OCR A Level Maths June 2022 Paper 3: question paper and mark scheme (H240/03)

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

   Solve the simultaneous equations $x-y=2$ and $x^2-xy+y^2=7$.

   Worked solution

   M1 Rearranges the linear equation to make $y$ the subject: $y=x-2$.

   1. M1 Substitutes this expression into the quadratic equation: $x^2-x(x-2)+(x-2{)}^2=7$.
   2. M1 Expands the brackets correctly: $x^2-x^2+2x+(x^2-4x+4)=7$.
   3. A1 Simplifies to a 3-term quadratic equation: $x^2-2x-3=0$.
   4. M1 Factorises the quadratic: $(x-3)(x+1)=0$.
   5. A1 Finds the $x$-values: $x=3$ and $x=-1$.
   6. A1 Substitutes back into the linear equation to find the corresponding $y$-values, giving the final solution pairs $x=3,y=1$ and $x=-1,y=-3$.

2. Integration · 6 marks

   The curve with equation $y=x^3-4x$ intersects the x-axis at $x=-2$, $x=0$, and $x=2$. Find the total exact area enclosed between the curve and the x-axis.

   Worked solution

   1. M1 for recognising that the region between $0$ and $2$ is below the x-axis, so the integrals must be computed separately.
   2. M1 for integrating to find the generic primitive: $[\frac{1}{4}{x}^4-2x^2]$.
   3. M1 for evaluating the integral from $-2$ to $0$: $(0)-(4-8)=4$.
   4. M1 for evaluating the integral from $0$ to $2$: $(4-8)-(0)=-4$.
   5. M1 for taking the absolute value of the negative area before combining.
   6. A1 for the total correct area: $4+|-4|=8$.

3. Kinematics · 8 marks

   Ship A starts at position $(0,5)$ relative to a port and moves with a constant velocity vector of $\left(\begin{array}{c}3\\ -1\end{array}\right){\text{ km h}}^{-1}$. Ship B starts at $(4,0)$ and moves with a constant velocity vector of $\left(\begin{array}{c}1\\ 2\end{array}\right){\text{ km h}}^{-1}$.

   1. Write down the position vectors of Ship A and Ship B at time $t$ hours.
   2. Find an expression for the square of the distance between the ships at time $t$, denoted $D^2(t)$.
   3. Find the exact time $t$ when the two ships are closest to each other.
   Worked solution

   1. M1 for writing the vector equations of motion.
      A1 for ${\mathbf{r}}_{\mathbf{A}}=\left(\begin{array}{c}3t\\ 5-t\end{array}\right)$ and ${\mathbf{r}}_{\mathbf{B}}=\left(\begin{array}{c}4+t\\ 2t\end{array}\right)$.
   2. M1 for finding the relative displacement vector ${\mathbf{r}}_{\mathbf{B}}-{\mathbf{r}}_{\mathbf{A}}=\left(\begin{array}{c}4-2t\\ 3t-5\end{array}\right)$.
      M1 for setting up the square of the magnitude: $D^2(t)=(4-2t{)}^2+(3t-5{)}^2$.
      A1 for expanding and simplifying to a quadratic: $D^2(t)=16-16t+4t^2+9t^2-30t+25=13t^2-46t+41$.
   3. M1 for recognizing that minimum distance occurs when the derivative of $D^2(t)$ is zero (or using the vertex formula $t=-b/2a$).
      M1 for solving $26t-46=0⟹26t=46$.
      A1 for finding exactly $t=\frac{23}{13}$ hours.

More on these topics: Algebra and functions · Integration · Kinematics

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 2 to 14 marks.
• Biggest question: Q5, 14 marks: tangent to a curve and an enclosed area.
• Marks by main topic: Kinematics 33, Differentiation 14, Forces and Newton's laws 10, Algebra and functions 8, Sequences and series 8, Integration 8, Trigonometry 8, Moments 7, Coordinate geometry 4.
• More Differentiation in other sittings: November 2021 Q8 (11 marks), November 2020 Q4 (11 marks), June 2024 Q5 (13 marks).
• Against the other 7 sittings we mapped: more marks on Kinematics (33 marks here, 24 on average) and Sequences and series (8 marks here, 3 on average); fewer on Forces and Newton's laws (10 marks here, 20 on average) and Algebra and functions (8 marks here, 13 on average).
• Not on this paper: Proof, Exponentials and logarithms, Numerical methods, Vectors.
• Calculator: allowed.

H240/03: question by question

Labels and topics are ours, read from the H240/03 paper (checked 2026-11-01); marks as printed. Tap a number to practise that topic.

1. Q1Modulus equation (3 marks) · Algebra and functions
2. Q2Transforming y = x^3 (5 marks) · Algebra and functions
3. Q3Perpendicular bisector of two points (4 marks) · Coordinate geometry
4. Q4Arithmetic progression of integers (8 marks) · Sequences and series
5. Q5Tangent to a curve and an enclosed area (14 marks) · Differentiation, Integration
6. Q6Area between a line and an exponential curve (8 marks) · Integration, Exponentials and logarithms
7. Q7Equation with sec, then a quadratic in tan (8 marks) · Trigonometry
8. Q8Child dragging a sledge (2 marks) · Forces and Newton's laws
9. Q9Velocity-time graph (6 marks) · Kinematics
10. Q10Particle on a block, friction (8 marks) · Forces and Newton's laws
11. Q11Rod leaning on a support (7 marks) · Moments
12. Q12Motion with i and j vectors (13 marks) · Kinematics
13. Q13Ball projected in a vertical plane (14 marks) · Kinematics