Cambridge A-Level Further Mathematics

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Further Pure Mathematics 1

Roots of polynomial equations

1.1 - Roots and coefficientsYou can discover the sum and product of an equation’s roots without solving the equation. In this first part of 1.1, you will see why the coefficients of quadratics and cubics contain that information, then use it to recover missing coefficients.
1.1 - Root relations for quarticsA quartic stores four combinations of its roots in its coefficients. You will derive those combinations, read them when coefficients are missing and use constraints on the roots to recover unknown coefficients.
1.1 - Symmetric expressions in rootsThe coefficients can reveal more than sums and products of roots. You will use identities and the original equation to find sums of squares, cubes, fourth powers and reciprocals without solving the equation, checking the conditions that make each expression meaningful.
1.1 - Shifting and scaling rootsYou can form an equation for shifted or scaled roots without finding the original roots. You will choose the substitution in the correct direction, expand it carefully and check the transformed equation using its roots and coefficients.
1.1 - Reciprocals and squares of rootsReciprocating or squaring roots changes the equation in a different way from a shift. You will derive both substitutions, handle zero and repeated transformed roots correctly and use symmetric relations to construct and check an equation for pair products of cubic roots.

Rational functions and graphs

1.2 - Rational functions and asymptotesA rational curve can have a missing point, an infinite branch or a line that it approaches far from the origin. You will distinguish these features, find intercepts and determine horizontal and oblique asymptotes using algebra.
1.2 - Sketching rational curvesA convincing sketch comes from a few exact features, rather than a table of plotted points. You will locate turning points, determine which way each branch approaches an asymptote, and combine that information with intercepts to sketch rational curves.
1.2 - Finding ranges with a discriminantWhich horizontal lines actually meet a rational curve? You will turn this geometric question into an equation in the input, use a discriminant to find possible outputs, and check exceptional values and holes before stating the range.
1.2 - Reciprocal and square-root graphsA known graph can reveal a reciprocal graph or the two branches of a square-root relation. You will track zeros, signs, asymptotes and domains, then use the related sketches to solve equations and inequalities.
1.2 - Modulus graphs and inequalitiesMoving the modulus signs changes the transformation: |f(x)| changes outputs, while f(|x|) changes inputs. You will sketch both, preserve the correct domains, and read equations and inequalities from the resulting curves.

Matrices and transformations

1.4 - Calculating with matricesA matrix keeps related numbers in a fixed arrangement, and that arrangement controls which calculations are possible. Learn to recognise matrix dimensions, combine entries and calculate products, including non-square matrices with at most three rows and columns.
1.4 - Matrix algebra and powersSome familiar algebraic laws survive for matrices, while others fail because order matters. Learn to keep products in the correct order, use identity and zero matrices safely, and recognise patterns in powers of square matrices.
1.4 - Determinants and inverses of two-by-two matricesAn inverse matrix reverses a matrix calculation, but some matrices cannot be reversed. Use the determinant to distinguish these cases, derive the two-by-two inverse formula, and solve matrix equations by multiplying on the correct side.
1.4 - Determinants and inverses of three-by-three matricesA three-by-three inverse uses the same idea as a two-by-two inverse: build a matrix whose product with the original is a multiple of the identity. Learn to evaluate determinants using cofactors, construct the inverse, and identify parameter values for which reversal is impossible.
1.4 - Inverting matrix productsReversing a sequence requires undoing its last step first. Derive the inverse-product rule, extend it to longer products, and use it to rearrange matrix equations while keeping every factor in its proper position.
1.4 - Matrices for plane transformationsA two-by-two matrix can transform an entire plane, yet two carefully chosen images determine the whole transformation. Construct these matrices, recognise rotations, reflections, enlargements, stretches and shears, and use the determinant to calculate how areas change.
1.4 - Composing and reversing transformationsA sequence of transformations is represented by a matrix product, with the first transformation acting from the right. Use that order to build composite matrices, reverse a sequence, and recover an unknown transformation from its overall effect.
1.4 - Invariant points and linesA transformation can leave a point exactly where it was, or leave a whole line unchanged while moving its points along it. Find both kinds of invariance, including vertical lines and lines away from the origin, and distinguish an unchanged line from one that collapses to a point.

Vectors and planes

1.6 - The vector productThe vector product turns two directions into a vector perpendicular to both. You will learn its direction and component form, then use its magnitude to find areas, calculate point-to-line distances and recognise parallel vectors.
1.6 - Equations of planesA plane can be described by a perpendicular direction or by two directions along its surface. You will build Cartesian, scalar-product and parametric equations, then change between them while checking that they describe the same points.
1.6 - Lines meeting planesA line can cross a plane once, miss it completely or lie entirely within it. You will use one substitution to distinguish these cases, find intersections, and construct the perpendicular foot from a point to a plane.
1.6 - Angles and distances involving planesNormals turn spatial angle and distance questions into scalar products. You will distinguish a line-plane angle from a normal angle, find the angle between planes, and calculate the shortest distance from a point to a plane.
1.6 - The line where two planes meetTwo nonparallel planes meet along a line. You will find its direction from the plane normals, find a point that satisfies both equations, and combine the two pieces into a verified vector equation.
1.6 - The shortest distance between skew linesSkew lines are neither parallel nor intersecting, so their separation cannot be found by looking for a meeting point. You will construct a common normal, project onto it to find the shortest distance, and locate both ends of the common perpendicular.

Proof by induction

1.7 - Proof by induction for sumsChecking several values can suggest a sum formula, but it cannot establish an infinite collection of cases. In this part of 1.7, you will turn a successful check into a proof, use the next term to prove power sums, and investigate a factorial sum before proving your conjecture.
1.7 - Proof by induction for sequencesA recurrence tells you how to produce the next term; a closed formula tells you a term directly from its index. In this part of 1.7, you will prove that these two descriptions agree, take care over the first index, and use the same logic to establish patterns in repeated derivatives.
1.7 - Proof by induction for divisibilityA divisibility proof must show an integer multiple, not just an expression that looks factorisable. In this part of 1.7, you will write precise hypotheses, connect successive exponential expressions, and justify the parity and domain statements that complete a proof.
1.7 - Proof by induction for matrix powersA few matrix powers can reveal a pattern, but a proof must establish that every entry continues correctly. In this part of 1.7, you will check the appropriate first power, use matrix multiplication in an induction step, and turn patterns in small powers into proved formulae.

Further Pure Mathematics 2

Hyperbolic functions

2.1 - Hyperbolic functions and graphsA small change from adding exponentials to subtracting them produces two very different curves. You will define the six hyperbolic functions, build their graphs from their formulae, and solve equations while keeping track of the real values each function can take.
2.1 - Hyperbolic identitiesHyperbolic functions satisfy identities resembling trigonometric identities, but a change of sign matters. You will prove the fundamental relation, derive addition and multiple-angle formulae, and choose identities to simplify expressions or solve equations without losing domain restrictions.
2.1 - Inverse hyperbolic functionsAn inverse hyperbolic function returns an argument rather than a reciprocal. You will choose real inverse branches, derive the logarithmic formulae from exponentials, and use those formulae to evaluate expressions and solve equations with their domains intact.

Matrix systems and eigenvectors

2.2 - Three equations as a matrix equationThree linear equations can be collected into one matrix equation without losing any information. You will translate between the two forms, solve nonsingular systems with an inverse, and interpret the solution as the common point of three planes.
2.2 - Singular systems and three planesA zero determinant does not tell you whether equations contradict one another or leave a choice of solutions. You will reduce singular systems, distinguish these outcomes, and express a whole line or plane of common solutions using parameters.
2.2 - Eigenvalues and eigenvectorsMost vectors change direction under a matrix transformation, but some special directions are preserved. You will identify these eigenvectors, find their scale factors from a characteristic equation, and solve for every corresponding eigendirection of a two-by-two matrix.
2.2 - Eigenvectors in three dimensionsA three-by-three matrix has a characteristic cubic, but its eigenvectors still come from linear equations. You will use determinant structure, find vectors for real distinct eigenvalues, and prove consequences of the defining equation for powers and inverses.
2.2 - Diagonalising a matrixA matrix becomes especially simple when vectors are described using its eigendirections. You will build matched eigenvector and eigenvalue matrices, justify the factorisation A=QDQ⁻¹, and use it to calculate powers and polynomial expressions efficiently.
2.2 - Using the characteristic equationA matrix satisfies its own characteristic equation, with the identity taking the place of scalar constant terms. You will turn that fact into a working relation, reduce high powers to lower ones, and find an inverse without constructing cofactors.

Further differentiation

2.3 - Differentiating inverse and hyperbolic functionsHyperbolic and inverse functions have useful derivative patterns, but their signs and domains matter. You will derive the patterns, apply the chain rule and distinguish a function's domain from the interval on which its derivative formula is valid.
2.3 - Second derivatives in other formsA second derivative measures how a curve's gradient changes with its horizontal coordinate. You will obtain it from implicit and parametric equations, then use its sign at points where the curve can be treated locally as a function of x.
2.3 - Maclaurin expansionsDerivatives at zero can build a polynomial that follows a function near zero. You will derive initial Maclaurin coefficients, obtain them by successive implicit differentiation, and retain enough terms when using an expansion for an approximation or a limit.

Further integration

2.4 - Hyperbolic and inverse-form integralsRecognising a derivative backwards can turn a complicated-looking integral into a short calculation. You will integrate hyperbolic functions, distinguish three inverse-function patterns and complete the square to reveal those patterns in quadratics.
2.4 - Trigonometric and hyperbolic substitutionsA substitution can replace a square root by a trigonometric or hyperbolic identity. You will choose substitutions from the sign of a quadratic, keep the correct real branch and transform limits to obtain exact definite integrals.
2.4 - Choosing an integration methodSome integrals need a sequence of familiar methods rather than one new formula. You will solve cyclic integrations by parts, turn trigonometric expressions into rational functions using a half-angle substitution, and choose a second method after a first substitution has simplified the problem.
2.4 - Reduction formulaeA high power inside an integral need not mean a long expansion. You will derive relations between neighbouring integrals, choose the starting values they need, and use those relations to find exact definite integrals.
2.4 - Bounding sums with integralsA curve can tell you how large a sum is without adding all its terms. You will compare rectangles with integrals, derive finite bounds, and recognise sums whose shrinking rectangles approach an exact area.
2.4 - Lengths of curvesThe distance along a curve is longer than the straight line between its endpoints. You will build its length from small straight segments, then calculate lengths from Cartesian, parametric and polar equations without counting a traced arc twice.
2.4 - Surface areas of revolutionRotating a curve sweeps out a surface, and the distance from the axis changes the area swept out by each small piece. You will combine radius with arc length to find curved surface areas about either coordinate axis, using Cartesian or parametric equations.

Complex numbers

2.5 - De Moivre's theoremPowers of a complex number become much simpler when you track its modulus and argument. You will interpret multiplication and division on an Argand diagram, prove de Moivre's theorem for positive integer powers, and extend the result to zero and negative integer powers with the necessary nonzero...
2.5 - Multiple-angle identitiesDe Moivre's theorem turns one binomial expansion into two trigonometric identities. You will compare real and imaginary parts, derive tangent multiple-angle formulae with their restrictions, and use the identities to solve exact equations without losing solutions.
2.5 - Trigonometric powers and sumsComplex numbers can combine a sine calculation and a cosine calculation into one expression. You will express powers of sine and cosine as multiple angles, sum trigonometric series by the C plus i S method, and recognise binomial coefficients as the terms of a complex binomial expansion.
2.5 - Complex roots and roots of unityA nonzero complex number has several distinct nth roots, arranged at equal angles on a circle. You will find every branch of positive and negative rational powers, locate the roots of unity, and use their geometry and polynomial factorisation to evaluate exact sums and products.

Differential equations

2.6 - First-order linear differential equationsSome differential equations cannot be separated, yet become a single product derivative after multiplication by a suitable function. You will identify these linear equations, derive an integrating factor and use an initial condition to select a solution on a valid interval.
2.6 - Complementary functionsA linear differential equation separates into a response to its right-hand side and a family of solutions to the corresponding zero-right-hand-side equation. You will combine these parts and find the complementary function when the auxiliary roots are distinct, repeated or complex.
2.6 - Finding particular integralsThe right-hand side of a linear differential equation suggests the shape of a particular integral. You will determine coefficients for polynomial, exponential and trigonometric trials, then use supplied forms when an ordinary trial overlaps the complementary function or needs extra terms.
2.6 - Substitutions in differential equationsA given substitution can turn variable coefficients into constant coefficients, or turn a first-order equation into one with separable variables. You will derive the transformed derivatives, preserve solutions that division might lose, and transfer initial conditions back to the original variables.
2.6 - Differential equations in contextAn algebraic solution becomes a model only when its constants, units and interval have a meaning. You will use values and rates as initial data, interpret transient and long-term behaviour, and check whether a solution remains physically valid.

Further Mechanics

Projectiles

3.1 - Modelling projectile motionA moving projectile curves even though its acceleration keeps the same direction. You will build a model using horizontal and vertical motion, then find the projectile’s position, speed and direction at a specified time or position.
3.1 - Flight time, range and heightTwo projectiles launched at the same speed can have the same range but very different flight times and greatest heights. You will derive those quantities, then adapt the equations when the landing surface is above or below the launch point.
3.1 - The equation of a trajectoryThe path of a projectile can be described without referring to time. You will derive its Cartesian equation, recover unknown launch information from points on the path, and decide which possible launches reach a target while clearing an obstacle.

Rigid-body equilibrium

3.2 - Moments of forcesA force can turn a body as well as move it. In this part of 3.2, you will calculate moments using perpendicular distances, keep track of their directions and choose a reference point that makes a problem simpler.
3.2 - Centres of mass and symmetryA body's weight is spread throughout its material, yet one force can represent its gravitational effect. In this part of 3.2, you will place that force at the centre of mass, use symmetry to locate it and select the appropriate standard result for a lamina, wire, shell or solid.
3.2 - Centres of mass of composite bodiesAn L-shaped plate need not balance at the centre of its enclosing rectangle. In this part of 3.2, you will replace simple pieces by particles at their centres of mass, then combine their moments. The same method handles holes, joined solids and different densities.
3.2 - Equilibrium of a rigid bodyBalanced forces can still turn a body. In this part of 3.2, you will combine force and moment balance, draw the forces permitted by each support and solve coplanar equilibrium problems. You will also check that the resulting contact forces are physically possible.
3.2 - Toppling and slidingA block can lose equilibrium because friction is insufficient or because its support can no longer balance its turning effect. In this final part of 3.2, you will identify each limiting condition, take moments about a toppling edge and compare the thresholds to find which motion begins first.

Circular motion

3.3 - Horizontal circular motionA particle can have constant speed and still accelerate. Learn how angular speed describes a turn, why circular motion needs an inward resultant force, and how friction or a sloping string can supply that force. You will use these ideas for horizontal circles and conical pendulums.
3.3 - Vertical circular motionA particle usually slows down as it rises around a vertical circle and speeds up as it falls. Use energy to find its speed, resolve the actual forces towards the centre, and decide whether a string or surface can maintain the assumed circular path. A zero-force calculation is useful only when the...
3.3 - Completing a vertical circleReaching the height of a circle's top is only part of the problem. Find the energy needed to keep a string taut or a particle against the inside of a track, compare those conditions with a rigid rod, and check force limits over the whole motion. The limiting point depends on what the constraint c...

Elastic strings and springs

3.4 - Forces in elastic strings and springsAn elastic string pulls harder as it stretches, while a spring can also push when compressed. You will use natural length and modulus of elasticity to calculate these forces, then combine them with force balance in vertical and inclined systems.
3.4 - Elastic potential energyAn elastic element can store energy even when its ends are momentarily at rest. You will calculate that energy, track its transfer to kinetic and gravitational potential energy, and find greatest extensions without assuming that a turning point is an equilibrium position.
3.4 - Elastic systems in motionWhen several forces act, geometry decides each extension and energy tracks the resulting motion. You will model two elastic elements, motion along inclined planes, and a conical pendulum whose string changes length as it rotates.

Momentum and impacts

3.6 - Direct impacts and restitutionA collision can send both spheres forwards while still making them separate. In this part of 3.6, you will use relative velocities to describe a direct impact, interpret the coefficient of restitution and predict how a sphere rebounds from a fixed surface.
3.6 - Direct impact of two spheresRestitution tells us how two spheres separate, but it does not determine their individual velocities. In this part of 3.6, you will combine it with conservation of momentum, then use the final velocities to check directions and kinetic-energy loss.
3.6 - Oblique impact with a fixed surfaceA smooth wall changes the velocity towards it but leaves motion along it untouched. In this part of 3.6, you will resolve an oblique velocity, apply restitution only in the normal direction, then find the rebound speed and angle.
3.6 - Oblique impact of smooth spheresIn an oblique collision, the line joining the sphere centres plays the role of a wall's normal. In this part of 3.6, you will conserve momentum and apply restitution along that line, preserve each sphere's perpendicular component, then reconstruct its final velocity.

Further Probability and Statistics

Continuous random variables

4.1 - Piecewise probability densitiesA probability density can change formula, jump, or even rise above one. In this lesson you will decide whether a piecewise function is a valid density, calculate probabilities across its pieces, and find expectations and variances of functions of a continuous random variable.
4.1 - Cumulative distribution functionsA cumulative distribution function records the probability accumulated up to any chosen value. You will construct it from a piecewise density, recover a density by differentiation, and use cumulative probabilities to find intervals and percentiles.
4.1 - Distributions of transformed variablesChanging a random variable changes both its range and the way probability is spread across it. You will find a transformed CDF by rewriting its probability event in terms of the original variable, differentiate to obtain the new density, and combine inverse branches when a transformation is many-...

Inference for means

4.2 - The one-sample t-testA small sample can test a claim about a population mean even when the population variance is unknown. You will estimate that variance, use a t-distribution with the correct degrees of freedom and tail, and express the conclusion as evidence about the population.
4.2 - Pooled variance and the two-sample t-testTwo independent samples can provide evidence about a difference between population means. You will combine their within-sample variation into a pooled estimate, identify when that estimate is justified, and carry out a two-sample t-test.
4.2 - Paired-sample t-testsBefore-and-after measurements belong together when they come from the same individual. You will turn matched observations into a single sample of differences, use a one-sample t-test on those differences, and keep the sign consistent from hypotheses to conclusion.
4.2 - Comparing means with a normal distributionA normal test can compare independent population means when variances are known, or approximately when samples are sufficiently large. You will calculate the standard error of a difference, choose the correct tail, and distinguish these situations from pooled and paired t-tests.
4.2 - Confidence intervals for one meanA sample estimate is more useful when accompanied by a measure of its uncertainty. You will use a t-distribution to construct a confidence interval for the mean of a normal population with unknown variance, then interpret the confidence level through repeated sampling.
4.2 - Confidence intervals for differences of meansA confidence interval estimates the size and direction of a population mean difference, rather than only deciding whether there is evidence of a difference. You will choose a pooled t interval, a paired-difference interval or a normal interval from the sampling design and variance information.

Chi-squared tests

4.3 - Fitting distributions to frequency dataA probability model predicts proportions, but a frequency table records counts. You will turn a prescribed distribution into expected frequencies, estimate a parameter when required, and keep complete tails when combining small classes for a chi-squared test.
4.3 - Chi-squared goodness of fitA fitted model will rarely reproduce every observed frequency exactly. You will measure the overall discrepancy, choose degrees of freedom after grouping and parameter estimation, and use an upper-tail chi-squared test to judge whether the data give evidence against the model.
4.3 - Chi-squared tests of independenceA contingency table records two categorical variables for each observation. You will predict cell counts under independence, combine categories when expected counts are too small, and use a chi-squared test to assess evidence of association in the population.

Non-parametric tests

4.4 - Non-parametric tests and the sign testHow can you test a typical value when the population cannot be assumed normal? Learn how a sign test uses a population median, why its probability calculation is binomial, and how to carry out exact and approximate tests.
4.4 - Wilcoxon signed-rank testsA sign records which side of a median an observation lies on; a signed rank also records its relative distance from that median. Learn why symmetry makes this extra information usable, how to calculate signed ranks, and how to carry out a one-sample Wilcoxon test using the supplied tables.
4.4 - Non-parametric tests for paired dataWhen two measurements belong to the same person or matched unit, their pairing contains useful information. Learn how to test for a systematic population difference using signs or signed ranks, and how the assumptions determine which method is suitable.
4.4 - The Wilcoxon rank-sum testTwo independent samples cannot be matched by subtracting the first entry in one list from the first in the other. Learn how to pool their ranks, calculate the Wilcoxon rank-sum statistic, and test for a difference between symmetric populations using the supplied table.
4.4 - Normal approximations for rank testsFor samples beyond the supplied Wilcoxon tables, a normal approximation turns a rank sum into a familiar standardised statistic. Learn the two different mean–variance formulae, apply continuity corrections, and keep the tail consistent with the rank sum you chose.

Probability generating functions

4.5 - Probability generating functionsA probability distribution can be stored in a single algebraic expression: the powers record the possible values, and the coefficients record their probabilities. You will construct these probability generating functions, derive the standard forms and recover probabilities without confusing a coe...
4.5 - Means and variances from PGFsDifferentiating a PGF changes its coefficients into weighted probabilities. You will see why the first derivative gives the mean, why the second derivative needs a correction, and how the two derivatives give the variance and determine unknown parameters.
4.5 - PGFs of independent sumsAdding independent counts corresponds to multiplying their probability generating functions. You will explain that connection, recognise binomial and Poisson sum distributions, and use coefficients and derivatives to find probabilities and moments of a total.