OCR A Level Maths November 2021 Paper 1: question paper and mark scheme (H240/01)

Try 3 A Level questions like these

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1. Coordinate geometry · 6 marks

   The circle $C$ has centre $(a, b)$ and passes through the points $P(0, 0)$, $Q(6, 0)$, and $R(0, 8)$. Find the values of $a$, $b$, and the radius of $C$.

   Worked answer

   $P$ and $Q$ both on circle ⇒ centre lies on the perpendicular bisector of $PQ$, so $a = 3$   M1 A1. Similarly $P$ and $R$ give $b = 4$   M1 A1. Radius $= |CP| = \sqrt{9 + 16} = 5$   M1 A1.

2. Differentiation · 6 marks

   The normal to the curve $y = \sqrt{x}$ at the point where $x = 4$ cuts the x-axis at $A$ and the y-axis at $B$. Calculate the exact area of the triangle $AOB$, where $O$ is the origin.

   Worked answer

   1. M1 for finding the y-coordinate at $x=4$: $y=2$.
   2. M1 for finding $\frac{dy}{dx} = \frac{1}{2}x^{-\frac{1}{2}}$ and evaluating at $x=4$ to get gradient $m_t = \frac{1}{4}$.
   3. M1 for finding the normal gradient $m_n = -4$ and its equation: $y - 2 = -4(x - 4) \implies y = -4x + 18$.
   4. M1 for finding the axis intercepts of the normal: at $y=0, 4x = 18 \implies x = 4.5$ (point $A$) and at $x=0, y=18$ (point $B$).
   5. M1 for using the area of a triangle formula $\frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 4.5 \times 18$.
   6. A1 for the final area of $40.5$ square units.

3. Proof · 4 marks

   Prove by contradiction that there are infinitely many prime numbers.

   Worked answer

   1. M1 for assuming that there is a finite number of prime numbers, listing them as $p_1, p_2, \dots, p_n$.
   2. M1 for constructing a new number $N = (p_1 \times p_2 \times \dots \times p_n) + 1$.
   3. M1 for arguing that dividing $N$ by any of the primes $p_i$ leaves a remainder of $1$, meaning $N$ has no prime factors from the finite list.
   4. A1 for concluding that $N$ must either be a new prime itself or have a prime factor not in the list, contradicting the assumption that the list contains all primes.

More on these topics: Coordinate geometry · Differentiation · Proof

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.