AQA A Level Maths June 2018 Paper 2: question paper and mark scheme (7357/2)

3 A Level questions like these, with worked solutions

Written and checked by us, not taken from the AQA paper. Try each one, then tap "Worked solution" to see every step and its marks.

1. 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 $P(2,-5)$.

2. Exponentials and logarithms · 5 marks

   The variables $x$ and $y$ satisfy the equation $y=a{b}^x$. A graph of $\ln y$ against $x$ is plotted and forms a straight line passing through $(0,2)$ and $(4,10)$. Find the exact values of the constants $a$ and $b$.

   Worked solution

   1. M1 for taking natural logarithms of both sides to get $\ln y=\ln a+x\ln b$.
   2. M1 for calculating the gradient of the line: $\ln b=\frac{10-2}{4-0}=2$.
   3. A1 for solving to find exactly $b=e^2$.
   4. M1 for identifying the y-intercept from the coordinate $(0,2)$, so $\ln a=2$.
   5. A1 for solving to find exactly $a=e^2$.

3. Kinematics · 5 marks

   Car A travels along a straight road at a constant speed of $25{\text{ m s}}^{-1}$. Car B starts from rest at the same point and accelerates uniformly at $3{\text{ m s}}^{-2}$ until it reaches a speed of $30{\text{ m s}}^{-1}$, after which it maintains this speed. Find the distance Car B travels before it catches up to Car A.

   Worked solution

   1. M1 for finding the time taken for Car B to reach $30{\text{ m s}}^{-1}$: $t=\frac{30}{3}=10\text{ s}$.
   2. M1 for finding the distances covered in the first $10\text{ s}$: Car B covers $\frac{1}{2}\times 10\times 30=150\text{ m}$. Car A covers $25\times 10=250\text{ m}$.
   3. M1 for setting up an equation for the remaining distance where Car B catches A. Let $t_2$ be the additional time: $150+30t_2=250+25t_2$.
   4. M1 for solving to find $t_2$: $5t_2=100⟹t_2=20\text{ s}$. Total time = $30\text{ s}$.
   5. A1 for finding the total distance: $25\times 30=750\text{ m}$.

More on these topics: Differentiation · Exponentials and logarithms · Kinematics

What this paper covers

What this paper is like

• Shape: 17 questions, 100 marks in 2 h (about 1.2 minutes a mark); questions from 1 to 14 marks.
• Biggest question: Q9, 14 marks: market trader's sales model and rates.
• Marks by main topic: Differentiation 29, Forces and Newton's laws 22, Kinematics 21, Trigonometry 10, Algebra and functions 6, Vectors 6, Proof 3, Sequences and series 1, Integration 1, Moments 1.
• More Differentiation in other sittings: November 2020 Q8 (10 marks), November 2021 Q2 (1 marks), June 2022 Q2 (1 marks).
• Against the other 5 sittings we mapped: more marks on Differentiation (29 marks here, 6 on average) and Forces and Newton's laws (22 marks here, 15 on average); fewer on Exponentials and logarithms (0 marks here, 8 on average) and Moments (1 marks here, 7 on average).
• Not on this paper: Coordinate geometry, Exponentials and logarithms, Numerical methods.
• Calculator: allowed.

7357/2: question by question

Labels and topics are ours, read from the 7357/2 paper (checked 2026-11-01); marks as printed. Tap a number to practise that topic.

1. Q1Implication and equivalence (tick one) (1 marks) · Proof
2. Q2Binomial coefficient (multiple choice) (1 marks) · Sequences and series
3. Q3Total shaded area for y = x^3 (multiple choice) (1 marks) · Integration
4. Q4Quadratic with a constant k: sketch and range of k (6 marks) · Algebra and functions
5. Q5Proving a number is prime (2 marks) · Proof
6. Q6Stationary point of an implicit curve with e^y (7 marks) · Differentiation, Exponentials and logarithms
7. Q7Graph of a function from its gradient (8 marks) · Differentiation, Algebra and functions
8. Q8Transformations of a sine graph and R form (10 marks) · Trigonometry
9. Q9Market trader's sales model and rates (14 marks) · Differentiation
10. Q10Snail with constant acceleration (1 marks) · Kinematics
11. Q11Rod balanced on a support (1 marks) · Moments
12. Q12Velocity-time graph and displacement (5 marks) · Kinematics
13. Q13Pushing and pulling a crate with friction (8 marks) · Forces and Newton's laws
14. Q14Position vectors and a parallelogram (6 marks) · Vectors
15. Q15Comparing two minibus motion models (9 marks) · Kinematics
16. Q16Projectile with a launch angle (6 marks) · Kinematics
17. Q17Buggy towing a skater: connected motion (14 marks) · Forces and Newton's laws