OCR A Level Maths June 2022 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. Coordinate geometry · 7 marks

   A curve has parametric equations $x=t+1$, $y=t^3-12t$.
   (a) Find the values of $t$ at the stationary points on the curve. (3)
   (b) Find the coordinates of the stationary points. (2)
   (c) Determine the nature of each stationary point. (2)

   Worked solution

   (a) $\frac{dy}{dx}=\frac{3t^2-12}{1}=3t^2-12$   M1 A1. Set $=0$: $t^2=4\Rightarrow t=\pm 2$   A1.
   (b) $t=2$: $(3,-16)$; $t=-2$: $(-1,16)$   A1 A1.
   (c) $\frac{d^{2}y}{d{x}^2}=6t\cdot \frac{dt}{dx}=6t$ (since $\frac{dt}{dx}=1$)   M1. $t=2$: $12>0$ ⇒ minimum; $t=-2$: $-12<0$ ⇒ maximum   A1.

2. Differentiation · 5 marks

   A curve has parametric equations $x=\ln t$ and $y=t^2$, $t>0$. Find the equation of the normal to the curve at the point where $t=1$. Give your answer in the form $ax+by+c=0$.

   Worked solution

   1. M1 for finding the coordinates at $t=1$: $x=0$, $y=1$.
   2. M1 for calculating $\frac{dx}{dt}=\frac{1}{t}$ and $\frac{dy}{dt}=2t$, giving $\frac{dy}{dx}=2t^2$.
   3. M1 for evaluating the tangent gradient at $t=1$ to get $m_t=2$.
   4. M1 for determining the normal gradient $m_n=-\frac{1}{2}$ and using $(0,1)$ to form the equation $y-1=-\frac{1}{2}(x-0)$.
   5. A1 for rearranging to the required integer format: $x+2y-2=0$.

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: Coordinate geometry · Differentiation · Probability

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 14 marks.
• Biggest question: Q9, 14 marks: heights of 150 plants: grouped data.
• Marks by main topic: Sampling and data 24, Algebra and functions 14, Differentiation 11, Integration 10, Probability 10, Proof 8, Statistical distributions 7, Hypothesis testing 6, Vectors 5, Sequences and series 5.
• More Sampling and data in other sittings: November 2021 Q11 (12 marks), June 2023 Q13 (10 marks), November 2020 Q11 (9 marks).
• Against the other 7 sittings we mapped: more marks on Algebra and functions (14 marks here, 7 on average) and Integration (10 marks here, 4 on average); fewer on Statistical distributions (7 marks here, 20 on average) and Trigonometry (0 marks here, 6 on average).
• Not on this paper: Coordinate geometry, Trigonometry, 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. Q1Solving three equations (8 marks) · Algebra and functions, Exponentials and logarithms
2. Q2Length and angle with 3D vectors (5 marks) · Vectors
3. Q3Two methods for one integral (10 marks) · Integration
4. Q4Steps in a design as an arithmetic series (5 marks) · Sequences and series
5. Q5A cubic with no stationary points (4 marks) · Differentiation
6. Q6Identifying functions from graphs (6 marks) · Algebra and functions
7. Q7Proof using the form 3m + r (8 marks) · Proof
8. Q8Rates of change in a conical tank (7 marks) · Differentiation
9. Q9Heights of 150 plants: grouped data (14 marks) · Sampling and data
10. Q10Large data set: age structure (10 marks) · Sampling and data
11. Q11Normal test for babies' masses (7 marks) · Statistical distributions, Hypothesis testing
12. Q12Testing underweight sugar packets (6 marks) · Hypothesis testing
13. Q13Students' subjects: Venn diagram (10 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.