OCR A Level Maths June 2018 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 · 5 marks

   Express $\frac{x+2}{x^2-5x+6}-\frac{x-1}{x^2-4x+3}$ as a single fraction in its simplest form.

   Worked solution

   1. M1 for factorising $x^2-5x+6$ to $(x-2)(x-3)$.
   2. M1 for factorising $x^2-4x+3$ to $(x-1)(x-3)$.
   3. M1 for simplifying the second fraction: $\frac{x-1}{(x-1)(x-3)}$ cancels to $\frac{1}{x-3}$.
   4. M1 for creating a common denominator $(x-2)(x-3)$ and subtracting: $\frac{x+2}{(x-2)(x-3)}-\frac{x-2}{(x-2)(x-3)}$.
   5. A1 for reaching the final simplified answer $\frac{4}{(x-2)(x-3)}$.

2. Coordinate geometry · 8 marks

   A curve $C$ has parametric equations $x=t^2$, $y=t^3-3t$.
   (a) Find ${\displaystyle \frac{dy}{dx}}$ in terms of $t$. (3)
   (b) Find the coordinates of the points on $C$ where the tangent is horizontal. (3)
   (c) Find the Cartesian equation of the tangent at $t=2$. (2)

   Worked solution

   (a) $\frac{dx}{dt}=2t$, $\frac{dy}{dt}=3t^2-3$   M1 A1. $\frac{dy}{dx}=\frac{3t^2-3}{2t}=\frac{3(t^2-1)}{2t}$   A1.
   (b) Horizontal iff $\frac{dy}{dt}=0$: $3t^2=3\Rightarrow t=\pm 1$   M1. $t=1$: $(1,-2)$; $t=-1$: $(1,2)$   A1 A1.
   (c) At $t=2$: $(x,y)=(4,2)$; gradient $=\frac{3\cdot 3}{4}=\frac{9}{4}$; equation: $y-2=\frac{9}{4}(x-4)$, i.e. $y=\frac{9}{4}x-7$   M1 A1.

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 · Coordinate geometry · Kinematics

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 3 to 14 marks.
• Biggest question: Q12, 14 marks: connected particles over a pulley.
• Marks by main topic: Forces and Newton's laws 25, Algebra and functions 17, Kinematics 16, Numerical methods 13, Integration 9, Moments 9, Trigonometry 8, Coordinate geometry 3.
• More Forces and Newton's laws in other sittings: June 2019 Q9 (9 marks), November 2020 Q9 (13 marks), November 2021 Q14 (11 marks).
• Against the other 7 sittings we mapped: more marks on Numerical methods (13 marks here, 1 on average) and Forces and Newton's laws (25 marks here, 17 on average); fewer on Differentiation (0 marks here, 12 on average) and Kinematics (16 marks here, 26 on average).
• Not on this paper: Proof, Sequences and series, Exponentials and logarithms, Differentiation, 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. Q1Circle centre and radius (3 marks) · Coordinate geometry
2. Q2Modulus equation (3 marks) · Algebra and functions
3. Q3Flower bed design with inequalities (6 marks) · Algebra and functions
4. Q4Composite and inverse functions (8 marks) · Algebra and functions
5. Q5Trapezium rule and an exact integral (13 marks) · Numerical methods, Integration
6. Q6Equation with sin and cos of a compound angle (8 marks) · Trigonometry
7. Q7Separable differential equation for a curve (9 marks) · Integration
8. Q8Particle acted on by a force vector (6 marks) · Kinematics, Forces and Newton's laws
9. Q9Plank resting on two supports (9 marks) · Moments
10. Q10Three forces in equilibrium (11 marks) · Forces and Newton's laws
11. Q11Variable velocity of a car (10 marks) · Kinematics
12. Q12Connected particles over a pulley (14 marks) · Forces and Newton's laws