OCR A Level Maths June 2018 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. Differentiation · 4 marks

   Find the exact coordinates of the stationary point on the curve $y=\sqrt{x}-2x$, for $x>0$.

   Worked solution

   1. M1 for rewriting as $y=x^{0.5}-2x$ and differentiating: $\frac{dy}{dx}=0.5x^{-0.5}-2$.
   2. M1 for setting to zero and rearranging: $\frac{1}{2\sqrt{x}}=2⟹\sqrt{x}=0.25$.
   3. A1 for solving to get $x=0.0625$ or $\frac{1}{16}$.
   4. A1 for finding the y-coordinate: $\sqrt{\frac{1}{16}}-2(\frac{1}{16})=\frac{1}{4}-\frac{1}{8}=\frac{1}{8}$, giving $(\frac{1}{16},\frac{1}{8})$.

2. Exponentials and logarithms · 4 marks

   Given that ${\log }_{2}x=p$ and ${\log }_{2}y=q$, express ${\log }_2(8x^2{y}^3)$ in terms of $p$ and $q$.

   Worked solution

   1. M1 for splitting the logarithm into a sum: ${\log }_{2}8+{\log }_2{x}^2+{\log }_2{y}^3$.
   2. B1 for evaluating ${\log }_{2}8=3$.
   3. M1 for applying the power law: $2{\log }_{2}x+3{\log }_{2}y$.
   4. A1 for the final expression $3+2p+3q$.

3. Sampling and data · 4 marks

   A dataset $x$ has mean $\overline{x}=5$ and standard deviation ${\sigma }_x=2$. A new dataset $y$ is formed using the linear transformation $y=ax+b$, where $a>0$. The new dataset has mean $\overline{y}=15$ and standard deviation ${\sigma }_y=6$. Find the values of $a$ and $b$.

   Worked solution

   1. M1 for linking the standard deviations: ${\sigma }_y=a{\sigma }_x$.
   2. A1 for solving for $a$: $6=a(2)⟹a=3$.
   3. M1 for linking the means: $\overline{y}=a\overline{x}+b$.
   4. A1 for substituting $a=3$ to find $b$: $15=3(5)+b⟹b=0$.

More on these topics: Differentiation · Exponentials and logarithms · Sampling and data

What this paper covers

What this paper is like

• Shape: 13 questions, 100 marks in 2 h (about 1.2 minutes a mark); questions from 4 to 13 marks.
• Biggest question: Q6, 13 marks: curve y = ln x / x: stationary point and area.
• Marks by main topic: Statistical distributions 31, Algebra and functions 18, Sampling and data 15, Differentiation 13, Hypothesis testing 7, Vectors 6, Proof 6, Trigonometry 4.
• More Differentiation in other sittings: June 2019 Q1 (11 marks), November 2020 Q1 (9 marks), November 2021 Q1 (4 marks).
• Against the other 7 sittings we mapped: more marks on Statistical distributions (31 marks here, 16 on average) and Algebra and functions (18 marks here, 7 on average); fewer on Probability (0 marks here, 7 on average) and Sequences and series (0 marks here, 6 on average).
• Not on this paper: Coordinate geometry, Sequences and series, Exponentials and logarithms, Integration, Numerical methods, Probability.
• 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 no real roots (7 marks) · Algebra and functions
2. Q2Column vectors and a ratio (6 marks) · Vectors
3. Q3Choosing a proportional model for revision time (4 marks) · Algebra and functions
4. Q4Proving an identity with sin^2 and cos^2 (4 marks) · Trigonometry
5. Q5Counter-example to a claim about squares and primes (6 marks) · Proof
6. Q6Curve y = ln x / x: stationary point and area (13 marks) · Differentiation, Integration, Exponentials and logarithms
7. Q7Reflected parabola pieces and an area (7 marks) · Algebra and functions, Integration
8. Q8Normal distribution probabilities (8 marks) · Statistical distributions
9. Q9Testing whether a dice is biased (7 marks) · Hypothesis testing
10. Q10Sampling trees in a forest (9 marks) · Sampling and data
11. Q11Large data set: correlation and public transport (6 marks) · Sampling and data
12. Q12Discrete distribution with a constant (11 marks) · Statistical distributions
13. Q13Late trains: binomial probabilities (12 marks) · Statistical distributions, 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.