OCR A Level Maths November 2020 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. Coordinate geometry · 8 marks

   Three points define the vertices of a triangle: $A(-2,-1)$, $B(4,7)$, and $C(8,4)$.

   1. By calculating the gradients of $[AB]$ and $[BC]$, prove that triangle $ABC$ is a right-angled triangle.

   2. Calculate the exact lengths of $[AB]$ and $[BC]$.

   3. Hence, find the exact area of the triangle $ABC$.

   Worked solution

   1. Gradient $m_{AB}=\frac{7-(-1)}{4-(-2)}=\frac{8}{6}=\frac{4}{3}$. A1
      Gradient $m_{BC}=\frac{4-7}{8-4}=\frac{-3}{4}$. A1
      Since $m_{AB}\times m_{BC}=\frac{4}{3}\times -\frac{3}{4}=-1$, $[AB]$ and $[BC]$ are perpendicular. M1
      Thus, triangle $ABC$ has a right angle at $B$. B1

   2. Length $AB=\sqrt{(4-(-2){)}^2+(7-(-1){)}^2}=\sqrt{36+64}=\sqrt{100}=10$. A1
      Length $BC=\sqrt{(8-4{)}^2+(4-7{)}^2}=\sqrt{16+9}=\sqrt{25}=5$. A1

   3. Since the triangle is right-angled at $B$, $AB$ and $BC$ act as the base and height. M1
      Area $=\frac{1}{2}\times \text{base}\times \text{height}=\frac{1}{2}\times 10\times 5=25$ sq units. A1

2. Sequences and series · 5 marks

   The first, third, and seventh terms of an arithmetic sequence (with a non-zero common difference) form three consecutive terms of a geometric sequence. Show that the first term is twice the common difference ($a=2d$).

   Worked solution

   1. M1 for writing the 1st, 3rd, and 7th terms as $a,a+2d,a+6d$.
   2. M1 for setting up the geometric ratio condition: $\frac{a+2d}{a}=\frac{a+6d}{a+2d}$.
   3. M1 for cross-multiplying and expanding: $(a+2d{)}^2=a(a+6d)⟹a^2+4ad+4d^2=a^2+6ad$.
   4. M1 for rearranging to $4d^2-2ad=0⟹2d(2d-a)=0$.
   5. A1 for concluding that since $d\ne 0$, $2d-a=0⟹a=2d$.

3. Proof · 4 marks

   Prove by contradiction that if $a$ and $b$ are integers, then $a^2-4b\ne 2$.

   Worked solution

   1. M1 for assuming that there exist integers $a$ and $b$ such that $a^2-4b=2$, which rearranges to $a^2=4b+2=2(2b+1)$.
   2. M1 for deducing that since $a^2$ is an even number, $a$ must also be an even number.
   3. M1 for letting $a=2k$, substituting to get $(2k{)}^2=4b+2⟹4k^2=4b+2$, and dividing by $2$ to get $2k^2=2b+1⟹2(k^2-b)=1$.
   4. A1 for observing that the left-hand side $2(k^2-b)$ is even while the right-hand side $1$ is odd, which is a contradiction, so $a^2-4b\ne 2$.

More on these topics: Coordinate geometry · Sequences and series · Proof

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 3 to 11 marks.
• Biggest question: Q7, 11 marks: two revision plans as arithmetic and geometric series.
• Marks by main topic: Integration 20, Algebra and functions 17, Differentiation 17, Sequences and series 11, Coordinate geometry 10, Exponentials and logarithms 9, Vectors 8, Trigonometry 5, Proof 3.
• More Sequences and series in other sittings: November 2021 Q6 (9 marks), June 2019 Q7 (8 marks), June 2022 Q6 (8 marks).
• Against the other 7 sittings we mapped: more marks on Integration (20 marks here, 10 on average) and Vectors (8 marks here, 3 on average); fewer on Trigonometry (5 marks here, 14 on average).
• Not on this paper: 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. Q1Small-angle approximation (5 marks) · Trigonometry
2. Q2Simplifying index expressions (8 marks) · Algebra and functions
3. Q3Minimising the surface area of a tin (10 marks) · Differentiation
4. Q4Proof by contradiction about multiples of 5 (3 marks) · Proof
5. Q5Vectors and a ratio on a line (8 marks) · Vectors
6. Q6Phone sales modelled by S = ab^t (9 marks) · Exponentials and logarithms
7. Q7Two revision plans as arithmetic and geometric series (11 marks) · Sequences and series
8. Q8Product rule with e^(2x) and an increasing function (7 marks) · Differentiation, Exponentials and logarithms
9. Q9Modulus graph and an equation (9 marks) · Algebra and functions
10. Q10Rectangles to bound an integral of a sine curve (11 marks) · Integration, Numerical methods
11. Q11Circle and a tangent from a point (10 marks) · Coordinate geometry
12. Q12Differential equation solved with partial fractions (9 marks) · Integration, Algebra and functions