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A-Level Mathematics formula sheet
Look up and print the algebra, calculus, geometry and statistics results provided in A-Level Mathematics examinations, including Further Mathematics.
H1 Mathematics revision reference
Look up 2027 H1 Mathematics formulas, model conditions, Paper 1 format and how to show calculator working.
H2 Further Mathematics revision reference
Look up the taught Further Mathematics results, from polar form and linear algebra to calculus, numerical methods, distributions and tests.
H2 Mathematics revision reference
Look up 2027 H2 Mathematics functions, sequences, vectors, complex numbers, calculus and statistics.
H3 Mathematics revision reference
Look up statement forms, negations, proof methods, classical inequalities, congruences and problem-solving heuristics.
Primary 1 Mathematics revision reference
Quick reminders for counting, number sentences, shapes, money, time and picture graphs.
Primary 2 Mathematics revision reference
Look up place value, times-table facts, fraction words, units, shapes and scaled picture graphs.
Primary 3 Mathematics revision reference
Recall division with remainders, equivalent fractions, measurement conversions, area, perimeter and bar graphs.
Primary 4 Mathematics revision reference
Check rounding, factors, fractions, decimal methods, composite rectangles, angles and symmetry.
Primary 5 Foundation Mathematics revision reference
Find reminders for whole numbers, fractions, decimals, rates, time, rectangles and angles.
Primary 5 Mathematics revision reference
Look up fraction products, percentages, rates, triangle area, cuboid volume and angle facts.
Primary 6 Foundation Mathematics revision reference
Check fraction division, decimals, percentages, triangle area, volume, angles and averages.
Primary 6 Mathematics revision reference
Revise fraction division, reverse percentages, ratios, algebra, circles, volume and averages.
G1 Mathematics revision reference
Find number, algebra, geometry, trigonometry, statistics and probability reminders for 2027 G1 Mathematics.
G2 Additional Mathematics revision reference
Revise 2027 G2 Additional Mathematics algebra, trigonometry, coordinates and polynomial calculus.
G2 Mathematics revision reference
Look up 2027 G2 Mathematics formulas and notation, with Paper 2 Section B distinctions.
G3 Additional Mathematics revision reference
Revise 2027 G3 Additional Mathematics algebra, binomial expansions, logarithms, trigonometry and calculus.
G3 Mathematics revision reference
Look up 2027 G3 Mathematics formulas, notation, method conditions and common mistakes.
H1 Mathematics foundations check
Try a short H1 Mathematics sample to choose what to study next. This check does not measure course mastery.
H2 Mathematics foundations check
Try a short H2 Mathematics sample to choose what to study next. This check does not measure course mastery.
Primary 1 Mathematics foundations check
Try a short Primary 1 Mathematics sample to choose what to study next. This check does not measure course mastery.
Primary 2 Mathematics foundations check
Try a short Primary 2 Mathematics sample to choose what to study next. This check does not measure course mastery.
Primary 3 Mathematics foundations check
Try a short Primary 3 Mathematics sample to choose what to study next. This check does not measure course mastery.
Primary 4 Mathematics foundations check
Try a short Primary 4 Mathematics sample to choose what to study next. This check does not measure course mastery.
Primary 5 Foundation Mathematics foundations check
Try a short Primary 5 Foundation Mathematics sample to choose what to study next. This check does not measure course mastery.
Primary 5 Mathematics foundations check
Try a short Primary 5 Mathematics sample to choose what to study next. This check does not measure course mastery.
Primary 6 Foundation Mathematics foundations check
Try a short Primary 6 Foundation Mathematics sample to choose what to study next. This check does not measure course mastery.
Primary 6 Mathematics foundations check
Try a short Primary 6 Mathematics sample to choose what to study next. This check does not measure course mastery.
G1 Mathematics foundations check
Try a short G1 Mathematics sample to choose what to study next. This check does not measure course mastery.
G2 Additional Mathematics foundations check
Try a short G2 Additional Mathematics sample to choose what to study next. This check does not measure course mastery.
G2 Mathematics foundations check
Try a short G2 Mathematics sample to choose what to study next. This check does not measure course mastery.
G3 Additional Mathematics foundations check
Try a short G3 Additional Mathematics sample to choose what to study next. This check does not measure course mastery.
G3 Mathematics foundations check
Try a short G3 Mathematics sample to choose what to study next. This check does not measure course mastery.
H1 Mathematics
Work through 7 H1 Mathematics topics, from Functions and graphs to Correlation and regression, with clear explanations, worked examples and practice.
Correlation and regression
Use scatter, correlation and directional least-squares fits to make contextual predictions and recognise their limits.
Paired-data regression results · H1 Mathematics
Practise paired-data regression results in H1 Mathematics, check independently, and return later to review. This check covers selected skills, not the whole course topic.
Correlation, regression and prediction
Interpret scatter and residuals, fit each prediction direction, and assess range, outliers and causal claims.
Differentiation
Connect tangent gradients and chain rules to stationary-point tests, numerical methods and constrained optimisation.
Exact derivatives, tangents and polynomial extrema · H1 Mathematics
Practise exact derivatives, tangents and polynomial extrema in H1 Mathematics, check independently, and return later to review. This check covers selected skills, not the whole course topic.
Derivatives and the chain rule
Distinguish height from gradient, find first and second derivatives with their input conditions, and recover tangents from point or slope conditions.
Stationary values and optimisation
Classify stationary points, resolve inconclusive tests and compare feasible candidates, endpoints and ties in optimisation.
Functions and graphs
Connect exponential and logarithmic models with graph features, numerical solutions and exact quadratic methods.
Logarithms, quadratics and linear systems · H1 Mathematics
Practise logarithms, quadratics and linear systems in H1 Mathematics, check independently, and return later to review. This check covers selected skills, not the whole course topic.
Function rules and natural logarithms
Connect function rules, exponential/logarithm graphs and valid log algebra to growth, decay and fitted model parameters.
Graph features and numerical solutions
Interpret graph features, numerical boundaries and model equations, then check precision and feasible solutions.
Linear systems from context
Turn three conditions into a system of linear equations, solve it with a graphing calculator and check for repeated or contradictory conditions.
Integration
Reverse differentiation, evaluate exact or numerical integrals and choose the correct boundaries for non-negative regions.
Integration and positive-strip area models · H1 Mathematics
Practise integration and positive-strip area models in H1 Mathematics, check independently, and return later to review. This check covers selected skills, not the whole course topic.
Exponential integrals and between-curve areas
Check inner factors, exact and numerical integrals, and changing upper/lower boundaries for regions on or above the axis.
Probability
Choose counting structures, update probability models and use a condition to identify the relevant sample space.
Probability models
Choose binomial and normal models, interpret their parameters and translate the event before calculating its probability.
Binomial models and calculator tails
Choose a binomial model, translate exact and conditional tails, recover parameters and combine component and box counts.
Binomial and normal results · H1 Mathematics
Practise binomial and normal results in H1 Mathematics, check independently, and return later to review. This check covers selected skills, not the whole course topic.
Normal models and linear combinations
Interpret normal areas and inverse tails, recover parameters, and compare transformed or independent measurements under a model.
Counting and finite-event results · H1 Mathematics
Practise counting and finite-event results in H1 Mathematics, check independently, and return later to review. This check covers selected skills, not the whole course topic.
Counting and conditional probability
Choose counting structures, interpret event regions and conditional trees, and use the observed condition as the reference set.
Restricted line arrangements
Count lines of distinct objects with together, apart, position and order restrictions, then turn the counts into probabilities.
Sampling and inference
Use sampling distributions, unbiased variance estimates and planned population-mean tests to draw contextual conclusions in H1 Mathematics.
Sampling results and planned mean tests · H1 Mathematics
Practise sampling results and planned mean tests in H1 Mathematics, check independently, and return later to review. This check covers selected skills, not the whole course topic.
Population-mean hypothesis tests
Carry out planned one- and two-tailed mean tests, distinguish exact and estimated-spread models, and interpret the evidence in context.
Sampling means and unbiased variance
Choose sampling distributions, estimate population variance from raw or coded data, and plan precision without confusing bias with spread.
H2 Further Mathematics
Lessons for H2 Further Mathematics cover every topic in its syllabus. Practice questions for this course are not available yet.
Applications of integration
Find arc lengths, volumes by discs or shells, and surface areas of revolution.
Arc length, volumes and surface area
Find arc lengths, choose discs or shells for volumes of revolution about either axis, and find surface areas of revolution.
Complex numbers in polar form
Write complex numbers in polar and exponential form, use de Moivre’s theorem for powers, roots and identities, and sketch loci and regions in the Argand diagram.
Loci in the Argand diagram
Read moduli and arguments of differences as distances and directions, then sketch circles, bisectors, half-lines and regions.
Polar form and de Moivre’s theorem
Multiply complex numbers by rotating and scaling, then use de Moivre’s theorem for powers, every nth root and multiple-angle identities.
Continuous random variables
Work with probability density and cumulative distribution functions, and uniform and exponential models.
Density functions and exponential models
Work with probability density and cumulative distribution functions, medians and modes, and uniform and exponential models.
Linear differential equations
Solve first- and second-order linear differential equations, use given substitutions, and analyse exponential and logistic growth with phase lines.
First-order equations and growth models
Solve first-order linear equations with an integrating factor, use given substitutions, and analyse growth models with phase lines and harvesting.
Second-order linear differential equations
Solve constant-coefficient equations from the auxiliary equation, add a particular integral and fit initial conditions.
Poisson and geometric distributions
Model counts and waiting times with Poisson and geometric distributions, including their conditions and properties.
Poisson and geometric models
Decide when Poisson and geometric models suit a situation, and use their probabilities, means, variances and special properties.
Functions of two variables
Use partial derivatives, gradients, tangent planes and stationary points of surfaces.
Partial derivatives and stationary points
Differentiate surfaces partially, use gradients, tangent planes and linear estimates, and classify maxima, minima and saddle points.
Hypothesis tests and confidence intervals
Carry out t-tests and χ² tests and construct confidence intervals.
χ² tests
Carry out χ² goodness-of-fit tests and contingency-table tests of independence.
Paired and two-sample tests
Test a mean difference with a paired t-test and compare two independent means with a normal test.
t-tests and confidence intervals
Test a mean from a small normal sample with a t-test, build confidence intervals and connect them to two-tailed tests.
Matrices and linear spaces
Solve linear systems by row reduction, represent linear transformations, work with linear spaces, rank and nullity, and diagonalise matrices.
Eigenvalues, eigenvectors and diagonalisation
Find the directions a matrix only stretches, diagonalise it as QDQ⁻¹ and use that form to find its powers.
Linear transformations and linear spaces
Represent linear transformations by matrices, find null spaces and ranges, and test subspaces, independence, span and bases.
Row reduction, rank and null space
Solve linear systems by row reduction, interpret unique, infinite and empty solution sets, find determinants and inverses, and relate rank to the null and column spaces.
Non-parametric tests
Use sign tests and Wilcoxon matched-pair signed-rank tests.
Sign and Wilcoxon tests
Test a median with the sign test and paired data with the Wilcoxon signed-rank test, and weigh them against a t-test.
Numerical methods
Locate and approximate roots, approximate integrals by the trapezium and Simpson's rules, and step through differential equations by Euler methods.
Locating and approximating roots
Locate roots by sign change, approximate them by interpolation, Newton–Raphson and fixed-point iteration, and explain when each fails.
Numerical integration and Euler methods
Approximate integrals with the trapezium rule and Simpson's rule, and differential equations with Euler and improved Euler steps.
Polar coordinates
Sketch simple polar curves and find the areas they enclose and their arc lengths.
Polar curves, area and arc length
Sketch polar curves with non-negative r, then find the area they sweep out and their arc length.
Recurrence relations
Solve first- and second-order linear recurrences, including those reduced by a given substitution, analyse their long-term behaviour and model with them.
Linear recurrence relations
Solve first-order relations through their fixed point and second-order relations through their characteristic equation, then model with them.
Recurrences solved by substitution
Use a given substitution to turn a recurrence into a linear second-order relation, solve it and return to the original sequence.
H2 Mathematics
Work through 12 H2 Mathematics topics, from Functions and graphs to Correlation and regression, with clear explanations, worked examples and practice.
Complex numbers
Calculate with complex numbers and conjugate roots, then interpret modulus, argument and transformations on an Argand diagram.
Argand diagrams, modulus and argument
Read complex numbers as points and distinguish reflection, negation, translation and multiplication by i.
Complex arithmetic, equations, arguments and polynomial roots · H2 Mathematics
Practise complex arithmetic, equations, arguments and polynomial roots in H2 Mathematics, check independently, and return later to review. This check covers selected skills, not the whole course topic.
Complex arithmetic and polynomial roots
Use i² = −1, compare real and imaginary parts and retain conjugate roots for real-coefficient polynomials.
Correlation and regression
Use directional least-squares fits and square, reciprocal or log transformations, then assess predictions in their original setting.