OCR A-Level Mathematics B (MEI)

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Pure Mathematics

Proof and mathematical language

Mp1, Ma1 - Proof, logic and mathematical argumentsGood mathematical writing makes a claim precise and then explains why it follows. Mp1 develops deduction and proof by exhaustion; Ma1 gives you the vocabulary and notation to express those arguments clearly. You will also read a short argument with a newly defined symbol and reconstruct what it s...
Mp2 - Conjectures and counterexamplesCounterexamples test the reach of a mathematical claim. You will choose admissible inputs, show an exact failure and explain what has been disproved. You will also read an argument critically: an unconvincing proof, a false conjecture and a proved replacement are three different outcomes.
Mp3 - Proof by contradictionContradiction turns a temporary assumption into a reason it cannot be true. You will prove that √(2) is irrational and that the primes never run out, then use a supplied definition to tackle an unfamiliar relationship between lengths. In each argument, the essential step is to identify th...

Algebra, equations and inequalities

Ma12, Ma13 - Rational indices and index lawsUse Ma12's laws to combine powers and Ma13's negative, fractional and zero indices to interpret what those powers mean. You will keep exact values, make restrictions visible and explain a simplification clearly enough for someone else to follow it.
Ma10, Ma11 - Surds and rationalising denominatorsUse Ma10 to simplify and calculate with exact surds, then Ma11 to rationalise their denominators. You will see why conjugates work, keep the domain of a variable expression and make short exact arguments without replacing roots by rounded decimals.
Ma3, MC3 - Quadratic graphs and the discriminantUse MC3's completing-the-square method to reveal a parabola's turning point, symmetry line and shape. Link the picture to Ma3's discriminant so you can classify real roots, interpret a parameter and explain why a boundary value matters.
Ma2 - Solving quadratic equations and disguised quadraticsLearn to solve quadratics by factorising, completing the square, using the formula and reading graphs. Then recognise the same structure inside another expression, and use its domain to decide which answers survive.
Ma4, Ma5 - Simultaneous equations and intersectionsUse elimination and substitution for two linear equations in Ma4, then extend those tools to a linear and a quadratic equation in Ma5. Learn to recover the correct pairs of values and justify how many common points the equations describe.
Ma7–Ma9 - Inequalities, sets and graphical regionsUse Ma7's linear and Ma8's quadratic inequalities to find intervals and describe regions in the plane. In Ma9, communicate the answer with “and”, “or” or set notation, making every included region and boundary clear.
Mf1, Mf2 - Polynomials, division and the factor theoremA polynomial equation becomes much friendlier when you can see its factors. Use Mf1 to expand and divide polynomials, then Mf2 to justify factors, recover unknown coefficients and find every real root of equations through degree four.
Ma16 - Rational expressions and algebraic divisionMake a complicated algebraic fraction simpler without changing the inputs it allows. You will factorise and cancel, combine fractions and use linear algebraic division to reveal a polynomial part and a remainder.
Ma15 - Partial fractions with linear factorsReplace an algebraic fraction by simpler fractions whose numerators are constants. Handle distinct and squared linear factors, then use supplied rules and a short mathematical passage to see what the decomposition can reveal.

Functions and graphs

Mf6, Mf7 - Modulus graphs, equations and inequalitiesRead a modulus as a distance, then connect that distance to a graph with two straight arms. You will find intersections, retain valid branches and express inequality solutions with the right intervals and endpoints.
MC1, MC2, MC4, Ma6 - Polynomial graphs, roots and interceptsRead the shape of a polynomial from its factors and leading term. You will distinguish a sketch from a plot, explain crossings and touches, and connect graph intersections with equation and simultaneous-equation solutions.
MC6, Mg8 - Reciprocal graphs, asymptotes and intersectionsRecognise reciprocal and inverse-square graphs from their signs, symmetry and proportional relationships. You will explain their asymptotes, find exact intersections and read a short mathematical definition without losing its domain restrictions.
Ma14 - Proportional relationships and modelsA proportional relationship might be a line, a reciprocal curve or a root curve. Learn to identify what stays constant, build the equation and use both observations and graph labels to judge a model.
Mf3 - Functions, mappings, domain and rangeA function is a rule with a specified set of permitted inputs, and its range is exactly the set of outputs it reaches. Use mapping language, formula restrictions and graphs to decide what changes when the input set changes.
Mf4, Mf5 - Composite and inverse functionsA composite must use its functions in the right order and respect both domains. Learn to determine composite rules and domains, decide when a function can be reversed, and find the inverse's formula, graph, domain and range.
MC7, MC8 - Transforming and combining graphsAn equation is one way to describe a curve; a supplied sketch or a newly defined profile works too. MC7 develops single transformations, including graphs with no algebraic definition. MC8 combines and recognises the movements, so you can form an equation, sketch its features and interpret an unfa...
Mf8 - Building, comparing and refining function modelsCan you trust a function simply because it matches the measurements? Mf8 uses functions to model a practical situation, with particular care over limitations and refinements. You will build and compare predictions, interpret their assumptions, and carry a decision through a complete problem-solvi...

Coordinate geometry

Mg1–Mg6 - Straight lines, distances and gradientsUse coordinates to determine straight lines, lengths and meeting points. You will choose an equation that suits the information, handle horizontal and vertical exceptions, and read a short geometric argument to reconstruct a line.
Mg7 - Straight line models and rates of changeConstruct and question straight-line models in practical settings. You will explain the assumptions behind constant rates, interpret parameters with units, change the input origin and decide what new observations say about a model.
Mg10 - Circle equations, centres and radiiConnect the fixed-distance meaning of a circle to its algebraic equation. You will complete squares to determine centres and radii, construct circles from points or diameters, and draw their important features accurately.
Mg9, Mg11 - Circle intersections, chords and tangentsUse the circle properties in Mg11 to connect geometric reasons with coordinate calculations. Then apply Mg9 to find line–circle intersections and recognise tangency through a repeated root, with exact working where a question requires it.
Mg12–Mg14 - Parametric curves and cartesian equationsUse Mg12 to understand a shared parameter, Mg13 to convert between equation forms with the correct restrictions, and Mg14 to describe circles by an angle. You will also read an unfamiliar curve definition and reconstruct its Cartesian relation without losing an endpoint condition.
Mg16 - Parametric models and restricted parametersConstruct and interpret parametric models, including the kinematics and projectile contexts in Mg16. You will choose physical domains, compare observations with predictions, and explain why a curve on its own does not specify the timing of a journey.

Sequences and series

Ms1, Ms2 - Integer binomial expansion and combinationsA long product can hide a very short coefficient calculation. You will use factorials and combinations to expand positive integer powers, find selected terms and explain the connection with binomial probabilities.
Ms3–Ms5 - Rational binomial expansions and approximationTurn a rational power into a polynomial that is easy to evaluate near a chosen input. You will expand fractional and negative powers, normalise a bracket, combine valid expansions and judge what a truncated approximation actually tells you.
Ms6–Ms11 - Sequences, recurrence and sigma notationRead a sequence rule carefully, generate its terms and describe how it behaves. You will distinguish terms from partial sums, interpret unfamiliar sigma notation and investigate limits without treating a few rounded values as a proof.
Ms12, Ms13 - Arithmetic sequences and sumsUse equal differences to find terms, totals and missing information. You will explain the arithmetic sum formula and read an unfamiliar endpoint estimate, deciding when a correction makes it exact.
Ms14–Ms16 - Geometric series and sums to infinityUse a fixed multiplier to find geometric terms and finite sums. You will justify the convergence condition for an infinite sum and distinguish a term limit from a partial-sum limit, including the awkward boundary cases.
Ms17 - Modelling growth, decay and finance with sequencesTurn repeated changes into models with a clear time index. You will compare simple and compound interest, find integer-time thresholds, and build savings and loan models whose predictions respect the timing of transactions.

Trigonometry

Mt1, Mt2, Mt8 - The unit circle, exact values and trigonometric graphsUse a rotating radius to define trigonometric functions for any angle, obtain exact values with the correct signs, and connect these values to the shape and transformations of their graphs.
Mt3, Mt4 - Triangle area, sine rule and cosine ruleChoose and use trigonometric rules for triangles that need not contain a right angle, derive their area from perpendicular height, and translate bearings into the angles a calculation needs.
Mt10, Mt11 - Radians, arc length and sector areaDefine angles through arc length, convert between radians and degrees, and derive circle formulae that distinguish curved boundaries, sectors and segments.
Mt12 - Small angle approximationsUnderstand the standard small angle approximations in Mt12 and use them with their radian conditions intact. You will simplify expressions, explain which powers matter and judge the error in a geometric measurement model.
Mt9, Mt13–Mt15 - Reciprocal and inverse trigonometric functionsUse the inverse functions in Mt9 and the reciprocal definitions in Mt13 without confusing their notation. Mt14 connects their graphs, domains and ranges to sine, cosine and tangent; Mt15 supplies two squared identities for exact values and algebra.
Mt5, Mt6 - Fundamental trigonometric identitiesUse the tangent ratio in Mt5 and the Pythagorean identity in Mt6 to connect exact values, prove equalities and solve equations. The emphasis is on choosing a useful identity while keeping signs, denominators and all permitted solutions under control.
Mt16, Mt17 - Compound and double angle formulaeUse a supplied geometric starting point to justify angle identities, derive double-angle results, and read a short argument that turns a multiple angle into an algebraic ratio.
Mt18 - Combining sine and cosine into harmonic formChoose equivalent sine and cosine forms by expanding them, interpret a combined wave on full and restricted domains, and solve a shifted equation with a non-acute phase.
Mt7, Mt19 - Trigonometric equations on given intervalsUse inverse principal values to find every solution on a stated interval, transform complete arguments, and justify which roots survive identity-based rearrangements.
Mt20 - Proving trigonometric identitiesBuild a general argument that explains why two trigonometric expressions agree. Learn to choose a productive starting side, handle fractions and shifted angles, and read a proposed proof critically. The algebra must preserve the domain as well as the answer.
Mt21 - Trigonometric models in contextA trigonometric function can describe a repeating measurement even when its input is time rather than a geometrical angle. Construct and interpret these models, find all threshold times, and use signed components in motion and force problems. Judge a prediction against the observations and assump...

Exponentials and logarithms

ME1, ME8, ME9 - Exponential functions and the number eConnect exponential graphs with repeated multiplication and interpret their intercepts, ranges and asymptotes. Use the special gradient of e^(kx) to explain when an exponential is a sensible model for change.
ME2–ME5, ME10 - Logarithms, natural logs and their lawsRead a logarithm as an exponent and sketch it as the inverse of an exponential. Derive and apply the logarithm laws, including fractional and negative powers, while keeping every argument inside its real domain.
ME6 - Solving exponential and logarithmic equationsSolve exponential equations by choosing logarithms, equal bases or a positive substitution. Check logarithm domains, interpret numerical intersections, and solve related inequalities using the direction of the exponential or logarithm.
ME7 - Linearising power and exponential modelsChoose coordinates that make a power or exponential model look straight, then recover the model from the line. Use fitted experimental graphs, read logarithmic scales and follow a short argument in newly defined coordinates, keeping the meaning of every parameter in view.
ME11 - Exponential growth and decay modelsUse exponential models to describe proportional growth and decay, estimate rates and solve threshold problems. Interpret half-lives, compounding and long-term values, then compare predictions with evidence and explain sensible refinements.

Differentiation

Mc1–Mc4 - Derivatives as gradients and rates of changeTurn gradients between pairs of points into the gradient at one point. Mc1–Mc3 connect that tangent direction to the derivative, its notation and rates with units; Mc4 develops the gradient graph from the shape of a curve, including cases where flattening does not reverse its direction.
Mc5 - First principles and the power ruleUse a limiting difference quotient to explain derivatives of small positive integer powers, then differentiate rational powers much faster with the power rule. Mc5 also covers constant multiples, sums and differences, with roots and reciprocals rewritten carefully and their real domains preserved.
Mc9 - Tangents and normals to curvesConstruct tangent and normal equations from a curve point and its derivative, as required by Mc9. You will deal with horizontal and vertical directions, find contacts from a specified gradient, and read a short mathematical passage which develops a new relation from familiar normals.
Mc6–Mc8, MC5 - Second derivatives, stationary points and monotonicityUse the second derivative to track changing gradient (Mc6), locate and classify stationary points (Mc7), and find increasing or decreasing intervals (Mc8). You will combine these decisions in curve sketches (MC5), including cases where a horizontal tangent does not mean a turn.
Mc17, Mc18, MC9 - Concavity and points of inflectionUse increasing and decreasing gradients to describe concavity (Mc17), and establish stationary or non-stationary points of inflection (Mc18). You will show how a stationary inflection differs from a maximum or minimum when sketching a curve (MC9).
Mc10 - Differentiating exponential and logarithmic functionsDifferentiate e^(kx), a^(kx) and x, together with related sums, differences and constant multiples, as required by Mc10. You will keep the input-rate factors and logarithm domains, and interpret a derivative as an instantaneous rate rather than an average over an interval.
Mc11 - Differentiating trigonometric functionsA wave can pass through zero while changing as quickly as it ever does. Use first principles to explain the derivatives of sine and cosine, then differentiate linear combinations of sine, cosine and tangent as Mc11 requires. Keep the input in radians and track each function's domain.
Mc12, Mc13 - Product and quotient rulesA derivative must account for every changing factor. Learn the product rule in Mc12 and the quotient rule in Mc13, then choose between rules and simpler algebra. Equivalent expressions share derivatives only where they describe the same function.
Mc14, Mc15 - The chain rule, inverse derivatives and connected ratesA rate may pass through several dependent quantities before reaching the one you want. Mc14 supplies the chain rule for composite functions; Mc15 uses it for inverse derivatives and connected rates. Learn to read the variable, sign and unit of each derivative before calculating.
Mc16, Mg15 - Implicit and parametric differentiationA relation can reveal a gradient without being rearranged into y=f(x). Mc16 develops implicit differentiation; Mg15 finds gradients from two parameter rates. Use first derivatives to match points, construct tangents and normals, and distinguish a genuine tangent from a failed derivative ratio.

Integration

Mc19–Mc21 - Integration, antiderivatives and constantsRecover a function from its derivative, integrate powers and select a particular curve using given information. Explain how the fundamental theorem connects reverse differentiation with accumulated signed area.
Mc22, Mc23, Mc25 - Definite integrals, signed area and limits of sumsEvaluate definite integrals, separate signed accumulation from geometric area and interpret an integral as a limit of rectangles. Read a finite-sum argument and explain its bounds, error and exact limit.
Mc24 - Integrating exponential, logarithmic and trigonometric formsIntegrate exponential, reciprocal, sine and cosine functions and their sums. Reverse input multipliers accurately, handle logarithms on real intervals and use identities to turn trigonometric powers into standard forms.
Mc26 - Areas between curves and parametric areasDescribe a region with vertical or horizontal strips, then calculate its exact area. Use intersections and signs to choose the bounds, and interpret a supplied parametric strip conversion geometrically.
Mc27, Mc28 - Integration by substitutionRecognise reverse-chain integrals and choose one useful change of variable in less obvious cases. Rewrite every factor, preserve domains and convert definite bounds consistently.
Mc29 - Integration by partsChoose factors that turn a difficult product into an easier integral. Derive the parts rule, repeat it for polynomial products, integrate logarithms and justify exact definite values.
Mc30 - Integrating rational functions with partial fractionsUse algebraic decomposition to turn rational integrals into logarithms and powers. Keep repeated factors complete, handle non-unit linear scales and evaluate exact integrals only on valid domain intervals.

Numerical methods

Me1, Me2 - Root intervals, accuracy and failed sign testsCan a table of function values miss two roots between consecutive integers? You will locate roots and choose useful starting intervals under Me1, then explain touching roots, discontinuities and missed crossings under Me2. You will also distinguish a plausible decimal from a justified approximation.
Me3 - Fixed point iteration and cobweb diagramsCan two equivalent rearrangements send a calculator in opposite directions? Under Me3, you will build a fixed point iteration, use the ANS key or a spreadsheet accurately, and connect the resulting sequence to a staircase or cobweb diagram. A final check will distinguish a numerical suggestion fr...
Me4, Me5 - Newton–Raphson and convergence failuresCan a tangent give an excellent approximation and still lead a calculation astray? Under Me4, you will use Newton–Raphson and represent its steps on a graph. Under Me5, you will explain Newton and fixed point failures through the actual paths, starting values and domains.
Mc34, Mc35 - Trapezium rule and area boundsCould a zero estimate hide a positive area? Under Mc34, you will construct and interpret trapezium estimates and justify their error direction on an interval of consistent concavity. Under Mc35, you will use rectangle sums for bounds, then read an unfamiliar expression for the same trapezium method.
Me6 - Numerical solutions in contextWhen will a generator use its planned fuel allowance, and what can a numerical answer promise? Under Me6, you will solve a contextual root problem, estimate an accumulated quantity from a rate, and evaluate the numerical result against the model's domain and data.

Vectors

Mv1–Mv3 - Vectors, components, magnitude and directionUse the language and geometry of vectors to describe movements, combine arrows and calculate lengths and directions. You will move between column vectors, unit-vector notation and magnitude-direction form, with signs that agree with your sketch.
Mv4–Mv6 - Position vectors, distances and vector problemsConnect Cartesian coordinates with position vectors, calculate distances and locate points in a given ratio. Build geometric arguments from general vectors and use the same addition rule to find and balance force resultants.
Mv7 - Vectors in three dimensionsUse three-dimensional vector language, operations and position calculations, including spatial directions described by perpendicular projections. Extend distance, ratio, geometric-proof and force-resultant methods by checking all three components.

Statistics

Data presentation and interpretation

MD1–MD5 - Univariate diagrams, histograms and distributionsChoose and read single-variable displays in MD1, use histogram area in MD2 and cumulative frequency in MD3, and describe distribution shape in MD4. MD5 explains what larger unbiased samples can improve. The current health large data set gives us real observations to investigate.
MD6–MD8 - Scatter diagrams, regression and correlationUse supplied straight and curved best-fit models in MD6, recognise distinct groups and outliers in MD7, and describe correlation without assuming causation in MD8. You will work with the current health large data set and see why pooling different groups can change a scatter diagram's message.
MD10, MD11 - Averages, percentiles and spreadChoose and interpret means, medians, modes, midranges and weighted means in MD10, then use percentiles, quartiles, range and interquartile range in MD11. Real pulse readings from the current health large data set show how a summary can change while the middle observation stays put.
MD12 - Variance, standard deviation and grouped estimatesUse MEI's sample variance and sample standard deviation to describe spread from raw data, frequency tables and grouped estimates. You will interpret calculator outputs, keep the sample divisor clear and explain what grouping conceals.
MD9, MD13, MD14 - Outliers, missing values and data cleaningApply MEI's inclusive outlier rules and make cleaning decisions that respect missing values, genuine extremes and paired observations. You will use current LDS9 measurements and choose displays whose scales and time order support a fair conclusion.

Probability

Mu1–Mu5 - Events, independence and probability diagramsUse the event rules in Mu1–Mu4 to distinguish mutual exclusivity from independence and calculate combined probabilities. You will organise outcomes in grids and trees, then use the overlapping Venn regions of Mu5, including diagrams with three events.
Mu6, Mu7 - Conditional probability and two-way dataCalculate conditional probabilities from the tables, trees, Venn regions and sample-space information in Mu6. Then use Mu7 to decide whether a probability stays unchanged after conditioning, giving a numerical argument for independence.
MR13 - Probability modelling and assumptionsBuild models for clearly defined populations, compare dependence and group assumptions, and judge when binomial or Normal distributions are unsuitable. MR13 asks you to connect the numerical prediction with the situation it is meant to describe, including the health records in the large data set.

Statistical distributions

MR6, MR7 - Finite discrete probability distributionsUse tables and algebraic probability functions to describe finite discrete random variables. MR6 connects the variable, its values and its probability rule; MR7 uses that rule to calculate events, including uniform selections and independent observations.
MR1–MR5 - Binomial models and probabilitiesRecognise a binomial situation and choose its success probability (MR1–MR2), then calculate exact and cumulative probabilities (MR3). Use the mean count and expected event frequencies to connect the model with observed batches (MR4–MR5), keeping those two kinds of average distinct.
MR9–MR12 - Normal models, standardisation and inverse probabilitiesConnect a Normal probability curve with sample histograms (MR9) and locate its symmetry and inflections (MR11). Standardise and transform a Normal variable (MR10), then calculate probabilities and inverse boundaries, including unknown parameters (MR12).
MR8 - Choosing distributions and the normal approximationA count can have a positive probability at one integer, while a continuous Normal variable has probability zero at every single value. How can the second model approximate the first? In MR8 you will judge Normal models by their shape and use continuity correction to turn integer events into areas.

Statistical hypothesis testing

MH1–MH3 - Hypotheses, significance and statistical decisionsCan a sample distinguish a real improvement from ordinary variation? Use a binomial model to set population hypotheses, choose one or two tails, and interpret a significance test as evidence with a controlled risk of a wrong decision.
MH4–MH6 - Testing a binomial proportionUse exact binomial probabilities to investigate a population probability. Set up the hypotheses, find critical and acceptance regions, and express what the evidence suggests without treating a statistical decision as certainty.
MH7–MH9 - The sample mean and normal mean testsHow can an average reveal a change that is hard to see in individual measurements? Use the Normal distribution of a sample mean, then carry out mean tests with a known population variance or, for a large sample, an estimated variance.
MH10, MH11 - Correlation coefficients and significance testsTwo coefficients can describe the same pairs differently because they measure different features. MH10 develops linear and rank correlation; MH11 uses supplied critical values and explained software probabilities to make cautious population inferences about the relationship actually being investi...

Mechanics

Kinematics

Mk1–Mk4 - Motion language and kinematics graphsDistinguish motion quantities and connect position, displacement, distance, velocity, speed and acceleration graphs. Use signed quantities to describe journeys and find one object's velocity as observed from another on the same line.
Mk7, Mk8 - Constant acceleration formulaeRecognise a constant-acceleration interval, derive its five scalar relationships and choose a useful equation. Solve staged straight-line and vertical-motion problems while checking directions, time roots and the limits of the model.
Mk5, Mk6 - Variable acceleration and calculus in motionConnect position, velocity and acceleration by differentiating and integrating with respect to time. Recover motion from conditions, distinguish signed change from total travel, and justify predictions from a variable-acceleration model.
Mk9–Mk12 - Motion in two dimensions with vectorsDescribe motion in a plane by connecting position, velocity and acceleration component by component. Distinguish distance travelled from displacement, find a Cartesian path with its physical domain, and compare particles' positions at a common instant.
My1–My5 - Projectile motion and trajectoriesForm a projectile's vector equations from its launch and gravity, then use them to find its flight, launch velocity and Cartesian trajectory. Keep the landing surface explicit, including a specified slope, and decide when the idealised model remains useful.

Forces and Newton's laws

Mn1–Mn3, Mn6 - Newton's laws and equations of motionConnect Newton's three laws with equations of motion for a particle. You will distinguish a force from a resultant, model motion under gravity, and use one equation for a train whose components have no relative motion.
MF4, MF6, Mn7 - Resolving forces and dynamics in a planeFind resultants of concurrent forces and choose useful directions for resolution. Then form equations of motion in a line or a plane, including gravity, and check the constraints on a particle moving along a surface.
MF5, MF7–MF9 - Particle equilibrium and polygons of forcesSolve coplanar equilibrium problems by components or by a closed force polygon. You will connect zero vector sum with concurrent forces, choose suitable axes for strings and inclines, and check that the resulting tensions and reactions are physically possible.
Mn4, Mn5 - Connected particles, strings and pulleysChoosing a system is part of the mathematics. You will model connected particles, form equations for individual bodies, and use the connection to link their motions. Comparing whole-system and separate-body answers gives a useful check on tension, acceleration and changes of stage.
Mn7, Mk12 - Variable forces and vector dynamicsA vector equation of motion describes how a resultant changes a particle's velocity. You will integrate time-dependent forces, use conditions at specified times, find relative positions at a common instant, and recover missing forces from prescribed motion.
MF10, MF11 - Contact forces, friction and limiting equilibriumContact with a rough surface supplies a normal push and friction along the surface. You will interpret their combined vector, check whether a proposed contact force is physically possible, and use different friction rules at rest, at the limit and during sliding.
MF12 - Static and limiting equilibrium on rough surfacesUse Newton's laws to decide whether rough contact can keep a particle at rest. Resolve applied forces, distinguish actual friction from its limiting value, and find both boundaries of an equilibrium force range.
MF12 - Motion on rough surfacesApply Newton's laws to sliding on rough surfaces, including a force that changes normal contact. Separate uphill motion, first rest and any reversal, then extend the force model to connected particles.