AQA A Level Maths June 2022 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. Algebra and functions · 4 marks

   A cubic curve crosses the x-axis at $x=-2$, $x=1$, and $x=4$. It crosses the y-axis at the point $(0,16)$. Find the equation of the curve in the form $y=a{x}^3+b{x}^2+cx+d$.

   Worked solution

   1. M1 for writing the equation in the factored form $y=k(x+2)(x-1)(x-4)$.
   2. M1 for substituting $(0,16)$ to find $k$: $16=k(2)(-1)(-4)⟹16=8k$.
   3. A1 for finding $k=2$.
   4. A1 for expanding to get $y=2(x^3-3x^2-6x+8)=2x^3-6x^2-12x+16$.

2. Differentiation · 5 marks

   A curve has the equation $y=e^{-x}\sin x$. Find the exact x-coordinate of the first positive turning point of the curve.

   Worked solution

   1. M1 for using the product rule to differentiate: $\frac{dy}{dx}=-e^{-x}\sin x+e^{-x}\cos x$.
   2. M1 for setting the derivative equal to 0 to find turning points: $e^{-x}(\cos x-\sin x)=0$.
   3. M1 for dividing by $e^{-x}$ (since $e^{-x}\ne 0$) to get $\cos x=\sin x$.
   4. M1 for dividing by $\cos x$ to obtain $\tan x=1$.
   5. A1 for identifying the first positive solution as $x=\frac{\pi }{4}$.

3. Kinematics · 5 marks

   A particle moving with constant acceleration passes three points $A$, $B$ and $C$, in that order, on a straight line. The distance $AB$ is $20\text{ m}$ and the distance $AC$ is $84\text{ m}$. The speed of the particle at $B$ is $12{\text{ m s}}^{-1}$ and its speed at $C$ is $20{\text{ m s}}^{-1}$. Find the acceleration of the particle and its speed at $A$.

   Worked solution

   1. M1 for $BC=84-20=64\text{ m}$.
   2. M1 for $v^2=u^2+2as$ from $B$ to $C$: ${20}^2={12}^2+2a(64)$.
   3. A1 for $128a=256\Rightarrow a=2{\text{ m s}}^{-2}$.
   4. M1 for $v^2=u^2+2as$ from $A$ to $B$: ${12}^2=u_A^2+2(2)(20)$.
   5. A1 for $u_A^2=64\Rightarrow u_A=8{\text{ m s}}^{-1}$.

More on these topics: Algebra and functions · Differentiation · Kinematics

What this paper covers

What this paper is like

• Shape: 19 questions, 100 marks in 2 h (about 1.2 minutes a mark); questions from 1 to 15 marks.
• Biggest question: Q10, 15 marks: tomato plant disease model.
• Marks by main topic: Kinematics 26, Forces and Newton's laws 20, Exponentials and logarithms 19, Integration 9, Sequences and series 6, Proof 5, Algebra and functions 5, Moments 4, Trigonometry 3, Differentiation 2, Coordinate geometry 1.
• More Exponentials and logarithms in other sittings: November 2021 Q6 (4 marks), June 2023 Q6 (6 marks), June 2019 Q8 (11 marks).
• Against the other 5 sittings we mapped: more marks on Exponentials and logarithms (19 marks here, 4 on average) and Integration (9 marks here, 3 on average); fewer on Differentiation (2 marks here, 12 on average) and Vectors (0 marks here, 5 on average).
• Not on this paper: Numerical methods, Vectors.
• 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. Q1Equation of a circle (tick one) (1 marks) · Coordinate geometry
2. Q2Derivative of sin from first principles (multiple choice) (1 marks) · Differentiation, Trigonometry
3. Q3Concave graph (multiple choice) (1 marks) · Differentiation
4. Q4Sine rule in a triangle (3 marks) · Trigonometry
5. Q5Binomial expansion with unknown coefficients (6 marks) · Sequences and series
6. Q6Counter-example to a digit rule (5 marks) · Proof
7. Q7Isosceles triangle under a parabola, maximum area (9 marks) · Integration, Differentiation
8. Q8Transforming the graph of y = x^2 (5 marks) · Algebra and functions
9. Q9Log equation to express x in terms of y (4 marks) · Exponentials and logarithms
10. Q10Tomato plant disease model (15 marks) · Exponentials and logarithms, Integration
11. Q11Weight on the moon (multiple choice) (1 marks) · Forces and Newton's laws
12. Q12Braking car (multiple choice) (1 marks) · Kinematics
13. Q13Projectile from horizontal ground (6 marks) · Kinematics
14. Q14Coin balanced on a rod (4 marks) · Moments
15. Q15Velocity-time graph of a car (4 marks) · Kinematics
16. Q16Two particles moving in a plane (8 marks) · Kinematics, Vectors
17. Q17Position vector with exponentials (7 marks) · Kinematics
18. Q18Object hanging from two strings (8 marks) · Forces and Newton's laws
19. Q19Particle on a rough ramp (11 marks) · Forces and Newton's laws