Function notation
f : x ↦ f(x); (f ∘ g)(x) = f(g(x)) Pearson Edexcel International GCSE 4PM0
📐 Edexcel International GCSE Further Pure Mathematics Formula Sheet 2026
Pure extension topics for Pearson’s International GCSE Further Pure Mathematics — calculus, series, trigonometry, vectors, and matrices with examiner-style reminders.
Pure focus International GCSE Exam-ready
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.
Further Pure, Clearly Organised
Use this reference alongside your specification: it groups the relationships most often needed for non-routine Pearson questions — from indices and logarithms through integration and counting.
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Algebra, functions & proof-style reasoning
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Trigonometry, circular measure & coordinate geometry
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Differentiation & integration foundations
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Permutations, combinations & probability
Functions, roots & inequalities
Mappings, domain and range, composite and inverse functions, and quadratic inequalities.
Inverse function
Reflection in y = x; domain/range swap.
f⁻¹(f(x)) = x Discriminant (quadratic)
For ax² + bx + c = 0: Δ = b² − 4ac Quadratic roots
x = (−b ± √(b² − 4ac)) / (2a) Sum and product of roots
α + β = −b/a, αβ = c/a Linear & quadratic inequalities
Sketch graphs; critical values; test intervals.
Topic Focus
Exam technique
- State domain exclusions (e.g. division by zero) before manipulating expressions.
- For |ax + b| inequalities, split into cases or square both sides when both sides ≥ 0.
Indices, surds & logarithms
Laws of indices, rationalising denominators, and logarithmic identities.
Index laws
a^m × a^n = a^{m+n} (a^m)^n = a^{mn} a^{-m} = 1/a^m Surd conjugate
(a + √b)(a − √b) = a² − b
Log definition
If a^x = N then x = log_a N Log laws
log_a (xy) = log_a x + log_a y log_a (x/y) = log_a x − log_a y log_a (x^k) = k log_a x Change of base
log_a x = ln x / ln a Topic Focus
Accuracy
- Rationalise denominators with surds unless asked otherwise.
- Check log arguments stay positive.
Polynomials & series
Factor and remainder theorems, binomial expansion for positive integer n, and arithmetic/geometric progressions.
Remainder theorem
P(x) divided by (x − a) has remainder P(a) Factor theorem
(x − a) is a factor ⟺ P(a) = 0 Binomial (positive integer n)
C(n,r) = n! / (r!(n−r)!)
(a + b)^n = Σ_{r=0}^{n} C(n,r) a^{n−r} b^r Arithmetic progression
T_k = a + (k − 1)d S_n = n/2 (2a + (n − 1)d) = n/2 (a + l) Geometric progression
T_k = ar^{k−1} S_n = a(1 − r^n)/(1 − r) (r ≠ 1) Infinite sum (|r| < 1)
S_∞ = a / (1 − r) Topic Focus
Binomial
- Identify a and b so the first term is a^n; watch signs in (a − b)^n.
- C(n,r) row on calculator or Pascal’s triangle.
Coordinate geometry
Straight lines, perpendicular distances, and the circle.
Distance
√((x₂ − x₁)² + (y₂ − y₁)²) Midpoint
((x₁ + x₂)/2 , (y₁ + y₂)/2) Gradient
m = (y₂ − y₁)/(x₂ − x₁) Parallel / perpendicular
parallel: m₁ = m₂ perpendicular: m₁ m₂ = −1 Line forms
y − y₁ = m(x − x₁) ax + by + c = 0 Circle centre (a,b), radius r
(x − a)² + (y − b)² = r² General circle
x² + y² + 2gx + 2fy + c = 0; centre (−g, −f), r = √(g² + f² − c) Topic Focus
Tangents
- Tangent to circle: perpendicular from centre to tangent equals radius.
- Substitute line into circle → equal roots for tangency.
Circular measure & trigonometry
Radians, arc length, sector area, and full trigonometric toolkit.
Arc length
s = r θ (θ in radians) Sector area
A = ½ r² θ Pythagorean identity
sin² θ + cos² θ = 1 tan in terms of sin/cos
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 of triangle
½ ab sin C Heron: √(s(s−a)(s−b)(s−c)) Double angle (selection)
sin 2θ = 2 sin θ cos θ cos 2θ = cos² θ − sin² θ Topic Focus
Mode
- Ensure calculator in correct mode (radians vs degrees) to match the question.
- Oblique triangles: often sine or cosine rule.
Vectors (2D)
Position vectors, magnitude, unit vectors, and the scalar product.
Vector form
OP⃗ = p = xi + yj Magnitude
|p| = √(x² + y²) Unit vector
p̂ = p / |p| Scalar product
a · b = |a| |b| cos θ = a₁b₁ + a₂b₂ Parallel vectors
a = k b for scalar k Line equation
r = a + λ d Topic Focus
Geometry
- Perpendicular vectors: a · b = 0.
- Angle: cos θ = (a · b)/(|a||b|).
Matrices (0606 / FM Pure)
Addition, multiplication, determinants, and inverses for 2×2 matrices.
Product
Rows × columns; (AB)_{ij} = row i of A · column j of B.
Determinant (2×2)
det [[a,b],[c,d]] = ad − bc Inverse (2×2)
A⁻¹ = (1/det A) [[d, −b], [−c, a]] Simultaneous equations
AX = B ⇒ X = A⁻¹B when A invertible.
Topic Focus
Check
- det A = 0 ⇒ no unique inverse.
- Verify A A⁻¹ = I.
Differentiation
Gradients, tangents, stationary points, and basic kinematics.
First principles
f′(x) = lim_{h→0} (f(x+h) − f(x))/h Power rule
d/dx (x^n) = n x^{n−1} Chain rule
dy/dx = dy/du · du/dx Product rule
d/dx (uv) = u′v + uv′ Quotient rule
d/dx (u/v) = (u′v − uv′)/v² Stationary points
Solve f′(x) = 0; use f″(x) or sign change to classify Connected rates
dy/dt = dy/dx · dx/dt Topic Focus
Context
- Interpret derivative as gradient or rate of change with units.
- Sketch behaviour using stationary points and asymptotes.
Integration
Antiderivatives, definite integrals, and area under curves.
Fundamental link
If F′ = f then ∫ f(x) dx = F(x) + C Definite integral
∫_a^b f(x) dx = F(b) − F(a) Area under curve
∫_a^b y dx (y ≥ 0) Power rule (n ≠ −1)
∫ x^n dx = x^{n+1}/(n+1) + C Linear substitution
∫ f(ax + b) dx — adjust for constant factor.
Topic Focus
Area
- Area below axis contributes negatively unless using |y|.
- Between curves: ∫ (f − g) dx where f ≥ g.
Kinematics & relative velocity
Constant-acceleration equations and vector relative motion.
Constant acceleration
v = u + at s = ut + ½ at² v² = u² + 2as s = ½ (u + v) t Relative velocity
v_{A rel B} = v_A − v_B Interception
Equal position vectors at same time; or parallel relative velocity with aligned paths.
Topic Focus
Vectors
- Draw clear diagrams; resolve into components i, j.
- Check units (m s⁻¹, m s⁻²).
Permutations, combinations & probability
Counting arrangements and basic probability rules.
Permutations (no repetition)
n! / (n − r)! = nP r Combinations
nC r = n! / (r!(n − r)!) With repetition (order matters)
n^r Probability
P(A) = favourable outcomes / total (equally likely) Complement
P(A′) = 1 − P(A) Independent events
P(A ∩ B) = P(A) P(B) Topic Focus
Structure
- Identify whether order matters (permutation) or not (combination).
- Use tree diagrams for multi-step problems.
How to Use This Formula Sheet
Boost your Cambridge exam confidence with these proven study strategies from our tutoring experts.
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Match the Spec
Confirm which topics your centre entered; omit or extend sections according to your final teaching route.
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Show Working
Pearson mark schemes reward method: quote formulas, sketch graphs, and justify steps in multi-mark questions.
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Time Strategy
Further pure questions can be long — practise finishing routine calculus and trig under timed conditions.
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Link to IGCSE Maths
Revise core algebra and graphs from 4MA1/4MB1; further pure builds directly on that fluency.
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 Edexcel International GCSE Further Pure 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. Pearson Edexcel International GCSE Further Pure Mathematics (4PM0) formula sheet for 2026: advanced algebra, trigonometry, coordinate geometry, calculus, matrices, and discrete topics beyond International GCSE Mathema… 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.
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