OCR A-Level Mathematics A

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Pure Mathematics

Proof and mathematical language

1.01a, 1.01b - Proof, logic and mathematical argumentsA proof turns a convincing pattern into a conclusion you can trust for every allowed case. In 1.01a you will build deductions and complete case arguments; in 1.01b you will use implication, equivalence and congruence precisely. Along the way, the language of expressions, equations and functions w...
1.01c - Conjectures and counterexamplesA single well-chosen example can defeat a claim that survives hundreds of successful trials. Learn to read the domain, keep a conditional claim's premise true, and demonstrate exactly where its conclusion fails. You will also distinguish disproving a claim from proving a repaired version of it.
1.01d - Proof by contradictionSome claims become easier when you temporarily assume their opposite. Learn to turn that assumption into a precise conflict, including the proofs that √(2) is irrational and that there are infinitely many primes. You will then use familiar rational-number properties to build fresh contrad...

Algebra, equations and inequalities

1.02a - Rational indices and index lawsTurn powers, roots and reciprocals into a single language. You will use rational indices accurately, explain their restrictions and simplify expressions that combine several laws, so a forbidding calculation can become a short exact argument.
1.02b - Surds and rationalising denominatorsKeep roots exact while you simplify, expand and divide expressions. You will connect surd and index notation, use conjugates to clear denominators and preserve the restrictions that make each step valid.
1.02d, 1.02e - Quadratic graphs and the discriminantUse 1.02e's completing-the-square method to uncover a quadratic's turning point and symmetry. Then connect those features to 1.02d's graphs and discriminant, so you can classify real roots and determine conditions on an unknown coefficient.
1.02f - Solving quadratic equations and disguised quadraticsA complicated-looking equation may hide a familiar quadratic. Learn to choose a solving method, explain why it works and separate genuine solutions from candidates introduced by your algebra.
1.02c - Simultaneous equations and intersectionsTwo equations can describe the same pair of unknowns, but only their common solutions satisfy both. Learn elimination and substitution, handle brackets and fractions, and explain why a line and a quadratic curve can meet twice, once or not at all.
1.02g–1.02i - Inequalities, sets and graphical regionsSolve linear and quadratic inequalities in 1.02g, express complete solution sets in 1.02h, and identify graphical regions and boundaries in 1.02i. Learn why roots give boundaries rather than the whole answer, and why an allowed curve can still have excluded endpoints.
1.02j - Polynomials, division and the factor theoremTurn an awkward cubic into a product you can understand and solve. You will expand and factorise polynomials, divide by a linear expression and use the factor theorem to find roots and unknown coefficients.
1.02k - Rational expressions and algebraic divisionSimplify algebraic fractions while keeping track of their allowed inputs. Combine and divide rational expressions, separate a quotient from its remainder and use factorisation to solve polynomial equations through degree four.
1.02y - Partial fractions with linear factorsSplit a complicated fraction into simple pieces, including repeated linear factors, and check the result by recombining. Then use the split with supplied expansion and integration rules to make longer problems manageable.

Functions and graphs

1.02l, 1.02s, 1.02t - Modulus graphs, equations and inequalitiesTurn modulus notation into distances, straight-line branches and clear solution sets. You will sketch a folded line, solve equations at its intersections and decide which intervals satisfy an inequality.
1.02m, 1.02n, 1.02p - Polynomial graphs, roots and interceptsConnect a polynomial's factors to its intercepts, crossings and end behaviour. You will turn those clues into a sketch, use stationary-point information when needed and interpret algebraic solutions as graph intersections.
1.02o, 1.02q - Reciprocal graphs, asymptotes and intersectionsUse signs, excluded inputs and asymptotes to sketch reciprocal and inverse-square curves. You will solve their intersections exactly, then use supplied exponential and trigonometric graphs when algebra does not give a convenient answer.
1.02r - Proportional relationships and modelsWhen does doubling an input double the output, and when does it halve it? You will turn proportional statements into equations, interpret their graphs and decide whether a calibrated model deserves to be trusted.
1.02u - Functions, mappings, domain and rangeCan you remove some permitted inputs without losing any possible outputs? Learn what makes a mapping a function, distinguish its domain from its range, and use formulas and graphs to track the effect of a restriction.
1.02v - Composite and inverse functionsPutting two functions together is sensitive to order; reversing one needs a carefully chosen domain. Learn to compose rules, check both stages and find inverses algebraically and by reflection in a graph.
1.02w, 1.02x - Transforming and combining graphsMove a graph by changing its inputs and outputs, even when you have no formula for the curve. You will describe single transformations in 1.02w, then combine them, form equations and recover important sketch features in 1.02x.
1.02z - Building, comparing and refining function modelsTurn a practical decision into a function, then test what its predictions mean. In 1.02z you will choose variables, assumptions and a useful domain, compare models with fresh evidence, and make specific refinements when a model stops describing the situation.

Coordinate geometry

1.03a, 1.03b - Straight lines, distances and gradientsTurn a line's direction and position into an equation you can use. You will connect gradients, distances and midpoints, construct parallel and perpendicular lines, and find exactly where lines meet.
1.03c - Straight line models and rates of changeBuild straight-line models from constant-change assumptions and observations. You will interpret rates and intercepts with their units, choose sensible domains and time origins, and test predictions against further evidence.
1.03d, 1.03e - Circle equations, centres and radiiRead a circle equation as a statement about distance. You will construct equations from geometric information, uncover centres and radii by completing squares, and draw circles with the correct position, size and scale.
1.03f - Circle intersections, chords and tangentsTurn circle geometry into exact coordinate arguments. You will use diameters, chords and radii to find right angles and tangents, then decide exactly where a line or another circle meets a circle.
1.03g - Parametric curves and cartesian equationsDescribe a curve by giving both coordinates in terms of one shared variable. You will convert between parametric and Cartesian equations, retain the correct restrictions, and sketch the path with its direction of traversal.
1.03h - Parametric models and restricted parametersUse a shared parameter to connect changing quantities in geometry and motion. You will build equations, translate physical restrictions, and compare predictions with observations while keeping units and timing clear.

Sequences and series

1.04a, 1.04b - Integer binomial expansion and combinationsFind a chosen coefficient without multiplying out every bracket. You will connect factorials and Pascal's triangle to integer binomial expansions, then see why the same numbers count success patterns in probability.
1.04c, 1.04d - Rational binomial expansions and approximationA short polynomial can approximate a square root or reciprocal surprisingly well near a suitable input. You will build rational binomial expansions, factor the constant correctly, state where the expansion is valid and distinguish that condition from numerical accuracy.
1.04e–1.04g - Sequences, recurrence and sigma notationGenerate sequence terms from an explicit rule or a recurrence, describe their behaviour and add them using sigma notation. You will distinguish a convincing general reason from a pattern suggested by a few displayed terms.
1.04h - Arithmetic sequences and sumsFind terms and totals when each step adds the same amount. You will derive the arithmetic sum formula, recover missing information and decide whether a proposed term or total is actually possible.
1.04i, 1.04j - Geometric series and sums to infinityFind geometric terms and finite totals by tracking a constant multiplier. You will derive the sum formula, decide when an infinite sum exists and separate the behaviour of terms from the behaviour of their totals.
1.04k - Modelling growth, decay and finance with sequencesChoose a discrete model for equal changes or percentage changes, and use it for interest, savings and repayments. You will interpret integer-time thresholds, payment timing and the limits of a financial prediction.

Trigonometry

1.05a, 1.05f, 1.05g - 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.
1.05b, 1.05c - Triangle area, sine rule and cosine ruleChoose and use trigonometric rules for triangles that need not contain a right angle, check when the same data allow two triangles, and translate bearings into the angles a calculation needs.
1.05d - 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.
1.05e - Small angle approximationsReplace trigonometric functions by simple expressions near zero, and understand why the angle must be in radians. You will keep the right powers, handle cancellation and decide whether an approximation is accurate enough for a geometric model.
1.05h, 1.05i, 1.05k - Reciprocal and inverse trigonometric functionsDistinguish a reciprocal value from an inverse function. The definitions in 1.05h lead to the graphs, domains and principal branches in 1.05i; the identities in 1.05k then connect exact reciprocal values without losing their signs.
1.05j - Fundamental trigonometric identitiesDerive the two fundamental identities in 1.05j from unit-circle coordinates. You will recover exact values with the correct signs, prove equalities on their allowed domains and solve equations without dividing away valid solutions.
1.05l, 1.05m - Compound and double angle formulaeBuild angle-addition formulae from a geometric picture, derive double-angle identities, and choose the right form for exact values, simplification and bounded equations.
1.05n - Combining sine and cosine into harmonic formCombine sine and cosine into one shifted wave, select a phase with the correct signs, sketch its features and solve equations over a stated interval.
1.05o - Trigonometric equations on given intervalsFind every permitted branch in a degree or radian interval, solve equations with transformed arguments and quadratic functions, and use identities without losing or adding roots.
1.05p - Proving trigonometric identitiesA trigonometric identity can hide a very simple expression inside a complicated fraction. Learn to uncover it with a general proof, choose identities that fit the target, and keep track of the inputs your algebra permits. You will also repair arguments that reach a plausible answer by an invalid...
1.05q - Trigonometric models in contextUse sine and cosine to describe repeating changes, predict when a condition is met, and resolve directed quantities into components. Turn observations into parameters, then interpret the answers in hours, metres or newtons. A model's neat arithmetic is useful only while its assumptions remain sen...

Exponentials and logarithms

1.06a, 1.06b - Exponential functions and the number eRead exponential graphs as repeated multiplication, distinguish growth from decay, and explain the special role of the number e. Use its gradient property to connect a graph's height with its rate of change.
1.06c–1.06f - Logarithms, natural logs and their lawsInterpret logarithms as exponents, connect the natural logarithm with its inverse exponential, and derive the laws for combining logarithms. Learn why a shorter expression must keep the domain of the expression it came from.
1.06g - Solving exponential and logarithmic equationsChoose equal bases, logarithms or a substitution to solve exponential equations. Preserve logarithm domains when checking candidates, and recognise when a graph intersection gives a useful numerical answer instead of a simple algebraic one.
1.06h - Linearising power and exponential modelsTurn curved power and exponential relationships into straight lines by choosing suitable logarithmic coordinates. Recover the model's parameters, use experimental data and fitted graphs sensibly, and read logarithmic scales without confusing addition with multiplication.
1.06i - Exponential growth and decay modelsBuild growth and decay models from equal proportional changes, fit their rates and find meaningful threshold times. Use half-lives, continuous compounding and nonzero baselines, then check predictions against observations and refine the assumptions when needed.

Differentiation

1.07a–1.07c - Derivatives as gradients and rates of changeFind a curve's gradient at one point by bringing two points together. You will connect the tangent and derivative in 1.07a–1.07b to instantaneous rates and their units, then sketch gradient functions in 1.07c without confusing height with slope.
1.07g, 1.07i - First principles and the power ruleDerive polynomial gradients from the definition before using the faster power rule. In 1.07g you prove the small positive integer cases from first principles; in 1.07i you differentiate rational powers, constant multiples, sums and differences, keeping real domains in view.
1.07m - Tangents and normals to curvesUse a derivative and a curve point to construct tangent and normal equations in 1.07m. You will handle horizontal and vertical exceptions, then work backwards from a specified direction to all possible points of contact.
1.07d, 1.07e, 1.07n, 1.07o - Second derivatives, stationary points and monotonicityA horizontal tangent can mark a peak, a trough or neither. You will find second derivatives (1.07d), interpret changes of gradient (1.07e), classify stationary points (1.07n) and use derivative signs to describe increasing and decreasing intervals (1.07o).
1.07f, 1.07p - Concavity and points of inflectionA curve can flatten without turning, and it can change its bending without flattening. You will use second-derivative signs to describe convex and concave sections (1.07f), then locate and classify stationary and non-stationary points of inflection (1.07p).
1.07j, 1.07l - Differentiating exponential and logarithmic functionsFind gradients of natural and other positive-base exponentials with linear exponents (1.07j), and of the natural logarithm on its real domain (1.07l). You will combine these rules in sums and distinguish an instantaneous rate from a change over a whole time interval.
1.07h, 1.07k - Differentiating trigonometric functionsWhy does a sine wave have its steepest slope when its height is zero? Derive the sine and cosine derivatives from first principles for 1.07h, then use them with scaled inputs and tangent for 1.07k. Radians, signs and excluded inputs make the rules work.
1.07q - Product and quotient rulesTwo changing factors need two contributions to their derivative. Learn the product and quotient rules in 1.07q, and decide when algebra gives a shorter route. You will keep domain restrictions even when a fraction simplifies to something familiar.
1.07r - The chain rule, inverse derivatives and connected ratesOne changing quantity can drive another through several stages. Use 1.07r to differentiate composite functions, reverse a derivative on a valid inverse branch and connect instantaneous rates. The variable, sign and unit of each rate matter as much as the arithmetic.
1.07s - Implicit and parametric differentiationCan you find a tangent without first writing the curve as y=f(x)? Use 1.07s to differentiate relations between coordinates and equations in a shared parameter. Find first derivatives, tangents, normals and special tangent points while keeping the correct branch and point.

Integration

1.08a, 1.08b - Integration, antiderivatives and constantsRecover a function from its derivative, integrate powers and use a point to choose one curve from a family. Connect this reverse process to accumulated area and the fundamental theorem of calculus.
1.08d, 1.08e, 1.08g - Definite integrals, signed area and limits of sumsEvaluate definite integrals, distinguish signed accumulation from geometric area and split at changes of sign. Explain why narrow rectangle sums approach an exact integral.
1.08c - Integrating exponential, logarithmic and trigonometric formsIntegrate exponential, reciprocal, sine and cosine forms, controlling signs, input multipliers and logarithmic domains. Rewrite trigonometric powers with identities and use a supplied inverse-function integral correctly.
1.08f - Areas between curves and parametric areasFind the area of a region by choosing its boundaries and the direction of its strips. Locate intersections, split crossing curves and use horizontal or parametric descriptions without losing the geometry.
1.08h - Integration by substitutionTurn a complicated-looking integral into a familiar one by choosing a new variable. Recognise reverse-chain forms, rewrite every factor, transform limits and keep logarithm and square-root domains intact.
1.08i - Integration by partsReverse the product rule to integrate products that substitution does not simplify. Choose factors purposefully, repeat the method when needed, handle logarithms and keep both endpoints in definite integrals.
1.08j - Integrating rational functions with partial fractionsBreak a rational integrand into simple linear-factor fractions, then integrate the logarithmic and power terms. Handle repeated factors, linear scaling and valid intervals without losing excluded inputs.

Numerical methods

1.09a, 1.09b - Root intervals, accuracy and failed sign testsHow can you trust a root that you cannot find exactly? You will locate roots and justify their accuracy under 1.09a, then investigate touching roots, asymptotes and hidden crossings under 1.09b.
1.09c - Fixed point iteration and cobweb diagramsCan you turn an awkward cubic into a calculation that improves itself? Under 1.09c, you will rearrange an equation, compute successive approximations and draw the staircase or cobweb that explains their movement. You will also check what the resulting decimal actually establishes.
1.09d, 1.09e - Newton–Raphson and convergence failuresCan a tangent find a root faster than repeated rearrangement? Under 1.09d, you will derive and use Newton–Raphson; under 1.09e, you will diagnose failed steps and apply the local slope condition for fixed point convergence. You will learn why a convincing decimal display still needs checking.
1.09f - Trapezium rule and area boundsCan straight-edged shapes give a trustworthy answer for a curved area? Under 1.09f, you will build and refine a trapezium estimate, use curvature to decide its error direction, and construct rectangle bounds that state what the integral must lie between.
1.09g - Numerical solutions in contextWhen will a pump deliver its target volume, and how accurately can a calculation answer that question? Under 1.09g, you will turn a physical target into a numerical root, estimate totals from rates, and separate numerical precision from the assumptions and measurements behind a model.

Vectors

1.10a, 1.10c, 1.10d - Vectors, components, magnitude and directionRead vectors as directed movements, combine their components and convert between component form and magnitude with direction. You will distinguish a signed component from a length and use a sketch to keep an inverse-tangent answer in the right quadrant.
1.10e–1.10g - Position vectors, distances and vector problemsTurn positions into displacements, distances and points along a segment. Use scalar multiples to prove geometric relationships, then interpret a vector sum as the resultant of forces on one particle.
1.10b - Vectors in three dimensionsExtend vector operations, position calculations and geometric arguments to three perpendicular components. Calculate spatial lengths with two applications of Pythagoras, locate points and check every component when proving collinearity or balancing forces.

Statistics

Data presentation and interpretation

2.02a, 2.02b - Univariate diagrams, histograms and distributionsRead tables and single-variable diagrams in 2.02a, then use histogram area and evaluate display choices in 2.02b. You will learn why a taller bar can represent fewer people, and how to recover the information a graph actually contains.
2.02c–2.02e - Scatter diagrams, regression and correlationInterpret paired data and supplied regression lines in 2.02c, describe correlation informally in 2.02d, and distinguish association from cause in 2.02e. You will see how a convincing overall trend can hide the opposite pattern within each group.
2.02f - Averages, percentiles and spreadUse 2.02f to calculate and interpret averages, percentiles, quartiles and measures of variation. You will choose summaries for the question being asked, and compare distributions without pretending that one number describes every observation.
2.02g - Variance, standard deviation and grouped estimatesCalculate mean and standard deviation from individual values, summary statistics and frequency tables. You will also estimate them from grouped data, while keeping the calculator's divisor, the units and the lost information clear.
2.02h–2.02j - Outliers, missing values and data cleaningIdentify possible outliers with H240's precise boundary rules, clean missing or erroneous observations without changing their meaning, and choose displays that answer the statistical question. You will use the census large data set to distinguish a large count from a large proportion.

Probability

2.03a, 2.03b - Events, independence and probability diagramsDescribe events precisely and decide when probabilities may be added or multiplied. You will connect the event notation of 2.03a with the sample spaces, trees and Venn diagrams of 2.03b, so that a picture supports each calculation.
2.03c, 2.03d - Conditional probability and two-way dataUse a given event to choose the right denominator, then connect two-way tables, trees and Venn regions. You will calculate conditional probabilities from first principles in 2.03d and use the diagram and probability rules of 2.03c to distinguish a branch, a complete path and independence.
2.03e - Probability modelling and assumptionsBuild probability models for clearly defined selections, compare their predictions and explain what their assumptions leave out. You will use real census data, conditional paths and weighted groups to practise the modelling and criticism required by 2.03e.

Statistical distributions

2.04a - Finite discrete probability distributionsUse a table or formula to describe a finite discrete random variable, then turn a verbal event into the values you need to add. You will check that a proposed distribution is valid and combine independent observations without losing an order or counting it twice.
2.04b, 2.04c - Binomial models and probabilitiesRecognise when a binomial model fits a count and explain its assumptions in context (2.04b). Calculate exact, cumulative and interval probabilities with the formula or a calculator, keeping the event's integer boundaries clear (2.04c).
2.04e–2.04g - Normal models, standardisation and inverse probabilitiesUse the mean and variance to interpret a normal model (2.04e), calculate probabilities and inverse boundaries (2.04f), and connect its curve with histograms, spread and familiar proportions (2.04g). The aim is to turn a measurement question into the right area, rather than trust an unexplained ca...
2.04d, 2.04h - Choosing distributions and the normal approximationA thousand trials sounds like plenty for a smooth bell curve. Can that still be a poor model? In 2.04h you will choose distributions from the process and the shape; in 2.04d you will match a normal approximation's centre and spread to a binomial count.

Statistical hypothesis testing

2.05a, 2.05c - Hypotheses, significance and statistical decisionsHow unusual must a sample be before we question a claim? Build the language of hypothesis testing through a binomial model, then distinguish a chosen significance level from the actual chance of a false rejection.
2.05b - Testing a binomial proportionTurn a claim about a population probability into an exact binomial test. Calculate the relevant tail, justify critical boundaries and recognise how a two-tail test changes the evidence threshold.
2.05d, 2.05e - The sample mean and normal mean testsWhy can a small change in an average be convincing when individual measurements vary widely? Treat a sample mean as a random variable, then test a normal population's mean using a known, given or assumed population variance.
2.05f, 2.05g - Correlation coefficients and significance testsA scatter diagram can look convincing without settling what happens in the population. Under 2.05f you will interpret Pearson's coefficient; under 2.05g you will use supplied critical values and p-values to test population linear correlation, then explain what the evidence does and does not estab...

Mechanics

Kinematics

3.02a–3.02c - Motion language and kinematics graphsDescribe motion along a straight line using position, displacement, distance, velocity, speed and acceleration. Use graph gradients and signed areas to reconstruct a journey, including stops and reversals.
3.02d - Constant acceleration formulaeDerive and use the five constant-acceleration equations for motion along a line. Choose signs and intervals carefully to solve staged journeys and vertical motion, including the instant when a particle turns.
3.02f - Variable acceleration and calculus in motionDifferentiate and integrate to connect position, velocity and acceleration along a line. Use conditions to fix constants, check reversals and distinguish a signed displacement integral from total distance travelled.
3.02e, 3.02g, 1.10h - Motion in two dimensions with vectorsTrack a particle in a plane using two components that share one clock. Use constant-acceleration equations or calculus to find position and velocity, then distinguish the route travelled from the displacement between its endpoints.
3.02h, 3.02i - Projectile motion and trajectoriesUse independent horizontal and vertical components to predict a projectile's flight under gravity. Find position, speed and direction, choose the correct landing level, recover a launch velocity and derive the Cartesian trajectory with the limits of the model in view.

Forces and Newton's laws

3.03b–3.03d, 3.03h - Newton's laws and equations of motionUse all three Newton laws to connect a body's forces with its motion. You will complete force diagrams, form directed equations for vehicles and lifts, work with force vectors, and decide when several bodies can be treated as one particle.
3.03e, 3.03l, 3.03p - Resolving forces and dynamics in a planeResolve angled forces into useful perpendicular directions, add their components and apply Newton's second law in a plane. You will calculate resultants, distinguish force from motion direction, and handle horizontal surfaces, inclines and resolved third-law pairs.
3.03m, 3.03n - Particle equilibrium and polygons of forcesUse zero resultant to solve particle-equilibrium problems with vectors, strings, inclines and simple ideal connections. You will select effective axes, interpret closed force polygons, and check whether the calculated forces can exist in the stated model.
3.03k, 3.03o - Connected particles, strings and pulleysA string links motions as well as forces. You will use separate particle equations to find acceleration and tension, then adapt the model when a particle reaches the floor or a string runs along an incline.
3.03q - Variable forces and vector dynamicsA changing force needs a changing-acceleration model. You will combine vector force equations with calculus to recover velocity and position, use conditions at the correct times, and distinguish where a particle is moving from the direction in which its velocity is changing.
3.03r–3.03t - Contact forces, friction and limiting equilibriumRough contact has two components: a normal push and friction along the surface. You will separate them, calculate the overall contact force, and distinguish the friction actually needed at rest from its limiting or sliding value.
3.03u - Static and limiting equilibrium on rough surfacesFind the friction needed to keep a particle at rest, then decide whether the surface can provide it. Solve limiting cases, allow for a force that changes normal contact, and find complete ranges of forces that permit equilibrium.
3.03v - Motion on rough surfacesUse friction in directed equations of motion, with its direction set by sliding. Find accelerations on level ground and inclines, reconsider the forces when motion stops, and solve connected-particle problems with rough contact.