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AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 SPS SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
AQA Further AS Paper 2 Mechanics 2021 June Q1
1 A light spring of natural length 0.6 metres is compressed to a length of 0.4 metres by a force of 20 newtons. The stiffness of the spring is \(k \mathrm { Nm } ^ { - 1 }\)
Find \(k\) Circle your answer. 2050100200
AQA Further AS Paper 2 Mechanics 2021 June Q2
1 marks
2 State the dimensions of force. Circle your answer.
[0pt] [1 mark]
MLT
\(M L ^ { 2 } T\)
\(M L T ^ { - 1 }\)
\(M L T ^ { - 2 }\)
AQA Further AS Paper 2 Mechanics 2021 June Q3
3 Use \(g\) as \(9.8 \mathrm {~ms} ^ { - 2 }\) in this question. A pump is used to pump water out of a pool.
The pump raises the water through a vertical distance of 5 metres and then ejects it through a pipe. The pump works at a constant rate of 400 W
Over a period of 50 seconds, 300 litres of water are pumped out of the pool and the water is ejected with speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) The mass of 1 litre of water is 1 kg
3
  1. Find the gain in the potential energy of the 300 litres of water.
    3
  2. \(\quad\) Calculate \(v\)
AQA Further AS Paper 2 Mechanics 2021 June Q4
4 A cyclist in a road race is travelling around a bend on a horizontal circular path of radius 15 metres and is prevented from skidding by a frictional force. The frictional force has a maximum value of 500 newtons. The total mass of the cyclist and his cycle is 75 kg
Assume that the cyclist travels at a constant speed.
4
  1. Work out the greatest speed, in \(\mathrm { km } \mathrm { h } ^ { - 1 }\), at which the cyclist can travel around the bend.
    4
  2. With reference to the surface of the road, describe one limitation of the model.
    \(5 \quad\) A ball is thrown vertically upwards with speed \(u\) so that at time \(t\) its displacement \(s\) is given by the formula $$s = u t - \frac { g t ^ { 2 } } { 2 }$$ Use dimensional analysis to show that this formula is dimensionally consistent.
    Fully justify your answer.
    \includegraphics[max width=\textwidth, alt={}, center]{a12155cc-cd07-40e0-af69-6b2590e4ea7c-06_2488_1730_219_141}
AQA Further AS Paper 2 Mechanics 2021 June Q6
5 marks
6 A ball of mass 0.15 kg is hit directly by a vertical cricket bat. Immediately before the impact, the ball is travelling horizontally with speed \(28 \mathrm {~ms} ^ { - 1 }\) Immediately after the impact, the ball is travelling horizontally with speed \(14 \mathrm {~ms} ^ { - 1 }\) in the opposite direction. 6
  1. Find the magnitude of the impulse exerted by the bat on the ball.
    [0pt] [2 marks]
    6
  2. In a simple model the force, \(F\) newtons, exerted by the bat on the ball, \(t\) seconds after the initial impact, is given by $$F = 10 k t ( 0.05 - t )$$ where \(k\) is a constant.
    Given the ball is in contact with the bat for 0.05 seconds, find the value of \(k\)
    [0pt] [3 marks]
    \(7 \quad\) Use \(g\) as \(9.81 \mathrm {~ms} ^ { - 2 }\) in this question. A light elastic string has one end attached to a fixed point \(A\) on a smooth plane inclined at \(25 ^ { \circ }\) to the horizontal. The other end of the string is attached to a wooden block of mass 2.5 kg , which rests on the plane. The elastic string has natural length 3 metres and modulus of elasticity 125 newtons.
    The block is pulled down the line of greatest slope of the plane to a point 4.5 metres from \(A\) and then released.
AQA Further AS Paper 2 Mechanics 2021 June Q7
7
  1. Find the elastic potential energy of the string at the point when the block is released.
    7
  2. Calculate the speed of the block when the string becomes slack.
    7
  3. Determine whether the block reaches the point \(A\) in the subsequent motion, commenting on any assumptions that you make.
AQA Further AS Paper 2 Mechanics 2021 June Q8
2 marks
8 Two spheres \(A\) and \(B\) are free to move on a smooth horizontal surface. The masses of \(A\) and \(B\) are 2 kg and 3 kg respectively.
Both \(A\) and \(B\) are initially at rest.
Sphere \(A\) is set in motion directly towards sphere \(B\) with speed \(4 \mathrm {~ms} ^ { - 1 }\) and subsequently collides with sphere \(B\) The coefficient of restitution between the spheres is \(e\)
8
    1. Show that the speed of \(B\) immediately after the collision is $$\frac { 8 ( 1 + e ) } { 5 }$$ 8
  2. (ii) Find an expression, in terms of \(e\), for the velocity of \(A\) immediately after the collision.
    8
  3. It is given that the spheres both move in the same direction after the collision. Find the range of possible values of \(e\)
    [0pt] [2 marks]
    8
    1. The impulse of sphere \(A\) on sphere \(B\) is \(I\)
      The impulse of sphere \(B\) on sphere \(A\) is \(J\)
      Given that the collision is perfectly inelastic, find the value of \(I + J\)
      8
  5. (ii) State, giving a reason for your answer, whether the value found in part (c)(i) would change if the collision was not perfectly inelastic.
    \includegraphics[max width=\textwidth, alt={}, center]{a12155cc-cd07-40e0-af69-6b2590e4ea7c-12_2488_1732_219_139}
    \includegraphics[max width=\textwidth, alt={}]{a12155cc-cd07-40e0-af69-6b2590e4ea7c-16_2496_1721_214_148}
AQA Further AS Paper 2 Mechanics Specimen Q1
1 marks
1 A child, of mass 40 kg , moves at constant speed of \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on a fairground ride.
The path of the child is a circle of radius 4 metres.
Find the magnitude of the resultant force acting on the child.
Circle your answer.
[0pt] [1 mark]
6.3 N
50 N
130 N
250 N
AQA Further AS Paper 2 Mechanics Specimen Q2
1 marks
2 The graph shows how a force, \(F\), varies with time during a period of 0.8 seconds.
\includegraphics[max width=\textwidth, alt={}, center]{18522f4c-4aa2-4ef5-898f-5ad2b06e287c-03_440_960_568_516} Find the magnitude of the impulse of \(F\) during the 0.8 seconds.
Circle your answer.
[0pt] [1 mark]
1.0 Ns
1.6 Ns
2.2 Ns
3.2 Ns Turn over for the next question
AQA Further AS Paper 2 Mechanics Specimen Q3
4 marks
3 A tank full of liquid has a hole made in its base.
Two students, Sarah and David, propose two different models for the speed, \(v\), at which liquid exits the tank. David thinks that \(v\) will depend on the height of the liquid in the tank, \(h\), the acceleration due to gravity, \(g\), and the density of the liquid, \(\rho\), such that \(v \propto g ^ { a } h ^ { b } \rho ^ { c }\) where \(a\), \(b\) and \(c\) are constants. Sarah thinks that \(v\) will not depend on the density of the liquid and suggests the model \(v \propto g ^ { a } h ^ { b }\) 3
  1. By considering dimensions, explain which student's model should be rejected.
    [0pt] [2 marks]
    3
  2. Find the values of the constants in order for the model that you did not reject in part (a) to be dimensionally consistent.
    [0pt] [2 marks]
AQA Further AS Paper 2 Mechanics Specimen Q4
5 marks
4 A cricket ball of mass 156 grams is thrown from a point which is 1.5 metres above the ground, with a speed of \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) A tennis ball of mass 58 grams is thrown from the same point, with the same speed.
Prove that both balls hit the ground with the same speed.
Clearly state any assumptions you have made and how you have used them.
[0pt] [5 marks]
AQA Further AS Paper 2 Mechanics Specimen Q5
4 marks
5 Two small smooth discs, \(C\) and \(D\), have equal radii and masses of 2 kg and 3 kg respectively. The discs are sliding on a smooth horizontal surface towards each other and collide directly. Disc \(C\) has speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and disc \(D\) has speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) as they collide. The coefficient of restitution between \(C\) and \(D\) is 0.6 The diagram shows the discs, viewed from above, before the collision.
\includegraphics[max width=\textwidth, alt={}, center]{18522f4c-4aa2-4ef5-898f-5ad2b06e287c-06_343_712_868_753} 5
  1. Show that the speed of \(D\) immediately after the collision is \(1.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), correct to 2 significant figures.
    5
  2. Find the speed of \(C\) immediately after the collision.
    [0pt] [2 marks]
    5
  3. In fact the horizontal surface on which the discs are sliding is not smooth.
    Explain how the introduction of friction will affect your answer to part (b).
    [0pt] [2 marks]
    Turn over for the next question
AQA Further AS Paper 2 Mechanics Specimen Q6
4 marks
6 A car, of mass 1200 kg , moves on a straight horizontal road where it has a maximum speed of \(40 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) When the car travels at a speed of \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) it experiences a resistance force which can be modelled as being of magnitude 30 v newtons. 6
  1. Show that the power output of the car is 48000 W , when it is travelling at its maximum speed. 6
  2. Find the maximum acceleration of the car when it is travelling at a speed of \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
    [0pt] [4 marks]
AQA Further AS Paper 2 Mechanics Specimen Q7
3 marks
7 A disc, of mass 0.15 kg , slides across a smooth horizontal table and collides with a vertical wall which is perpendicular to the path of the disc. The disc is in contact with the wall for 0.02 seconds and then rebounds.
A possible model for the force, \(F\) newtons, exerted on the disc by the wall, whilst in contact, is given by $$F = k t ^ { 2 } ( t - b ) ^ { 2 } \quad \text { for } \quad 0 \leq t \leq 0.020$$ where \(k\) and \(b\) are constants.
The force is initially zero and becomes zero again as the disc loses contact with the wall. 7
  1. State the value of \(b\).
    7
  2. Find the magnitude of the impulse on the disc, giving your answer in terms of \(k\).
    7
  3. The disc is travelling at \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it hits the wall.
    The disc rebounds with a speed of \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
    Find \(k\).
    [0pt] [3 marks]
AQA Further AS Paper 2 Mechanics Specimen Q8
6 marks
8 In this question use \(g = 10 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
A particle, of mass 2 kg , is attached to one end of a light elastic string of natural length 0.2 metres. The other end of the string is attached to a fixed point \(O\).
The particle is pulled down and released from rest at a point 0.6 metres directly below \(O\).
The particle then moves vertically and next comes to rest when it is 0.1 metres below \(O\).
Assume that no air resistance acts on the particle.
8
  1. Find the maximum speed of the particle.
    [0pt] [6 marks]
    8
  2. Describe one way in which the model you have used could be refined.
AQA Further AS Paper 2 Discrete 2024 June Q1
1 marks
1 A connected planar graph has \(v\) vertices, \(e\) edges and \(f\) faces.
Which one of the formulae below is correct? Circle your answer.
[0pt] [1 mark]
\(v + e + f = 2\)
\(v - e + f = 2\)
\(v - e - f = 2\)
\(v + e - f = 2\)
AQA Further AS Paper 2 Discrete 2024 June Q2
2 Find an expression for the number of edges in the complete bipartite graph, \(K _ { m , n }\) Circle your answer.
\(\frac { m } { n }\)
\(m - n\)
\(m + n\)
\(m n\)
AQA Further AS Paper 2 Discrete 2024 June Q3
1 marks
3 Which one of the graphs shown below is semi-Eulerian? Tick ( ✓ ) one box.
[0pt] [1 mark]
\includegraphics[max width=\textwidth, alt={}, center]{2e397d7b-b751-4f2c-aa0e-31dd4a071b56-03_352_335_459_333}
\includegraphics[max width=\textwidth, alt={}, center]{2e397d7b-b751-4f2c-aa0e-31dd4a071b56-03_113_113_612_849}
\includegraphics[max width=\textwidth, alt={}, center]{2e397d7b-b751-4f2c-aa0e-31dd4a071b56-03_254_254_918_370}
\includegraphics[max width=\textwidth, alt={}, center]{2e397d7b-b751-4f2c-aa0e-31dd4a071b56-03_113_113_973_849}
\includegraphics[max width=\textwidth, alt={}, center]{2e397d7b-b751-4f2c-aa0e-31dd4a071b56-03_257_254_1272_370}
\includegraphics[max width=\textwidth, alt={}, center]{2e397d7b-b751-4f2c-aa0e-31dd4a071b56-03_113_113_1329_849}
\includegraphics[max width=\textwidth, alt={}, center]{2e397d7b-b751-4f2c-aa0e-31dd4a071b56-03_250_254_1631_370}
AQA Further AS Paper 2 Discrete 2024 June Q4
4 The set \(S\) is defined as \(S = \{ 1,2,3,4 \}\) 4
  1. Complete the Cayley Table shown below for \(S\) under the binary operation multiplication modulo 5
    \(\times _ { 5 }\)1234
    1
    2
    3
    4
    4
  2. State the identity element for \(S\) under multiplication modulo 5 4
  3. State the self-inverse elements of \(S\) under multiplication modulo 5
AQA Further AS Paper 2 Discrete 2024 June Q5
5 A network of roads connects the villages \(A , B , C , D , E , F\) and \(G\) The weight on each arc in the network represents the distance, in miles, between adjacent villages. The network is shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{2e397d7b-b751-4f2c-aa0e-31dd4a071b56-05_769_983_543_511} 5
  1. Draw, in the space below, the spanning tree of minimum total length for this road network. 5
  2. Find the total length of the spanning tree drawn in part (a). A Young Enterprise Company decides to sell two types of cakes at a breakfast club. The two types of cakes are blueberry and chocolate. From its initial market research, the company knows that it will:
    • sell at most 200 cakes in total
    • sell at least twice as many blueberry cakes as they will chocolate cakes
    • make 20 p profit on each blueberry cake they sell
    • make 15p profit on each chocolate cake they sell.
    The company's objective is to maximise its profit. Formulate the Young Enterprise Company's situation as a linear programming problem.
AQA Further AS Paper 2 Discrete 2024 June Q7
7 The binary operation ∇ is defined as $$a \nabla b = a + b + a b \text { where } a , b \in \mathbb { R }$$ 7
  1. Determine if \(\nabla\) is commutative on \(\mathbb { R }\) Fully justify your answer. 7
  2. Prove that ∇ is associative on \(\mathbb { R }\)
AQA Further AS Paper 2 Discrete 2024 June Q8
1 marks
8 The diagram below shows a network of pipes.
\includegraphics[max width=\textwidth, alt={}, center]{2e397d7b-b751-4f2c-aa0e-31dd4a071b56-08_764_1009_317_497} The uncircled numbers on each arc represent the capacity of each pipe in \(\mathrm { m } ^ { 3 } \mathrm {~s} ^ { - 1 }\)
The circled numbers on each arc represent an initial feasible flow, in \(\mathrm { m } ^ { 3 } \mathrm {~s} ^ { - 1 }\), through the network. The initial flow through pipe \(S D\) is \(x \mathrm {~m} ^ { 3 } \mathrm {~s} ^ { - 1 }\)
The initial flow through pipe \(D C\) is \(y \mathrm {~m} ^ { 3 } \mathrm {~s} ^ { - 1 }\)
The initial flow through pipe \(C B\) is \(\mathrm { z } ^ { 3 } \mathrm {~s} ^ { - 1 }\) 8
  1. By considering the flows at the source and the sink, explain why \(x = 7\)
    8
    		3. 8
  3. Prove that the maximum flow through the network is at most \(27 \mathrm {~m} ^ { 3 } \mathrm {~s} ^ { - 1 }\)
    4. \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)
    [1 mark] \(\_\_\_\_\) \(\_\_\_\_\)
AQA Further AS Paper 2 Discrete 2024 June Q9
9 Robert, a project manager, and his team of builders are working on a small building project. Robert has divided the project into ten activities labelled \(A , B , C , D , E , F , G , H , I\) and \(J\) as shown in the precedence table below:
ActivityImmediate Predecessor(s)Duration (Days)
ANone1
BNone1
CA10
DA2
EB, D5
F\(E\)6
G\(E\)1
H\(F\)1
\(I\)\(F\)2
JC, G, H, I4
9
  1. On the opposite page, construct an activity network for the project and fill in the earliest start time and latest finish time for each activity. 9
  2. Robert claims that the project can be completed in 20 days.
    Comment on the validity of Robert's claim.
    \includegraphics[max width=\textwidth, alt={}, center]{2e397d7b-b751-4f2c-aa0e-31dd4a071b56-11_467_440_239_534}
AQA Further AS Paper 2 Discrete 2024 June Q10
10 Bilal and Mayon play a zero-sum game. The game is represented by the following pay-off matrix for Bilal, where \(x\) is an integer.
Mayon
\cline { 2 - 5 }Strategy\(\mathbf { M } _ { \mathbf { 1 } }\)\(\mathbf { M } _ { \mathbf { 2 } }\)\(\mathbf { M } _ { \mathbf { 3 } }\)
Bilal\(\mathbf { B } _ { \mathbf { 1 } }\)- 2- 11
\cline { 2 - 5 }\(\mathbf { B } _ { \mathbf { 2 } }\)4- 31
\cline { 2 - 5 }\(\mathbf { B } _ { \mathbf { 3 } }\)- 1\(x\)0
The game has a stable solution. 10
  1. Show that there is only one possible value for \(x\)
    Fully justify your answer.
    13 10
  2. State the value of the game for Bilal.
AQA Further Paper 1 Specimen Q1
1
- 1
1 \end{array} \right] \quad \left[ \begin{array} { l } 3
0