OCR A Level Maths June 2018 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. Trigonometry · 6 marks

   Solve $3{\cot }^2\theta +5\text{cosec}\theta +1=0$ for $0^{\circ }\le \theta \le {360}^{\circ }$.

   Worked solution

   1. M1 for substituting ${\cot }^2\theta ={\text{cosec}}^2\theta -1$ to form $3({\text{cosec}}^2\theta -1)+5\text{cosec}\theta +1=0$.
   2. M1 for simplifying to the quadratic $3{\text{cosec}}^2\theta +5\text{cosec}\theta -2=0$.
   3. M1 for factorising to $(3\text{cosec}\theta -1)(\text{cosec}\theta +2)=0$.
   4. M1 for showing $\text{cosec}\theta =\frac{1}{3}⟹\sin \theta =3$ yields no real solutions.
   5. M1 for solving $\text{cosec}\theta =-2⟹\sin \theta =-0.5$.
   6. A1 for finding the two correct angles: ${210}^{\circ }$ and ${330}^{\circ }$.

2. Differentiation · 6 marks

   The tangents to the curve $y=x^2-4x$ at the points where $x=1$ and $x=3$ intersect at a point $P$. Find the coordinates of $P$.

   Worked solution

   1. M1 for finding the y-coordinates: at $x=1,y=-3$ and at $x=3,y=-3$.
   2. M1 for differentiating $\frac{dy}{dx}=2x-4$ to find the gradients: $m_1=-2$ (at $x=1$) and $m_3=2$ (at $x=3$).
   3. M1 for finding the equation of the first tangent: $y-(-3)=-2(x-1)⟹y=-2x-1$.
   4. M1 for finding the equation of the second tangent: $y-(-3)=2(x-3)⟹y=2x-9$.
   5. M1 for equating the two tangent equations: $-2x-1=2x-9⟹4x=8⟹x=2$.
   6. A1 for finding the corresponding y-coordinate $y=-5$, giving $P(2,-5)$.

3. Sequences and series · 4 marks

   The first three terms of the expansion $(a+bx{)}^6$ are $64-192x+c{x}^2$. Find the values of the constants $a$, $b$, and $c$.

   Worked solution

   1. M1 for equating the first term: $a^6=64⟹a=2$.
   2. M1 for equating the $x$ coefficient to find $b$: $(\genfrac{}{}{0ex}{}{6}{1})a^{5}b=-192⟹6(32)b=-192⟹192b=-192$.
   3. A1 for obtaining $b=-1$.
   4. A1 for finding the $x^2$ coefficient: $c=(\genfrac{}{}{0ex}{}{6}{2})a^4{b}^2=15(16)(-1{)}^2=240$.

More on these topics: Trigonometry · Differentiation · Sequences and series

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 4 to 13 marks.
• Biggest question: Q13, 13 marks: modelling an insect colony.
• Marks by main topic: Exponentials and logarithms 24, Algebra and functions 20, Differentiation 20, Sequences and series 16, Coordinate geometry 12, Numerical methods 4, Proof 4.
• More Exponentials and logarithms in other sittings: November 2020 Q6 (9 marks), November 2021 Q12 (13 marks), June 2022 Q8 (9 marks).
• Against the other 7 sittings we mapped: more marks on Exponentials and logarithms (24 marks here, 9 on average) and Coordinate geometry (12 marks here, 6 on average); fewer on Trigonometry (0 marks here, 14 on average) and Integration (0 marks here, 12 on average).
• Not on this paper: Trigonometry, Integration, Vectors.
• 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. Q1Line through two points (4 marks) · Coordinate geometry
2. Q2Trapezium rule with an exponential (4 marks) · Numerical methods, Exponentials and logarithms
3. Q3Quartic solved as a quadratic in x^2 (4 marks) · Algebra and functions
4. Q4Proof that an expression is odd (4 marks) · Proof
5. Q5Circle centre, radius and a line (8 marks) · Coordinate geometry
6. Q6Factor theorem and a cubic inequality (9 marks) · Algebra and functions
7. Q7Half-marathon times as a sequence (9 marks) · Sequences and series
8. Q8Binomial expansion with a negative power (7 marks) · Sequences and series
9. Q9Quadratic with a constant c and its maximum (7 marks) · Algebra and functions, Differentiation
10. Q10Parametric curve: gradient and a Cartesian equation (10 marks) · Differentiation, Coordinate geometry
11. Q11Exponential decay in an experiment (11 marks) · Exponentials and logarithms
12. Q12Normal to a trig curve and an area (10 marks) · Differentiation, Trigonometry
13. Q13Modelling an insect colony (13 marks) · Exponentials and logarithms, Integration