OCR A Level Maths June 2019 Paper 1: question paper and mark scheme (H240/01)

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

   When $f(x)=2x^3+p{x}^2+qx+1$ is divided by $(2x-1)$ the remainder is 2. When $f(x)$ is divided by $(x+1)$ the remainder is -4. Find the values of $p$ and $q$.

   Worked solution

   1. M1 for applying the Remainder Theorem with $x=0.5$: $f(0.5)=2⟹2(\frac{1}{8})+\frac{p}{4}+\frac{q}{2}+1=2$.
   2. M1 for simplifying the first equation: $\frac{1}{4}+\frac{p}{4}+\frac{2q}{4}=1⟹p+2q=3$.
   3. M1 for applying the Remainder Theorem with $x=-1$: $f(-1)=-4⟹-2+p-q+1=-4⟹p-q=-3$.
   4. M1 for solving the simultaneous equations (e.g., subtracting them gives $3q=6$).
   5. A1 for stating $q=2$ and $p=-1$.

2. Trigonometry · 5 marks

   Two identical circles of radius $8$ cm intersect such that the circumference of each passes through the centre of the other. Find the exact area of the overlapping region.

   Worked solution

   1. M1 for recognising that lines drawn from the intersection points to the centres form two equilateral triangles, hence the central angle in each circle covering the overlap is $\frac{2\pi }{3}$.
   2. M1 for establishing the overlapping region consists of two identical segments, so Area $=2\times \frac{1}{2}(8^2)(\frac{2\pi }{3}-\sin \frac{2\pi }{3})$.
   3. M1 for substituting $\sin \frac{2\pi }{3}=\frac{\sqrt{3}}{2}$.
   4. M1 for expanding the expression $64(\frac{2\pi }{3}-\frac{\sqrt{3}}{2})$.
   5. A1 for the exact simplified area $\frac{128\pi }{3}-32\sqrt{3}{\text{ cm}}^2$.

3. Numerical methods · 6 marks

   The equation $\cos x=x$ has a unique root $\alpha$ in $[0,1]$. Using the iteration $x_{n+1}=\cos x_n$ with $x_0=0.5$, find $x_1$, $x_2$, $x_3$, and $x_4$ each to $4$ d.p. (Angle in radians.)

   Worked solution

   $x_1=\cos 0.5=0.8776$   M1 A1. $x_2=\cos 0.8776=0.6390$   A1. $x_3=\cos 0.6390=0.8027$   A1. $x_4=\cos 0.8027=0.6948$   A1 A1.

More on these topics: Algebra and functions · Trigonometry · Numerical methods

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 4 to 12 marks.
• Biggest question: Q12, 12 marks: differentiating a^(3x).
• Marks by main topic: Differentiation 26, Trigonometry 22, Integration 18, Sequences and series 15, Algebra and functions 11, Vectors 8.
• More Differentiation in other sittings: June 2018 Q10 (10 marks), November 2020 Q3 (10 marks), November 2021 Q7 (9 marks).
• Against the other 7 sittings we mapped: more marks on Differentiation (26 marks here, 14 on average) and Trigonometry (22 marks here, 11 on average); fewer on Exponentials and logarithms (0 marks here, 12 on average) and Algebra and functions (11 marks here, 21 on average).
• Not on this paper: Proof, Coordinate geometry, Exponentials and logarithms, Numerical methods.
• Calculator: allowed.

H240/01: question by question

Labels and topics are ours, read from the H240/01 paper (checked 2026-11-01); marks as printed. Tap a number to practise that topic.

1. Q1Quadratic inequality (4 marks) · Algebra and functions
2. Q2Vector from a magnitude and a bearing (8 marks) · Vectors
3. Q3Restricted domain and an inverse function (7 marks) · Algebra and functions
4. Q4Salary rises as a geometric series (7 marks) · Sequences and series
5. Q5Implicit differentiation and a stationary point (8 marks) · Differentiation
6. Q6Differentiation from first principles for a cubic (6 marks) · Differentiation
7. Q7Sequence with powers of 0.6 (8 marks) · Sequences and series
8. Q8Differential equation for a draining tank (8 marks) · Integration
9. Q9R cos(3x - a) form and an equation (11 marks) · Trigonometry
10. Q10Sector: arc, area and an angle (11 marks) · Trigonometry
11. Q11Integration by parts with ln(x - 4) (10 marks) · Integration, Exponentials and logarithms
12. Q12Differentiating a^(3x) (12 marks) · Differentiation, Exponentials and logarithms