AQA · A-Level

Mathematics

Papers, samples and curriculum documents for this course.

Qualification code: 7357

Recent past papers

25 paper and mark-scheme pairs

Course units and learning goals

These lessons teach selected course objectives. Check the remaining coverage gaps; the material is not a complete preparation programme.

Lessons →Vocabulary practice →

A · Proof

Deduction, contradiction and induction: A single valid case that disproves a universal claim.
Complete cases and counterexamples: A proof that checks every possibility in a finite, complete set of cases.
Irrational roots and infinitely many primes: A proof that assumes the negation of a claim and derives an impossibility.

counterexample: A single valid case that disproves a universal claim.
proof by exhaustion: A proof that checks every possibility in a finite, complete set of cases.
proof by contradiction: A proof that assumes the negation of a claim and derives an impossibility.

B · Algebra and functions

Indices, surds and standard form: The power to which a base is raised.
Quadratics and inequalities: The quantity b²-4ac that determines the real roots of ax²+bx+c=0.
Domains, inverses and composition: The set of allowed inputs to a function.
Polynomial division, remainders and factors: The statement that x−a is a factor of polynomial P(x) exactly when P(a)=0.
Algebraic fractions and excluded values: An input for which an original denominator or divisor is zero.
Partial fractions with distinct linear factors: A simpler fraction in a sum that equals a rational expression on its original domain.
Repeated linear factors in partial fractions: A denominator factor occurring more than once, requiring a term for each power.
Linear and quadratic simultaneous equations: An ordered pair that satisfies every equation in a system at the same time.
Rational indices and real domains: An exponent written as a fraction of integers.
Surds and conjugate denominators: A paired expression with the sign between two terms reversed.
Completing the square and quadratic structure: A quadratic written as a multiple of a squared bracket plus a constant.
Equations quadratic in another expression: Replacing a repeated expression with a new variable to reveal a simpler equation.
Inequalities, fractions and solution sets: The values common to two solution sets, corresponding to AND.
Regions between linear and quadratic boundaries: The line or curve separating points that satisfy an inequality from those that do not.
Modulus graphs and piecewise solutions: The nonnegative distance of a real number from zero.
Reciprocal graphs and asymptotes: A line that a curve approaches in a limiting direction.
Inverse graphs and restricted branches: A function for which different inputs always give different outputs.
Composite functions and intermediate domains: Applying one function to the output of another, in a specified order.
Combined graph transformations and order: The factor multiplying the original horizontal coordinates when a graph is stretched or compressed.
Fitting and refining a function model: An observed output minus the output predicted by a model.
Polynomial sketches, repeated roots and signs: The number of times a root’s linear factor appears in a polynomial.

index: The power to which a base is raised.
discriminant: The quantity b²-4ac that determines the real roots of ax²+bx+c=0.
domain: The set of allowed inputs to a function.
factor theorem: The statement that x−a is a factor of polynomial P(x) exactly when P(a)=0.
excluded value: An input for which an original denominator or divisor is zero.
partial fraction: A simpler fraction in a sum that equals a rational expression on its original domain.
repeated factor: A denominator factor occurring more than once, requiring a term for each power.
simultaneous solution: An ordered pair that satisfies every equation in a system at the same time.
rational exponent: An exponent written as a fraction of integers.
conjugate: A paired expression with the sign between two terms reversed.
completed square: A quadratic written as a multiple of a squared bracket plus a constant.
substitution: Replacing a repeated expression with a new variable to reveal a simpler equation.
intersection: The values common to two solution sets, corresponding to AND.
boundary: The line or curve separating points that satisfy an inequality from those that do not.
modulus: The nonnegative distance of a real number from zero.
asymptote: A line that a curve approaches in a limiting direction.
one-to-one: A function for which different inputs always give different outputs.
composition: Applying one function to the output of another, in a specified order.
horizontal scale factor: The factor multiplying the original horizontal coordinates when a graph is stretched or compressed.
residual: An observed output minus the output predicted by a model.
multiplicity: The number of times a root’s linear factor appears in a polynomial.

C · Coordinate geometry

Coordinate geometry and tangents: The change in y divided by the corresponding change in x.
Parametric curves, domains and direction: A variable that determines both coordinates of a point.
Parametric modelling in motion and design: A model expressing two quantities using a common variable and stated assumptions.
General-form lines and contextual models: The form ax+by+c=0 for a straight line, with a and b not both zero.
Shifted circles, chords and tangents: The point at equal distance from every point on a circle.

gradient: The change in y divided by the corresponding change in x.
parameter: A variable that determines both coordinates of a point.
parametric model: A model expressing two quantities using a common variable and stated assumptions.
general form: The form ax+by+c=0 for a straight line, with a and b not both zero.
centre: The point at equal distance from every point on a circle.

D · Sequences and series

Sequences, series and recurrence: The constant multiplier between consecutive terms of a geometric sequence.
Binomial expansion and valid approximations: The number of ways to choose a specified number of objects from a set.
Factorials, combinations and binomial coefficients: A selection in which order does not distinguish different choices.
Increasing, decreasing and periodic recurrences: A sequence whose terms repeat after a fixed positive number of steps.
Sigma notation, indices and changed limits: The variable that takes each integer value between a sum’s stated limits.
Arithmetic series and inverse problems: The fixed amount added to each term to obtain the next.
Geometric sums, alternating ratios and convergence: The fixed factor multiplying each term to obtain the next.
Repeated-deposit sequence models and timing: An amount added to an account at a stated time.

common ratio: The fixed factor multiplying each term to obtain the next.
binomial coefficient: The number of ways to choose a specified number of objects from a set.
combination: A selection in which order does not distinguish different choices.
periodic sequence: A sequence whose terms repeat after a fixed positive number of steps.
summation index: The variable that takes each integer value between a sum’s stated limits.
common difference: The fixed amount added to each term to obtain the next.
deposit: An amount added to an account at a stated time.

E · Trigonometry

Right triangles and non-right triangles: The side opposite the right angle in a right-angled triangle.
Radians, identities and trigonometric equations: The angle subtended by an arc whose length equals the radius.
Radian measure, arcs, sectors and segments: The region bounded by two radii and their circular arc.
Small-angle approximations and their limits: A nearby simpler value used with a stated accuracy limit.
All-angle definitions, exact values and periodic graphs: The circle of radius one centred at the coordinate origin.
Reciprocal trigonometric functions and excluded angles: The reciprocal of cosine wherever cosine is nonzero.
Inverse trigonometric branches and principal values: The single inverse output chosen from a stated restricted interval.
Reciprocal identities and one-sided proof chains: An equality true for every input in its stated common domain.
Compound-angle formulae and geometric proofs: An angle expressed as the sum or difference of two angles.
Double-angle identities, quadrant signs and squared forms: An angle equal to twice the original angle.
Auxiliary-angle forms, ranges and interval solutions: The nonnegative size of an oscillation about its centre line.
Quadratic trigonometric equations and complete root sets: A possible solution that still needs checking in the original equation and interval.
Multiple-angle equations and transformed intervals: The interval for a substituted angle after applying the same input change to its endpoints.
Trigonometric components and motion models: The signed projection of a vector along a chosen coordinate direction.
Sine rule, cosine rule and triangle area: The angle between the two named sides.
The ambiguous sine-rule case and valid triangle counts: Given triangle data that can produce two different valid triangles.
Key-point sketches and transformed tangent domains: A line approached by a graph while its values grow without bound or tend toward a limit.

hypotenuse: The side opposite the right angle in a right-angled triangle.
radian: The angle subtended by an arc whose length equals the radius.
sector: The region bounded by two radii and their circular arc.
approximation: A nearby simpler value used with a stated accuracy limit.
unit circle: The circle of radius one centred at the coordinate origin.
secant: The reciprocal of cosine wherever cosine is nonzero.
principal value: The single inverse output chosen from a stated restricted interval.
identity: An equality true for every input in its stated common domain.
compound angle: An angle expressed as the sum or difference of two angles.
double angle: An angle equal to twice the original angle.
amplitude: The nonnegative size of an oscillation about its centre line.
candidate: A possible solution that still needs checking in the original equation and interval.
transformed interval: The interval for a substituted angle after applying the same input change to its endpoints.
component: The signed projection of a vector along a chosen coordinate direction.
included angle: The angle between the two named sides.
ambiguous case: Given triangle data that can produce two different valid triangles.
asymptote: A line approached by a graph while its values grow without bound or tend toward a limit.

F · Exponentials and logarithms

Exponentials, logarithms and modelling: The time for a decaying quantity to fall to half its initial value.
Exponential graphs and logarithmic inverses: The exponent to which a stated valid base must be raised to produce a positive input.
Logarithm laws, negative powers and equation domains: The input inside a logarithm, which must be positive for a real logarithm.
Logarithmic graphs for power and exponential fits: The transformed vertical value when the chosen horizontal variable is zero.
Proportional rates, half-life and model limits: A rate of change equal to a constant multiplied by the current amount.

half-life: The time for a decaying quantity to fall to half its initial value.
logarithm: The exponent to which a stated valid base must be raised to produce a positive input.
argument: The input inside a logarithm, which must be positive for a real logarithm.
intercept: The transformed vertical value when the chosen horizontal variable is zero.
proportional rate: A rate of change equal to a constant multiplied by the current amount.

G · Differentiation

Derivatives and stationary points: The instantaneous rate of change, also the gradient of a tangent.
Chain, product, quotient and implicit differentiation: The rule that multiplies the outer derivative by the inner derivative for a composite function.
Secants, limits and first-principles power derivatives: The change in function output divided by a nonzero change in input.
First-principles sine and cosine derivatives: A small-angle limit used with angles measured in radians.
Gradient graphs, concavity and inflection tests: A point where a continuous curve changes its direction of concavity.
Rational-power derivatives and valid domains: An exponent that can be written as a fraction of integers, interpreted with a valid real root.
Exponential, logarithmic and trigonometric derivatives: The derivative of the changing input inside a composite function.
Product and quotient rules with retained exclusions: The rule for differentiating a ratio of two changing functions when the denominator is nonzero.
Chain rules for nested functions and restricted inputs: The outer derivative evaluated at the inner output, multiplied by the derivative of that inner input.
Inverse derivatives at corresponding points: The rate of the inverse output with respect to its input, evaluated at the corresponding original point.
Connected rates, geometry and time units: Rates linked by a relationship between quantities changing with the same time variable.
Implicit first derivatives and tangent domains: An equation relating x and y without isolating one as an explicit function of the other.
Parametric first derivatives and vertical tangents: The ratio of the y-rate to the x-rate along a parametrised curve, when the x-rate is nonzero.
Tangent and normal equations and gradient conditions: A line through the curve point perpendicular to its local tangent.
Optimisation with constraints, boundaries and rejected roots: The allowed input values after the original physical or mathematical constraints are applied.
Constructing differential equations from rate laws: A relationship involving an unknown function and one or more of its derivatives.

derivative: The instantaneous rate of change, also the gradient of a tangent.
chain rule: The rule that multiplies the outer derivative by the inner derivative for a composite function.
difference quotient: The change in function output divided by a nonzero change in input.
radian limit: A small-angle limit used with angles measured in radians.
point of inflection: A point where a continuous curve changes its direction of concavity.
rational power: An exponent that can be written as a fraction of integers, interpreted with a valid real root.
inner rate: The derivative of the changing input inside a composite function.
quotient rule: The rule for differentiating a ratio of two changing functions when the denominator is nonzero.
composite derivative: The outer derivative evaluated at the inner output, multiplied by the derivative of that inner input.
inverse derivative: The rate of the inverse output with respect to its input, evaluated at the corresponding original point.
connected rates: Rates linked by a relationship between quantities changing with the same time variable.
implicit relation: An equation relating x and y without isolating one as an explicit function of the other.
parametric gradient: The ratio of the y-rate to the x-rate along a parametrised curve, when the x-rate is nonzero.
normal line: A line through the curve point perpendicular to its local tangent.
feasible domain: The allowed input values after the original physical or mathematical constraints are applied.
differential equation: A relationship involving an unknown function and one or more of its derivatives.

H · Integration

Integrals, area and accumulation: A function whose derivative equals the given integrand.
Substitution, parts and partial fractions: An integration method based on the derivative of a product.
Differential equations and numerical solutions: A specified value that selects a particular solution from a family.
The Fundamental Theorem and accumulated change: A function whose value is the signed integral from a fixed starting point to a variable endpoint.
Integration as the limit of rectangle sums: A sum of signed rectangle contributions that approaches an integral as the partition becomes finer.
Elementary antiderivatives, coefficients and domains: A function whose derivative is the specified integrand on the stated interval.

antiderivative: A function whose derivative is the specified integrand on the stated interval.
integration by parts: An integration method based on the derivative of a product.
initial condition: A specified value that selects a particular solution from a family.
accumulation function: A function whose value is the signed integral from a fixed starting point to a variable endpoint.
Riemann sum: A sum of signed rectangle contributions that approaches an integral as the partition becomes finer.

I · Numerical methods

Root finding and numerical integration: A repeated update in which each new approximation is calculated from the previous one.

iteration: A repeated update in which each new approximation is calculated from the previous one.

J · Vectors

Vectors and transformation geometry: The vector sum representing the combined displacement or force.
Reflections, rotations and enlargements: The point from which each point's displacement is multiplied by a scale factor.

resultant: The vector sum representing the combined displacement or force.
centre of enlargement: The point from which each point's displacement is multiplied by a scale factor.

K · Statistical sampling

Sampling and a large data set: The list or population definition from which a sample is selected.

sampling frame: The list or population definition from which a sample is selected.

L · Data presentation and interpretation

Data summaries, histograms and interpretation: Frequency divided by class width, used as histogram height.
Regression, financial models and residuals: The observed value minus the value predicted by a fitted model.

frequency density: Frequency divided by class width, used as histogram height.
residual: The observed value minus the value predicted by a fitted model.

M · Probability

Probability, trees and conditional reasoning: The probability of an event after restricting the sample space to a stated condition.

conditional probability: The probability of an event after restricting the sample space to a stated condition.

N · Statistical distributions

Binomial, normal and Poisson models: The probability-weighted mean of a random variable.

expected value: The probability-weighted mean of a random variable.

O · Statistical hypothesis testing

Hypothesis testing and contextual conclusions: The chosen probability threshold for rejecting a null hypothesis.

significance level: The chosen probability threshold for rejecting a null hypothesis.

P · Quantities and units in mechanics

Accuracy, bounds and compound measures: The smallest possible value consistent with a stated rounding rule.
Kinematics and Newton's laws: The vector sum of all external forces acting on the modelled object.

lower bound: The smallest possible value consistent with a stated rounding rule.
resultant force: The vector sum of all external forces acting on the modelled object.

Q · Kinematics

Kinematics and Newton's laws: The vector sum of all external forces acting on the modelled object.
Integrals, area and accumulation: A function whose derivative equals the given integrand.

resultant force: The vector sum of all external forces acting on the modelled object.
antiderivative: A function whose derivative equals the given integrand.

R · Forces and Newton's laws

Kinematics and Newton's laws: The vector sum of all external forces acting on the modelled object.

resultant force: The vector sum of all external forces acting on the modelled object.

S · Moments

Moments, equilibrium and centres of mass: Force multiplied by the perpendicular distance from its line of action to a pivot.

moment: Force multiplied by the perpendicular distance from its line of action to a pivot.

Preparing for this qualification

Paper 1 covers A–I. Paper 2 adds J and P–S to Paper 1 content. Paper 3 adds K–O to Paper 1 content.
All students study pure, mechanics and statistics. Use the AQA large data set and the 7357 formula booklet.
Calculators need statistical distribution functions; symbolic algebra/differentiation/integration facilities are prohibited under the examination rules.

Teaching coverage still needed

Official question-bank boundaries, scheme alignment and all objective-level teaching coverage require the recorded review; no practice registry promotion.
The authored diagnostic assessments are not full-length qualification mocks.