Mathematics (AS & A Level) revision notes

One chapter for each section of the 9709 syllabus, with worked examples. Read a chapter, then practise it with real past-paper questions.

1. 1QuadraticsPaper 1. Completing the square, the discriminant, quadratic equations and inequalities, simultaneous equations with one linear and one quadratic, and equations that are quadratic in a function of x.7 min read
2. 2FunctionsPaper 1. Domain and range, composite functions, one-one functions and their inverses, and transformations of the graph of y = f(x).6 min read
3. 3Coordinate geometryPaper 1. Straight lines in all their forms, parallel and perpendicular lines, the equation of a circle, tangents, and where lines and curves meet.5 min read
4. 4Circular measurePaper 1. Radians, converting to and from degrees, arc length, sector area, and how to find the area of a segment.4 min read
5. 5TrigonometryPaper 1. Graphs of sine, cosine and tangent, exact values, inverse notation and principal values, the two basic identities, and solving trigonometric equations in an interval.6 min read
6. 6SeriesPaper 1. The binomial expansion for a positive integer power, arithmetic and geometric progressions, their sums, and the sum to infinity of a convergent geometric progression.5 min read
7. 7DifferentiationPaper 1. The gradient as a limit, the power rule and chain rule, tangents and normals, increasing and decreasing functions, rates of change, and stationary points.5 min read
8. 8IntegrationPaper 1. Integration as reverse differentiation, finding the constant, definite integrals, areas under and between curves, and volumes of revolution.4 min read
9. 9AlgebraPapers 2 and 3. The modulus function, polynomial division, the factor and remainder theorems. Paper 3 only, partial fractions and the binomial expansion for any rational power.8 min read
10. 10Logarithms and exponentialsPapers 2 and 3. Logarithms and indices, the laws of logarithms, e to the x and ln x, equations with the unknown in a power, and turning a relationship into a straight line.5 min read
11. 11Further trigonometryPapers 2 and 3. Secant, cosecant and cotangent, the identities that use them, the addition and double angle formulae, and writing a sin x + b cos x as a single wave.5 min read
12. 12Further differentiationPapers 2 and 3. Derivatives of exponentials, logarithms and trigonometric functions, the product and quotient rules, and curves defined parametrically or implicitly. Paper 3 adds the inverse tangent.4 min read
13. 13Further integrationPapers 2 and 3. Standard integrals, using trigonometric identities, and the trapezium rule. Paper 3 adds partial fractions, integrals of f'(x)/f(x), integration by parts and substitution.5 min read
14. 14Numerical solution of equationsPapers 2 and 3. Locating a root with a graph or a sign change, and using an iterative formula to find a root to a required accuracy.4 min read
15. 15VectorsPaper 3. Vector notation and operations, magnitude and unit vectors, the equation of a line, whether lines meet, and the scalar product.5 min read
16. 16Differential equationsPaper 3. Writing a rate of change as a differential equation, separating the variables, using an initial condition, and interpreting the solution in context.4 min read
17. 17Complex numbersPaper 3. Real and imaginary parts, arithmetic in Cartesian form, conjugate roots, the Argand diagram, modulus and argument, polar and exponential forms, square roots, and loci.6 min read
18. 18Forces and equilibriumPaper 4. Identifying forces, components and resultants, equilibrium of a particle, the normal contact force and friction, limiting equilibrium, and Newton's third law.6 min read
19. 19KinematicsPaper 4. Distance, speed, displacement, velocity and acceleration, motion graphs, the constant acceleration formulae, and using calculus when the acceleration varies.6 min read
20. 20MomentumPaper 4. Linear momentum as a vector, conservation of momentum in a direct impact, particles that coalesce or rebound, and collisions in a chain.4 min read
21. 21Newton's laws of motionPaper 4. Newton's second law, mass and weight, motion on slopes and in lifts, and connected particles joined by strings and rods.6 min read
22. 22Energy, work and powerPaper 4. Work done by a force, kinetic and gravitational potential energy, the work-energy principle, conservation of energy, and power with cars on hills.5 min read
23. 23Representation of dataPaper 5. Stem-and-leaf diagrams, box-and-whisker plots, histograms, cumulative frequency graphs, measures of centre and spread, and the mean and standard deviation from totals, including coded totals.6 min read
24. 24Permutations and combinationsPaper 5. Arranging objects in a line, with repeated items and with restrictions, and selecting groups with and without conditions.6 min read
25. 25ProbabilityPaper 5. Probability by counting, the addition and multiplication rules, exclusive and independent events, tree diagrams, and conditional probability.4 min read
26. 26Discrete random variablesPaper 5. Probability distribution tables, expectation and variance, the binomial distribution, and the geometric distribution.5 min read
27. 27The normal distributionPaper 5. Modelling continuous data with a normal distribution, using tables, finding probabilities, working back to the mean or standard deviation, and approximating the binomial with a continuity correction.4 min read
28. 28The Poisson distributionPaper 6. Poisson probabilities, the mean and variance, when the model fits, and approximations using the Poisson for the binomial and the normal for the Poisson.4 min read
29. 29Linear combinations of random variablesPaper 6. The expectation and variance of aX + b and aX + bY, and how normal and Poisson variables combine, including the difference between X1 + X2 and 2X.4 min read
30. 30Continuous random variablesPaper 6. Probability density functions and their properties, finding probabilities, the mean, the variance and the median by integration.4 min read
31. 31Sampling and estimationPaper 6. Samples and populations, why randomness matters, the distribution of the sample mean, the Central Limit Theorem, unbiased estimates, and confidence intervals for a mean and a proportion.5 min read
32. 32Hypothesis testsPaper 6. The language of testing, one-tailed and two-tailed tests, tests using binomial and Poisson probabilities and normal approximations, tests for a mean, and Type I and Type II errors.7 min read