AQA A Level Maths June 2019 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. Coordinate geometry · 8 marks

   Three points define the vertices of a triangle: $A(-2,-1)$, $B(4,7)$, and $C(8,4)$.

   1. By calculating the gradients of $[AB]$ and $[BC]$, prove that triangle $ABC$ is a right-angled triangle.

   2. Calculate the exact lengths of $[AB]$ and $[BC]$.

   3. Hence, find the exact area of the triangle $ABC$.

   Worked solution

   1. Gradient $m_{AB}=\frac{7-(-1)}{4-(-2)}=\frac{8}{6}=\frac{4}{3}$. A1
      Gradient $m_{BC}=\frac{4-7}{8-4}=\frac{-3}{4}$. A1
      Since $m_{AB}\times m_{BC}=\frac{4}{3}\times -\frac{3}{4}=-1$, $[AB]$ and $[BC]$ are perpendicular. M1
      Thus, triangle $ABC$ has a right angle at $B$. B1

   2. Length $AB=\sqrt{(4-(-2){)}^2+(7-(-1){)}^2}=\sqrt{36+64}=\sqrt{100}=10$. A1
      Length $BC=\sqrt{(8-4{)}^2+(4-7{)}^2}=\sqrt{16+9}=\sqrt{25}=5$. A1

   3. Since the triangle is right-angled at $B$, $AB$ and $BC$ act as the base and height. M1
      Area $=\frac{1}{2}\times \text{base}\times \text{height}=\frac{1}{2}\times 10\times 5=25$ sq units. A1

2. Exponentials and logarithms · 4 marks

   Given that ${\log }_{2}x=p$ and ${\log }_{2}y=q$, express ${\log }_2(8x^2{y}^3)$ in terms of $p$ and $q$.

   Worked solution

   1. M1 for splitting the logarithm into a sum: ${\log }_{2}8+{\log }_2{x}^2+{\log }_2{y}^3$.
   2. B1 for evaluating ${\log }_{2}8=3$.
   3. M1 for applying the power law: $2{\log }_{2}x+3{\log }_{2}y$.
   4. A1 for the final expression $3+2p+3q$.

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: Coordinate geometry · 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 16 marks.
• Biggest question: Q16, 16 marks: variable acceleration of a sprinter.
• Marks by main topic: Kinematics 22, Algebra and functions 17, Exponentials and logarithms 11, Forces and Newton's laws 11, Sequences and series 9, Vectors 9, Moments 8, Integration 7, Trigonometry 6.
• More Kinematics in other sittings: June 2018 Q15 (9 marks), November 2020 Q19 (8 marks), November 2021 Q18 (11 marks).
• Against the other 5 sittings we mapped: more marks on Algebra and functions (17 marks here, 8 on average) and Vectors (9 marks here, 3 on average); fewer on Differentiation (0 marks here, 12 on average) and Forces and Newton's laws (11 marks here, 17 on average).
• Not on this paper: Proof, Coordinate geometry, Differentiation, 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. Q1Modulus graph (tick one) (1 marks) · Algebra and functions
2. Q2Index laws (multiple choice) (1 marks) · Algebra and functions
3. Q3Which function has no inverse (multiple choice) (1 marks) · Algebra and functions
4. Q4Common factor of two quadratics (4 marks) · Algebra and functions
5. Q5Separable differential equation with ln x (7 marks) · Integration, Exponentials and logarithms
6. Q6a sin x + b cos x from its maximum (6 marks) · Trigonometry
7. Q7Cubic graphs and roots (10 marks) · Algebra and functions
8. Q8House price growth model (11 marks) · Exponentials and logarithms
9. Q9Binomial expansion and its validity (9 marks) · Sequences and series
10. Q10Velocity-time graph (multiple choice) (1 marks) · Kinematics
11. Q11Crate and friction (multiple choice) (1 marks) · Forces and Newton's laws
12. Q12Particle under two constant forces (1 marks) · Forces and Newton's laws
13. Q13Free fall and modelling assumptions (5 marks) · Kinematics
14. Q14Metal rod overhanging a shelf (8 marks) · Moments
15. Q15Buoys on a lake with position vectors (9 marks) · Vectors, Kinematics
16. Q16Variable acceleration of a sprinter (16 marks) · Kinematics
17. Q17Sledge pulled on rough ground (9 marks) · Forces and Newton's laws