OCR A Level Maths June 2019 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

   The points $A(1,2)$ and $B(7,10)$ lie on a circle $C$ with $AB$ as a diameter.
   (a) Find the equation of $C$. (4)
   (b) The tangent to $C$ at $A$ meets the $y$-axis at $T$. Find the coordinates of $T$. (3)

   Worked solution

   (a) Centre $=$ midpoint $=(4,6)$   B1. Radius $=\frac{1}{2}|AB|=\frac{1}{2}\sqrt{36+64}=5$   M1 A1. Equation: $(x-4{)}^2+(y-6{)}^2=25$   A1.
   (b) Radius from centre to $A$ has gradient $\frac{2-6}{1-4}=\frac{4}{3}$   B1. Tangent gradient $=-\frac{3}{4}$; equation: $y-2=-\frac{3}{4}(x-1)$. Set $x=0$: $y=2+\frac{3}{4}=\frac{11}{4}$   M1 A1.

2. Differentiation · 5 marks

   Find the equation of the tangent to the curve $2x^2-xy+y^2=8$ at the point $(2,2)$. Give your answer in the form $y=mx+c$.

   Worked solution

   1. M1 for differentiating $2x^2$ to $4x$ and $y^2$ to $2y\frac{dy}{dx}$.
   2. M1 for differentiating $-xy$ using the product rule to get $-y-x\frac{dy}{dx}$.
   3. M1 for setting the entire derivative to zero: $4x-y-x\frac{dy}{dx}+2y\frac{dy}{dx}=0$.
   4. M1 for substituting $x=2$ and $y=2$ to find the gradient $m$: $8-2-2\frac{dy}{dx}+4\frac{dy}{dx}=0⟹2\frac{dy}{dx}=-6⟹m=-3$.
   5. A1 for using $y-2=-3(x-2)$ to establish the final equation $y=-3x+8$.

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 solution

   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: $w\times h=12w⟹h=12\text{ cm}$.

More on these topics: Coordinate geometry · Differentiation · 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 12 marks.
• Biggest question: Q12, 12 marks: discrete distribution with kx.
• Marks by main topic: Statistical distributions 30, Differentiation 20, Sampling and data 14, Coordinate geometry 9, Sequences and series 8, Exponentials and logarithms 5, Algebra and functions 5, Hypothesis testing 5, Proof 4.
• More Statistical distributions in other sittings: June 2018 Q13 (12 marks), November 2020 Q15 (10 marks), November 2021 Q14 (11 marks).
• Against the other 7 sittings we mapped: more marks on Statistical distributions (30 marks here, 17 on average) and Differentiation (20 marks here, 9 on average); fewer on Probability (0 marks here, 7 on average) and Sampling and data (14 marks here, 20 on average).
• Not on this paper: Trigonometry, Integration, Numerical methods, Vectors, 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. Q1Quotient and chain rules, then a substitution (11 marks) · Differentiation, Integration
2. Q2Binomial coefficient and a fractional expansion (8 marks) · Sequences and series
3. Q3Parametric circle: centre, radius and a line (9 marks) · Coordinate geometry
4. Q4Animal population on an island (5 marks) · Exponentials and logarithms
5. Q5Cone: minimising the slant height (9 marks) · Differentiation
6. Q6Counter-example to a claim about 2^n - 1 (4 marks) · Proof
7. Q7Quartic solved with a substitution (5 marks) · Algebra and functions
8. Q8Stem-and-leaf diagram of plant heights (6 marks) · Sampling and data
9. Q9Normal model for plum masses (11 marks) · Statistical distributions
10. Q10Pollutant levels: a normal model and a test (7 marks) · Statistical distributions, Hypothesis testing
11. Q11Large data set: population profiles (8 marks) · Sampling and data
12. Q12Discrete distribution with kx (12 marks) · Statistical distributions
13. Q13Binomial test for app users (5 marks) · Hypothesis testing

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.