Edexcel IGCSE Further Pure Maths 4PM1 Help and Topics

Q: How many papers are in IGCSE Further Pure Maths 4PM1?

Two, Paper 1 and Paper 2. Each is a two-hour written exam worth 100 marks, with about eleven questions, and both are sat in one series.

Q: Is there a Foundation tier for 4PM1?

No. The specification is offered through a single tier, aimed at grades 9 to 4, though a 3 can be awarded. A performance below that is reported as U, unclassified.

Q: Is a calculator allowed in IGCSE Further Pure?

Yes, on both papers. It needs ln x, e to the x and trigonometry in radians, and any model that does algebra symbolically, stores text or has a full typing keyboard is banned.

Q: What topics are in 4PM1?

Ten sections: logarithms and indices, the quadratic function, identities and inequalities, graphs, series, the binomial series, vectors, Cartesian coordinates, calculus and trigonometry. Statistics and matrices are not required.

Q: What formulae do I need to memorise for 4PM1?

Appendix 5 of the specification lists them. They include the laws of logarithms, the sum and product of the roots of a quadratic, arithmetic and geometric series, coordinate formulae and the standard derivatives.

Q: Is IGCSE Further Pure Maths hard?

It is harder than International GCSE Mathematics A, because every question is pure maths at a level close to the first year of A level, and questions often combine topics. It assumes the Higher tier 4MA1 content already.

Q: Should I take 4PM1 with 4MA1?

Most candidates do, because 4PM1 assumes the Higher 4MA1 content. They are different subjects, so 4PM1 adds a second maths qualification rather than repeating the first.

Q: Is IGCSE Further Pure Maths regulated in the UK?

No. Pearson's specification says no UK regulator accredits or oversees it. It exists to prepare pupils for sixth-form maths and further maths.

Q: When do your 4PM1 classes run?

Weekdays or weekends, flexible: choose your batch, or for a very personalised plan choose 1-on-1 at your convenience, weekend mornings or weekdays. We confirm every time in UK time.

Q: Are you part of Pearson Edexcel?

No. Modern Age Coders is independent of Pearson; we follow the published specification, and every example here is our own work.

The short answer

Pearson Edexcel International GCSE Further Pure Mathematics, code 4PM1, is a single-tier qualification for able pupils, usually taken alongside International GCSE Mathematics A. Candidates sit Paper 1 and Paper 2 in the same January or June series; each runs for two hours, with 100 marks spread over roughly eleven questions. Grades 9 to 4 are targeted, with 3 allowed. Content runs from logarithms and series to calculus and radians. Modern Age Coders runs 4PM1 classes live online as part of its IGCSE Mathematics course; families pick a group of 15 to 20 or 1-on-1 lessons, and the first class is free.

Start here

Three courses for a Further Pure candidate

The IGCSE course carries 4PM1. The A level course continues every topic in it, and Further Maths is where a keen 4PM1 pupil often ends up.

FP / 01

IGCSE Mathematics

Further Pure 4PM1 alongside the Higher 4MA1 content it assumes.

Open the syllabus →

FP / 02

A Level Maths

The pure mathematics that picks up where 4PM1 calculus and series stop.

Open the syllabus →

FP / 03

A Level Further Maths

For the pupil whose favourite lesson of the week is 4PM1.

Open the syllabus →

The papers

How is Edexcel IGCSE Further Pure Maths assessed?

Two long papers, one tier, one series. Any topic can appear on either paper, and questions often join two topics together.

The specification describes a short, demanding format. "This specification is offered through a single tier." Both papers draw on the whole content, and Pearson notes that "the solution of any questions may require knowledge of more than one section of the specification content." With "around 11 questions" in two hours, each question is long: a typical one carries several parts and asks for a chain of reasoning, not a single technique.

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1: the assessment
Paper	Time	Marks	Questions	Weight	Calculator
4PM1/01	2 hours	100	Around 11	50 per cent	Allowed
4PM1/02	2 hours	100	Around 11	50 per cent	Allowed

Source: Pearson Edexcel International GCSE in Further Pure Mathematics specification, Issue 1, January 2016, "Qualification at a glance", from the Pearson 4PM1 qualification page.

Grades on one tier

Every question aims at grades 9 down to 4, and a grade 3 can still be awarded. Roughly two fifths of the marks sit over grades 4 and 5, and three fifths over grades 6 to 9.

Below the minimum standard the result is an unclassified U. There is no Foundation option to fall back on.

Status in the UK

The specification states: "This qualification is not accredited or regulated by any UK regulatory body." Pearson designs it as a lead-in to its AS and A levels in Mathematics, Further Mathematics or Pure Mathematics.

For a UK pupil that makes it a signal of ability and a head start on sixth-form maths, rather than a qualification that replaces anything.

Ten topics

What is in the IGCSE Further Pure Maths syllabus?

Ten sections, all pure mathematics. Statistics and matrices are explicitly left out.

The 4PM1 content, with the detail from Pearson's notes that changes how each is taught
#	Section	A detail from the specification notes
1	Logarithmic functions and indices	Laws of logarithms, change of base, equations of the form a^x = b, simple surds
2	The quadratic function	Completing the square, the discriminant, and functions of the roots of a quadratic
3	Identities and inequalities	Algebraic division, the factor and remainder theorems, quadratic inequalities and simple linear programming
4	Graphs	Polynomials and rational functions with asymptotes, and graphical solution of equations
5	Series	Arithmetic and geometric series and sum to infinity; "Proofs of the above are not required."
6	The binomial series	Positive integer n, and rational n for \|x\| < 1; "The validity condition for (ii) is expected."
7	Scalar and vector quantities	Vectors with i and j, position vectors, and dividing a line in a ratio
8	Rectangular Cartesian coordinates	Distance, gradient, dividing a line in a given ratio, parallel and perpendicular lines
9	Calculus	Product, quotient and chain rules, kinematics, areas, volumes of revolution, connected rates of change
10	Trigonometry	Radians, arc and sector, addition formulae, and equations on a given interval

The calculus section is the one that surprises families. It includes differentiating products and quotients, finding volumes by rotating a curve about an axis ("The volumes will be obtained only by revolution about the coordinate axes"), and connected rates of change, where one quantity's rate is found from another's. For maxima and minima, "Justification of maxima and minima will be expected", so a pupil must show why a stationary point is a maximum, not just find it.

What 4PM1 assumes

Pearson expects "a thorough knowledge of the content common to" International GCSE Mathematics A at Higher tier, or Mathematics B. Questions can lean on that content without teaching it again.

In practice a pupil needs secure Higher algebra, surds, trigonometry in triangles and straight-line geometry before starting 4PM1.

What 4PM1 leaves out

The specification says "knowledge of statistics and matrices will not be required." There is no data handling and no probability to collect easier marks on.

For trigonometry, Pearson adds: "Long questions, explicitly involving excessive manipulation, will not be set." The difficulty is in combining ideas, not in pages of algebra.

Formulae to learn

Which formulae must you memorise for 4PM1?

Pearson prints a formula sheet in the exam, and then publishes a separate list of formulae that will not be on it.

Appendix 5 of the specification is titled "Formulae to learn", and it "gives formulae that students are expected to remember and will not be included on the formula sheet provided in the examination papers." It is the single most useful page in the specification for revision, because it tells a pupil exactly what has to be carried into the exam room.

Areas covered by the 4PM1 "Formulae to learn" list, with examples in our words
Area	Examples of what must be remembered
Logarithms and indices	log_a xy = log_a x + log_a y, the quotient and power laws, log_a a = 1 and log_a 1 = 0
Quadratic equations	The quadratic formula, and for roots α and β: α + β = −b/a and αβ = c/a
Series	The nth term and the sum to n terms of arithmetic and geometric series
Coordinate geometry	Gradient, distance between two points, and the point dividing a line in the ratio m : n
Calculus	The standard derivatives of powers of x, sin ax, cos ax and e^ax
Trigonometry	Arc length s = rθ, sector area ½r²θ, the area ½ab sin C and cos²θ + sin²θ = 1

Some trigonometry is given rather than learned. The specification notes that the addition formulae for sin(A + B), cos(A + B) and tan(A + B) "will be on the formula sheet", and that tan θ = sin θ ÷ cos θ is provided too. The cosine rule is given; the sine rule is expected to be known. We give pupils a one-page list split into "on the sheet" and "learn it" in the first month, and test the second column every week.

Calculator rules for 4PM1, from the specification
Rule	Detail
Use	Expected on both papers
Minimum functions	Including ln x, e^x, standard form, and trigonometric functions in degrees or radians
Prohibited	Stored data or text, a typewriter-style keyboard, or symbolic algebra and calculus

Source: 4PM1 specification, Appendix 5 "Formulae to learn", the notes to sections 9 and 10, and "Calculators". The examples in the first table are our summary; the appendix gives the formulae in full.

Worked examples, ours

Three 4PM1 style problems, worked in full

We wrote these to show the three habits 4PM1 tests most: rates in calculus, the binomial series with its validity condition, and the roots of a quadratic.

Our first problem, connected rates. A spherical balloon is inflated so that its volume V cm³ increases at 50 cm³ per second. Given V = (4/3)πr³, find the rate at which the radius is increasing when r = 5 cm.

Write down what is known and what is wanted: dV/dt = 50, and we want dr/dt when r = 5.
Differentiate the volume with respect to the radius: dV/dr = 4πr². At r = 5 this is 100π.
Link the rates with the chain rule: dV/dt = (dV/dr) × (dr/dt), so 50 = 100π × dr/dt.
Solve: dr/dt = 50 ÷ 100π = 1/(2π), which is about 0.159 cm per second. Leaving 1/(2π) as the exact answer first, then rounding, protects the accuracy mark.

Our second problem, the binomial series

Expand (1 + x)^1/2 up to the term in x³, state the values of x for which it is valid, and use it to estimate √1.02.

With n = 1/2: 1 + (1/2)x + [(1/2)(−1/2)/2]x² + [(1/2)(−1/2)(−3/2)/6]x³ = 1 + x/2 − x²/8 + x³/16. Because n is not a positive integer, the series is valid only for |x| < 1. Putting x = 0.02 gives 1 + 0.01 − 0.00005 + 0.0000005 = 1.0099505, which agrees with a calculator to seven decimal places.

Our third problem, functions of the roots

The roots of 2x² − 5x + 1 = 0 are α and β. Without solving the equation, find α² + β².

From the formulae to learn, α + β = 5/2 and αβ = 1/2. Then α² + β² = (α + β)² − 2αβ = 25/4 − 1 = 21/4. The identity in the middle step is the whole question; pupils who try to find the roots first waste minutes on surds.

Practice set: six questions we wrote for 4PM1 candidates, each answer verified in Python
#	Our question	Section	Answer
1	Solve 2^x+1 = 3^x, giving x to 3 significant figures.	Logarithms	1.71
2	Find the sum to infinity of the geometric series 12 + 6 + 3 + ...	Series	24
3	Differentiate x²e^3x.	Calculus	e^3x(2x + 3x²)
4	Find the equation of the tangent to y = x³ − 2x at the point where x = 1.	Calculus	y = x − 2
5	Solve sin 2x = 0.5 for 0° < x < 180°.	Trigonometry	x = 15° or 75°
6	A is (2, 1) and B is (10, 5). Find the point that divides AB in the ratio 1 : 3.	Coordinates	(4, 2)

We wrote the three worked problems and the six practice questions ourselves on 31 October 2026; none comes from Pearson. Pearson's specimen papers sit on the Pearson 4PM1 page, and past papers are available to schools and exam centres through Pearson.

Who it suits

Is IGCSE Further Pure Maths worth taking?

For a pupil heading towards A level maths and enjoying algebra, often yes. For a pupil still consolidating Higher 4MA1, not yet.

Pearson positions 4PM1 as "a solid basis for students wishing to progress to Edexcel AS and Advanced GCE Level, or equivalent qualifications." Much of its content (logarithms, the binomial expansion, radians, differentiation rules and integration) reappears in the first year of A level Mathematics. A pupil who has met it at fifteen arrives in the sixth form with a term or more of pure maths already familiar.

A readiness check we use before recommending 4PM1 (our criteria, not Pearson's)
Sign	Ready for 4PM1	Wait a term
Higher 4MA1 algebra	Rearranges and factorises without prompts	Still needs worked examples for each type
Surds and indices	Simplifies and rationalises reliably	Makes sign or index-law slips
Unfamiliar questions	Tries an approach and checks it	Stops when the method is not obvious
Time available	Can give 4PM1 its own weekly slot	Already stretched across other subjects

One point from the 4MA1 specification is worth repeating here. Pearson warns that taking two qualifications "in the same subject" is very likely to be treated as one by colleges and universities. 4PM1 is a different subject from Mathematics A, so it adds a second maths qualification rather than duplicating the first.

Families thinking about the sixth form can read whether A level Further Maths is worth it and our further maths tuition page. For the main IGCSE across boards, start from IGCSE maths tuition online.

Progression

Four rungs through Further Pure

The order we teach follows dependency: algebra and logarithms first, calculus once those hold, mixed papers last.

Milestones for a Further Pure candidate
Stage	When	Evidence of readiness
1. Higher 4MA1 secure	Before starting	All Higher algebra, surds, trigonometry and coordinate work done reliably
2. Algebraic topics	First two terms	Logarithms, quadratics and their roots, identities, inequalities, series and the binomial expansion
3. Calculus, vectors, radians	Following two terms	All four rules of differentiation, integration with areas and volumes, rates, vectors and radian trigonometry
4. Mixed long questions	Before the series	Two-hour papers where one question joins two or three sections, with every step written

Rung two decides the pace

Logarithms and the binomial series feed into calculus questions later, so slips here repeat everywhere.

Pupils who like seeing series converge can try them in code with coding for IGCSE students.

Rung four is the real exam

About eleven questions in two hours means long, joined-up problems. We practise those in full, timed.

Weekly classes slow down during the exam weeks.

The catalogue

Nine courses near Further Pure

Grouped by stage. A free first class shows which one fits.

I

The IGCSE pair

4MA1 and 4PM1

UK / FP1 / 01

IGCSE mathematics

Higher 4MA1 and Further Pure 4PM1 taught as one programme.

Open the syllabus

UK / FP1 / 02

Algebra foundations

For a pupil whose Higher algebra needs securing first.

Open the syllabus

UK / FP1 / 03

GCSE mathematics

For UK school pupils adding 4PM1 to a GCSE.

Open the syllabus

II

The sixth form

Where 4PM1 leads

UK / FP2 / 01

A level maths

Pure, mechanics and statistics building on 4PM1 calculus.

Open the syllabus

UK / FP2 / 02

A level further maths

Complex numbers, matrices and further pure.

Open the syllabus

UK / FP2 / 03

IB mathematics

For pupils heading into the IB Diploma.

Open the syllabus

III

Problem solving

Beyond the specification

UK / FP3 / 01

Competition mathematics

Hard problems for pupils who finish 4PM1 papers early.

Open the syllabus

UK / FP3 / 02

Maths through coding

Series, functions and rates explored in Python.

Open the syllabus

UK / FP3 / 03

Statistics and probability

The data topics 4PM1 leaves out, for A level.

Open the syllabus

How classes run

Two topics in every question

Weekdays or weekends, flexible: choose your batch, or for a very personalised plan choose 1-on-1 at your convenience, weekend mornings or weekdays. The teaching team sits in India, which has no daylight saving, and every time we send you is in UK time.

After school

A common slot for Year 10 and 11.

Evening

For pupils with clubs after school.

Weekend morning

A full two-hour paper fits here.

Formulae drill

The Appendix 5 list tested weekly, so nothing has to be derived in the exam.

Exact first

Answers kept exact, as in 1/(2π), before any rounding.

Justified maxima

A reason for every maximum and minimum, as the specification expects.

Validity conditions

The |x| < 1 condition stated whenever the binomial series uses a fractional power.

Joined-up questions

Practice where logarithms meet calculus or vectors meet coordinates.

Group or 1-on-1

A group of 15 to 20 4PM1 pupils, or one teacher for one pupil.

Student work

What our students built in class

Pieces from normal lessons, not marked for any exam. Browse the rest in student labs.

AI and ML

NutriLife

An AI nutrition coach that reads what you eat and works you toward a target.

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AI and ML

Misti

A chatbot that answers mathematics and programming questions, built and deployed by a student.

by Harshit · Open it

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GuardianX

An assistant that helps a young person recognise unsafe situations online.

by Vivaan · Open it

Web app

SkyCast

A weather forecasting site with live conditions for any location.

by Krish

Fees

Fees for Further Pure 4PM1 classes

A monthly fee in US dollars applies to families outside India. Your first class is free, and we only bill once a course and a fixed weekly slot are agreed.

Free first class

USD 0

no card required

A first class on a 4PM1 topic
A readiness view against Higher 4MA1
A phone number books it

Book it

Group batch

USD 100

a month, billed in US dollars

A group of 15 to 20 4PM1 pupils
One teacher through the course
Formulae tests most weeks
Lighter weeks around each series

Start here

One to one

USD 150

a month, billed in US dollars

A teacher for one pupil
Planned around both IGCSEs
Suits a pupil starting 4PM1 late

Enquire

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Questions about 4PM1

What families ask about IGCSE Further Pure Maths

How many papers are in IGCSE Further Pure Maths 4PM1?

Two, Paper 1 and Paper 2. Each is a two-hour written exam worth 100 marks, with about eleven questions, and both are sat in one series.

Is there a Foundation tier for 4PM1?

No. The specification is offered through a single tier, aimed at grades 9 to 4, though a 3 can be awarded. A performance below that is reported as U, unclassified.

Is a calculator allowed in IGCSE Further Pure?

Yes, on both papers. It needs ln x, e to the x and trigonometry in radians, and any model that does algebra symbolically, stores text or has a full typing keyboard is banned.

What topics are in 4PM1?

Ten sections: logarithms and indices, the quadratic function, identities and inequalities, graphs, series, the binomial series, vectors, Cartesian coordinates, calculus and trigonometry. Statistics and matrices are not required.

What formulae do I need to memorise for 4PM1?

Appendix 5 of the specification lists them. They include the laws of logarithms, the sum and product of the roots of a quadratic, arithmetic and geometric series, coordinate formulae and the standard derivatives.

Is IGCSE Further Pure Maths hard?

It is harder than International GCSE Mathematics A, because every question is pure maths at a level close to the first year of A level, and questions often combine topics. It assumes the Higher tier 4MA1 content already.

Should I take 4PM1 with 4MA1?

Most candidates do, because 4PM1 assumes the Higher 4MA1 content. They are different subjects, so 4PM1 adds a second maths qualification rather than repeating the first.

Is IGCSE Further Pure Maths regulated in the UK?

No. Pearson's specification says no UK regulator accredits or oversees it. It exists to prepare pupils for sixth-form maths and further maths.

When do your 4PM1 classes run?

Weekdays or weekends, flexible: choose your batch, or for a very personalised plan choose 1-on-1 at your convenience, weekend mornings or weekdays. We confirm every time in UK time.

Are you part of Pearson Edexcel?

No. Modern Age Coders is independent of Pearson; we follow the published specification, and every example here is our own work.

Elsewhere on this site

More for able young mathematicians

The main IGCSE, the sixth-form choices, and problem solving beyond any syllabus.

IGCSE maths tuition online

Both IGCSE boards explained on one page.

Further maths tuition online

Where a 4PM1 pupil often heads at sixteen.

Is A level Further Maths worth it?

A clear look at the choice many 4PM1 families face.

A level maths tuition online

The course that continues 4PM1 calculus and series.

British Mathematical Olympiad preparation

For pupils who want harder problems than any exam.

Coding classes in the UK

All of our UK pages, listed together.

Related pages in this guide

Part of the guide: UK maths exam boards and specification codes

International GCSE and Cambridge International

GCSE maths

A level maths and further maths

Scotland

International Baccalaureate

More guides: UK maths year by year, Reception to Year 11 · UK selective school maths tests: 7+, 11+, 13+ and 16+ · UK university admissions: maths tests and computing applications · UK maths enrichment, competitions and able pupils

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Book a free 4PM1 class

Leave a number and we will call at a UK time that suits you. The class works one of our 4PM1 problems with your child and checks the Higher 4MA1 base underneath it.

Would you like to read first? Every course page shows its full syllabus, how we teach walks through a class, and student labs has examples of student work.

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Three courses for a Further Pure candidate

IGCSE Mathematics

A Level Maths

A Level Further Maths

How is Edexcel IGCSE Further Pure Maths assessed?

Grades on one tier

Status in the UK

What is in the IGCSE Further Pure Maths syllabus?

What 4PM1 assumes

What 4PM1 leaves out

Which formulae must you memorise for 4PM1?

Three 4PM1 style problems, worked in full

Our second problem, the binomial series

Our third problem, functions of the roots

Is IGCSE Further Pure Maths worth taking?

Four rungs through Further Pure

Rung two decides the pace

Rung four is the real exam

Nine courses near Further Pure

The IGCSE pair

IGCSE mathematics

Algebra foundations

GCSE mathematics

The sixth form

A level maths

A level further maths

IB mathematics

Problem solving

Competition mathematics

Maths through coding

Statistics and probability

Two topics in every question

Formulae drill

Exact first

Justified maxima

Validity conditions

Joined-up questions

Group or 1-on-1

What our students built in class

NutriLife

Misti

GuardianX

SkyCast

Fees for Further Pure 4PM1 classes

Rated 4.9 across 547 Google reviews

What families ask about IGCSE Further Pure Maths

How many papers are in IGCSE Further Pure Maths 4PM1?

Is there a Foundation tier for 4PM1?

Is a calculator allowed in IGCSE Further Pure?

What topics are in 4PM1?

What formulae do I need to memorise for 4PM1?

Is IGCSE Further Pure Maths hard?

Should I take 4PM1 with 4MA1?

Is IGCSE Further Pure Maths regulated in the UK?

When do your 4PM1 classes run?

Are you part of Pearson Edexcel?

More for able young mathematicians

IGCSE maths tuition online

Further maths tuition online

Is A level Further Maths worth it?

A level maths tuition online

British Mathematical Olympiad preparation

Coding classes in the UK

Related pages in this guide

International GCSE and Cambridge International

GCSE maths

A level maths and further maths

Scotland

International Baccalaureate

Book a free 4PM1 class

Want a class sooner? Book a priority demo

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