England · AQAQ&A
Further MathsQ&A by dot point
A short Q&A bank for every England Further Maths syllabus dot point. Each question and answer is drawn directly from our worked dot-point page, so you can scan key concepts before opening the long-form answer.
Core pure
- 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.0Q&A pairs
- 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.0Q&A pairs
- Roots of polynomials and their relationships to coefficients, summation of series using standard results, the method of differences, partial fractions and the Maclaurin series.1Q&A pairs
- 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.0Q&A pairs
- 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.0Q&A pairs
- Definitions of hyperbolic functions in terms of exponentials, their graphs and identities, inverse hyperbolic functions in logarithmic form, and differentiation and integration involving them.0Q&A pairs
- Matrix arithmetic, determinants, inverses of 2x2 and 3x3 matrices, matrices as linear transformations, invariant points and lines, and solving systems of linear equations.0Q&A pairs
- Polar coordinates and the relationship with Cartesian coordinates, sketching polar curves, and finding areas enclosed by polar curves using integration.1Q&A pairs
- 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.1Q&A pairs
Discrete mathematics
- Activity networks, forward and backward passes to find earliest and latest times, the critical path, float, and resource scheduling with Gantt charts.0Q&A pairs
- 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.0Q&A pairs
- 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.0Q&A pairs
- Formulating linear programming problems, the feasible region and graphical solution, the vertex (objective line) method, slack variables, and the simplex algorithm for maximisation.0Q&A pairs
- 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.0Q&A pairs
Further mechanics
- 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.0Q&A pairs
- 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.0Q&A pairs
- 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.0Q&A pairs
- 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.0Q&A pairs
- 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.0Q&A pairs
Further statistics
- 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.0Q&A pairs
- 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.0Q&A pairs
- 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.0Q&A pairs
- 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.0Q&A pairs
- 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.1Q&A pairs