---
title: "Edexcel IGCSE Further Pure Maths 4PM1 Help and Topics"
description: "Edexcel IGCSE Further Pure Maths 4PM1 help: one tier, two 2-hour papers of about 11 questions, calculus, logs and series, and formulae to learn. Live online classes."
canonical: https://learn.modernagecoders.com/edexcel-igcse-further-pure-maths-4pm1-help
source: src/pages/edexcel-igcse-further-pure-maths-4pm1-help.html
---
> Edexcel IGCSE Further Pure Maths 4PM1 help: one tier, two 2-hour papers of about 11 questions, calculus, logs and series, and formulae to learn. Live online classes.

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.

[![IGCSE Maths course thumbnail](/images/igcse-maths.webp)  FP / 01 IGCSE Mathematics Further Pure 4PM1 alongside the Higher 4MA1 content it assumes. Open the syllabus →](/courses/igcse-mathematics-mastery)[![A-Level Maths course thumbnail](/images/a-level-maths.webp)  FP / 02 A Level Maths The pure mathematics that picks up where 4PM1 calculus and series stop. Open the syllabus →](/courses/a-level-maths-course-pure-mechanics-statistics)[![A Level Further Maths course thumbnail](/images/a-level-further-maths.webp)  FP / 03 A Level Further Maths For the pupil whose favourite lesson of the week is 4PM1. Open the syllabus →](/courses/a-level-further-maths-course)

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.

| 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](https://qualifications.pearson.com/en/qualifications/edexcel-international-gcses/international-gcse-further-pure-mathematics-2017.html).

### 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.

| # | Section | A detail from the specification notes |
| --- | --- | --- |
| 1 | Logarithmic functions and indices | Laws of logarithms, change of base, equations of the form ax = 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.

| Area | Examples of what must be remembered |
| --- | --- |
| Logarithms and indices | loga xy = loga x + loga y, the quotient and power laws, loga a = 1 and loga 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 eax |
| 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.

| Rule | Detail |
| --- | --- |
| Use | Expected on both papers |
| Minimum functions | Including ln x, ex, 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.

1. Write down what is known and what is wanted: dV/dt = 50, and we want dr/dt when r = 5.
2. Differentiate the volume with respect to the radius: dV/dr = 4πr2. At r = 5 this is 100π.
3. Link the rates with the chain rule: dV/dt = (dV/dr) × (dr/dt), so 50 = 100π × dr/dt.
4. 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.

| # | Our question | Section | Answer |
| --- | --- | --- | --- |
| 1 | Solve 2x+1 = 3x, 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 x2e3x. | Calculus | e3x(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](https://qualifications.pearson.com/en/qualifications/edexcel-international-gcses/international-gcse-further-pure-mathematics-2017.html), 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.

| 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](/is-a-level-further-maths-worth-it) and our [further maths tuition page](/further-maths-tuition-online). For the main IGCSE across boards, start from [IGCSE maths tuition online](/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.

| 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](/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](/courses/igcse-mathematics-mastery)

UK / FP1 / 02

#### Algebra foundations

For a pupil whose Higher algebra needs securing first.

[Open the syllabus](/courses/algebra-foundations-masterclass)

UK / FP1 / 03

#### GCSE mathematics

For UK school pupils adding 4PM1 to a GCSE.

[Open the syllabus](/courses/gcse-mathematics-mastery)II

### The sixth form

Where 4PM1 leads

UK / FP2 / 01

#### A level maths

Pure, mechanics and statistics building on 4PM1 calculus.

[Open the syllabus](/courses/a-level-maths-course-pure-mechanics-statistics)

UK / FP2 / 02

#### A level further maths

Complex numbers, matrices and further pure.

[Open the syllabus](/courses/a-level-further-maths-course)

UK / FP2 / 03

#### IB mathematics

For pupils heading into the IB Diploma.

[Open the syllabus](/courses/ib-mathematics-aa-ai-masterclass)III

### Problem solving

Beyond the specification

UK / FP3 / 01

#### Competition mathematics

Hard problems for pupils who finish 4PM1 papers early.

[Open the syllabus](/courses/olympiad-competition-mathematics-mastery)

UK / FP3 / 02

#### Maths through coding

Series, functions and rates explored in Python.

[Open the syllabus](/courses/maths-through-coding)

UK / FP3 / 03

#### Statistics and probability

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

[Open the syllabus](/courses/statistics-probability-maths-course)

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](/student-labs).

![NutriLife AI nutrition coach project screenshot](/images/projects/nutrilife.webp)

AI and ML

#### NutriLife

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

by Bhavya · [Open it](https://nutrilife-bhavya.lovable.app)

![Misti AI chatbot for maths and coding screenshot](/images/projects/misti.webp)

AI and ML

#### Misti

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

by Harshit · [Open it](https://misti.modernagecoders.com)

![GuardianX AI internet safety assistant screenshot](/images/projects/guardianx.webp)

AI and ML

#### GuardianX

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

by Vivaan · [Open it](https://guardianx-cybersaver.lovable.app)

![SkyCast weather forecast application screenshot](/images/projects/skycast.webp)

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

What families say

## Rated 4.9 across 547 Google reviews

Parents' Google reviews, reproduced as written.

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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](/igcse-maths-tuition-online)

Both IGCSE boards explained on one page.

### [Further maths tuition online](/further-maths-tuition-online)

Where a 4PM1 pupil often heads at sixteen.

### [Is A level Further Maths worth it?](/is-a-level-further-maths-worth-it)

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

### [A level maths tuition online](/a-level-maths-tuition-online)

The course that continues 4PM1 calculus and series.

### [British Mathematical Olympiad preparation](/british-mathematical-olympiad-bmo-preparation)

For pupils who want harder problems than any exam.

### [Coding classes in the UK](/coding-classes-in-united-kingdom)

All of our UK pages, listed together.

## Related pages in this guide

Part of the guide: [UK maths exam boards and specification codes](/uk-maths-exam-boards-and-specification-codes)

### International GCSE and Cambridge International

- [Cambridge IGCSE Maths 0580 help](/cambridge-igcse-maths-0580-help)
- [IGCSE Additional Maths 0606 help](/cambridge-igcse-additional-maths-0606-help)
- [Edexcel International GCSE Maths A 4MA1 help](/edexcel-international-gcse-maths-a-4ma1-help)
- [IGCSE maths for home-educated students](/igcse-maths-for-home-educated-students-uk)
- [Cambridge A Level Maths 9709 help](/cambridge-a-level-maths-9709-help)
- [Cambridge A level Further Maths 9231 help](/cambridge-a-level-further-maths-9231-help)
- [IGCSE maths tuition online](/igcse-maths-tuition-online)

### GCSE maths

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- [GCSE Foundation maths tutor online](/gcse-foundation-maths-tutor-online)
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- [GCSE Statistics tutor online](/gcse-statistics-tutor-online)
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- [GCSE maths tuition online](/gcse-maths-tuition-online)
- [AQA vs Edexcel vs OCR GCSE maths](/aqa-vs-edexcel-vs-ocr-gcse-maths)

### A level maths and further maths

- [AQA A level Maths 7357 help](/aqa-a-level-maths-7357-help)
- [Edexcel A level Maths 9MA0 help](/edexcel-a-level-maths-9ma0-help)
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- [OCR MEI A level Maths H640 help](/ocr-mei-a-level-maths-h640-help)
- [CCEA A level Maths help](/ccea-a-level-maths-help)
- [WJEC A level Maths help (Wales)](/wjec-a-level-maths-help-wales)
- [AQA A level Further Maths 7367 help](/aqa-a-level-further-maths-7367-help)
- [Edexcel A level Further Maths 9FM0 help](/edexcel-a-level-further-maths-9fm0-help)
- [Further Maths when your school does not offer it](/further-maths-when-your-school-does-not-offer-it)
- [A level maths tuition online](/a-level-maths-tuition-online)

### Scotland

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- [National 5 Maths tuition online](/national-5-maths-tuition-online)
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- [Higher Maths tuition online](/higher-maths-tuition-online)
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### International Baccalaureate

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- [IB Maths IA guidance](/ib-maths-ia-guidance)
- [IB Extended Essay in maths or computer science](/ib-extended-essay-maths-computer-science-mentoring)
- [IB maths tuition online](/ib-maths-tuition-online)

More guides: [UK maths year by year, Reception to Year 11](/uk-maths-year-by-year-guide) · [UK selective school maths tests: 7+, 11+, 13+ and 16+](/uk-selective-school-maths-tests-guide) · [UK university admissions: maths tests and computing applications](/uk-university-admissions-maths-and-computing-guide) · [UK maths enrichment, competitions and able pupils](/uk-maths-enrichment-and-competitions-guide)

Start here

## 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](/courses) shows its full syllabus, [how we teach](/how-we-teach) walks through a class, and [student labs](/student-labs) has examples of student work.

[WhatsApp us](https://wa.me/919123366161?text=Hello%20Modern%20Age%20Coders%2C%20I%20would%20like%20a%20free%20first%20class%20for%20Edexcel%20IGCSE%20Further%20Pure%20Maths%204PM1.) · [+91 91233 66161](tel:+919123366161) · [connect@modernagecoders.com](mailto:connect@modernagecoders.com)

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