A Level Mathematics

This topic covers the laws of indices, manipulation of surds, and standard algebraic simplification and rearrangement. Exams typically test fluency with exact forms, valid use of index laws, and accurate manipulation of algebraic expressions.

• Apply the laws of indices for positive, zero, negative and fractional powers.
• Simplify algebraic expressions involving indices and write answers in equivalent exact forms.
• Simplify surds, including expressions of the form $a\sqrt{b}$, by extracting square factors.
• Rationalise denominators involving monomial and binomial surds.
• Expand, factorise and simplify algebraic expressions, including common-factor and quadratic forms.
• Rearrange formulae to make a specified subject.

This topic covers quadratic expressions and equations, their graphical and algebraic properties, and systems of simultaneous equations including linear-quadratic systems. Exams test solving methods, interpretation of roots, and links between equations and graphs.

• Factorise quadratic expressions and solve quadratic equations by factorisation, completing the square and the quadratic formula.
• Determine the discriminant and interpret the nature and number of roots of a quadratic equation.
• Complete the square to write a quadratic function in vertex form and identify turning points.
• Solve simultaneous equations involving one linear and one quadratic equation algebraically.
• Interpret the solutions of simultaneous equations as points of intersection of graphs.
• Form and solve quadratic equations arising from contextual or geometric problems.

This topic covers the solution of algebraic equations and inequalities, including polynomial, rational and modulus forms. Exam questions often require interval-based reasoning, sign analysis, and careful handling of piecewise definitions from modulus expressions.

• Solve polynomial and rational equations, identifying and rejecting invalid roots where necessary.
• Solve linear and quadratic inequalities and express solutions using set or interval notation.
• Determine solution sets for inequalities involving products or quotients by sign analysis.
• Interpret and solve equations and inequalities involving modulus, including forms such as $|f(x)|=a$, $|f(x)|<a$ and $|f(x)|>a$.
• Represent modulus functions as piecewise-defined relationships and use this to solve related problems.
• Solve systems of equations or inequalities involving modulus or algebraic constraints.

This topic covers polynomial functions and the use of the factor theorem and remainder theorem. Exams test exact substitution, identification of factors, and construction or simplification of polynomials using given conditions.

• Evaluate polynomial functions for given values of the variable.
• Apply the remainder theorem to find the remainder when a polynomial is divided by a linear factor.
• Apply the factor theorem to test whether a given linear expression is a factor of a polynomial.
• Use known factors or roots to factorise cubic and higher-degree polynomials.
• Determine unknown coefficients in a polynomial using factor or remainder conditions.
• Interpret repeated roots and related factor structure where required.

This topic covers division of polynomials and decomposition of rational expressions into partial fractions. In exams, students must manipulate improper and proper rational functions accurately, often as preparation for integration or algebraic solution.

• Divide one polynomial by another using long division or algebraic division.
• Express an improper rational function as a polynomial plus a proper rational remainder.
• Decompose proper rational functions into partial fractions when the denominator has distinct linear factors.
• Decompose rational functions into partial fractions when the denominator includes repeated linear factors.
• Decompose rational functions into partial fractions when the denominator includes irreducible quadratic factors, where required by the course route.
• Use partial fractions in algebraic simplification and related pure-mathematics problems.

This topic covers the formal idea of a function, notation, domain and range, and the composition of functions. Exams assess whether students can interpret mappings, determine valid inputs and outputs, and combine functions correctly.

• Use function notation, including $f(x)$, $fg$-style mapping language where appropriate, and mapping diagrams or set-based descriptions.
• Determine the domain and range of functions defined algebraically or graphically.
• Evaluate functions for specified inputs and find the images of sets under a function.
• Form and simplify composite functions such as $f(g(x))$ and $g(f(x))$.
• Determine the domain of a composite function by considering restrictions from each stage of the mapping.
• Interpret functions in context and distinguish functions from non-functions where appropriate.

This topic covers one-to-one functions and their inverses, together with transformations of graphs. Exams test algebraic and graphical understanding of inverses and accurate description of translations, stretches and reflections.

• Determine whether a function is one-to-one on a stated domain and explain when an inverse exists.
• Find inverse functions algebraically and state any necessary domain or range restrictions.
• Verify inverse relationships using composition.
• Sketch or interpret the graph of an inverse function as a reflection in the line $y=x$.
• Apply and describe graph transformations, including translations, stretches parallel to the axes, and reflections in the axes.
• Relate algebraic forms such as $y=f(x)+a$, $y=f(x+a)$, $y=af(x)$ and $y=f(ax)$ to their graph transformations.

This topic covers coordinate geometry of straight lines, including gradients, equations, distances and midpoints. Exams commonly require interpretation of geometric conditions and formation of equations from points and gradients.

• Calculate the gradient of a line from two points or from its equation.
• Form the equation of a straight line in gradient-intercept, point-slope or general form.
• Determine whether two lines are parallel or perpendicular using gradients.
• Find the distance between two points in the Cartesian plane.
• Find the midpoint of a line segment.
• Solve geometric problems involving straight lines, intersections, distances and midpoints.

This topic covers the equation and geometry of circles and the properties of tangents in coordinate geometry. Exams test recognition of circle forms, determination of centres and radii, and use of tangent conditions.

• Use the equation of a circle in forms such as $(x-a)^2+(y-b)^2=r^2$ and the expanded general form.
• Determine the centre and radius of a circle from its equation, including by completing the square.
• Find the equation of a tangent to a circle at a given point.
• Use the fact that a radius is perpendicular to a tangent to solve coordinate geometry problems.
• Determine points of intersection of a line and a circle and interpret tangent conditions using repeated roots or discriminants.
• Solve problems involving circles, chords and tangents in the coordinate plane.

This topic covers patterns in sequences, sums of finite series, and the use of sigma notation. Exams assess term-finding, summation formulae, and modelling with arithmetic or geometric progression structures.

• Generate and interpret terms of sequences defined explicitly or recursively.
• Use the formulae for the $n$th term and the sum of an arithmetic series.
• Use the formulae for the $n$th term and the sum of a finite geometric series.
• Determine whether a sequence or series is arithmetic or geometric and find unknown parameters.
• Use sigma notation to represent and manipulate finite sums.
• Solve contextual and algebraic problems involving sequences and series.

This topic covers the binomial expansion of expressions of the form $(a+b)^n$ for positive integer $n$. Exams typically test use of binomial coefficients, identification of specific terms, and approximation or coefficient problems.

• Expand $(a+b)^n$ for positive integer values of $n$ using the binomial theorem.
• Use and interpret binomial coefficients, including notation such as $\binom{n}{r}$ where used by the course route.
• Find the general or specific term in a binomial expansion.
• Determine a required coefficient or unknown constant from a binomial expansion.
• Use binomial expansions in algebraic simplification and approximation where appropriate.

AQA onlyPearson Edexcel onlyOCR Mathematics A onlyOCR Mathematics B (MEI) onlyWJEC (Wales) onlyCCEA (Northern Ireland) onlyPearson Edexcel International A Level onlyPearson Edexcel International A Level Pure Mathematics only

This topic covers binomial series expansions for fractional or negative powers and the conditions under which they are valid. Exams test formation of the first few terms, interval of validity, and approximation using truncated series.

• Expand expressions of the form $(1+x)^n$ where $n$ is rational or negative, to a specified number of terms.
• State and use the condition of validity for a binomial series expansion, typically in the form $|x|<1$ after suitable substitution.
• Obtain expansions for expressions reducible to the form $(1+u)^n$.
• Use a truncated binomial series to approximate numerical values.
• Determine or simplify coefficients in binomial series expansions.

This topic covers the radian measure of angle and its use in circular measure formulae. Exams test conversion between degrees and radians and application to arc length, sector area and related geometric problems.

• Convert angles between degrees and radians.
• Interpret one radian as the angle subtended by an arc equal in length to the radius.
• Use the formula $s=r\theta$ for arc length with $\theta$ in radians.
• Use the formula $A=\tfrac12 r^2\theta$ for sector area with $\theta$ in radians.
• Solve problems involving sectors, segments and combined shapes using radian measure.

This topic covers the sine, cosine and tangent functions, their exact values, graphs and key features. Exams assess knowledge of periodicity, symmetry, transformations and use of trigonometric graphs to model or solve problems.

• Use exact values of $\sin\theta$, $\cos\theta$ and $\tan\theta$ for standard angles.
• Sketch and interpret the graphs of $y=\sin x$, $y=\cos x$ and $y=\tan x$, including amplitude, period and asymptotes where appropriate.
• Use the unit-circle definition and symmetries of trigonometric functions.
• Apply graph transformations to trigonometric functions.
• Determine the period, amplitude, phase shift and vertical translation of trigonometric graphs in forms such as $a\sin(bx+c)+d$ and $a\cos(bx+c)+d$.
• Use trigonometric graphs to interpret solutions and behaviour in pure or contextual settings.

This topic covers standard trigonometric identities and the solution of trigonometric equations over specified intervals. Exams test algebraic manipulation, selection of suitable identities, and complete solution sets.

• Use the identities $\sin^2 x+\cos^2 x\equiv 1$, $\tan x\equiv \dfrac{\sin x}{\cos x}$, and related forms.
• Use compound-angle formulae for $\sin(A\pm B)$, $\cos(A\pm B)$ and $\tan(A\pm B)$ where required by the course route.
• Use double-angle formulae and related identities where required by the course route.
• Simplify trigonometric expressions and prove simple identities.
• Solve trigonometric equations in a given interval, including equations reducible to quadratic form in a trigonometric function.
• Select solutions correctly using graph knowledge, periodicity and the unit circle.

This topic covers exponential and logarithmic functions, their laws, and their use in modelling growth and decay. Exams test algebraic manipulation, solving equations, graph interpretation and application in contexts such as population, finance or radioactive decay.

• Use and manipulate the laws of logarithms, including change of base where appropriate.
• Sketch and interpret graphs of exponential and logarithmic functions.
• Solve exponential equations and logarithmic equations exactly or numerically as appropriate.
• Use the relationship between exponential and logarithmic functions as inverses.
• Construct and use exponential models for growth and decay, including repeated percentage change and continuous models where specified.
• Interpret parameters and solutions in contextual exponential models.

This topic covers the derivative as a rate of change and gradient function, together with the standard rules for differentiating algebraic, exponential, logarithmic and trigonometric functions. Exams test routine differentiation, correct notation and application of product, quotient and chain rules.

• Differentiate from first principles in simple cases and interpret the derivative as a gradient or rate of change.
• Differentiate powers of $x$, polynomials and functions involving constants and sums.
• Differentiate composite functions using the chain rule.
• Differentiate products and quotients using the product rule and quotient rule.
• Differentiate exponential, logarithmic and trigonometric functions within the specification route.
• Use function notation and Leibniz notation correctly, including second derivatives where required.

This topic covers the use of derivatives to analyse graphs and solve optimisation and kinematics-type rate problems. Exams usually test stationary points, increasing/decreasing behaviour, tangents, normals and interpretation of derivatives in context.

• Find equations of tangents and normals to curves at given points.
• Determine stationary points and classify them using first-derivative or second-derivative methods.
• Identify intervals where a function is increasing or decreasing.
• Use differentiation to locate local maxima and minima and solve optimisation problems.
• Apply differentiation to rates of change in pure or contextual problems.
• Interpret first and second derivatives in relation to the shape of a graph, including concavity where required.

This topic covers indefinite integration as the reverse of differentiation and definite integrals as signed area and accumulation. Exams test standard antiderivatives, evaluation of areas, and interpretation of integral expressions.

• Integrate powers of $x$ and standard algebraic functions using the reverse power rule.
• Integrate exponential and trigonometric functions within the specification route.
• Evaluate indefinite integrals and include constants of integration where appropriate.
• Evaluate definite integrals using antiderivatives and limits.
• Interpret a definite integral as the signed area under a curve or as accumulated change.
• Use integration to find the area of a region bounded by a curve, the axes or other lines.

This topic covers methods needed to integrate less direct forms, especially substitution, integration by parts and partial fractions. Exams test recognition of the appropriate method and accurate execution in both indefinite and definite settings.

• Use substitution to evaluate integrals, including changes of limits where appropriate for definite integrals.
• Use integration by parts to integrate products of functions.
• Integrate rational functions by first expressing them in partial fractions.
• Integrate functions involving simple algebraic rearrangement to standard forms.
• Select and justify an appropriate integration technique for a given integral.

This topic covers numerical solution of equations using iteration and related methods, together with graphical interpretation of convergence. Exams test construction of iterative formulae, execution to a required accuracy, and reasoning about whether a method converges.

• Locate approximate roots of equations by change of sign or graphical methods.
• Apply fixed-point iteration using a rearrangement of an equation.
• Determine an interval containing a root and refine approximations using an iterative numerical method such as the Newton-Raphson method where specified by the route.
• Use iteration to a stated degree of accuracy or number of decimal places.
• Interpret convergence or divergence of an iterative process graphically or algebraically.
• Choose a suitable initial approximation and comment on the validity of the method in context.

This topic covers curves defined parametrically and differentiation and integration linked to parameter form. Exams test conversion between forms, sketching, and calculation of gradients, tangents and areas.

• Use parametric equations to generate coordinates and eliminate the parameter where possible.
• Sketch or interpret curves given in parametric form.
• Differentiate parametric equations using $\dfrac{dy}{dx}=\dfrac{dy/dt}{dx/dt}$.
• Find equations of tangents and normals to parametric curves.
• Evaluate areas related to parametric curves using integration where required by the specification route.

This topic covers vector notation, operations, equations of lines, and geometric reasoning in two and three dimensions. Exams assess algebraic fluency with vectors and the ability to solve geometric problems involving parallelism, intersection and position.

• Use vector notation and perform addition, subtraction and multiplication by a scalar.
• Calculate the magnitude of a vector and use unit vectors where appropriate.
• Determine vector equations of lines in two and three dimensions.
• Test whether vectors or lines are parallel and determine points of intersection where appropriate.
• Use vectors to solve geometric problems involving collinearity, ratios, midpoints and displacement.
• Apply the scalar product to find angles or test perpendicularity where required by the specification route.

This topic covers first-order differential equations and their use in mathematical modelling. Exams test formation of differential equations from contexts, separation of variables, use of initial conditions and interpretation of solutions.

• Form a first-order differential equation to model a stated situation.
• Solve separable differential equations by integration.
• Apply initial or boundary conditions to find particular solutions.
• Interpret the constants and behaviour of solutions in context.
• Use differential equations to model exponential growth or decay and related rate processes.
• Assess the validity or limitations of a simple differential-equation model where appropriate.

Cambridge International only

This topic covers complex numbers in Cartesian form and their representation on the Argand diagram. Exams test algebra with $i$, solving quadratic equations with complex roots, and interpretation of modulus and conjugate.

• Use the relation $i^2=-1$ to simplify powers and expressions involving $i$.
• Add, subtract, multiply and divide complex numbers in the form $a+bi$.
• Find the conjugate and modulus of a complex number.
• Solve quadratic equations with negative discriminant and express roots in complex form.
• Represent complex numbers on an Argand diagram and interpret modulus geometrically.