AQA A Level Maths Specimen 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 · 5 marks

   Express $\frac{x+2}{x^2-5x+6}-\frac{x-1}{x^2-4x+3}$ as a single fraction in its simplest form.

   Worked solution

   1. M1 for factorising $x^2-5x+6$ to $(x-2)(x-3)$.
   2. M1 for factorising $x^2-4x+3$ to $(x-1)(x-3)$.
   3. M1 for simplifying the second fraction: $\frac{x-1}{(x-1)(x-3)}$ cancels to $\frac{1}{x-3}$.
   4. M1 for creating a common denominator $(x-2)(x-3)$ and subtracting: $\frac{x+2}{(x-2)(x-3)}-\frac{x-2}{(x-2)(x-3)}$.
   5. A1 for reaching the final simplified answer $\frac{4}{(x-2)(x-3)}$.

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 · 7 marks

   At time $t$ hours after 09:00, a boat $P$ has position vector $((1+2t)\mathbf{i}+(5-t)\mathbf{j})$ km relative to a fixed origin. A lighthouse $L$ is at the point with position vector $(8\mathbf{i}+7\mathbf{j})$ km.

   (a) Find $\overrightarrow{PL}$ in terms of $t$. [2]

   (b) Show that $P{L}^2=5t^2-24t+53$. [2]

   (c) Find the shortest distance between the boat and the lighthouse, and the time at which it occurs. [3]

   Worked solution

   (a) $\overrightarrow{PL}=(8\mathbf{i}+7\mathbf{j})-((1+2t)\mathbf{i}+(5-t)\mathbf{j})$ M1

   $(7-2t)\mathbf{i}+(2+t)\mathbf{j}$ A1

   (b) $P{L}^2=(7-2t{)}^2+(2+t{)}^2$ M1

   $=49-28t+4t^2+4+4t+t^2=5t^2-24t+53$ A1 (cso)

   (c) $5t^2-24t+53=5(t-2.4{)}^2+24.2$ (or any complete method for the minimum) M1

   $t=2.4$, i.e. 11:24 A1

   Shortest distance $\sqrt{24.2}=4.92$ km A1

More on these topics: Algebra and functions · 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 12 marks.
• Biggest question: Q16, 12 marks: box on a rough board.
• Marks by main topic: Kinematics 30, Forces and Newton's laws 18, Differentiation 14, Sequences and series 11, Trigonometry 9, Numerical methods 6, Algebra and functions 5, Proof 4, Moments 2, Exponentials and logarithms 1.
• More Forces and Newton's laws in other sittings: June 2018 Q17 (14 marks), June 2019 Q17 (9 marks), November 2020 Q18 (13 marks).
• Against the other 6 sittings we mapped: more marks on Kinematics (30 marks here, 24 on average) and Numerical methods (6 marks here, 0 on average); fewer on Exponentials and logarithms (1 marks here, 7 on average) and Algebra and functions (5 marks here, 9 on average).
• Not on this paper: Coordinate geometry, Integration, 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. Q1Validity of a binomial expansion (multiple choice) (1 marks) · Sequences and series
2. Q2Squirrel population model (1 marks) · Exponentials and logarithms
3. Q3Parametric curve gradient (6 marks) · Differentiation, Coordinate geometry
4. Q4Locating a root of a cubic (6 marks) · Numerical methods
5. Q5Transforming cos to an R form graph (9 marks) · Trigonometry
6. Q6Quadratic with a constant k (5 marks) · Algebra and functions
7. Q7Proof about consecutive odd numbers (4 marks) · Proof
8. Q8Product rule with trig functions (8 marks) · Differentiation, Trigonometry
9. Q9Arithmetic sequence with exponentials (10 marks) · Sequences and series, Exponentials and logarithms
10. Q10Force and acceleration (multiple choice) (1 marks) · Forces and Newton's laws
11. Q11Rod with an attached mass (2 marks) · Moments
12. Q12Constant acceleration (4 marks) · Kinematics
13. Q13Three forces in equilibrium (5 marks) · Forces and Newton's laws, Vectors
14. Q14Velocity-time model for a train (7 marks) · Kinematics
15. Q15Parachutist's velocity model (11 marks) · Kinematics
16. Q16Box on a rough board (12 marks) · Forces and Newton's laws
17. Q17Projectile from the origin (8 marks) · Kinematics