England · Pearson EdexcelSyllabus
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 represent, manipulate and find roots of them?Arithmetic of complex numbers, the Argand diagram, modulus-argument form, de Moivre's theorem, nth roots, complex roots of polynomials and loci.13 min answer →
- How do you solve first and second order differential equations, and model oscillations with them?First order linear equations by integrating factor, second order constant-coefficient equations using the auxiliary equation, complementary function and particular integral, and modelling damped and forced oscillations and coupled systems.13 min answer →
- How do you sum series, relate roots to coefficients, and use the method of differences?Summing series of powers of integers, relationships between roots and coefficients of polynomials, transforming equations with new roots, and the method of differences.12 min answer →
- How do you extend integration to improper integrals, volumes, arc lengths and mean values?Improper integrals, volumes of revolution, the mean value of a function, integration using partial fractions, and the derivation of standard inverse trig and hyperbolic integrals.12 min answer →
- How do you describe lines and planes in three dimensions and find angles, intersections and distances?Vector and Cartesian equations of lines and planes, the scalar and vector products, angles between lines and planes, intersections, and shortest distances including between skew lines.13 min answer →
- What are the hyperbolic functions, and how do you differentiate, integrate and invert them?Definitions of sinh, cosh and tanh from exponentials, hyperbolic identities, logarithmic forms of the inverse functions, and differentiation and integration of hyperbolic functions.12 min answer →
- How do matrices encode linear 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 linear systems.13 min answer →
- How do polar coordinates describe curves, and how do you find areas they enclose?Polar coordinates and curves, conversion to and from Cartesian form, sketching cardioids and spirals, tangents parallel and perpendicular to the initial line, and areas enclosed by polar curves.12 min answer →
- How does mathematical induction prove a statement for all positive integers?The structure of proof by induction, applied to summation formulae, divisibility results, recurrence relations and powers of matrices, with rigorous base case, inductive step and conclusion.12 min answer →
Further Mechanics
Module overview →- How do you analyse motion in a circle at constant and varying speed?Angular speed, acceleration towards the centre, motion in a horizontal circle, the conical pendulum, and motion in a vertical circle with energy conservation.12 min answer →
- How does the coefficient of restitution determine the outcome of collisions?Newton's experimental law of restitution, direct and oblique impact of smooth spheres, impact with a fixed surface, and kinetic energy lost in a collision.12 min answer →
- How do momentum and impulse describe collisions and forces acting over time?Momentum and impulse in one and two dimensions, the impulse-momentum principle, conservation of momentum, and impulse as the area under a force-time graph.12 min answer →
- How do work, energy and power connect forces, motion and time?Work done by a force, kinetic and potential energy, the work-energy principle, the conservation of mechanical energy, and power as the rate of doing work.12 min answer →
Further Pure options
Module overview →- How are the conic sections defined, and how do you work with their tangents, normals and parametric forms?The parabola, ellipse and hyperbola in Cartesian and parametric form, foci and directrices, tangents and normals, and the rectangular hyperbola.12 min answer →
- How do you approximate roots and integrals numerically when no exact method works?Solving equations numerically by interval bisection, linear interpolation and the Newton-Raphson method, and approximating definite integrals using Simpson's rule and the mid-ordinate rule.12 min answer →
- How do you sum trigonometric series and work with the t-substitution and inverse trig functions?The t-substitution for trigonometric integrals and equations, summing series of sines and cosines, the general solution of trigonometric equations, and inverse trigonometric functions.12 min answer →
- How do you build power series approximations of functions and use them to solve differential equations?Maclaurin and Taylor series of standard functions, finding series solutions of differential equations, and using series to approximate functions and limits.12 min answer →
Further Statistics
Module overview →- How do you test goodness of fit and independence using the chi-squared distribution?Goodness of fit tests, contingency tables and tests for independence using the chi-squared statistic, expected frequencies, degrees of freedom, and Yates' correction.12 min answer →
- How do you describe a discrete random variable and compute its expectation and variance?Discrete random variables and probability distributions, expectation and variance, the effect of linear coding, and expectation and variance of functions of a discrete variable.12 min answer →
- How do the geometric and negative binomial distributions model waiting times until successes?The geometric distribution as a model for the trial of the first success, the negative binomial distribution for the rth success, and their means and variances.12 min answer →
- How do the Poisson and binomial distributions model counts, and when does one approximate the other?The Poisson distribution as a model for random events, its mean and variance, the binomial distribution, the additive property of Poisson variables, and the Poisson approximation to the binomial.12 min answer →