartesian and polar operations, conjugate polynomial roots, Argand geometry, both square roots, transformations, distance loci and argument rays with combined regions. All nine Paper 3 sections now link teaching for their mapped requirements. Mechanics section 4.1 now has nine lessons covering all seven Forces and equilibrium requirements. It includes force identification, numerical components and resultants, two-direction equilibrium, contact components, smooth-model limits, static and limiting friction, and Newton’s third law. Mechanics section 4.2 now has nine lessons on one-dimensional kinematics, including initial-condition integration, multiple reversals and delayed starts with two valid meetings. Mechanics section 4.3 now has six Momentum lessons with both official requirements linked to teaching and staged practice. Mechanics section 4.4 now has twelve Newton’s laws lessons with all four official requirements linked to teaching and staged practice. Mechanics section 4.5 now has nine Energy, work and power lessons linked to all five official requirements. All five Mechanics sections now link teaching and staged practice. Statistics 1 section 5.1 now has thirteen Representation of data lessons with all five official requirements linked to teaching and staged practice. Statistics 1 section 5.2 now has ten Permutations and combinations lessons with both official requirements linked to teaching and staged practice. Statistics 1 section 5.3 now has eleven Probability lessons with all four official requirements linked to teaching and staged practice. Statistics 1 section 5.4 now has twelve Discrete random variables lessons with all three official requirements linked to teaching and staged practice. Statistics 1 section 5.5 now has twelve Normal distribution lessons with all three official requirements linked to teaching and staged practice. All five Statistics 1 sections now have linked teaching and staged practice. Statistics 2 section 6.1 now has fourteen Poisson distribution lessons linked to all five official requirements. Statistics 2 section 6.2 now has fourteen Linear combinations lessons linked to the six listed results in its official requirement. Statistics 2 section 6.3 now has fourteen Continuous random variables lessons linked to both official requirements, including median and percentile areas and infinite-support densities. Statistics 2 section 6.4 now has eighteen Sampling and estimation lessons linked to all eight official requirements, including random-number selection, normal and CLT means, unbiased estimates and mean and proportion confidence intervals. Statistics 2 section 6.5 now has eighteen Hypothesis tests lessons linked to all five official requirements, with exact binomial and Poisson decisions, suitable continuity-corrected normal approximations, known-variance and large-sample mean tests, contextual conclusions and Type I and Type II error probabilities. All 38 component sections now link teaching and staged practice. A linked section shows available teaching, not certified mastery, teacher approval or full assessment coverage.
Pure Mathematics 1 · Paper 1
- Quadratics1.1Complete a square and read a quadratic vertexComplete the square when the leading coefficient is not oneUse the discriminant to classify roots and parameter valuesSolve quadratic equations with a method you can justifySolve quadratic inequalities using signs and endpointsSolve a linear and a quadratic equation togetherRecognise equations that become quadratic after substitution
- Functions1.2Define a function and find its domain and rangeCompose functions in the correct order and check the inputsFind an inverse rule and exchange domain with rangeRestrict a quadratic inverse and reflect its graphTranslate, stretch and reflect a graph verticallyTransform graph inputs and combine horizontal moves
- Coordinate geometry1.3Build a straight-line equation from points and gradientsUse distance, midpoint and perpendicular gradients togetherConnect graph intersections to simultaneous equationsRead and construct a circle equation by completing squaresFind a circle tangent from its perpendicular radiusClassify line-circle intersections with substitutionUse circle symmetry, chords and diameters in coordinate problems
- Circular measure1.4
- Trigonometry1.5Sketch trigonometric graphs and track their transformationsUse exact trigonometric values and related anglesInterpret principal inverse trigonometric valuesProve and simplify trigonometric identities with restrictionsFind every trigonometric solution in a stated intervalSolve quadratic trigonometric equations and reject invalid ratios
- Series1.6Expand a positive-integer binomial with correct coefficients and signsSelect binomial coefficients and solve for unknown parametersFind arithmetic progression terms and finite sumsFind geometric progression terms and finite sumsConnect arithmetic and geometric progressions in one problemCheck geometric convergence and interpret a sum to infinity
- Differentiation1.7Understand a curve gradient as a limit of chord gradientsDifferentiate rational powers, sums and differencesUse the chain rule for composite power functionsBuild tangent and normal equations at a curve pointUse derivative signs to locate increasing and decreasing intervalsConnect rates of change between related quantitiesLocate stationary points and determine their nature
- Integration1.8Find antiderivatives of rational and linear-composite powersUse an initial condition to determine the integration constantEvaluate definite integrals and interpret signed accumulationEvaluate simple improper integrals using endpoint limitsFind total areas by splitting at axis crossingsFind areas between curves with the correct boundary orderFind volumes about the x axis with disks and washersSet up volumes about the y axis using horizontal slices
Pure Mathematics 2 · Paper 2 (AS only)
- Algebra2.1Interpret modulus and sketch a linear absolute-value graphSolve modulus equations and reject invalid signed candidatesSolve modulus inequalities using intervals and valid sign casesDivide polynomials and verify quotient and remainderUse remainder values and conditions to find unknown coefficientsUse factors to solve cubic and quartic equations completely
- Logarithmic and exponential functions2.2Use logarithm laws with valid arguments and basesSketch natural exponential and logarithmic graphsSolve exponential equations using logarithms and positive substitutionsSolve exponential inequalities with the correct directionLinearise a power model and recover its constantsLinearise an exponential model and interpret growth or decay
- Trigonometry2.3Interpret reciprocal trigonometric functions and their domainsSketch secant cosecant and cotangent with separate branchesUse squared reciprocal identities to solve trigonometric equationsExpand compound angles and evaluate exact trigonometric valuesUse double-angle identities without losing valid solutionsExpress a sine cosine combination in amplitude phase formSolve phase shifted trigonometric equations over a full interval
- Differentiation2.4Differentiate natural exponentials and logarithms with their inner ratesDifferentiate trigonometric composites using radian measureDifferentiate products while retaining both changing factorsDifferentiate quotients and preserve the original denominator exclusionsFind parametric gradients from two derivatives with respect to the parameterDifferentiate implicit curves using product and chain rulesHandle horizontal vertical and indeterminate tangent cases
- Integration2.5Integrate natural exponential composites by reversing their inner scaleIntegrate linear reciprocals with logarithms on valid intervalsIntegrate linear trigonometric composites with radian inputUse trigonometric identities to expose integrable termsApply the trapezium rule with correct spacing and endpoint weightsUse curve shape to judge trapezium overestimates and underestimatesChoose exact integration or data estimates and explain their limits
- Numerical solution of equations2.6Locate equation roots with graphs and valid sign changesRefine root brackets and justify the requested roundingInterpret iteration notation and verify an equivalent fixed-point equationRun given iterations with retained precision and original-equation checksRecognise oscillating failing and undefined iterationsVerify numerical accuracy without mistaking stable digits for proof
Pure Mathematics 3 · Paper 3 (A Level)
- Algebra3.1Interpret modulus and sketch a linear absolute-value graphSolve modulus equations and reject invalid signed candidatesSolve modulus inequalities using intervals and valid sign casesDivide polynomials and verify quotient and remainderUse remainder values and conditions to find unknown coefficientsUse factors to solve cubic and quartic equations completelyDecompose distinct linear factors and equal-degree rational functionsInclude every power of a repeated linear factor in partial fractionsUse a linear numerator over an irreducible quadratic factorExpand rational powers of one plus x with correct coefficientsAdapt a binomial expansion and transform its validity intervalCombine binomial expansions and check a numerical approximation
- Logarithmic and exponential functions3.2Use logarithm laws with valid arguments and basesSketch natural exponential and logarithmic graphsSolve exponential equations using logarithms and positive substitutionsSolve exponential inequalities with the correct directionLinearise a power model and recover its constantsLinearise an exponential model and interpret growth or decay
- Trigonometry3.3Sketch trigonometric graphs and track their transformationsInterpret reciprocal trigonometric functions and their domainsSketch secant cosecant and cotangent with separate branchesUse squared reciprocal identities to solve trigonometric equationsExpand compound angles and evaluate exact trigonometric valuesUse double-angle identities without losing valid solutionsExpress a sine cosine combination in amplitude phase formSolve phase shifted trigonometric equations over a full interval
- Differentiation3.4Differentiate natural exponentials and logarithms with their inner ratesDifferentiate trigonometric composites using radian measureDifferentiate products while retaining both changing factorsDifferentiate quotients and preserve the original denominator exclusionsFind parametric gradients from two derivatives with respect to the parameterDifferentiate implicit curves using product and chain rulesHandle horizontal vertical and indeterminate tangent casesDifferentiate inverse tangent with its inner derivative
- Integration3.5Integrate natural exponential composites by reversing their inner scaleIntegrate linear reciprocals with logarithms on valid intervalsIntegrate linear trigonometric composites with radian inputUse trigonometric identities to expose integrable termsIntegrate positive quadratic reciprocals with inverse tangentIntegrate linear partial fractions including repeated factorsSplit quadratic partial fractions into logarithmic and arctangent partsRecognise a logarithmic derivative inside an integralUse integration by parts to simplify a productRepeat integration by parts and combine it with inverse tangentUse a given substitution and transform every integration limit
- Numerical solution of equations3.6Locate equation roots with graphs and valid sign changesRefine root brackets and justify the requested roundingInterpret iteration notation and verify an equivalent fixed-point equationRun given iterations with retained precision and original-equation checksRecognise oscillating failing and undefined iterationsVerify numerical accuracy without mistaking stable digits for proof
- Vectors3.7Read vector notation and distinguish position from displacementAdd vectors and use midpoint and parallelogram geometryFind vector magnitudes and unit directions in two and three dimensionsWrite vector equations of lines from points and directionsClassify parallel intersecting and skew lines with independent parametersUse scalar products to find angles between vectors and linesFind the foot of a perpendicular from a point to a lineSolve vector geometry inside cuboids and tetrahedra
- Differential equations3.8Formulate a differential equation from a stated rate of changeSeparate variables and retain solutions lost by divisionApply an initial condition and verify the solution branchSolve separable equations using logarithms partial fractions and inverse tangentUse integration by parts and given substitutions in differential equationsModel exponential growth decay and threshold timesModel an approach to a limiting value from above or belowRespect physical boundaries and singular times in changing-quantity modelsInterpret a saturating model and check it against further data
- Complex numbers3.9Understand complex numbers and equate real and imaginary partsCalculate Cartesian products quotients and conjugatesUse conjugate pairs to solve real coefficient polynomialsPlot Argand points and find modulus and argumentConvert between Cartesian and polar complex formsMultiply and divide polar forms by combining moduli and anglesFind both square roots with exact Cartesian workingInterpret conjugation translation rotation and complex divisionDescribe modulus circles disks and annular regionsUse equal distances for bisectors and nearer point half planesDraw argument rays and combine angular and radial conditions
Mechanics · Paper 4
- Forces and equilibrium4.1Identify forces and draw a particle free body diagramResolve forces and calculate a resultant with its directionSolve particle equilibrium in two perpendicular directionsSeparate contact into normal reaction and frictionUse smooth contact on inclined planes and vertical wallsDistinguish static friction from its limiting valueFind the equilibrium range on a rough inclined planeCalculate limiting pulls and pushes when reaction changesApply Newtons third law and separate interacting bodies
- Kinematics of motion in a straight line4.2Distinguish distance displacement speed and directed velocityRead and sketch displacement time graphs using their gradientsUse velocity time gradients and signed areasDifferentiate displacement to interpret variable motionIntegrate acceleration and velocity using initial conditionsCalculate total distance when a particle changes directionSelect constant acceleration formulae and keep directed signsJoin constant acceleration stages without resetting velocitySolve two particle meetings with a shared position and clock
- Momentum4.3Calculate signed linear momentum and combine moving particlesChoose a momentum system and test conservation across an impactFind an unknown outgoing velocity after direct impactFind the common velocity when bodies coalesceRecover unknown masses or incoming velocities from an impactTrack successive impacts and distinguish momentum from kinetic energy
- Newton’s laws of motion4.4Use the resultant force to find signed accelerationDistinguish mass weight and an accelerating support forceJoin vertical force stages with constant acceleration motionAccelerate a particle on a rough horizontal surfaceResolve Newton equations along a smooth inclined planeChange friction direction on uphill and downhill motionInfer a missing force mass or friction coefficientSolve two hanging particles joined over a smooth pulleyConnect a rough horizontal particle to a hanging massSolve a rough incline linked to a hanging particleDistinguish towing tension from braking thrustRebuild connected motion after a string goes slack
- Energy, work and power4.5Calculate signed work by a constant forceCalculate kinetic energy and changes of speedUse vertical height in gravitational potential energyApply net work to kinetic energy on a level pathConserve mechanical energy on smooth slopes and slidesInclude friction and driving work on a rough slopeUse energy for a taut connected particle systemDistinguish average power from instantaneous driving powerFind instantaneous acceleration of a powered vehicle on a hill
Probability & Statistics 1 · Paper 5
- Representation of data5.1Choose a representation that answers a statistical questionBuild ordered stem-and-leaf diagrams with an unambiguous keyCompare two raw datasets with back-to-back stemsFind raw-data centre, quartiles and spreadDraw and interpret common-scale box-and-whisker plotsUse histogram density and area for unequal classesDraw cumulative frequencies and estimate quartiles and proportionsCompare distributions using suitable centre and variationCalculate descriptive standard deviation from raw and frequency dataEstimate mean and standard deviation from grouped intervalsDecode shifted and scaled totals without losing unitsCombine two datasets by restoring and adding momentsCorrect totals and infer missing observations consistently
- Permutations and combinations5.2Build a count using product and sum rulesCount ordered selections with permutationsCount unordered selections with combinationsSelect groups with exact and minimum requirementsArrange repeated objects without counting identical copies twiceUse blocks for objects that must stay togetherKeep objects apart with complements and gapsHandle fixed positions and leading-zero restrictionsSeat people in two or more labelled rowsCheck mixed restrictions with disjoint cases and small lists
- Probability5.3Define equiprobable outcomes before calculating probabilityUse selections and ordered assignments to calculate probabilityUse complements for at-least and not eventsCombine events using disjoint regions and overlapDistinguish independence from mutual exclusionCondition on a restricted sample spaceRead conditional probabilities from two-way tablesFollow probability trees without replacementModel replacement and independent repeated trialsReverse a tree condition using joint and total probabilitiesSolve unknown probability parameters and mixed conditions
- Discrete random variables5.4Build a discrete probability distribution from outcomesNormalise probability tables and check parameter boundsCalculate expectation and variance from a probability tableInfer table parameters without overclaiming a unique distributionRecognise when a binomial model is justifiedCalculate exact binomial probabilities with the order factorCalculate binomial tails and intervals with exact boundariesUse binomial moments and infer valid parametersDefine geometric waiting times starting at the first trialCalculate geometric first-success probabilitiesUse geometric tails and conditions after early failuresInterpret geometric expectation and choose attempt thresholds
- The normal distribution5.5Interpret a normal model and probability as areaRead cumulative normal tables and reflect negative valuesStandardise measurements using the standard deviationChoose normal tails and check the boundary signCalculate normal intervals, outside regions and conditional areasFind normal percentiles and central probability limitsInfer a normal mean from a probability conditionInfer normal spread and reject incompatible conditionsInfer both normal parameters from two percentilesCheck when a normal approximation to binomial is suitableApply continuity correction to one-sided binomial eventsCorrect binomial intervals, point events and probability thresholds
Probability & Statistics 2 · Paper 6 (A Level)
- The Poisson distribution6.1Define a Poisson count and its modelling assumptionsScale event rates to the required exposureCalculate exact Poisson count probabilitiesCalculate Poisson tails and inclusive intervalsCondition Poisson probabilities on a known count eventUse Poisson mean variance and standard deviationInfer a Poisson parameter from a probability or ratioChoose an exposure using a Poisson probability targetUse independent disjoint intervals and reject overlap productsCheck a Poisson approximation to a binomial countCalculate Poisson approximations and compare exact binomial answersMatch a Poisson count with a suitable normal approximationCorrect normal approximations to Poisson eventsCheck exact Poisson probabilities before a threshold decision
- Linear combinations of random variables6.2Transform a random variable and track its momentsHandle negative scales and constant random variablesCombine expectations without assuming independenceAdd squared variance contributions for independent variablesModel totals with fixed fees and random componentsDistinguish independent copies from repeating one observationTransform a normal distribution with a nonzero scaleReverse normal inequalities under a negative scaleCombine independent normal sums and differencesCalculate weighted normal probabilities and conditional areasInfer linear coefficients from moment conditionsAdd independent Poisson counts with exact distribution closureCombine Poisson exposures without double counting shared eventsCheck distribution claims for weighted counts and dependent inputs
- Continuous random variables6.3Distinguish density height from continuous probabilityNormalise a density over its stated supportCheck density non-negativity and parameter boundsIntersect a continuous event with the density supportUse continuous complements and central density areasCondition on a continuous density regionCalculate a continuous expectation by weighting densityCalculate continuous variance from the second momentFind medians and percentiles from density areasUse density symmetry without equating every mean and medianInfer density parameters and reject invalid moment solutionsIntegrate an infinite-support density with limitsUse exponential density areas and check divergent momentsCombine density probabilities moments and model interpretation
- Sampling and estimation6.4Distinguish populations samples and sampling framesSelect a random sample using labelled random numbersExplain why a sampling method may be unsatisfactoryTreat the sample mean as a random variableCalculate probabilities for means from a normal populationUse the Central Limit Theorem with clear limitsCompare sample sizes standard errors and dependenceCalculate unbiased estimates from raw sample dataEstimate population moments from summarised sample totalsRecover unbiased estimates from coded sample dataConstruct a mean confidence interval with known varianceInterpret confidence as repeated-sampling coverageChoose sample size for a mean interval precision targetConstruct a large-sample mean interval with estimated varianceConstruct an approximate population proportion intervalCheck proportion interval conditions and extreme outcomesPlan a proportion sample size for a target marginConnect sampling estimates and confidence claims in context
- Hypothesis tests6.5Describe hypothesis tests and their decision regionsChoose one-tailed and two-tailed alternatives before testingTest a binomial increase using an inclusive upper tailTest a binomial decrease using an inclusive lower tailAllocate both tails in a two-sided binomial testFind discrete critical regions and their actual significanceTest a Poisson increase at a fixed exposureTest a Poisson decrease with an exact cumulative tailConstruct an equal-tail two-sided Poisson decisionApproximate a binomial hypothesis tail with continuity correctionApproximate a Poisson test when the null mean is largeTest a normal population mean in a specified directionConnect two-sided normal mean tests with confidence intervalsTest a mean with a large sample and estimated varianceDistinguish Type I and Type II testing errorsCalculate binomial error probabilities for a fixed testCalculate Poisson error probabilities at specified ratesCalculate normal test errors and compare sample-size power
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- 01
Complete a square and read a quadratic vertex
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Not started - 02
Complete the square when the leading coefficient is not one
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Not started - 03
Use the discriminant to classify roots and parameter values
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Not started - 04
Solve quadratic equations with a method you can justify
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Not started - 05
Solve quadratic inequalities using signs and endpoints
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Not started - 06
Solve a linear and a quadratic equation together
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Not started - 07
Recognise equations that become quadratic after substitution
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Not started - 08
Define a function and find its domain and range
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Not started - 09
Compose functions in the correct order and check the inputs
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Not started - 10
Find an inverse rule and exchange domain with range
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Not started - 11
Restrict a quadratic inverse and reflect its graph
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Translate, stretch and reflect a graph vertically
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Not started - 13
Transform graph inputs and combine horizontal moves
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Not started - 14
Build a straight-line equation from points and gradients
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Not started - 15
Use distance, midpoint and perpendicular gradients together
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Not started - 16
Connect graph intersections to simultaneous equations
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Not started - 17
Read and construct a circle equation by completing squares
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Not started - 18
Find a circle tangent from its perpendicular radius
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Not started - 19
Classify line-circle intersections with substitution
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Use circle symmetry, chords and diameters in coordinate problems
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Not started - 21
Define a radian and convert angle units accurately
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Not started - 22
Calculate arc lengths, sector areas and sector perimeters
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Not started - 23
Connect sectors, chords, triangles and circular segments
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Not started - 24
Sketch trigonometric graphs and track their transformations
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Not started - 25
Use exact trigonometric values and related angles
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Not started - 26
Interpret principal inverse trigonometric values
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Not started - 27
Prove and simplify trigonometric identities with restrictions
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Not started - 28
Find every trigonometric solution in a stated interval
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Not started - 29
Solve quadratic trigonometric equations and reject invalid ratios
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Not started - 30
Expand a positive-integer binomial with correct coefficients and signs
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Not started - 31
Select binomial coefficients and solve for unknown parameters
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Not started - 32
Find arithmetic progression terms and finite sums
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Not started - 33
Find geometric progression terms and finite sums
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Not started - 34
Connect arithmetic and geometric progressions in one problem
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Not started - 35
Check geometric convergence and interpret a sum to infinity
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Not started - 36
Understand a curve gradient as a limit of chord gradients
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Not started - 37
Differentiate rational powers, sums and differences
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Not started - 38
Use the chain rule for composite power functions
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Not started - 39
Build tangent and normal equations at a curve point
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Not started - 40
Use derivative signs to locate increasing and decreasing intervals
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Not started - 41
Connect rates of change between related quantities
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Not started - 42
Locate stationary points and determine their nature
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Not started - 43
Find antiderivatives of rational and linear-composite powers
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Not started - 44
Use an initial condition to determine the integration constant
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Not started - 45
Evaluate definite integrals and interpret signed accumulation
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Not started - 46
Evaluate simple improper integrals using endpoint limits
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Not started - 47
Find total areas by splitting at axis crossings
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Not started - 48
Find areas between curves with the correct boundary order
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Not started - 49
Find volumes about the x axis with disks and washers
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Not started - 50
Set up volumes about the y axis using horizontal slices
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Not started - 51
Interpret modulus and sketch a linear absolute-value graph
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Not started - 52
Solve modulus equations and reject invalid signed candidates
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Not started - 53
Solve modulus inequalities using intervals and valid sign cases
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Not started - 54
Divide polynomials and verify quotient and remainder
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Not started - 55
Use remainder values and conditions to find unknown coefficients
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Not started - 56
Use factors to solve cubic and quartic equations completely
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Not started - 57
Use logarithm laws with valid arguments and bases
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Not started - 58
Sketch natural exponential and logarithmic graphs
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Not started - 59
Solve exponential equations using logarithms and positive substitutions
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Not started - 60
Solve exponential inequalities with the correct direction
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Not started - 61
Linearise a power model and recover its constants
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Not started - 62
Linearise an exponential model and interpret growth or decay
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Not started - 63
Interpret reciprocal trigonometric functions and their domains
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Not started - 64
Sketch secant cosecant and cotangent with separate branches
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Not started - 65
Use squared reciprocal identities to solve trigonometric equations
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Not started - 66
Expand compound angles and evaluate exact trigonometric values
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Not started - 67
Use double-angle identities without losing valid solutions
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Not started - 68
Express a sine cosine combination in amplitude phase form
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Not started - 69
Solve phase shifted trigonometric equations over a full interval
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Not started - 70
Differentiate natural exponentials and logarithms with their inner rates
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Not started - 71
Differentiate trigonometric composites using radian measure
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Not started - 72
Differentiate products while retaining both changing factors
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Not started - 73
Differentiate quotients and preserve the original denominator exclusions
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Not started - 74
Find parametric gradients from two derivatives with respect to the parameter
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Not started - 75
Differentiate implicit curves using product and chain rules
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Not started - 76
Handle horizontal vertical and indeterminate tangent cases
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Not started - 77
Integrate natural exponential composites by reversing their inner scale
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Not started - 78
Integrate linear reciprocals with logarithms on valid intervals
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Not started - 79
Integrate linear trigonometric composites with radian input
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Not started - 80
Use trigonometric identities to expose integrable terms
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Not started - 81
Apply the trapezium rule with correct spacing and endpoint weights
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Not started - 82
Use curve shape to judge trapezium overestimates and underestimates
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Not started - 83
Choose exact integration or data estimates and explain their limits
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Not started - 84
Locate equation roots with graphs and valid sign changes
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Not started - 85
Refine root brackets and justify the requested rounding
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Not started - 86
Interpret iteration notation and verify an equivalent fixed-point equation
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Not started - 87
Run given iterations with retained precision and original-equation checks
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Not started - 88
Recognise oscillating failing and undefined iterations
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Not started - 89
Verify numerical accuracy without mistaking stable digits for proof
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Not started - 90
Decompose distinct linear factors and equal-degree rational functions
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Not started - 91
Include every power of a repeated linear factor in partial fractions
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Not started - 92
Use a linear numerator over an irreducible quadratic factor
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Not started - 93
Expand rational powers of one plus x with correct coefficients
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Not started - 94
Adapt a binomial expansion and transform its validity interval
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Not started - 95
Combine binomial expansions and check a numerical approximation
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Not started - 96
Differentiate inverse tangent with its inner derivative
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Not started - 97
Integrate positive quadratic reciprocals with inverse tangent
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Not started - 98
Integrate linear partial fractions including repeated factors
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Not started - 99
Split quadratic partial fractions into logarithmic and arctangent parts
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Not started - 100
Recognise a logarithmic derivative inside an integral
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Not started - 101
Use integration by parts to simplify a product
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Not started - 102
Repeat integration by parts and combine it with inverse tangent
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Not started - 103
Use a given substitution and transform every integration limit
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Not started - 104
Read vector notation and distinguish position from displacement
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Not started - 105
Add vectors and use midpoint and parallelogram geometry
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Not started - 106
Find vector magnitudes and unit directions in two and three dimensions
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Not started - 107
Write vector equations of lines from points and directions
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Not started - 108
Classify parallel intersecting and skew lines with independent parameters
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Not started - 109
Use scalar products to find angles between vectors and lines
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Not started - 110
Find the foot of a perpendicular from a point to a line
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Not started - 111
Solve vector geometry inside cuboids and tetrahedra
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Not started - 112
Formulate a differential equation from a stated rate of change
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Not started - 113
Separate variables and retain solutions lost by division
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Not started - 114
Apply an initial condition and verify the solution branch
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Not started - 115
Solve separable equations using logarithms partial fractions and inverse tangent
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Not started - 116
Use integration by parts and given substitutions in differential equations
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Not started - 117
Model exponential growth decay and threshold times
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Not started - 118
Model an approach to a limiting value from above or below
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Not started - 119
Respect physical boundaries and singular times in changing-quantity models
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Not started - 120
Interpret a saturating model and check it against further data
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Not started - 121
Understand complex numbers and equate real and imaginary parts
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Not started - 122
Calculate Cartesian products quotients and conjugates
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Not started - 123
Use conjugate pairs to solve real coefficient polynomials
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Not started - 124
Plot Argand points and find modulus and argument
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Not started - 125
Convert between Cartesian and polar complex forms
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Not started - 126
Multiply and divide polar forms by combining moduli and angles
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Not started - 127
Find both square roots with exact Cartesian working
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Not started - 128
Interpret conjugation translation rotation and complex division
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Not started - 129
Describe modulus circles disks and annular regions
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Not started - 130
Use equal distances for bisectors and nearer point half planes
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Not started - 131
Draw argument rays and combine angular and radial conditions
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Not started - 132
Identify forces and draw a particle free body diagram
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Not started - 133
Resolve forces and calculate a resultant with its direction
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Not started - 134
Solve particle equilibrium in two perpendicular directions
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Not started - 135
Separate contact into normal reaction and friction
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Not started - 136
Use smooth contact on inclined planes and vertical walls
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Not started - 137
Distinguish static friction from its limiting value
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Not started - 138
Find the equilibrium range on a rough inclined plane
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Not started - 139
Calculate limiting pulls and pushes when reaction changes
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Not started - 140
Apply Newtons third law and separate interacting bodies
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Not started - 141
Distinguish distance displacement speed and directed velocity
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Not started - 142
Read and sketch displacement time graphs using their gradients
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Not started - 143
Use velocity time gradients and signed areas
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Not started - 144
Differentiate displacement to interpret variable motion
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Not started - 145
Integrate acceleration and velocity using initial conditions
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Not started - 146
Calculate total distance when a particle changes direction
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Not started - 147
Select constant acceleration formulae and keep directed signs
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Not started - 148
Join constant acceleration stages without resetting velocity
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Not started - 149
Solve two particle meetings with a shared position and clock
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Not started - 150
Calculate signed linear momentum and combine moving particles
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Not started - 151
Choose a momentum system and test conservation across an impact
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Not started - 152
Find an unknown outgoing velocity after direct impact
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Not started - 153
Find the common velocity when bodies coalesce
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Not started - 154
Recover unknown masses or incoming velocities from an impact
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Not started - 155
Track successive impacts and distinguish momentum from kinetic energy
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Not started - 156
Use the resultant force to find signed acceleration
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Not started - 157
Distinguish mass weight and an accelerating support force
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Not started - 158
Join vertical force stages with constant acceleration motion
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Not started - 159
Accelerate a particle on a rough horizontal surface
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Not started - 160
Resolve Newton equations along a smooth inclined plane
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Not started - 161
Change friction direction on uphill and downhill motion
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Not started - 162
Infer a missing force mass or friction coefficient
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Not started - 163
Solve two hanging particles joined over a smooth pulley
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Not started - 164
Connect a rough horizontal particle to a hanging mass
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Not started - 165
Solve a rough incline linked to a hanging particle
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Not started - 166
Distinguish towing tension from braking thrust
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Not started - 167
Rebuild connected motion after a string goes slack
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Not started - 168
Calculate signed work by a constant force
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Not started - 169
Calculate kinetic energy and changes of speed
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Not started - 170
Use vertical height in gravitational potential energy
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Not started - 171
Apply net work to kinetic energy on a level path
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Not started - 172
Conserve mechanical energy on smooth slopes and slides
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Not started - 173
Include friction and driving work on a rough slope
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Not started - 174
Use energy for a taut connected particle system
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Not started - 175
Distinguish average power from instantaneous driving power
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Not started - 176
Find instantaneous acceleration of a powered vehicle on a hill
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Not started - 177
Choose a representation that answers a statistical question
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Not started - 178
Build ordered stem-and-leaf diagrams with an unambiguous key
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Not started - 179
Compare two raw datasets with back-to-back stems
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Not started - 180
Find raw-data centre, quartiles and spread
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Not started - 181
Draw and interpret common-scale box-and-whisker plots
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Not started - 182
Use histogram density and area for unequal classes
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Not started - 183
Draw cumulative frequencies and estimate quartiles and proportions
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Not started - 184
Compare distributions using suitable centre and variation
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Not started - 185
Calculate descriptive standard deviation from raw and frequency data
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Not started - 186
Estimate mean and standard deviation from grouped intervals
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Not started - 187
Decode shifted and scaled totals without losing units
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Not started - 188
Combine two datasets by restoring and adding moments
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Not started - 189
Correct totals and infer missing observations consistently
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Not started - 190
Build a count using product and sum rules
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Not started - 191
Count ordered selections with permutations
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Not started - 192
Count unordered selections with combinations
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Not started - 193
Select groups with exact and minimum requirements
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Not started - 194
Arrange repeated objects without counting identical copies twice
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Not started - 195
Use blocks for objects that must stay together
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Not started - 196
Keep objects apart with complements and gaps
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Not started - 197
Handle fixed positions and leading-zero restrictions
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Not started - 198
Seat people in two or more labelled rows
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Not started - 199
Check mixed restrictions with disjoint cases and small lists
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Not started - 200
Define equiprobable outcomes before calculating probability
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Not started - 201
Use selections and ordered assignments to calculate probability
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Not started - 202
Use complements for at-least and not events
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Not started - 203
Combine events using disjoint regions and overlap
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Not started - 204
Distinguish independence from mutual exclusion
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Not started - 205
Condition on a restricted sample space
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Not started - 206
Read conditional probabilities from two-way tables
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Not started - 207
Follow probability trees without replacement
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Not started - 208
Model replacement and independent repeated trials
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Not started - 209
Reverse a tree condition using joint and total probabilities
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Not started - 210
Solve unknown probability parameters and mixed conditions
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Not started - 211
Build a discrete probability distribution from outcomes
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Not started - 212
Normalise probability tables and check parameter bounds
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Not started - 213
Calculate expectation and variance from a probability table
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Not started - 214
Infer table parameters without overclaiming a unique distribution
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Not started - 215
Recognise when a binomial model is justified
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Not started - 216
Calculate exact binomial probabilities with the order factor
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Not started - 217
Calculate binomial tails and intervals with exact boundaries
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Not started - 218
Use binomial moments and infer valid parameters
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Not started - 219
Define geometric waiting times starting at the first trial
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Not started - 220
Calculate geometric first-success probabilities
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Not started - 221
Use geometric tails and conditions after early failures
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Not started - 222
Interpret geometric expectation and choose attempt thresholds
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Not started - 223
Interpret a normal model and probability as area
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Not started - 224
Read cumulative normal tables and reflect negative values
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Not started - 225
Standardise measurements using the standard deviation
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Not started - 226
Choose normal tails and check the boundary sign
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Not started - 227
Calculate normal intervals, outside regions and conditional areas
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Not started - 228
Find normal percentiles and central probability limits
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Not started - 229
Infer a normal mean from a probability condition
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Not started - 230
Infer normal spread and reject incompatible conditions
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Not started - 231
Infer both normal parameters from two percentiles
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Not started - 232
Check when a normal approximation to binomial is suitable
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Not started - 233
Apply continuity correction to one-sided binomial events
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Not started - 234
Correct binomial intervals, point events and probability thresholds
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Not started - 235
Define a Poisson count and its modelling assumptions
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Not started - 236
Scale event rates to the required exposure
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Not started - 237
Calculate exact Poisson count probabilities
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Not started - 238
Calculate Poisson tails and inclusive intervals
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Not started - 239
Condition Poisson probabilities on a known count event
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Not started - 240
Use Poisson mean variance and standard deviation
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Not started - 241
Infer a Poisson parameter from a probability or ratio
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Not started - 242
Choose an exposure using a Poisson probability target
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Not started - 243
Use independent disjoint intervals and reject overlap products
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Not started - 244
Check a Poisson approximation to a binomial count
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Not started - 245
Calculate Poisson approximations and compare exact binomial answers
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Not started - 246
Match a Poisson count with a suitable normal approximation
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Not started - 247
Correct normal approximations to Poisson events
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Not started - 248
Check exact Poisson probabilities before a threshold decision
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Not started - 249
Transform a random variable and track its moments
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Not started - 250
Handle negative scales and constant random variables
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Not started - 251
Combine expectations without assuming independence
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Not started - 252
Add squared variance contributions for independent variables
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Not started - 253
Model totals with fixed fees and random components
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Not started - 254
Distinguish independent copies from repeating one observation
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Not started - 255
Transform a normal distribution with a nonzero scale
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Not started - 256
Reverse normal inequalities under a negative scale
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Not started - 257
Combine independent normal sums and differences
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Not started - 258
Calculate weighted normal probabilities and conditional areas
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Not started - 259
Infer linear coefficients from moment conditions
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Not started - 260
Add independent Poisson counts with exact distribution closure
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Not started - 261
Combine Poisson exposures without double counting shared events
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Not started - 262
Check distribution claims for weighted counts and dependent inputs
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Not started - 263
Distinguish density height from continuous probability
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Not started - 264
Normalise a density over its stated support
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Not started - 265
Check density non-negativity and parameter bounds
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Not started - 266
Intersect a continuous event with the density support
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Not started - 267
Use continuous complements and central density areas
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Not started - 268
Condition on a continuous density region
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Not started - 269
Calculate a continuous expectation by weighting density
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Not started - 270
Calculate continuous variance from the second moment
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Not started - 271
Find medians and percentiles from density areas
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Not started - 272
Use density symmetry without equating every mean and median
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Not started - 273
Infer density parameters and reject invalid moment solutions
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Not started - 274
Integrate an infinite-support density with limits
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Not started - 275
Use exponential density areas and check divergent moments
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Not started - 276
Combine density probabilities moments and model interpretation
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Not started - 277
Distinguish populations samples and sampling frames
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Not started - 278
Select a random sample using labelled random numbers
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Not started - 279
Explain why a sampling method may be unsatisfactory
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Not started - 280
Treat the sample mean as a random variable
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Not started - 281
Calculate probabilities for means from a normal population
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Not started - 282
Use the Central Limit Theorem with clear limits
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Not started - 283
Compare sample sizes standard errors and dependence
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Not started - 284
Calculate unbiased estimates from raw sample data
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Not started - 285
Estimate population moments from summarised sample totals
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Not started - 286
Recover unbiased estimates from coded sample data
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Not started - 287
Construct a mean confidence interval with known variance
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Not started - 288
Interpret confidence as repeated-sampling coverage
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Not started - 289
Choose sample size for a mean interval precision target
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Not started - 290
Construct a large-sample mean interval with estimated variance
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Not started - 291
Construct an approximate population proportion interval
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Not started - 292
Check proportion interval conditions and extreme outcomes
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Not started - 293
Plan a proportion sample size for a target margin
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Not started - 294
Connect sampling estimates and confidence claims in context
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Not started - 295
Describe hypothesis tests and their decision regions
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Not started - 296
Choose one-tailed and two-tailed alternatives before testing
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Not started - 297
Test a binomial increase using an inclusive upper tail
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Not started - 298
Test a binomial decrease using an inclusive lower tail
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Not started - 299
Allocate both tails in a two-sided binomial test
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Not started - 300
Find discrete critical regions and their actual significance
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Not started - 301
Test a Poisson increase at a fixed exposure
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Not started - 302
Test a Poisson decrease with an exact cumulative tail
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Not started - 303
Construct an equal-tail two-sided Poisson decision
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Not started - 304
Approximate a binomial hypothesis tail with continuity correction
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Not started - 305
Approximate a Poisson test when the null mean is large
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Not started - 306
Test a normal population mean in a specified direction
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Not started - 307
Connect two-sided normal mean tests with confidence intervals
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Not started - 308
Test a mean with a large sample and estimated variance
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Not started - 309
Distinguish Type I and Type II testing errors
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Not started - 310
Calculate binomial error probabilities for a fixed test
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Not started - 311
Calculate Poisson error probabilities at specified rates
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Not started - 312
Calculate normal test errors and compare sample-size power
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Not started - 313
Quadratics and repeated roots
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Not started - 314
Binomial coefficients
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Not started - 315
Radians and circular measure
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Not started - 316
Trigonometric equations
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Not started - 317
Differentiation and integration
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Not started - 318
- 319
Counting and combined probability
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Not started - 320
Discrete and binomial distributions
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Browse 38 topics →AS & A Level Mathematics 9709 · Pure 1 Mini
6 questions · 45 min · Timed practice
AS & A Level Mathematics 9709 · Statistics 1 Mini
5 questions · 35 min · Timed practice
Official papers
Historical and specimen papers may follow a different syllabus. Check the year and component before practising.
Paper 1 · Pure Mathematics 1 · Timezone 1
June 2024 · 9709/11
Paper 2 · Pure Mathematics 2 · Timezone 1
June 2024 · 9709/21
Paper 3 · Pure Mathematics 3 · Timezone 1
June 2024 · 9709/31
Paper 4 · Mechanics · Timezone 1
June 2024 · 9709/41
Paper 5 · Probability & Statistics 1 · Timezone 1
June 2024 · 9709/51
Paper 6 · Probability & Statistics 2 · Timezone 1
June 2024 · 9709/61