OCR A Level Maths Specimen Paper 3: question paper (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. Differentiation · 5 marks

   A curve has parametric equations $x=\ln t$ and $y=t^2$, $t>0$. Find the equation of the normal to the curve at the point where $t=1$. Give your answer in the form $ax+by+c=0$.

   Worked solution

   1. M1 for finding the coordinates at $t=1$: $x=0$, $y=1$.
   2. M1 for calculating $\frac{dx}{dt}=\frac{1}{t}$ and $\frac{dy}{dt}=2t$, giving $\frac{dy}{dx}=2t^2$.
   3. M1 for evaluating the tangent gradient at $t=1$ to get $m_t=2$.
   4. M1 for determining the normal gradient $m_n=-\frac{1}{2}$ and using $(0,1)$ to form the equation $y-1=-\frac{1}{2}(x-0)$.
   5. A1 for rearranging to the required integer format: $x+2y-2=0$.

2. Sequences and series · 5 marks

   The second term of a geometric series is 12 and its sum to infinity is 64. Find the two possible values of the common ratio, $r$, and the corresponding values of the first term, $a$.

   Worked solution

   1. M1 for setting up the two equations: $ar=12$ and $\frac{a}{1-r}=64$.
   2. M1 for substituting $a=\frac{12}{r}$ to get $\frac{12/r}{1-r}=64⟹12=64r(1-r)$.
   3. M1 for forming the quadratic $16r^2-16r+3=0$.
   4. A1 for finding $r=0.25$ and $r=0.75$.
   5. A1 for finding the corresponding first terms: $a=48$ and $a=16$.

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: Differentiation · Sequences and series · Kinematics

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 14 marks.
• Biggest question: Q12, 14 marks: netball projectile.
• Marks by main topic: Kinematics 23, Forces and Newton's laws 19, Trigonometry 14, Integration 10, Sequences and series 8, Moments 8, Algebra and functions 7, Differentiation 7, Numerical methods 4.
• More Kinematics in other sittings: June 2018 Q11 (10 marks), June 2019 Q10 (13 marks), November 2020 Q11 (13 marks).
• Against the other 8 sittings we mapped: more marks on Sequences and series (8 marks here, 3 on average) and Trigonometry (14 marks here, 9 on average); fewer on Algebra and functions (7 marks here, 12 on average).
• Not on this paper: Proof, Coordinate geometry, Exponentials and logarithms, 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 and inequalities (7 marks) · Algebra and functions
2. Q2Trapezium rule estimate (4 marks) · Numerical methods
3. Q3Exact value from a trig equation (4 marks) · Trigonometry
4. Q4Small-angle approximation (4 marks) · Trigonometry
5. Q5Binomial expansion with two constants (8 marks) · Sequences and series
6. Q6Curve with a minimum point (7 marks) · Differentiation
7. Q7Integrating a fraction and tan^2 (10 marks) · Integration, Trigonometry
8. Q8Exact trig values from a triangle (6 marks) · Trigonometry
9. Q9Resultant of two forces (4 marks) · Forces and Newton's laws
10. Q10Body on a rough inclined plane (7 marks) · Forces and Newton's laws
11. Q11Motion with i and j vectors (9 marks) · Kinematics
12. Q12Netball projectile (14 marks) · Kinematics
13. Q13Connected particles on two planes (8 marks) · Forces and Newton's laws
14. Q14Ladder on rough ground (8 marks) · Moments, Forces and Newton's laws