A level Further Mathematics formula sheet 2026: What is given and what you must memorise
A Level Further Mathematics formula sheets vary by exam board, but every student receives some form of official reference in the exam. This guide covers what Cambridge, AQA, and Edexcel provide, which formulae you still need to memorise, and a topic-by-topic checklist to sharpen your revision before exam season. Let’s get started.
- Does A level Further Mathematics provide a formula sheet or booklet in the exam?
- What formulae Cambridge A level Further Mathematics provides in the exam
- What formulae AQA and Edexcel A level Further Mathematics provide in their exams
- Which formulae you must memorise for A level Further Mathematics that are not given
- Topic-by-topic formula checklist for A level Further Mathematics students
- How to memorise A level Further Mathematics formulae effectively for exam season
- Frequently asked questions
Does A level Further Mathematics provide a formula sheet or booklet in the exam?
Yes, all three major exam boards provide an official formula booklet during A Level Further Mathematics [1] examinations. The booklet is not optional reading material; it is a working reference that you are expected to know how to use quickly and accurately under timed conditions.
A common mistake we see at Times Edu is students treating the formula booklet as a safety net. They assume that because a formula is listed, they do not need to understand it deeply. That assumption costs marks, because the booklet gives you the formula but never tells you when or how to apply it.
The practical reality is that you still need to be fluent with every formula in the booklet, and you must additionally memorise a significant number of results that are deliberately left out. Knowing exactly what falls into each category is the first step toward efficient revision.
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What formulae Cambridge A level Further Mathematics provides in the exam
Cambridge International A Level Further Mathematics follows the 9231 specification, and the official reference document is the Cambridge International List MF19. This is the same booklet used across Cambridge A Level Mathematics and Further Mathematics, so students sitting both subjects work from a single, unified source.
The Cambridge 9231 formula booklet covers the following major areas:
- Mensuration: Surface area and volume of standard solids, arc length, and sector area.
- Trigonometry: Compound angle identities, double angle results, and the t-formulae where $t = \tan(\theta/2)$.
- Hyperbolic functions: Definitions of sinh, cosh, and tanh in exponential form, plus inverse hyperbolic results.
- Algebra: Summation formulae for $\sum r$, $\sum r^2$, and $\sum r^3$.
- Calculus: Standard derivatives and integrals, including those for inverse trigonometric and hyperbolic functions.
- Complex numbers: Euler’s relation $e^{i\theta} = \cos\theta + i\sin\theta$, the exponential/polar form $z = re^{i\theta}$, and De Moivre’s theorem.
- Matrices: No specific matrix formulae are listed; matrix algebra is largely procedural.
- Probability and statistics: Probability generating functions and moment generating functions are referenced for Further Statistics components.
- Mechanics: Standard results for motion and energy in the Further Mechanics options.
MF19 does not include conic section parametric forms explicitly for every curve. Students are expected to know standard parametric representations of parabolas, ellipses, and rectangular hyperbolas from memory, even though eccentricity relations may appear in outline form.
>>> Read more: A Level Further Mathematics 9231 books 2026: Complete guide for students and teachers
What formulae AQA and Edexcel A level Further Mathematics provide in their exams
AQA and Edexcel each publish their own versions of the A Level Further Maths formula sheet, and the differences between them are real enough to affect your revision strategy if you switch boards or sit mock papers from multiple sources.
AQA Further Maths formula sheet
The AQA Further Maths formula sheet accompanies the Level 3 Certificate and A Level specifications. It includes trigonometric identities, Maclaurin series expansions for standard functions, summation results, and calculus tables. AQA provides expansions for $e^x$, $\ln(1+x)$, $\sin x$, $\cos x$, and $(1+x)^n$ within its Maclaurin section.
A point many students miss: AQA separates the formula sheet for Mathematics from the one for Further Mathematics. If you are sitting both qualifications, confirm which sheet applies to which paper before your first practice run.
Edexcel Further Maths formulae
The Pearson Edexcel A Level Mathematics Formulae Book is a notably comprehensive document. It covers Core Pure, Mechanics, and Statistics in one volume, and it includes statistical distribution tables alongside mathematical formulae. The Edexcel Core Pure formula sheet features:
- Complex numbers: Rectangular form $z = x + iy$, polar form, Euler’s relation, and De Moivre’s theorem.
- Calculus: Derivatives and integrals for inverse trigonometric functions and hyperbolic functions.
- Series: Maclaurin expansions for key functions.
- Hyperbolic identities and inverse hyperbolic integrals.
In our experience working with Edexcel students at Times Edu, the Edexcel formulae book is the most generous in scope. That can ironically create a false sense of security. Students who rely heavily on looking up results during timed practice tend to underperform because the seconds add up.
The table below summarises what each exam board provides at a glance:
| Area | Cambridge MF19 | AQA Formula Sheet | Edexcel Formulae Book |
|---|---|---|---|
| Euler’s relation and De Moivre’s theorem | Yes | Yes | Yes |
| Maclaurin series expansions | Partial | Yes (key functions) | Yes (key functions) |
| Hyperbolic function definitions | Yes | Yes | Yes |
| Inverse hyperbolic integrals | Yes | Yes | Yes |
| Summation formulae | Yes | Yes | Yes |
| Conic section forms | Partial | Partial | Partial |
| Matrix formulae | No | No | No |
| Statistical tables | Separate document | Separate document | Included in booklet |
>>> Read more: A Level Further Maths Mock Improvement Plan : 4-Week Recovery from Bad Mock
Which formulae you must memorise for A level Further Mathematics that are not given
Drawing on years of experience at Times Edu working with Further Mathematics students, this section is where revision effort genuinely separates the A from the A*. The formula booklet is curated, not exhaustive, and the gaps are deliberate.
Complex numbers: While Euler’s relation and De Moivre’s theorem are given, the applications of De Moivre’s theorem to find roots of unity, to expand $\cos n\theta$ and $\sin n\theta$ in terms of powers, and to derive trigonometric series are not spelled out. You must know the method, not just the formula.
Matrix formulae: No exam board provides a formula for the inverse of a $2 \times 2$ or $3 \times 3$ matrix. The general result for the inverse of a $2 \times 2$ matrix, the condition for singularity (det = 0), the Cayley-Hamilton theorem, and eigenvalue/eigenvector methods are all expected from memory.
Conic sections: Standard Cartesian and parametric forms for the parabola $y^2 = 4ax$, ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, and hyperbola $\frac{x^2}{a^2} – \frac{y^2}{b^2} = 1$ must be memorised. Focal chord properties and directrix equations are not given.
Differential equations: No board provides the integrating factor method as a formula. You must recall the form $\frac{dy}{dx} + P(x)y = Q(x)$ and derive the integrating factor $e^{\int P(x)\,dx}$ independently.
Proof by induction structures: The logical scaffold for induction proofs, especially for matrix and divisibility problems, is procedural knowledge you carry into the exam without any reference.
Polar coordinates: Arc length and area formulae in polar form are sometimes included (Cambridge MF19 lists them), but the method for finding tangents, normals, and intersection points in polar curves requires procedural memory.
>>> Read more: A Level Further Maths Topic Order for 2026: What to Study First for Smarter Revision
Topic-by-topic formula checklist for A level Further Mathematics students
Use this checklist during revision to track which formulae are given and which require active memorisation. Mark each item as G (given in booklet) or M (must memorise).
| Topic | Formula / Result | Status |
|---|---|---|
| Complex numbers | Euler’s relation: $e^{i\theta} = \cos\theta + i\sin\theta$ | G |
| Complex numbers | De Moivre’s theorem | G |
| Complex numbers | Roots of unity method | M |
| Complex numbers | Loci conditions (circle, half-line, perpendicular bisector) | M |
| Matrices | Inverse of $2 \times 2$ matrix | M |
| Matrices | Determinant of $3 \times 3$ matrix | M |
| Matrices | Eigenvalue equation $\det(A – \lambda I) = 0$ | M |
| Matrices | Cayley-Hamilton theorem | M |
| Further calculus | Maclaurin series (standard functions) | G (most boards) |
| Further calculus | Integrating factor for first-order ODEs | M |
| Further calculus | Reduction formulae | M (derive in exam) |
| Hyperbolic functions | Definitions of sinh, cosh, tanh | G |
| Hyperbolic functions | Osborn’s rule application | M |
| Conic sections | Standard Cartesian forms | M |
| Conic sections | Parametric forms and eccentricity | M |
| Polar coordinates | Area: $\frac{1}{2}\int r^2\,d\theta$ | G (Cambridge) / M (others) |
| Series | $\sum r$, $\sum r^2$, $\sum r^3$ | G |
| Series | Method of differences | M |
| Proof | Induction scaffold for matrices and divisibility | M |
| t-formulae | $\sin\theta$, $\cos\theta$, $\tan\theta$ in terms of $t$ | G |
>>> Read more: A Level Further Maths Mark Scheme Tips for 2026: How to Pick Up More Marks in Every Paper
How to memorise A level Further Mathematics formulae effectively for exam season
A common mistake we see with Further Maths students is treating formula memorisation as a reading task. Passive re-reading of a formula list almost never produces exam-ready recall. The following approaches are practical and proven.
Spaced repetition with active recall. Write a formula from memory, check it, identify any error, then wait 24 hours and repeat. This is far more effective than highlighting. Apps like Anki allow you to digitise your formula cards.
Derive, do not just copy. For results like the integrating factor or reduction formulae, work through the derivation once per week during the lead-up to exams. Understanding where a formula comes from makes it harder to forget and easier to reconstruct if memory fails.
Categorise by exam board provision. Create two separate lists: Formulae your booklet gives you, and formulae you must produce independently. Spend revision time in proportion to the second list.
Timed formula reproduction tests. Set a ten-minute timer and write down every formula you can recall for a single topic. Check against your list. Gaps under time pressure reveal the exact formulae that need more repetition.
Link formulae to question types. Every formula exists to solve a specific class of problem. Each time you learn or review a formula, immediately solve one exam-style question that uses it. This builds retrieval pathways grounded in context, not isolation.
In our experience, the students who score highest are not necessarily those who memorised the most. They are the ones who trained themselves to retrieve the right formula quickly, apply it accurately, and move on without hesitation.
Frequently asked questions
Does Cambridge A level Further Mathematics provide a formula booklet in the exam?
Yes. Cambridge provides the MF19 formula booklet for both Mathematics and Further Mathematics examinations under the 9231 specification. This document covers trigonometry, hyperbolic functions, calculus, series, and complex number formulae, among others.
Which complex number and matrix formulae are given in A level Further Mathematics?
Complex number formulae including Euler’s relation, polar and exponential forms, and De Moivre’s theorem are provided by all major exam boards. Matrix formulae, including inverse calculations and eigenvalue methods, are not given by any board. Students must memorise all matrix procedures independently.
Which calculus formulae must you memorise for A level Further Mathematics?
The integrating factor formula for first-order ordinary differential equations is not given and must be memorised. Reduction formulae are typically expected to be derived in the exam from integration by parts. Maclaurin series for standard functions are generally provided, but the structure of the method of differences is procedural and must be known.
Which further mechanics and statistics formulae are provided in the exam?
This depends on which optional components you are sitting. Edexcel includes mechanics and statistics formulae alongside statistical tables in its formulae book. Cambridge includes relevant mechanics results in MF19. Check your specific component specification before assuming a formula will or will not appear.
How many formulae do you need to memorise for A level Further Mathematics in total?
There is no fixed number, because the answer depends on your exam board and optional modules. As a working estimate, students typically need to memorise between 30 and 50 individual results or procedures that are not covered by the formula booklet, covering matrices, conic sections, proof, and differential equations at minimum.
Are the formulae given in Further Mathematics the same across all exam boards?
No. There is significant overlap, particularly for complex numbers, hyperbolic functions, and series, but the scope and formatting differ. Edexcel’s booklet is generally the most comprehensive. Cambridge’s MF19 is shared with A Level Mathematics. AQA’s formula sheet follows its own structure. Always use the official document from your specific board.
Where can I find a complete formula list for A level Further Mathematics revision?
Each board publishes its formula booklet on its official website. Edexcel’s formulae book is accessible through Pearson’s qualifications portal. Cambridge’s MF19 is available through the Cambridge International portal. AQA’s formula sheet is downloadable from the AQA website. Physics and Maths Tutor (PMT) also hosts condensed versions that are widely used for revision.
Conclusion
At Times Edu, our 1-on-1 Further Mathematics tutors work with students to build exactly this kind of structured, board-specific formula literacy, alongside deep problem-solving practice. If you or your child is preparing for 9231, AQA, or Edexcel Further Maths and wants a clear revision roadmap built around their specific exam board and timeline, reach out to our academic consultants for a personalised session. Knowing what to memorise, when to memorise it, and how to use it under exam conditions is the kind of strategic advantage that consistently produces top results.
