OCR A Level Maths June 2023 Paper 2: question paper and mark scheme (H240/02)

3 A Level questions like these, with worked solutions

Written and checked by us, not taken from the OCR paper. Try each one, then tap "Worked solution" to see every step and its marks.

1. Integration · 4 marks

   Find $\int (\sin x+\cos x{)}^{2}dx$.

   Worked solution

   1. M1 for expanding the bracket to ${\sin }^{2}x+2\sin x\cos x+{\cos }^{2}x$.
   2. M1 for simplifying using ${\sin }^{2}x+{\cos }^{2}x\equiv 1$ and $2\sin x\cos x\equiv \sin 2x$.
   3. M1 for integrating $1+\sin 2x$.
   4. A1 for $x-\frac{1}{2}\cos 2x+c$.

2. Sequences and series · 5 marks

   Expand $(2-x{)}^{-2}$ up to the term in $x^2$. Use your expansion to find an approximate value for $\frac{1}{{1.98}^2}$.

   Worked solution

   1. M1 for rewriting as $2^{-2}(1-\frac{x}{2}{)}^{-2}=\frac{1}{4}(1-\frac{x}{2}{)}^{-2}$.
   2. M1 for expanding correctly: $\frac{1}{4}(1+2(\frac{x}{2})+3(\frac{x}{2}{)}^2)=\frac{1}{4}+\frac{1}{4}x+\frac{3}{16}{x}^2$.
   3. M1 for deducing the appropriate value of $x$ to substitute is $x=0.02$.
   4. M1 for substituting $x=0.02$ into the expansion: $\frac{1}{4}+\frac{1}{4}(0.02)+\frac{3}{16}(0.0004)$.
   5. A1 for arriving at the correct approximation: $0.25+0.005+0.000075=0.255075$.

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 solution

   1. M1 for identifying the three scenarios: $(A\cap B^{\prime }\cap C^{\prime })$, $(A^{\prime }\cap B\cap C^{\prime })$, and $(A^{\prime }\cap B^{\prime }\cap C)$.
   2. M1 for calculating the probabilities of these scenarios: $(0.5\times 0.6\times 0.7)+(0.5\times 0.4\times 0.7)+(0.5\times 0.6\times 0.3)$.
   3. A1 for evaluating and summing: $0.21+0.14+0.09=0.44$.

More on these topics: Integration · Sequences and series · Probability

What this paper covers

What this paper is like

• Shape: 14 questions, 100 marks in 2 h (about 1.2 minutes a mark); questions from 3 to 12 marks.
• Biggest question: Q5, 12 marks: r sin(x + a) and solving equations.
• Marks by main topic: Sampling and data 23, Statistical distributions 17, Trigonometry 12, Coordinate geometry 10, Differentiation 9, Probability 7, Algebra and functions 5, Vectors 5, Proof 5, Hypothesis testing 4, Integration 3.
• More Trigonometry in other sittings: June 2024 Q6 (10 marks), June 2025 Q4 (6 marks), November 2020 Q4 (5 marks).
• Against the other 7 sittings we mapped: more marks on Trigonometry (12 marks here, 4 on average) and Coordinate geometry (10 marks here, 4 on average); fewer on Sequences and series (0 marks here, 6 on average).
• Not on this paper: Sequences and series, Exponentials and logarithms, Numerical methods.
• Calculator: allowed.

H240/02: question by question

Labels and topics are ours, read from the H240/02 paper (checked 2026-11-01); marks as printed. Tap a number to practise that topic.

1. Q1Completing the square and a sketch (5 marks) · Algebra and functions
2. Q2Vectors and a point dividing a line (5 marks) · Vectors
3. Q3Exact area under 1/(x + 2) (3 marks) · Integration, Exponentials and logarithms
4. Q4Normal to y = x^2 and an enclosed area (9 marks) · Differentiation, Integration
5. Q5R sin(x + a) and solving equations (12 marks) · Trigonometry
6. Q6Circle on the x-axis and the line y = x (10 marks) · Coordinate geometry
7. Q7Proof about a^2 - 4b (5 marks) · Proof
8. Q8Stem-and-leaf diagram of plant heights (7 marks) · Sampling and data
9. Q9Stratified sample in a school (6 marks) · Sampling and data
10. Q10Normal test for fish masses (8 marks) · Statistical distributions, Hypothesis testing
11. Q11Normal distribution with unknown mean and sd (9 marks) · Statistical distributions
12. Q12Testing a dice for bias against six (4 marks) · Hypothesis testing
13. Q13Large data set: scatter diagram of LAs (10 marks) · Sampling and data
14. Q14Tree disease: conditional probability (7 marks) · Probability

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.