Edexcel A-Level Further Mathematics
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Core Pure Mathematics
Complex numbers
CP 2.2, 2.3 - Complex number arithmetic and the conjugateThe equation x² = -1 has no real solution, so mathematicians created a number that solves it. In this lesson you build the complex numbers from that one idea: you add, subtract, multiply and divide numbers of the form x + iy (CP 2.2), use the complex conjugate z to make div...
CP 2.1, 2.3 - Complex roots of polynomial equationsSolve quadratics even when their discriminant is negative, then use a known root or factor to solve cubics and quartics completely (CP 2.1). The conjugate-pair result (CP 2.3) turns one non-real root into a real quadratic factor, often saving a great deal of algebra.
CP 2.4 - Argand diagramsAn Argand diagram gives each complex number a position in a plane. You will plot numbers, interpret conjugation and show sums and differences as vectors (CP 2.4), so that the geometry explains the arithmetic.
CP 2.5 - Modulus-argument formDescribe a complex number by its distance from the origin and its direction, then convert between this modulus-argument form and Cartesian form (CP 2.5). You will choose angles in the correct quadrant, keep exact values when possible and recognise the exceptional case at zero.
CP 2.6 - Multiplying and dividing in modulus-argument formMultiply and divide complex numbers by working with their lengths and directions (CP 2.6). You will derive the rules from compound angles, return calculated directions to the principal range and interpret a complex multiplication as a rotation and a change of scale.
CP 2.7 - Loci in the Argand diagramTurn complex equations into geometric sets of points: circles, perpendicular bisectors and rays. This lesson covers the equality loci in CP 2.7; you will interpret their centres, directions and endpoints, and use a diagram to find points satisfying two conditions together.
CP 2.7 - Regions in the Argand diagramInterpret complex inequalities as regions of an Argand diagram: inside or outside circles, nearer to one point than another, and between two argument rays. You will combine conditions using intersections and decide exactly which boundaries belong to the region (CP 2.7).
CP 2.8 - De Moivre's theoremFind integer powers of a complex number by raising its modulus and multiplying its angle. You will see why de Moivre's theorem works, use it for large positive powers and extend it carefully to zero and negative powers.
CP 2.8 - Multiple-angle formulaeUse de Moivre's theorem to turn the sine, cosine or tangent of a multiple angle into powers of trigonometric functions of the original angle. You will choose real or imaginary parts, simplify the resulting polynomial and keep the restrictions that matter when dividing.
CP 2.8 - Powers of sine and cosineTurn powers of sine and cosine into sums of multiple-angle terms by expanding reciprocal pairs of complex numbers. You will also express powers of tangent using multiple angles and use the expansions to integrate trigonometric powers.
CP 2.8 - Sums of series with de MoivreSum trigonometric series by treating their cosine and sine terms as the real and imaginary parts of one complex geometric progression. You will derive finite-sum formulae, prove a useful half-circle identity and handle geometric weights as well as unit-length terms.
CP 2.9 - Exponential form of a complex numberAn exponential can describe a turn as neatly as it describes growth. You will use the definition of e^(iθ) to write complex numbers in exponential form, calculate with them, and recover sine and cosine from pairs of exponentials.
CP 2.10 - nth roots of a complex numberTaking a complex root produces a whole set of answers, arranged with striking regularity. You will find all the distinct solutions of zⁿ=a for non-zero a, give the answers in a requested form, and explain why their Argand diagram is a regular polygon.
CP 2.11 - Roots of unity in geometric problemsRoots of unity give you a ready-made regular polygon, which you can turn, enlarge and move using complex arithmetic. You will use this structure to find lengths and areas, describe transformed polygons, and recover missing vertices from geometric information.
Matrices
CP 3.1, 3.2 - Matrix arithmetic, zero and identity matricesA matrix keeps a whole collection of numbers in a useful order. You will add, subtract, scale and multiply matrices (CP 3.1), then use the zero and identity matrices to see which familiar algebraic rules survive and which need care (CP 3.2).
CP 3.3 - Matrices for 2-D transformationsA transformation matrix turns a geometric instruction into a calculation on coordinates. You will build and recognise matrices for reflections, rotations, stretches and enlargements in two dimensions, and use the images of two simple vectors to explain their entries.
CP 3.3 - Successive and 3-D transformationsTwo transformations combine by matrix multiplication, with the first action written on the right. You will explain that order, calculate matrices for successive transformations, and extend the column-image method to reflections in coordinate planes and rotations about coordinate axes in three dim...
CP 3.4 - Invariant points and linesA transformation can move points while leaving a whole line in the same place. You will distinguish invariant points from invariant lines, find each algebraically, and check vertical lines and non-zero intercepts so that your answer includes every possibility.
CP 3.5 - Determinants and scale factorsA matrix moves points, but its determinant tells you what happens to an entire shape. You will calculate determinants of two by two and three by three matrices, then interpret their size and sign as information about area, volume and orientation.
CP 3.6 - Singular and inverse matricesAn inverse matrix reverses a transformation, but only when the original transformation has kept enough information. You will recognise singular matrices, calculate two by two and three by three inverses, and use inverse properties without losing the order of multiplication.
CP 3.7 - Solving three equations with matricesThree simultaneous linear equations can be written as one matrix equation. You will use an inverse matrix to recover the three unknowns, check the result in the original equations, and solve systems whose coefficients contain a parameter.
CP 3.8 - How three planes intersectEach linear equation in three variables describes a plane, so solving three equations means finding points that lie on three planes at once. You will distinguish a unique intersection, a common line, and inconsistent arrangements, including a prism, then use algebra to justify the geometry.
Proof
CP 1.1 - Induction for sums of seriesA sum can follow a pattern for thousands of terms without that pattern being proved. You will use mathematical induction to turn a promising formula into a proof for every positive integer, then make the algebra of the next term do the work.
CP 1.1 - Induction for divisibilityDivisibility proofs let you say something exact about numbers far too large to write down. You will use the induction structure you already know, express the assumed case as an integer multiple and rearrange the next case until that multiple appears.
CP 1.1 - Induction for powers of matricesA formula for a matrix power can replace thousands of multiplications with one substitution. You will prove such formulae by induction, using the whole assumed matrix in one further row-by-column product, then use the proved result to calculate or determine a power.
Further algebra and functions
CP 4.1 - Roots and coefficients of polynomialsA polynomial's coefficients contain information about all its roots together, even when the individual roots are awkward. You will extract sums and products for equations up to degree four, then use them to evaluate expressions involving powers, reciprocals and shifted roots.
CP 4.2 - Equations with transformed rootsYou can move every root of a polynomial without finding the roots individually. You will form equations for shifted and scaled roots by reversing the transformation, substituting into the original equation and clearing numerical denominators.
CP 4.3 - Sums of integers, squares and cubesThree standard sums can handle a much wider family of series than their names suggest. You will expand polynomial summands into integers, squares and cubes, then adjust the limits when a series starts part-way along the sequence.
CP 4.4 - The method of differencesSome long sums simplify because almost all their terms cancel. You will expose that cancellation with partial fractions, keep the correct terms at both ends and use a finite partial sum to decide whether an infinite series has a sum.
CP 4.5 - Maclaurin series and the general termA function's derivatives at zero can reveal an entire power series. You will build Maclaurin expansions from those derivative values, distinguish a truncated approximation from the full series and describe the general term, including when many coefficients vanish.
CP 4.6 - Standard and compound Maclaurin seriesStandard expansions let you build new series without repeatedly differentiating a complicated function. You will choose and adapt the five main series, keep their validity conditions and derive expansions of products and functions inside other functions.
Further calculus
CP 5.1 - Volumes of revolution from Cartesian equationsRotating a region around an axis builds a solid from circular slices. You will derive the volume formulae for rotation about both coordinate axes, choose the radius and limits correctly, and subtract inner discs when the solid has a hole.
CP 5.1 - Volumes of revolution from parametric equationsA parametric curve supplies the radius and the axial position separately. You will convert both volume formulae into the parameter, handle decreasing coordinates without a negative volume, and calculate volumes for a curved body joined to a cylindrical section.
CP 5.2 - Improper integralsAn infinite interval or an undefined integrand needs a limiting process before an integral can have a value. You will evaluate improper integrals at infinite and finite endpoints, split at interior singularities, and decide whether each required limit converges.
CP 5.3 - Mean value of a functionThe mean value of a function is its average height across an interval. You will calculate it using an integral, interpret positive and negative contributions, and use a stated mean to find an unknown constant.
CP 5.5 - Differentiating inverse trigonometric functionsInverse trigonometric functions turn ratios back into angles. You will derive their derivatives, use the chain and product rules in compound expressions, and keep track of the branches and domains that make the answers valid.
CP 5.6 - Integrals giving inverse trigonometric functionsSome algebraic integrands have inverse trigonometric antiderivatives. You will recognise the arcsine and arctangent forms, choose sine or tangent substitutions for related integrals, and distinguish these from a reciprocal difference of squares, which gives a logarithm.
CP 5.4 - Partial fractions with quadratic factorsA quadratic denominator can produce both a logarithm and an inverse tangent. You will choose the correct partial-fraction numerators, integrate the resulting terms, and handle improper fractions and exact definite integrals.
Further vectors
CP 6.1 - Equations of lines in 3-DA straight line in space needs a point and a direction. You will describe it in vector and Cartesian forms, handle directions with a zero component, and decide whether two lines meet, run parallel or miss each other in three dimensions.
CP 6.2 - Equations of a planeTwo independent directions let you move anywhere on a flat surface. You will construct a plane through three points, convert between its vector and Cartesian descriptions, and recognise when the information describes only a line.
CP 6.3, 6.4 - The scalar product and normal vectorsThe scalar product turns two vectors into a number that measures their alignment. You will calculate it, use zero scalar products to test perpendicularity (CP 6.4), and express a plane using a normal vector and the equation r.n=k (CP 6.3).
CP 6.3 - Angles between lines and planesOne scalar-product calculation can describe several different angles, so choosing the right vectors matters. You will find angles between two lines, between two planes, and between a line and a plane, including the complementary angle that is easy to miss.
CP 6.5 - Where a line meets a planeSubstituting a line into a plane reduces a spatial intersection to one equation. You will find the intersection point, distinguish a parallel line from one contained in the plane, and determine constants from a specified intersection condition.
CP 6.5 - Perpendicular distances to lines and planesThe shortest join to a line or plane is perpendicular, but the points where it lands are often unknown. You will locate those feet using scalar products, calculate point-to-line and point-to-plane distances, and find the shortest distance between skew or parallel lines.
Polar coordinates
CP 7.1 - Converting between polar and CartesianDescribe a point by its distance and direction from a fixed origin, then convert between that description and Cartesian coordinates. You will choose angles in the required range and translate simple curve equations without losing their domain restrictions.
CP 7.2 - Sketching polar curvesSketch the standard polar curve families by combining admissible angles, symmetry and carefully chosen radius values. You will distinguish lines, circles, spirals, cardioids, dimpled curves and two different kinds of paired loops, using non-negative radius throughout.
CP 7.3 - Areas enclosed by polar curvesFind exact polar areas by adding narrow sectors, choosing an interval that sweeps the intended region once. You will integrate whole curves and individual loops, then compare boundaries to find areas between curves.
CP 7.3 - Tangents to polar curvesFind tangents parallel or perpendicular to the initial line by differentiating Cartesian components of a polar curve. You will check both parametric derivatives, find every valid angle and handle tangents at the pole using the direction of approach.
Hyperbolic functions
CP 8.1 - The functions sinh, cosh and tanhBuild three new functions from exponentials, understand their domains, ranges and graphs, and use their identities to solve exact equations (CP 8.1). Their names resemble trigonometric functions, but their behaviour comes from exponentials rather than repeated motion round a circle.
CP 8.2 - Hyperbolic derivatives and integralsDifferentiate and integrate hyperbolic functions, combining their standard results with the chain, product and quotient rules (CP 8.2). You will see why both fundamental derivatives have positive signs and recognise integrals whose inner derivative is already present.
CP 8.3, 8.4 - Inverse hyperbolic functions as logarithmsDefine the inverse hyperbolic functions with their correct domains and ranges (CP 8.3), then derive and use their logarithmic forms (CP 8.4). The logarithms make exact values accessible, while branch restrictions explain which solutions are allowed.
CP 8.5 - Integrals giving inverse hyperbolic functionsRecognise reciprocal square-root integrals giving arsinh or arcosh, and choose hyperbolic substitutions for associated integrals (CP 8.5). Completing the square reveals the structure; the real domain and the sign of a square root determine the correct branch.
Differential equations
CP 9.1 - The integrating factor methodA carefully chosen multiplier turns a first-order linear differential equation into one product derivative. You will recognise when the integrating factor method applies, put the equation into the right form, and use constant, algebraic and trigonometric factors to obtain its solution.
CP 9.2 - General and particular solutionsA differential equation usually describes a family of curves, while an initial condition selects a member. You will find both kinds of solution and sketch representative curves using intercepts, gradients, stationary points and limiting behaviour.
CP 9.3 - Modelling with differential equationsA differential equation turns a statement about rates into a prediction over time. You will form and solve first-order models for mixing, motion and cooling, use initial or measured data, and interpret what their solutions say about limits and the assumptions behind them.
CP 9.4, 9.6 - Second-order homogeneous equationsAn auxiliary quadratic turns a second-order differential equation into a root-finding problem. You will solve the homogeneous equation in CP 9.4, use the discriminant to choose the correct solution form in CP 9.6, and apply two conditions to select a particular solution.
CP 9.5 - Complementary functions and particular integralsA non-zero right-hand side supplies an input to a differential equation. You will add a particular integral to the complementary function, choose trials for polynomial, exponential and trigonometric inputs, and adjust a trial when it overlaps the complementary function.
CP 9.7 - Simple harmonic motionSimple harmonic motion is governed by an acceleration directed towards equilibrium and proportional to displacement. You will solve its differential equation, use initial displacement and velocity, and connect the solution to amplitude, period, speed and turning points.
CP 9.8 - Damped and forced oscillationsResistance changes both the shape and long-term behaviour of an oscillator's motion. You will model resistance proportional to velocity, distinguish light, critical and heavy damping, and solve forced-motion models by combining a transient response with a steady response.
CP 9.9 - Coupled first-order equationsTwo changing quantities can affect each other's rates of change. You will build and interpret linear coupled models, eliminate one variable to obtain a second-order equation, and recover the other variable with the same two linked constants.
Further Pure Mathematics 1
Further trigonometry
FP1 1.1 - The t-formulaeDerive and use the half-angle formulae that express sine, cosine and tangent as rational functions of t=(θ/2) (FP1 1.1). You will track their domains and recover the correct angles, including the points the substitution cannot represent.
FP1 1.2, 1.3 - Identities and equations with t-formulaeUse the t-formulae to prove trigonometric identities (FP1 1.2) and convert equations into algebra (FP1 1.3). You will preserve domain restrictions through cancellation, recover angles in their stated ranges and check the half-angle cases that no finite parameter represents.
Further calculus
FP1 2.1 - Taylor seriesBuild a power series around a point of your choice, rather than always around zero. You will derive the Taylor coefficients, expand explicit functions and obtain a local series from a differential equation without first solving it.
FP1 2.2 - Limits using series expansionsEvaluate limits by expanding far enough to reveal the first terms that survive cancellation. You will compare orders of vanishing, handle products of expansions and shift the expansion variable when the limit is taken at a non-zero point.
FP1 2.3 - Leibniz's theoremDifferentiate a product many times using binomial coefficients instead of repeatedly expanding the product rule. You will calculate higher derivatives, exploit polynomial factors that eventually disappear and use a general derivative to solve a high-order condition.
FP1 2.4 - L'Hospital's ruleEvaluate indeterminate limits by differentiating the numerator and denominator separately. You will check when the rule applies, repeat it when necessary and turn differences, products and powers into suitable quotients.
FP1 2.5 - The Weierstrass substitutionTurn rational expressions in sine and cosine into rational functions of t=(x/2), then integrate using familiar algebra. You will handle indefinite and definite integrals, retain original exclusions and split intervals when the half-angle tangent jumps.
Further differential equations
FP1 3.1 - Series solutions of differential equationsConstruct a local Taylor solution from a differential equation and its initial conditions, even when an explicit solution is unavailable (FP1 3.1). You will generate higher derivatives, expand about the correct centre and use Leibniz's theorem to turn repeated differentiation into a recurrence.
FP1 3.2 - Solving differential equations by substitutionUse a given substitution to turn an unfamiliar differential equation into a first-order linear or constant-coefficient second-order equation (FP1 3.2). You will transform derivatives and initial conditions, solve using Core Pure methods and return to the original variables with the correct domain.
Coordinate systems
FP1 4.1 - Parabolas, ellipses and hyperbolasRecognise and use Cartesian and parametric forms of the parabola, ellipse, hyperbola and rectangular hyperbola. You will eliminate parameters, identify key features and choose parameter domains that trace the intended curve or branch.
FP1 4.2 - Foci, directrices and eccentricityDescribe a conic using the ratio of distances from a focus and a directrix. You will connect eccentricity with the Cartesian parameters, find foci and directrices, and use shared geometric information to determine unknown conics.
FP1 4.3 - Tangents and normals to conicsUse algebraic differentiation to find tangents and normals to the four standard conics. You will handle vertical tangents, derive the conditions for a line to touch a conic and find tangents through an external point.
FP1 4.4 - Loci problems with conicsFind the path of a point built from a moving conic point or its tangents. You will eliminate parameters, prove which points are attainable and keep the restrictions that an equation alone can hide.
Further vectors
FP1 5.1 - The vector productThe vector product makes a perpendicular vector from two directions, while its magnitude measures an area (FP1 5.1). You will choose its orientation, calculate its components and use it to find exact areas in three dimensions.
FP1 5.2 - Scalar triple product and volumesCombine a vector product with a scalar product to measure volume (FP1 5.2). You will calculate the signed scalar triple product, distinguish it from geometric volume and find volumes of parallelepipeds and tetrahedra, including degenerate cases.
FP1 5.3 - Points, lines and planes in 3-DUse the vector product to describe lines, construct plane normals and calculate perpendicular distances (FP1 5.3). You will also use direction ratios and direction cosines, including the cases where more than one line fits the angle information.
Further numerical methods
FP1 6.1 - Numerical solution of differential equationsReplace derivatives by finite differences to approximate first- and second-order differential equations (FP1 6.1). You will use forward and central differences, generate solution values on a grid and distinguish starting-value calculations from boundary-value equations.
FP1 6.2 - Simpson's ruleEstimate definite integrals by fitting parabolas through equally spaced ordinates (FP1 6.2). You will apply Simpson's rule with the correct weights, check that the grid is valid and reuse an estimate through algebraic relationships between integrals.
Further Pure Mathematics 2
Number theory
FP2 5.1, 5.2 - The Euclidean algorithm and Bezout's identityUse division with remainder to find highest common factors and express them as integer combinations (FP2 5.1, 5.2). You will run the Euclidean algorithm forwards, then use its own equations backwards.
FP2 5.3 - Modular arithmetic and congruencesCalculate with remainders using congruence notation (FP2 5.3). You will use the equivalence, addition, subtraction, multiplication and power laws, while keeping the modulus fixed and checking before cancelling.
FP2 5.4 - Fermat's little theoremUse Fermat's little theorem to reduce large powers modulo a prime (FP2 5.4). You will check its conditions, reduce exponents, use convenient signed bases and finish with the requested residue.
FP2 5.5 - Divisibility testsDerive and use the decimal divisibility tests for 2,3,4,5,6,9,10,11. Choose a suitable test, justify it with congruences, and use digit restrictions without confusing a divisibility test with ordinary equality.
FP2 5.6 - Solving congruence equationsDecide whether a linear congruence has solutions, find multiplicative inverses using Bezout's identity, and solve with the correct modulus. Describe every solution, including the multiple residue classes that occur when the coefficient and modulus are not coprime.
FP2 5.7 - Counting principles and subsetsUse set notation and the multiplication, addition and subtraction principles to count outcomes. Count subsets, and choose between direct counting and counting a complement without omitting or repeating cases.
FP2 5.7 - Permutations and combinationsDistinguish ordered selections from unordered selections, use permutations and combinations, and combine them with restrictions and case counts. Explain why each arrangement is counted once, especially when a code's characters can move between positions.
Groups
FP2 1.1 - Binary operations and group axiomsTest a set and an operation against the group axioms (FP2 1.1). You will check closure, associativity, an identity and inverses, and distinguish changing brackets from changing order.
FP2 1.2 - Cayley tables and cyclic groupsRecognise groups in modular arithmetic, symmetry, matrices and permutations (FP2 1.2). You will construct and read Cayley tables, compose operations in the right order and test whether one element generates a whole group.
FP2 1.3 - Orders of elements and subgroupsFind the order of a group and of an element, and recognise subgroups (FP2 1.3). You will locate the first return to the identity, test a subset using the original operation and construct the subgroup generated by one element.
FP2 1.4 - Lagrange's theoremUse Lagrange's theorem to restrict subgroup and element orders (FP2 1.4). You will see why subgroup sizes divide a finite group size, separate necessary conditions from existence claims, and explain why a group of prime order must be cyclic.
FP2 1.5 - Isomorphic groupsDecide whether small groups have the same structure and construct an isomorphism when they do (FP2 1.5). You will preserve every product, use element orders to reject impossible matches and connect cyclic groups, symmetry, matrices and permutations.
Further calculus
FP2 2.1 - Reduction formulaeDerive and use reduction formulae, keeping endpoint terms, index ranges and starting values under control. Connect an integral with a simpler member of the same family instead of expanding a high power or starting the integration again.
FP2 2.2 - Arc length and surface areaCalculate arc length and curved surface area from Cartesian, parametric and polar equations. Choose a once-only trace, identify the distance from the rotation axis, and retain the correct positive square root and integration bounds.
Further matrix algebra
FP2 3.1 - Eigenvalues and eigenvectorsFind the directions that a matrix stretches without turning away from their own line. Calculate eigenvalues and eigenvectors of two by two and three by three matrices, including zero, repeated and complex eigenvalues, and give normalised vectors when requested.
FP2 3.2 - Diagonalising matricesBuild a diagonal form from independent eigenvectors, keeping columns and eigenvalues in the same order. Recognise when diagonalisation fails, construct orthogonal diagonalisation for real symmetric matrices, and use a diagonal form to calculate powers and inverses.
FP2 3.3 - The Cayley-Hamilton theoremUse a matrix's characteristic equation as an identity involving the matrix itself. Reduce high powers to lower ones and find inverses when they exist, for two by two and three by three matrices, including matrices with repeated or complex eigenvalues.
Further complex numbers
FP2 4.1 - Further loci and regionsFind complete complex loci and regions involving distance ratios, argument quotients and intersecting restrictions (FP2 4.1). You will select the correct circle or arc, identify boundary intersections and retain every domain exclusion.
FP2 4.2 - Transformations from z-plane to w-planeFind complete images of points, loci and regions under complex transformations (FP2 4.2). You will square in Cartesian or polar form, invert fractional transformations and preserve poles, missing points and boundary restrictions.
Further sequences and series
FP2 6.1, 6.2 - First-order recurrence relationsTurn a rule for the next term into a formula for any term (FP2 6.1, 6.2). You will combine complementary functions and particular solutions, handle changing coefficients, and interpret a population model.
FP2 6.1, 6.2 - Second-order recurrence relationsSolve second-order recurrence relations using auxiliary roots, complementary functions and particular solutions (FP2 6.1, 6.2). You will handle all three root types, repeated-root overlap and a population model with a one-year delay.
FP2 6.3 - Proving closed forms by inductionProve that a proposed formula gives every term of a recurrence (FP2 6.3). You will choose enough base cases, state the right inductive assumptions and use the recurrence to build the next closed form.
Further Statistics 1
Poisson and binomial distributions
FS1 2.1 - The Poisson distributionRandom arrivals do not have to arrive in a regular pattern for their counts to have a useful model. You will decide when a Poisson distribution is reasonable, calculate point and cumulative probabilities and keep the endpoints of discrete probability events exact.
FS1 2.1 - Adding independent Poisson variablesIndependent Poisson counts combine by adding their parameters. You will use that property across time intervals and separate event streams, distinguish a new total from a scaled old count and calculate probabilities when a combined total is already known.
FS1 2.2, 2.3 - Means, variances and the Poisson approximationA binomial and a Poisson model can have the same mean while predicting different spreads and probabilities. You will use their mean and variance formulae (FS1 2.2), recover binomial parameters and justify when a Poisson distribution is a useful approximation (FS1 2.3).
Geometric and negative binomial distributions
FS1 3.1 - Geometric and negative binomial probabilitiesSome experiments stop when a success arrives, rather than after a fixed number of trials. You will model the trial of the first or a later success, calculate exact and cumulative probabilities, and see why a long run of failures does not make the next success more likely under an independent-tria...
FS1 3.2, 3.3 - Means and variances of both distributionsWaiting times can have a much larger spread than their most likely value suggests. You will use the geometric mean and variance (FS1 3.2), use the corresponding negative binomial results (FS1 3.3), recover parameters from moments, and translate requirements about an average wait into conditions o...
Hypothesis testing
FS1 4.1 - Testing a Poisson meanA surprisingly large or small count can challenge a claimed Poisson mean. You will choose hypotheses, calculate the correct tail, construct discrete critical regions and check their actual significance levels (FS1 4.1).
FS1 4.2 - Testing a geometric parameterA long wait for success can challenge a claimed success probability. You will test the population parameter of a geometric distribution, choose the correct tail, construct integer critical regions and check their actual significance levels (FS1 4.2).
Chi-squared tests
FS1 6.1 - Goodness of fit testsA model predicts a whole pattern of frequencies, not just a mean. You will compare observed and expected counts using a chi-squared goodness of fit test, pool small expected cells and adjust the degrees of freedom when parameters are estimated from the data.
FS1 6.1 - Contingency table testsTwo categorical variables can be associated even when neither is numerical. You will calculate expected frequencies under independence, carry out a chi-squared contingency table test, combine sparse categories appropriately and explain why real counts cannot be replaced by percentages.
Probability generating functions
FS1 7.1 - Generating functions of standard distributionsA probability generating function stores an entire discrete distribution in one expression. You will construct and check these functions, derive the four standard forms, recover exact probabilities and transform the values of a random variable (FS1 7.1).
FS1 7.2 - Mean and variance from generating functionsDifferentiating a PGF turns its probability coefficients into moments. You will justify the mean and variance formulae, prove the four standard distributions' moment results and use calculus with less familiar PGFs (FS1 7.2).
FS1 7.3 - Generating functions of sumsIndependent observations combine by multiplying their PGFs. You will extract sum probabilities, recognise distributions after simplifying products, and handle shifted or weighted totals with the correct independence assumptions (FS1 7.3).
Quality of tests
FS1 8.1 - Type I and Type II errorsA statistical decision can be wrong in two different ways: rejecting a true claim or missing a real change. You will identify each error in context and calculate its probability using the correct population model and the test's fixed rejection rule.
FS1 8.1 - Size and power of a testAn effective test must control false alarms while detecting changes that matter. You will calculate size and power, derive power functions, choose a feasible rejection rule and compare tests using detection, error rates and sampling cost.
Further Statistics 2
Linear regression
FS2 1.1 - Least squares regression linesA regression line is chosen by a precise measure of its errors, rather than by eye. You will explain least squares, calculate the regression of a response on an explanatory variable and recover a model when the data have been coded.
FS2 1.2 - Residuals and residual sum of squaresA fitted line needs checking as well as calculating. You will interpret signed residuals, calculate the residual sum of squares, use residual patterns to question a model, investigate possible outliers and compare fits on a consistent response scale.
Continuous probability distributions
FS2 2.1, 2.2 - Density and cumulative distribution functionsA continuous model assigns probabilities to intervals by measuring area. You will use a probability density function to find probabilities, construct a cumulative distribution function with the correct branches, and move between the two functions by integration and differentiation.
FS2 2.3 - Mean and variance of continuous variablesA density describes the whole distribution; its mean and variance summarise location and spread. You will calculate these summaries by integration, find the expected value of a function of a continuous variable, and use your results to judge how well a model describes observations.
FS2 2.3 - Mode, median, percentiles and skewnessA peak, a halfway point and a long tail describe different features of a distribution. You will locate modes from a density, find medians and percentiles from a CDF, and justify whether a continuous distribution has positive, negative or zero skewness.
FS2 2.4 - The continuous uniform distributionA uniform density gives equal probabilities to equal-length intervals within its support. You will derive its CDF, mean and variance, use interval lengths for probabilities and conditional probabilities, and calculate expectations of quantities that depend on a uniform random measurement.
Correlation
FS2 3.1 - Product moment correlation coefficientThe product moment correlation coefficient measures linear association on a scale that does not depend on ordinary measurement units. You will calculate it from summaries, recognise when it is informative and track the effect of coding either variable.
FS2 3.2 - Spearman's rank correlationRanks compare ordering without treating score differences as equally meaningful. You will calculate Spearman's coefficient with and without ties, keep objects correctly paired and interpret what agreement in rank does and does not establish.
FS2 3.3 - Testing for zero correlationA sample coefficient is evidence about a population, not a conclusion by itself. You will state zero-correlation hypotheses, select the correct Pearson critical-table column and complete one- or two-tailed tests using PMCC or Spearman's coefficient.
Estimation and tests using a normal distribution
FS2 5.1 - Estimators, bias and standard errorA sample gives one answer, but an estimator is a rule that would produce different answers from fresh samples. You will distinguish these ideas, calculate bias and standard error, estimate a population variance with the correct divisor, and compare the quality of estimators.
FS2 5.2, 5.3 - Confidence intervals for a normal meanA confidence interval turns a sample mean into a range of plausible population means. You will interpret confidence correctly, calculate intervals when the population variance is known, recover information from an interval, plan a sample size and connect an interval with a two-sided hypothesis test.
FS2 5.4, 5.5 - Testing the difference between two meansTwo sample means vary even when their population means have a stated difference. You will test that difference using independent normal samples with known variances, choose the correct tail, and extend the method to large samples with estimated variances.
Tests and confidence intervals for variances
FS2 6.1 - Testing and estimating a normal varianceA process can keep its average while becoming much less consistent. You will test a normal population variance with a chi-squared statistic, find critical regions and construct confidence intervals for variance, keeping standard deviations and squared units distinct.
FS2 6.2 - The F-test for two variancesTwo processes can have similar average outputs but very different consistency. You will compare two normal population variances with an F-test, keep the two degrees of freedom in order and select one- or two-tailed critical regions without confusing variance with standard deviation.
Confidence intervals and tests using the t-distribution
FS2 7.1 - A normal mean with unknown varianceAn estimated standard deviation introduces uncertainty of its own. You will use the t distribution to test a normal population mean and construct confidence intervals when the population variance is unknown, using the unbiased sample variance and the correct tail.
FS2 7.2 - The paired t-testWhen observations belong together, analyse their differences together. You will recognise paired data, test a population mean difference and construct an interval for it, using the number of pairs and the spread of the differences.
FS2 7.3 - Two-sample t-tests with pooled varianceTwo independent samples can estimate one common variance while comparing different population means. You will pool their within-sample variation, test a mean difference and construct its confidence interval under the normal and equal-variance assumptions.
Further Mechanics 1
Momentum and impulse
FM1 1.1 - Impulse and conservation of momentumA collision can change a velocity in a fraction of a second. Momentum lets you calculate that change without knowing every detail of the contact force. You will use signed velocities, connect impulse to force and time, and find unknown velocities when two spheres collide directly.
FM1 1.2 - Momentum as a vectorA particle can keep the same speed and still receive a substantial impulse. You will retain both velocity components when finding momentum changes, recover velocities from vector impulses, and use an impulse's magnitude to find possible final motions. You will also apply equal and opposite impuls...
Work, energy and power
FM1 2.1 - Work-energy principle and energy conservationEnergy can replace a long calculation of forces and accelerations with one equation linking the start and finish of a journey. You will calculate signed work, use kinetic and gravitational potential energy, and decide when mechanical energy is conserved. You will also handle friction on slopes an...
FM1 2.1 - Power and driving forcePower tells you how quickly a force transfers energy. You will connect power to driving force and speed, then combine that relationship with Newton's second law on level and inclined roads. You will also find steady speeds when resistance changes with speed and explain the limits of a constant-po...
Elastic strings, springs and elastic energy
FM1 3.1 - Hooke's law for strings and springsAn elastic force depends on how far a string or spring has changed from its natural length. You will use Hooke's law to calculate that force, find unknown elastic parameters, and combine it with equilibrium or Newton's second law. You will distinguish a slack string from a compressed spring throu...
FM1 3.2 - Elastic potential energyAn elastic string or spring can store energy and release it into motion. You will derive its stored energy from Hooke's law, track the difference between two elastic states, and combine elastic energy with kinetic energy, height changes and friction. You will use those balances to find speeds and...
Elastic collisions in one dimension
FM1 4.1 - Newton's law of restitutionConservation of momentum alone usually leaves two unknown velocities after a collision. Newton's law of restitution supplies the missing relationship through relative speeds. You will combine the two laws, use the permitted coefficient range to interpret directions, and calculate the total kineti...
FM1 4.2 - Successive direct impactsA sequence of impacts is a sequence of short problems, each with its own immediate before-and-after velocities. You will handle a direct impact with a fixed wall, carry velocities into later sphere collisions, and decide whether two spheres meet again. You will also combine constant-speed travel...
Elastic collisions in two dimensions
FM1 5.1 - Oblique impact with a fixed surfaceResolve an oblique collision into components parallel and perpendicular to a smooth fixed surface (FM1 5.1). You will recover rebound velocities, infer an unknown wall direction and coefficient of restitution, and calculate impulse and kinetic-energy loss.
FM1 5.1 - Oblique impact of two spheresResolve an oblique sphere collision along and perpendicular to the line of centres (FM1 5.1). You will combine momentum and restitution, retain each sphere's tangential component, calculate kinetic-energy loss and use a prescribed deflection to constrain the coefficient.
FM1 5.2 - Successive oblique impactsApply the smooth-wall component laws one collision at a time (FM1 5.2). You will check the order of impacts, change tangent-normal frames when surfaces are not perpendicular, and calculate the final motion and combined energy loss.
Further Mechanics 2
Motion in a circle
FM2 1.1 - Angular speed and horizontal circlesA particle can have constant speed and still accelerate: its direction keeps changing. You will connect angular speed to speed along a circle, find the inward acceleration, and use actual forces to produce it. You will also find the speed limits imposed by a string or by friction.
FM2 1.1 - Conical pendulums and banked surfacesHorizontal circles often hide inside three-dimensional arrangements. You will find the actual circle radius, resolve forces vertically and towards its centre, and handle conical pendulums, elastic strings and banked surfaces. You will also check when contact, friction or two taut strings can supp...
FM2 1.2 - Motion in a vertical circleIn a vertical circle, gravity changes the speed as well as the force needed to turn the particle. You will combine energy with radial and tangential equations, find complete-circle conditions, and decide whether a particle turns back, a string becomes slack, or contact with a surface ends. Each c...
Centres of mass of plane figures
FM2 2.1 - Centre of mass of particlesA collection of particles has one point that summarises how its mass is distributed. You will locate that centre of mass in one and two dimensions by matching moments, and use a prescribed centre to find an unknown mass. You will also distinguish a force's perpendicular moment arm from the distan...
FM2 2.2 - Composite laminas and frameworksA plate or a framework can be replaced by a collection of simpler pieces, each acting at its own centre of mass. You will use symmetry and the formula booklet, combine pieces with different densities, and account for removed material. You will also distinguish mass proportional to area from mass...
FM2 2.2 - Equilibrium of laminas and frameworksKnowing a body's centre of mass lets you replace its distributed weight by one force at one point. You will combine force balance with moments to find support reactions, hanging angles and forces that hold a lamina or framework in a prescribed position. You will distinguish what a smooth support,...
Further centres of mass
FM2 3.1 - Centres of mass by integrationIntegration extends mass-weighted averages to a continuous body. You will choose a mass element for a wire, lamina, solid or thin shell, integrate its mass moment, and use symmetry to remove unnecessary coordinates. You will also see why the same outline can give different centres when its densit...
FM2 3.1 - Composite rigid bodiesA composite body may join solids of different densities, combine a shell with a solid, or contain a cavity. You will replace each piece by its mass at its own centre, express every centre in one coordinate system, and combine signed mass moments. You will also work backwards from a required centr...
FM2 3.2 - Equilibrium of rigid bodiesA rigid body can balance its forces yet still have a turning effect. You will use its centre of mass to solve suspension and contact problems, taking moments as well as resolving forces. You will also check that a calculated contact force can actually be supplied by the supporting surface.
FM2 3.3 - Toppling and slidingA supported rigid body can lose equilibrium because the available friction is too small or because the required contact force reaches the edge of its base. You will calculate both boundaries, decide which occurs first, and check that the proposed event is physically possible. You will work with i...
Further dynamics
FM2 4.1 - Variable forces and gravitationA force may change with time, speed or position, so Newton's second law produces a differential equation rather than a constant acceleration. You will build that equation from a signed force diagram, choose a suitable acceleration form, and use the initial motion to select a physical solution. In...
FM2 4.2 - Simple harmonic motion of a particleCore Pure supplied the differential-equation solution for SHM. Here you will establish it from a force model, identify an unknown oscillation from measured motion, and choose times using the actual starting direction. You will also calculate durations within a cycle and connect the speed relation...
FM2 4.2 - Oscillations on strings and springsElastic oscillations begin with a force law and a carefully chosen equilibrium. You will derive the elastic results needed here, prove SHM for horizontal and vertical systems, combine two restoring forces, and use physical energy contributions. You will also identify when a string becomes slack a...
Decision Mathematics 1
Algorithms and graph theory
D1 1.1 - Algorithms and their orderAn algorithm is a precise set of instructions that turns inputs into outputs. You will follow instructions given in text or a flowchart, record changing values in a trace table, and use an algorithm's order to estimate how its running time grows.
D1 1.2 - Bubble sort and quick sortSorting algorithms reach an ordered list by different sequences of comparisons. You will carry out bubble sort and Pearson's middle-pivot quick sort, record complete passes, and distinguish comparisons, swaps and order of growth.
D1 1.2 - Bin packing algorithmsBin packing places items into containers of fixed capacity without splitting an item. You will apply first-fit and first-fit decreasing, distinguish a heuristic packing from an optimum, and infer a capacity from the placements an algorithm makes.
D1 1.3 - Graph terminology and Eulerian graphsYou will describe graphs accurately, count node orders, decide whether every arc can be traced exactly once, and recognise complete, planar and isomorphic graphs.
D1 1.4 - The planarity algorithmYou will recognise Hamiltonian cycles and apply the planarity algorithm by assigning the remaining arcs to two lists. You will justify both a successful assignment and a contradiction.
Algorithms on graphs
D1 2.1 - Minimum spanning treesYou will find a minimum connector using Kruskal's and Prim's algorithms, show the order of selected arcs, and move between a weighted network and its matrix.
D1 2.2 - Dijkstra's algorithmYou will apply Dijkstra's algorithm with Pearson's working and permanent labels, recover shortest paths, and use a specified starting node to find a shortest route that must pass through it.
D1 2.2 - Floyd's algorithmYou will construct directed distance and route tables, perform Floyd's iterations in the specified order, and read complete shortest routes from the final next-hop table.
Route inspection and travelling salesman problems
D1 3.1 - The route inspection algorithmYou will minimise a route that covers every arc, compare all relevant odd-node pairings, identify the actual repeated arcs, and adapt the calculation for specified endpoints or closed roads.
D1 3.2 - Travelling salesman upper boundsYou will distinguish the practical and classical travelling salesman problems, create a complete network of shortest distances, and find upper bounds using nearest neighbour and doubled minimum spanning trees with shortcuts.
D1 3.2 - Travelling salesman lower boundsYou will derive lower bounds by deleting a node, finding a residual minimum spanning tree and restoring two connections. You will compare bounds correctly and prove optimality when a lower bound meets an actual tour length.
Critical path analysis
D1 4.1, 4.2 - Activity networks and precedence tablesBuild activity-on-arc networks from immediate-precedence tables and read those tables back from networks. You will use dummy activities only where needed, preserving the dependencies without accidentally creating new ones.
D1 4.3 - Finding the critical pathUse forward and backward passes to calculate event times, activity start and finish times, and every critical path. You will also calculate a lower bound for the workers needed to meet the minimum project duration.
D1 4.4 - Total float and Gantt chartsCalculate total float and display activity timings in a Gantt, or cascade, chart. You will distinguish a flexibility window from a chosen schedule and check how delaying one activity affects its successors.
D1 4.5, 4.6 - Resource histograms and schedulingConstruct and interpret resource histograms, move non-critical work to reduce peaks, and produce schedules using the least possible number of workers. You will check both the timing constraints and the allocation of individual workers.
Linear programming
D1 5.1 - Formulating linear programsA useful linear program starts with precise variables and faithful translations of its conditions. You will formulate objectives and constraints, handle proportions and exact totals, and explain the roles of slack, surplus and artificial variables (D1 5.1).
D1 5.2 - Solving linear programs graphicallyA feasible-region diagram turns a two-variable optimisation problem into geometry. You will use vertices and parallel objective lines, find genuine integer optima, and distinguish finite, multiple and unbounded solutions (D1 5.2).
D1 5.3 - The Simplex algorithmSimplex improves a feasible linear programming solution through exact tableau row operations. You will choose pivots, solve maximisation and minimisation problems with upper-bound constraints, reconstruct models from tableaux, and check optimality and exceptional outcomes (D1 5.3).
D1 5.4 - Two-stage Simplex and big-M methodsLower-bound and equality constraints may need temporary artificial variables before a feasible basis is available. You will use two-stage Simplex and big-M for maximisation and minimisation, remove artificial variables correctly and interpret parameterised final tableaux (D1 5.4).
Decision Mathematics 2
Transportation problems
D2 1.1 - The north-west corner methodYou will read a transportation table, use the north-west corner method to obtain an initial basic feasible allocation, and calculate its cost without confusing quantities sent with unit costs.
D2 1.2 - Stepping-stone method and improvement indicesYou will calculate shadow costs and improvement indices, make stepping-stone improvements with explicit entering and exiting cells, recognise optimality, and handle degenerate and unbalanced transportation problems.
D2 1.3 - Transportation as a linear programYou will turn transportation tables into complete linear programs, define shipment variables with units, write source and destination constraints, and handle unused stock. You will also see why one balance equation is redundant in a balanced problem.
Allocation (assignment) problems
D2 2.1 - The Hungarian algorithmYou will reduce cost matrices, obtain an optimal one-to-one allocation, adapt the table for idle workers and forbidden assignments, and convert maximum profit into minimum cost.
D2 2.2 - Allocation as a linear programYou will formulate one-to-one allocation problems using binary variables, complete worker and task constraints, and objectives that respect dummy and forbidden assignments.
Flows in networks
D2 3.1 - Cuts and their capacityYou will distinguish flow from capacity, describe directed cuts precisely, and use cut capacities to bound the flow through a network.
D2 3.2, 3.3 - Labelling and the max-flow min-cut theoremYou will label spare capacity and potential backflow, augment a feasible directed flow, and prove that a final flow is maximum by finding a matching cut.
D2 3.4, 3.5 - Multiple sources, sinks and restricted capacitiesYou will combine several sources and sinks, represent a restricted vertex, and find feasible flow patterns with upper and lower arc capacities.
Game theory
D2 5.1, 5.2 - Zero-sum games and stable solutionsYou will interpret a pay-off matrix, find each player's play-safe strategy, and identify stable solutions and game values.
D2 5.3, 5.4 - Dominance and graphical mixed strategiesYou will reduce games by dominance, draw expected-pay-off lines, and find and check optimal mixed strategies for either player.
D2 5.5 - Games as linear programsYou will formulate a game as a linear program, use Simplex tableaux, and interpret and verify optimal mixed strategies in the original game.
Recurrence relations
D2 6.1, 6.2 - Recurrence models and first-order solutionsA recurrence describes how a system changes one step at a time. You will build models with clear timing, solve first-order linear relations using complementary and particular solutions, apply conditions and recover a recurrence from a closed formula.
D2 6.2 - Solving second-order recurrence relationsA second-order relation remembers two previous terms. You will use its auxiliary equation, handle distinct and repeated roots, add particular solutions for forcing terms and apply two conditions, including conditions that control the signs of later terms.