AQA A-Level Further Mathematics

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Core Pure

Complex arithmetic

B2 - Introducing complex numbersCan two numbers that are not real multiply to give an ordinary positive number? You will meet a number whose square is negative, then learn to read, add, subtract and multiply complex numbers. You will see how familiar algebra works in a larger number system.
B3 - The complex conjugateCan a number and its partner reveal its real and imaginary parts separately? You will learn how conjugation changes a complex number, why conjugate products are real and nonnegative, and how conjugation interacts with addition and multiplication.
B2, B3 - Dividing complex numbersHow can you divide by a number containing i and still give the answer as x+iy? You will use conjugation to make denominators real, divide by any nonzero complex number, and solve linear equations by comparing real and imaginary parts.
B1, B3 - Quadratics with non-real rootsWhen a quadratic has no real roots, where do its two solutions go? B1 extends familiar quadratic methods to complex answers; B3 explains why non-real solutions come in conjugate pairs when the coefficients are real. You will solve quadratics by completing the square and using the quadratic formul...
B1, B3 - Complex roots of cubic and quartic equationsHow much can one known root tell you about an entire polynomial equation? B3 supplies a second root when the coefficients are real, and B1 uses the resulting factors to solve cubics and quartics. You will recover all the roots from the information supplied, including zero and repeated roots.

Complex geometry

B4 - Points on the Argand diagramWhat does adding a complex number look like? You will represent complex numbers as points, interpret their sums as movements, and see conjugation and sign changes as reflections and turns. The geometry gives you a second way to check the algebra.
B5 - Modulus and argumentCan you locate a complex number using a distance and a direction instead of two coordinates? You will calculate its modulus and argument, handle the axes and zero carefully, and convert between Cartesian and modulus-argument forms. These descriptions prepare you to interpret multiplication geomet...
B6 - Multiplication as rotation and scalingWhat does multiplying by i do to a point? You will multiply and divide complex numbers in modulus-argument form, explain why their angles combine, and interpret the result as rotation and scaling. You will also adjust angles into a required principal interval.
B7 - Loci and regions in the complex planeWhat shape is hidden in an equation such as |z-(2+i)|=3? You will read moduli as distances and arguments as directions, then use them to describe circles, regions, perpendicular bisectors and rays. Combining conditions will let you select just the points that satisfy them all.

Complex powers and roots

B9 - Exponential formAn exponential can describe a direction as well as a real growth factor. You will connect e^(iθ) to the unit circle, convert between Cartesian, trigonometric and exponential forms, and use index laws to combine complex numbers. You will also see why zero needs special c...
B8 - De Moivre’s theoremRaising a complex number to a power can be easier than multiplying out its brackets. You will use de Moivre’s theorem, explain it through repeated angle addition, include zero and negative integer powers where they are defined, and derive multiple-angle identities by comparing real and imaginary...
B8 - Sums of trigonometric seriesA sum of sines can be hidden inside a much simpler geometric series of complex numbers. You will sum finite complex geometric series, extract their real or imaginary parts, and handle the special cases where the common ratio is one. The distinction between a finite sum and a sum to infinity will...
B10 - Finding all complex rootsTaking one root on a calculator does not necessarily find every complex solution. You will use de Moivre’s theorem to find all the roots of a nonzero complex number, explain why the list has exactly the required number of distinct values, and recognise the regular polygons formed by those roots.
B11 - Roots of unity in geometryRoots of unity put equally spaced points on the unit circle. You will use them to calculate polygon lengths and angles, explain symmetric sums, and turn the geometry of a translated or rotated polygon into a complex equation. A single equation can then describe every vertex at once.

Matrix arithmetic

C1, C2 - Calculating with matricesA matrix keeps several numbers in a fixed arrangement, so that you can calculate with the whole arrangement at once. C1 introduces matrix size, addition, subtraction and scalar multiplication; C2 introduces the zero and identity matrices. You will see why the positions of the numbers matter just...
C1, C2 - Matrix multiplicationMultiplying matrices combines whole rows with whole columns, rather than multiplying entries in matching positions. C1 develops that rule, its size conditions and the importance of order; C2 explains how zero and identity matrices behave in products. You will use the same rule for rectangular pro...
C5 - Determinants in two and three dimensionsA determinant turns a square matrix into one number that detects whether its columns retain all their dimensions. You will calculate determinants of order 2 and 3 by expansion, then see why a zero determinant signals a collapse onto a line, a plane or a point. This gives a useful warning before y...
C9 - Factorising determinantsExpanding a determinant immediately can hide its structure. You will use row and column operations to create zeros, keep track of changes to the determinant and expose polynomial factors before expanding. The aim is to make the calculation shorter while preserving exactly what the determinant equ...
C6 - Inverse matricesAn inverse matrix reverses a matrix calculation, recovering the original input from its output. You will decide when an inverse exists, use its algebraic properties and calculate inverses of order 2 and 3. Checking both multiplication orders will distinguish a true inverse from an array that mere...
C7, C8 - Three simultaneous equationsThree linear equations can determine one point, leave a whole family of solutions or contradict one another. C7 uses an inverse matrix to solve the equations when the coefficient matrix is non-singular. C8 connects each algebraic outcome to the way three planes intersect, including what a zero de...

Matrix transformations

C3 - Transformations in two dimensionsHow can four matrix entries describe what happens to every point in a plane? You will use the images of two unit vectors to build a transformation matrix, recognise rotations, reflections, stretches and enlargements, and recover a matrix from other known images. The geometry explains why the colu...
C3, C5 - Combining transformationsStretch a shape, then rotate it: does reversing those instructions change the result? C3 connects successive transformations to ordered matrix products and shows how to undo them. C5 adds a geometric meaning to the determinant: its magnitude scales area or volume, while its sign records orientation.
C3 - Transformations in three dimensionsA mirror plane changes one coordinate, while a rotation about an axis leaves that axis coordinate unchanged. You will represent a single reflection in a coordinate plane or a single rotation about a coordinate axis using a 3×3 matrix. Choosing the unchanged coordinate and checking the tu...
C4 - Invariant points and linesA line can stay in the same place even while its individual points move along it. You will find fixed points, distinguish a line of fixed points from an invariant line, and find invariant lines through or away from the origin. Keeping the original and image coordinates separate is the key to the...
C10 - Eigenvalues and eigenvectorsA matrix usually changes both the length and direction of a vector. Eigenvectors identify the special directions in which it acts as a simple scale. You will find eigenvalues from a characteristic equation, find their corresponding vectors in two and three dimensions, and interpret the results ge...
C11 - Diagonalisation and matrix powersRepeated matrix multiplication can become a calculation with ordinary powers. You will choose independent eigenvectors, place their eigenvalues in a matching diagonal matrix, and use the resulting diagonalisation to find matrix powers. You will also decide when this method is available over the r...

Polynomials and inequalities

D1 - Roots and coefficientsYou can find the sum of several unknown roots without solving the equation. Learn how a polynomial records sums and products of its roots, how to keep the signs correct, and how to recover sums of powers from that information.
D2 - Transforming polynomial rootsMove every root of a polynomial by the same scaling and translation, and you can find the new equation without solving the old one. Learn to reverse the root transformation, substitute safely and check the result using symmetric sums.
D8 - Polynomial inequalitiesAn equation finds the inputs where a polynomial is zero; an inequality asks what happens between those inputs. You will use factors to determine the signs of cubics and quartics, explain what repeated roots do, and give complete exact solution sets with the right endpoints.
D9 - Rational inequalitiesMultiplying an inequality by a negative number reverses its direction, so a denominator containing x needs care. You will bring rational inequalities to one fraction, separate zeros from undefined inputs, and use signs to find complete solution sets safely.
D10 - Modulus functions and inequalitiesA modulus turns a signed value into a distance from zero. You will use its piecewise definition to build graphs, solve inequalities by distance or by cases, and keep the sign and domain restrictions that prevent false solutions.

Graphs and curves

D11 - Modulus and reciprocal graphsThe same graph can generate two very different new graphs by replacing each output with its modulus or its reciprocal. You will sketch these from a given function, tracking fixed levels, zeros, excluded inputs and asymptotes rather than relying on a table of unrelated points.
D12 - Linear rational functionsA fraction of two linear expressions often hides a translated reciprocal graph. You will expose that structure, find asymptotes and intersections, and solve the associated inequalities while distinguishing genuine vertical asymptotes from removable holes.
D13 - Quadratic rational functionsA rational graph can have several branches, a sloping asymptote or a single missing point. Learn to identify these features from its numerator and denominator, keeping the original domain visible whenever factors cancel.
D14 - Ranges of rational functionsThe range is the set of outputs a function actually attains. Learn to turn an output question into an equation in the input, use quadratic theory to decide which outputs are possible, and locate stationary points without calculus.
D15 - Parabolas, ellipses and hyperbolasRecognise four standard curve equations and sketch them from symmetry, intercepts and limiting behaviour. You will see why changing one plus sign to a minus sign can turn a closed ellipse into two unbounded branches.
D16 - Transforming curvesTransform a curve by following its points, then use the inverse coordinate map to find its new equation. This method handles translations, stretches, reflections, rotations and enlargements, including combinations whose order matters.

Series and induction

D3 - Sums of powersHow can you add the first hundred cubes without calculating a hundred separate cubes? You will learn to read sigma notation, use the sums of integers, squares and cubes, and reshape other finite sums so these formulae apply. One striking connection is that a sum of cubes is itself a perfect square.
D4 - The method of differencesCan a long sum of fractions collapse to just a few endpoint terms? The method of differences makes this possible when each term can be written as a difference that cancels with its neighbours. You will learn to display that cancellation, create it with partial fractions, and decide whether the fi...
A1 - Proof by inductionChecking a formula for ten, a thousand or a million cases still leaves infinitely many cases unchecked. Proof by induction replaces that endless task with a starting case and a rule that carries truth to the next integer. You will use it to prove finite sums and recurrence formulae, and learn to...
A1 - Induction for divisibilityWhy is 5ⁿ-1 divisible by 4 for every positive integer n, even when the power is too large to calculate? Induction lets you prove divisibility by exposing an assumed integer multiple inside the next expression. You will learn to choose that rewrite, state the integer domain carefully,...
A1 - Induction for matrix powersA few matrix products can suggest a formula, but they cannot prove it for every positive integer power. You will check a starting case, state an induction hypothesis precisely, and multiply matrices to prove the next case. You will then compare this proof with diagonalisation, including a matrix...

Power series and limits

D5 - Building Maclaurin seriesHow can a polynomial imitate an exponential or a sine curve near zero? You will use derivatives at zero to build a Maclaurin expansion, identify its general term and distinguish a finite approximation from an infinite series.
D6 - Using standard Maclaurin seriesStandard series turn exponentials, logarithms, trigonometric functions and powers into algebra near zero. You will recognise their patterns, substitute new expressions, multiply expansions and keep the correct validity range alongside the answer.
D7 - Limits from power seriesA quotient can approach a definite number even when direct substitution gives zero divided by zero. You will use Maclaurin series to expose the first surviving powers, distinguish finite and infinite behaviour, and decide how many terms and which domain a limit needs.
D7 - L’Hôpital’s ruleDifferentiation offers another way to evaluate certain indeterminate limits. You will recognise when l’Hôpital’s rule applies, differentiate the numerator and denominator separately, and check the conditions again before repeating the rule.

Vectors in space

F1 - Equations of lines in spaceHow can one equation describe a line when every point needs three coordinates? F1 uses a starting point, a direction and one shared parameter to do exactly that. You will connect vector and Cartesian equations, handle zero direction components and decide whether two equations describe the same li...
F3, F4 - Scalar products and anglesCan multiplying two vectors give a negative number? F3 introduces the scalar product and connects that number to an angle; F4 uses a zero scalar product to detect perpendicular directions. You will distinguish the angle between directed vectors from the smaller angle between unoriented lines.
F2, F3 - Equations of planesWhy does a line need one parameter, while a plane needs two? F2 describes planes using two independent directions or one Cartesian equation. F3 connects those descriptions through a normal vector and the scalar product, so you can convert forms and test whether a point belongs.
F3 - Angles involving planesCan the same dot-product calculation describe two different geometric angles? F3 uses normals to find angles between planes and uses a complementary angle for a line and a plane. You will choose the smaller unoriented angle carefully and interpret the parallel and perpendicular boundary cases.
F5 - The vector productTwo vectors can determine a third vector perpendicular to both. You will calculate this vector product, explain its size and direction, use it to find triangle areas, and express a line without a parameter. The same product will later measure shortest distances in space.
F6 - Intersections in spaceLines that cross on a drawing may miss each other in space. You will find line-line and line-plane intersections by solving for parameters, then use direction and consistency checks to distinguish a single intersection, parallel objects, coincident lines and skew lines.
F6 - Perpendicular distancesThe shortest route to a line or plane is perpendicular to it. You will find perpendicular feet and distances, then extend the same idea to two lines in space. Choosing the right normal matters: skew lines and parallel lines require different calculations.

Hyperbolic functions

H1 - Hyperbolic functions from exponentialsAdding and subtracting the same two exponential functions creates three very different shapes: an unbounded curve, a curve with a minimum, and a curve trapped between two horizontal lines. You will define the six hyperbolic functions, work with exact values, and connect their domains and ranges t...
H6, H7 - Hyperbolic identitiesThe point ( t, t) moves along a hyperbola, which explains the name “hyperbolic”. You will derive the identities behind that connection, use double-angle formulae, and build proofs from exponential definitions with the necessary domain restrictions.
H2 - Differentiating and integrating hyperbolic functionsDifferentiating sinh gives cosh, and differentiating cosh gives sinh: there is no alternating minus sign. You will derive the rules for all six hyperbolic functions, reverse them in integration, include chain factors, and construct the less obvious logarithmic antiderivatives with their domains.
H3, H4 - Inverse hyperbolic functionsAn inverse must return one input for each output. Cosh and sech need chosen branches; the other four hyperbolic functions already give unique inputs throughout their ranges. You will define all six inverses, derive their logarithmic forms, and solve equations while keeping the original domain and...
H5 - Hyperbolic substitutionsA carefully chosen hyperbolic substitution turns a square root into a single sinh or cosh factor. You will integrate reciprocal square roots and related functions, choose substitutions from the completed square, and handle the separate positive and negative branches of a difference of squares.

Further integration

E5, E6 - Inverse trigonometric functionsAn integral containing a square root can have an exact answer involving an angle. In E5 you will derive inverse trigonometric derivatives by undoing familiar functions. In E6 you will reverse those derivatives to integrate, keeping track of scales, branches and domains.
E6 - Trigonometric substitutionsA trigonometric substitution can turn a square root into a familiar trigonometric factor. You will choose sine, tangent or secant by matching an identity, transform differentials and limits, and use a deliberate angle range to control the sign of a root.
E4 - Partial fractions with quadratic factorsPartial fractions can split a rational integrand even when a quadratic factor has no real roots. You will choose the right numerator over each factor, find its coefficients, then separate logarithmic and inverse tangent integrals.
E1, E9 - Improper integralsAn infinite interval can enclose a finite area, while a finite interval can give an infinite integral. You will replace troublesome endpoints by limits, decide whether those limits exist, and use exponential and logarithmic limits to finish exact evaluations.
E3 - Mean value of a functionA changing quantity can be replaced by one constant with the same accumulated total. You will calculate that constant using an integral, explain the equal-area idea, and see why the value halfway through an interval is not generally the average.
E8 - Reduction formulaeA family of difficult integrals can sometimes be reduced to one or two simple starting values. You will derive these links by integration by parts, retain the endpoint terms, and use the correct base case to calculate higher powers.

Calculus in geometry

E2 - Volumes of revolutionRotating a plane region creates a solid whose circular cross-sections can be added by integration. You will derive the disc formula, choose radii about either coordinate axis, subtract holes using washers, and change variables for parametric curves.
E7 - Arc lengthA curve's length can be found by adding tiny straight segments. You will derive the short-element formula, use Cartesian and parametric descriptions, and choose bounds and square-root signs that measure the required arc exactly once.
E7 - Surface area of revolutionRotating a curve produces a surface, whose thin bands have area equal to circumference times slant length in the limit. You will derive this rule, use Cartesian and parametric curves about either coordinate axis, and select nonnegative radii and branches that count the surface once.
G1 - Polar coordinatesA distance and a direction can locate a point just as well as horizontal and vertical coordinates. You will use polar coordinates, explain why their labels are not unique, convert points and equations into Cartesian form, and connect this geometry to complex numbers.
G2 - Sketching polar curvesA polar graph is traced by changing an angle and travelling the corresponding radius. You will use key values, symmetry and admissible angles to sketch trigonometric curves, then find intersections and tangents at the pole. You will see why setting two radius formulae equal can miss a genuine int...
G3 - Areas inside polar curvesPolar areas are built from thin sectors rather than vertical strips. You will derive the area formula, choose bounds from the actual tracing, and calculate regions inside or between curves. Careful sketches will help you avoid counting a region twice or subtracting the wrong radii.

First-order differential equations

I1, I2 - Integrating factorsAn equation can tell you how a quantity changes without giving the quantity itself. I1 introduces a multiplier that turns a first-order linear differential equation into a single product derivative; I2 uses it to find a whole family of solutions. You will recognise the right form, prepare the equ...
I2, I3 - Initial conditions and modelsA differential equation gives a rate law; an initial condition selects the solution that describes a particular situation. I2 develops this choice of constants, and I3 turns statements about motion and temperature into first-order models. You will interpret the resulting functions, including thei...
I9 - Coupled differential equationsTwo quantities can change together, with each rate depending on both quantities. I9 represents this interaction with coupled first-order equations. You will solve constant-coefficient systems using real eigenmodes, see how elimination produces one second-order equation, and interpret initial cond...

Second-order differential equations

I2, I4, I6 - Auxiliary equations with real rootsA second-order differential equation links a function to both its slope and the rate at which its slope changes. I4 uses an auxiliary quadratic to find exponential solutions; I6 explains why distinct and repeated roots give different forms. I2 then uses two initial conditions to select a particul...
I2, I4, I6 - Oscillatory solutionsComplex auxiliary roots can describe a completely real curve. I4 gives the oscillatory solution form, I6 connects it to a negative discriminant, and I2 uses initial conditions to choose one curve. You will separate the oscillation from its exponential envelope and check both the equation and the...
I5 - Forced differential equationsA nonzero right side changes what a differential equation is asking the curve to do. I5 combines the homogeneous solution with one response to polynomial, exponential or trigonometric forcing. You will choose a suitable trial, find its coefficients and recognise when overlap with the homogeneous...
I9 - Solving coupled models by eliminationA coupled model needs two functions of the same independent variable, with each function satisfying its own rate equation. I9 extends elimination to oscillatory and forced systems, reconstructs the second variable and interprets a predator–prey approximation. You will check the full pair and dist...
I3, I7, I10 - Simple harmonic motionA restoring acceleration can keep a particle moving backwards and forwards around equilibrium. I7 connects that motion to an oscillatory differential equation; I10 derives it from Hooke's law; I3 interprets the resulting displacement, velocity and acceleration. You will find the amplitude, phase...
I8, I11 - Damped oscillationsResistance changes a spring's ideal oscillation into a return towards equilibrium with a shrinking amplitude or no repeated crossings. I11 models a damping force proportional to velocity, and I8 connects the differential equation to light, critical and heavy damping. You will retain force signs,...

Numerical methods

J1 - Mid-ordinate and Simpson’s rulesNumerical integration estimates a definite integral from a finite set of function values. You will build the mid-ordinate and Simpson’s rules, choose the right points and weights, and judge what a change of step size actually tells you about accuracy.
J2 - Euler’s methodA differential equation tells you a curve’s gradient even when you do not have its equation. Euler’s method uses those gradients to build an approximate solution from an initial value. You will organise the updates, work backwards from a given estimate, and compare step sizes without confusing nu...
J3 - Improved Euler’s methodA slope at the middle of an interval can estimate the change across the whole interval. You will use AQA’s two-step improved Euler recurrence, obtain its missing starting value with Euler’s method, and compare estimates at the same endpoint. Careful indexing matters as much as the arithmetic.

Mechanics

Dimensions and momentum

MA1 - Dimensions and consistencyDimensions describe the kind of quantity a formula represents, independently of the units used to measure it. You will build dimensions from mass, length and time, test proposed equations and see why a formula can pass that test yet still be wrong.
MA2 - Predicting formulaeDimensional analysis can narrow a proposed formula to a particular combination of powers. You will equate exponents, interpret the resulting model and recognise what dimensions cannot determine about constants, functions and missing variables.
MB1 - Conservation of momentumMomentum combines mass with velocity and retains its direction. You will conserve signed or vector momentum during an interaction, choose a suitable system and interpret the direction of the resulting motion.
MB2 - Direct collisionsA collision law describes how rapidly two bodies separate compared with how rapidly they approached. You will combine this law with momentum, model rebounds from a fixed wall and check that calculated velocities fit the physical collision.
MB1, MB2 - Oblique impactsAn angled collision is easier to understand when velocity is split along and across the contact surface. You will resolve those components, apply smooth-impact laws to walls and pairs of bodies, and reconstruct the outgoing velocity, speed and direction.
MB3, MB4 - ImpulseImpulse measures the accumulated effect of a force and equals the change in momentum. You will retain its vector direction, connect constant or average force with contact time, and integrate a variable force in one dimension using signed areas.

Work, energy and power

MC1–MC3 - Work and mechanical energyA force can act on a moving object without doing any work on it. MC1 explains how the direction of a force determines its work; MC2 and MC3 connect that work to gravitational potential energy and kinetic energy. You will use an energy balance to find speeds while keeping resistance and modelling...
MC4 - Hooke’s lawPull an elastic string further and it pulls back harder. MC4 describes this response using stiffness or modulus of elasticity. You will measure extension from natural length, calculate elastic forces and distinguish a slack string from a compressed spring.
MC5, MC6 - Variable forces and elastic energyA force that changes as an object moves cannot usually be handled by multiplying one force value by the whole distance. MC5 uses integration to calculate its work; MC6 applies that idea to elastic potential energy. You will track kinetic, gravitational and elastic energies together, including fri...
MC7 - PowerTwo engines can do the same work but take different times. MC7 measures that rate using power and connects it to force and speed. You will distinguish driving force from resultant force, analyse steady and changing speeds, and use a supplied efficiency to connect input power with useful output.

Circular motion

MD1–MD3 - Speed in circular motionA particle can keep the same speed while accelerating throughout its journey. Learn how its velocity changes around a circle, convert between angular-speed units, and connect radius, speed and inward acceleration to the real forces acting.
MD4 - Circular motion with vectorsDescribe a circular path using vector components, differentiate to find velocity and acceleration, and prove that uniform circular motion has tangential velocity and inward acceleration. See why the centre's position matters when interpreting a position vector.
MD5 - Conical pendulumsModel horizontal circular motion supported by one or two strings. Separate vertical equilibrium from the horizontal inward resultant, use the geometry to find the circle's radius, and decide whether calculated tensions describe strings that can actually stay taut.
MD6 - Vertical circlesCombine the radial equation of motion with conservation of energy to follow a particle around a vertical circle. Decide whether a string stays taut or a surface stays in contact, and explain why a rod can complete some circles that a string cannot.

Centres of mass and equilibrium

ME1 - Centres of mass of particlesLocate the single point that represents the mass distribution of several particles. You will use weighted coordinates in one, two and three dimensions, recognise genuine mass symmetry and work backwards to find an unknown mass.
ME2 - Composite bodiesReplace each simple component of a body by its mass at its own centre, then combine those masses. You will handle joined pieces, removed material and different densities without confusing length, area and volume weights.
ME3 - Laminas by integrationFind the centre of a uniform curved sheet by adding the masses and moments of thin strips. You will choose the strip's area and centre carefully, derive both coordinate integrals and use symmetry to simplify the work.
ME4 - Solids of revolutionLocate the centre of a solid made by rotating a region about the horizontal axis. You will turn thin strips into discs or washers, weight their positions by volume and explain exactly when uniform density cancels.
ME6 - Moments and couplesDetermine forces that keep a rigid body at rest by balancing both force and turning effect. You will use couples, model supports and friction, and choose moment points that simplify the algebra while checking that the resulting forces are physically possible.
ME5 - Sliding, toppling and suspensionDecide which equilibrium condition fails first when a body is pushed or a supporting plane is tilted. You will compare sliding and toppling thresholds, check contact and friction constraints, and use a centre of mass to predict how a freely suspended body hangs.

Statistics

Discrete probability

SA1, SA2 - Discrete probability distributionsA probability table describes what a random quantity can do before you observe it. SA1 introduces tables and probability functions; SA2 uses them to find probabilities, including events with a condition. You will learn to check a proposed model before trusting its predictions.
SA3, SA4 - Averages and spreadThe most likely outcome and the long-run average need not be the same. SA3 introduces mean, mode, median, variance and standard deviation for a discrete random variable; SA4 calculates them using expectation and second moments. You will also derive the shortcut that makes variance practical to ca...
SA5 - Functions of random variablesA fixed change to every outcome changes the whole probability distribution. SA5 uses expectation and variance rules for linear changes, then calculates moments of nonlinear functions, including inverse powers. You will see why transforming an average is usually different from averaging transforme...
SA6, SA7 - The discrete uniform modelEqual probabilities make some calculations remarkably simple, provided the possible values are identified correctly. SA6 models a uniformly selected integer from 1 to n and checks when that model is reasonable. SA7 proves its mean and variance using finite sums, then shows how relabelling the sam...

Poisson models and testing

SB1–SB3 - The Poisson distributionA mean count does not tell you exactly what will happen next. SB1 introduces a model for random events, SB2 turns it into probabilities, and SB3 connects its mean to its spread. You will also learn when the model deserves to be questioned.
SB4 - Combining Poisson variablesSeparate random event streams can sometimes be treated as one. You will combine independent Poisson counts, match parameters to observation intervals, and recognise why overlap or dependence can invalidate an apparently simple calculation.
SB5 - Testing a Poisson meanAn unusually large count may indicate a higher event rate, but random variation can also produce surprises. You will formulate a claim about a population mean, evaluate an exact Poisson tail, and make a cautious decision in context. Critical regions show how that decision would change for other p...
SC1 - Type I errorsA careful test can still lead to a wrong decision. You will distinguish Type I and Type II errors in context, calculate the risk of a Type I error from a complete rejection region, and explain the difference between a chosen significance level and the actual risk. Binomial and normal calculations...
SC2 - Type II errors and powerA test designed to avoid false alarms can also miss a real change. You will fix a decision rule using the null model, calculate its chance of missing a specified alternative, and find its power. The same event-based method works for Poisson, binomial and normal tests.

Continuous probability

SD1, SD2 - Probability density functionsA measurement can take any value in an interval, so a list of separate probabilities is no longer enough. SD1 introduces probability density and its connection to area; SD2 uses that area to calculate interval probabilities. You will also see why a density can exceed 1 even though a probability c...
SD6 - Cumulative distribution functionsA cumulative distribution function records all the probability collected up to a chosen value. SD6 connects it to the density through integration and differentiation. You will build complete piecewise functions, handle tails and gaps, and subtract cumulative probabilities to find intervals.
SD3 - Medians and quartilesA median splits probability into equal halves, while quartiles mark cumulative probabilities of one quarter and three quarters. SD3 finds these positions from a density or CDF. You will solve area equations, see how an asymmetric shape moves the median, and find numerical quantiles when an exact...
SD4, SD5 - Continuous expectation and varianceSD4 uses a density to find a continuous variable's mean, variance and standard deviation. SD5 extends those calculations to functions of the variable, including changes of scale and inverse powers. You will connect weighted sums to integrals and check that the moments you calculate actually exist.
SD1 - Mixed distributionsA random quantity can put some probability at an exact value and spread the rest across an interval. You will combine point masses with continuous area, recognise jumps in a cumulative distribution function and handle strict and inclusive boundaries correctly.
SD7 - The rectangular distributionThe rectangular distribution models a continuous value spread evenly across a bounded interval. You will find its density and probabilities, decide when equal likelihood is reasonable, and prove its mean, variance and standard deviation.
SD8 - Sums of independent variablesTotals and differences of random quantities have their own means and spreads. You will combine discrete or continuous variables, see why independence matters for variance, and distinguish repeated observations from repeated use of the same variable.

Association and inference

SE1–SE3 - Chi-squared tests for associationLearn to organise categorical observations, calculate the counts predicted by independence and use a chi-squared test to investigate association. You will check the expected-count condition and combine categories when that is meaningful.
SE4, SE5 - Interpreting associationUse cell contributions to describe the pattern behind a significant association, distinguish evidence from causation and apply Yates' continuity correction to a two-by-two table. You will compare corrected and uncorrected tests without overstating their conclusions.
SG1 - One-sample t-testsTest a claim about the mean of a normal population when its variance is unknown. You will estimate the variance correctly, use the appropriate t distribution and match the rejection tail to the contextual alternative hypothesis.
SH1, SH3 - Confidence intervals with known varianceConstruct symmetric confidence intervals for a normal population mean when its variance is known. You will connect the interval to the sampling distribution, interpret confidence correctly and use a constructed or given interval to assess a proposed mean.
SH2–SH4 - Confidence intervals with estimated varianceConstruct confidence intervals for a normal population mean when its variance must be estimated. You will distinguish the large-sample normal approximation from a small-sample t interval, choose degrees of freedom and interpret changes in interval width.

Discrete Mathematics

Graphs and planarity

DA1, DA6 - The language of graphsA graph keeps track of which objects are connected, without needing their sizes or positions. DA1 gives the language for connections and routes; DA6 uses that language to recognise simple graphs and trees. You will see why a small change to a drawing can change a graph, while a large change of sh...
DA5 - Complete and bipartite graphsHow many connections does a graph need when everyone can meet everyone, or when connections are allowed only between two groups? Learn to recognise complete and bipartite graphs, encode their edges in adjacency matrices, and find the connections missing from a graph. Matrix entries will also reve...
DA7 - Graph isomorphismTwo graph drawings can look different while describing exactly the same pattern of connections. Learn to rule out false matches using properties that relabelling preserves, and to prove a match by giving a complete correspondence of vertices. You will distinguish evidence that suggests a match fr...
DA2 - Eulerian and Hamiltonian graphsA route can be designed to use every edge once, or to visit every vertex once. Those are different demands. Learn to construct and justify Eulerian and semi-Eulerian trails using connectedness and vertex degrees, then decide Hamiltonian questions by explicit cycles or structural obstructions.
DA3 - Euler’s formula for planar graphsA planar graph can be drawn with no crossing edges, and such a drawing divides the plane into faces. Learn to count those faces, including the outside region, and use Euler's formula to link them to vertices and edges. You will then derive edge bounds that can prove a graph non-planar.
DA4 - Proving non-planaritySome graphs remain non-planar even when extra vertices disguise the familiar complete-graph pattern. Learn to find subdivisions of K5 and K3,3, trace their internally disjoint paths, and use Kuratowski's theorem to prove non-planarity. You will also see why crossing lines in one sketch...

Networks and routes

DB1, DB2 - Minimum spanning treesHow can five buildings share a cable network at the smallest total cost? DB1 introduces the language of weighted networks; DB2 uses minimum spanning trees to connect every node efficiently. You will use Prim's and Kruskal's algorithms, handle ties and decide whether the mathematical answer meets...
DB3 - Route inspectionA route inspector must cover every road, rather than merely reach every junction. You will use odd vertex degrees to identify the necessary repetition, compare pairings of shortest paths and find minimum closed or open inspection routes. You will also justify that a proposed walk really attains t...
DB4 - Travelling salesperson boundsVisiting every location once and returning home is a different task from inspecting every road. You will construct travelling salesperson tours, use nearest neighbours for upper bounds and deleted-vertex minimum spanning trees for lower bounds. Comparing the bounds tells you what has been proved...
DB5 - Refining network modelsA network answer is only as useful as the assumptions behind its nodes, arcs and weights. You will translate a practical objective into weights, identify missing restrictions and refine a model when costs or available links change. Earlier spanning-tree and route methods will let you assess which...

Network flows

DC1, DC2 - Flows and cutsA network can have plenty of capacity near its entrance and still struggle to deliver material at its exit. DC1 introduces directed capacities, feasible flows and the roles of sources and sinks. DC2 uses cuts to place an upper limit on delivery, while keeping capacity separate from the flow actua...
DC3, DC5 - Maximum flow and minimum cutA poor first route need not prevent a good final flow: you can cancel part of an earlier choice and redirect it. DC5 develops residual networks and flow augmentation. DC3 explains how a feasible flow and an equal-valued cut give a complete proof that delivery is as large as possible.
DC4, DC7 - Multiple sources and restricted nodesReal networks may collect material from several places and deliver it to several destinations. DC4 combines those entrances and exits using a supersource and supersink. DC7 refines the model to respect a shared processing limit at a node and explains which contextual restrictions a static flow ne...
DC6 - Lower capacity constraintsSome routes must carry a minimum flow as well as staying below an upper limit. DC6 combines these bounds with conservation, adjusts flows without cancelling too much, and proves a maximum using the correct cut bound. Compulsory flow back across a cut can reduce the greatest possible net delivery.

Linear programming

DD1 - Formulating linear programmesA profitable product can still use too much of a scarce ingredient. You will turn a practical decision into variables, an objective and constraints, then check the units and assumptions that make the model meaningful. A good formulation says exactly what can be chosen and what must be respected.
DD2 - Graphical optimisationFor two decision variables, the constraints become a picture of all permitted choices. You will locate an optimum using vertices and parallel objective lines, recognise when an optimum is absent or shared, and check whole-number decisions carefully. The geometry explains why trying every possible...
DD3, DD4 - The simplex algorithmGraphical optimisation becomes awkward when there are more than two decision variables. DD3 introduces the simplex algorithm, which improves a feasible solution using row operations; DD4 explains what its tableaux mean. You will build a tableau, choose safe pivots and read the resulting optimum i...
DD3, DD4 - Minimisation and tableau interpretationA minimum-cost problem can be solved by maximising a carefully chosen quantity. For DD3 you will make an equivalent transformation and apply simplex from a feasible start; for DD4 you will interpret intermediate and final tableaux. The final answer must describe the original cost and decisions, i...

Project scheduling

DE1 - Activity-on-node networksA project can contain jobs that run together and jobs that must wait. You will turn a list of dependencies into an activity-on-node network, read the information in each node and recognise whether the network represents the intended project.
DE2, DE3 - Critical paths and floatYou will calculate the earliest and latest times in an activity-on-node project, identify the chains which determine completion and measure how far an activity can move without delaying the project. The calculations assume fixed durations and enough resources for all permitted parallel work.
DE4 - Changing project schedulesA project plan changes when an estimate, working requirement or dependency changes. You will rebuild the affected timing calculations, recognise when the critical path changes and distinguish what the mathematical model predicts from what needs checking in practice.
DE5 - Gantt charts and resource histogramsA network shows which jobs must wait; a Gantt chart shows when they run. You will draw time bars with clearly distinguished float, combine simultaneous resource requirements into a histogram and interpret its peak, area and periods of unused capacity.
DE6 - Scheduling with limited resourcesPrecedence allows activities to overlap, but workers or machines may be unavailable. You will move flexible activities to reduce peaks, build schedules under a hard resource limit and use bounds to distinguish an improved schedule from a proved optimum.

Zero-sum games

DF1–DF3 - Pay-off matrices and stable gamesA good decision in a competitive game depends on what the other player can do. DF1 turns the rules into a pay-off matrix; DF2 finds each player's play-safe strategy; DF3 proves whether those choices give a stable solution. You will keep the pay-offs in one player's viewpoint and explain what the...
DF4 - Dominated strategiesSome choices can be replaced by another choice that is at least as good against every possible response. You will identify dominated rows and columns, reduce a game carefully, and explain why the reduction preserves its optimal value. Equality cases matter: a removable strategy need not be worse...
DF5 - Mixed strategiesA game without a saddle point can reward a player who randomises their choices. You will turn probabilities into expected pay-offs, find optimal two-by-two mixtures, and use lower and upper envelopes for larger games with two strategies on one side. You will also recognise pure boundary solutions...
DF6 - Games as linear programmesA higher-order game can require more probabilities than a single graph can show. You will express each opponent response as an inequality, preserve the probability conditions, and use a valid simplex formulation to solve a three-by-three game. Shifting every pay-off by a constant provides a safe...

Binary operations and groups

DG1, DG4 - Binary operations and Cayley tablesWhat changes when an operation acts on a particular set rather than on any numbers you like? You will learn what makes a rule a binary operation, calculate with modular arithmetic and matrices, and construct and read Cayley tables without reversing the operands.
DG2, DG3 - Commutativity and associativityDoes a familiar-looking operation obey familiar algebraic laws? You will distinguish changing the order of two inputs from changing the brackets around three inputs, prove laws for every allowed input, and use a single valid counterexample to disprove a claim.
DG5, DG6 - Identity and inversesAn identity leaves every element unchanged; an inverse combines with an element to return that identity. You will find and prove identities, check whether inverses belong to the given set, and keep left and right calculations separate when order matters.
DG7, DG8 - The group axiomsA group is a set with an operation that combines and undoes elements consistently. You will prove the four group axioms, distinguish a group's order from an element's period, and use Cayley tables while recognising what a table pattern does not prove.
DG7, DG9 - Groups and subgroupsA smaller collection of moves can sometimes form a group inside a larger one. You will distinguish finite and infinite groups, combine polygon symmetries, recognise cyclic and abelian groups, and prove when a subset is a subgroup. The words proper and non-trivial describe different restrictions.
DG10 - Lagrange’s theoremA subgroup cannot have an arbitrary size inside a finite group. You will use Lagrange’s theorem to restrict subgroup orders and element periods, then separate those necessary restrictions from proof that a particular subset is a subgroup.
DG11 - Generators of groupsGenerators build a group from a small collection of elements. You will list generated elements, identify generators of cyclic groups, and combine rotations and reflections to generate polygon symmetry groups. You will distinguish an individual generator from a generating set.
DG12 - Isomorphism of finite groupsDifferent-looking groups can have exactly the same operation structure. You will construct bijections that preserve the operation, write an isomorphism using the correct symbol, and rule out proposed matches using element periods or commutativity. Equal group orders alone are not enough.