AQA A Level Maths June 2023 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

   Given that $f(x)=\frac{x+1}{x-2}$, $x\ne 2$, find an expression for $ff(x)$ in its simplest form.

   Worked solution

   1. M1 for substituting $f(x)$ into itself: $ff(x)=\frac{\frac{x+1}{x-2}+1}{\frac{x+1}{x-2}-2}$.
   2. M1 for multiplying the numerator and the denominator by $(x-2)$ to clear the nested fractions.
   3. M1 for obtaining the expanded fraction $\frac{(x+1)+(x-2)}{(x+1)-2(x-2)}$.
   4. A1 for fully simplifying to $\frac{2x-1}{5-x}$ (or $\frac{1-2x}{x-5}$).

2. Coordinate geometry · 7 marks

   A curve has parametric equations $x=t+1$, $y=t^3-12t$.
   (a) Find the values of $t$ at the stationary points on the curve. (3)
   (b) Find the coordinates of the stationary points. (2)
   (c) Determine the nature of each stationary point. (2)

   Worked solution

   (a) $\frac{dy}{dx}=\frac{3t^2-12}{1}=3t^2-12$   M1 A1. Set $=0$: $t^2=4\Rightarrow t=\pm 2$   A1.
   (b) $t=2$: $(3,-16)$; $t=-2$: $(-1,16)$   A1 A1.
   (c) $\frac{d^{2}y}{d{x}^2}=6t\cdot \frac{dt}{dx}=6t$ (since $\frac{dt}{dx}=1$)   M1. $t=2$: $12>0$ ⇒ minimum; $t=-2$: $-12<0$ ⇒ maximum   A1.

3. Kinematics · 6 marks

   A drone is programmed to fly in a straight line. Its position vector $r$ (in km) at time $t$ (in hours) is $r=\left(\begin{array}{c}-5\\ 12\end{array}\right)+t\left(\begin{array}{c}8\\ -6\end{array}\right)$.
   Calculate the exact speed of the drone in km h${}^{-1}$, and find its distance from the origin at $t=2$.

   Worked solution

   Velocity vector $v=\left(\begin{array}{c}8\\ -6\end{array}\right)$. Speed $=|v|=\sqrt{8^2+(-6{)}^2}=\sqrt{64+36}=10$ km h${}^{-1}$. M1A1
   Position at $t=2$: $r(2)=\left(\begin{array}{c}-5\\ 12\end{array}\right)+2\left(\begin{array}{c}8\\ -6\end{array}\right)=\left(\begin{array}{c}11\\ 0\end{array}\right)$. M1A1
   Distance from origin $=\sqrt{{11}^2+0^2}=11$ km. M1A1

More on these topics: Algebra and functions · Coordinate geometry · Kinematics

What this paper covers

What this paper is like

• Shape: 20 questions, 100 marks in 2 h (about 1.2 minutes a mark); questions from 1 to 12 marks.
• Biggest question: Q19, 12 marks: toy train and trailer connected by a rod.
• Marks by main topic: Kinematics 23, Forces and Newton's laws 21, Sequences and series 13, Trigonometry 10, Differentiation 7, Algebra and functions 6, Exponentials and logarithms 6, Proof 6, Moments 6, Integration 1, Coordinate geometry 1.
• More Forces and Newton's laws in other sittings: June 2022 Q19 (11 marks), November 2021 Q15 (5 marks), November 2020 Q18 (13 marks).
• Against the other 5 sittings we mapped: more marks on Sequences and series (13 marks here, 4 on average) and Forces and Newton's laws (21 marks here, 15 on average); fewer on Vectors (0 marks here, 5 on average) and Algebra and functions (6 marks here, 10 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. Q1Quadratic from its roots (multiple choice) (1 marks) · Algebra and functions
2. Q2Properties of definite integrals (multiple choice) (1 marks) · Integration
3. Q3Radius of a circle (multiple choice) (1 marks) · Coordinate geometry
4. Q4Tangent to a curve with a root (7 marks) · Differentiation
5. Q5Swimming lengths as an arithmetic series (7 marks) · Sequences and series
6. Q6Online sales model fitted with logs (6 marks) · Exponentials and logarithms
7. Q7Composite and inverse functions (5 marks) · Algebra and functions
8. Q8Proving a cosec identity, then solving (10 marks) · Trigonometry
9. Q9Binomial expansion with a negative power (6 marks) · Sequences and series
10. Q10Disproving a claim about a number and its reciprocal (6 marks) · Proof
11. Q11Hanging decoration (multiple choice) (1 marks) · Forces and Newton's laws
12. Q12Displacement after 4 seconds (multiple choice) (1 marks) · Kinematics
13. Q13Ball falling past two points (5 marks) · Kinematics
14. Q14Variable acceleration of a car (4 marks) · Kinematics
15. Q15Particle on a rough horizontal surface (4 marks) · Forces and Newton's laws
16. Q16Resultant of two force vectors (4 marks) · Forces and Newton's laws, Vectors
17. Q17Plank resting on two supports (6 marks) · Moments
18. Q18Motion with i and j vectors (6 marks) · Kinematics, Vectors
19. Q19Toy train and trailer connected by a rod (12 marks) · Forces and Newton's laws
20. Q20Projectile on a beach (7 marks) · Kinematics