Mathematics A – H230, H240 AS & A-level Past Papers ( OCR – Oxford Cambridge and RSA)

Course guide: H230 AS and H240 A Level

Verified against official Cambridge OCR materials on 6 October 2026. This guide covers AS Mathematics A H230 and A Level Mathematics A H240. It does not cover Mathematics B (MEI). The lessons and questions below are original Helovesmath practice, not OCR examination questions or an official mark scheme.

Choose the right qualification and paper

AS Level Mathematics A: H230

• H230/01: Pure mathematics and statistics — 75 marks, 90 minutes, 50% of the qualification. Approximately 50 marks assess pure mathematics and 25 assess statistics.
• H230/02: Pure mathematics and mechanics — 75 marks, 90 minutes, 50%. Approximately 50 marks assess pure mathematics and 25 assess mechanics.
• Both components are required.

A Level Mathematics A: H240

• H240/01: Pure mathematics — 100 marks, 2 hours, one third of the qualification.
• H240/02: Pure mathematics and statistics — 100 marks, 2 hours, one third; approximately 50 marks in each section.
• H240/03: Pure mathematics and mechanics — 100 marks, 2 hours, one third; approximately 50 marks in each section.
• All three components are required.

The paper number matters: H230/01 includes statistics, while H240/01 is pure mathematics. Match the qualification, component, year and series before downloading the question paper, answer booklet, mark scheme and examiner report.

What should you revise?

Pure mathematics: proof; algebra and functions; coordinate geometry; sequences and series; trigonometry; exponentials and logarithms; differentiation; integration; vectors. H240 also includes numerical methods.

Statistics: sampling; presenting and interpreting data; probability; distributions; hypothesis tests.

Mechanics: quantities and units; motion; forces and Newton’s laws. H240 also includes moments.

These are topic families, not a statement that AS and A Level assess every technique within them.

Use H240’s Stage 1 and Stage 2 columns to check depth: Stage 1 corresponds to H230 content. For example, the AS sequences-and-series section contains positive-integer binomial expansion, while geometric series, integration by parts, normal-distribution work and moments belong to the additional A Level content.

A quick lesson in each strand

Pure mathematics: connect representations. An expression, graph and derivative describe different aspects of the same function. Before calculating, decide what the question needs: x-intercepts come from y = 0, stationary points from dy/dx = 0, and accumulated area from integration. A sketch is a useful check on signs and the number of solutions.

Statistics: define the model before using the calculator. Define the random variable and its units. Explain why a distribution is suitable. In a hypothesis test, define the population parameter, state the hypotheses, choose the appropriate tail, then interpret the result in the original setting. A probability alone is not a conclusion.

Mechanics: translate the situation into a model. Draw the body or motion diagram, choose a positive direction and list known quantities with units. Forces are measured in newtons, mass in kilograms and acceleration in m s⁻². Check which assumptions make the equations valid: for example, constant acceleration, negligible air resistance or a uniform beam.

Turn a past paper into a revision plan

1. Set a baseline

Choose the correct qualification and component. Work without notes under the paper’s stated time conditions. Keep a record of questions you skipped and places where you guessed. Both specifications permit suitable scientific or graphical calculators, subject to current examination rules.

2. Mark the reasoning

Use the matching official mark scheme. Record which step lost marks, not only the final score. Read the corresponding examiner report for the questions that caused difficulty. OCR’s public assessment page links papers, mark schemes, reports and student resources.

3. Give each error a next action

• Concept gap: explain the underlying idea in two sentences, then complete a short topic set.
• Setup error: redraw the diagram or redefine the random variable before calculating.
• Algebra or arithmetic slip: redo the calculation and add an independent substitution or estimate.
• Interpretation error: write the answer in context with units, assumptions or a statistical conclusion.
• Time pressure: try a short mixed set with a time limit after the method is secure.

4. Rebuild the solution without copying

Close the mark scheme. Rework the question from a blank page and explain why each equation is valid. If you get stuck, identify the first missing idea rather than transcribing the whole answer. Use one similar original practice question to check that the method transfers.

5. Revisit after a gap

Return to the error a few days later, then again in a mixed set. Mark it secure only when you can recognise and execute the method without a topic label or a worked example beside you. Schedule short regular sessions alongside full papers.

6. Include the large data set and calculator practice

Use the official data-set workbook and its information sheets to practise interpreting variables, units, samples and unusual values. The current specifications still describe data-set familiarity as relevant to the statistics assessment; they do not require memorising the individual data values.

Practise the actual calculator operations you need: cumulative binomial probabilities, summary statistics and, for H240, normal probabilities and inverse normal. Write the mathematical probability or equation before entering numbers. Retain unrounded working, then use the precision requested.

Six original worked examples

W1. Differentiation and the shape of a quadratic

Level: H230 AS and H240 A Level · Pure mathematics

Question. For y = x² − 4x + 3, determine the stationary point and its nature. Find the x-intercepts and explain how they fit a sketch.

1. Differentiate: dy/dx = 2x − 4. A stationary point has zero gradient, so 2x − 4 = 0 and x = 2.
2. Substitute into the original function: y = 2² − 4(2) + 3 = −1. The stationary point is (2, −1).
3. The second derivative is d²y/dx² = 2, which is positive. Therefore the point is a minimum. Alternatively, the gradient changes from negative to positive as x passes through 2.
4. For the x-intercepts, set y = 0: x² − 4x + 3 = (x − 1)(x − 3) = 0. The intercepts are (1, 0) and (3, 0).
5. Completing the square gives y = (x − 2)² − 1. This independently confirms the minimum and the symmetry line x = 2. The y-intercept is (0, 3).

Why this method works. The derivative measures the slope of the curve. The roots answer a different question: where the curve meets the horizontal axis. Here the minimum lies below that axis, so two distinct roots are consistent with the sketch.

Check yourself. A stationary point needs both coordinates. The roots 1 and 3 are not stationary x-values.

W2. Exact area using integration by parts

Level: H240 A Level · Pure mathematics

Question. Find the exact area between y = x e⁻ˣ, the x-axis, x = 0 and x = ln 2.

1. On this interval, x ≥ 0 and e⁻ˣ > 0, so the function is non-negative. The area is A = ∫₀ˡⁿ² x e⁻ˣ dx.
2. Use integration by parts. Choose u = x and dv/dx = e⁻ˣ, so du/dx = 1 and v = −e⁻ˣ.
3. Then ∫x e⁻ˣ dx = −x e⁻ˣ + ∫e⁻ˣ dx = −(x + 1)e⁻ˣ + C.
4. Apply the limits: A = [−(x + 1)e⁻ˣ]₀ˡⁿ² = −(ln 2 + 1)/2 − (−1).
5. Hence A = (1 − ln 2)/2 square units, approximately 0.153426.

Why this method works. Differentiating the factor x simplifies it to 1. The remaining exponential is straightforward to integrate. This makes integration by parts a good choice for this product.

Independent check. Differentiating −(x + 1)e⁻ˣ gives −e⁻ˣ + (x + 1)e⁻ˣ = x e⁻ˣ. A numerical integral also gives 0.1534264097.

Watch the sign. The antiderivative of e⁻ˣ is −e⁻ˣ. Evaluate the lower limit before subtracting it. Keep the logarithm in the final answer when an exact value is requested.

W3. A one-sided binomial hypothesis test

Level: H230 AS and H240 A Level · Statistics

Question. A fictional production process historically produces a red component with probability 0.25. After a change, a random sample of 20 components contains 9 red components. Assuming independent outcomes and a common red-component probability, test at the 5% level whether the probability has increased.

1. Let p be the probability that a component is red under the changed process. State H₀: p = 0.25 and H₁: p > 0.25.
2. Let X be the number of red components in the 20-component sample. Under H₀, X ~ B(20, 0.25).
3. An increase corresponds to the upper tail. We need the probability of the observed result or a more extreme result: P(X ≥ 9).
4. P(X ≥ 9) = 1 − P(X ≤ 8) = 0.0409251677. This is about 0.0409, below 0.05.
5. Reject H₀. There is sufficient evidence at the 5% significance level that the red-component probability has increased. This is evidence under the model, not proof that the process changed.

Critical-region check. P(X ≥ 8) = 0.1018118569, which is above 0.05. Therefore the upper-tail critical region is X ≥ 9; its actual significance level is approximately 4.09% because a discrete distribution does not usually allow an exact 5% tail.

Why the assumptions matter. A fixed sample size, two outcome categories, independent trials and constant p justify the binomial model. Sampling components in batches with strongly linked colours could undermine independence.

Common trap. P(X = 9) alone is not the p-value. Do not decide whether to use an upper or lower tail after seeing which one gives a smaller probability.

W4. Normal probability and an inverse-normal cutoff

Level: H240 A Level · Statistics

Question. In a fictional model, a task’s completion time X minutes follows N(60, 8²). Find P(X > 72) and the time exceeded by only 10% of tasks.

1. The mean is 60 minutes and the standard deviation is 8 minutes. The second parameter 8² = 64 is the variance.
2. Standardise 72: z = (72 − 60)/8 = 1.5. Thus P(X > 72) = P(Z > 1.5) = 0.0668072, approximately 0.0668.
3. If only 10% of tasks exceed c, then P(X ≤ c) = 0.90. This is the 90th percentile, not the 10th percentile.
4. The standard-normal 90th percentile is z = 1.2815516. Therefore c = 60 + 8(1.2815516) = 70.2524 minutes, or 70.3 minutes to three significant figures.

Sense check. A cutoff above the mean should have an upper-tail probability below one half. The 90th percentile is below 72 because more than 90% of the distribution is below 72.

Model check. A normal model is symmetric and theoretically includes negative values. Check whether that approximation is plausible for the real setting; it is a mathematical model rather than a guarantee about each task.

W5. Reading acceleration and displacement from velocity

Level: H230 AS and H240 A Level · Mechanics

Question. A particle moves along a straight line with velocity v = 2t m s⁻¹ for 0 ≤ t ≤ 4, where t is measured in seconds. Find its acceleration, displacement and mean velocity during this interval.

1. Acceleration is the rate of change of velocity: a = dv/dt = 2 m s⁻².
2. Displacement is the integral of velocity: s(4) − s(0) = ∫₀⁴ 2t dt = [t²]₀⁴ = 16 m.
3. Alternatively, the velocity–time graph is a straight line from (0, 0) to (4, 8). Its triangular area is ½ × 4 × 8 = 16 m.
4. Mean velocity = displacement ÷ elapsed time = 16/4 = 4 m s⁻¹. The final velocity is 8 m s⁻¹, which is a different quantity.
5. Since v ≥ 0 throughout this interval, the distance travelled also equals 16 m.

Why this method works. A velocity–time graph’s gradient is acceleration; its signed area is displacement. The units confirm this: (m s⁻¹)/s = m s⁻², and (m s⁻¹) × s = m.

Boundary of the model. The rule is stated only for the first four seconds. It gives a change in position, not an absolute position unless the starting position is also specified. If velocity changes sign, split the integral at each zero when calculating distance.

W6. Moments and support reactions

Level: H240 A Level · Mechanics

Question. A uniform horizontal beam AB is 4 m long and has mass 12 kg. It rests on supports at A and B. A small load of mass 8 kg is placed 1 m from A. Find the upward support reactions. Take g = 9.8 m s⁻².

1. Draw a force diagram. Let R_A and R_B be the upward reactions. The beam’s weight is 12g N and acts 2 m from A because the beam is uniform. The load’s weight is 8g N and acts 1 m from A.
2. Vertical equilibrium gives R_A + R_B = 12g + 8g = 196 N.
3. Take moments about A. R_A has zero moment about A, so 4R_B = (12g)(2) + (8g)(1) = 313.6 N m.
4. Therefore R_B = 78.4 N and R_A = 196 − 78.4 = 117.6 N.
5. Check moments about B: 4(117.6) = (12g)(2) + (8g)(3) = 470.4 N m. Both moment equations agree.

Why this method works. Equilibrium requires zero resultant force and zero resultant moment. Taking moments about a support removes its unknown reaction from that equation.

Sense check. The additional load is nearer A, so a larger reaction at A is reasonable. The reactions add to the total weight. Treating a non-uniform beam’s weight as acting at its midpoint would require justification.

Eight practice questions

Try each question before opening its answer. These are learning exercises with worked solutions, not official OCR questions or allocated exam marks. Use g = 9.8 m s⁻² where needed.

Misconceptions to catch before your next paper

1. Stationary point and root mean the same thing. A root satisfies y = 0; a stationary point satisfies dy/dx = 0. Calculate the requested feature and label both coordinates where appropriate.
2. A definite integral is always a positive area. It is signed accumulation. Split at axis crossings and add positive area contributions when finding total geometric area or distance.
3. A calculator output is enough for every command. OCR distinguishes commands that require justification. When a question asks you to determine, prove, show a result or provide detailed reasoning, make the mathematical argument visible.
4. The normal notation contains the standard deviation. In N(μ, σ²), the second parameter is the variance. Read a calculator’s input labels carefully.
5. A hypothesis test proves its conclusion. State the strength of evidence at the chosen significance level. Failure to reject H₀ does not establish that H₀ is true.
6. “At least k” means 1 − P(X ≤ k). For an integer-valued X, it means 1 − P(X ≤ k − 1).
7. The velocity at the highest point of a projectile is zero. Its vertical component is zero. Its horizontal component can remain non-zero.
8. Constant-acceleration formulae work for every motion. Check the acceleration assumption; otherwise use the relevant derivative or integral.
9. Mass can be inserted directly as a weight. A mass m kg has weight mg newtons. Moment calculations also need the perpendicular distance from the pivot to the force’s line of action.
10. Each new past paper should immediately follow the last. First repair the errors you have identified. Repeating an uncorrected method often repeats the same lost marks.

Quick questions

Can an AS student use every example here? Start with the H230/H240 labels. The H240-only examples require additional A Level methods. Shared topic names do not mean identical content.

Should I work through papers in date order? Use whichever unattempted paper fits your current purpose. Preserve some unseen papers for timed practice. Revisit corrected questions before accumulating more scores.

What if I get the answer right by a different method? Check the mark scheme’s accepted approaches and the question’s command words. A sound alternative method can be mathematically valid; the reasoning must still satisfy what was asked.

Is the example set a complete course checklist? No. It demonstrates selected techniques across all three strands. Use the official specification and your teacher’s guidance for full coverage.

Are these official OCR questions? No. The six worked examples and eight practice questions are original educational material. Use the linked official assessment page for OCR’s own papers and associated documents.