A Level Maths Notes (9709)

A Level Maths is where mathematics stops being a list of methods and starts being an argument. These notes follow the Cambridge International AS & A Level Mathematics syllabus (9709) paper by paper: Pure Mathematics 1 and 3, Mechanics, and Probability & Statistics 1 and 2. Each note explains why a method works, not only how to use it, then shows exam-standard worked examples with the M, A and B marks the examiners award. Results given in the formula list (MF19) are tagged, so you know what to learn and what you can look up.

What every note gives you

• Organised by paper: P1, P3, Mechanics (P4), Statistics 1 (P5) and Statistics 2 (P6), with the syllabus code on every topic.
• Why, not just how. Short derivations and proofs, so methods make sense and you can adapt them to unfamiliar questions.
• Marked like the mark scheme. Worked examples show the M1, A1 and B1 marks, including the working you must show.
• MF19 tags. Formulas in the list of formulae are tagged, so you know what is given and what to learn.
• Examiner-report traps. Each note flags the errors Cambridge examiners report every session.
• Exam-style questions with answers at the end of every note.

Paper 1 · Pure Mathematics 1

1. 1.1QuadraticsCompleting the square, the discriminant, quadratic inequalities, line–curve intersections and equations that are quadratics in disguise, with the working Cambridge examiners expect to see.~13 min read · AS & A Level
2. 1.2Functions and transformationsDomain and range, one-one functions, inverse functions and their graphs, composite functions and when they exist, and the four graph transformations described in the exact words examiners credit.~11 min read · AS & A Level
3. 1.3Coordinate geometry: lines and circlesEquations of straight lines, parallel and perpendicular gradients, perpendicular bisectors, the equation of a circle in both forms, tangents, chords and where lines meet circles.~8 min read · AS & A Level
4. 1.4Circular measure: radians, arcs and sectorsWhat a radian really is, converting between degrees and radians, arc length, sector area and segment area, and the multi-step sector problems that combine them with triangles.~6 min read · AS & A Level
5. 1.5Trigonometry: graphs, identities and equationsGraphs of sine, cosine and tangent in degrees and radians, exact values, principal values of sin⁻¹, cos⁻¹ and tan⁻¹, the two key identities, and how to find every solution of a trig equation in an interval.~10 min read · AS & A Level
6. 1.6Series: binomial expansion, AP and GPThe binomial expansion for positive integer powers, specific terms and terms independent of x, arithmetic and geometric progressions, sums to infinity, and problems that link the two.~8 min read · AS & A Level
7. 1.7DifferentiationThe gradient as the limit of chord gradients, differentiating xⁿ for any rational n, the chain rule, tangents and normals, increasing and decreasing functions, stationary points and connected rates of change.~9 min read · AS & A Level
8. 1.8IntegrationIntegration as reverse differentiation, integrating (ax + b)ⁿ, finding the constant from a point, definite and improper integrals, areas between curves and lines, and volumes of revolution about either axis.~8 min read · AS & A Level

Paper 3 · Pure Mathematics 3

1. 3.1Algebra: modulus, polynomials and partial fractionsThe modulus function and its equations and inequalities, polynomial division, the factor and remainder theorems, partial fractions for the three syllabus forms, and the binomial expansion for any rational power with its range of validity.~11 min read · A Level
2. 3.2Logarithms and exponential functionsLogarithms as inverses of indices, the laws of logarithms, eˣ and ln x and their graphs, solving equations and inequalities with the unknown in an index, and turning y = kxⁿ and y = kaˣ into straight lines to find constants.~8 min read · A Level
3. 3.3Further trigonometrysec, cosec and cot and their graphs, the identities 1 + tan²θ ≡ sec²θ and 1 + cot²θ ≡ cosec²θ, the addition and double-angle formulae, and expressing a sin θ + b cos θ as R sin(θ ± α) to solve equations and find maxima.~8 min read · A Level
4. 3.4Further differentiationDerivatives of eˣ, ln x, sin x, cos x, tan x and tan⁻¹x, the chain, product and quotient rules together, implicit differentiation and parametric differentiation, with tangents, normals and stationary points.~6 min read · A Level
5. 3.5Further integrationIntegrating eˣ, 1/x, trig functions and 1/(a² + x²), using identities and partial fractions, the f′(x)/f(x) pattern, integration by parts, a given substitution, and the trapezium rule with over- and under-estimates.~8 min read · A Level
6. 3.6Numerical solution of equationsLocating a root by a change of sign or by sketching two graphs, rearranging an equation into x = F(x), running the iteration xₙ₊₁ = F(xₙ) to a stated accuracy, and understanding why some rearrangements converge and others do not.~6 min read · A Level
7. 3.7VectorsVectors in two and three dimensions, magnitude and unit vectors, the scalar product and angles, the vector equation of a line, and deciding whether two lines are parallel, intersect or are skew, with the foot of a perpendicular and shortest distances.~6 min read · A Level
8. 3.8Differential equationsForming a differential equation from a rate-of-change statement, solving first-order equations by separating the variables, using a condition to find the particular solution, and interpreting the result in context.~5 min read · A Level
9. 3.9Complex numbersArithmetic with complex numbers, conjugates, modulus and argument, the Argand diagram, square roots of a complex number, conjugate roots of real polynomials, the forms r(cos θ + i sin θ) and re^{iθ}, and loci and regions.~6 min read · A Level

Paper 4 · Mechanics

1. 4.1Forces and equilibriumIdentifying the forces on a particle, resolving them into components, using the conditions for equilibrium, finding resultants, and modelling friction with F ≤ μR, including limiting equilibrium on rough horizontal and inclined surfaces.~6 min read · AS & A Level
2. 4.2Kinematics of motion in a straight lineDistance, displacement, speed, velocity and acceleration, displacement-time and velocity-time graphs, the constant-acceleration (suvat) equations including vertical motion under gravity, and calculus for motion with variable acceleration.~5 min read · AS & A Level
3. 4.3MomentumMomentum as mass × velocity, the principle of conservation of momentum for direct collisions in one dimension, particles that coalesce, and deciding whether further collisions happen.~4 min read · AS & A Level
4. 4.4Newton's laws of motionNewton's three laws, F = ma along the line of motion, weight, motion on rough and smooth slopes, lifts, and connected particles: towing systems and particles joined by a string over a pulley.~5 min read · AS & A Level
5. 4.5Energy, work and powerWork done by a force, kinetic and gravitational potential energy, the work–energy principle with resistances and friction, conservation of energy on smooth surfaces, and power as force × velocity for vehicles on level roads and hills.~5 min read · AS & A Level

Paper 5 · Probability & Statistics 1

1. 5.1Representation of dataStem-and-leaf diagrams, box-and-whisker plots, histograms with unequal class widths and cumulative frequency graphs; median, quartiles and IQR; mean and standard deviation from raw, grouped and coded data, and choosing the right measures.~5 min read · AS & A Level
2. 5.2Permutations and combinationsCounting arrangements with factorials, repeated items, items kept together or apart, and selections with nCr, including "at least" conditions and probabilities found by counting.~4 min read · AS & A Level
3. 5.3ProbabilityEvents, the addition rule, mutually exclusive and independent events, conditional probability, tree diagrams with and without replacement, and reversing a tree to find a "given" probability.~4 min read · AS & A Level
4. 5.4Discrete random variablesProbability distribution tables, expectation and variance of a discrete random variable, and the binomial and geometric distributions: when to use them, their probabilities, means and variances.~4 min read · AS & A Level
5. 5.5The normal distributionThe shape and properties of the normal distribution, standardising with z = (x − μ)/σ, using tables forwards and backwards, finding unknown μ and σ, and the normal approximation to the binomial with a continuity correction.~4 min read · AS & A Level

Paper 6 · Probability & Statistics 2

1. 6.1The Poisson distributionModelling random events with Po(λ), its conditions, scaling λ to a new interval, mean equal to variance, and using the Poisson to approximate the binomial and the normal to approximate the Poisson.~4 min read · A Level
2. 6.2Linear combinations of random variablesExpectation and variance of aX + b and of sums and differences of independent random variables, the difference between 3X and X₁ + X₂ + X₃, and the facts that sums of independent normal (or Poisson) variables are normal (or Poisson).~4 min read · A Level
3. 6.3Continuous random variablesProbability density functions, the area condition ∫f(x) dx = 1, probabilities as areas, the mean and variance by integration, and the median and other percentiles of a continuous random variable.~3 min read · A Level
4. 6.4Sampling and estimationPopulations and random samples, the distribution of the sample mean, the Central Limit Theorem, unbiased estimates of the mean and variance, and confidence intervals for a population mean and a population proportion.~4 min read · A Level
5. 6.5Hypothesis testsNull and alternative hypotheses, significance levels, one- and two-tailed tests, critical regions, tests for a binomial proportion, a Poisson mean and a normal mean, and Type I and Type II errors with their probabilities.~5 min read · A Level

Exam special notes

Built from the Cambridge examiner reports and mark schemes: where students actually lose marks, and how to stop it.

1. ★A Level Maths 9709: Exam Technique and the MF19 Formula ListHow 9709 is examined: the paper routes, timings and marks, the accuracy rules (3 s.f., angles to 1 d.p.), what the MF19 formula list gives you and what you must learn, calculator rules, and the errors the examiner reports flag most.~6 min read · Exam guide

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