OCR A-Level Further Mathematics B (MEI)

Last Accessed

Recently visited lessons will appear here for quick access.Start exploring your first lesson today!

Core Pure

Complex numbers

Pj1, Pj2 - Complex number terms and quadratic rootsSome quadratic equations have no real roots, yet they still have roots. In this lesson you meet the imaginary unit i and the language of complex numbers: real and imaginary parts, the conjugate, the modulus and the argument, and the real and imaginary axes. You then solve any quadratic equati...
Pj4, Pj5 - Complex arithmetic and equating partsIn this lesson you add, subtract, multiply and divide complex numbers given in the form x + yi, and you use the fact that a complex number is zero only when its real and imaginary parts are both zero to turn one complex equation into two real ones. Along the way you find the square roots of \...
Pj3 - Solving cubics and quarticsUse the conjugate-pair rule to find complex roots of polynomials with real coefficients. You will reduce a cubic to a quadratic using a real root, and turn a given non-real root of a quartic into a real quadratic factor.
Pj6–Pj8 - Modulus-argument formDescribe a complex number by its size and direction, then use those two pieces of information to multiply and divide. Pj6 supplies exact radian angles, Pj7 connects Cartesian and modulus-argument forms, and Pj8 turns a long expansion into a short calculation with lengths and angles.
Pj9, Pj10 - Using the Argand diagramRead complex arithmetic as geometry. Pj9 connects a number and its conjugate to points on an Argand diagram; Pj10 shows how addition, subtraction, multiplication and division move those points.
Pj11 - Loci on the Argand diagramTranslate a complex equation into a set of points. You will recognise circles, half-lines and perpendicular bisectors, locate them accurately, and state the features needed to describe each locus completely.
Pj11 - Regions on the Argand diagramUse complex inequalities to describe regions, including exactly which boundaries belong. You will combine circles, half-planes and angular sectors using intersections and unions, and check the result with particular points.
Pj12, Pj13 - De Moivre's theorem and multiple anglesDe Moivre's theorem turns powers of complex numbers into multiples of angles. Pj12 gives the theorem and its meaning; Pj13 uses it to derive multiple-angle identities and to express powers of sine and cosine in terms of multiple angles.
Pj13 - Summing trigonometric seriesAdd a sine series and a cosine series together as one complex series, sum it, then extract the part you need. This second use of Pj13 covers binomially weighted sums, finite geometric sums and convergent infinite series, including series whose angles advance by more than one step.
Pj14–Pj17 - Exponential form and nth rootsPj14 introduces the compact form re^(iθ). Pj15–Pj17 use it to find every n-th root of a non-zero complex number, explain their regular-polygon arrangement, and prove why their sum is zero when n≥2.
Pj18–Pj20 - Geometry on the Argand diagramPj18 turns multiplication into an enlargement and a rotation, while Pj19 places roots of unity on the Argand diagram. Pj20 combines these ideas with complex differences and moduli to solve geometrical problems, including the side and diagonal lengths of regular polygons.

Matrices and transformations

Pm1–Pm3 - Adding and multiplying matricesA matrix lets you organise several numbers and calculate with them as one object. You learn the arithmetic in Pm1, equality and the zero and identity matrices in Pm2, and the multiplication rules in Pm3. The important change from ordinary algebra is that the order of a product can change its value.
Pm4 - Matrices of 2-D transformationsTwo columns can describe what a transformation does to every point in the plane. You learn how to build and recognise matrices for reflections, rotations, enlargements, stretches and shears, and how to describe each transformation precisely. The images of just two unit vectors will do most of the...
Pm4, Pm5 - Successive and 3-D transformationsCombining two transformations means multiplying their matrices in the correct order. You learn that connection in Pm5, then extend Pm4 to reflections in coordinate planes and quarter-turn multiples about coordinate axes in three dimensions. Unit-vector images keep the larger matrices manageable.
Pm6 - Invariant points and linesA transformation can leave a point fixed or leave a whole line in the same place while moving its points. You learn to distinguish these two meanings of invariance and use coordinate equations to find every invariant point and line of a plane transformation. Checking vertical lines will be just a...
Pm7–Pm10 - Determinants and scale factorsA transformation can push a square sideways, make it much longer and still leave its area unchanged. Pm7 introduces the determinant and singular matrices; Pm8 and Pm9 connect its size and sign to area, volume and orientation. Pm10 lets you combine scale factors without multiplying every matrix en...
Pm11–Pm14 - Inverse matricesAn inverse matrix reverses a transformation and recovers its original input. Pm11 explains what that means, Pm12 develops inverses of orders 2 and 3, and Pm13 uses them to find pre-images and solve matrix equations. Pm14 explains why undoing a product reverses the order of its factors.
Pm15 - Determinants and inverses by handA numerical matrix can be inverted by calculator, but an unknown entry calls for algebra. Pm15 develops the hand methods for determinants and inverses of order 3, including matrices containing parameters. Minors, signed cofactors and one transpose turn a large calculation into a sequence of s...

Vectors and 3-D space

Pv1 - Scalar product and angles between vectorsTurn two vectors into one number that reveals the angle between them. You will calculate scalar products in components, use their geometric form to find angles, and recognise perpendicular directions without drawing a three-dimensional picture.
Pv2, Pv3 - Equations of planesDescribe a flat surface using a point and a perpendicular direction. You will explain what a normal vector tells you, form and use vector and Cartesian equations of planes, and find a normal from two non-parallel directions in a plane.
Pv4–Pv6 - Intersections and angles of planesDecide how three planes fit together, find their common point using an inverse matrix, and calculate the angle between two planes from their normals. You will distinguish a failed inverse calculation from a geometrical conclusion and see why the angle between planes needs no subtraction from a ri...
Pv7, Pv8 - The vector productProduce a vector perpendicular to two given directions, rather than a single scalar. You will calculate the vector product, interpret its direction and zero cases, and use its magnitude to find an angle when the acute or obtuse choice is known.
Pv9, Pv10 - Lines and the angle between themDescribe a straight line using a point and a direction, convert between vector and Cartesian forms, and calculate the smaller angle between two lines from their direction vectors. You will handle zero direction components without dividing by zero and recognise different equations of the same line.
Pv11, Pv12 - Intersecting, parallel or skew linesTwo lines in space can miss each other even when they are not parallel. Pv11 names the possible relationships; Pv12 gives you a reliable calculation to distinguish them and find an intersection when it exists. You will compare directions first, then require one pair of parameter values to satisfy...
Pv14, Pv15 - Line–plane intersections and anglesA line can pierce a plane, run alongside it or lie entirely in it. Pv14 lets you distinguish these cases and locate an intersection; Pv15 gives the angle of the line to the surface. You will use a direction vector for the line and a normal vector for the plane, keeping their different roles clear.
Pv16, Pv17 - Point-to-line and point-to-plane distancesDistance to a line or plane means the shortest distance, measured along a perpendicular connector. Pv16 covers a point and a line in two or three dimensions; Pv17 covers a point and a plane. You will use the supplied formulae where they apply and find a perpendicular foot when a three-dimensional...
Pv13 - Distances between parallel or skew linesThe distance between two lines is the shortest gap between them, not the length between their displayed starting points. For parallel lines you reduce the problem to a point and a line. For skew lines you use the supplied formula and understand why its direction is perpendicular to both lines.

Series

Ps1 - Standard summation formulaeSum expressions involving integers, squares and cubes without adding every term. You will combine the standard formulae, handle a changed starting index, and prove the square and cube formulae when a question requires justification.
Ps1 - The method of differencesAdd a series by making most of its terms cancel. You will recognise consecutive differences, keep the correct boundary terms when the index gap changes, and find sums to infinity by taking the limit of a finite partial sum.
Ps2 - Summing series with partial fractionsUse partial fractions to uncover cancellation inside a rational series. You will choose a decomposition that exposes shifted terms, retain the correct finite boundaries, and deduce exact sums to infinity.
Ps3, Ps4 - Maclaurin series and their validityBuild a function's expansion from its derivatives at zero. You will find initial terms and a general term, use a short polynomial for a nearby numerical value, and distinguish a function's domain from the values where its Maclaurin series represents it.
Ps5 - Standard Maclaurin seriesUse the five standard expansions in the formula booklet rather than differentiating afresh. You will substitute and simplify exponential, logarithmic, trigonometric and binomial series, track their validity, and calculate a signed approximation error.

Calculus

Pc1 - Improper integralsEvaluate integrals with an infinite interval or an integrand that becomes undefined within the interval. Pc1 replaces the troublesome boundary by a variable, evaluates an ordinary integral first, and then tests whether the required limit is finite.
Pc2, Pc3 - Volumes of revolution and mean valuesPc2 builds volumes of revolution from thin cylinders, for rotation about either coordinate axis. Pc3 uses a definite integral to find a function's mean value, connecting the area under a changing height to an equivalent constant height.
Pc5 - Differentiating inverse trigonometric functionsDifferentiate x, x and x, then combine their derivatives with the chain and product rules. Pc5 explains the signs and domains from the principal inverse-function branches, so the formulae have a reason behind them.
Pc6 - Integrals giving inverse trigonometric functionsRecognise integrals that reverse the derivatives of arcsine and arctangent, including scaled inputs. Pc6 also uses sine and tangent substitutions for related integrals, with careful choices of angle range, transformed limits and square-root signs.
Pc4 - Integrating with partial fractionsExtend partial-fraction integration to a denominator containing one linear factor and a quadratic factor of the form ax²+c. Pc4 combines logarithmic integrals with the inverse-tangent integrals from the previous lesson, keeping decomposition, integration and endpoint evaluation as distinct s...

Hyperbolic functions

Pa3, Pa4 - Sinh, cosh and tanhHyperbolic functions are built from exponentials, despite having names that sound like trigonometry. In Pa3 you learn their definitions, graphs, domains and ranges; in Pa4 you establish the identity that links them. Their graphs and signs will help you choose valid answers before you calculate.
Pa5 - Calculus of sinh, cosh and tanhThe exponential definitions make hyperbolic differentiation and integration surprisingly direct. You derive the derivatives, use the chain and product rules, and recognise the corresponding antiderivatives, including a logarithm when you integrate tanh. Differentiating your answers will catch bot...
Pa6–Pa8 - Inverse hyperbolic functions and integralsInverse hyperbolic functions undo sinh, cosh and tanh, with a branch restriction where two inputs give the same output. You establish their domains and ranges in Pa6, derive and use their logarithmic forms in Pa7, and recognise the radical integrals in Pa8. Careful sign choices will keep every lo...

Differential equations

Pp19–Pp22 - Forming differential equation modelsA differential equation describes how a quantity changes, rather than giving its value directly. Pp19 defines the variables; Pp20 translates rates and changing rates into derivatives; Pp21 supplies the language of motion; Pp22 uses these ideas to build and interpret models. You will turn physical...
Pc7–Pc10 - First order equations and integrating factorsAn integrating factor turns a first-order linear differential equation into one product derivative that you can integrate. Pc7 distinguishes general and particular solutions; Pc8 identifies the right equation form; Pc9 explains the factor; Pc10 uses it to solve equations and apply conditions. You...
Pc11, Pc12 - Second order homogeneous equationsA quadratic equation can determine the whole family of solutions of a second-order differential equation. Pc11 uses this auxiliary equation; Pc12 explains how distinct real, repeated real and complex roots lead to different solution forms. You will choose the correct form and use two conditions t...
Pc13, Pc14 - Second order non-homogeneous equationsA non-zero right side adds a forcing term to a second-order equation. Pc13 combines the homogeneous solution with one particular integral; Pc14 finds that integral for polynomial, exponential and trigonometric forcing, including cases where the first trial overlaps the complementary function. You...
Pc15 - Simple harmonic motionSimple harmonic motion turns a second-order differential equation into a repeating movement about equilibrium. Pc15 connects the equation to its sine-and-cosine solution, uses initial displacement and velocity to find the constants, and reads amplitude and period from the result. The equation als...
Pc16, Pc17 - Damped oscillationsDamping changes a repeating motion into one that settles towards equilibrium. Pc16 builds second-order models from restoring and resistive effects; Pc17 connects their auxiliary roots to under-damping, critical damping and over-damping. You will interpret graphs, distinguish actual peaks from an...
Pc18 - Coupled first order equationsWhen two changing quantities affect one another, their differential equations must be solved together. Pc18 turns a coupled pair of first-order linear equations into one second-order equation, recovers the other variable and applies consistent initial data. You will also interpret population-rate...

Mechanics (Major and Minor)

Forces

Md1–Md5 - Modelling frictionMd1–Md3 describe normal contact, friction and their resultant, with a model that distinguishes resting from sliding. Md4 derives the condition for impending slipping on a slope, and Md5 applies Newton's laws with friction in the correct direction.
Md6–Md10 - Resolving forces and particle equilibriumMd6–Md7 resolve forces in useful directions and add concurrent forces as vectors. Md8–Md10 use a zero resultant to solve particle-equilibrium problems, either through component equations or a closed triangle or polygon of forces.
Md11–Md14 - Moments and couplesMd11–Md12 retain the positions of forces on a rigid body so that their turning effects can be modelled. Md13–Md14 introduce couples and calculate moments using perpendicular distances or resolved force components, including the effect of a given couple.
Md15, Md16 - Equilibrium, sliding or topplingMd15 combines zero resultant force with zero resultant moment to decide whether a rigid body can remain in equilibrium. Md16 checks the physical limits of that equilibrium: friction may run out, or the contact force may reach the edge of the supporting base, producing impending sliding or toppling.

Work, energy and power

Mw1–Mw5 - Work done and mechanical energyWork measures energy transferred by a force acting through a displacement. You learn the language in Mw1, calculate work along and at an angle to a force in Mw2–Mw3, and calculate kinetic and gravitational potential energy in Mw4–Mw5. Signs, directions and the chosen zero height matter throughout.
Mw6, Mw7 - Energy conservation and the work-energy principleEnergy methods link the beginning and end of a motion without finding every intermediate acceleration. In Mw6 you identify when mechanical energy is conserved; in Mw7 you include the work of driving and resistive forces when it is not. The main habit is to account for each energy transfer exactly...
Mw8 - Power as rate of workingPower measures the rate at which a force does work. You calculate average and instantaneous power, connect a vehicle's useful power to its driving force, and find steady-speed limits when driving and resistance balance. Constant power will not usually mean constant force or constant acceleration.

Momentum and impulse

Mi1–Mi3 - Impulse and momentum as vectorsCalculate momentum and impulse with their directions intact. You will use the total external impulse to find a velocity change, handle a rebound and a mean contact force, and use relative velocity to describe approach and separation along a line.
Mi4–Mi6 - Conservation of momentum in direct impactsChoose a system and conserve its total momentum when external impulse is zero or negligible. You will state the direct-impact assumptions, calculate velocities when bodies join or separate after a collision, and handle a body dividing into parts using vector components.
Mi7–Mi12 - Restitution and energy loss in impactsUse Newton's experimental law to complete a direct-impact model. You will interpret the coefficient of restitution, solve wall and two-body collisions, and calculate kinetic energy lost while distinguishing perfectly elastic impact from coalescence.

Centre of mass

MG1–MG3 - Centres of mass of particles and shapesA centre of mass replaces a whole mass distribution by one point for calculations involving its weight. You will locate it for particles in one, two and three dimensions, use symmetry, and learn the standard positions for rods and laminas.
MG4 - Centres of mass of composite bodiesA complicated uniform body can often be split into simple shapes whose centres of mass you already know. Treat each part as a particle at its own centre, using area, length or volume to find the appropriate mass ratios.
MG5 - Centre of mass and equilibriumThe centre of mass tells you where a body's weight acts when calculating moments. You will use it to find a suspended body's orientation, test whether supports can hold a horizontal body, and decide whether an inclined body first slides or topples.

Circular motion (Major only)

Mr1–Mr3 - Angular speed, forces and accelerationA body can keep the same speed while accelerating throughout its motion. Mr1 gives the language of a circle; Mr2 and Mr3 connect its inward acceleration to the actual forces. You will convert between angular and tangential speed, calculate radial acceleration and build equations of motion.
Mr4 - Horizontal circles at constant speedMr4 brings circular acceleration into familiar force models: a car on a flat bend, a conical pendulum and a banked track. You will use zero vertical acceleration alongside a non-zero horizontal radial acceleration, choose the correct circle radius and test whether friction or contact can sustain...
Mr5–Mr8 - Varying speed and vertical circlesA changing speed adds tangential acceleration to the inward radial component. Mr5 and Mr6 separate those jobs; Mr7 combines forces and energy in vertical circles; Mr8 checks when a string or surface can no longer maintain the circle and follows the motion after departure.

Vectors and variable forces (Major only)

Mk1, Mk2, Mv1, Mv2 - Position and velocity vectorsDescribe motion in two and three dimensions without confusing a vector with its magnitude. You will use constant-acceleration formulae, differentiate and integrate components for variable motion, infer the resultant force, and compare positions of moving particles.
Mv3–Mv5 - Equation of a particle's pathEliminate time to describe where a particle moves. You will retain the physical section of a curve, derive a projectile trajectory, and use a quadratic in the launch angle to decide whether a target is reachable.
Mv6, Mv7 - Projectiles on an inclined planeFind where a projectile next meets a uniform slope and choose the launch direction that maximises its range. You will distinguish angles from the horizontal and from the slope, resolve gravity into rotated coordinates, and measure range along the plane.
Mv8–Mv10 - Differential equations for variable forcesTranslate a variable-force model into differential equations, verify a supplied solution, and apply initial or boundary conditions. You will choose between time and displacement as the independent variable and keep track of the physical branch of the motion.
Mv11, Mv12 - Modelling with simple harmonic motionTurn restoring-force equations into simple harmonic motion about the correct equilibrium. You will apply the Core Pure solution to springs and a small-angle pendulum, use the speed–displacement relation, and stop an SHM model when a string goes slack or contact is lost.

Statistics (Major and Minor)

Discrete random variables

SR1–SR3 - Discrete distributions, mean and varianceDescribe a finite random variable by its possible values and their probabilities. SR1 turns tables, formulae and graphs into probability statements; SR2 and SR3 use the same information to measure the distribution's centre and spread.
SR4–SR6 - Expectation and variance of combinationsWork out how changing a scale, adding a constant or combining independent observations changes a distribution. SR4 and SR5 transform its mean and variance; SR6 combines variables and explains why averaging fresh observations reduces variation.
SR7–SR9 - The discrete uniform distributionUse equally likely values to build a discrete uniform model. SR7 and SR8 identify the model and count event probabilities; SR9 calculates its mean and variance, including derivations and distributions whose labels do not start at one.
SR10 - The binomial distributionConnect the familiar binomial model to independent zero-or-one trials. Check its assumptions in context, calculate probabilities and prove its mean and variance in particular cases instead of simply quoting the results.
SR11–SR15 - The Poisson distributionModel event counts in time or space and calculate their probabilities. SR11 and SR12 choose a suitable model, SR13 handles point and cumulative probabilities, and SR14 and SR15 connect the parameter to moments, interval lengths and independent totals.
SR16–SR18 - The geometric distributionModel the number of trials up to and including a first success. SR16 checks the stopping rule and assumptions, SR17 turns event wording into point and cumulative probabilities, and SR18 uses the supplied mean and variance to interpret or recover the success probability.

Bivariate data

Sb1–Sb3 - Scatter diagramsPlot paired observations, describe their pattern and interpret the extra information software puts on a scatter diagram. You will distinguish an unusual observation from a trend, and a straight-line summary from the relationship itself.
Sb4–Sb7 - Pearson's product moment correlationCalculate Pearson's product moment correlation coefficient and use a valid population test, with either critical values or software output. You will also interpret the size of a correlation, which answers a different question from statistical significance.
Sb8–Sb10 - Spearman's rank and choosing a testCalculate Spearman's coefficient from untied ranks, test for population association and choose between the Pearson and Spearman procedures. The shape of the relationship matters as much as the availability of a calculator button.
Sb11, Sb12 - Regression on a non-random variableFit and use a least squares line when an experimenter controls the input and measures a variable response. Calculate residuals, judge a linear model informally and keep predictions within the role and range the model supports.
Sb13–Sb15 - Regression lines for two random variablesCalculate the two regression lines for random-on-random data and choose the line that estimates the required variable. Interpret the predicted conditional mean, the common mean point and the model's fit without treating either line as an exact inverse relationship.

Continuous random variables (Major only)

SR19–SR21 - Probability density functionsSR19–SR21 model measurements using continuous random variables. You will interpret a density graph, check that a proposed density is valid, sketch its support and calculate probabilities as areas, including when the formula changes partway through an interval.
SR22, SR23 - Averages and spread from a pdfSR22 uses a density to calculate the mean, variance and standard deviation, including the continuous uniform model. SR23 finds the median by splitting probability in half and the mode by locating the density's maximum. The three measures of centre answer different questions.
SR24–SR26 - Cumulative distribution functionsSR24 accumulates density into a cumulative distribution function and sketches its full shape. SR25 recovers density by differentiation. SR26 uses cumulative probabilities to find intervals, medians and percentiles, including selecting the correct branch of a piecewise function.
SR27–SR29 - Combining Normal random variablesSR27 extends expectation algebra to continuous variables. SR28 uses calculator probabilities and percentiles for a Normal model; SR29 adds the crucial distribution result for independent Normal combinations. You will state the resulting distribution before calculating a probability, and distingui...
SR30, SR31 - Checking a Normal modelSR30 distinguishes a Normal model for individual observations from a Normal approximation for a sampling distribution. SR31 uses supplied Normal probability plots and Kolmogorov-Smirnov output to judge a proposed fit. You will interpret evidence cautiously, without constructing the plots or calcu...

Inference (Major only)

SI1–SI6 - Sample means and the Central Limit TheoremSI1–SI2 turn sample data into estimates of a population's mean and variance. SI3–SI6 then treat the sample mean itself as a random variable: you will calculate its standard error, distinguish exact Normal results from the Central Limit Theorem, and approximate probabilities for means and totals.
SI7–SI9, SR32 - Confidence intervals for a meanSI7 explains what confidence describes; SI8 controls interval width. SI9 and SR32 construct intervals for a single population mean using a Normal or t distribution, with the sample size and knowledge of the population variance deciding which is appropriate.
SI10, SI11 - Confidence intervals for paired samplesSI10 identifies a genuine pairing between observations. SI11 then treats the differences as one sample, constructing a Normal or t confidence interval for the mean difference and interpreting its sign in context.
SI12, SI13 - Confidence intervals from softwareSI12 reads confidence intervals from software, including output for a parameter or distribution whose calculation has been supplied. SI13 uses the reported interval to assess a hypothesised parameter value, with a conclusion that reflects sampling uncertainty.
SH5 - The Wilcoxon signed rank testSH5 tests a single population median using the signs and ranks of deviations from its hypothesised value. You will check the symmetry assumption, handle zero deviations and ties, choose the correct one- or two-tailed table column, and reach a non-assertive conclusion in context.
SH6 - Hypothesis tests for a meanSH6 tests a single population mean using a Normal or t reference distribution. You will choose the distribution from the sampling conditions, distinguish one- and two-sided alternatives, use the right critical-value convention, and state what the evidence suggests in context.

Modelling with Algorithms

Algorithms

AA1–AA3 - Using and adapting algorithmsAn algorithm turns a task into instructions precise enough to follow without guesswork. You will read written instructions, pseudocode and flowcharts, trace changing variables, and repair or adapt a procedure so that it produces the intended output and stops.
AA4–AA6 - Efficiency, correctness and heuristicsAn algorithm needs more than a correct-looking answer on one input. You will count its work, compare how that work grows, distinguish a proof from a collection of tests, and explain why a useful heuristic need not guarantee an optimal solution.
AA7–AA9 - Sorting algorithmsSorting puts a list into ascending or descending order while retaining every item. You will trace quick sort, apply another fully specified sort, count comparisons and swaps using its actual rules, and explain why those rules produce an ordered list.
AA10, AA11 - Bin packing algorithmsBin packing places indivisible items into containers of fixed capacity, usually aiming to use as few containers as possible. You will apply first fit, first fit decreasing and full-bin strategies, justify optimal results using bounds, and count the comparisons made by a precisely stated packing p...

Networks

AN1–AN4 - Graphs and networks as modelsReplace a complicated setting by points, connections and, when needed, weights. AN1 supplies the graph language, AN2 models relationships and legal moves, and AN3 and AN4 turn those graphs into networks that describe distances, costs or capacities.
AN5 - Minimum connectors by Kruskal and PrimConnect every vertex for the least total weight using Kruskal's or Prim's algorithm. Learn which arcs are eligible at each step, record ties and rejected cycles clearly, and apply Prim's algorithm to both a diagram and a table.
AN6, AN7 - Dijkstra's algorithm and complexityFind shortest paths with Dijkstra's algorithm, keeping working labels, permanent labels and predecessor routes distinct. AN6 handles undirected and directed networks; AN7 compares the prescribed quadratic complexity of Dijkstra, Kruskal and Prim using the correct measure of input size.
AN8, AN9 - Critical path analysisUse activity-on-arc networks to turn precedence requirements into a project model. AN8 constructs the network; AN9 finds event times, critical activities and float, then tests whether a resource schedule can achieve the minimum completion time.
AN10–AN12 - Network flows and cutsModel transmission through capacity-limited directed arcs. AN10 checks feasible flows and introduces super sources and sinks, AN11 calculates cut capacities, and AN12 certifies a maximum flow by matching it to a cut of the same value.

Linear programming

AL1–AL5 - Formulating linear programsTranslate a practical decision into numerical variables, a linear objective and constraints. Recognise standard and augmented forms, interpret slack variables and state when fractions of a decision are not allowed.
AL6, AN13 - Network problems as linear programsExpress paths, flows, project durations, matchings, allocations and transportation as linear models suitable for a computer. The network picture remains useful, but the decision variables and constraints tell the software what its arrows mean.
AL7–AL11 - Graphical solution of linear programsUse feasible regions and objective lines to solve continuous and integer problems. Explore changed constraints, recognise special cases, interpret a supplied three-dimensional region and reduce an equality-constrained problem to two variables.
AL12, AL13 - The simplex methodBuild, pivot and interpret simplex tableaux for a maximisation problem in augmented form. Connect the algebra to movement between feasible vertices, and distinguish optimality, degeneracy and other possible outcomes.
AL14–AL16 - Simplex for non-standard problemsUnderstand why lower bounds can prevent an ordinary slack start, introduce surplus and artificial variables, and construct the initial two-stage tableau. Reformulate equalities as pairs of inequalities and recognise other cases that need an initial reformulation.
AL17, AL18 - Solving linear programs with softwareSpecify an LP or ILP for a solver, read its decision values and objective in context, and distinguish an optimal result from a feasible candidate or a failed model. Software provides computing power; the formulation determines what its answer means.

Numerical Methods

Solution of equations

Ne1, Ne2 - Bisection, false position and secant methodsLocate a root by halving an interval or by intersecting a chord with the horizontal axis. You will distinguish false position from the secant method, implement their different updates, and justify the decimal places claimed for a numerical root.
Ne1, Ne2 - Fixed point iteration and Newton-RaphsonGenerate root approximations by repeated substitution or tangent intercepts. You will draw staircase and cobweb paths, implement the correct iteration, choose meaningful starting values, and use the accuracy checks from the previous lesson.
Ne3, Ne4, NU7 - Order of convergence and failureInterpret error and difference columns, distinguish sequence convergence from step-length order, and recognise when an iteration is slow or fails. You will compare computational costs and investigate why Newton-Raphson is generally, rather than invariably, second order.
Ne5 - Relaxation of fixed point iterationUse relaxation to change the behaviour of a fixed-point iteration. You will compare supplied parameters, accelerate a convergent sequence and turn a repelling fixed point into an attracting one, while checking accuracy and domains.

Extra Pure

Recurrence relations

Xs1–Xs3 - Recurrence models and their behaviourBuild a recurrence from a repeated process, describe its structure and investigate its terms. Distinguish evidence for a long-run pattern from a proof, including for non-linear recurrences.
Xs4–Xs6 - Solving first order recurrencesVerify a solution, solve first order homogeneous recurrences and add a trial solution for constant, polynomial or exponential forcing. Keep the starting index and the initial condition visible.
Xs7 - Second order homogeneous recurrencesSolve second order homogeneous recurrences using an auxiliary equation. Handle distinct, repeated and complex roots, and use two sequence values to determine the constants.
Xs8, Xs9 - Second order non-homogeneous recurrencesCombine a homogeneous family with an appropriate trial solution. Handle polynomial and exponential forcing, resolve overlap, and use complete solutions to justify limits, model validity and the behaviour of associated sequences.

Groups

XS1, XS2, Xa1, Xa2 - Sets, group axioms and examplesA group records a collection of objects together with a way to combine them. XS1 and XS2 make the set and number language precise; Xa1 gives the four group axioms, and Xa2 tests them in arithmetic, complex numbers, matrices and geometric symmetries.
Xa3–Xa6 - Cyclic groups, subgroups and Lagrange's theoremSome finite groups can be reached by repeatedly using just one element. Xa3 and Xa4 distinguish a generator's cycle from the order of the whole group; Xa5 tests subgroups, and Xa6 uses Lagrange's theorem to restrict their sizes and the orders of elements.
Xa7, Xa8 - Isomorphic groupsGroups can describe different objects while obeying exactly the same combination rules. Xa7 recognises that common structure; Xa8 specifies an isomorphism, proves it preserves the operation, and uses element orders to rule out impossible correspondences.

Matrices

Xm1, Xm2 - Eigenvalues and eigenvectorsFind the directions a matrix preserves, calculate their eigenvalues, and form and solve characteristic equations of order 2 and 3. You will distinguish a vector's scale from its direction, handle zero and repeated eigenvalues, and recognise when some eigenvalues are not real.
Xm3, Xm4 - Diagonalisation and matrix powersUse a matrix of eigenvectors to put a matrix with distinct real eigenvalues into diagonal form. Then calculate powers of matrices of order 2 and 3, keeping each eigenvector matched to its eigenvalue and stating when negative powers exist.
Xm5, Xm6 - Cayley-Hamilton and transformation eigenvectorsUse the Cayley-Hamilton theorem to reduce powers and calculate inverses, then interpret eigenvectors as geometric directions in two and three dimensions. You will distinguish an invariant line from fixed points, locate a reflection plane, and find a rotation axis from a given matrix.

Further Pure with Technology

Investigation of curves

TC1–TC5 - Families of curves with softwareInvestigate cartesian, polar and parametric families using sliders and CAS. Turn observations into checked general statements, describe the curves precisely, and convert their equations without losing domains, branches or exceptional points.
TC6, TC7 - Tangents, normals and chordsFind tangent gradients and equations from cartesian, parametric and polar curves. Use CAS derivatives without losing their conditions, handle horizontal and vertical directions, and connect a chord between distinct points with a limiting tangent.
TC8, TC9 - Arc length and envelopesSet up and evaluate arc-length integrals in cartesian, parametric and polar form. Find candidate envelopes by differentiating a family equation with respect to its parameter, then verify the contacts, tangencies and permitted parameter range.
TC10–TC12 - Limits, asymptotes and cuspsUse CAS limits and nearby behaviour to distinguish holes from vertical asymptotes, find horizontal and oblique asymptotes, and identify asymptotic approach to another curve. Recognise MEI cusps by the two branches and their common limiting tangent.

Exploring differential equations

Tc1, Tc2, Tc4–Tc6 - Software solutions and tangent fieldsExplore first-order differential equations with a computer algebra system and a tangent-field plotter. You will control parameters, select and verify particular solutions, use isoclines to sketch a field, and read the domain and shape of a solution from its local directions.
Tc3, Tc7, Tc8 - Euler methods in a spreadsheetConstruct and adapt spreadsheets for Euler's method and the modified Euler method. You will use the correct starting and predicted gradients, retain precision, compare step lengths at the same endpoint and support an appropriate numerical answer.
Tc9, Tc10 - Runge-Kutta methodsUse four-stage Runge–Kutta calculations and implement them in a spreadsheet. You will explain the stage points and weights, adapt the differential equation and step length, and read a supplied variation carefully rather than force every method into the standard pattern.

Number theory

TT1–TT3 - Programs and prime factorisationWrite, trace and refine short number-theory programs. Use conditions, loops, functions, local and global variables, input and output; build a reliable prime test and use unique prime factorisation.
TT4–TT7 - Theorems of Fermat, Euler and WilsonCalculate and search with congruences. Use Fermat's little theorem, Euler's totient and theorem, and Wilson's theorem with their exact conditions; distinguish a necessary primality condition from an exact test.
TT8–TT10 - Diophantine and Pell's equationsConstruct Pythagorean triples and related power identities. Solve Pell equations, translate their solutions back into practical integer problems, and use given identities, exact searches and modular obstructions for other Diophantine equations.