OCR A-Level Mathematics A (H240) is assessed by three written papers, each 100 marks, each 2 hours and each worth one third of the A Level: Paper 01 Pure Mathematics, Paper 02 Pure Mathematics and Statistics, and Paper 03 Pure Mathematics and Mechanics. Marks are split across three assessment objectives, with AO1 (standard techniques) the largest at roughly 60%, and AO2 (reasoning and communication) and AO3 (problem solving and modelling) each around 20%. The most common reasons for lost marks are careless algebra, incomplete reasoning on "show that" and "prove" questions, and answers that do not match the command word used.
Who this guide is for
This is a guide for students sitting OCR A-Level Mathematics A, specification code H240, and for parents trying to understand how the exam works. It assumes you are partway through the course and want to convert the content you already know into marks. The structural details below reflect the current OCR A (H240) specification; the worked questions are written in OCR's style but are entirely original, not reproduced from any past paper. For one-to-one help see A-Level maths tuition.
1. How H240 is assessed
There is no coursework in OCR A-Level Maths. The whole grade comes from three written papers sat at the end of the course. They are equally weighted, so no single paper can be treated as the soft option.
100 marks, 2 hours, one third of the A Level. A single section of pure mathematics with a deliberate gradient of difficulty: short, accessible questions early on building to longer multi-step problems. Covers proof, algebra and functions, coordinate geometry, sequences and series, trigonometry, exponentials and logarithms, differentiation, integration, numerical methods and vectors.
100 marks, 2 hours, one third of the A Level. Two sections of roughly 50 marks each: pure mathematics, then statistics. The statistics section covers sampling, data presentation and interpretation, probability, statistical distributions and hypothesis testing, and some questions are set on the OCR pre-release large data set.
100 marks, 2 hours, one third of the A Level. Two sections of roughly 50 marks each: pure mathematics, then mechanics. The mechanics section covers quantities and units, kinematics, forces and Newton's laws, and moments.
A practical consequence: pure mathematics is examined on all three papers and makes up the clear majority of the qualification, while statistics and mechanics each appear on only one. Strong, fluent pure technique is the single highest-leverage thing to drill, because it pays off three times over.
2. The assessment objectives, and why they matter
Every mark in H240 is tagged to one of three assessment objectives (AOs). Understanding them changes how you read a question, because the AO tells you what kind of response earns the marks.
The largest objective. Selecting and carrying out routine procedures accurately: differentiating, integrating, solving equations, manipulating algebra. These are the marks you secure by being fast and accurate at the mechanics.
Constructing arguments, proofs and chains of reasoning, and communicating them with correct notation and language. "Show that", "prove" and "explain" questions live here. Sloppy notation costs AO2 marks even when the maths is right.
Translating an unstructured or real-world situation into mathematics, solving it, and interpreting the result. OCR reports AO3 in two strands, AO3(PS) for problem solving and AO3(M) for modelling. These are the questions where you must decide the method yourself.
The exact percentages move a little between the three papers within OCR's published ranges, but the headline is stable: roughly three fifths of the qualification is AO1, with the remaining two fifths split between reasoning (AO2) and problem solving and modelling (AO3). Students who are excellent at technique but treat reasoning as optional typically cap out in the low grade bands, because they are conceding two marks in five.
3. Where OCR candidates lose marks
OCR examiner reports, published after each series, are strikingly consistent year on year about what separates strong scripts from weak ones. The recurring themes are these.
- Careless algebra. The single most cited issue. Sign errors, dropped terms, and in particular "cancelling" a single term from a numerator and denominator that is not a common factor. Method marks are earned but accuracy marks bleed away.
- Not answering the command word. Examiners repeatedly note candidates who produce correct mathematics that does not respond to what was actually asked. "Show that", "prove", "determine", "verify" and "hence" each demand a specific kind of answer; ignoring the command word forfeits marks even with correct work.
- Insufficient evidence in "show that" questions. When the answer is given to you, every step of the reasoning has to be visible. Jumping to the printed result without showing the intermediate line is the classic way to lose the final mark.
- Imprecise notation and language. Reports note that more concise, correct notation would have unlocked accuracy marks. Confusing \(\Rightarrow\) with \(=\), omitting \(dx\), or vague verbal justification all cost marks under AO2.
- Calculator inefficiency. Strong candidates use the calculator well, for example to evaluate a definite integral or a binomial probability quickly, freeing time for the questions that need written reasoning. Weak candidates either over-rely on it where working is required, or fail to use it where it would save time.
- Conclusions out of context in statistics. In hypothesis testing, stating "reject \(H_0\)" without interpreting what that means for the original claim loses the final communication mark.
None of these is about not knowing the maths. They are about exam discipline. That is exactly why they are recoverable with deliberate practice.
4. Four worked exam-style questions
The following four questions are written in OCR's H240 style and format, one from each of the main flavours you will meet: a pure "show that" with calculus, a binomial hypothesis test from the statistics section, a mechanics problem with connected particles, and a problem-solving question on sequences. They are original, not past-paper questions. Work each one before reading the solution, then read the examiner note to see where the marks actually sit.
Question 1 — Differentiation and a "show that" (Pure)
A curve has equation \(y = x^3 - 6x^2 + 9x + 2\).
(a) Find \(\dfrac{dy}{dx}\) and hence find the coordinates of the two stationary points. [4]
(b) Show that the curve has a point of inflection at \(x = 2\), and determine whether the stationary point at the larger value of \(x\) is a minimum or a maximum. [3]
Solution, part (a). Differentiate term by term:
Stationary points occur where \(\frac{dy}{dx} = 0\), so \(x = 1\) or \(x = 3\). Substituting back into the original equation:
The stationary points are \((1, 6)\) and \((3, 2)\).
Solution, part (b). Differentiate again:
At \(x = 2\), \(\frac{d^2y}{dx^2} = 12 - 12 = 0\). To confirm a point of inflection we check that the second derivative changes sign: for \(x < 2\) we have \(6x - 12 < 0\) and for \(x > 2\) we have \(6x - 12 > 0\), so the concavity changes at \(x = 2\). Hence there is a point of inflection at \(x = 2\), as required. At the stationary point with the larger \(x\), namely \(x = 3\),
so \((3, 2)\) is a minimum.
Part (a) is pure AO1: one method mark for a correct derivative, one for setting it to zero and solving, and accuracy marks for both correct coordinate pairs. The most common slip is an arithmetic error substituting back, so the marks for \(x=1\) and \(x=3\) are awarded independently. Part (b) is AO2 reasoning. For the "show that" you must do more than state \(\frac{d^2y}{dx^2}=0\) at \(x=2\): a second-derivative of zero alone does not prove inflection, so the sign-change argument (or equivalent) is the load-bearing step examiners want to see. Asserting the printed result without it typically loses the final mark. The minimum/maximum decision is one accuracy mark, and stating "minimum" without the supporting \(\frac{d^2y}{dx^2}>0\) value is penalised.
Question 2 — Binomial hypothesis test (Statistics)
A manufacturer claims that at most 15% of the components it produces are faulty. A quality inspector takes a random sample of 20 components and finds that 6 are faulty. Test, at the 5% significance level, whether there is evidence that the proportion of faulty components exceeds 15%.
Solution. Let \(p\) be the proportion of faulty components and let \(X\) be the number of faulty components in the sample, so under the null hypothesis \(X \sim B(20, 0.15)\). Set up the hypotheses:
This is a one-tailed test at the 5% level. We find the probability of observing a result at least as extreme as \(6\) faulty, assuming \(H_0\) is true:
Using the binomial cumulative distribution (calculator) with \(n = 20\), \(p = 0.15\):
Compare with the significance level: \(0.0673 > 0.05\). The result is not in the critical region, so we do not reject \(H_0\).
There is insufficient evidence at the 5% level to conclude that the proportion of faulty components exceeds 15%. The manufacturer's claim is not contradicted by this sample.
This question spans AO1 (computing the probability), AO2 (communicating the conclusion) and AO3(M) (setting up the model). The marks break down roughly: one for both hypotheses stated in terms of \(p\) — writing them in words or in terms of \(X\) is a frequent way to lose it; one for identifying \(X\sim B(20,0.15)\); method and accuracy marks for the correct tail probability \(P(X\ge 6)\); and crucially a final mark for a conclusion in context. OCR reports flag candidates who compute \(0.0673\) correctly, then write only "accept \(H_0\)". You must say what it means for the faulty-component claim. A second classic error is comparing the wrong tail or testing \(P(X\ge 5)\); the observed value is \(6\), so the inclusive tail is \(X\ge 6\).
Question 3 — Connected particles (Mechanics)
Two particles \(A\) and \(B\), of masses \(3\,\text{kg}\) and \(2\,\text{kg}\) respectively, are connected by a light inextensible string that passes over a smooth fixed pulley. The system is released from rest with the string taut and both particles hanging vertically. Take \(g = 9.8\,\text{m s}^{-2}\).
(a) Find the acceleration of the system and the tension in the string. [5]
(b) State one modelling assumption you have used and explain its effect. [2]
Solution, part (a). The heavier particle \(A\) accelerates downwards and \(B\) upwards with the same magnitude of acceleration \(a\), and the tension \(T\) is the same throughout the string because it is light and the pulley smooth. Apply Newton's second law to each particle, taking the direction of motion as positive.
For \(A\) (moving down):
For \(B\) (moving up):
Adding the two equations eliminates \(T\):
Substitute back into the equation for \(B\):
The acceleration is \(1.96\,\text{m s}^{-2}\) and the tension is \(23.52\,\text{N}\) (or \(23.5\,\text{N}\) to 3 s.f.).
Solution, part (b). The string is modelled as light (massless), so the tension is the same at both ends and we do not have to account for the weight of the string. If the string had mass, the tension would differ along its length and both the acceleration and the two tensions would change.
Part (a) is AO1 and AO3(M). The marks reward two correct equations of motion (one each), eliminating \(T\) to find \(a\), and substituting back for \(T\). The most common error is a sign slip — writing \(T - 3g = 3a\) for the descending particle — which examiners treat as a method error that loses the accuracy marks downstream. A clear force diagram, even though not always demanded, protects against this. Part (b) is AO2: a single isolated word such as "light" earns little; the mark is for naming an assumption and explaining its consequence ("so tension is equal throughout"). The "smooth pulley" assumption is an equally valid answer. Leaving \(g\) symbolic until the end and only then substituting \(9.8\) reduces rounding errors and keeps the working transparent.
Question 4 — Sequences and problem solving (Pure)
An arithmetic sequence has first term \(a\) and common difference \(d\). The sum of the first \(10\) terms is \(155\), and the sum of the first \(20\) terms is \(610\).
Determine the values of \(a\) and \(d\), and hence find the first term of the sequence that exceeds \(100\). [6]
Solution. The sum of the first \(n\) terms of an arithmetic sequence is \(S_n = \tfrac{n}{2}\big(2a + (n-1)d\big)\). Apply this to both pieces of information.
Subtract the first equation from the second to eliminate \(a\):
Substitute \(d = 3\) into \(2a + 9d = 31\):
The \(n\)th term is \(u_n = a + (n-1)d = 2 + 3(n-1) = 3n - 1\). We need the first term exceeding \(100\):
So the smallest integer is \(n = 34\), giving \(u_{34} = 3(34) - 1 = 101\). The first term to exceed \(100\) is \(101\), the 34th term.
This is AO1 with a strong AO3(PS) element, since you must decide to form and solve simultaneous equations. Marks: one for a correct sum formula applied to give a usable equation, one for the second equation, method and accuracy for solving to get \(d\) and then \(a\), and the final two marks for the problem-solving step. A common trap is the last part: solving \(3n-1>100\) gives \(n>33.67\), and candidates either round down to \(33\) (which gives \(98\), not exceeding \(100\)) or quote \(n=34\) but forget to state the actual term value \(101\). Read the command: "determine the values" expects both \(a\) and \(d\) shown clearly, and "hence" signals that you must use your values, not start a fresh method. Stating \(n=34\) without the term, or the term without identifying which it is, leaves a mark on the table.
5. OCR-specific exam technique
Beyond knowing the content, a handful of habits are worth more on OCR papers specifically.
- Read the command word first. Before writing anything, identify whether the question says "show that", "prove", "determine", "verify", "hence", "find" or "state". Each has a contract. "Hence" in particular means you must use the previous result; a fresh method can be refused the marks even if correct.
- Treat "show that" as a full proof. When the answer is printed, the marks are entirely for the reasoning. Show every intermediate line; never collapse two steps into the given answer.
- Use the gradient of difficulty. Each paper, and each section within Papers 02 and 03, ramps up. Bank the early marks quickly and accurately, then spend your remaining time on the longer problems, rather than getting stuck early.
- Exploit your calculator, but show working where reasoning is assessed. Definite integrals, binomial and normal probabilities and equation solving can be done on the calculator to save time, but if a question asks you to "show" or "prove", the written steps are the marks.
- Carry exact values, round at the end. Premature rounding is a recurring accuracy-mark loss. Keep surds, fractions and \(g\) symbolic for as long as you can, then round once to the requested precision.
- Always conclude in context. In statistics, finish a hypothesis test by saying what the decision means for the real-world claim. In mechanics, state units. These final communication marks are easy to earn and easy to forget.
- Know the large data set. Some Paper 02 statistics questions are set on OCR's pre-release data set. Familiarity with its variables, units and quirks turns these into quick marks rather than time sinks.
A-Level Maths tuition
Most marks lost on OCR A-Level Maths are lost to exam technique, not unknown content. For focused one-to-one help with H240 — pure technique, statistics, mechanics, or exam discipline on "show that" and problem-solving questions — see A-Level maths tuition, explore A-Level and admissions tutoring, or book a free consultation.