Questions — AQA Paper 2 (149 questions)

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AQA Paper 2 2023 June Q13
5 marks Moderate -0.3
13 A ball falls freely towards the Earth.
The ball passes through two different fixed points \(M\) and \(N\) before reaching the Earth's surface. At \(M\) the ball has velocity \(u \mathrm {~ms} ^ { - 1 }\) At \(N\) the ball has velocity \(3 u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) It can be assumed that:
  • the motion is due to gravitational force only
  • the acceleration due to gravity remains constant throughout.
13
  1. Show that the time taken for the ball to travel from \(M\) to \(N\) is \(\frac { 2 u } { g }\) seconds.
    [0pt] [2 marks] 13
  2. Point \(M\) is \(h\) metres above the Earth. Show that \(h > \frac { 4 u ^ { 2 } } { g }\) Fully justify your answer.
    The car is moving in a straight line.
    The acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) of the car at time \(t\) seconds is given by $$a = 3 k t ^ { 2 } - 2 k t + 1$$ where \(k\) is a constant.
    When \(t = 3\) the car has a velocity of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) Show that \(k = \frac { 1 } { 3 }\)
AQA Paper 2 2023 June Q14
4 marks Moderate -0.8
14 A car has an initial velocity of \(1 \mathrm {~ms} ^ { - 1 }\) A particle, \(Q\), moves in a straight line across a rough horizontal surface.
A horizontal driving force of magnitude \(D\) newtons acts on \(Q\) \(Q\) moves with a constant acceleration of \(0.91 \mathrm {~ms} ^ { - 2 }\) \(Q\) has a weight of 0.65 N
The only resistance force acting on \(Q\) is due to friction.
The coefficient of friction between \(Q\) and the surface is 0.4 Find \(D\)
AQA Paper 2 2023 June Q15
4 marks Standard +0.3
15 In this question use \(g = 9.8 \mathrm {~ms} ^ { - 2 }\) In this question use \(g = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\)
AQA Paper 2 2023 June Q16
4 marks Moderate -0.8
16 A particle moves under the action of two forces, \(\mathbf { F } _ { 1 }\) and \(\mathbf { F } _ { 2 }\) It is given that $$\begin{aligned} & \mathbf { F } _ { 1 } = ( 1.6 \mathbf { i } - 5 \mathbf { j } ) \mathrm { N } \\ & \mathbf { F } _ { 2 } = ( k \mathbf { i } + 5 k \mathbf { j } ) \mathrm { N } \end{aligned}$$ where \(k\) is a constant.
The acceleration of the particle is \(( 3.2 \mathbf { i } + 12 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }\) Find \(k\) \includegraphics[max width=\textwidth, alt={}, center]{de8a7d38-a665-4feb-854e-ac83f413d133-25_2488_1716_219_153}
AQA Paper 2 2023 June Q17
6 marks Standard +0.3
17 A uniform plank \(P Q\), of length 7 metres, lies horizontally at rest, in equilibrium, on two fixed supports at points \(X\) and \(Y\) The distance \(P X\) is 1.4 metres and the distance \(Q Y\) is 2 metres as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{de8a7d38-a665-4feb-854e-ac83f413d133-26_56_689_534_762} \includegraphics[max width=\textwidth, alt={}, center]{de8a7d38-a665-4feb-854e-ac83f413d133-26_225_830_607_694} 17
  1. The reaction force on the plank at \(X\) is \(4 g\) newtons.
    17
    1. (i) Show that the mass of the plank is 9.6 kilograms.
      17
    2. (ii) Find the reaction force, in terms of \(g\), on the plank at \(Y\) 17
    3. The support at \(Y\) is moved so that the distance \(Q Y = 1.4\) metres. The plank remains horizontally at rest in equilibrium.
      It is claimed that the reaction force at \(Y\) remains unchanged.
      Explain, with a reason, whether this claim is correct.
AQA Paper 2 2023 June Q18
6 marks Moderate -0.3
18 In this question \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors representing due east and due north respectively. A particle, \(T\), is moving on a plane at a constant speed.
The path followed by \(T\) makes the exact shape of a triangle \(A B C\). \(T\) moves around \(A B C\) in an anticlockwise direction as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{de8a7d38-a665-4feb-854e-ac83f413d133-28_447_366_671_925} On its journey from \(A\) to \(B\) the velocity vector of \(T\) is \(( 3 \mathbf { i } + \sqrt { 3 } \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) 18
  1. Find the speed of \(T\) as it moves from \(A\) to \(B\) 18
  2. On its journey from \(B\) to \(C\) the velocity vector of \(T\) is \(( - 3 \mathbf { i } + \sqrt { 3 } \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) Show that the acute angle \(A B C = 60 ^ { \circ }\) 18
  3. It is given that \(A B C\) is an equilateral triangle. \(T\) returns to its initial position after 9 seconds.
    Vertex \(B\) lies at position vector \(\left[ \begin{array} { l } 1 \\ 0 \end{array} \right]\) metres with respect to a fixed origin \(O\) Find the position vector of \(C\)
AQA Paper 2 2023 June Q19
12 marks Moderate -0.3
19 A wooden toy comprises a train engine and a trailer connected to each other by a light, inextensible rod. The train engine has a mass of 1.5 kilograms.
The trailer has a mass 0.7 kilograms.
A string inclined at an angle of \(40 ^ { \circ }\) above the horizontal is attached to the front of the train engine. The tension in the string is 2 newtons.
As a result the toy moves forward, from rest, in a straight line along a horizontal surface with acceleration \(0.06 \mathrm {~ms} ^ { - 2 }\) as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{de8a7d38-a665-4feb-854e-ac83f413d133-30_373_789_904_756} As it moves the train engine experiences a total resistance force of 0.8 N
19
  1. Show that the total resistance force experienced by the trailer is approximately 0.6 N
    19
  2. At the instant that the toy reaches a speed of \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) the string breaks. As a result of this the train engine and trailer decelerate at a constant rate until they come to rest, having travelled a distance of \(h\) metres. It can be assumed that the resistance forces remain unchanged.
    19 (b) (i) Find the tension in the rod after the string has broken.
    19 (b) (ii) Find \(h\)Do not write outside the box
    \includegraphics[max width=\textwidth, alt={}]{de8a7d38-a665-4feb-854e-ac83f413d133-33_2488_1716_219_153}
    Nell and her pet dog Maia are visiting the beach.
    The beach surface can be assumed to be level and horizontal. Nell and Maia are initially standing next to each other.
    Nell throws a ball forward, from a height of 1.8 metres above the surface of the beach, at an angle of \(60 ^ { \circ }\) above the horizontal with a speed of \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) Exactly 0.2 seconds after the ball is thrown, Maia sets off from Nell and runs across the surface of the beach, in a straight line with a constant acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) Maia catches the ball when it is 0.3 metres above ground level as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{de8a7d38-a665-4feb-854e-ac83f413d133-34_778_1287_1027_463}
AQA Paper 2 2023 June Q20
7 marks Moderate -0.8
20 In this question use \(g = 9.8 \mathrm {~m \mathrm {~s} ^ { - 2 }\)} Find \(a\)
\includegraphics[max width=\textwidth, alt={}]{de8a7d38-a665-4feb-854e-ac83f413d133-36_2488_1719_219_150}
AQA Paper 2 2018 June Q1
1 marks Easy -2.5
Which of these statements is correct? Tick one box. [1 mark] \(x = 2 \Rightarrow x^2 = 4\) \(x^2 = 4 \Rightarrow x = 2\) \(x^2 = 4 \Leftrightarrow x = 2\) \(x^2 = 4 \Rightarrow x = -2\)
AQA Paper 2 2018 June Q2
1 marks Easy -1.8
Find the coefficient of \(x^2\) in the expansion of \((1 + 2x)^7\) Circle your answer. [1 mark] 42 4 21 84
AQA Paper 2 2018 June Q3
1 marks Easy -1.8
The graph of \(y = x^3\) is shown. \includegraphics{figure_1} Find the total shaded area. Circle your answer. [1 mark] \(-68\) 60 68 128
AQA Paper 2 2018 June Q4
6 marks Standard +0.3
A curve, C, has equation \(y = x^2 - 6x + k\), where \(k\) is a constant. The equation \(x^2 - 6x + k = 0\) has two distinct positive roots.
  1. Sketch C on the axes below. [2 marks]
  2. Find the range of possible values for \(k\). Fully justify your answer. [4 marks]
AQA Paper 2 2018 June Q5
2 marks Easy -2.0
Prove that 23 is a prime number. [2 marks]
AQA Paper 2 2018 June Q6
7 marks Challenging +1.2
Find the coordinates of the stationary point of the curve with equation \((x + y - 2)^2 = e^y - 1\) [7 marks]
AQA Paper 2 2018 June Q7
8 marks Standard +0.8
A function f has domain \(\mathbb{R}\) and range \(\{y \in \mathbb{R} : y \geq c\}\) The graph of \(y = f(x)\) is shown. \includegraphics{figure_2} The gradient of the curve at the point \((x, y)\) is given by \(\frac{dy}{dx} = (x - 1)e^x\) Find an expression for f(x). Fully justify your answer. [8 marks]
AQA Paper 2 2018 June Q8
10 marks Standard +0.8
  1. Determine a sequence of transformations which maps the graph of \(y = \sin x\) onto the graph of \(y = \sqrt{3} \sin x - 3 \cos x + 4\) Fully justify your answer. [7 marks]
    1. Show that the least value of \(\frac{1}{\sqrt{3} \sin x - 3 \cos x + 4}\) is \(\frac{2 - \sqrt{3}}{2}\) [2 marks]
    2. Find the greatest value of \(\frac{1}{\sqrt{3} \sin x - 3 \cos x + 4}\) [1 mark]
AQA Paper 2 2018 June Q9
14 marks Challenging +1.2
A market trader notices that daily sales are dependent on two variables: number of hours, \(t\), after the stall opens total sales, \(x\), in pounds since the stall opened. The trader models the rate of sales as directly proportional to \(\frac{8 - t}{x}\) After two hours the rate of sales is £72 per hour and total sales are £336
  1. Show that $$x \frac{dx}{dt} = 4032(8 - t)$$ [3 marks]
  2. Hence, show that $$x^2 = 4032t(16 - t)$$ [3 marks]
  3. The stall opens at 09.30.
    1. The trader closes the stall when the rate of sales falls below £24 per hour. Using the results in parts (a) and (b), calculate the earliest time that the trader closes the stall. [6 marks]
    2. Explain why the model used by the trader is not valid at 09.30. [2 marks]
AQA Paper 2 2018 June Q10
1 marks Easy -2.0
A garden snail moves in a straight line from rest to 1.28 cm s\(^{-1}\), with a constant acceleration in 1.8 seconds. Find the acceleration of the snail. Circle your answer. [1 mark] 2.30 m s\(^{-2}\) 0.71 m s\(^{-2}\) 0.0071 m s\(^{-2}\) 0.023 m s\(^{-2}\)
AQA Paper 2 2018 June Q11
1 marks Easy -1.8
A uniform rod, AB, has length 4 metres. The rod is resting on a support at its midpoint C. A particle of mass 4 kg is placed 0.6 metres to the left of C. Another particle of mass 1.5 kg is placed \(x\) metres to the right of C, as shown. \includegraphics{figure_3} The rod is balanced in equilibrium at C. Find \(x\). Circle your answer. [1 mark] 1.8 m 1.5 m 1.75 m 1.6 m
AQA Paper 2 2018 June Q12
5 marks Standard +0.3
The graph below shows the velocity of an object moving in a straight line over a 20 second journey. \includegraphics{figure_4}
  1. Find the maximum magnitude of the acceleration of the object. [1 mark]
  2. The object is at its starting position at times 0, \(t_1\) and \(t_2\) seconds. Find \(t_1\) and \(t_2\) [4 marks]
AQA Paper 2 2018 June Q13
8 marks Moderate -0.3
In this question use \(g = 9.8\) m s\(^{-2}\) A boy attempts to move a wooden crate of mass 20 kg along horizontal ground. The coefficient of friction between the crate and the ground is 0.85
  1. The boy applies a horizontal force of 150 N. Show that the crate remains stationary. [3 marks]
  2. Instead, the boy uses a handle to pull the crate forward. He exerts a force of 150 N, at an angle of 15° above the horizontal, as shown in the diagram. \includegraphics{figure_5} Determine whether the crate remains stationary. Fully justify your answer. [5 marks]
AQA Paper 2 2018 June Q14
6 marks Moderate -0.8
A quadrilateral has vertices A, B, C and D with position vectors given by $$\overrightarrow{OA} = \begin{pmatrix} 3 \\ 5 \\ 1 \end{pmatrix}, \overrightarrow{OB} = \begin{pmatrix} -1 \\ 2 \\ 7 \end{pmatrix}, \overrightarrow{OC} = \begin{pmatrix} 0 \\ 7 \\ 6 \end{pmatrix} \text{ and } \overrightarrow{OD} = \begin{pmatrix} 4 \\ 10 \\ 0 \end{pmatrix}$$
  1. Write down the vector \(\overrightarrow{AB}\) [1 mark]
  2. Show that ABCD is a parallelogram, but not a rhombus. [5 marks]
AQA Paper 2 2018 June Q15
9 marks Standard +0.8
A driver is road-testing two minibuses, A and B, for a taxi company. The performance of each minibus along a straight track is compared. A flag is dropped to indicate the start of the test. Each minibus starts from rest. The acceleration in m s\(^{-2}\) of each minibus is modelled as a function of time, \(t\) seconds, after the flag is dropped: The acceleration of A = \(0.138 t^2\) The acceleration of B = \(0.024 t^3\)
  1. Find the time taken for A to travel 100 metres. Give your answer to four significant figures. [4 marks]
  2. The company decides to buy the minibus which travels 100 metres in the shortest time. Determine which minibus should be bought. [4 marks]
  3. The models assume that both minibuses start moving immediately when \(t = 0\) In light of this, explain why the company may, in reality, make the wrong decision. [1 mark]
AQA Paper 2 2018 June Q16
6 marks Standard +0.3
In this question use \(g = 9.81\) m s\(^{-2}\) A particle is projected with an initial speed \(u\), at an angle of 35° above the horizontal. It lands at a point 10 metres vertically below its starting position. The particle takes 1.5 seconds to reach the highest point of its trajectory.
  1. Find \(u\). [3 marks]
  2. Find the total time that the particle is in flight. [3 marks]
AQA Paper 2 2018 June Q17
14 marks Moderate -0.3
A buggy is pulling a roller-skater, in a straight line along a horizontal road, by means of a connecting rope as shown in the diagram. \includegraphics{figure_6} The combined mass of the buggy and driver is 410 kg A driving force of 300 N and a total resistance force of 140 N act on the buggy. The mass of the roller-skater is 72 kg A total resistance force of R newtons acts on the roller-skater. The buggy and the roller-skater have an acceleration of 0.2 m s\(^{-2}\)
    1. Find R. [3 marks]
    2. Find the tension in the rope. [3 marks]
  2. State a necessary assumption that you have made. [1 mark]
  3. The roller-skater releases the rope at a point A, when she reaches a speed of 6 m s\(^{-1}\) She continues to move forward, experiencing the same resistance force. The driver notices a change in motion of the buggy, and brings it to rest at a distance of 20 m from A.
    1. Determine whether the roller-skater will stop before reaching the stationary buggy. Fully justify your answer. [5 marks]
    2. Explain the change in motion that the driver noticed. [2 marks]