OCR A Level Maths June 2019 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. Trigonometry · 5 marks

   Solve ${\text{cosec}}^{2}x+{\cot }^{2}x=3$ for $0^{\circ }\le x\le {360}^{\circ }$.

   Worked solution

   1. M1 for using the identity ${\text{cosec}}^{2}x=1+{\cot }^{2}x$ to obtain $1+2{\cot }^{2}x=3$.
   2. M1 for rearranging to ${\cot }^{2}x=1$.
   3. M1 for taking the square root to find $\tan x=\pm 1$.
   4. A1 for finding the principal values ${45}^{\circ }$ and ${135}^{\circ }$.
   5. A1 for stating all four correct solutions: ${45}^{\circ },{135}^{\circ },{225}^{\circ },{315}^{\circ }$.

2. Proof · 4 marks

   Prove by contradiction that there are infinitely many prime numbers.

   Worked solution

   1. M1 for assuming that there is a finite number of prime numbers, listing them as $p_1,p_2,\dots ,p_n$.
   2. M1 for constructing a new number $N=(p_1\times p_2\times \cdots \times p_n)+1$.
   3. M1 for arguing that dividing $N$ by any of the primes $p_i$ leaves a remainder of $1$, meaning $N$ has no prime factors from the finite list.
   4. A1 for concluding that $N$ must either be a new prime itself or have a prime factor not in the list, contradicting the assumption that the list contains all primes.

3. Kinematics · 6 marks

   A drone is programmed to fly in a straight line. Its position vector $r$ (in km) at time $t$ (in hours) is $r=\left(\begin{array}{c}-5\\ 12\end{array}\right)+t\left(\begin{array}{c}8\\ -6\end{array}\right)$.
   Calculate the exact speed of the drone in km h${}^{-1}$, and find its distance from the origin at $t=2$.

   Worked solution

   Velocity vector $v=\left(\begin{array}{c}8\\ -6\end{array}\right)$. Speed $=|v|=\sqrt{8^2+(-6{)}^2}=\sqrt{64+36}=10$ km h${}^{-1}$. M1A1
   Position at $t=2$: $r(2)=\left(\begin{array}{c}-5\\ 12\end{array}\right)+2\left(\begin{array}{c}8\\ -6\end{array}\right)=\left(\begin{array}{c}11\\ 0\end{array}\right)$. M1A1
   Distance from origin $=\sqrt{{11}^2+0^2}=11$ km. M1A1

More on these topics: Trigonometry · Proof · Kinematics

What this paper covers

What this paper is like

• Shape: 11 questions, 100 marks in 2 h (about 1.2 minutes a mark); questions from 2 to 14 marks.
• Biggest question: Q4, 14 marks: curve y = 3x sin 2x: stationary point and area.
• Marks by main topic: Kinematics 27, Algebra and functions 17, Differentiation 14, Moments 14, Trigonometry 11, Forces and Newton's laws 9, Coordinate geometry 8.
• More Differentiation in other sittings: November 2020 Q4 (11 marks), November 2021 Q8 (11 marks), June 2022 Q5 (14 marks).
• Against the other 7 sittings we mapped: more marks on Moments (14 marks here, 8 on average) and Algebra and functions (17 marks here, 11 on average); fewer on Forces and Newton's laws (9 marks here, 20 on average) and Integration (0 marks here, 7 on average).
• Not on this paper: Proof, Sequences and series, 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. Q1Sine rule in a triangle (2 marks) · Trigonometry
2. Q2Circle centre, radius and tangents (8 marks) · Coordinate geometry
3. Q3Modulus inequality and graph transformations (7 marks) · Algebra and functions
4. Q4Curve y = 3x sin 2x: stationary point and area (14 marks) · Differentiation, Integration
5. Q5Proving a cot and cosec identity, then solving (9 marks) · Trigonometry
6. Q6Partial fractions and an exact area (10 marks) · Algebra and functions, Integration
7. Q7Cyclist with uniform acceleration (4 marks) · Kinematics
8. Q8Projectile landing after 2.4 seconds (10 marks) · Kinematics
9. Q9Block and particle connected by a string (9 marks) · Forces and Newton's laws
10. Q10Motion with i and j vectors (13 marks) · Kinematics
11. Q11Ladder in limiting equilibrium (14 marks) · Moments, Forces and Newton's laws