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AQA Further Paper 3 Mechanics 2024 June Q1
1 marks Easy -1.8
A particle moves in a circular path so that at time \(t\) seconds its position vector, \(\mathbf{r}\) metres, is given by $$\mathbf{r} = 4\sin(2t)\mathbf{i} + 4\cos(2t)\mathbf{j}$$ Find the velocity of the particle, in m s\(^{-1}\), when \(t = 0\) Circle your answer. [1 mark] \(8\mathbf{i}\) \quad \(-8\mathbf{j}\) \quad \(8\mathbf{j}\) \quad \(8\mathbf{i} - 8\mathbf{j}\)
AQA Further Paper 3 Mechanics 2024 June Q2
1 marks Easy -1.2
As a particle moves along a straight horizontal line, it is subjected to a force \(F\) newtons that acts in the direction of motion of the particle. At time \(t\) seconds, \(F = \frac{t}{5}\) Calculate the magnitude of the impulse on the particle between \(t = 0\) and \(t = 3\) Circle your answer. [1 mark] 0.3 N s \quad 0.6 N s \quad 0.9 N s \quad 1.8 N s
AQA Further Paper 3 Mechanics 2024 June Q3
1 marks Moderate -0.8
A conical pendulum consists of a light string and a particle of mass \(m\) kg The conical pendulum completes horizontal circles with radius \(r\) metres and angular speed \(\omega\) radians per second. The string makes an angle \(\theta\) with the downward vertical. The tension in the string is \(T\) newtons. The conical pendulum and the forces acting on the particle are shown in the diagram. \includegraphics{figure_3} Which one of the following statements is correct? Tick (\(\checkmark\)) one box. [1 mark] \(T \cos \theta = mr\omega^2\) \quad \(\square\) \(T \sin \theta = mr\omega^2\) \quad \(\square\) \(T \cos \theta = \frac{m\omega^2}{r}\) \quad \(\square\) \(T \sin \theta = \frac{m\omega^2}{r}\) \quad \(\square\)
AQA Further Paper 3 Mechanics 2024 June Q4
5 marks Moderate -0.8
A particle of mass 3 kg is attached to one end of a light inextensible string. The other end of the string is attached to a fixed point on a smooth horizontal surface. The particle is set into motion so that it moves with a constant speed 4 m s\(^{-1}\) in a circular path with radius 0.8 metres on the horizontal surface.
  1. Find the acceleration of the particle. [2 marks]
  2. Find the tension in the string. [1 mark]
  3. Show that the angular speed of the particle is 48 revolutions per minute correct to two significant figures. [2 marks]
AQA Further Paper 3 Mechanics 2024 June Q5
4 marks Standard +0.3
When a sphere of radius \(r\) metres is falling at \(v\) m s\(^{-1}\) it experiences an air resistance force \(F\) newtons. The force is to be modelled as $$F = kr^\alpha v^\beta$$ where \(k\) is a constant with units kg m\(^{-2}\)
  1. State the dimensions of \(F\) [1 mark]
  2. Use dimensional analysis to find the value of \(\alpha\) and the value of \(\beta\) [3 marks]
AQA Further Paper 3 Mechanics 2024 June Q6
10 marks Standard +0.3
In this question use \(g = 9.8\) m s\(^{-2}\) A light elastic string has natural length 3 metres and modulus of elasticity 18 newtons. One end of the elastic string is attached to a particle of mass 0.25 kg The other end of the elastic string is attached to a fixed point \(O\) The particle is released from rest at a point \(A\), which is 4.5 metres vertically below \(O\)
  1. Calculate the elastic potential energy of the string when the particle is at \(A\) [2 marks]
  2. The point \(B\) is 3 metres vertically below \(O\) Calculate the gravitational potential energy gained by the particle as it moves from \(A\) to \(B\) [2 marks]
  3. Find the speed of the particle at \(B\) [3 marks]
  4. The point \(C\) is 3.6 metres vertically below \(O\) Explain, showing any calculations that you make, why the speed of the particle is increasing the first time that the particle is at \(C\) [3 marks]
AQA Further Paper 3 Mechanics 2024 June Q7
10 marks Standard +0.3
A sphere, of mass 0.2 kg, moving on a smooth horizontal surface, collides with a fixed wall. Before the collision the sphere moves with speed 5 m s\(^{-1}\) at an angle of 60° to the wall. After the collision the sphere moves with speed \(v\) m s\(^{-1}\) at an angle of \(\theta\)° to the wall. The velocities are shown in the diagram below. \includegraphics{figure_7} The coefficient of restitution between the wall and the sphere is 0.7
  1. Assume that the wall is smooth.
    1. Find the value of \(v\) Give your answer to two significant figures. [4 marks]
    2. Find the value of \(\theta\) Give your answer to the nearest whole number. [2 marks]
    3. Find the magnitude of the impulse exerted on the sphere by the wall. Give your answer to two significant figures. [2 marks]
  2. In reality the wall is not smooth. Explain how this would cause a change in the magnitude of the impulse calculated in part (a)(iii). [2 marks]
AQA Further Paper 3 Mechanics 2024 June Q8
10 marks Challenging +1.2
The finite region enclosed by the line \(y = kx\), the \(x\)-axis and the line \(x = 5\) is rotated through 360° around the \(x\) axis to form a solid cone.
    1. Use integration to show that the position of the centre of mass of the cone is independent of \(k\) [4 marks]
    2. State the distance between the base of the cone and its centre of mass. [1 mark]
  2. State one assumption that you have made about the cone. [1 mark]
  3. The plane face of the cone is placed on a rough inclined plane. The coefficient of friction between the cone and the plane is 0.8 The angle between the plane and the horizontal is gradually increased from 0° Find the range of values of \(k\) for which the cone slides before it topples. [4 marks]
AQA Further Paper 3 Mechanics 2024 June Q9
8 marks Challenging +1.8
A small sphere, of mass \(m\), is attached to one end of a light inextensible string of length \(a\) The other end of the string is attached to a fixed point \(O\) The sphere is at rest in equilibrium directly below \(O\) when it is struck, giving it a horizontal impulse of magnitude \(mU\) After the impulse, the sphere follows a circular path in a vertical plane containing the point \(O\) until the string becomes slack at the point \(C\) At \(C\) the string makes an angle of 30° with the upward vertical through \(O\), as shown in the diagram below. \includegraphics{figure_9}
  1. Show that $$U^2 = \frac{ag}{2}\left(4 + 3\sqrt{3}\right)$$ where \(g\) is the acceleration due to gravity. [6 marks]
  2. With reference to any modelling assumptions that you have made, explain why giving your answer as an inequality would be more appropriate, and state this inequality. [2 marks]
AQA Further Paper 3 Discrete 2022 June Q1
1 marks Moderate -0.8
The graph \(G\) has a subgraph isomorphic to \(K_5\), the complete graph with 5 vertices. Which of the following statements about \(G\) must be true? Tick \((\checkmark)\) one box. [1 mark] \(G\) is not connected \(G\) is not Hamiltonian \(G\) is not planar \(G\) is not simple
AQA Further Paper 3 Discrete 2022 June Q2
1 marks Easy -1.8
Graph \(A\) is a connected planar graph with 12 vertices, 18 edges and \(n\) faces. Find the value of \(n\) Circle your answer. [1 mark] 4 8 28 32
AQA Further Paper 3 Discrete 2022 June Q3
1 marks Standard +0.3
A company undertakes a project which consists of 12 activities, \(A\), \(B\), \(C\), \(\ldots\), \(L\) Each activity requires one worker. The resource histogram below shows the duration of each activity. Each activity begins at its earliest start time. The path \(ADGJL\) is critical. \includegraphics{figure_1} The company only has two workers available to work on the project. Which of the following could be a correctly levelled histogram? Tick \((\checkmark)\) one box. [1 mark] \includegraphics{figure_2} \includegraphics{figure_3} \includegraphics{figure_4} \includegraphics{figure_5}
AQA Further Paper 3 Discrete 2022 June Q4
6 marks Standard +0.3
Ben and Jadzia play a zero-sum game. The game is represented by the following pay-off matrix for Ben.
Jadzia
StrategyXYZ
A-323
Ben B60-4
C7-11
D6-21
  1. State, with a reason, which strategy Ben should never play. [1 mark]
  2. Determine whether or not the game has a stable solution. Fully justify your answer. [3 marks]
  3. Ben knows that Jadzia will always play her play-safe strategy. Explain how Ben can maximise his expected pay-off. [2 marks]
AQA Further Paper 3 Discrete 2022 June Q5
6 marks Standard +0.8
A council wants to convert all of the street lighting in a village to use LED lighting. The network below shows each street in the village. Each node represents a junction and the weight of each arc represents the length, in metres, of the street. The street lights are only positioned on one side of each street in the village. \includegraphics{figure_6} The total length of all of the streets in the village is 2250 metres. In order to determine the total number of street lights in the village, a council worker is required to walk along every street in the village at least once, starting and finishing at the same junction. The shortest possible distance the council worker can walk in order to determine the total number of street lights in the village is \(x\) metres.
  1. Find the value of \(x\) Fully justify your answer. [4 marks]
  2. A new council regulation requires that the mean distance along a street between adjacent LED street lights in a village be less than 25 metres. The council worker counted 91 different street lights on their journey around the village. Determine whether or not the village will meet the new council regulation. [2 marks]
AQA Further Paper 3 Discrete 2022 June Q6
6 marks Standard +0.3
Bill Durrh Ltd undertake a construction project. The activity network for the project is shown below. The duration of each activity is given in weeks. \includegraphics{figure_7}
    1. Find the earliest start time and the latest finish time for each activity and show these values on the activity network above. [3 marks]
    2. Identify all of the critical activities. [1 mark]
  2. The manager of Bill Durrh Ltd recruits some additional temporary workers in order to reduce the duration of one activity by 2 weeks. The manager wants to reduce the minimum completion time of the project by the largest amount. State, with a reason, which activity the manager should choose. [2 marks]
AQA Further Paper 3 Discrete 2022 June Q7
8 marks Standard +0.3
The group \(G\) has binary operation \(*\) and order \(p\), where \(p\) is a prime number.
  1. Determine the number of distinct subgroups of \(G\) Fully justify your answer. [2 marks]
  2. \(G\) contains an element \(g\) which has period \(p\)
    1. State the general name given to elements such as \(g\) [1 mark]
    2. State the name of a group that is isomorphic to \(G\) [1 mark]
  3. \(G\) contains an element \(g^r\), where \(r < p\) Find, in terms of \(g\), \(r\) and \(p\), the inverse of \(g^r\) [2 marks]
  4. In the case when \(p = 5\) and the binary operation \(*\) represents addition modulo 5, \(G\) contains the elements 0, 1, 2, 3 and 4
    1. Explain why \(G\) is closed. [1 mark]
    2. Complete the Cayley table for \((G, *)\) [1 mark]
      \(*\)
AQA Further Paper 3 Discrete 2022 June Q8
10 marks Standard +0.8
Figure 1 shows a network of gas pipes. The numbers on each arc represent the lower and upper capacity for each pipe in \(\text{m}^3 \text{s}^{-1}\) The numbers in the circles represent an initial feasible flow of 73 \(\text{m}^3 \text{s}^{-1}\) through the network. \includegraphics{figure_8}
  1. On Figure 1 above, add a supersink \(T\) to the network. [2 marks]
  2. Using flow augmentation, find the maximum flow through the network. You must indicate any flow augmenting paths clearly in the table below. You may use Figure 2, on the page opposite, in your solution. [4 marks]
    Augmenting PathFlow
    Maximum flow \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_
  3. Prove that the flow found in part (b) is the maximum flow through the network. [2 marks]
  4. A trainee engineer claims that increasing the upper capacity of the pipe \(AG\) will increase the maximum flow through the network, as the flow through this pipe cannot currently be increased. Comment on the validity of the trainee's claim. [2 marks]
AQA Further Paper 3 Discrete 2022 June Q9
6 marks Standard +0.8
The binary operation \(\oplus\) acts on the positive integers \(x\) and \(y\) such that $$x \oplus y = x + y + 8 \pmod{k^2 - 16k + 74}$$ where \(k\) is a positive integer.
    1. Show that \(\oplus\) is commutative. [1 mark]
    2. Determine whether or not \(\oplus\) is associative. Fully justify your answer. [2 marks]
  2. Find the values of \(k\) for which 3 is an identity element for the set of positive integers under \(\oplus\) [3 marks]
AQA Further Paper 3 Discrete 2022 June Q10
5 marks Standard +0.3
Kira and Julian play a zero-sum game that does not have a stable solution. Kira has three strategies to choose from: \(\mathbf{K_1}\), \(\mathbf{K_2}\) and \(\mathbf{K_3}\) To determine her optimal mixed strategy, Kira begins by defining the following variables: \(v =\) value of the game for Kira \(p_1 =\) probability of Kira playing strategy \(\mathbf{K_1}\) \(p_2 =\) probability of Kira playing strategy \(\mathbf{K_2}\) \(p_3 =\) probability of Kira playing strategy \(\mathbf{K_3}\) Kira then formulates the following linear programming problem. Maximise \(v\) subject to \(7p_1 + p_2 + 8p_3 \geq v\) \(3p_1 + 7p_2 + 2p_3 \geq v\) \(9p_1 + 2p_2 + 4p_3 \geq v\) and \(p_1 + p_2 + p_3 \leq 1\) \(p_1, p_2, p_3 \geq 0\)
    1. Explain why the condition \(p_1 + p_2 + p_3 \leq 1\) is necessary in Kira's linear programming problem. [1 mark]
    2. Explain why the condition \(p_1, p_2, p_3 \geq 0\) is necessary in Kira's linear programming problem. [1 mark]
  2. Julian has three strategies to choose from: \(\mathbf{J_1}\), \(\mathbf{J_2}\) and \(\mathbf{J_3}\) Complete the following pay-off matrix which represents the game for Kira. [3 marks]
    Julian
    Strategy\(\mathbf{J_1}\)\(\mathbf{J_2}\)\(\mathbf{J_3}\)
    \(\mathbf{K_1}\)7
    Kira \(\mathbf{K_2}\)
    \(\mathbf{K_3}\)
AQA Further Paper 3 Discrete 2024 June Q1
1 marks Moderate -0.5
Which one of the following sets forms a group under the given binary operation? Tick \((\checkmark)\) one box. [1 mark]
SetBinary Operation
\(\{1, 2, 3\}\)Addition modulo 4\(\square\)
\(\{1, 2, 3\}\)Multiplication modulo 4\(\square\)
\(\{0, 1, 2, 3\}\)Addition modulo 4\(\square\)
\(\{0, 1, 2, 3\}\)Multiplication modulo 4\(\square\)
AQA Further Paper 3 Discrete 2024 June Q2
1 marks Easy -2.0
A student is trying to find the solution to the travelling salesperson problem for a network. They correctly find two lower bounds for the solution: 15 and 19 They also correctly find two upper bounds for the solution: 48 and 51 Based on the above information only, which of the following pairs give the best lower bound and best upper bound for the solution of this problem? Tick \((\checkmark)\) one box. [1 mark]
Best Lower BoundBest Upper Bound
1548\(\square\)
1551\(\square\)
1948\(\square\)
1951\(\square\)
AQA Further Paper 3 Discrete 2024 June Q3
1 marks Moderate -0.8
The simple-connected graph \(G\) has the adjacency matrix $$\begin{array}{c|cccc} & A & B & C & D \\ \hline A & 0 & 1 & 1 & 1 \\ B & 1 & 0 & 1 & 0 \\ C & 1 & 1 & 0 & 1 \\ D & 1 & 0 & 1 & 0 \\ \end{array}$$ Which one of the following statements about \(G\) is true? Tick \((\checkmark)\) one box. [1 mark] \(G\) is a tree \(\square\) \(G\) is complete \(\square\) \(G\) is Eulerian \(\square\) \(G\) is planar \(\square\)
AQA Further Paper 3 Discrete 2024 June Q4
4 marks Standard +0.8
Daniel and Jackson play a zero-sum game. The game is represented by the following pay-off matrix for Daniel. Jackson
StrategyWXYZ
\multirow{4}{*}{Daniel}A3\(-2\)14
B51\(-4\)1
C2\(-1\)12
D\(-3\)02\(-1\)
Neither player has any strategies which can be ignored due to dominance.
  1. Prove that the game does not have a stable solution. Fully justify your answer. [3 marks]
  2. Determine the play-safe strategy for each player. [1 mark] Play-safe strategy for Daniel _______________________________________________ Play-safe strategy for Jackson ______________________________________________
AQA Further Paper 3 Discrete 2024 June Q5
4 marks Moderate -0.8
The owners of a sports stadium want to install electric car charging points in each of the stadium's nine car parks. An engineer creates a plan which requires installing electrical connections so that each car park is connected, directly or indirectly, to the stadium's main electricity power supply. The engineer produces the network shown below, where the nodes represent the stadium's main electricity power supply \(X\) and the nine car parks \(A\), \(B\), \(\ldots\), \(I\) \includegraphics{figure_5} Each arc represents a possible electrical connection which could be installed. The weight on each arc represents the time, in hours, it would take to install the electrical connection. The electrical connections can only be installed one at a time. To reduce disruption, the owners of the sports stadium want the required electrical connections to be installed in the minimum possible total time.
    1. Determine the electrical connections that should be installed. [2 marks]
    2. Find the minimum possible total time needed to install the required electrical connections. [1 mark]
  2. Following the installation of the electrical connections, some of the car parks have an indirect connection to the stadium's main electricity power supply. Give one limitation of this installation. [1 mark]
AQA Further Paper 3 Discrete 2024 June Q6
6 marks Standard +0.8
A company delivers parcels to houses in a village, using a van. The network below shows the roads in the village. Each node represents a road junction and the weight of each arc represents the length, in miles, of the road between the junctions. \includegraphics{figure_6} The total length of all of the roads in the village is 31.4 miles. On one particular day, the driver is due to make deliveries to at least one house on each road, so the van must travel along each road at least once. However, the driver has forgotten to add fuel to the van and it only has 4.5 litres of fuel to use to make its deliveries. The van uses, on average, 1 litre of fuel to travel 7.8 miles along the roads of this village. Whilst making each delivery, the driver turns off the van's engine so it does not use any fuel. Determine whether the van has enough fuel for the driver to make all of the deliveries to houses on each road of the village, starting and finishing at the same junction. Fully justify your answer. [6 marks]