A Level · AQA 7367

A Level Further Maths

Ten sections of compulsory Core Pure, then two optional application modules from Mechanics, Statistics or Discrete Mathematics. Every section of the AQA 7367 specification is covered here — complex numbers, matrices, differential equations and beyond — written by a current AQA examiner with a master’s in applied mathematics.

Four papers minimum · 1h 30 or 2h each Core Pure 1 & 2 compulsory for all Two option papers chosen by centre Strand owner: Mr Poore

Core Pure — Compulsory Content (Sections A to J)

Ten sections. Every Further Maths student takes these.

Core Pure is the heart of the qualification — the content that defines A Level Further Maths and the foundation all optional applications build on. It extends A Level Maths in every direction: deeper algebra, new functions, matrix methods and calculus techniques that genuinely demand mathematical maturity.

A — Proof

Proof by mathematical induction applied to series summation results (e.g. Σr²), divisibility proofs and results involving matrices. The technique that defines the rigour of the whole qualification.

Planned

B — Complex Numbers

Operations in Cartesian form a + bi. Argand diagrams and loci. Modulus–argument form and multiplication in that form. De Moivre’s theorem and its applications. Euler’s formula eiθ. The nth roots of a complex number and roots of unity. Roots of polynomials over ℂ.

In production

C — Matrices

Matrix arithmetic and scalar multiplication. 2D and 3D linear transformations, compositions of transformations. Determinants and their geometric interpretation. Inverse matrices, simultaneous equations via matrix inversion. Eigenvalues, eigenvectors and diagonalisation.

In production

D — Further Algebra and Functions

Roots and coefficients of quadratic, cubic and quartic polynomials. Transformations of roots. Series summation using standard formulae and the method of differences. Maclaurin series for standard functions. Inequalities involving rational functions and modulus. Graphs of rational functions (linear and quadratic denominators) and conic sections.

Planned

E — Further Calculus

Improper integrals. Volumes of revolution about both axes. Mean values of functions. Integration of inverse trigonometric forms. Reduction formulae. Arc length and surface area of revolution. Limits of indeterminate forms.

Planned

F — Further Vectors

Vector and Cartesian equations of straight lines and planes in 3D. The scalar (dot) product and vector (cross) product and their applications. Angles between lines and planes. Perpendicular distances from points to lines and planes. Intersection of lines and planes.

Planned

G — Polar Coordinates

The polar coordinate system (r, θ). Converting between Cartesian and polar forms. Sketching polar curves r = f(θ) including cardioids, limaçons and rose curves. Finding areas enclosed by polar curves using A = ½∫r² dθ.

Planned

H — Hyperbolic Functions

Definitions and graphs of sinh, cosh and tanh via exponentials. Hyperbolic identities (cosh² − sinh² = 1 etc.). Differentiation and integration of hyperbolic functions. Inverse hyperbolic functions in logarithmic form. Integration using hyperbolic substitutions.

Planned

I — Differential Equations

First-order: integrating factor method for dy/dx + P(x)y = Q(x). Second-order homogeneous and non-homogeneous (constant coefficients): complementary function and particular integral. Discriminant cases and solution forms. Simple harmonic motion, damped oscillations. Coupled first-order systems.

Planned

J — Numerical Methods

The mid-ordinate rule and Simpson’s rule for numerical integration. Euler’s step-by-step method and the improved Euler (Runge–Kutta style) method for solving first-order ordinary differential equations numerically.

Planned

Optional Application 1 — Mechanics (Sections MA to ME)

Five sections of applied mathematics beyond A Level.

The Mechanics option extends A Level Maths mechanics into territory that requires proper mathematical treatment: impulse from variable forces, circular motion in vertical planes, elastic energy and the full theory of centres of mass. These five sections cover the complete AQA Mechanics application.

MA — Dimensional Analysis

Finding the dimensions of physical quantities (force, energy, pressure etc.), checking formulae for dimensional consistency, and using dimensional analysis to predict the form of a physical relationship from first principles.

Planned

MB — Momentum and Collisions

Conservation of linear momentum including problems requiring vector resolution. Newton’s Experimental Law and coefficient of restitution e. Direct collisions and impacts with fixed smooth surfaces. Impulse and the relationship I = mv − mu. Impulse from variable forces: I = ∫F dt.

Planned

MC — Work, Energy and Power

Work done by a constant and a variable force. Kinetic and gravitational potential energy. Hooke’s Law: T = λx/L. Elastic potential energy: E = λx²/2L. Conservation of energy in elastic string and spring problems. Work done using WD = ∫F dx.

Planned

MD — Circular Motion

Constant-speed circular motion. Angular speed ω in rad s−1. Centripetal acceleration v²/r = ω²r. Motion in a vertical circle: tension and normal reaction conditions, completing the circle. Conical pendulums with single and double strings.

Planned

ME — Centres of Mass and Moments

Centre of mass of systems of particles and composite bodies. Integration to find centres of mass of plane laminas and solids of revolution. Equilibrium under concurrent forces and couples. Toppling and sliding conditions on inclined planes.

Planned

Optional Application 2 — Statistics (Sections SA to SH)

Eight sections of rigorous statistical inference.

The Statistics option builds a complete framework for inference — from discrete and continuous random variables through Poisson and exponential distributions to chi-squared tests, the t-distribution and confidence intervals. Technically demanding and high reward for students who engage with the theory properly.

SA — Discrete Random Variables

Probability distributions given by table or formula. Expectation E(X), E(X²) and variance Var(X). Linear transformation rules: E(aX + b) and Var(aX + b). The discrete uniform distribution on {1, 2, …, n} and when to use it as a model.

Planned

SB — Poisson Distribution

Conditions for Poisson modelling. X ∼ Po(λ): probabilities from the formula, mean and variance both equal λ. Distribution of the sum of independent Poisson variables. Hypothesis testing for a population mean using a single Poisson observation.

Planned

SC — Type I and Type II Errors

Definition and classification of Type I and Type II errors. Calculating the probability of each across Poisson, binomial and normal distributions. The power of a test (1 − P(Type II error)) and how it varies with the hypothesised parameter value.

Planned

SD — Continuous Random Variables

Probability density functions and the distinction from discrete distributions. Calculating probabilities, medians, quartiles, E(X) and Var(X) by integration. Cumulative distribution functions and their relationship to f(x). The rectangular (uniform) distribution. Independence of combined variables.

Planned

SE — Chi-Squared Tests

Constructing contingency tables (n × m). Calculating observed and expected frequencies. The χ² statistic and degrees of freedom. The requirement Ei > 5 for all cells. Yates’ correction for 2 × 2 tables. Interpreting the result in terms of association.

Planned

SF — Exponential Distribution

The exponential distribution as a model for waiting times. PDF f(x) = λe−λx and CDF F(x) = 1 − e−λx. Mean = 1/λ, variance = 1/λ². The fundamental connection between a Poisson process and exponential inter-event times.

Planned

SG — Inference: t-Distribution

Testing the mean of a normal distribution when the variance is unknown. One-sample t-test. Degrees of freedom ν = n − 1. Assumptions, critical values from t-tables, and interpreting the conclusion in context.

Planned

SH — Confidence Intervals

Symmetric confidence intervals for a normal mean with known variance. Large-sample intervals with unknown variance. Using the t-distribution for small-sample intervals. Interpreting a confidence interval correctly and making inferences from it.

Planned

Optional Application 3 — Discrete Mathematics (Sections DA to DG)

Seven sections where structure beats computation.

Discrete Maths is the most conceptually distinct of the three options — graph theory, network algorithms, linear programming and abstract algebra. It rewards precise logical reasoning and clean argument more than any calculator skill. These seven sections cover the complete AQA Discrete application.

DA — Graphs

Terminology: vertex, edge, degree, trail, path, cycle. Eulerian and semi-Eulerian (traversable) graphs. Hamiltonian graphs. Euler’s formula V − E + F = 2 for connected planar graphs. Kuratowski’s theorem for planarity. Complete, bipartite and complement graphs. Adjacency matrices. Graph isomorphism.

Planned

DB — Networks

Network terminology (node, arc, weight). Minimum spanning tree: Prim’s and Kruskal’s algorithms. Route inspection (Chinese Postman) for Eulerian and semi-Eulerian networks. Travelling Salesperson Problem: nearest-neighbour upper bounds and minimum spanning tree lower bounds.

Planned

DC — Network Flows

Directed networks and feasible flows. Cuts: definition and value. The Max-Flow Min-Cut theorem. Supersources and supersinks for multiple sources/sinks. Flow augmentation to find maximum flows. Problems with lower as well as upper capacity constraints on arcs.

Planned

DD — Linear Programming

Formulating optimisation problems with linear constraints and objective function. Graphical methods: feasible region, objective line and vertex search. The simplex algorithm with slack variables. Interpreting maximisation and minimisation from the simplex tableau.

Planned

DE — Critical Path Analysis

Precedence networks using activity-on-node. Forward and backward passes for earliest and latest event times. Critical activities, critical path and total float. Gantt (cascade) diagrams and resource histograms. Resource levelling and scheduling with restricted resources.

Planned

DF — Game Theory

Zero-sum two-player games and pay-off matrices. Play-safe (maximin and minimax) strategies. Stable solutions (saddle points). Identifying dominated strategies and reducing the game. Optimal mixed strategies for 2 × n games via graphical methods. Formulating as a linear programming problem.

Planned

DG — Binary Operations and Groups

Binary operations including modular arithmetic and matrix multiplication. Commutativity and associativity. Cayley tables and Latin squares. Group axioms: closure, identity, inverses, associativity. Order of a group and period of an element. Subgroups and Lagrange’s theorem. Cyclic and abelian groups. Group isomorphism for finite groups.

Planned

Papers and packs

What’s coming, and when.

Every resource is original — written from the specification and past-paper patterns, never recycled exam questions — and comes with a full mark scheme showing how marks are actually awarded.

ResourceIncludesStatus
Core Pure topic packsSections A–J · compulsory for all students All ten Core Pure sections with original questions and full mark schemes — complex numbers and matrices first In production
Optional Mechanics packsSections MA–ME · Application 1 Five sections of Further Mechanics: impulse, circular motion, energy methods and centres of mass Planned
Optional Statistics packsSections SA–SH · Application 2 Eight sections of Further Statistics: from discrete random variables through to chi-squared, t-tests and confidence intervals Planned
Optional Discrete packsSections DA–DG · Application 3 Seven sections of Discrete Mathematics: graph theory, algorithms, linear programming and abstract algebra with group theory Planned
Predicted papers — summer 2027Core Pure 1 & 2 plus chosen option papers Full predicted paper set with mark schemes, built from past-paper gap analysis across all examined sections Coming spring 2027