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Year 12: AQA A-level Mathematics 7357

Listed Mathematics content statements at A-level, in the words of the AQA, with the practice on this site that covers it. Checked against the source on 7 September 2026.

Content statements from the AQA A-level Mathematics specification (7357), version 1.3, which covers the full two-year course. Superscripts the PDF text layer does not carry have been restored.

  • 88content statements
  • 155skills mapped
  • 86%have practice

Coverage is the percentage of listed content statements with at least one linked practice skill; it does not measure how fully every part of an entry is practised. A skill can practise more than one content statement, so some appear more than once below.

A: Proof

  1. A1Understand and use the structure of mathematical proof, proceeding from given assumptions through a series of logical steps to a conclusion; use methods of proof, including proof by deduction, proof by exhaustion. Disproof by counter example. Proof by contradiction (including proof of the irrationality of √2 and the infinity of primes, and application to unfamiliar proofs).

B: Algebra and functions

  1. B1Understand and use the laws of indices for all rational exponents.

  2. B2Use and manipulate surds, including rationalising the denominator.

  3. B3Work with quadratic functions and their graphs; the discriminant of a quadratic function, including the conditions for real and repeated roots; completing the square; solution of quadratic equations including solving quadratic equations in a function of the unknown.

  4. B4Solve simultaneous equations in two variables by elimination and by substitution, including one linear and one quadratic equation.

  5. B5Solve linear and quadratic inequalities in a single variable and interpret such inequalities graphically, including inequalities with brackets and fractions. Express solutions through correct use of ‘and’ and ‘or’, or through set notation. Represent linear and quadratic inequalities such as y>x+1 and y>ax²+bx+c graphically.

  6. B6Manipulate polynomials algebraically, including expanding brackets and collecting like terms, factorisation and simple algebraic division; use of the factor theorem. Simplify rational expressions including by factorising and cancelling, and algebraic division (by linear expressions only).

  7. B7Understand and use graphs of functions; sketch curves defined by simple equations including polynomials, the modulus of a linear function, y=a/x and y=a/x² (including their vertical and horizontal asymptotes); interpret algebraic solution of equations graphically; use intersection points of graphs to solve equations. Understand and use proportional relationships and their graphs.

  8. B8Understand and use composite functions; inverse functions and their graphs.

  9. B9Understand the effect of simple transformations on the graph of y=f(x) including sketching associated graphs: y=af(x), y=f(x)+a, y=f(x+a), y=f(ax), and combinations of these transformations.

  10. B10Decompose rational functions into partial fractions (denominators not more complicated than squared linear terms and with no more than 3 terms, numerators constant or linear).

  11. B11Use of functions in modelling, including consideration of limitations and refinements of the models.

C: Coordinate geometry in the (x, y) plane

  1. C1Understand and use the equation of a straight line, including the forms y-y_1=m(x-x_1) and ax+by+c=0; gradient conditions for two straight lines to be parallel or perpendicular. Be able to use straight line models in a variety of contexts.

  2. C2Understand and use the coordinate geometry of the circle including using the equation of a circle in the form (x-a)²+(y-b)²=r²; completing the square to find the centre and radius of a circle; use of the following properties:the angle in a semicircle is a right anglethe perpendicular from the centre to a chord bisects the chordthe radius of a circle at a given point on its circumference is perpendicular to the tangent to the circle at that point.

  3. C3Understand and use the parametric equations of curves and conversion between Cartesian and parametric forms.

    No practice for this one yet.

  4. C4Use parametric equations in modelling in a variety of contexts.

    No practice for this one yet.

D: Sequences and series

  1. D1Understand and use the binomial expansion of (a+bx)ⁿ for positive integer n; the notations n!, nCr and (nr); link to binomial probabilities. Extend to any rational n, including its use for approximation; be aware that the expansion is valid for |bx/a|<1 . (Proof not required.)

  2. D2Work with sequences including those given by a formula for the n th term and those generated by a simple relation of the form x_n+1=f(x_n); increasing sequences; decreasing sequences; periodic sequences.

  3. D3Understand and use sigma notation for sums of series.

  4. D4Understand and work with arithmetic sequences and series, including the formulae for n th term and the sum to n terms.

  5. D5Understand and work with geometric sequences and series including the formulae for the n th term and the sum of a finite geometric series; the sum to infinity of a convergent geometric series, including the use of |r|<1; modulus notation.

  6. D6Use sequences and series in modelling.

E: Trigonometry

  1. E1Understand and use the definitions of sine, cosine and tangent for all arguments; the sine and cosine rules; the area of a triangle in the form 1/2absin C Work with radian measure, including use for arc length and area of sector.

  2. E2Understand and use the standard small angle approximations of sine, cosine and tangent sinθ≈θ, cosθ≈1-θ²/2, tanθ≈θ where θ is in radians.

  3. E3Understand and use the sine, cosine and tangent functions; their graphs, symmetries and periodicity. Know and use exact values of sin and cos for 0, π/6, π/4, π/3, π/2, π and multiples thereof, and exact values of tan for 0, π/6, π/4, π/3, π and multiples thereof.

  4. E4Understand and use the definitions of secant, cosecant and cotangent and of arcsin, arccos and arctan; their relationships to sine, cosine and tangent; understanding of their graphs; their ranges and domains.

  5. E5Understand and use tanθ≡(sinθ)/(cosθ) Understand and use sin²θ+cos²θ≡1; sec²θ≡1+tan²θ and cosec²θ≡1+cot²θ

  6. E6Understand and use double angle formulae; use of formulae for sin(A±B), cos(A±B) and tan(A±B); understand geometrical proofs of these formulae. Understand and use expressions for acosθ+bsinθ in the equivalent forms of rcos(θ±α) or rsin(θ±α)

  7. E7Solve simple trigonometric equations in a given interval, including quadratic equations in sin, cos and tan and equations involving multiples of the unknown angle.

  8. E8Construct proofs involving trigonometric functions and identities.

  9. E9Use trigonometric functions to solve problems in context, including problems involving vectors, kinematics and forces.

F: Exponentials and logarithms

  1. F1Know and use the function a^x and its graph, where a is positive. Know and use the function e^x and its graph.

    No practice for this one yet.

  2. F2Know that the gradient of e^(kx) is equal to ke^(kx) and hence understand why the exponential model is suitable in many applications.

  3. F3Know and use the definition of log_ax as the inverse of a^x, where a is positive and x≥0 Know and use the function lnx and its graph. Know and use lnx as the inverse function of e^x

  4. F4Understand and use the laws of logarithms: log_ax+log_ay≡log_a(xy); log_ax-log_ay≡log_a(x/y); klog_ax≡log_ax^k (including, for example, k=-1 and k=-1/2)

  5. F5Solve equations of the form a^x=b

    No practice for this one yet.

  6. F6Use logarithmic graphs to estimate parameters in relationships of the form y=axⁿ and y=kb^x, given data for x and y .

  7. F7Understand and use exponential growth and decay; use in modelling (examples may include the use of e^ in continuous compound interest, radioactive decay, drug concentration decay, exponential growth as a model for population growth); consideration of limitations and refinements of exponential models.

G: Differentiation

  1. G1Understand and use the derivative of f(x) as the gradient of the tangent to the graph of y = f(x) at a general point (x, y); the gradient of the tangent as a limit; interpretation as a rate of change; sketching the gradient function for a given curve; second derivatives; differentiation from first principles for small positive integer powers of x and for sin x and cos x Understand and use the second derivative as the rate of change of gradient; connection to convex and concave sections of curves and points of inflection.

  2. G2Differentiate xⁿ, for rational values of n, and related constant multiples, sums and differences. Differentiate e^(kx) and a^(kx), sin kx, cos kx, tan kx and related sums, differences and constant multiples. Understand and use the derivative of lnx

  3. G3Apply differentiation to find gradients, tangents and normals, maxima and minima and stationary points, points of inflection. Identify where functions are increasing or decreasing.

  4. G4Differentiate using the product rule, the quotient rule and the chain rule, including problems involving connected rates of change and inverse functions.

  5. G5Differentiate simple functions and relations defined implicitly or parametrically, for first derivative only.

  6. G6Construct simple differential equations in pure mathematics and in context, (contexts may include kinematics, population growth and modelling the relationship between price and demand).

    No practice for this one yet.

H: Integration

  1. H1Know and use the Fundamental Theorem of Calculus.

    No practice for this one yet.

  2. H2Integrate xⁿ (excluding n = -1), and related sums, differences and constant multiples. Integrate e^(kx), 1/x, sin kx, cos kx and related sums, differences and constant multiples.

  3. H3Evaluate definite integrals; use a definite integral to find the area under a curve and the area between two curves.

  4. H4Understand and use integration as the limit of a sum.

  5. H5Carry out simple cases of integration by substitution and integration by parts; understand these methods as the inverse processes of the chain and product rules respectively.(Integration by substitution includes finding a suitable substitution and is limited to cases where one substitution will lead to a function which can be integrated; integration by parts includes more than one application of the method but excludes reduction formulae).

  6. H6Integrate using partial fractions that are linear in the denominator.

  7. H7Evaluate the analytical solution of simple first order differential equations with separable variables, including finding particular solutions (Separation of variables may require factorisation involving a common factor).

    No practice for this one yet.

  8. H8Interpret the solution of a differential equation in the context of solving a problem, including identifying limitations of the solution; includes links to kinematics.

    No practice for this one yet.

I: Numerical methods

  1. I1Locate roots of f(x) = 0 by considering changes of sign of f(x) in an interval of x on which f(x) is sufficiently well-behaved. Understand how change of sign methods can fail.

    No practice for this one yet.

  2. I2Solve equations approximately using simple iterative methods; be able to draw associated cobweb and staircase diagrams. Solve equations using the Newton-Raphson method and other recurrence relations of the form x_n+1=g(x_n) Understand how such methods can fail.

  3. I3Understand and use numerical integration of functions, including the use of the trapezium rule and estimating the approximate area under a curve and limits that it must lie between.

  4. I4Use numerical methods to solve problems in context.

    No practice for this one yet.

J: Vectors

  1. J1Use vectors in two dimensions and in three dimensions.

  2. J2Calculate the magnitude and direction of a vector and convert between component form and magnitude/direction form.

  3. J3Add vectors diagrammatically and perform the algebraic operations of vector addition and multiplication by scalars, and understand their geometrical interpretations.

  4. J4Understand and use position vectors; calculate the distance between two points represented by position vectors.

  5. J5Use vectors to solve problems in pure mathematics and in context, including forces and kinematics.

K: Statistical sampling

  1. K1Understand and use the terms ‘population’ and ‘sample’. Use samples to make informal inferences about the population. Understand and use sampling techniques, including simple random sampling and opportunity sampling. Select or critique sampling techniques in the context of solving a statistical problem, including understanding that different samples can lead to different conclusions about the population.

L: Data presentation and interpretation

  1. L1Interpret diagrams for single-variable data, including understanding that area in a histogram represents frequency. Connect to probability distributions.

  2. L2Interpret scatter diagrams and regression lines for bivariate data, including recognition of scatter diagrams which include distinct sections of the population (calculations involving regression lines are excluded). Understand informal interpretation of correlation. Understand that correlation does not imply causation.

  3. L3Interpret measures of central tendency and variation, extending to standard deviation. Be able to calculate standard deviation, including from summary statistics.

  4. L4Recognise and interpret possible outliers in data sets and statistical diagrams. Select or critique data presentation techniques in the context of a statistical problem. Be able to clean data, including dealing with missing data, errors and outliers.

M: Probability

  1. M1Understand and use mutually exclusive and independent events when calculating probabilities. Link to discrete and continuous distributions.

  2. M2Understand and use conditional probability, including the use of tree diagrams, Venn diagrams, two-way tables. Understand and use the conditional probability formula. P (AB)=(P(A∩B))/(P(B))

  3. M3Modelling with probability, including critiquing assumptions made and the likely effect of more realistic assumptions.

N: Statistical distributions

  1. N1Understand and use simple, discrete probability distributions (calculation of mean and variance of discrete random variables is excluded), including the binomial distribution, as a model; calculate probabilities using the binomial distribution.

  2. N2Understand and use the Normal distribution as a model; find probabilities using the Normal distribution. Link to histograms, mean, standard deviation, points of inflection and the binomial distribution.

  3. N3Select an appropriate probability distribution for a context, with appropriate reasoning, including recognising when the binomial or Normal model may not be appropriate.

O: Statistical hypothesis testing

  1. O1Understand and apply the language of statistical hypothesis testing, developed through a binomial model: null hypothesis, alternative hypothesis, significance level, test statistic, 1-tail test, 2-tail test, critical value, critical region, acceptance region, p -value; extend to correlation coefficients as measures of how close data points lie to a straight line and be able to interpret a given correlation coefficient using a given p -value or critical value (calculation of correlation coefficients is excluded).

    No practice for this one yet.

  2. O2Conduct a statistical hypothesis test for the proportion in the binomial distribution and interpret the results in context. Understand that a sample is being used to make an inference about the population and appreciate that the significance level is the probability of incorrectly rejecting the null hypothesis.

    No practice for this one yet.

  3. O3Conduct a statistical hypothesis test for the mean of a Normal distribution with known, given or assumed variance and interpret the results in context.

P: Quantities and units in mechanics

  1. P1Understand and use fundamental quantities and units in the SI system: length, time, mass. Understand and use derived quantities and units: velocity, acceleration, force, weight, moment.

Q: Kinematics

  1. Q1Understand and use the language of kinematics: position; displacement; distance travelled; velocity; speed; acceleration.

  2. Q2Understand, use and interpret graphs in kinematics for motion in a straight line: displacement against time and interpretation of gradient; velocity against time and interpretation of gradient and area under the graph.

  3. Q3Understand, use and derive the formulae for constant acceleration for motion in a straight line; extend to 2 dimensions using vectors.

  4. Q4Use calculus in kinematics for motion in a straight line: v=(dr)/(dt), a=(dv)/(dt)=(d²r)/(dt²), r=∫vdt, v=∫adt; extend to 2 dimensions using vectors.

  5. Q5Model motion under gravity in a vertical plane using vectors; projectiles.

R: Forces and Newton's laws

  1. R1Understand the concept of a force; understand and use Newton’s first law.

  2. R2Understand and use Newton’s second law for motion in a straight line (restricted to forces in two perpendicular directions or simple cases of forces given as 2D vectors); extend to situations where forces need to be resolved (restricted to 2 dimensions).

  3. R3Understand and use weight and motion in a straight line under gravity; gravitational acceleration, g, and its value in SI units to varying degrees of accuracy.(The inverse square law for gravitation is not required and g may be assumed to be constant, but students should be aware that g is not a universal constant but depends on location).

  4. R4Understand and use Newton’s third law; equilibrium of forces on a particle and motion in a straight line (restricted to forces in two perpendicular directions or simple cases of forces given as 2D vectors); application to problems involving smooth pulleys and connected particles; resolving forces in 2 dimensions; equilibrium of a particle under coplanar forces.

  5. R5Understand and use addition of forces; resultant forces; dynamics for motion in a plane.

  6. R6Understand and use the F≤μR model for friction; coefficient of friction; motion of a body on a rough surface; limiting friction and statics.

S: Moments

  1. S1Understand and use moments in simple static contexts.

Other Year 12 skills

Year 12 skills not currently linked to an entry on this page. These may include revision, extension practice, or skills still awaiting a curriculum mapping.

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