OCR A Level Maths June 2022 Paper 1: question paper and mark scheme (H240/01)

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 · 4 marks

   Find the x-coordinates of all the points of intersection between the curves $y=x(x+3)$ and $y=x^3(x+3)$.

   Worked solution

   1. M1 for equating the functions: $x^3(x+3)=x(x+3)$.
   2. M1 for bringing all terms to one side and factorising: $x(x+3)(x^2-1)=0$.
   3. M1 for further factorising $x^2-1$ to $(x-1)(x+1)$.
   4. A1 for listing all four correct x-coordinates: $x=0,-3,1,-1$.

2. Differentiation · 5 marks

   Find the equation of the normal to the curve $y=x^2{e}^{-x}$ at the point where $x=1$. Give your answer in the form $y=mx+c$.

   Worked solution

   1. M1 for finding the y-coordinate at $x=1$: $y=e^{-1}$.
   2. M1 for applying the product rule to find $\frac{dy}{dx}=2x{e}^{-x}-x^2{e}^{-x}$.
   3. M1 for evaluating the gradient at $x=1$: $m_t=2e^{-1}-e^{-1}=e^{-1}$.
   4. M1 for finding the perpendicular gradient of the normal: $m_n=-e$.
   5. A1 for forming the equation $y-e^{-1}=-e(x-1)$ and rearranging to $y=-ex+e+e^{-1}$.

3. Sequences and series · 5 marks

   Expand $(2-x{)}^{-2}$ up to the term in $x^2$. Use your expansion to find an approximate value for $\frac{1}{{1.98}^2}$.

   Worked solution

   1. M1 for rewriting as $2^{-2}(1-\frac{x}{2}{)}^{-2}=\frac{1}{4}(1-\frac{x}{2}{)}^{-2}$.
   2. M1 for expanding correctly: $\frac{1}{4}(1+2(\frac{x}{2})+3(\frac{x}{2}{)}^2)=\frac{1}{4}+\frac{1}{4}x+\frac{3}{16}{x}^2$.
   3. M1 for deducing the appropriate value of $x$ to substitute is $x=0.02$.
   4. M1 for substituting $x=0.02$ into the expansion: $\frac{1}{4}+\frac{1}{4}(0.02)+\frac{3}{16}(0.0004)$.
   5. A1 for arriving at the correct approximation: $0.25+0.005+0.000075=0.255075$.

More on these topics: Algebra and functions · Differentiation · Sequences and series

What this paper covers

What this paper is like

• Shape: 12 questions, 100 marks in 2 h (about 1.2 minutes a mark); questions from 6 to 12 marks.
• Biggest question: Q10, 12 marks: sector with a midpoint: angles and area.
• Marks by main topic: Exponentials and logarithms 17, Integration 16, Algebra and functions 15, Trigonometry 12, Coordinate geometry 12, Sequences and series 8, Differentiation 8, Numerical methods 6, Proof 6.
• More Trigonometry in other sittings: November 2021 Q10 (11 marks), June 2023 Q7 (9 marks), November 2020 Q1 (5 marks).
• Against the other 7 sittings we mapped: more marks on Exponentials and logarithms (17 marks here, 10 on average) and Coordinate geometry (12 marks here, 6 on average); fewer on Differentiation (8 marks here, 17 on average) and Algebra and functions (15 marks here, 20 on average).
• Not on this paper: Vectors.
• Calculator: allowed.

H240/01: question by question

Labels and topics are ours, read from the H240/01 paper (checked 2026-11-01); marks as printed. Tap a number to practise that topic.

1. Q1Trapezium rule and its accuracy (6 marks) · Numerical methods
2. Q2Counter-example with real numbers (6 marks) · Proof
3. Q3Intersection of two curves (7 marks) · Algebra and functions
4. Q4Completing the square and the range (8 marks) · Algebra and functions
5. Q5Transformations of y = 2^x (8 marks) · Exponentials and logarithms, Algebra and functions
6. Q6Binomial expansion and an approximation (8 marks) · Sequences and series
7. Q7Implicit curve with no horizontal tangents (8 marks) · Differentiation
8. Q8Exponential decay of two substances (9 marks) · Exponentials and logarithms
9. Q9Substitution x = 2 sin u for an exact value (7 marks) · Integration
10. Q10Sector with a midpoint: angles and area (12 marks) · Trigonometry
11. Q11Separable differential equation with e^(3y) (9 marks) · Integration, Exponentials and logarithms
12. Q12Parametric curve: tangent and normal (12 marks) · Coordinate geometry, Differentiation