AQA A Level Maths November 2021 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 · 5 marks

   A curve has parametric equations $x=\ln t$ and $y=t^2$, $t>0$. Find the equation of the normal to the curve at the point where $t=1$. Give your answer in the form $ax+by+c=0$.

   Worked solution

   1. M1 for finding the coordinates at $t=1$: $x=0$, $y=1$.
   2. M1 for calculating $\frac{dx}{dt}=\frac{1}{t}$ and $\frac{dy}{dt}=2t$, giving $\frac{dy}{dx}=2t^2$.
   3. M1 for evaluating the tangent gradient at $t=1$ to get $m_t=2$.
   4. M1 for determining the normal gradient $m_n=-\frac{1}{2}$ and using $(0,1)$ to form the equation $y-1=-\frac{1}{2}(x-0)$.
   5. A1 for rearranging to the required integer format: $x+2y-2=0$.

2. 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 $(3,1,2)$.

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 · Vectors · 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 13 marks.
• Biggest question: Q16, 13 marks: uniform rod with a weight attached.
• Marks by main topic: Algebra and functions 21, Kinematics 18, Moments 13, Vectors 11, Trigonometry 9, Coordinate geometry 8, Forces and Newton's laws 8, Proof 6, Exponentials and logarithms 4, Differentiation 1, Sequences and series 1.
• More Moments in other sittings: June 2022 Q14 (4 marks), November 2020 Q13 (3 marks), June 2023 Q17 (6 marks).
• Against the other 5 sittings we mapped: more marks on Algebra and functions (21 marks here, 7 on average) and Moments (13 marks here, 4 on average); fewer on Differentiation (1 marks here, 12 on average) and Forces and Newton's laws (8 marks here, 18 on average).
• Not on this paper: Integration, 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. Q1Quadratic graphs with a zero discriminant (multiple choice) (1 marks) · Algebra and functions
2. Q2Point of inflection (multiple choice) (1 marks) · Differentiation
3. Q3Sum of a periodic sequence (multiple choice) (1 marks) · Sequences and series
4. Q4Two modulus graphs and their intersections (6 marks) · Algebra and functions
5. Q5Partial fractions (3 marks) · Algebra and functions
6. Q6Exponential equation solved with logs (4 marks) · Exponentials and logarithms
7. Q7Circle with a constant p and a tangent (8 marks) · Coordinate geometry
8. Q8Checking a proof about multiples of 3 (6 marks) · Proof
9. Q9Robotic arm drawing with a trig model (9 marks) · Trigonometry, Coordinate geometry
10. Q10Domain of a function and its maximum (11 marks) · Algebra and functions, Differentiation
11. Q11Displacement as a function of time (1 marks) · Kinematics
12. Q12Velocity from a speed and a direction (1 marks) · Vectors, Kinematics
13. Q13Driving force and resistance on a vehicle (3 marks) · Forces and Newton's laws
14. Q14Scooter velocity-time graph (4 marks) · Kinematics
15. Q15Cyclist towing a trailer (5 marks) · Forces and Newton's laws
16. Q16Uniform rod with a weight attached (13 marks) · Moments
17. Q17Two velocity models as t gets large (2 marks) · Kinematics
18. Q18Two particles projected at once (11 marks) · Kinematics
19. Q19Two skaters moving with vectors (10 marks) · Vectors, Kinematics