England · AQASyllabus
Further Maths syllabus, dot point by dot point
Every dot point in the England Further Mathssyllabus, with a focused answer for each one. Click any dot point for a worked explainer, past exam questions, and links to related dot points. Written by Claude Opus 4.8, Anthropic's latest AI.
Core pure
Module overview →- How do complex numbers extend the real numbers, and how do you add, multiply, divide and represent them geometrically?Solving quadratic, cubic and quartic equations with complex roots, arithmetic of complex numbers, the Argand diagram, modulus-argument form, de Moivre's theorem and loci.11 min answer →
- How do you solve first and second order differential equations, and how do you model real situations with them?First order linear differential equations using an integrating factor, second order equations with constant coefficients including the complementary function and particular integral, and modelling with damped and forced systems.12 min answer →
- How do the roots of a polynomial relate to its coefficients, and how do you sum series and split fractions?Roots of polynomials and their relationships to coefficients, summation of series using standard results, the method of differences, partial fractions and the Maclaurin series.11 min answer →
- How do you find improper integrals, arc lengths, surface areas and the mean value of a function, and how do you use the Maclaurin series?Improper integrals, volumes of revolution, mean value of a function, arc length, surface area of revolution, integration using partial fractions and the Maclaurin series of standard functions.11 min answer →
- How do you describe lines and planes in three dimensions, and how do you find angles, intersections and distances between them?Vector and Cartesian equations of lines and planes, the scalar and vector products, angles between lines and planes, intersections and shortest distances in three dimensions.11 min answer →
- What are the hyperbolic functions, how are they defined, and how do you differentiate, integrate and invert them?Definitions of hyperbolic functions in terms of exponentials, their graphs and identities, inverse hyperbolic functions in logarithmic form, and differentiation and integration involving them.10 min answer →
- How do matrices represent transformations and systems of equations, and how do you invert them?Matrix arithmetic, determinants, inverses of 2x2 and 3x3 matrices, matrices as linear transformations, invariant points and lines, and solving systems of linear equations.11 min answer →
- How do polar coordinates describe curves, and how do you sketch them and find the area they enclose?Polar coordinates and the relationship with Cartesian coordinates, sketching polar curves, and finding areas enclosed by polar curves using integration.10 min answer →
- How does proof by induction work, and how do you use it for sums, divisibility and matrices?Proof by mathematical induction applied to summation formulae, divisibility results, recurrence relations and powers of matrices, with a clearly stated base case, inductive step and conclusion.10 min answer →
Discrete mathematics
Module overview →- How do you schedule a project, find its shortest completion time and identify the activities that cannot be delayed?Activity networks, forward and backward passes to find earliest and latest times, the critical path, float, and resource scheduling with Gantt charts.11 min answer →
- How do you analyse a two-player zero-sum game, find a stable solution, and handle games with no saddle point?Two-player zero-sum games, the pay-off matrix, play-safe strategies, saddle points and stable solutions, dominance to reduce a game, and mixed strategies including conversion to linear programming.11 min answer →
- What is a graph, what vocabulary describes it, and how are graphs represented and classified?Graph terminology including vertices, edges, degree, paths and cycles, special graphs such as trees and complete graphs, the adjacency matrix representation, and Eulerian and Hamiltonian graphs.10 min answer →
- How do you formulate a linear programming problem and solve it graphically or with the simplex method?Formulating linear programming problems, the feasible region and graphical solution, the vertex (objective line) method, slack variables, and the simplex algorithm for maximisation.11 min answer →
- How do standard algorithms find minimum spanning trees, shortest paths and good routes through a network?Kruskal's and Prim's algorithms for minimum spanning trees, Dijkstra's algorithm for the shortest path, and the route inspection and travelling salesperson problems.11 min answer →
Further mechanics
Module overview →- How do you find the centre of mass of a system of particles or a lamina, and how does it determine equilibrium?Centre of mass of a system of particles, of uniform laminae and standard shapes, of composite bodies, and using the centre of mass to analyse suspended and toppling bodies.10 min answer →
- How do you analyse motion in a circle, both at constant speed and in a vertical plane?Angular speed, the relationship between linear and angular speed, centripetal acceleration and force, horizontal circular motion such as the conical pendulum, and motion in a vertical circle.11 min answer →
- How do you use the dimensions of physical quantities to check equations and predict relationships?The dimensions of physical quantities in terms of mass, length and time, checking equations for dimensional consistency, and using dimensional analysis to find the form of a relationship.9 min answer →
- How do you use conservation of momentum and the coefficient of restitution to analyse collisions and impulses?Conservation of linear momentum, impulse as change in momentum, the coefficient of restitution and Newton's experimental law, direct and oblique impacts, and successive collisions.11 min answer →
- How do work, energy and power relate, and how do you use the work-energy principle and conservation of energy?Work done by a force, kinetic and potential energy, the work-energy principle, conservation of mechanical energy, power as the rate of doing work, and work done against resistance.10 min answer →
Further statistics
Module overview →- How do you use the chi-squared test for goodness of fit and for independence in a contingency table?The chi-squared statistic, goodness of fit tests for given distributions, contingency tables and tests for independence, degrees of freedom, and Yates' correction for a two by two table.11 min answer →
- How do you build a confidence interval for a population mean, and what does the confidence level actually mean?Confidence intervals for a population mean with known variance, the meaning of a confidence level, the effect of sample size and confidence level on width, and using the t distribution when the variance is unknown.10 min answer →
- How do you describe a discrete random variable, and how do you find its expectation and variance?Probability distributions of discrete random variables, the expectation and variance, the effect of linear coding, and expectation and variance of functions of a random variable.10 min answer →
- How do you carry out hypothesis tests on a Poisson mean and on a population mean, and what do Type I and Type II errors mean?Hypothesis tests for the mean of a Poisson distribution, tests for a population mean using the normal distribution, one-tailed and two-tailed tests, and the meaning of Type I and Type II errors.11 min answer →
- When is the Poisson distribution the right model, and how do you calculate and combine Poisson probabilities?The Poisson distribution as a model for random events, its mean and variance, calculating probabilities, the sum of independent Poisson variables, and the Poisson approximation to the binomial.10 min answer →