Free A-level Maths and Further Maths revision

Written explainers for all 65 core topics. Topics are grouped by mathematical content. Use the exam-board maps to match them to your qualification.

Further Maths: the Further Pure core

Core Pure: the compulsory Pearson Edexcel 9FM0 Core Pure content. 24 topics.

Further Maths: the option papers

Option papers: all eight Pearson Edexcel 9FM0 option papers. A candidate takes two. 82 topics.

PROOF

• The structure of proof: deduction and exhaustion
• Disproof and proof by contradiction

ALGEBRA AND FUNCTIONS

• Indices and surds
• Quadratic functions
• Simultaneous equations and inequalities
• Polynomials and the factor theorem
• Graphs, proportion and transformations
• Functions, inverses and the modulus
• Partial fractions
• Functions in modelling

COORDINATE GEOMETRY

• Straight lines
• Circles
• Parametric equations

SEQUENCES AND SERIES

• The binomial expansion
• The general binomial expansion
• Sequences and sigma notation
• Arithmetic series
• Geometric series

TRIGONOMETRY

• Triangles and the sine and cosine rules
• Trigonometric graphs and equations
• Radians, arcs and small angles
• Reciprocal and inverse trigonometric functions
• Compound angles and the harmonic form
• Trigonometric modelling

EXPONENTIALS AND LOGARITHMS

• Exponential functions and e
• Logarithms and their laws
• Log graphs and exponential models

DIFFERENTIATION

• The derivative from first principles
• Differentiating powers of x
• Tangents, turning points and curve behaviour
• Differentiating trig, exponentials and logs
• The product, quotient and chain rules
• Implicit and parametric differentiation
• Rates of change and building differential equations

INTEGRATION

• Integration as antidifferentiation
• Definite integrals and areas
• Integrating standard functions
• Integration by substitution and by parts
• Integrating rational functions
• Areas, parametric curves and the limit of a sum
• Solving differential equations

NUMERICAL METHODS

• Locating roots and iteration
• The Newton-Raphson method
• The trapezium rule

VECTORS

• Vectors in two dimensions
• Vectors in three dimensions

STATISTICS

• Sampling and the large data set
• Measures of location and spread
• Representing and interpreting data
• Correlation and regression
• Probability and Venn diagrams
• Conditional probability
• The binomial distribution
• The normal distribution
• Hypothesis testing with the binomial
• Hypothesis testing: correlation and the normal

MECHANICS

• Modelling, quantities and units
• Kinematics with constant acceleration
• Kinematics with variable acceleration
• Forces and Newton's laws
• Connected particles and pulleys
• Projectiles
• Friction and inclined planes
• Statics of a particle
• Moments

FURTHER PROOF

• Proof by induction: sums and series
• Induction: divisibility and matrices

COMPLEX NUMBERS

• Complex arithmetic and the Argand diagram
• Modulus, argument and loci
• De Moivre's theorem and trigonometric identities
• Roots of unity and complex roots

MATRICES

• Matrix algebra and transformations
• Determinants and inverses
• Systems of equations and invariance

FURTHER ALGEBRA AND SERIES

• Roots of polynomials
• Summing series and the method of differences
• Maclaurin series

FURTHER VECTORS

• Lines and planes in three dimensions
• Intersections, angles and distances

FURTHER CALCULUS

• Volumes of revolution
• Mean values and improper integrals
• Calculus with inverse trigonometric functions

HYPERBOLIC FUNCTIONS

• Hyperbolic functions and identities
• Calculus with hyperbolic functions

POLAR COORDINATES

• Polar curves
• Areas with polar coordinates

DIFFERENTIAL EQUATIONS

• First order equations and integrating factors
• Second order equations
• Modelling with differential equations

FURTHER PURE 1

• The t-formulae
• Taylor series
• Limits and L'Hôpital's rule
• Leibniz's theorem and the Weierstrass substitution
• Series solutions of differential equations
• Conic sections
• Tangents, normals and loci of conics
• The vector product and the scalar triple product
• Numerical methods for differential equations
• Inequalities and inequations

FURTHER STATISTICS 1

• Discrete random variables and expectation
• The Poisson distribution
• Geometric and negative binomial distributions
• Hypothesis tests for Poisson and geometric models
• The Central Limit Theorem
• Goodness-of-fit tests
• Contingency tables
• Probability generating functions
• Type I and Type II errors, test size and power

FURTHER PURE 2

• Groups and their axioms
• Subgroups, Lagrange's theorem and isomorphism
• Reduction formulae
• Arc length and surface area
• Functions of two variables
• Eigenvalues and eigenvectors
• Diagonalisation and the Cayley-Hamilton theorem
• Further loci and regions in the Argand diagram
• Transformations of the complex plane
• The Euclidean algorithm and Bézout's identity
• Modular arithmetic and Fermat's little theorem
• Combinatorics
• Recurrence relations

FURTHER STATISTICS 2

• Least squares regression and residuals
• Continuous random variables: density and distribution functions
• Mean, variance and skewness of continuous variables
• The continuous uniform distribution
• Correlation coefficients: product moment and Spearman
• Testing a correlation coefficient
• Combinations of normal random variables
• Estimators, standard error and confidence intervals
• Comparing two normal means
• Testing variances: chi-squared and the F-distribution
• Confidence intervals and tests with the t-distribution
• Non-parametric tests

FURTHER MECHANICS 1

• Momentum and impulse
• Impulse and momentum as vectors
• Work, energy and power
• Hooke's law and elastic strings
• Elastic potential energy
• Direct impact and Newton's law of restitution
• Successive impacts and impacts with a wall
• Oblique impact and impact with a smooth surface

FURTHER MECHANICS 2

• Angular speed and horizontal circular motion
• Motion in a vertical circle
• Centre of mass of a discrete distribution
• Centres of mass of plane figures and frameworks
• Centres of mass by integration
• Equilibrium, suspension, toppling and sliding
• Newton's laws with a variable force
• Simple harmonic motion
• Oscillations on strings and springs
• Further kinematics: acceleration as a function of x, t or v

DECISION MATHEMATICS 1

• Algorithms, sorting and bin packing
• Further packing and the knapsack problem
• Graphs: order, Eulerian paths and planarity
• Minimum spanning trees: Prim and Kruskal
• Shortest paths: Dijkstra and Floyd
• Route inspection
• The travelling salesman problem
• Critical path analysis
• Float, Gantt charts and scheduling
• Linear programming: formulation and graphical solution
• The Simplex algorithm

DECISION MATHEMATICS 2

• Transportation problems and the north-west corner method
• The stepping-stone method
• Allocation and the Hungarian algorithm
• Flows in networks: cuts and capacity
• Maximum flow and the labelling procedure
• Dynamic programming and Bellman's principle
• Game theory: play safe and stable solutions
• Mixed strategies in game theory
• Decision trees, expected monetary value and utility