A Level Further Maths Topics by Exam Board: Full List

Several different exam boards offer A level Further Maths qualifications. This can make finding information about the content that is covered confusing for students and parents. This article looks at the A level Further Maths topics that are covered by the main exam boards, so you can rest assured that you know what you need to know for success in your maths A level.

The most popular A level Further Maths courses are:

• Edexcel A Level Further Maths (9FM0)

• AQA A Level Maths (7367)

• OCR A Level Further Maths A (H245)

• OCR A Level Further Maths B (MEI) (H645)

In this article, I will be focusing on the topics as they are organised in the Edexcel 9FM0 specification. The content of the different specifications is broadly similar, however. The main differences between the exam boards are presented in a later section of the article.

How is the Edexcel A Level Further Maths (9FM0) course organised?

Every student sits the Core Pure Mathematics 1 and Core Pure Mathematics 2 papers

• Each paper is 1 hour and 30 minutes

• Each paper contains a total of 75 marks

• Each paper is worth 25% of the total course grade

In addition, each student sits two options papers

• Each paper is 1 hour and 30 minutes

• Each paper contains a total of 75 marks

• Each paper is worth 25% of the total course grade

There are 8 options papers to choose from

You may choose either any two Option 1 papers, or else a matching pair of Option 1 and Option 2 papers. This means that there are a total of 10 different option pairs.

Note that it is also possible to sit more than two of the option papers. In this case, the combination of papers that result in the best course grade will be used by the exam board to determine your mark.

Paper 1 & 2: Core Pure Mathematics 

These papers lay the essential groundwork for Further Mathematics, covering fundamental topics such as proof, complex numbers, matrices, and calculus. They help students develop a deeper understanding of abstract mathematical structures, logical reasoning, and problem-solving techniques that are crucial for advanced study. 

The topics included in these papers are highly applicable in physics, engineering, and computer science, making them an indispensable part of the curriculum. By mastering these core concepts, students build a strong foundation that prepares them for tackling more specialised areas of mathematics.

Proof

• Proof by mathematical induction (including sums of series, divisibility and matrix powers)

Complex Numbers

• Solving quadratic, cubic, and quartic equations with real coefficients

• Addition, subtraction, multiplication, and division of complex numbers in x+iy form

• Understanding and using real parts, imaginary parts and the complex conjugate

• Representation on an Argand diagram

• Conversion between Cartesian and modulus-argument forms

• Multiplication and division using modulus-argument form

• Loci and simple transformations in the Argand diagram

• de Moivre’s theorem (including use with multiple angle formulas and series sums)

• Euler’s formula and exponential form of complex numbers

• Finding and interpreting nth roots of complex numbers

• Applications of complex roots in geometry

Matrices

• Matrix addition, subtraction, and multiplication

• The zero and identity matrices

• Matrix representation of linear transformations in 2D and 3D

• Invariant points and lines under transformations

• Determinants and inverses of 2×2 and 3×3 matrices

• Singular and non-singular matrices

• Solving simultaneous linear equations in three variables using matrices

• Geometric interpretation of solutions of simultaneous linear equations in three variables

Further Algebra & Functions

• Relationships between roots and coefficients of polynomial equations (up to quartics) 

• New equations from linear transformations of roots of polynomial equations

• Formulas for sums and products of polynomial roots

• Method of differences for series sums (including use of partial fractions)

• Finding Maclaurin series of a function

• Recognising and using Maclaurin series for common functions

Further Calculus

• Volumes of revolution using integration

• Improper integrals and convergence

• Mean value of a function

• Integration using partial fractions (including quadratic factors in the denominator)

• Differentiation of inverse trigonometric functions

• Integration using inverse trigonometric functions and trigonometric substitution 

Further Vectors

• Vector and Cartesian equations of lines in three dimensions

• Vector and Cartesian equations of planes

• Calculation and use of the scalar product

• Perpendicular vectors and the scalar product

• Intersection of a line and a plane; distances between two lines or between a point and a line or a plane

Polar Coordinates

• Use of polar coordinates and conversion between Cartesian and polar forms

• Sketching curves in polar coordinates

• Calculating areas enclosed by polar curves

Hyperbolic Functions

• Definitions and graphs of sinhx, coshx and tanhx

• Differentiation and integration of hyperbolic functions

• Definitions and use of the inverse hyperbolic functions

• Derivation and use of logarithmic forms of the inverse hyperbolic functions

• Integration using inverse hyperbolic functions and related substitutions

Differential Equations

• Solving first-order differential equations using integrating factors

• General and particular solutions to differential equations and families of solution curves

• Differential equation for kinematics and other modelling contexts

• Solving homogeneous second-order differential equations using auxiliary equations

• Solving non-homogeneous second-order differential equations using Complementary functions and particular integrals

• Relationship between discriminant of the auxiliary equation and form of solutions

• Finding and understanding the solutions for simple harmonic motion models

• Finding and understanding the solutions for damped harmonic motion models

• Modelling systems using coupled first-order differential equations

Paper 3A: Further Pure Mathematics 1 

This paper explores more advanced pure mathematical techniques, introducing students to deeper levels of abstraction in areas such as further trigonometry, calculus, and number theory. 

The topics covered here, including Taylor series, parametric equations, and inequalities, enhance problem-solving skills and logical reasoning. 

These mathematical tools are not only essential for theoretical applications but also have practical uses in scientific computing, finance, and cryptography. By studying these advanced concepts, students gain insight into the versatility and power of mathematical modelling.

Further Trigonometry

• The t-formulae for trigonometric functions

• Applying t-formulae to trigonometric identities

• Solving trigonometric equations using t-formulae

Further Calculus

• Derivation and use of Taylor series

• Using series expansions to find limits of functions

• Leibniz’s theorem for differentiating products

• L'Hospital’s rule for evaluating indeterminate limits

• Weierstrass substitution for integration

Further Differential Equations

• Solving differential equations using Taylor series expansions

• Using substitution to reduce differential equations to standard forms

Coordinate Systems

• Cartesian and parametric equations for conic sections

• Focus-directrix properties of conic sections

• Tangents and normals to conic section curves

• Loci problems involving conic sections

Further Vectors

• The vector (‘cross’) product and its geometric interpretation

• The scalar triple product and its geometric interpretation

• Applications of vectors to three-dimensional geometry

Further Numerical Methods

• Numerical solutions to first- and second-order differential equations

• Simpson’s Rule

Inequalities

• Manipulation and solving of algebraic inequalities (including those involving the modulus)

Paper 4A: Further Pure Mathematics 2 

Building upon the foundations of Paper 3A, this paper expands into abstract mathematical concepts such as group theory, advanced calculus, and combinatorics. These topics are highly relevant in pure mathematics and have significant applications in physics, economics, and coding theory. 

By studying areas such as eigenvalues, recurrence relations, and modular arithmetic, students develop strong analytical skills that are useful in both theoretical and applied contexts. This paper challenges students to think critically and logically, strengthening their ability to construct and analyse mathematical arguments.

Groups

• The axioms of a group

• Examples of groups including Cayley tables and cyclic groups

• The order of a group and the order of an element, including subgroups

• Lagrange’s theorem

• Isomorphism (for groups of order up to 8)

Further Calculus

• Derivation and use of reduction formulae for integration

• Calculation of arc length and area of a surface of revolution

Further Matrix Algebra

• Eigenvalues and eigenvectors of 2×2 and 3×3 matrices

• Reduction of matrices to diagonal form

• Use of the Cayley-Hamilton theorem

Further Complex Numbers

• Further loci and regions in the Argand plane

• Transformations from the z-plane to the w-plane

Number Theory

• The division theorem, with application to Euclidean algorithm and congruences

• Bezout’s identity

• Modular arithmetic and properties of congruences

• Fermat’s Little Theorem

• Divisibility tests

• Solution of congruence equations

• Combinatorics (counting problems, permutations and combinations)

Further Sequences & Series

• First and second order recurrence relations

• Solving recurrence relations to obtain closed forms

• Proof by induction of closed forms

Paper 3B: Further Statistics 1 

This paper introduces advanced statistical methods that are fundamental for data analysis, probability theory, and real-world decision-making. Topics such as probability distributions, hypothesis testing, and statistical modelling help students develop quantitative reasoning skills. 

These techniques are widely applied in fields like finance, engineering, and artificial intelligence, making this paper highly relevant for students interested in data-driven careers. Mastering these concepts allows students to interpret and manipulate data effectively, a crucial skill in today’s digital world.

Discrete Probability Distributions

• Mean and variance of discrete probability distributions, with expected values of functions of random variables

Poisson & Binomial Distributions

• The Poisson distribution

• Mean and variance of the Poisson and binomial distributions

• Approximating binomial distributions by Poisson distributions

Geometric & Negative Binomial Distributions

• Geometric and negative binomial distributions

• Mean and variance of a geometric distribution

• Mean and variance of a negative binomial distribution

Hypothesis Testing

• Testing for the mean of a Poisson distribution

• Testing for the parameter of a geometric distribution

Central Limit Theorem

• Applying the central limit theorem to other distributions

Chi Squared Tests

• Chi squared goodness of fit tests

Probability Generating Functions

• Definition and derivation of probability generating functions, and application to standard distributions

• Use of probability generating functions to find mean and variance

• Probability generating functions of sums of independent random variables

Quality of Tests

• Type I and II errors, size and power of tests, and the power function

Paper 4B: Further Statistics 2 

Expanding upon the concepts in Paper 3B, this paper delves deeper into statistical modelling, correlation analysis, and probability distributions. 

Advanced topics such as confidence intervals, regression analysis, and hypothesis testing provide students with essential tools for research and data science. 

This paper equips students with the abilities to critically assess data and apply statistical methods to complex problems, skills that are invaluable in scientific research and business analytics.

Linear Regression

• Least squares regression and residuals

• Use of residuals and the residual sum of squares

Continuous Probability Distributions

• Probability density functions and cumulative distribution functions of continuous random variables

• Relationship between PDFs and CDFs

• Mean and variance, mode, median and percentiles, and skewness of continuous random variables

• The continuous uniform distribution

Correlation

• Calculation and use of the product moment correlation coefficient

• Use and interpretation of Spearman’s rank correlation coefficient

• Hypothesis testing for correlation

Combinations of Random Variables

• Distribution of linear combinations of independent normal variables

Estimation, Confidence Intervals & Tests Using a Normal Distribution

• Concepts of standard error, estimator and bias (including quality of estimators)

• Confidence intervals and their interpretation

• Confidence limits for a normal mean with known variance

• Hypothesis test for difference of two means

• Use of large sample results for unknown population variances

Other Hypothesis Tests & Confidence Intervals

• Hypothesis test and confidence interval for the variance of a normal distribution

• Hypothesis test that two independent random samples are from normal populations with equal variances

Confidence Intervals & Tests Using the t-Distribution

• Hypothesis test and confidence interval for the mean of a normal distribution with unknown variance

• Paired t-tests

• Hypothesis test and confidence interval for the difference of means between two independent normal distributions

Paper 3C: Further Mechanics 1 

This paper extends classical mechanics concepts, covering topics such as momentum, energy conservation, and collisions. The principles studied here are foundational for physics and engineering, particularly in understanding motion, forces, and impacts. 

By exploring real-world applications of Newtonian mechanics, students develop an appreciation for how mathematical models can describe physical systems with precision and accuracy.

Momentum & Impulse

• Impulse-momentum principle and collision of spheres

• Impulse-momentum principle in vector form

Work, Energy & Power

• Kinetic and potential energy, work and power, and the conservation of mechanical energy

Elastic Strings & Springs & Elastic Energy

• Elastic strings and springs and Hooke’s Law

• Energy stored in an elastic string or spring

Elastic Collisions in One Dimension

• Direct impact of elastic spheres and Newton’s law of restitution

• Successive direct impacts

Elastic Collisions in Two Dimensions

• Oblique impact of smooth elastic spheres

• Successive oblique impacts

Paper 4C: Further Mechanics 2 

This paper builds on concepts introduced in Paper 3C, delving into more advanced topics in mechanics. It extends knowledge of motion in circles, equilibrium, and impulse-momentum principles, which are crucial in understanding real-world physics problems. 

The topics studied here are particularly relevant for engineering, aerodynamics, and applied physics, where forces, energy, and motion are key to solving complex challenges.

Motion in a Circle

• Angular speed and motion in a horizontal circle (including conical pendulums)

• Motion in a vertical circle

Centres of Mass of Plane Figures

• Moment of a force and centre of mass of discrete 1D and 2D mass distributions

• Centres of mass of uniform and composite plane figures and of frameworks (including equilibrium problems)

Further Centres of Mass

• Centres of mass of uniform and non-uniform rigid bodies and composite bodies

• Equilibrium of rigid bodies

• Toppling and sliding of rigid bodies

Further Dynamics

• Newton’s laws in one dimension for variable forces

• Simple harmonic motion (including kinetic, potential and elastic energy)

Further Kinematics

• Kinematics of straight-line motion with acceleration as a function of displacement, time or velocity

Paper 3D: Decision Mathematics 1 

This paper introduces fundamental techniques in algorithms, graph theory, and optimisation, which are essential for problem-solving in fields such as logistics, computing, and operational research. 

Students learn how to apply mathematical strategies to real-world problems, including network design, scheduling, and efficient resource allocation. Understanding decision mathematics equips students with tools to analyse and optimise systems in business, economics, and engineering.

Algorithms and Graph Theory

• Ideas of algorithms including implementation of text- or flow chart based algorithms

• Bin packing, bubble sort and quick sort

• Order of nodes and Eulerian and semi-Eulerian graphs

• Planarity algorithm for planar graphs

Algorithms on Graphs

• Prim’s and Kruskal’s algorithms for minimum spanning trees

• Dijkstra’s and Floyd’s algorithms for shortest paths

Algorithms on Graphs II

• The route inspection algorithm

• The practical and classical travelling salesman problems (including the nearest neighbour algorithm)

Critical Path Analysis

• Modelling of a project by an activity network from a precedence table

• Completion of a precedence table

• Algorithm for finding the critical path (including identifying critical activities)

• Calculating total float of an activity and constructing Gantt charts

• Constructing resource histograms

• Scheduling project activities using the least number of workers required

Linear Programming

• Formulating problems as linear programs (including slack, surplus and artificial variables)

• Objective line and vertex methods for graphical solution of two variable problems

• The simplex algorithm and tableau for max/min problems

• The two-stage simplex and big-M methods for max/min problems

Paper 4D: Decision Mathematics 2 

Building on Paper 3D, this paper explores more advanced topics in decision mathematics, including complex network flows, game theory, and dynamic programming. These concepts are widely used in economics, artificial intelligence, and strategic planning, helping students develop a deep understanding of optimisation techniques. 

By mastering these topics, students enhance their ability to formulate and solve real-world decision-making problems in a systematic and logical manner.

Transportation Problems

• The north-west corner method for finding an initial basic feasible solution

• The steppingstone method for obtaining an improved solution (including improvement indices)

• Formulating the transportation problem as a linear programming problem

Allocation (Assignment) Problems

• Cost matrix reduction and use of the Hungarian algorithm

• Formulating the Hungarian algorithm as a linear programming problem

Flows in Networks

• Cuts and their capacity

• Use of the labelling procedure in determining the maximum flow in a network

• Use of the max–flow min–cut theorem to prove that a flow is a maximum flow

• Multiple sources and sinks, and vertices with restricted capacity

• Determining the optimal flow rate in a network

Dynamic Programming

• Principles of dynamic programming (including Bellman’s principle, stage and state variables, and use of tabulation)

Game Theory

• Two person zero-sum games and the pay-off matrix

• Identification of play safe strategies and stable solutions

• Reduction of pay-off matrices using dominance arguments

• Optimal mixed strategies for a game with no stable solution using graphical methods

• Optimal mixed strategies for a game with no stable solution by converting games to linear programming problems

Recurrence Relations

• Use of recurrence relations to model problems

• Solution of first and second order linear homogeneous and non-homogeneous recurrence relations

Decision Analysis

• Using, constructing and interpreting decision trees

• Use of expected monetary values (EMVs) and utility to compare alternative courses of action

How Do the Other Exam Boards Differ from Edexcel?

The topics included in the different A level Further Maths courses are broadly similar. In this section, I will outline the different ways that the other Further Maths courses are organised.

How is the AQA A Level Further Maths (7367) course organised?

Every student sits the same Paper 1 and Paper 2, covering the core pure maths content of the course

• Each paper is 2 hours

• Each paper contains a total of 100 marks

• Each paper is worth 33⅓% of the total course grade

• Either or both of the papers may assess the following topics

  • A: Proof

  • B: Complex numbers

  • C: Matrices

  • D: Further algebra and functions

  • E: Further calculus

  • F: Further vectors

  • G: Polar coordinates

  • H: Hyperbolic functions

  • I: Differential equations

  • J: Numerical methods

In addition, each student sits a Paper 3 that is based on the options they have chosen

• The paper is 2 hours

• The paper contains a total of 100 marks

• The paper is worth 33⅓% of the total course grade

• Each student will choose two out of the following three options

  • Mechanics

  • Statistics

  • Discrete Mathematics

See the course specification for further details.

How is the OCR A Level Further Maths A (H245) course organised?

Every student sits two mandatory Pure Core papers, Pure Core 1 and Pure Core 2

• Each paper is 1 hour and 30 minutes

• Each paper contains a total of 75 marks

• Each paper is worth 25% of the total course grade

In addition, each student sits two optional papers, based on the options they have chosen

• Each paper is 1 hour and 30 minutes

• Each paper contains a total of 75 marks

• Each paper is worth 25% of the total course grade

• Each student will choose two out of the following four options

  • Statistics

  • Mechanics

  • Discrete Mathematics

  • Additional Pure Mathematics

See the course specification for further details.

How is the OCR A Level Further Maths B (MEI) (H645) course organised?

Every student sits a mandatory Core Pure paper

• This paper is 2 hours and 40 minutes

• The paper contains a total of 144 raw marks (180 scaled marks)

• The paper is worth 50% of the total course grade

In addition, each student may sit a Major Option paper and up to three Minor Option papers, based on the options they have chosen

• The Major Option paper is 2 hours and 15 minutes

  • It contains a total of 120 raw marks (120 scaled marks)

  • It is worth 33⅓% of the total course grade

  • There are two Major Options to choose from

    • Mechanics Major

    • Statistics Major

• Each Minor Option paper is 1 hour and 15 minutes

  • Each contains a total of 60 raw marks (60 scaled marks)

  • Each is worth 16⅔% of the total course grade

  • There are 6 minor options to choose from

    • Mechanics Minor

    • Statistics Minor

    • Modelling with Algorithms

    • Numerical Methods

    • Extra Pure

    • Further Pure with Technology

  • (Note that the Further Pure with Technology paper is 1 hour and 45 minutes instead of 1 hour and 15 minutes)

There are three different ‘routes’ for choosing options and completing the A Level Further Mathematics qualification:

• Route A

  • Mandatory Core Pure paper

  • Mechanics Major paper

  • One optional minor paper (but not Mechanics Minor)

• Route B

  • Mandatory Core Pure paper

  • Statistics Major paper

  • One optional minor paper (but not Statistics Minor)

• Route C

  • Mandatory Core Pure paper

  • Any three optional minor papers

See the course specification for further details.

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References

Edexcel A Level Further Maths Specification
AQA A Level Further Maths Specification
OCR A Level Further Maths A Specification
OCR A Level Further Maths B Specification