Quadratic formula
x = (−b ± √(b² − 4ac)) / (2a) AQA Level 2 Certificate 8365
🔢 AQA Level 2 Further Mathematics Formula Sheet 2026
Stretch GCSE mathematics with matrices, deeper trigonometry, coordinate geometry, and calculus foundations — tailored to AQA’s Level 2 Further Maths certificate.
UK Level 2 Algebra & geometry Calculus intro
Our formula sheets are free to download — save this one as PDF for offline revision.
Aligned with the latest 2026 syllabus and board specifications. This sheet is prepared to match your exam board’s official specifications for the 2026 exam series.
Bridge to A-Level Mathematics
This sheet supports students taking AQA’s Level 2 Certificate in Further Mathematics alongside GCSE Maths: it emphasises fluency, reasoning, and the multi-step problems that distinguish further maths papers.
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Quadratics, inequalities & coordinate geometry
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Matrices & simultaneous equations
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Differentiation & integration basics
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Arithmetic & geometric sequences
Algebra & functions
Quadratics, inequalities, simultaneous equations, and function ideas.
Discriminant
Δ = b² − 4ac Completing the square
x² + px = (x + p/2)² − (p/2)² Simultaneous (linear–quadratic)
Substitute; or eliminate; check solutions in context.
Composite function
(f ∘ g)(x) = f(g(x)) Inverse (simple cases)
Swap x and y; rearrange; state domain restrictions.
Topic Focus
Accuracy
- Show factorisation or formula clearly before decimal answers.
- Reject extraneous roots from squaring.
Coordinate geometry & circles
Lines, distances, midpoints, and circle equations.
Distance
√((x₂ − x₁)² + (y₂ − y₁)²) Midpoint
((x₁ + x₂)/2 , (y₁ + y₂)/2) Gradient
m = (y₂ − y₁)/(x₂ − x₁) Perpendicular lines
m₁ m₂ = −1 Circle
(x − a)² + (y − b)² = r² Tangent condition
Distance from centre to line equals radius.
Topic Focus
Sketching
- Mark centre and radius before algebraic work.
- Tangents from external point: equal lengths or right angle at point of contact.
Trigonometry
Identities, equations, sine/cosine rules, and area.
Pythagorean identity
sin² θ + cos² θ = 1 tan
tan θ = sin θ / cos θ Sine rule
a / sin A = b / sin B = c / sin C Cosine rule
a² = b² + c² − 2bc cos A cos A = (b² + c² − a²)/(2bc) Area
½ ab sin C Simple equations
Use identities; find all solutions in given interval.
Topic Focus
Ambiguous case
- Sine rule can yield two triangles — check with diagram.
- Use degrees or radians consistently.
Matrices
Addition, multiplication, determinants, and inverses (typically 2×2).
Matrix product
Row × column; check dimensions (m×n)(n×p) = (m×p).
Determinant
det [[a,b],[c,d]] = ad − bc Inverse
A⁻¹ = (1/det A) [[d, −b], [−c, a]] Linear systems
AX = B ⇒ X = A⁻¹B Topic Focus
Non-invertible
- det A = 0: no unique solution to AX = B in general.
- Show matrix multiplication is not commutative.
Differentiation
Gradients, tangents, and turning points.
Power rule
d/dx (x^n) = n x^{n−1} Constant multiple & sum
d/dx (k f + g) = k f′ + g′ Chain rule
dy/dx = dy/du · du/dx Stationary points
f′(x) = 0 Second derivative test
f″ > 0 minimum; f″ < 0 maximum (simple cases).
Topic Focus
Interpretation
- Gradient of tangent; rate of change with correct units.
- Sketch curve using turning points and intercepts.
Integration
Antiderivatives and definite integrals for area.
Antiderivative
∫ x^n dx = x^{n+1}/(n+1) + C (n ≠ −1) Definite integral
∫_a^b f(x) dx = F(b) − F(a) Area under curve
∫_a^b y dx Constant of integration
+C for indefinite integrals.
Topic Focus
Signed area
- Negative region below x-axis; split at roots if needed.
- Verify limits match the shaded region in the diagram.
Sequences & series
Arithmetic and geometric progressions.
Arithmetic nth term
a + (n − 1)d Arithmetic sum
S_n = n/2 (2a + (n − 1)d) Geometric nth term
ar^{n−1} Geometric sum
S_n = a(1 − r^n)/(1 − r), r ≠ 1 Infinite sum
S_∞ = a/(1 − r) for |r| < 1 Topic Focus
Modelling
- Identify first term and common difference/ratio.
- Check whether n is the number of terms or index of last term.
Number & proof
Indices, surds, and simple deductive reasoning.
Index laws
a^m × a^n = a^{m+n} (a^m)^n = a^{mn} Rationalising
Multiply numerator and denominator by conjugate surd.
Contradiction (structure)
Assume opposite; deduce impossible conclusion ⇒ original true.
Topic Focus
Presentation
- Each step should follow logically with reasons.
- State ‘assume …’ clearly in proof by contradiction.
How to Use This Formula Sheet
Boost your Cambridge exam confidence with these proven study strategies from our tutoring experts.
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Precision
AQA rewards clear reasoning: label diagrams, state methods, and avoid skipping algebraic steps in proof-style items.
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GCSE + Further
Keep standard GCSE formulas fast — further maths papers assume you can apply them while handling harder structure.
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Pace
Practise timed papers; matrix and trig questions can absorb time if you hesitate on method choice.
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Check Answers
Substitute back, verify matrix products with dimensions, and sense-check areas and gradients.
Formula sheet FAQ
Quick answers about this free PDF, how to use it for exam revision, and how it relates to your official syllabus.
Is the AQA Level 2 Further Mathematics Formula Sheet 2026 free to download as a PDF?
Yes. This Tutopiya formula sheet is free to use and you can download it as a PDF from this page for offline revision. There is no payment or account required for the PDF download.
What Mathematics topics and equations does this formula sheet cover?
This page groups key Mathematics formulas in one place for revision. AQA Level 2 Certificate in Further Mathematics (8365) formula sheet for 2026: advanced algebra, coordinate geometry, trigonometry, matrices, introductory calculus, and sequences. Always cross-check with your official syllabus and past papers for your exam session.
Can I use this instead of the official exam formula booklet in the exam?
No. In the exam you must follow only what your exam board allows in the hall—usually the official formula booklet or data sheet where provided. This page is a revision and teaching aid, not a replacement for board-issued materials.
Who is this formula sheet for (Secondary)?
It is written for students preparing for assessments at Secondary in Mathematics, including classroom revision, homework support, and independent study. Teachers and tutors can also share it as a quick reference.
How should I revise with this formula sheet?
Work through past paper questions, quote the correct formula before substituting values, and check units and notation every time. Pair this sheet with timed practice and mark schemes so you see how examiners expect working to be set out.
Where can I get more help with Mathematics revision?
Explore Tutopiya’s study tools, past paper finder, and revision checklists linked from our tools hub, or book a trial lesson with a subject specialist for personalised support alongside this formula reference.
Excel in AQA Level 2 Further Maths
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Content reflects common Level 2 Further Mathematics themes; always use the official AQA 8365 specification and formulae you are given in the exam.
This qualification is distinct from GCSE Mathematics (8300) — check your centre’s entries.