OCR A Level Maths June 2025 Paper 3: question paper and mark scheme (H240/03)

Try 3 A Level questions like these

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1. Sequences and series · 4 marks

   The first three terms of the expansion $(a+bx)^6$ are $64 - 192x + cx^2$. Find the values of the constants $a$, $b$, and $c$.

   Worked answer

   1. M1 for equating the first term: $a^6 = 64 \implies a=2$.
   2. M1 for equating the $x$ coefficient to find $b$: $\binom{6}{1}a^5 b = -192 \implies 6(32)b = -192 \implies 192b = -192$.
   3. A1 for obtaining $b = -1$.
   4. A1 for finding the $x^2$ coefficient: $c = \binom{6}{2}a^4 b^2 = 15(16)(-1)^2 = 240$.

2. Proof · 4 marks

   Prove by contradiction that if $a$ and $b$ are integers, then $a^2 - 4b \ne 2$.

   Worked answer

   1. M1 for assuming that there exist integers $a$ and $b$ such that $a^2 - 4b = 2$, which rearranges to $a^2 = 4b + 2 = 2(2b + 1)$.
   2. M1 for deducing that since $a^2$ is an even number, $a$ must also be an even number.
   3. M1 for letting $a = 2k$, substituting to get $(2k)^2 = 4b + 2 \implies 4k^2 = 4b + 2$, and dividing by $2$ to get $2k^2 = 2b + 1 \implies 2(k^2 - b) = 1$.
   4. A1 for observing that the left-hand side $2(k^2 - b)$ is even while the right-hand side $1$ is odd, which is a contradiction, so $a^2 - 4b \ne 2$.

3. Kinematics · 5 marks

   Car A travels along a straight road at a constant speed of $25 \text{ m s}^{-1}$. Car B starts from rest at the same point and accelerates uniformly at $3 \text{ m s}^{-2}$ until it reaches a speed of $30 \text{ m s}^{-1}$, after which it maintains this speed. Find the distance Car B travels before it catches up to Car A.

   Worked answer

   1. M1 for finding the time taken for Car B to reach $30 \text{ m s}^{-1}$: $t = \frac{30}{3} = 10 \text{ s}$.
   2. M1 for finding the distances covered in the first $10 \text{ s}$: Car B covers $\frac{1}{2} \times 10 \times 30 = 150 \text{ m}$. Car A covers $25 \times 10 = 250 \text{ m}$.
   3. M1 for setting up an equation for the remaining distance where Car B catches A. Let $t_2$ be the additional time: $150 + 30t_2 = 250 + 25t_2$.
   4. M1 for solving to find $t_2$: $5t_2 = 100 \implies t_2 = 20 \text{ s}$. Total time = $30 \text{ s}$.
   5. A1 for finding the total distance: $25 \times 30 = 750 \text{ m}$.

More on these topics: Sequences and series · Proof · Kinematics

What this paper covers

Pure Mathematics and Mechanics: a pure section, then mechanics (3.01 to 3.04).

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

Mechanics

• Kinematics
• Forces and Newton's laws
• Moments

Practise these topics (opens the practice engine)

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

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.