OCR A Level Maths November 2021 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. 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.

2. Probability · 3 marks

   A bag contains 5 green counters and 3 yellow counters. A counter is drawn at random, its colour is noted, and it is replaced. This is done 3 times in total. Find the probability that exactly 2 green counters are drawn.

   Worked answer

   1. M1 for identifying the 3 ways to draw exactly 2 green counters: GGY, GYG, YGG.
   2. M1 for calculating the probability of one such arrangement, e.g., $(\frac{5}{8}) \times (\frac{5}{8}) \times (\frac{3}{8}) = \frac{75}{512}$.
   3. A1 for multiplying by 3 to find the total probability: $\frac{225}{512}$.

3. Sampling and data · 5 marks

   A histogram is drawn. The class $0 \le x < 20$ has frequency $10$, and the class $20 \le x < 30$ has frequency $30$. If the height of the bar for the first class is $2\text{ cm}$, find the height of the bar for the second class, assuming the width of the bars is proportional to the class width.

   Worked answer

   1. M1 for stating the class widths: $20$ and $10$.
   2. M1 for establishing the width ratio: if class width $20$ has a certain drawn width, class width $10$ has half that drawn width. Let drawn width of first bar be $2w$, then second bar is $w$.
   3. M1 for using Area $\propto$ Frequency for the first bar: Area $= 2w \times 2 = 4w$. So $4w$ represents frequency $10$, meaning $w$ represents frequency $2.5$.
   4. M1 for applying to the second bar: Area $= w \times h$. Frequency $= 30$. So $w \times h$ represents $30$. Since $w$ represents $2.5$, Area must be $12w$.
   5. A1 for solving for height: $w \times h = 12w \implies h = 12\text{ cm}$.

More on these topics: Numerical methods · Probability · Sampling and data

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.