Specification Map & Revision Strategy

This lesson maps out the full AQA A-Level Mathematics specification, identifies high-frequency topics from past papers, highlights cross-topic links, and provides a structured revision strategy to maximise your exam performance.

---

Full AQA A-Level Maths Specification Overview

The AQA A-Level Mathematics specification (7357) is divided into three overarching content areas:

1. Pure Mathematics (assessed across Papers 1, 2, and 3)
2. Mechanics (assessed on Paper 2 only)
3. Statistics (assessed on Paper 3 only)

The specification also includes overarching themes that apply to all areas:

• Mathematical argument, language and proof — used across all topics
• Mathematical problem-solving — applying known techniques to novel problems
• Mathematical modelling — translating real-world problems into mathematical form

---

Pure Mathematics: Complete Topic Breakdown

Pure mathematics content can appear on any of the three papers. The following table gives the full specification with sub-topics.

Topic 1: Proof

Sub-topic	Key content
Understanding of proof	Know the difference between a proof and a verification
Proof by deduction	Start from known results and deduce the required statement
Proof by exhaustion	Check all finite cases
Disproof by counter-example	Find one case that contradicts the statement
Proof by contradiction	Assume the opposite and derive a contradiction

Topic 2: Algebra and Functions

Sub-topic	Key content
Laws of indices	Simplify expressions using index laws
Surds	Simplify and rationalise denominators
Quadratic equations	Solve, complete the square, use the discriminant
Simultaneous equations	Linear-linear and linear-quadratic
Inequalities	Linear, quadratic, graphical interpretation
Polynomials	Factor theorem, algebraic division
Partial fractions	Decompose rational expressions
Functions	Domain, range, composite functions, inverse functions
Transformations of graphs	Translations, reflections, stretches
Modulus function

Topic 3: Coordinate Geometry

Sub-topic	Key content
Straight lines	Gradient, y = mx + c, perpendicular lines, midpoints, distances
Circles	Equation (x − a)² + (y − b)² = r², tangents, chords, properties
Parametric equations	Convert between parametric and Cartesian, differentiate

Topic 4: Sequences and Series

Sub-topic	Key content
Arithmetic sequences and series	nth term, sum of n terms
Geometric sequences and series	nth term, sum of n terms, sum to infinity
Sigma notation	Evaluate sums using sigma notation
Binomial expansion (positive integer)	(a + b)ⁿ, finding specific terms
Binomial expansion (general)	(1 + x)ⁿ for non-integer n, validity condition
Recurrence relations	Define and use sequences of the form uₙ₊₁ = f(uₙ)

Topic 5: Trigonometry

Sub-topic	Key content
Radians	Convert between degrees and radians
Arc length and sector area	s = rθ, A = ½r²θ
Exact trigonometric values	sin, cos, tan of 0°, 30°, 45°, 60°, 90°
Trigonometric identities	sin²θ + cos²θ = 1, tan θ = sin θ/cos θ, sec, cosec, cot identities
Trigonometric equations	Solve in given intervals, multiple solutions
Addition and double angle formulae	sin(A ± B), cos(A ± B), tan(A ± B), sin 2A, cos 2A, tan 2A
R cos(θ ± α) and R sin(θ ± α)	Harmonic form for solving equations and finding max/min
Small angle approximations	sin θ ≈ θ, cos θ ≈ 1 − θ²/2, tan θ ≈ θ
Inverse trigonometric functions	arcsin, arccos, arctan — domains and ranges

Topic 6: Exponentials and Logarithms

Sub-topic	Key content
Exponential functions	eˣ, graphs, growth and decay
Logarithms	Definition, laws of logarithms, change of base
Natural logarithm (ln)	ln x as inverse of eˣ
Solving exponential equations	Using logarithms
Modelling with exponentials	Growth, decay, Newton's law of cooling

Topic 7: Differentiation

Sub-topic	Key content
First principles	Limit definition of the derivative
Differentiating xⁿ, sin x, cos x, eˣ, ln x, aˣ	Standard results
Product rule	d/dx [uv]
Quotient rule	d/dx [u/v]
Chain rule	d/dx [f(g(x))]
Implicit differentiation	Differentiate equations not solved for y
Parametric differentiation	dy/dx = (dy/dt)/(dx/dt)
Second derivatives	d²y/dx², concavity, points of inflection
Stationary points	Maxima, minima, classification using second derivative or nature table
Connected rates of change	Using the chain rule with time
Tangents and normals	Equations of tangent and normal lines
Optimisation	Finding maximum or minimum values in context

Topic 8: Integration

Sub-topic	Key content
Indefinite integrals	Reverse of differentiation, +C
Definite integrals	Area under a curve, evaluating with limits
Integration of xⁿ, sin, cos, eˣ, 1/x	Standard results
Integration by substitution	Change of variable
Integration by parts	∫u dv = uv − ∫v du
Partial fractions integration	Decompose and integrate term by term
Trapezium rule	Numerical integration
Area between curves	∫[f(x) − g(x)] dx
Differential equations	Separate variables and integrate

Topic 9: Numerical Methods

Sub-topic	Key content
Change of sign	Locating roots using the intermediate value theorem
Fixed-point iteration	xₙ₊₁ = g(xₙ), convergence and divergence
Newton-Raphson	xₙ₊₁ = xₙ − f(xₙ)/f'(xₙ)
Limitations	When methods fail (e.g., stationary point near root, discontinuity)

Topic 10: Vectors

Sub-topic	Key content
2D vectors	Magnitude, direction, position vectors
3D vectors	Extension to three dimensions
Vector arithmetic	Addition, subtraction, scalar multiplication
Unit vectors	Finding unit vectors in a given direction
Collinearity	Showing three points are collinear
Geometric problems	Using vectors to solve problems about shapes and positions

---

Mechanics Topics (Paper 2 Only)

Topic 11: Quantities and Units in Mechanics

Sub-topic	Key content
SI units	metres (m), kilograms (kg), seconds (s), newtons (N)
Scalar and vector quantities	Distinguish between quantities with and without direction

Topic 12: Kinematics

Sub-topic	Key content
Displacement-time and velocity-time graphs	Interpret gradient, area under the curve
SUVAT equations	Constant acceleration in a straight line
Variable acceleration	Using calculus: v = ds/dt, a = dv/dt, s = ∫v dt
Projectiles	Horizontal and vertical components, time of flight, range, maximum height

Topic 13: Forces and Newton's Laws

Sub-topic	Key content
Force diagrams	Weight, normal reaction, tension, friction, driving force, resistance
Newton's first law	Equilibrium — no resultant force means no acceleration
Newton's second law	F = ma — resolving forces, connected particles, pulleys
Newton's third law	Action-reaction pairs
Friction	F ≤ μR, limiting friction
Inclined planes	Resolving parallel and perpendicular to the plane

Topic 14: Moments

Sub-topic	Key content
Moments about a point	Moment = Force × perpendicular distance
Equilibrium	Sum of clockwise moments = sum of anticlockwise moments
Non-uniform rods	Centre of mass not at the geometric centre
Tilting and toppling	When objects are on the point of tilting

---

Statistics Topics (Paper 3 Only)

Topic 15: Statistical Sampling

Sub-topic	Key content
Types of sampling	Simple random, stratified, systematic, opportunity, quota
Advantages and disadvantages	When each method is appropriate
Using the Large Data Set	Selecting samples from the AQA data

Topic 16: Data Presentation and Interpretation

Sub-topic	Key content
Histograms	Frequency density, unequal class widths
Box plots and cumulative frequency	Median, quartiles, IQR, outliers
Scatter diagrams	Correlation, lines of best fit, interpolation vs extrapolation
Measures of central tendency	Mean, median, mode — from raw data and grouped data
Measures of spread	Range, IQR, variance, standard deviation
Coding	Linear transformations of data

Topic 17: Probability

Sub-topic	Key content
Venn diagrams	Representing events and their probabilities
Tree diagrams	Sequential events, conditional probabilities
Addition rule	P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
Conditional probability	P(A
Independence	P(A ∩ B) = P(A) × P(B)
Mutually exclusive events	P(A ∩ B) = 0

Topic 18: Statistical Distributions

Sub-topic	Key content
Binomial distribution	X ~ B(n, p), conditions, P(X = r), mean and variance
Normal distribution	X ~ N(μ, σ²), standardisation, using tables
Approximating binomial with normal	When n is large and p is close to 0.5

Topic 19: Statistical Hypothesis Testing

Sub-topic	Key content
Binomial hypothesis test	Testing a proportion, one-tailed and two-tailed
Normal hypothesis test	Testing a mean with known variance
Significance levels	1%, 5%, 10%
Critical values and critical regions	Finding the boundary for rejection
Conclusions in context	Always interpret in terms of the original problem

The Large Data Set (revisited)

Sub-topic	Key content
Structure	8 stations, weather variables, two time periods
Missing data	"n/a", "tr" (trace), how to handle
Seasonal patterns	Temperature, rainfall, sunshine vary by month
Comparing locations	UK vs overseas, north vs south
Outliers	Identification and interpretation

---

High-Frequency Topics from AQA Past Papers

Analysis of AQA A-Level Mathematics past papers reveals certain topics that appear with very high frequency. Prioritise these in your revision.

Pure Mathematics — Most Frequently Examined

Topic	Frequency	Typical marks
Differentiation (including chain, product, quotient rules)	Almost every paper	8–15 marks
Integration (substitution, by parts, areas)	Almost every paper	8–15 marks
Trigonometry (equations, identities, R-formula)	Almost every paper	6–12 marks
Algebra (quadratics, inequalities, partial fractions)	Almost every paper	5–10 marks
Exponentials and logarithms	Most papers	4–8 marks
Coordinate geometry (circles, parametric equations)	Most papers	5–10 marks
Sequences and series (binomial expansion)	Most papers	4–8 marks
Proof	Most papers	3–6 marks
Numerical methods (iteration, Newton-Raphson)	Appears regularly	4–8 marks
Vectors	Appears regularly	4–6 marks

Mechanics — Most Frequently Examined

Topic	Frequency	Typical marks
Newton's second law (F = ma, connected particles)	Almost every Paper 2	8–15 marks
Kinematics (SUVAT, variable acceleration)	Almost every Paper 2	6–12 marks
Inclined planes and friction	Most Paper 2s	5–10 marks
Projectiles	Appears regularly	5–8 marks
Moments	Appears regularly	4–8 marks

Statistics — Most Frequently Examined

Topic	Frequency	Typical marks
Normal distribution (standardisation, probability)	Almost every Paper 3	8–12 marks
Hypothesis testing (binomial and normal)	Almost every Paper 3	6–10 marks
Probability (conditional, tree/Venn diagrams)	Most Paper 3s	5–8 marks
Large Data Set questions	Most Paper 3s	3–6 marks
Data presentation (histograms, box plots, measures)	Appears regularly	4–8 marks
Binomial distribution	Appears regularly	4–8 marks

---

Cross-Topic Links and Common Multi-Step Problems

AQA frequently sets questions that combine multiple topics. Being aware of these links helps you prepare for the types of multi-step problems that appear on the exam.

Common cross-topic combinations

Combination	Example
Differentiation + Trigonometry	Differentiate trigonometric functions, find stationary points of trig curves
Integration + Exponentials	Integrate eᵏˣ, solve differential equations involving exponential growth/decay
Algebra + Coordinate Geometry	Use the discriminant to determine tangency conditions for circles
Sequences + Proof	Prove properties of arithmetic/geometric sequences
Trigonometry + Integration	Use trig identities to rewrite integrands (e.g., ∫sin²x dx using cos 2x)
Differentiation + Integration	Use differentiation to check integration results; areas and volumes
Logs + Exponentials + Modelling	Model real-world growth/decay and solve for time/quantity
Parametric equations + Differentiation	Find dy/dx for parametric curves, tangent/normal equations
Statistics + Probability	Use probability rules within hypothesis testing or distribution questions
Kinematics + Calculus	Variable acceleration — differentiate/integrate position, velocity, acceleration
Forces + Trigonometry	Resolve forces on inclined planes at angle θ
Vectors + Coordinate Geometry	Position vectors, distances, midpoints in 2D and 3D

Example multi-step problem

A curve is defined parametrically by x = 2cos t, y = 3sin t for 0 ≤ t ≤ 2π.

(a) Find dy/dx in terms of t.                    [Parametric differentiation]
(b) Find the equation of the tangent at t = π/4. [Coordinate geometry + trig exact values]
(c) Find the area enclosed by the curve.          [Integration + parametric + trig identities]
(d) Show that the Cartesian equation is an        [Algebra + trig identity sin²t + cos²t = 1]
    ellipse of the form x²/4 + y²/9 = 1.

This single question spans four topics. Being prepared for these connections is essential.

---

Revision Checklist

Use this checklist to track your progress across the entire specification. Rate each topic as:

• R (Red): Not confident — needs significant revision
• A (Amber): Partially confident — some gaps remain
• G (Green): Confident — can answer exam questions accurately

Pure Mathematics

Topic	Sub-topic	Rating
Proof	By deduction, exhaustion, contradiction, counter-example
Algebra	Indices, surds, quadratics, discriminant
Algebra	Simultaneous equations, inequalities
Algebra	Polynomials, factor theorem, algebraic division
Algebra	Partial fractions
Algebra	Functions, composite, inverse, modulus
Algebra	Graph transformations
Coordinate Geometry	Straight lines, circles
Coordinate Geometry	Parametric equations
Sequences	Arithmetic and geometric series
Sequences	Binomial expansion (integer and general)
Trigonometry	Radians, arc length, sector area
Trigonometry	Identities and equations
Trigonometry	Addition and double angle formulae
Trigonometry	R-formula
Trigonometry	Small angle approximations
Exponentials	eˣ, ln x, solving equations
Exponentials	Modelling growth and decay
Differentiation	Power rule, trig, exponentials, ln x
Differentiation	Product, quotient, chain rules
Differentiation	Implicit and parametric differentiation
Differentiation	Stationary points, optimisation
Differentiation	Connected rates of change
Integration	Standard integrals
Integration	Substitution and by parts
Integration	Partial fractions, areas, trapezium rule
Integration	Differential equations
Numerical Methods	Change of sign, iteration, Newton-Raphson
Vectors	2D and 3D, arithmetic, geometric problems

Mechanics

Topic	Sub-topic	Rating
Kinematics	SUVAT, graphs, variable acceleration
Kinematics	Projectiles
Forces	Force diagrams, Newton's laws
Forces	Friction, inclined planes
Forces	Connected particles, pulleys
Moments	Equilibrium, non-uniform rods, tilting

Statistics

Topic	Sub-topic	Rating
Sampling	Types, advantages, disadvantages
Data	Histograms, box plots, cumulative frequency
Data	Mean, variance, standard deviation, coding
Probability	Venn/tree diagrams, conditional, independence
Distributions	Binomial — conditions, calculation, mean, variance
Distributions	Normal — standardisation, probabilities
Hypothesis Testing	Binomial and normal tests, critical regions
Large Data Set	Structure, missing data, patterns, outliers

---

Practice Strategy

Past Papers vs Topic Practice

Both are important, but they serve different purposes:

Topic practice (first phase of revision):

• Work through questions on a single topic from textbooks or topic-sorted past paper questions
• Builds fluency in specific techniques
• Identifies gaps in knowledge
• Best done under "open book" conditions initially, then without notes

Past papers (second phase of revision):

• Work through complete papers under timed conditions
• Builds exam stamina and time management
• Practises topic selection (identifying which technique to use)
• Exposes you to multi-step and cross-topic questions
• Should be done under exam conditions: 2 hours, with formula book, no notes

Recommended progression

Phase	Duration	Activity
Phase 1: Topic Review	6–8 weeks before exam	Review notes, work through textbook exercises, fill knowledge gaps
Phase 2: Topic Practice	4–6 weeks before exam	Topic-sorted past paper questions, mark using mark schemes
Phase 3: Full Papers	3–4 weeks before exam	Complete past papers under timed conditions
Phase 4: Targeted Review	1–2 weeks before exam	Revisit weak topics identified from full papers
Phase 5: Final Preparation	Last few days	Review formula book, key facts, common mistakes

Timing advice for full papers

When practising full papers, aim for:

First attempt: Complete the paper with no time limit, but record how long it takes.
Second attempt (different paper): Set a timer for 2 hours 15 minutes (slight margin).
Third attempt onwards: Strict 2-hour time limit.

This builds your speed gradually rather than overwhelming you with time pressure from the start.

After each practice paper

1. Mark your paper using the official AQA mark scheme
2. Identify errors — categorise them:
   • Knowledge gap (did not know the method) → revise the topic
   • Careless error (wrong sign, arithmetic mistake) → slow down and check
   • Time pressure (did not finish) → practise speed on routine questions
   • Misread question (answered wrong thing) → read more carefully, underline key words
3. Log your errors in a revision journal — track patterns over time
4. Redo the questions you got wrong one week later to check retention

How many past papers should you do?

Aim for a minimum of 6 full papers (2 per paper type) under timed conditions. Ideally, complete all available past papers and specimen papers.

Paper	Available papers (as of current specification)
Paper 1	Specimen, specimen set 2, plus annual papers from first sitting
Paper 2	Same
Paper 3	Same

AQA also publishes practice papers and topic tests on their website — use these for Phase 2 revision.

---

Strategies for the Exam Day

Before the exam

• Review your formula card (key facts not in the formula book)
• Ensure your calculator is in the correct mode (radians for most pure questions, degrees for some mechanics)
• Bring spare batteries and a spare calculator if possible
• Bring a ruler and pencil for graph sketching

During the exam

1. Read through the entire paper in the first 5 minutes
2. Identify quick wins — start with questions you are confident about
3. Allocate time — roughly 1.2 minutes per mark
4. Show all working — method marks are earned through clear working
5. Answer every question — even partial attempts earn marks
6. Check your answers — especially signs, units, and whether you answered what was asked
7. Re-read "show that" questions — ensure every step is explicit

Common exam-day errors to avoid