OCR A Level Maths June 2023 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. Differentiation · 6 marks

   The normal to the curve $y=\sqrt{x}$ at the point where $x=4$ cuts the x-axis at $A$ and the y-axis at $B$. Calculate the exact area of the triangle $AOB$, where $O$ is the origin.

   Worked solution

   1. M1 for finding the y-coordinate at $x=4$: $y=2$.
   2. M1 for finding $\frac{dy}{dx}=\frac{1}{2}{x}^{-\frac{1}{2}}$ and evaluating at $x=4$ to get gradient $m_t=\frac{1}{4}$.
   3. M1 for finding the normal gradient $m_n=-4$ and its equation: $y-2=-4(x-4)⟹y=-4x+18$.
   4. M1 for finding the axis intercepts of the normal: at $y=0,4x=18⟹x=4.5$ (point $A$) and at $x=0,y=18$ (point $B$).
   5. M1 for using the area of a triangle formula $\frac{1}{2}\times \text{base}\times \text{height}=\frac{1}{2}\times 4.5\times 18$.
   6. A1 for the final area of $40.5$ square units.

2. Sequences and series · 5 marks

   The binomial expansion of $(a+bx{)}^{-2}$, where $a>0$, begins $\frac{1}{9}-\frac{2}{27}x+\dots$. Find the values of the constants $a$ and $b$.

   Worked solution

   1. M1 for writing $(a+bx{)}^{-2}=a^{-2}(1+\frac{b}{a}x{)}^{-2}$.
   2. M1 for equating the constant term: $a^{-2}=\frac{1}{9}$.
   3. A1 for finding $a=3$ (since $a>0$).
   4. M1 for equating the $x$ coefficient: $a^{-2}(-2)(\frac{b}{a})=-\frac{2}{27}$.
   5. A1 for substituting $a=3$ to get $\frac{1}{9}(\frac{-2b}{3})=-\frac{2}{27}⟹-\frac{2b}{27}=-\frac{2}{27}⟹b=1$.

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: Differentiation · Sequences and series · 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 3 to 13 marks.
• Biggest question: Q12, 13 marks: projectile with g = 10.
• Marks by main topic: Kinematics 29, Forces and Newton's laws 25, Trigonometry 14, Algebra and functions 8, Moments 8, Coordinate geometry 7, Sequences and series 6, Exponentials and logarithms 3.
• More Kinematics in other sittings: June 2022 Q13 (14 marks), June 2024 Q12 (10 marks), November 2021 Q13 (13 marks).
• Against the other 7 sittings we mapped: more marks on Forces and Newton's laws (25 marks here, 17 on average) and Trigonometry (14 marks here, 9 on average); fewer on Differentiation (0 marks here, 12 on average) and Integration (0 marks here, 7 on average).
• Not on this paper: Proof, Differentiation, 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. Q1Exponential equation solved with logs (3 marks) · Exponentials and logarithms
2. Q2R sin(x - a) form (5 marks) · Trigonometry
3. Q3Cubic with constants p and q (8 marks) · Algebra and functions
4. Q4Circle with a constant k (7 marks) · Coordinate geometry
5. Q5Emblem design with sectors (9 marks) · Trigonometry
6. Q6Arithmetic progression terms (6 marks) · Sequences and series
7. Q7Car velocity with calculus (12 marks) · Kinematics
8. Q8Constant acceleration with vectors (4 marks) · Kinematics
9. Q9Block on a rough inclined plane (6 marks) · Forces and Newton's laws
10. Q10Particle under two constant forces (7 marks) · Forces and Newton's laws, Kinematics
11. Q11Rod resting against a wall (8 marks) · Moments
12. Q12Projectile with g = 10 (13 marks) · Kinematics
13. Q13Block and particle connected over a pulley (12 marks) · Forces and Newton's laws