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	Jadzia
Strategy	X	Y	Z
A	-3	2	3
Ben B	6	0	-4
C	7	-1	1
D	6	-2	1

State, with a reason, which strategy Ben should never play. [1 mark]
Determine whether or not the game has a stable solution. Fully justify your answer. [3 marks]
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.

Find the value of $x$ Fully justify your answer. [4 marks]
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]
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.

Determine the number of distinct subgroups of $G$ Fully justify your answer. [2 marks]
$G$ 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]
$G$ contains an element $g^r$, where $r < p$ Find, in terms of $g$, $r$ and $p$, the inverse of $g^r$ [2 marks]
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}

On Figure 1 above, add a supersink $T$ to the network. [2 marks]
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 Path Flow
Maximum flow \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_
Prove that the flow found in part (b) is the maximum flow through the network. [2 marks]
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]
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]
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]

Set	Binary 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 Bound	Best Upper Bound
15	48	$\square$
15	51	$\square$
19	48	$\square$
19	51	$\square$

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

	Strategy	W	X	Y	Z
\multirow{4}{*}{Daniel}	A	3	$-2$	1	4
	B	5	1	$-4$	1
	C	2	$-1$	1	2
	D	$-3$	0	2	$-1$

Neither player has any strategies which can be ignored due to dominance.

Prove that the game does not have a stable solution. Fully justify your answer. [3 marks]
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]
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]

AQA Further Paper 3 Discrete 2024 June Q7

12 marks Standard +0.3

By considering associativity, show that the set of integers does not form a group under the binary operation of subtraction. Fully justify your answer. [2 marks]
The group $G$ is formed by the set $$\{1, 7, 8, 11, 12, 18\}$$ under the operation of multiplication modulo 19
1. Complete the Cayley table for $G$ [3 marks]
  $\times_{19}$ 1 7 8 11 12 18
  1 1 7 8 11 12 18
  7 7 11
  8 8 7
  11 11 7
  12 12 11
  18 18 1
2. State the inverse of 11 in $G$ [1 mark]
1. State, with a reason, the possible orders of the proper subgroups of $G$ [2 marks]
2. Find all the proper subgroups of $G$ Give your answers in the form $\langle g \rangle, \times_{19}$ where $g \in G$ [3 marks]
3. The group $H$ is such that $G \cong H$ State a possible name for $H$ [1 mark]

AQA Further Paper 3 Discrete 2024 June Q8

8 marks Standard +0.8

Figure 1 shows a network of water pipes. The number on each arc represents the upper capacity for each pipe in litres per second. The numbers in the circles represent an initial feasible flow of 103 litres per second. \includegraphics{figure_1}

On Figure 1 above, add a supersource $S$ and a supersink $T$ to the network. [2 marks]
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 opposite page, in your solution. [4 marks]
Augmenting Path Extra Flow
Maximum Flow ______________ litres per second
While the flow through the network is at its maximum value, the pipe $EG$ develops a leak. To repair the leak, an engineer turns off the flow of water through $EG$ The engineer claims that the maximum flow of water through the network will reduce by 31 litres per second. Comment on the validity of the engineer's claim. [2 marks]

AQA Further Paper 3 Discrete 2024 June Q9

6 marks Challenging +1.2

Janet and Samantha play a zero-sum game. The game is represented by the following pay-off matrix for Janet. Samantha

	Strategy	$S_1$	$S_2$	$S_3$
\multirow{4}{*}{Janet}	$J_1$	2	7	6
	$J_2$	5	5	1
	$J_3$	4	3	8
	$J_4$	1	6	4

Explain why Janet should never play strategy $J_4$ [1 mark]
Janet wants to maximise her winnings from the game. She defines the following variables. $p_1 = $ the probability of Janet playing strategy $J_1$ $p_2 = $ the probability of Janet playing strategy $J_2$ $p_3 = $ the probability of Janet playing strategy $J_3$ $v = $ the value of the game for Janet Janet then formulates her situation as the following linear programming problem. Maximise $P = v$ subject to $2p_1 + 5p_2 + 4p_3 \geq v$ $7p_1 + 5p_2 + 3p_3 \geq v$ $6p_1 + p_2 + 8p_3 \geq v$ and $p_1 + p_2 + p_3 \leq 1$ $p_1, p_2, p_3 \geq 0$
1. Complete the initial Simplex tableau for Janet's situation in the grid below. [2 marks]
  $P$ $v$ $p_1$ $p_2$ $p_3$ value
2. Hence, perform one iteration of the Simplex algorithm, showing your answer on the grid below. [2 marks]
  $P$ $v$ $p_1$ $p_2$ $p_3$ value
Further iterations of the Simplex algorithm are performed until an optimal solution is reached. The grid below shows part of the final Simplex tableau.
$p_1$ $p_2$ value
1 0 $\frac{1}{12}$
0 1 $\frac{1}{2}$
Find the probability of Janet playing strategy $J_3$ when she is playing to maximise her winnings from the game. [1 mark]

AQA Further Paper 3 Discrete 2024 June Q10

7 marks Standard +0.3

A project is undertaken by Higton Engineering Ltd. The project is broken down into 11 separate activities $A$, $B$, $\ldots$, $K$ Figure 3 below shows a completed activity network for the project, along with the earliest start time, duration, latest finish time and the number of workers required for each activity. All times and durations are given in days. \includegraphics{figure_3}

Write down the critical path. [1 mark]
Using Figure 4 below, draw a resource histogram for the project to show how the project can be completed in the minimum possible time. Assume that each activity is to start as early as possible. [3 marks] \includegraphics{figure_4}
Higton Engineering Ltd only has four workers available to work on the project. Find the minimum completion time for the project. Use Figure 5 below in your answer. [3 marks] \includegraphics{figure_5} Minimum completion time _____________________________________

	Julian
Strategy	\(\mathbf{J_1}\)	\(\mathbf{J_2}\)	\(\mathbf{J_3}\)
\(\mathbf{K_1}\)	7
Kira \(\mathbf{K_2}\)
\(\mathbf{K_3}\)

Set	Binary 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	\(\square\)

	Strategy	W	X	Y	Z
\multirow{4}{*}{Daniel}	A	3	\(-2\)	1	4
	B	5	1	\(-4\)	1
	C	2	\(-1\)	1	2
	D	\(-3\)	0	2	\(-1\)

	Strategy	\(S_1\)	\(S_2\)	\(S_3\)
\multirow{4}{*}{Janet}	\(J_1\)	2	7	6
	\(J_2\)	5	5	1
	\(J_3\)	4	3	8
	\(J_4\)	1	6	4

\(P\)	\(v\)	\(p_1\)	\(p_2\)	\(p_3\)	value

\(*\)

Augmenting Path	Flow

\(\times_{19}\)	1	7	8	11	12	18
1	1	7	8	11	12	18
7	7	11
8	8		7
11	11			7
12	12				11
18	18					1

Augmenting Path	Extra Flow

Questions — AQA (3620 questions)

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AQA Further Paper 3 Discrete 2022 June Q1

AQA Further Paper 3 Discrete 2022 June Q2

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AQA Further Paper 3 Discrete 2022 June Q4

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AQA Further Paper 3 Discrete 2022 June Q10

AQA Further Paper 3 Discrete 2024 June Q1

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AQA Further Paper 3 Discrete 2024 June Q5

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AQA Further Paper 3 Discrete 2024 June Q7

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