OCR A Level Maths Specimen Paper 2: question paper (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. Algebra and functions · 7 marks

   Consider the rational function $f(x)=\frac{3x}{(x-1)(x+2)}$.

   1. Express $f(x)$ in partial fractions.
   2. Hence, evaluate exactly the definite integral ${\int }_2^{3}f(x)dx$, giving your answer in the form $\ln (k)$ where $k$ is a rational number.
   Worked solution

   1. M1 for setting up the partial fraction decomposition: $\frac{3x}{(x-1)(x+2)}=\frac{A}{x-1}+\frac{B}{x+2}$.
      M1 for clearing the denominators: $3x=A(x+2)+B(x-1)$.
      M1 for substituting strategic values of $x$ (e.g., $x=1$ and $x=-2$) to solve for constants.
      A1 for finding $A=1$ and $B=2$, yielding $\frac{1}{x-1}+\frac{2}{x+2}$.
   2. M1 for setting up the integral and determining the anti-derivatives: ${[\ln |x-1|+2\ln |x+2|]}_2^3$.
      M1 for substituting the limits: $(\ln 2+2\ln 5)-(\ln 1+2\ln 4)$.
      A1 for using logarithm laws to combine into a single logarithm: $\ln 2+\ln 25-0-\ln 16=\ln \left(\frac{50}{16}\right)=\ln \left(\frac{25}{8}\right)$.

2. Trigonometry · 5 marks

   Solve $2{\sin }^{2}x+3\cos x-3=0$ for $0^{\circ }\le x\le {360}^{\circ }$.

   Worked solution

   1. M1 for substituting ${\sin }^{2}x$ with $1-{\cos }^{2}x$ to get $2(1-{\cos }^{2}x)+3\cos x-3=0$.
   2. M1 for expanding and rearranging to form the quadratic $2{\cos }^{2}x-3\cos x+1=0$.
   3. M1 for factorising the quadratic as $(2\cos x-1)(\cos x-1)=0$.
   4. A1 for finding the solutions from $\cos x=0.5$ giving $x={60}^{\circ }$ and ${300}^{\circ }$.
   5. A1 for finding the solutions from $\cos x=1$ giving $x=0^{\circ }$ and ${360}^{\circ }$.

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: Algebra and functions · Trigonometry · 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 4 to 12 marks.
• Biggest question: Q6, 12 marks: savings growth as a differential equation.
• Marks by main topic: Sampling and data 21, Algebra and functions 20, Statistical distributions 14, Integration 12, Coordinate geometry 11, Probability 8, Differentiation 7, Hypothesis testing 7.
• More Integration in other sittings: November 2020 Q8 (9 marks), November 2021 Q6 (5 marks), June 2022 Q3 (10 marks).
• Against the other 8 sittings we mapped: more marks on Algebra and functions (20 marks here, 8 on average) and Integration (12 marks here, 5 on average); fewer on Trigonometry (0 marks here, 5 on average) and Sequences and series (0 marks here, 5 on average).
• Not on this paper: Proof, Sequences and series, Trigonometry, Exponentials and logarithms, Numerical methods, Vectors.
• 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. Q1Simplifying index expressions (4 marks) · Algebra and functions
2. Q2Stationary point of a quintic (7 marks) · Differentiation
3. Q3Quadratic profit model for a book (9 marks) · Algebra and functions, Differentiation
4. Q4Partial fractions, then integration (7 marks) · Algebra and functions, Integration
5. Q5Circle and a parabola: area between them (11 marks) · Coordinate geometry, Integration
6. Q6Savings growth as a differential equation (12 marks) · Integration, Exponentials and logarithms
7. Q7Normal distribution of heights (6 marks) · Statistical distributions
8. Q8Grouped data: masses of plums (7 marks) · Sampling and data
9. Q9Census data: cycling to work (4 marks) · Sampling and data
10. Q10Testing library visit times (7 marks) · Hypothesis testing
11. Q11Venn diagram of sports played (8 marks) · Probability
12. Q12Census data: population table (5 marks) · Sampling and data
13. Q13Census scatter diagrams (5 marks) · Sampling and data
14. Q14Discrete distribution and expectation (8 marks) · Statistical distributions

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.