OCR A Level Maths November 2020 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 $2{\sin }^{2}x+3\cos x-3=0$ for $0^{\circ }\le x\le {360}^{\circ }$.

   Worked solution

   1. M1 for substituting ${\sin }^{2}x$ with $1-{\cos }^{2}x$ to get $2(1-{\cos }^{2}x)+3\cos x-3=0$.
   2. M1 for expanding and rearranging to form the quadratic $2{\cos }^{2}x-3\cos x+1=0$.
   3. M1 for factorising the quadratic as $(2\cos x-1)(\cos x-1)=0$.
   4. A1 for finding the solutions from $\cos x=0.5$ giving $x={60}^{\circ }$ and ${300}^{\circ }$.
   5. A1 for finding the solutions from $\cos x=1$ giving $x=0^{\circ }$ and ${360}^{\circ }$.

2. Integration · 4 marks

   Find $\int (\cos x-\sin x{)}^{2}dx$.

   Worked solution

   1. M1 for expanding the bracket to ${\cos }^{2}x-2\sin x\cos x+{\sin }^{2}x$.
   2. M1 for simplifying to $1-\sin 2x$.
   3. M1 for integrating the expression term by term.
   4. A1 for $x+\frac{1}{2}\cos 2x+c$.

3. Kinematics · 5 marks

   A particle moving with constant acceleration passes three points $A$, $B$ and $C$, in that order, on a straight line. The distance $AB$ is $20\text{ m}$ and the distance $AC$ is $84\text{ m}$. The speed of the particle at $B$ is $12{\text{ m s}}^{-1}$ and its speed at $C$ is $20{\text{ m s}}^{-1}$. Find the acceleration of the particle and its speed at $A$.

   Worked solution

   1. M1 for $BC=84-20=64\text{ m}$.
   2. M1 for $v^2=u^2+2as$ from $B$ to $C$: ${20}^2={12}^2+2a(64)$.
   3. A1 for $128a=256\Rightarrow a=2{\text{ m s}}^{-2}$.
   4. M1 for $v^2=u^2+2as$ from $A$ to $B$: ${12}^2=u_A^2+2(2)(20)$.
   5. A1 for $u_A^2=64\Rightarrow u_A=8{\text{ m s}}^{-1}$.

More on these topics: Trigonometry · Integration · 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 13 marks.
• Biggest question: Q9, 13 marks: particles connected over a pulley.
• Marks by main topic: Kinematics 26, Differentiation 22, Forces and Newton's laws 13, Integration 12, Algebra and functions 11, Moments 11, Exponentials and logarithms 3, Trigonometry 2.
• More Forces and Newton's laws in other sittings: November 2021 Q14 (11 marks), June 2019 Q9 (9 marks), June 2022 Q10 (8 marks).
• Against the other 7 sittings we mapped: more marks on Differentiation (22 marks here, 8 on average) and Integration (12 marks here, 6 on average); fewer on Trigonometry (2 marks here, 10 on average) and Forces and Newton's laws (13 marks here, 19 on average).
• Not on this paper: Proof, Coordinate geometry, Sequences and series, 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. Q1Area of a triangle (2 marks) · Trigonometry
2. Q2Transforming y = e^x (3 marks) · Exponentials and logarithms, Algebra and functions
3. Q3Composite functions and an inequality (11 marks) · Algebra and functions
4. Q4Stationary point of a curve with ln(k - 3x) (11 marks) · Differentiation, Exponentials and logarithms
5. Q5Area under a parametric curve with e^(-2t) (12 marks) · Integration, Coordinate geometry
6. Q6Implicit differentiation and tangents (11 marks) · Differentiation
7. Q7Constant acceleration with vectors (6 marks) · Kinematics
8. Q8Car velocity with calculus (7 marks) · Kinematics
9. Q9Particles connected over a pulley (13 marks) · Forces and Newton's laws
10. Q10Moments on a wall-mounted light (11 marks) · Moments
11. Q11Projectile in a vertical plane (13 marks) · Kinematics