OCR A-Level Further Mathematics A

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Pure Core

Complex numbers

4.02a–4.02c - The language of complex numbersMeet i, the number whose square is -1, and the words and symbols used for every complex number: real part, imaginary part, conjugate, modulus and argument. You will write a complex number in cartesian form x + iy and in modulus-argument form r(θ + iθ), picture it...
4.02e - Arithmetic in cartesian formLearn to add, subtract, multiply and divide complex numbers written in cartesian form x + iy, using i² = -1 and the complex conjugate, and to solve equations in z and z by equating real and imaginary parts, giving exact answers in the form the exam asks for. Along the way you wi...
4.02g–4.02j - Solving equations with complex rootsSolve real quadratic equations with non-real roots, find both square roots of a complex number algebraically, and use conjugate pairs to factorise real cubics and quartics. You will see why a single given complex root can reveal a whole real quadratic factor, and how to keep the two different kin...
4.02k, 4.02l - Argand diagramsUse an Argand diagram to represent complex numbers as points and position vectors. You will interpret conjugation as a reflection, addition as a translation, and subtraction as a directed displacement, connecting the picture to the arithmetic you already know.
4.02e, 4.02f, 4.02m - Working in modulus-argument formConvert between cartesian and modulus-argument forms, choosing the correct quadrant and principal argument. Use the form that makes each calculation simpler, and see multiplication, division and positive integer powers as changes of length and direction on an Argand diagram.
4.02n - Euler's formulaUse Euler's formula to connect the complex exponential with cosine and sine. You will interpret an imaginary exponent as an angle on the unit circle, calculate exact values, and solve equations by identifying a direction and its full turns.
4.02d - Using exponential formWrite a complex number as re^(iθ), keeping its modulus and angle visible, and calculate with that form. You will convert between representations, simplify products and quotients, and use conjugate exponentials to separate real and imaginary contributions.
4.02o, 4.02p - Drawing loci and regionsTranslate complex equations and inequalities into circles, straight lines, half-lines and regions on an Argand diagram. Use set notation to state exactly which points belong, including boundaries, and combine conditions by taking their intersection.
4.02q - De Moivre's theorem for multiple anglesUse de Moivre's theorem to turn a complex power into a multiple-angle identity. You will derive formulae for sine, cosine and tangent, explain the conditions on a quotient, and use a trigonometric identity to solve a polynomial equation exactly.
4.02q - Rewriting powers of sine and cosineRewrite powers of sine and cosine as sums of trigonometric functions with multiple angles. Euler's formula supplies a short route: expand powers of a sum or difference of exponentials, pair opposite powers, and turn the pairs back into real functions.
4.02q - Summing trigonometric seriesAdd trigonometric terms by putting their cosine and sine parts back into a complex exponential. You will sum finite and convergent infinite geometric series, handle their exceptional cases, and recognise binomial coefficients hidden inside trigonometric sums.
4.02r - nth roots of complex numbersFind every root of a complex equation wⁿ=z, give the roots in modulus-argument, exponential or cartesian form, and explain their regular polygon on an Argand diagram. The key is to include all the arguments of z before dividing by n.
4.02s - Using roots of unityUse the roots of zⁿ=1 to describe regular polygons, prove sums of roots and their real parts, and calculate sides, diagonals and exact trigonometric values. Algebra and symmetry will explain why a regular pentagon contains the golden ratio.

Matrices

4.03a - The language of matricesA matrix keeps numbers in positions, so its shape and the order of its entries matter as much as the entries themselves. Learn to read real and complex matrices, recognise the special matrices, compare matrices and take a transpose. This vocabulary will let you describe calculations precisely bef...
4.03b, 4.03c - Adding and multiplying matricesAdd and subtract matching entries, multiply matrices by scalars, and build matrix products one row and column at a time. In 4.03b you will also use powers, zero matrices and identity matrices; in 4.03c you will see why brackets can move in a product but the matrices usually cannot. The sa...
4.03d - Matrices for 2D transformationsA transformation matrix tells you where every point goes, using only two columns. Find and use matrices for reflections, rotations, enlargements, stretches and shears, and describe each transformation with the details that identify it completely. Pictures will make the entries mean something befo...
4.03e, 4.03f - Successive and 3D transformationsCombine transformations in the right order, then recognise the single transformation their product represents. Extend the same column-vector idea to space: reflect in coordinate planes and rotate about coordinate axes, with the viewing direction made explicit. Successive transformations are the f...
4.03g - Invariant points and linesFind the points that a two-dimensional transformation leaves fixed and the lines that it maps onto themselves. These are different questions: a line can remain in place while its points move along it. Use coordinate equations to distinguish the two, including lines that do not pass through the or...
4.03h–4.03k - Calculating and interpreting determinantsCalculate determinants of matrices of order two and three, by hand and with a calculator. Use their magnitudes to find areas and volumes after a transformation, and their signs to decide whether orientation is preserved.
4.03l, 4.03m - Singular matrices and det(AB)Recognise singular matrices and explain the significance of a zero determinant. Find parameter values that make a matrix singular, and use the determinant of a product to analyse successive transformations without multiplying out every entry.
4.03n–4.03q - Finding and using inverse matricesFind inverses of non-singular matrices of order two and three, by hand and with a calculator. Use inverses to rearrange matrix equations and reverse transformations, keeping the order of multiplication under control.
4.03r, 4.03s - Solving simultaneous equationsWrite two or three simultaneous linear equations as one matrix equation and solve a system with a unique solution by using an inverse. When no unique solution exists, use the equations themselves to distinguish an infinite set of solutions from no solutions.

Further vectors

4.04a - Equations of straight linesDescribe a straight line by one point and one direction, in two dimensions or three. You will build vector equations from points, convert between vector and cartesian forms, and handle horizontal, vertical and other lines with a constant coordinate without dividing by zero.
4.04c - Scalar product and angles between linesUse the scalar product to measure the angle between vectors, find the acute angle between lines and test perpendicularity. You will connect the coordinate calculation to its geometric meaning, then use perpendicularity to determine unknown components.
4.04e - Intersecting, parallel or skew linesDetermine whether two lines meet, run parallel or are skew, and find their intersection when it exists. You will use separate parameters for the two lines, check all three coordinates in space, and distinguish separate parallel lines from two equations of the same line.
4.04g - Finding a common perpendicularFind a vector perpendicular to two given vectors using the vector product, then verify it with scalar products. You will use the supplied component formula carefully, recognise the parallel-vector case and solve problems where another perpendicularity condition determines a parameter.
4.04b - Equations of planesDescribe a plane using two directions within it or one direction perpendicular to it. You will construct and convert all four plane forms, and verify that apparently different equations describe the same set of points.
4.04d - Angles with planesMeasure the angle between two planes and between a line and a plane using their normals. You will distinguish the angle to a surface from the angle to its normal, choose the required acute angle and recognise parallel and perpendicular cases.
4.04f - Where a line meets a planeFind a line-plane intersection by substituting the line's general point into the plane. You will recognise a unique intersection, a line parallel to the plane and a line contained in it, then find the foot of a perpendicular from a point.
4.04h–4.04j - Finding shortest distancesFind perpendicular distances from points to lines and planes, and between parallel or skew lines. In 4.04h the objects are two lines; 4.04i covers a point and a line in two or three dimensions; 4.04j covers a point and a plane. You will choose the right normal or perpendicular rather than measuri...

Further algebra

4.05a - Roots and coefficients of polynomialsRead the sums and products of polynomial roots directly from the coefficients, even when the roots themselves are awkward or non-real. You will derive the relationships for quadratics, cubics and quartics, then combine them to find less obvious symmetric expressions without solving an equation.
4.05b - Transforming polynomial equationsFind an equation with prescribed changes to its roots without solving the original equation. You will translate and scale roots, take their reciprocals, and replace them by their squares, using a substitution that connects the old unknown to the new one.
4.05c - Quadratic and improper partial fractionsExtend partial fractions to denominators containing a quadratic such as x² + 4, and to fractions whose numerator has degree at least as large as the denominator. You will choose the right numerator, extract a polynomial part when needed, and check a decomposition by recombining it.

Hyperbolic functions

4.07a–4.07c - Hyperbolic definitions, graphs and identitiesBuild three new functions from exponentials: x, x and x. You will use their definitions and domains and ranges in 4.07a, sketch their graphs in 4.07b, and prove and use identities in 4.07c, including solving equations exactly.
4.07d - Differentiating and integrating hyperbolic functionsDifferentiate and integrate hyperbolic functions by connecting their exponential definitions to familiar calculus rules. You will handle composite functions and products, find exact integrals, and use derivatives and the ranges of the functions to locate a curve's minimum.
4.07e, 4.07f - Inverse hyperbolic functions as logarithmsUnderstand the inverse hyperbolic functions, including the domain and range of each, in 4.07e. In 4.07f you will derive their logarithmic expressions from exponentials and use them for exact values and equations, keeping careful track of the branch of inverse cosine hyperbolic.

Further calculus

4.08a, 4.08b - Maclaurin seriesReplace a function by a power series built from its derivatives at zero. In 4.08a you will calculate coefficients and general terms; in 4.08b you will adapt standard series, combine them, and distinguish an infinite series' interval of validity from a finite approximation's accuracy.
4.08c - Improper integralsUse limits to evaluate integrals with an infinite range or an undefined integrand. You will decide whether a finite value exists, handle singularities at either endpoint, and split an integral when the difficulty lies inside its interval.
4.08d - Volumes of revolutionBuild volume formulae from thin circular slices, then use them for rotation about either coordinate axis. You will choose the radius and integration variable, subtract a hollow centre correctly, and calculate volumes when a curve is defined parametrically.
4.08e - Mean value of a functionFind the average height of a curve over an interval, and use it to interpret a changing quantity. You will distinguish a function's mean value from the average of its endpoint values, evaluate exact means by integration, and work backwards from a given mean.
4.08g - Inverse trig and hyperbolic derivativesDerive the six inverse trigonometric and hyperbolic derivatives by differentiating their defining equations. You will use the correct branches to choose signs, apply the chain rule to composite functions, and find exact normal equations without leaving inverse functions in the final answer.
4.08h - Inverse trig and hyperbolic integralsRecognise the four standard integrals that give inverse trigonometric or hyperbolic functions, including after completing the square. You will explain them using substitutions and choose trigonometric or hyperbolic substitutions for related definite and indefinite integrals that contain square ro...
4.08f - Integrating with partial fractionsUse partial fractions to integrate rational functions with linear factors, repeated linear factors and irreducible quadratic factors. You will combine logarithms with inverse tangent terms, divide improper fractions before integrating, and check the whole antiderivative by differentiation.

Polar coordinates

4.09a - Using polar coordinatesDescribe a point by its distance and direction from a fixed point. You will convert points and curve equations between polar and cartesian coordinates, keeping the angle range and the convention that distance is never negative.
4.09b - Sketching polar curvesBuild a polar sketch from the rays the curve occupies, its symmetry, its visits to the pole and its least and greatest distances from the pole. You will use trigonometric values and differentiation to justify the shape, including curves whose largest radius is not obvious.
4.09c - Areas enclosed by polar curvesFind an area by adding thin sectors as a ray sweeps through a polar curve. You will choose limits from the geometry, integrate squared radii, handle shared regions and radial gaps, and use an area formula containing a parameter.

Differential equations

4.10a - General and particular solutionsA differential equation describes a relationship between a function and its derivatives. You will distinguish a whole family of solutions from the particular member selected by initial or boundary conditions, and check both the equation and the conditions.
4.10c - The integrating factor methodSome first order equations become integrable when you multiply by the right function. You will build that function from the product rule, recognise when the method applies, and use initial conditions without losing the interval on which your solution is valid.
4.10b - Modelling with differential equationsA differential equation turns a statement about rates or forces into a prediction. You will construct and interpret models in kinematics and other contexts, solve suitable first order models, and distinguish a model you can formulate from one you can solve analytically.
4.10d - Second order homogeneous equationsA quadratic equation can determine an entire family of functions. You will use the auxiliary equation to solve second order homogeneous differential equations, choose the right form for real, repeated or complex roots, and determine the constants from conditions.
4.10e - Second order non-homogeneous equationsSolve second order equations with a polynomial, exponential or trigonometric forcing term. You will choose a particular integral, combine it with the complementary function, and recognise when a familiar-looking trial must change because it already belongs to the complementary function.
4.10f - Simple harmonic motionConnect a restoring acceleration to a repeating motion. You will solve the simple harmonic motion equation, use displacement and velocity conditions, and interpret amplitude, phase, period and distance travelled in a model.
4.10g - Damped oscillationsModel a restoring force together with resistance proportional to velocity. You will solve the resulting second order equation and distinguish underdamping, critical damping and overdamping by what their solutions predict.
4.10h - Coupled first-order systemsModel two quantities whose rates depend on each other, then solve their coupled first-order equations by elimination. You will recover both variables, translate initial conditions correctly, and interpret the solution in its original setting.

Statistics

Discrete random variables

5.02a–5.02c - Discrete distributions, expectation and varianceBuild a finite discrete probability distribution, calculate its expectation and variance, and track what happens when its values are shifted or scaled. These tools turn a list of possible outcomes into a model you can calculate with and criticise.
5.02d, 5.02e - Binomial and discrete uniform modelsUse binomial means and variances, recognise a discrete uniform model, and calculate its probabilities and moments. Learn to distinguish a model for counting successes from a model in which each listed integer is equally likely.
5.02f–5.02h - The geometric distributionModel the number of independent trials up to and including the first success. Calculate exact, interval and conditional probabilities, and use the geometric mean and variance to describe waiting in discrete trials.
5.02i–5.02l - The Poisson distribution as a modelRecognise a Poisson count, scale its parameter to the stated interval, calculate exact and cumulative probabilities, and explain its modelling assumptions in context. The model describes random event counts over time, distance or area.
5.02m, 5.02n - Poisson mean, variance and sumsUse the equality of Poisson mean and variance, combine independent Poisson counts, and recognise when that addition rule cannot be used. Distinguish exact distribution properties from the approximate patterns expected in a finite sample.

Continuous random variables

5.03a–5.03d - Probability density, mean and varianceA continuous measurement assigns probability to intervals rather than individual values. In 5.03a–5.03b you will turn a density curve, including a piecewise curve, into probabilities; 5.03c–5.03d then use the same areas to find means, variances and expected values of functions.
5.03e–5.03g - Cumulative distribution functionsA cumulative distribution function stores every left-tail probability in one curve. In 5.03e–5.03f you will construct and use that curve, recover its density and locate percentiles; 5.03g then uses inequalities to find the distributions of related variables.
5.03a - Continuous uniform and exponential distributionsA constant density models equal chances across equal-length intervals; an exponential density models waiting for events arriving at a constant rate. This lesson develops the named continuous distributions in 5.03a, recalls how the normal model fits alongside them, and connects exponential waiting...

Hypothesis tests and confidence intervals

5.05a, 5.05b - Sample means and unbiased estimatesAverages are steadier than single observations, but averaging does not turn the original population into a normal distribution. In 5.05a you will distinguish the distribution of a sample mean from the population distribution; in 5.05b you will use sample data to estimate an unknown population mea...
5.05c - Hypothesis tests for a population meanUse a normal distribution to decide whether sample data provide significant evidence about a population mean. You will choose the correct tail, distinguish a known population variance from an estimated one, and write a conclusion that says what the evidence supports without claiming certainty.
5.05d - Confidence intervals for a population meanEstimate an unknown population mean by an interval that reflects sampling uncertainty. You will construct normal confidence intervals with a known or estimated variance, interpret the confidence level correctly, and work backwards from a desired precision to a sample size.

Chi-squared tests

5.06a - Chi-squared tests for independenceDecide whether two categorical variables are associated, using observed counts and the counts independence would predict. You will calculate expected frequencies and contributions, combine categories when necessary, choose the degrees of freedom and complete a chi-squared test, including Yates’s...
5.06b, 5.06c - Fitting a theoretical distributionTurn a proposed probability model into the frequencies it predicts for a sample. Given ratios and discrete uniform models provide the starting point in 5.06b; 5.06c extends the same idea to binomial, geometric, Poisson and continuous distributions, including a distribution specified by its density.
5.06d - Chi-squared goodness of fit testsUse expected frequencies to test whether a proposed probability distribution is compatible with observed data. You will combine adjacent classes, allow for parameters estimated from the sample, choose a critical value and give a conclusion that distinguishes lack of evidence from proof of a model.

Non-parametric tests

5.07a–5.07c - Single-sample non-parametric testsYou can test a population median without fitting a normal distribution. In 5.07a you will choose an appropriate non-parametric test; in 5.07b you will see why signs and ranks provide evidence; in 5.07c you will carry out single-sample sign and Wilcoxon signed-rank tests.
5.07d, 5.07e - Paired-sample and two-sample testsTwo columns of measurements can represent two very different experiments. In 5.07d you will distinguish genuine pairs from independent samples; in 5.07e you will use paired sign and matched-pairs signed-rank tests, or pool independent samples for a Wilcoxon rank-sum test.
5.07f - Normal approximations for Wilcoxon testsLarge rank sums can be tested using a normal approximation rather than a small-sample table. In 5.07f you will choose the correct signed-rank or rank-sum distribution, apply a continuity correction, and make one-sided or two-sided decisions without confusing a raw rank sum with a folded statistic.

Mechanics

Work, energy and power

6.02a–6.02c - Work done by a forceCalculate work by resolving a constant force, by taking a scalar product in two dimensions and by integrating a variable force along a straight line. You will keep the sign of the work and distinguish the work done by one force from the work done by the resultant.
6.02d–6.02f - Kinetic and potential energyCalculate kinetic energy from a speed or a velocity vector, and gravitational potential energy from a chosen reference level. You will interpret mechanical energy, compare energy changes and use vector constant-acceleration equations when the final velocity is not given directly.
6.02g, 6.02h - Hooke's law and elastic energyUse Hooke's law to relate an elastic string's tension to its extension, and distinguish a taut string from a compressed spring. You will derive and calculate elastic potential energy, then combine equilibrium, geometry and elasticity in a two-dimensional system.
6.02i, 6.02j - Energy conservation and the work-energy principleBuild an energy equation from initial and final states, with the right signs for work and energy loss. You will use conservation on smooth tracks, account for resistance through the work-energy principle, and handle spring and elastic-string systems, including a string that starts slack.
6.02k–6.02m - Power and driving forceInterpret power as a rate of work, calculate driving force at a given speed and find the greatest steady speed on a level or inclined road. You will also calculate instantaneous power from a variable force and a velocity vector in two dimensions, keeping the sign of the result.

Impulse and momentum

6.03a–6.03d - Momentum and its conservationUse signed velocities to calculate momentum in one dimension, then extend the idea to vectors in a plane. You will apply conservation of momentum to two particles, recognise the assumptions that make it valid, and distinguish a conserved vector from a conserved speed or kinetic energy.
6.03e–6.03h - Impulse and change in momentumCalculate the impulse in a sudden change of motion, using signed velocities in a line and vectors in a plane. You will include impulsive tension in an inelastic string, distinguish impulse from force, and calculate the impulse of a constant force or a one-dimensional variable force acting over time.
6.03i–6.03k - Restitution and direct impactUse the coefficient of restitution to relate approach and separation speeds in a direct impact. You will combine Newton's experimental law with momentum, handle a smooth fixed wall, connect the extreme coefficients with kinetic energy, and decide whether a sequence of impacts continues.
6.03l - Oblique impactsExtend Newton's experimental law to two dimensions by resolving along and perpendicular to the line of impact. You will predict the velocity after an oblique impact with a smooth fixed plane or another smooth sphere, assemble the result as a vector, and check the direction and impulse.

Centre of mass

6.04a–6.04c - Centres of mass of composite bodiesReplace a body's distributed weight by one force at its centre of mass. Use symmetry and mass-weighted coordinates to locate that point for particles, joined or cut-out laminae, solids and wires, choosing the correct standard centre from the formula booklet.
6.04d - Centres of mass by integrationBuild a centre-of-mass calculation from a thin strip or disc. Find both coordinates of a uniform lamina, including a region between curves, and the axial centre of a uniform solid of revolution. Explain why the mass element changes when the same profile is turned into a solid.
6.04e - Rigid bodies in equilibriumUse force and moment balance for one rigid body under coplanar forces. Find support forces and a suspended body's orientation, then decide whether a body on an incline can remain at rest, slide or topple. Relate every calculation to a clear force diagram and the body's actual centre of mass.

Motion in a circle

6.05a–6.05c - Uniform circular motionA particle can keep the same speed while accelerating all the time. Learn how angular velocity describes its turning, why its acceleration points towards the centre, and how real forces produce horizontal circles. You will model a conical pendulum, a banked track and a particle supported by a hor...
6.05d, 6.05e - Motion in a vertical circleUse energy to find speed on a vertical circular path, then resolve forces towards the centre and along the tangent. You will distinguish radial acceleration from the acceleration that changes speed, interpret a rod's force as tension or compression, and combine perpendicular acceleration components.
6.05f - Leaving a circular pathA circular path lasts only while the constraint can supply the force that it needs. Learn to find when a particle leaves the outside of a smooth circular surface or a string goes slack, then carry the correct position and tangential velocity into free fall. You will also determine the limiting sp...

Discrete Mathematics

Sets and counting

7.01a–7.01c - Problem types, sets and pigeonholesLearn to distinguish four kinds of mathematical problem, describe sets and partitions precisely, and prove that a repetition must occur without finding it. The pigeonhole principle turns a count of objects and categories into a guarantee, even when you know nothing about how the objects were chosen.
7.01d–7.01f - Permutations and combinationsCount outcomes by splitting a choice into stages. You will use factorials for complete arrangements, permutations when order matters and combinations when it does not, and explain exactly why the two latter counts differ.
7.01g, 7.01h - Counting arrangements of objectsCount arrangements when some objects look identical, a number cannot begin with zero, or specified objects must be together or apart. You will also arrange only part of a collection, splitting into cases when repeated objects or different number lengths make a single factorial unsuitable.
7.01i, 7.01j - Selections with constraintsChoose groups whose members must satisfy quotas, minimum requirements or restrictions on who may serve together. You will multiply choices from separate categories, use complements for “at least”, and organise several simultaneous constraints into disjoint cases.
7.01k–7.01m - Inclusion-exclusion and derangementsCount a union without counting overlaps twice, extending inclusion-exclusion from two sets to three or more. Then count small derangements directly: arrangements in which every object is away from its own position, including questions that prescribe an exact number of correctly placed objects.

Graphs and networks

7.02a–7.02c - Graph terminology and notationLearn the language that describes connections without measuring their lengths or angles. In 7.02a you will describe vertices, arcs and degrees; in 7.02b you will recognise simple graphs, connected graphs and trees; in 7.02c you will distinguish the different kinds of route through a g...
7.02d–7.02f - Complete and bipartite graphsRecognise graphs containing every possible connection, and graphs whose vertices can be divided into two sets with connections only between the sets. In 7.02d you will count the arcs of a complete graph; in 7.02e you will describe complete bipartite graphs; in 7.02f you will prove or...
7.02g–7.02i - Eulerian and Hamiltonian graphsDecide when a graph can be traversed using every arc exactly once, and distinguish this from visiting every vertex exactly once. In 7.02g you will use degrees to recognise Eulerian and semi-Eulerian graphs; in 7.02h you will recognise Hamiltonian paths and cycles; in 7.02i you will ap...
7.02j, 7.02k - Isomorphic graphs and digraphsDecide when different drawings encode the same connections, then describe connections that have a direction. In 7.02j you will construct an isomorphism and use structural differences to disprove one; in 7.02k you will work with digraphs, indegrees and outdegrees.
7.02l–7.02o - Planar graphsDecide whether connections can be drawn without crossings, and justify your answer. In 7.02l you will redraw graphs and use subdivision and contraction; in 7.02m you will count regions with Euler's formula; in 7.02n you will identify Kuratowski's obstructions; in 7.02o you will pr...
7.02p–7.02r - Networks, matrices and modellingTranslate a connection problem into a graph or weighted network, then store and use its information in a matrix. In 7.02p you will choose vertices, arcs, directions and weights; in 7.02q you will construct adjacency and weighted matrices; in 7.02r you will solve small problems and int...

Algorithms

7.03a–7.03d - Defining, tracing and timing algorithmsAn algorithm turns a problem into instructions that another person or a computer can follow. You will recognise its essential properties, trace and adapt short instructions, judge their uses and limitations, and estimate how running time grows when the problem gets larger.
7.03e–7.03i - Efficiency and complexityTwo algorithms can reach the same answer with very different amounts of work. You will compare specified operation counts, calculate best, worst and average costs, recognise the hierarchy of growth orders, and use polynomial, logarithmic and exponential models to estimate running times.
7.03j, 7.03k - Sorting algorithmsSorting is more than writing the final ordered list: the route matters. You will carry out OCR's bubble, shuttle and quick sorts, show the correct passes and stopping points, and explain how the input and the choice of pivot affect the work required.
7.03l, 7.03m - Packing algorithmsA packing rule must decide where each item goes, while respecting a capacity. You will use next-fit, first-fit, first-fit decreasing and full-bin packing, distinguish online from offline decisions, and extend the ideas to dimensions and knapsack problems where mass and profit compete.

Network algorithms

7.04a - Dijkstra's algorithmFind a shortest route without trying every possible journey. You will use temporary and permanent labels, recover a route from its distances, adapt the calculation to restrictions, and explain the algorithm's conditions and quadratic order.
7.04b - Minimum spanning treesConnect every location with the least total length of cable. You will construct minimum spanning trees using Prim's graphical and tabular algorithms and Kruskal's algorithm, then adapt a connector to practical restrictions.
7.04c, 7.04d - Bounds for the travelling salespersonConstruct a tour quickly, then judge how much shorter the best tour might be. You will find upper bounds by nearest neighbour, improve feasible routes, calculate lower bounds from reduced minimum spanning trees, and recognise when an incomplete network changes the argument.
7.04e, 7.04f - Route inspection and choosing algorithmsPlan a shortest inspection journey that covers every road, including necessary repetitions. You will pair odd vertices, compare all pairings of up to six odd vertices, count those pairings, adapt the endpoints, and choose an algorithm from the practical objective.

Graphical linear programming

7.06a–7.06c - Formulating linear programmesTurn a practical choice into a linear programme: define variables, write constraints and decide what to maximise or minimise. You will translate ratios, introduce slack variables and investigate numerical cases, so that every line of the model has a clear meaning.
7.06d - Solving linear programmes graphicallyUse a graph to find and interpret the optimum of a linear programme in two variables. You will leave the feasible region unshaded, compare its vertices, handle whole-number solutions and recognise when a problem has several optima or no optimum.
7.06e, 7.06f - Changing coefficients and integer programmingFind when a changed coefficient preserves an optimum and when it changes the answer. Then use continuous relaxations, bounds and branching to prove an integer optimum, including problems where nearby rounded points are not enough.

Additional Pure Mathematics

Sequences and series

8.01a, 8.01b - Recurrence relations and inductionDescribe a sequence either by a rule that produces the next term or by a formula that goes straight to a chosen position. In 8.01a you will move between these descriptions and handle starting indices carefully; in 8.01b you will use induction to prove formulae for sequences and their sums.
8.01c–8.01e - Sequence behaviour and Fibonacci numbersDescribe how a sequence behaves and decide what a claimed limit actually means. In 8.01c–8.01d you will distinguish convergence, divergence, oscillation, periodicity and monotonicity, including sequences of differences or ratios; in 8.01e you will explore Fibonacci and Lucas numbers and the golde...
8.01f - Solving first-order recurrence relationsSolve first-order linear recurrence systems with constant coefficients, turning an update rule into a closed form. You will use an auxiliary equation and a complementary function, add a particular solution for polynomial or exponential forcing, and adapt a trial when it overlaps the complementary...
8.01g - Solving second-order recurrence relationsSolve second-order linear recurrence systems with constant coefficients using auxiliary equations, complementary functions and particular solutions. You will handle distinct real, repeated and complex roots, polynomial and exponential forcing, and the two initial conditions that select one sequence.
8.01h, 8.01i - Modelling with recurrence relationsTurn descriptions of births, deaths, arrivals and delayed effects into recurrence systems, and interpret what their solutions predict. In 8.01h you will build first-order models and use the integer-part function for individual counts; in 8.01i you will extend the modelling to second-order systems...

Number theory

8.02a–8.02d - Bases, divisibility and the division algorithmRead and calculate with numbers in different bases, use familiar divisibility tests, and build a test for an unfamiliar prime. You will also express division precisely using a quotient and a remainder, so that a calculation becomes a statement you can prove.
8.02e–8.02h - Modular arithmetic and congruencesCalculate with remainders, solve linear congruences, and use quadratic residues to settle equations without searching through every integer. You will combine up to three congruences, including systems whose moduli share a factor, and express every solution rather than one successful example.
8.02i–8.02k - Primes, factors and Euclid's lemmaUse prime factorisation to recognise composites and find highest common factors. Then turn common divisors into divisors of small integer combinations, prove coprimality without factorising large numbers, and apply Euclid's lemma with its essential condition.
8.02l–8.02o - Fermat's little theorem and ordersUse both forms of Fermat's little theorem to reduce powers and prove divisibility, while keeping track of what the theorem does and does not establish. Find the order of a number modulo a prime, and use the binomial theorem to explain a powerful prime-modulus identity.

Groups

8.03a–8.03d - Binary operations and group axiomsA group is a set with a way of combining its elements so that every action can be undone. You will test binary operations, construct Cayley tables, use the group axioms and explain the Latin square property. Numbers, functions, matrices and symmetries will turn out to share the same underlying ru...
8.03e–8.03h - Orders, subgroups and cyclic groupsThe order of a group counts its elements; the order of an element counts how many repetitions return to the identity. You will find both, decide whether a subset is a subgroup, recognise cyclic groups and find generators. You will also see how two generators can reach elements that no single gene...
8.03i–8.03k - Group structures and Lagrange's theoremSmall groups have surprisingly few possible structures. You will recognise the groups of orders up to seven, use Lagrange's theorem to restrict subgroup orders, and analyse larger finite groups and infinite groups from their definitions. The key is to turn a statement about the whole group into a...
8.03l, 8.03m - Isomorphism and abstract groupsTwo groups can have different-looking elements but identical multiplication structure. You will establish or rule out an isomorphism, and use group laws to solve equations, prove identities and complete abstract Cayley tables. The reasoning will depend on the stated algebraic properties rather th...

Surfaces and partial differentiation

8.05a–8.05c - Surfaces, sections and contoursRead equations in three variables as surfaces in space. You will connect explicit and implicit equations with their pictures, sketch vertical sections and horizontal contours, and extend the same ideas to surfaces involving trigonometric functions, logarithms and exponentials.
8.05d - First and second partial derivativesDifferentiate a function of two variables while holding one variable fixed. You will use both standard notations, calculate pure and mixed second derivatives, and show that a surface satisfies an identity involving its partial derivatives.
8.05e–8.05g - Stationary points and tangent planesFind the stationary points of a surface and classify them using the Hessian matrix. You will handle coupled equations and an inconclusive determinant, then find tangent planes from the height and the two coordinate gradients.