OCR A-Level Further Mathematics A (H245): complete guide to the mandatory Core Pure and the optional papers

OCR A-Level Further Mathematics A (specification H245) is a two-year linear course taken in addition to A-Level Mathematics and assessed by four written papers at the end of Year 13. It pairs a large compulsory Pure Core (two papers) with two optional papers chosen from four. This page is the index: below is a map of the Pure Core content, the optional papers, the exam structure, and how to study each area.

The mandatory Pure Core

Every Further Mathematics student studies the Pure Core, the mathematical backbone of the qualification. It is assessed across both Pure Core papers, each of which can draw on the whole core.

Matrices and transformations

Matrix arithmetic, determinants of $2 \times 2$ and $3 \times 3$ matrices, inverses, solving systems of linear equations, and matrices as linear transformations of the plane and of space.

Complex numbers

Arithmetic and the conjugate, the Argand diagram, modulus-argument and exponential form, de Moivre's theorem, multiple-angle identities, the $n$th roots of unity and loci.

Further calculus

The Maclaurin series, improper integrals, volumes of revolution, and differential equations (first and second order, including simple harmonic motion and damping).

Further vectors and planes

The scalar and vector products, vector and Cartesian equations of lines and planes in three dimensions, intersections, and distances and angles.

Series and proof

Proof by mathematical induction, the standard summation formulae for $\sum r$, $\sum r^2$ and $\sum r^3$, the method of differences, and the relationships between the roots and coefficients of polynomials.

Polar coordinates and hyperbolic functions

Polar curves and the area they enclose, and the hyperbolic functions with their identities, inverses, and calculus.

The optional papers

A candidate takes two of the four optional papers, each examined as one 90-minute, 75-mark paper.

• Statistics (Y542) - discrete and continuous random variables, the Poisson and geometric distributions, chi-squared and non-parametric tests, and correlation and regression.
• Mechanics (Y543) - work, energy and power (including elastic strings and springs), momentum, impulse and collisions, circular motion, and centre of mass.
• Discrete Mathematics (Y544) - graphs and networks, network algorithms, critical path analysis, linear programming and game theory.
• Additional Pure Mathematics (Y545) - sequences and series, number theory, groups, and further vectors and calculus.

This library builds out the Pure Core in full, plus the Statistics and Mechanics options.

Exam structure

OCR A-Level Further Mathematics A is assessed by four written papers, all sat at the end of the course. A calculator is allowed in every paper and OCR provides a formulae booklet.

• Pure Core 1 (Y540). 90 minutes, 75 marks, 25 percent of the A-level. Assesses the whole Pure Core.
• Pure Core 2 (Y541). 90 minutes, 75 marks, 25 percent. Also assesses the whole Pure Core.
• Optional paper 1. 90 minutes, 75 marks, 25 percent. One of Statistics, Mechanics, Discrete or Additional Pure.
• Optional paper 2. 90 minutes, 75 marks, 25 percent. A different option from the same list.

Each paper assesses its content alongside the three overarching themes (mathematical argument and proof, problem solving, and mathematical modelling).

How to study OCR Further Maths A

Further Mathematics rewards fluent technique and clear presentation.

1. Master the Pure Core first. Both core papers and several optional techniques draw on it, so build it topic by topic.
2. Drill each method to automaticity. Inverting matrices, dividing complex numbers, evaluating improper integrals, solving differential equations and laying out induction proofs should be second nature.
3. Show every step. Method marks dominate, so write working an examiner can follow, especially in proofs.
4. Then revise your two optional areas. Practise each model or algorithm fully, in context.
5. Sit mixed timed papers. Combine topics under exam conditions from the start of Year 13.

The areas, topic by topic

Each area has specification-level answer pages with worked exam questions and cross-links, plus an overview guide and quiz:

• Matrices and transformations overview
• Complex numbers overview
• Further calculus overview
• Further vectors and planes overview
• Series and proof overview
• Polar coordinates and hyperbolic functions overview
• Further statistics overview
• Further mechanics overview

Browse the full set at /a-level-ocr/further-mathematics/syllabus.

For the official specification

OCR publishes the full specification (H245), past papers, mark schemes and the formulae booklet at ocr.org.uk. Always revise from the current specification and OCR's own past papers, because question style and the optional structure are board-specific.