A Level Further Maths: Core Pure and Applied Options
The second maths A level, taught as a subject of its own: complex numbers, matrices, proof and differential equations, then the applied options your board sets.
Syllabus updated October 2026
Questions? WhatsApp +91 91233 66161 or email connect@modernagecoders.com
Recordings are free with a Google sign-in. These recordings show how we teach, not the exact syllabus of this course.
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Book a Priority DemoHow long this course takes
This course runs 18 months (72 weeks), taught alongside A Level Maths at 2 live classes/week + 3-4 hours practice, rising to full timed papers in the last two months. The range allows for background and pace: a complete beginner uses the full span, and a student with prior knowledge finishes sooner. The syllabus below is planned to the shorter end, with margin kept for revision.
For personalised duration planning, call +91 91233 66161 and we'll map a schedule to your goals.
At a glance
- Who it is for
- Year 12 and 13 students (or equivalent) taking Further Maths with AQA, Edexcel or OCR.
- Prerequisites
- Studying A level Maths at the same time, or already holding it. A strong GCSE Maths grade, which we suggest is 7 or above, makes the first months far smoother. The week 1 diagnostic finds gaps and we close them inside the course
- Format
- Live, interactive classes with a real instructor, never pre-recorded videos. Live classes run about one hour.
- Language
- English or Hindi, depending on the batch.
- Duration
- 18 months (72 weeks), taught alongside A Level Maths
- Weekly commitment
- 2 live classes/week + 3-4 hours practice, rising to full timed papers in the last two months
- Class size
- Group batches of 15 to 20 students, mini batches of up to 6, or 1-on-1.
- Certificate
- Course-completion certificate from Modern Age Coders
- Watch first
- Free recordings of real classes for ages 13 and up, free with a Google sign-in. These recordings show how we teach, not the exact syllabus of this course.
- Live demo
- Optional. Enrol directly on this page, or book a Priority Demo: a full live class of about 45 to 60 minutes with a mentor reserved for you.
- Questions
- WhatsApp +91 91233 66161 or email connect@modernagecoders.com.
How we teach
Deep understanding. Real projects. Expert guidance.
Learn coding, AI and maths through careful explanations, practical demonstrations, and hands-on problem-solving. Our instructors guide you from the foundations to advanced ideas, helping you understand what happens, why it happens, and how to build it yourself.
You will explore concepts step by step, write and improve code, investigate mistakes, ask questions, and apply your knowledge to meaningful projects. Lessons are designed around active participation, thoughtful feedback, and growing independence.
Expert-led live teaching, with practical participation built into every lesson.
- Students explain their thinking.
- Instructors demonstrate, then guide practice.
- Learners code, solve, or build during lessons.
- Questions and misconceptions get attention.
- Assignments receive useful feedback.
- Progress is checked before advancing.
How to join A Level Further Maths: Core Pure and Applied Options
Four steps from this page to the first class. See a full class first with a Priority Demo, or choose a plan and enrol today.
Rated 4.9 across 547 Google reviews. Monthly billing, cancel any time.
How enrolment works
See a full class first (optional)
Book a Priority Demo: a live class of about 45 to 60 minutes with a mentor reserved for you, today or tomorrow. After the class the mentor sends a written skill report and a personal learning roadmap on WhatsApp. If you enrol, the demo fee is adjusted against your first month's fee. Share your demo Order ID with us on WhatsApp and we take it off.
Book a Priority DemoDemo fee ₹499Choose a plan
Group batch (15 to 20 students), mini batch (up to 6 students, India only) or 1-on-1 with your own mentor. Every plan is live online, and each class runs about one hour. The cards below show classes per week and the monthly fee in your currency.
Enrol and pay the first month
Press Enrol Now on your plan, fill in the student's name, email and phone, and pay through secure Razorpay checkout: UPI, cards or net banking in India, and debit or credit card from any other country.
Fix your class timings
After payment, message us on WhatsApp or call +91 91233 66161 with the student's name, the course and your preferred time slot. We confirm the batch, send the class link and schedule the first class at your chosen time.
Billing is monthly and you can cancel any time.
International Students (Outside India)
Billed monthly in US dollars, the same price in every country. Contact us with any questions.
Questions before you enrol? Talk to our team.
Ask about class timings, the right level for your child, what they need to know already, or payment. Parents and adult learners can reach us on WhatsApp, by email or by phone.
Program Overview
Further Maths is a full A level that sits on top of A level Maths. The boards are clear about the relationship: OCR states that learners are assumed to know the content of GCSE Mathematics and A Level Mathematics, and Pearson Edexcel says the students who benefit most are likely to hold A level Maths. In practice most students take the two side by side across Years 12 and 13, and this course is timed to run alongside our A Level Maths course or your school's.
Every board splits Further Maths into a compulsory pure core and a set of options. Edexcel 9FM0 has two Core Pure papers plus two options chosen from Further Pure, Further Statistics, Further Mechanics and Decision Maths. AQA 7367 has two papers on compulsory content and a third paper on two of mechanics, statistics and discrete mathematics. OCR A H245 has Pure Core papers Y540 and Y541 plus two optional papers, and OCR MEI H645 has the Core Pure paper Y420 with a major option or a set of minor options. We teach the shared core to everyone and the options your school or exam centre has entered you for.
The first nine months cover the AS-standard core and the first applied option. Months 10 to 16 take the core to full A level depth, from de Moivre's theorem to second-order differential equations, and finish the options. The final two months are revision and past papers under real timing, matched to your board's paper structure and marked against official schemes. Classes are live and small, so the teacher reads your working as you write it.
What Makes This Program Different
- Built from the boards' own specifications: Edexcel 9FM0, AQA 7367, OCR A H245 and OCR MEI H645, with your board's paper structure driving the practice phase
- Sequenced to run alongside A Level Maths, so each Further Maths topic arrives after the A level Maths skill it leans on, not before
- Proof treated as a skill in its own right: induction, contradiction and the writing of a complete argument are practised every month, not saved for one chapter
- Options taught to your entry, whether that is mechanics, statistics, decision or discrete maths, or further pure, rather than a generic mixture
- Calculator technique for matrices, complex numbers and distributions taught explicitly, alongside the work that must be done by hand
- Two full months of past papers under real timing, marked against official mark schemes, with an error clinic after every paper
Your Learning Journey
Career Progression
Detailed Course Curriculum
Explore the complete week-by-week breakdown of what you'll learn in this comprehensive program.
Topics Covered
- Diagnostic paper on the A level Maths algebra Further Maths assumes
- The structure of a proof by induction: basis, assumption, inductive step, conclusion
- Induction on sums of series
- Induction on divisibility statements
- Induction on matrix powers, previewed for month 3
- Writing the conclusion so it would earn the final mark
Projects You Build
- Proof portfolio opened: six induction proofs written to exam standard and checked line by line
Practice & Assignments
12 induction proofs of increasing difficulty, each with a written conclusion sentence
Topics Covered
- Sigma notation and the standard results for r, r squared and r cubed
- Combining standard results to sum new series
- The method of differences and when it applies
- Partial fractions used to set up a telescoping sum
- Checking a summation formula by substitution and by induction
- Common slips: index limits and the first term
Projects You Build
- Question bank entry: 15 solved series questions, each checked by a second method
Practice & Assignments
20 series problems mixing standard results and the method of differences
Assessment
Month 1 test: a 45-minute paper on induction and series, marked with method marks
Topics Covered
- The imaginary unit and the algebra of complex numbers
- Conjugates and division of complex numbers
- Solving quadratics, cubics and quartics with complex roots
- Representing complex numbers on an Argand diagram
- Modulus and argument, and the modulus-argument form
- Multiplying and dividing in modulus-argument form
Projects You Build
- Complex number reference sheet built by the student, with one worked example per rule
Practice & Assignments
25 complex arithmetic and equation problems, roots shown on an Argand diagram
Topics Covered
- Loci of the form |z - a| = r as circles
- Perpendicular bisectors from |z - a| = |z - b|
- Half-lines from arg(z - a) = theta
- Shading regions defined by inequalities
- Finding greatest and least values of |z| on a locus
- Sketching accurately enough to read answers from the diagram
Projects You Build
- Locus atlas: eight annotated sketches of standard loci and combined regions
Practice & Assignments
16 locus and region questions, each with a labelled sketch
Assessment
Month 2 test: a 45-minute paper on complex numbers
Topics Covered
- Matrix addition and multiplication, and why order matters
- The identity and zero matrices
- 2 by 2 matrices as transformations of the plane
- Rotations, reflections, enlargements and stretches
- Composite transformations and the product of their matrices
- Invariant points and invariant lines
Projects You Build
- Transformation catalogue: each standard 2 by 2 matrix with its effect drawn on the unit square
Practice & Assignments
20 problems on matrix products and transformations, with diagrams
Topics Covered
- Determinants of 2 by 2 and 3 by 3 matrices
- The determinant as an area or volume scale factor
- Inverse matrices and singular matrices
- Solving systems of three linear equations with matrices
- Geometric meaning of a system: three planes and how they can meet
- Calculator checking for 3 by 3 work, and what must still be shown
Projects You Build
- Three-planes casebook: every arrangement of three planes, with a system that produces it
Practice & Assignments
18 problems on determinants, inverses and linear systems
Assessment
Month 3 test: a 45-minute paper on matrices
Topics Covered
- Relationships between roots and coefficients for quadratics, cubics and quartics
- Symmetric functions of roots
- Forming a new equation whose roots are related to the original roots
- Substitution as a shortcut for transformed roots
- Complex roots of real polynomials and conjugate pairs
- Checking answers by reconstructing the polynomial
Projects You Build
- Question bank entry: 12 roots questions solved two ways, by sums of roots and by substitution
Practice & Assignments
20 problems on roots and coefficients
Topics Covered
- Rational functions and their asymptotes
- Sketching graphs of rational functions
- Solving inequalities involving rational expressions
- Graphs of y = |f(x)| and related transformations
- Conic sections in the forms your board specifies
- Using a sketch to check an algebraic answer
Projects You Build
- Graph file: ten sketches with asymptotes, intercepts and turning points labelled
Practice & Assignments
16 sketching and inequality problems
Assessment
Month 4 test: a 45-minute paper on roots and curves
Topics Covered
- Vector and Cartesian equations of a line in three dimensions
- The scalar product and the angle between lines
- Equations of a plane in vector and Cartesian form
- Angles between a line and a plane, and between two planes
- Intersections of lines and planes
- Shortest distances from a point to a line and to a plane
Projects You Build
- 3D geometry sheet: every distance and angle method on one page with a worked example
Practice & Assignments
18 problems on lines and planes
Topics Covered
- Volumes of revolution about the x-axis and the y-axis
- Volumes from parametric curves where your board includes them
- Mean value of a function over an interval
- Choosing the right integral from a description in words
- Modelling a real solid with a volume of revolution
- Accuracy and units in modelling answers
Projects You Build
- Modelling task: the volume of a real container estimated by a volume of revolution and compared with its label
Practice & Assignments
14 volume and mean value problems
Assessment
Month 5 test: a 50-minute paper on vectors and first further calculus
Topics Covered
- Momentum and impulse in one dimension
- Conservation of momentum
- Newton's law of restitution and the coefficient of restitution
- Direct collisions between two particles
- Collisions with a fixed wall
- Loss of kinetic energy in a collision
Projects You Build
- Collision file: worked solutions for every collision type, with a diagram for each
Practice & Assignments
18 momentum and restitution problems
Topics Covered
- Three-particle and successive collisions
- Deciding whether a further collision happens
- Impulse as a vector in two dimensions
- Oblique impacts with a smooth surface where your board includes them
- Setting up equations from a clear diagram
- Statistics route equivalent: discrete random variables, expectation and variance
Projects You Build
- Question bank entry: 10 multi-stage collision problems with the decision points marked
Practice & Assignments
14 multi-stage problems under 10-minute timing each
Assessment
Month 6 test: a 45-minute option paper
Topics Covered
- Work done by a constant force
- Kinetic and gravitational potential energy
- The work-energy principle
- Power, and driving force for a vehicle
- Problems on slopes with resistance
- Statistics route equivalent: the Poisson distribution and when it models a count
Projects You Build
- Energy method sheet: when to use energy and when to use Newton's second law
Practice & Assignments
18 work, energy and power problems
Topics Covered
- Hooke's law for strings and springs
- Elastic potential energy
- Energy problems with elastic strings
- Finding extensions and maximum speeds
- Combining elastic energy with gravity
- Decision route equivalent: algorithms, sorting and the order of an algorithm
Projects You Build
- Modelling task: a bungee jump modelled with energy methods and its assumptions criticised
Practice & Assignments
14 elastic energy problems
Assessment
Month 7 test: a 45-minute option paper
Topics Covered
- Angular speed and acceleration towards the centre
- Horizontal circles: conical pendulums and banked tracks
- Vertical circles and energy
- Conditions for completing a vertical circle
- Particles leaving a circular surface
- Statistics route equivalent: chi-squared tests for goodness of fit and contingency tables
Projects You Build
- Circular motion casebook: six standard set-ups with force diagrams
Practice & Assignments
16 circular motion problems
Topics Covered
- Dimensions of physical quantities
- Checking a formula by its dimensions
- Finding an unknown formula by dimensional analysis
- Criticising and refining a mechanics model
- Writing assumptions the way mark schemes expect
- Decision route equivalent: graphs, networks and minimum spanning trees
Projects You Build
- Model critique: one real situation modelled three ways, with the assumptions compared
Practice & Assignments
12 dimensional analysis and modelling questions
Assessment
Month 8 test: a 45-minute option paper
Topics Covered
- Mixed core problems that do not announce their topic
- Proof rehearsal: induction proofs rewritten from memory
- Option problems mixed across months 6 to 8
- Reading a long question and planning before writing
- Error log review with the teacher
- Targeted re-teach of the two weakest topics per student
Projects You Build
- Revision sheet set one: student-made single pages for every core topic so far
Practice & Assignments
Two 40-minute mixed papers, one core and one option
Topics Covered
- An AS-standard mock across Core Pure and the first option
- Marking against an official-style scheme
- Classifying every lost mark: knowledge, method, accuracy or reading
- Planning the second year from the mock results
- Students who sit the AS as a standalone qualification prepare for it here
- Summer bridging tasks set for the move to second-year content
Projects You Build
- The mock file: marked script, error classification and a written plan
Practice & Assignments
Mock review set: every lost mark reworked to full credit
Assessment
AS-standard mock, marked and reviewed one to one
Topics Covered
- The exponential form of a complex number
- De Moivre's theorem and its proof by induction
- Expressing cos n theta and sin n theta in powers
- Powers of sines and cosines in terms of multiple angles
- Sums of series using complex numbers
- Choosing between Cartesian, modulus-argument and exponential form
Projects You Build
- Proof portfolio entry: de Moivre's theorem proved by induction for positive integers
Practice & Assignments
18 problems on exponential form and de Moivre's theorem
Topics Covered
- The nth roots of unity and their symmetry
- The nth roots of any complex number
- Roots shown as regular polygons on the Argand diagram
- Geometric problems solved with complex roots
- Further loci and transformations where your board includes them
- Exam-style questions that chain several complex number techniques
Projects You Build
- Roots of unity poster: the roots for n = 3 to 8 drawn and their sums explained
Practice & Assignments
16 roots problems
Assessment
Month 10 test: a 50-minute paper on complex numbers
Topics Covered
- Improper integrals and whether they converge
- Derivatives of inverse trigonometric functions
- Integrals that lead to inverse sine and inverse tangent
- Partial fractions with quadratic factors
- Choosing an integration method from the form of the integrand
- Accuracy in exact answers
Projects You Build
- Calculus method map extended: every further integral form routed to its technique
Practice & Assignments
20 further integration problems
Topics Covered
- Maclaurin series from repeated differentiation
- Standard series for exponential, sine, cosine and log functions
- Combining and composing known series
- Range of validity of a series
- Using series to approximate and to find limits
- Checking a series against a calculator value
Projects You Build
- Series table: standard Maclaurin series with validity ranges and one application each
Practice & Assignments
16 Maclaurin series problems
Assessment
Month 11 test: a 50-minute paper on further calculus
Topics Covered
- Polar coordinates and converting to and from Cartesian form
- Sketching polar curves such as cardioids and spirals
- Areas enclosed by polar curves
- Areas between two polar curves
- Tangents parallel and perpendicular to the initial line
- Reading symmetry from a polar equation
Projects You Build
- Polar curve gallery: eight curves sketched by hand and checked against software
Practice & Assignments
16 polar coordinate problems
Topics Covered
- Definitions of sinh, cosh and tanh in terms of exponentials
- Hyperbolic identities and Osborn's rule
- Inverse hyperbolic functions in logarithmic form
- Differentiating and integrating hyperbolic functions
- Integrals that lead to inverse hyperbolic functions
- Solving equations involving hyperbolic functions
Projects You Build
- Hyperbolic reference sheet: definitions, identities, derivatives and integrals on one page
Practice & Assignments
18 hyperbolic function problems
Assessment
Month 12 test: a 50-minute paper on polar and hyperbolic work
Topics Covered
- First-order linear equations with an integrating factor
- Second-order linear equations with constant coefficients
- The auxiliary equation: real, repeated and complex roots
- Particular integrals for common right-hand sides
- General and particular solutions from boundary conditions
- Checking a solution by substituting it back
Projects You Build
- Question bank entry: one solved example for every case of the auxiliary equation
Practice & Assignments
18 differential equation problems
Topics Covered
- Simple harmonic motion as a second-order equation
- Damped and forced oscillations
- Coupled first-order systems
- Interpreting a solution in the context of the model
- Criticising the assumptions of a differential equation model
- Long questions that combine setting up, solving and interpreting
Projects You Build
- Modelling task: a damped oscillation modelled, solved and compared with measured data
Practice & Assignments
12 modelling questions with full written interpretation
Assessment
Month 13 test: a 50-minute paper on differential equations
Topics Covered
- Long questions that cross complex numbers, matrices and calculus
- Proof by induction across series, divisibility, matrices and derivatives
- Further vectors where your board extends them
- Numerical methods where your board includes them
- Choosing the efficient method under time pressure
- Error log review and targeted re-teach
Projects You Build
- Revision sheet set two: single pages for every second-year core topic
Practice & Assignments
Two 60-minute mixed Core Pure papers
Topics Covered
- A full Core Pure paper under timing for your board
- Marking against an official-style scheme
- Classifying every lost mark
- Rewriting proofs that lost marks to full credit
- Comparing performance with the AS-standard mock
- Setting priorities for the final months
Projects You Build
- The mock file updated with the Core Pure mock and a written plan
Practice & Assignments
Mock review set: every lost mark reworked
Assessment
Core Pure mock, marked and reviewed one to one
Topics Covered
- Statistics route: discrete and continuous random variables, and the geometric distribution
- Mechanics route: further work on collisions and energy at A level depth
- Decision route: route inspection, travelling salesperson bounds and linear programming
- Further pure route: further vectors, number theory or groups where your board sets them
- Calculator technique for the option
- Reading option questions carefully: what the context is really asking
Projects You Build
- Option reference sheet: every method in the second option on one page
Practice & Assignments
20 problems on the second option
Topics Covered
- Statistics route: hypothesis tests for the Poisson and geometric distributions
- Mechanics route: centres of mass of laminas and solids
- Decision route: critical path analysis and network flows
- Further pure route: further calculus and proof extensions
- Long option questions under timing
- Writing conclusions in context
Projects You Build
- Question bank entry: 12 long option questions with model conclusions
Practice & Assignments
16 option problems under timing
Assessment
Month 15 test: a 60-minute option paper
Topics Covered
- The first option's A level content beyond the AS checkpoint
- Mixed problems drawing on both options
- Modelling assumptions and their criticism
- Exam technique for multi-part option questions
- Error log review
- Targeted re-teach of the weakest option topics
Projects You Build
- Revision sheet set three: single pages for every option topic
Practice & Assignments
Two 45-minute option papers, one per option
Topics Covered
- A full options paper under timing for your board
- Marking against an official-style scheme
- Classifying every lost mark
- Comparing option performance with core performance
- Planning the final revision cycle
- Agreeing the past-paper timetable
Projects You Build
- The mock file completed with the options mock
Practice & Assignments
Mock review set: every lost mark reworked
Assessment
Options mock, marked and reviewed one to one
Topics Covered
- Rapid re-teach of each student's weakest core topics, chosen from the error log
- Mixed core sets that do not announce their topic
- Formula booklet fluency: what is given and what must be known
- Proof rehearsal from memory
- Accuracy audit: the slips each student makes, named and drilled
- Show-that questions worked backwards honestly
Projects You Build
- Final revision pack: the student's revision sheets combined and tidied
Practice & Assignments
One 75-mark core paper self-timed, plus a drill on two weak strands
Topics Covered
- Rapid re-teach of the weakest option topics
- Mixed option sets under timing
- Model assumptions and conclusions rehearsed
- Calculator checks for the option
- Long questions planned before writing
- Error log closed out topic by topic
Projects You Build
- Option revision pack completed
Practice & Assignments
One full option paper self-timed
Assessment
Revision checkpoint: a mixed 60-minute paper across core and options
Topics Covered
- Full Core Pure past papers for your board under real timing
- Marking against the official mark scheme
- Error clinic after every paper
- Time management across long questions
- Which questions to attempt first
- Checking strategies in the final minutes
Projects You Build
- Past paper score chart started
Practice & Assignments
Two full core papers under exam timing
Topics Covered
- Full option past papers for your board under real timing
- Marking and error clinic
- Final review of the error log
- Exam-day routine and equipment
- Last proof rehearsal
- A calm plan for the final weeks before the exams
Projects You Build
- Completed mock file: every marked paper, error classification and score chart
Practice & Assignments
Two full option papers under exam timing
Assessment
Final past paper series, marked and reviewed
Projects You'll Build
Build a professional portfolio with A complete revision system for Further Maths: proof portfolio, question bank, sketches, method map, error log and marked mock file real-world projects.
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Straight answers about A Level Further Maths: Core Pure and Applied Options
- Who should take this course?
- Year 12 and 13 students (or equivalent) taking Further Maths with AQA, Edexcel or OCR. It suits students starting further maths alongside a level maths who want both taught in a sensible order; students whose school or college does not teach further maths, who need the whole course taught and will sit the exams at an exam centre; students whose school offers further maths in a small, rushed or part-timetabled class and who want steady teaching beside it.
- What will they learn and build?
- Further Maths is a full A level that sits on top of A level Maths. The boards are clear about the relationship: OCR states that learners are assumed to know the content of GCSE Mathematics and A Level Mathematics, and Pearson Edexcel says the students who benefit most are likely to hold A level Maths.
- How deeply are topics covered?
- The course runs 18 months (72 weeks), taught alongside A Level Maths at 2 live classes/week + 3-4 hours practice, rising to full timed papers in the last two months, across 5 phases listed week by week in the syllabus below. A structured, well-paced curriculum taught step by step, with hands-on practice in every session.
- Who teaches it?
- Modern Age Coders mentors, who teach the live classes themselves. You can watch them at work in the free class recordings before you decide.
- How do practice, feedback and assessment work?
- Monthly tests, an AS-standard mock at month 9, Core Pure and options mocks, and a full marked past-paper cycle. Doubt support between classes over WhatsApp, so you are never left stuck. A course-completion certificate backed by a marked mock file that shows the real grade trajectory.
- What does it cost?
- Three monthly plans: a group batch, a mini batch and 1-on-1. Prices are shown in your currency in the plans section. Monthly billing, cancel any time.
- When can classes take place?
- Live classes are scheduled around your week. Group batches meet at a fixed weekly slot; 1-on-1 students choose their own. International students are scheduled in their own timezone. Ask on WhatsApp for the current slots.
- Can I watch the teaching before deciding?
- Yes. Full recordings of real 13 and up classes are free to watch, free with a Google sign-in. These recordings show how we teach, not the exact syllabus of this course. Some classes are in English and some in Hindi; everyone follows at their own pace. Watch a class end to end, like you are in it. Sessions are interactive and each moment builds on the last.
- Can I enroll without a live demo?
- Yes. Choose a plan on this page and enrol directly; a demo is not required. If you would like to see a full class with a mentor first, book a Priority Demo.
- How do I reach you with a question?
- WhatsApp +91 91233 66161, email connect@modernagecoders.com, or call +91 91233 66161. Tell us the student's age and what they have learned so far, and we will point you to the right plan and batch.
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