OCR A Level Maths June 2024 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. Algebra and functions · 7 marks

   Let $f(x)=5-4x-2x^2$.
   
   (a) Express $f(x)$ in the form $A-B(x+C{)}^2$, where $A$, $B$, and $C$ are integers.
   (b) State the maximum value of $f(x)$ and the value of $x$ at which it occurs.
   (c) Solve $f(x)=0$, giving your answers in exact form.

   Worked solution

   (a) M1 Factors out $-2$ from the $x$ terms: $5-2(x^2+2x)$.

   1. M1 Completes the square inside the bracket: $5-2((x+1{)}^2-1)$.
   2. A1 Expands and simplifies to obtain $7-2(x+1{)}^2$.

   (b) B1 States the maximum value is $7$.

   1. B1 States this occurs at $x=-1$.

   (c) M1 Sets $7-2(x+1{)}^2=0$ and rearranges to $(x+1{)}^2=\frac{7}{2}$.

   1. A1 Solves for $x$ by square rooting, yielding $x=-1\pm \sqrt{\frac{7}{2}}$ (or equivalent exact surd).

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. Sampling and data · 5 marks

   The data in a histogram is grouped into classes. A bar representing the class $100\le x<150$ has a width of $2.5\text{ cm}$ and a height of $4\text{ cm}$. The total area of all bars in the histogram is $90{\text{ cm}}^2$. If the bar for $100\le x<150$ represents a frequency of $20$, calculate the total frequency of the entire data set.

   Worked solution

   1. M1 for calculating the area of the specified bar: $2.5\times 4=10{\text{ cm}}^2$.
   2. M1 for establishing the relationship between area and frequency: $10{\text{ cm}}^2$ represents a frequency of $20$.
   3. A1 for calculating the scale factor: $1{\text{ cm}}^2$ represents $2$ items (since $20/10=2$).
   4. M1 for using the total area to find total frequency: $90{\text{ cm}}^2\times 2$.
   5. A1 for the final answer: $180$ items.

More on these topics: Algebra and functions · Trigonometry · Sampling and data

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 11 marks.
• Biggest question: Q10, 11 marks: store customers: correlation and regression.
• Marks by main topic: Statistical distributions 22, Sampling and data 19, Numerical methods 10, Trigonometry 10, Exponentials and logarithms 8, Sequences and series 8, Algebra and functions 6, Hypothesis testing 5, Differentiation 4, Vectors 4, Probability 4.
• More Sampling and data in other sittings: June 2023 Q13 (10 marks), June 2025 Q9 (11 marks), June 2022 Q9 (14 marks).
• Against the other 7 sittings we mapped: more marks on Numerical methods (10 marks here, 0 on average) and Exponentials and logarithms (8 marks here, 2 on average); fewer on Differentiation (4 marks here, 11 on average) and Integration (0 marks here, 6 on average).
• Not on this paper: Proof, Coordinate geometry, Integration.
• 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. Q1Differentiating powers and roots (4 marks) · Differentiation
2. Q2Vector from its magnitude and direction (4 marks) · Vectors
3. Q3Factor theorem and a repeated root (6 marks) · Algebra and functions
4. Q4Sign change and Newton-Raphson for x e^(1-3x) (10 marks) · Numerical methods, Differentiation
5. Q5Declining population model (8 marks) · Exponentials and logarithms
6. Q6Triple-angle formula and an equation (10 marks) · Trigonometry
7. Q7Two arithmetic progressions (8 marks) · Sequences and series
8. Q8Binomial model for orange sweets (6 marks) · Statistical distributions
9. Q9Normal model for flour bags (8 marks) · Statistical distributions
10. Q10Store customers: correlation and regression (11 marks) · Sampling and data
11. Q11Large data set: employee travel increases (8 marks) · Sampling and data
12. Q12Choosing a student fairly (4 marks) · Probability
13. Q13Planning a test on election votes (5 marks) · Hypothesis testing
14. Q14Discrete distribution with a constant p (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.

What the examiners said about this paper

Summarised in our own words from the board's examiner report (Examiner Insights).

• H240/02: "Detailed reasoning" (Q6); Rejecting a solution without a reason (Q14); "Explain" and "interpret" answers without context (Q5, Q10, Q11, Q12). Official examiner report ↗

Question-by-question topics, the top mistakes and what to focus on next →