OCR A Level Maths June 2023 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 · 5 marks

   Solve the equation $\tan 2x=4\tan x$ for $0^{\circ }\le x\le {180}^{\circ }$. Give your answers to 1 decimal place where appropriate.

   Worked solution

   1. M1 for substituting the identity $\tan 2x=\frac{2\tan x}{1-{\tan }^{2}x}$.
   2. M1 for rearranging to $2\tan x=4\tan x(1-{\tan }^{2}x)$.
   3. M1 for grouping terms on one side and factorising to get $2\tan x(2{\tan }^{2}x-1)=0$.
   4. A1 for finding solutions from $\tan x=0$, giving $x=0^{\circ }$ and ${180}^{\circ }$.
   5. A1 for finding solutions from ${\tan }^{2}x=0.5⟹\tan x=\pm \frac{1}{\sqrt{2}}$, giving $x={35.3}^{\circ }$ and $x={144.7}^{\circ }$.

2. Proof · 6 marks

   Prove that $n^4-n^2$ is a multiple of 12 for every integer $n$.

   Worked solution

   Factorises: $n^4-n^2=n^2(n-1)(n+1)$ M1

   $(n-1)n(n+1)$ is a product of three consecutive integers, so one factor is a multiple of 3 B1

   Considers $n$ even and $n$ odd M1

   $n$ even: $n=2k$, so $n^2=4k^2$ is a multiple of 4 A1

   $n$ odd: $n-1$ and $n+1$ are both even, so $(n-1)(n+1)$ is a multiple of 4 A1

   The number is a multiple of 3 and of 4, which have no common factor, so it is a multiple of 12 A1 (cso)

3. Vectors · 7 marks

   Line $L_1$ has vector equation ${\mathbf{r}}_{\mathbf{1}}=\left(\begin{array}{c}1\\ 0\\ 1\end{array}\right)+\lambda \left(\begin{array}{c}2\\ 1\\ 1\end{array}\right)$ and line $L_2$ has equation ${\mathbf{r}}_{\mathbf{2}}=\left(\begin{array}{c}5\\ -3\\ 0\end{array}\right)+\mu \left(\begin{array}{c}-1\\ 2\\ 1\end{array}\right)$.

   1. Show that $L_1$ and $L_2$ intersect.
   2. Find the exact coordinates of the point of intersection.
   Worked solution

   1. M1 for equating the parametric equations: $1+2\lambda =5-\mu$, $\lambda =-3+2\mu$, and $1+\lambda =\mu$.
      M1 for taking two equations to solve for the parameters, e.g., $\lambda =\mu -1$ into $2\lambda +\mu =4$.
      M1 for $2(\mu -1)+\mu =4⟹3\mu -2=4⟹3\mu =6$.
      A1 for finding $\mu =2$ and $\lambda =1$.
      B1 for verifying with the unused equation: $\lambda -2\mu =1-4=-3$, which matches the second component equation, showing the lines intersect.
   2. M1 for substituting $\lambda =1$ into $L_1$ (or $\mu =2$ into $L_2$).
      A1 for the intersection point $(3,1,2)$.

More on these topics: Trigonometry · Proof · Vectors

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 5 to 12 marks.
• Biggest question: Q11, 12 marks: online sales modelled by S = ab^t.
• Marks by main topic: Algebra and functions 22, Differentiation 16, Trigonometry 14, Exponentials and logarithms 12, Integration 10, Sequences and series 9, Numerical methods 9, Vectors 8.
• More Exponentials and logarithms in other sittings: June 2022 Q8 (9 marks), November 2021 Q12 (13 marks), June 2025 Q10 (12 marks).
• Against the other 7 sittings we mapped: more marks on Numerical methods (9 marks here, 2 on average) and Vectors (8 marks here, 3 on average); fewer on Coordinate geometry (0 marks here, 8 on average) and Proof (0 marks here, 5 on average).
• Not on this paper: Proof, Coordinate geometry.
• 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. Q1Cosine rule and triangle area (5 marks) · Trigonometry
2. Q2Rationalising surd fractions (8 marks) · Algebra and functions
3. Q3Differentiation from first principles for a quadratic (7 marks) · Differentiation
4. Q4Vectors and a quadrilateral (8 marks) · Vectors
5. Q5Linear functions, inverse and composite (8 marks) · Algebra and functions
6. Q6Stationary points of e^x + 3x (9 marks) · Differentiation, Exponentials and logarithms
7. Q7Proving compound-angle results (9 marks) · Trigonometry
8. Q8Binomial expansion and its validity (9 marks) · Sequences and series
9. Q9Model for bees and wildflowers (6 marks) · Algebra and functions
10. Q10Locating a root and an iteration (9 marks) · Numerical methods, Exponentials and logarithms
11. Q11Online sales modelled by S = ab^t (12 marks) · Exponentials and logarithms
12. Q12Integration by substitution with e^x (10 marks) · Integration, Exponentials and logarithms