OCR A Level Maths June 2025 Paper 1: question paper and mark scheme (H240/01)

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 · 4 marks

   Given that $f(x) = \frac{x+1}{x-2}$, $x \ne 2$, find an expression for $ff(x)$ in its simplest form.

   Worked answer

   1. M1 for substituting $f(x)$ into itself: $ff(x) = \frac{\frac{x+1}{x-2} + 1}{\frac{x+1}{x-2} - 2}$.
   2. M1 for multiplying the numerator and the denominator by $(x-2)$ to clear the nested fractions.
   3. M1 for obtaining the expanded fraction $\frac{(x+1) + (x-2)}{(x+1) - 2(x-2)}$.
   4. A1 for fully simplifying to $\frac{2x-1}{5-x}$ (or $\frac{1-2x}{x-5}$).

2. Numerical methods · 7 marks

   $f(x) = x^3 - 3x - 4$.
   (a) Show that $f(x) = 0$ has a root in $(2, 3)$. (2)
   (b) Rewrite the equation in the form $x = \sqrt[3]{3x + 4}$ and use this iteration with $x_0 = 2$ to find $x_1, x_2, x_3, x_4$ each to 4 d.p. State the value of the root to 3 d.p. (5)

   Worked answer

   (a) $f(2) = 8 - 6 - 4 = -2 < 0$; $f(3) = 27 - 9 - 4 = 14 > 0$   M1 A1.
   (b) $x_1 = \sqrt[3]{10} = 2.1544$; $x_2 = \sqrt[3]{10.4633} = 2.1872$; $x_3 = \sqrt[3]{10.5616} = 2.1940$; $x_4 = \sqrt[3]{10.5821} = 2.1955$   M1 A1 A1 A1. Root $\approx 2.196$   A1.

3. Vectors · 8 marks

   Consider the following two lines in 3D space: \[L_1: r = \begin{pmatrix} 1 \\ -1 \\ 2 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ 1 \\ -3 \end{pmatrix} \quad \text{and} \quad L_2: r = \begin{pmatrix} 4 \\ 2 \\ -1 \end{pmatrix} + \mu \begin{pmatrix} -1 \\ 2 \\ 1 \end{pmatrix}\] Show algebraically that these two lines are skew (they are neither parallel nor do they intersect).

   Worked answer

   Set the components equal:
   \(1 + 2\lambda = 4 - \mu \implies 2\lambda + \mu = 3\) (Eq 1). M1
   \(-1 + \lambda = 2 + 2\mu \implies \lambda - 2\mu = 3\) (Eq 2). M1
   \(2 - 3\lambda = -1 + \mu \implies -3\lambda - \mu = -3\) (Eq 3). M1
   Solve Eq 1 and Eq 2: \(2(3 + 2\mu) + \mu = 3 \implies 6 + 5\mu = 3 \implies \mu = -0.6\). M1
   Substitute \(\mu = -0.6 \implies \lambda = 1.8\). A1
   Check in Eq 3: \(-3(1.8) - (-0.6) = -5.4 + 0.6 = -4.8\). M1
   However, the RHS of Eq 3 is \(-3\). Since \(-4.8 \neq -3\), there is no consistent solution. B1
   Therefore, the lines do not intersect. Since their direction vectors are not scalar multiples, they are skew. A1

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

What this paper covers

Pure Mathematics only (OCR content 1.01 to 1.10).

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

Practise these topics (opens the practice engine)

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

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.