OCR A-level Further Maths A (H245): papers, options and grade boundaries

OCR A-level Further Maths at a glance

OCR has two further maths courses: Further Mathematics A (H245), covered here, and Further Mathematics B (MEI) (H645). Further Maths A is a linear qualification. All the papers must be taken in the same examination series, in May and June, and you take it alongside A-level Maths, since it assumes both GCSE Maths and the A-level Maths content.

You take Y540, Y541 and any two of the four options. You may take more than two options to broaden the course, and OCR’s specification explains how the grade is then awarded. Each paper is made up of short and long questions with an increasing gradient of difficulty, and every paper assesses the overarching themes.

What the papers cover

Pure Core (Y540 and Y541)

Both papers assess the whole Pure Core, so the content is not split between them. It extends the A-level Maths topics of proof, algebra, functions, calculus, vectors and differential equations and adds new areas:

• Proof: proof by induction, including sums of series, powers of matrices and divisibility.
• Complex numbers: modulus-argument form, loci on the Argand diagram, De Moivre’s theorem and roots of complex numbers.
• Matrices: arithmetic, linear transformations in 2-D and some in 3-D, determinants, inverses and invariance.
• Further vectors, algebra and series: equations of lines and planes, scalar and vector products, roots and coefficients of polynomials, partial fractions and the method of differences.
• Hyperbolic functions, further calculus and polar coordinates: Maclaurin series, improper integrals, volumes of revolution and areas in polar form.
• Differential equations: the integrating factor method, second order equations including SHM and damped oscillations, and simple systems.

The options

• Statistics (Y542): probability, discrete and continuous random variables, linear combinations, hypothesis tests and confidence intervals, chi-squared tests, non-parametric tests, correlation and regression.
• Mechanics (Y543): dimensional analysis, work, energy and power, impulse and momentum, centre of mass, motion in a circle and further dynamics and kinematics.
• Discrete Mathematics (Y544): graphs and networks, algorithms, network algorithms, project management, linear programming, the simplex algorithm and game theory.
• Additional Pure Mathematics (Y545): further pure topics such as groups, where group axioms and Lagrange’s theorem are studied.

Calculator and Formulae Booklet

OCR permits a scientific or graphical calculator in all papers, subject to the JCQ rules. You are given a Formulae Booklet in each assessment, which includes the formulae given for A-level Maths A. The specification contains the booklet, so use it throughout the course and learn where each result is, but do not rely on it for definitions or for proofs.

Worked examples in OCR style

Complex numbers: arithmetic and a quadratic

Multiply (4 − i)(2 + 3i), using i² = −1:

Now solve z² − 6z + 25 = 0. The discriminant is 36 − 100 = −64, so:

The roots of a polynomial with real coefficients come in conjugate pairs. Practise related ideas in our Group Theory course, which supports the Additional Pure option:

Matrices: determinant and inverse

Take the 2 × 2 matrix A whose first row is 5, 2 and whose second row is 7, 3. The determinant is 5 × 3 − 2 × 7 = 1. Swap the diagonal entries, change the signs of the other two and divide by the determinant: the inverse has first row 3, −2 and second row −7, 5. Check by multiplying: A × A⁻¹ should give the identity matrix. A determinant of 0 means there is no inverse, and the transformation squashes the plane onto a line.

Proof by induction: sum of odd numbers

Prove that 1 + 3 + 5 + … + (2n − 1) = n² for every positive integer n.

• Basis: for n = 1 the left-hand side is 1 and the right-hand side is 1², so it is true.
• Assumption: suppose it is true for n = k, so the sum of the first k odd numbers is k².
• Step: for n = k + 1, add the next odd number, 2k + 1, to get k² + 2k + 1 = (k + 1)².
• Conclusion: it is true for n = 1, and if it is true for n = k then it is true for n = k + 1, so it is true for all positive integers n.

Mark schemes reward the full structure: the basis, the assumption, the step and a clear closing sentence.

Grades and June 2026 grade boundaries

Each paper has its own boundaries, and the overall boundaries depend on which options you took. OCR adds your four paper marks into a total out of 300. The tables are from OCR’s document published on 14 September 2026 for the June 2026 series.

Worked through: marks of 50, 48, 52 and 46 on Y540, Y541, Y542 and Y543 give 196 out of 300. For that combination the A boundary is 206 and the B boundary is 174, so it is a B, 10 marks short of an A.

Using the daily paper as a short timed mock

Uni-Sine publishes a free daily Further Maths paper, with a new one every day. It is a short timed paper of 8 questions worth 25 to 35 marks, marked instantly with worked answers. It is not an OCR paper and it is much shorter than a 90 minute, 75 mark paper.

• It covers well: complex numbers and Argand diagrams, hyperbolic functions, matrices and transformations, roots of polynomials, series and Maclaurin expansions, proof by induction, polar coordinates, second order differential equations and volumes of revolution. That is the Pure Core.
• It does not cover: the four options: statistics, mechanics, discrete mathematics and additional pure. Use OCR’s past papers and your teacher’s materials for those.

For more on mocks, see our A-level Maths mocks guide and the A-level Further Maths guide. If you also take A-level Maths with OCR, read the OCR A-level Maths guide.

Exam answers

Uni-Sine does not publish official answers or mark schemes for OCR A-level Further Maths papers. The answers and a worked mark scheme for each question come with today’s paper once you have submitted it and it has been marked, so you can see where every mark was won or lost.

Common mistakes in OCR A-level Further Maths

• Sign errors with i. Replace i² by −1 straight away and keep real and imaginary parts apart.
• Skipping the structure of induction. Basis, assumption, step and conclusion each carry marks.
• Multiplying matrices in the wrong order. AB is not BA.
• Forgetting the Argand diagram. A sketch often shows the answer to loci and root problems.
• Treating options as an afterthought. Two options are half the A-level, so give them equal revision time.

Read next

UK · A-levelOCR A-level Further Maths formula bookletThe OCR A-level Further Maths A formula booklet for H245: what is on it, what is not, and worked examples of using it.