OCR A Level Maths June 2024 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

   Consider the rational function $f(x)=\frac{3x}{(x-1)(x+2)}$.

   1. Express $f(x)$ in partial fractions.
   2. Hence, evaluate exactly the definite integral ${\int }_2^{3}f(x)dx$, giving your answer in the form $\ln (k)$ where $k$ is a rational number.
   Worked solution

   1. M1 for setting up the partial fraction decomposition: $\frac{3x}{(x-1)(x+2)}=\frac{A}{x-1}+\frac{B}{x+2}$.
      M1 for clearing the denominators: $3x=A(x+2)+B(x-1)$.
      M1 for substituting strategic values of $x$ (e.g., $x=1$ and $x=-2$) to solve for constants.
      A1 for finding $A=1$ and $B=2$, yielding $\frac{1}{x-1}+\frac{2}{x+2}$.
   2. M1 for setting up the integral and determining the anti-derivatives: ${[\ln |x-1|+2\ln |x+2|]}_2^3$.
      M1 for substituting the limits: $(\ln 2+2\ln 5)-(\ln 1+2\ln 4)$.
      A1 for using logarithm laws to combine into a single logarithm: $\ln 2+\ln 25-0-\ln 16=\ln \left(\frac{50}{16}\right)=\ln \left(\frac{25}{8}\right)$.

2. Integration · 7 marks

   Find the exact total area of the finite regions bounded by the curves $y=x^3+x^2$ and $y=2x^2+2x$.

   Worked solution

   1. M1 for setting $x^3+x^2=2x^2+2x⟹x^3-x^2-2x=0$.
   2. M1 for factoring $x(x-2)(x+1)=0$ to find the intersection points $x=-1,0,2$.
   3. M1 for recognising two distinct regions and setting up integrals: ${\int }_{-1}^0(x^3-x^2-2x)dx$ and ${\int }_0^2(2x^2+2x-x^3-x^2)dx$.
   4. M1 for integrating the generic polynomial to $\pm [\frac{x^4}{4}-\frac{x^3}{3}-x^2]$.
   5. M1 for evaluating the area of the first region to $\frac{5}{12}$.
   6. M1 for evaluating the area of the second region to $\frac{8}{3}$.
   7. A1 for calculating the combined total area: $\frac{5}{12}+\frac{8}{3}=\frac{37}{12}$.

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 4 to 13 marks.
• Biggest question: Q5, 13 marks: curve with ln x: a point of inflection.
• Marks by main topic: Forces and Newton's laws 26, Differentiation 20, Kinematics 16, Trigonometry 14, Sequences and series 7, Moments 7, Coordinate geometry 6, Algebra and functions 4.
• More Differentiation in other sittings: June 2022 Q5 (14 marks), November 2021 Q8 (11 marks), November 2020 Q4 (11 marks).
• Against the other 7 sittings we mapped: more marks on Differentiation (20 marks here, 9 on average) and Forces and Newton's laws (26 marks here, 17 on average); fewer on Kinematics (16 marks here, 26 on average) and Algebra and functions (4 marks here, 13 on average).
• Not on this paper: Proof, Exponentials and logarithms, Integration, 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. Q1Simplifying index and algebraic expressions (4 marks) · Algebra and functions
2. Q2Sector with radius in x (5 marks) · Trigonometry
3. Q3Binomial expansion with a negative power (7 marks) · Sequences and series
4. Q4Equation with cot and cosec (9 marks) · Trigonometry
5. Q5Curve with ln x: a point of inflection (13 marks) · Differentiation, Exponentials and logarithms
6. Q6Parametric curve with a constant k (6 marks) · Coordinate geometry
7. Q7Implicit curve and tangents (7 marks) · Differentiation
8. Q8Constant acceleration with vectors (6 marks) · Kinematics
9. Q9Resultant of two horizontal forces (8 marks) · Forces and Newton's laws
10. Q10Blocks connected by a string (6 marks) · Forces and Newton's laws
11. Q11Lamina in equilibrium (7 marks) · Moments
12. Q12Variable velocity with calculus (10 marks) · Kinematics
13. Q13Particle on a line of greatest slope (12 marks) · Forces and Newton's laws, Kinematics

What the examiners said about this paper

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

• H240/03: "Detailed reasoning" (Q2, Q4); Binomial expansion (rational n) (Q3); Points of inflection (Q5). Official examiner report ↗

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