What does the foundations Pure content in OCR Maths A cover?

It covers the bedrock pure techniques: the four methods of proof, the laws of indices and surd manipulation, quadratics and the discriminant, simultaneous equations and inequalities, polynomial division and the factor theorem, the binomial expansion, curve sketching and graph transformations, coordinate geometry of lines and circles, and arithmetic and geometric sequences and series. Almost every later topic relies on these skills, so they are examined constantly across Papers 1, 2 and 3.

Which foundation topics carry the most marks in OCR Maths A?

Algebra and functions and coordinate geometry are the heaviest, because quadratics, the discriminant, simultaneous equations and the equation of a circle appear in nearly every paper and feed into calculus and the applied strands. Sequences and series and the binomial theorem also recur reliably. Proof threads through everything as overarching theme OT1. Build fluency in algebra first, since weak algebra leaks marks throughout the qualification.

What techniques recur across the OCR foundation content?

Completing the square, using the discriminant to control the number of roots, the factor theorem to factorise cubics, the binomial expansion to extract a coefficient, the perpendicular gradient rule and completing the square for a circle, rationalising surd denominators with the conjugate, and the arithmetic and geometric sum formulae. These are non-negotiable, automatic skills by exam day.

What is the most common foundation mistake in OCR Maths A?

Algebraic slips in the middle of multi-step questions: sign errors completing the square, forgetting to reverse an inequality when dividing by a negative, misreading a quadratic inequality region, using the wrong perpendicular gradient, and applying the sum to infinity when the common ratio is not less than one. Show every step so method marks survive an arithmetic slip.

How should I revise the OCR foundation Pure content?

Work topic by topic against the specification, drilling each technique until it is automatic, then practise mixed problems under timed conditions. Keep a sheet of standard results: the quadratic formula, the discriminant conditions, the factor theorem, the binomial general term, the circle equation, the transformation rules, and the sequence sum formulae. Sketch graphs to justify inequality and circle answers.

EnglandMaths

OCR A-Level Maths A Pure foundations: proof, algebra, polynomials, graphs, coordinate geometry and sequences

A deep-dive OCR A-Level Mathematics A guide to the foundational Pure content: proof, indices and surds, algebra and functions, polynomials and the binomial theorem, graphs and transformations, coordinate geometry, and sequences and series, with the techniques OCR repeats across all three papers.

Generated by Claude Opus 4.820 min readH240/1.01-1.04Updated 2026-06-02

Reviewed by: AI editorial process; not yet individually human-reviewed

Jump to a section

What the foundations Pure content demands
Proof
Indices, surds and algebra
Polynomials and the binomial theorem
Graphs and coordinate geometry
Sequences and series
How the foundation content is examined
Check your knowledge

What the foundations Pure content demands

The foundational pure topics are the backbone of OCR A-Level Mathematics A (H240). They develop the algebra, reasoning and graph work that every other part of the course depends on. Pure content appears on all three papers, so this material is examined again and again. The examiners test two linked skills: fluent technique with standard methods, and the judgement to choose and combine those methods in unfamiliar multi-step problems.

This guide walks through the seven foundation topics in specification order, then sets out the exam patterns OCR repeats. Each topic has a matching dot-point page with practice questions; this overview ties them together.

Proof

The content opens with proof (1.01): deduction, exhaustion, disproof by counter-example, and contradiction, including the classic results that $\sqrt{2}$ is irrational and that there are infinitely many primes. Clear logical layout earns marks across the whole paper, and proof is overarching theme OT1.

Indices, surds and algebra

Indices and surds (1.02) sets up exact non-calculator manipulation: the laws of indices for all rational powers, negative and fractional indices, simplifying surds, and rationalising denominators with the conjugate. Algebra and functions is the most reused topic: solving quadratics by factorising, completing the square and the formula, the discriminant and the nature of roots, simultaneous linear-and-quadratic equations, and linear and quadratic inequalities. Weak algebra leaks marks everywhere, so it is the first thing to master.

Polynomials and the binomial theorem

Polynomials and the binomial theorem covers polynomial manipulation, algebraic division, the factor theorem for finding and confirming roots, and the binomial expansion of $(a + b)^n$ for a positive integer $n$ using binomial coefficients $\binom{n}{r}$ . A frequent question type asks for one coefficient in an expansion, best found from the general term.

Graphs and coordinate geometry

Graphs and transformations covers sketching polynomial and reciprocal curves, asymptotes, intersections, and the four transformations $f(x) + a$ , $f(x + a)$ , $af(x)$ and $f(ax)$ . Coordinate geometry (1.03) covers straight lines, the parallel and perpendicular gradient conditions, the equation of a circle (via completing the square), the tangent-radius and chord-bisector facts, and parametric equations of curves.

Sequences and series

Sequences and series (1.04) introduces arithmetic and geometric sequences, the sum formulae, sigma notation, recurrence relations, increasing, decreasing and periodic sequences, and the sum to infinity of a convergent geometric series (which exists only when $|r| < 1$ ).

How the foundation content is examined

A typical OCR profile for the foundation pure topics:

Short technique questions. Solving quadratics, simplifying surds, expanding a bracket to find a coefficient, and finding the equation of a line.
Multi-step problems. Combining the discriminant with a parameter, completing the square for a circle then finding a tangent, or summing a series to meet a condition.
Proof and reasoning. Constructing deductive and contradiction proofs and disproving statements by counter-example.
Graph work. Sketching transformed curves and reading the effect of a transformation on a named point.

Check your knowledge

A mix of recall and technique questions covering the foundation content. Attempt them under timed conditions, then check against the solutions.

Express $x^2 - 10x + 7$ in completed-square form. (2 marks)
State the discriminant condition for two distinct real roots. (1 mark)
Find the coefficient of $x^2$ in $(1 + 3x)^4$ . (2 marks)
Find the centre and radius of $(x - 2)^2 + (y + 5)^2 = 9$ . (2 marks)
Find the sum to infinity of the geometric series with $a = 9$ , $r = \tfrac{1}{3}$ . (2 marks)
Rationalise $\dfrac{4}{\sqrt{2}}$ . (2 marks)

Solutions

Step 1: Complete the square

To complete the square for $x^2 - 10x + 7$ , halve the coefficient of $x$ to get $-5$ , write $(x - 5)^2$ , then subtract $5^2 = 25$ to compensate for the extra term introduced, and add the original constant.

(x - 5)^2 - 25 + 7 = (x - 5)^2 - 18

Step 2: State the condition for two distinct real roots

The discriminant $b^2 - 4ac$ determines the nature of the roots of $ax^2 + bx + c = 0$ . When it is strictly positive, the quadratic has two distinct real roots; zero gives a repeated root and negative gives no real roots.

b^2 - 4ac > 0

Step 3: Find the required coefficient using the binomial theorem

The general term in the expansion of $(1 + 3x)^4$ is $\binom{4}{r}(3x)^r$ . For the $x^2$ term, set $r = 2$ : the binomial coefficient is $\binom{4}{2} = 6$ and $(3)^2 = 9$ .

\binom{4}{2}(3)^2 = 6 \times 9 = 54

Step 4: Read the centre and radius from standard form

The equation $(x - h)^2 + (y - k)^2 = r^2$ represents a circle with centre $(h, k)$ and radius $r$ . Matching to $(x - 2)^2 + (y + 5)^2 = 9$ gives $h = 2$ , $k = -5$ , and $r = \sqrt{9} = 3$ .

Centre $(2, -5)$ , radius $3$ .

Step 5: Apply the sum to infinity formula

For a geometric series with first term $a$ and common ratio $|r| < 1$ , the sum to infinity is $S_\infty = \dfrac{a}{1 - r}$ . Here $a = 9$ and $r = \tfrac{1}{3}$ , so the denominator becomes $\tfrac{2}{3}$ .

S_\infty = \dfrac{9}{1 - 1/3} = \dfrac{9}{2/3} = 13.5

Step 6: Rationalise the denominator

Multiplying numerator and denominator by $\sqrt{2}$ removes the surd from the denominator. This works because $\sqrt{2} \times \sqrt{2} = 2$ , a rational number.

\dfrac{4}{\sqrt{2}} \times \dfrac{\sqrt{2}}{\sqrt{2}} = \dfrac{4\sqrt{2}}{2} = 2\sqrt{2}

Sources & how we know this

OCR A Level Mathematics A (H240) specification — OCR (2017)