OCR A Level Maths June 2025 Paper 2: question paper and mark scheme (H240/02)

Try 3 A Level questions like these

Written and checked by us, not taken from the OCR paper. Tap "Worked answer" to see every step.

1. Algebra and functions · 5 marks

   Let $f(x) = x^3 + cx^2 + dx - 4$. Given $f(x)$ has a factor $(x+2)$ and leaves a remainder of $-10$ when divided by $(x-3)$, find the values of $c$ and $d$.

   Worked answer

   1. M1 for applying the Factor Theorem for $(x+2)$: $f(-2) = 0 \implies -8 + 4c - 2d - 4 = 0 \implies 4c - 2d = 12$.
   2. M1 for applying the Remainder Theorem for $(x-3)$: $f(3) = -10 \implies 27 + 9c + 3d - 4 = -10 \implies 9c + 3d = -33$.
   3. M1 for simplifying both equations (e.g., $2c - d = 6$ and $3c + d = -11$).
   4. M1 for solving the simultaneous equations (e.g., adding them gives $5c = -5$).
   5. A1 for finding the correct values $c = -1$ and $d = -8$.

2. Numerical methods · 6 marks

   The equation $\cos x = x$ has a unique root $\alpha$ in $[0, 1]$. Using the iteration $x_{n+1} = \cos x_n$ with $x_0 = 0.5$, find $x_1$, $x_2$, $x_3$, and $x_4$ each to $4$ d.p. (Angle in radians.)

   Worked answer

   $x_1 = \cos 0.5 = 0.8776$   M1 A1. $x_2 = \cos 0.8776 = 0.6390$   A1. $x_3 = \cos 0.6390 = 0.8027$   A1. $x_4 = \cos 0.8027 = 0.6948$   A1 A1.

3. Probability · 3 marks

   Three independent events $A$, $B$, and $C$ have probabilities $P(A) = 0.5$, $P(B) = 0.4$, and $P(C) = 0.3$. Find the probability that exactly one of these events occurs.

   Worked answer

   1. M1 for identifying the three scenarios: $(A \cap B' \cap C')$, $(A' \cap B \cap C')$, and $(A' \cap B' \cap C)$.
   2. M1 for calculating the probabilities of these scenarios: $(0.5\times0.6\times0.7) + (0.5\times0.4\times0.7) + (0.5\times0.6\times0.3)$.
   3. A1 for evaluating and summing: $0.21 + 0.14 + 0.09 = 0.44$.

More on these topics: Algebra and functions · Numerical methods · Probability

What this paper covers

Pure Mathematics and Statistics: a pure section, then statistics (2.01 to 2.05), including OCR's pre-release large data set.

We haven't mapped this paper question by question yet, so these are the topics it can test, from the specification's paper structure. Each links to a page where you can learn the topic and then practise questions on it.

Pure

• Proof
• Algebra and functions
• Coordinate geometry
• Sequences and series
• Trigonometry
• Exponentials and logarithms
• Differentiation
• Integration
• Numerical methods
• Vectors

Statistics

• Sampling and data
• Probability
• Statistical distributions
• Hypothesis testing

Practise these topics (opens the practice engine)

• Pure I
• Pure II
• Pure III (calculus)
• Statistics

Some Paper 2 statistics questions use OCR's pre-release large data set. We don't write large data set questions, so use your class materials for those.

Formula booklet

A Level Mathematics A (H240): formulae printed in each question paper · OCR · H240 specification version 3.1 (current)

OCR has no separate formula booklet for H240: the formulae are printed on pages 2 and 3 of every question paper. The link opens OCR’s H240 assessment page, where the papers are. Checked against OCR’s own papers on 11 October 2026.