Questions Further Paper 3 Discrete

1. The mining company wants to maximise the amount of precious metal it can extract from the new mine. Determine the tunnels the mining company should use.
  3
(ii) Estimate the maximum amount of precious metal the mining company can extract from the new mine. 3
Comment on why the maximum amount of precious metal the mining company can extract from the new mine may be different from your answer to part (a)(ii).
[0pt] [2 marks]
3
Before the mining company begins work on the new mine, a government survey prevents the mining company drilling the tunnel represented by $C F$. Determine the effect, if any, the government survey has on your answers to part (a)(i) and part (a)(ii).

AQA Further Paper 3 Discrete 2021 June Q4

2 marks

4 Derrick, a tanker driver, is required to deliver fuel to 6 different service stations $A , B$, $C , D , E$ and $F$. Derrick needs to begin and finish his delivery journey at the refinery $O$.
The distances, in miles, between the 7 locations which have a direct road between them are shown in the network below.
\includegraphics[max width=\textwidth, alt={}, center]{59347089-ea4a-4ee6-b40e-1ab78aa7cdc3-06_921_1440_628_303} Derrick spends 30 minutes at each service station to complete the fuel delivery.
When driving, the tanker travels at an average speed of 40 miles per hour.
The minimum total time that it takes Derrick to travel to and deliver fuel to all 6 service stations, starting and finishing at the refinery, is $T$ minutes. 4

Using the nearest neighbour algorithm starting from the refinery, find an upper bound for $T$
4
Before setting off to make his fuel deliveries, Derrick is notified that, due to a low bridge, the road represented by CE is not suitable for tankers to travel along. State, with a reason, the effect this new information has on your answer to part (a).
[0pt] [2 marks]

AQA Further Paper 3 Discrete 2021 June Q5

Describe the conditions necessary for a set of elements, $S$, under a binary operation * to form a group.
5
In the multiplicative group of integers modulo 13, the group $G$ is defined as $$G = \left( \langle 10 \rangle , \times _ { 13 } \right)$$ 5
1. Explain why $G$ is an abelian group.
  5
(ii) Find the order of $G$.
5
State the identity element of $G$ and prove it is an identity element. Fully justify your answer.
5
Find all the proper non-trivial subgroups of $G$, giving your answers in the form $\left( \langle g \rangle , \times _ { 13 } \right)$, where $g$ is an integer less than 13

AQA Further Paper 3 Discrete 2021 June Q6

6 marks

A connected planar graph has $( x + 1 ) ^ { 2 }$ vertices, $( 25 + 2 x - 2 y )$ edges and $( y - 1 ) ^ { 2 }$ faces, where $x > 0$ and $y > 0$ Find the possible values for the number of vertices, edges and faces for the graph.
[0pt] [6 marks]
LL
6
Explain why $K _ { 6 }$, the complete graph with 6 vertices, is not planar. Fully justify your answer.

AQA Further Paper 3 Discrete 2021 June Q7

7 Avon and Roj play a zero-sum game. The game is represented by the following pay-off matrix for Avon. 7 (c) (i) Find the optimal mixed strategy for Avon.
7 (c) (ii) Find the value of the game for Avon.
7 (d) Roj thinks that his best outcome from the game is to play strategy $\mathbf { R } _ { \mathbf { 2 } }$ each time. Avon notices that Roj always plays strategy $\mathbf { R } _ { \mathbf { 2 } }$ and Avon wants to use this knowledge to maximise his expected pay-off from the game. Explain how your answer to part (c)(i) should change and find Avon's maximum expected pay-off from the game.
\includegraphics[max width=\textwidth, alt={}, center]{59347089-ea4a-4ee6-b40e-1ab78aa7cdc3-16_2490_1735_219_139}

AQA Further Paper 3 Discrete 2022 June Q1

1 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.
$G$ is not connected
\includegraphics[max width=\textwidth, alt={}, center]{bcb1dd40-4e54-4ac7-a623-3a4b46e3ea9d-03_104_108_872_973}
$G$ is not Hamiltonian
\includegraphics[max width=\textwidth, alt={}, center]{bcb1dd40-4e54-4ac7-a623-3a4b46e3ea9d-03_108_108_1005_973} G is not planar
\includegraphics[max width=\textwidth, alt={}, center]{bcb1dd40-4e54-4ac7-a623-3a4b46e3ea9d-03_108_108_1142_973}
$G$ is not simple □

AQA Further Paper 3 Discrete 2022 June Q2

2 Graph $A$ is a connected planar graph with 12 vertices, 18 edges and $n$ faces.
Find the value of $n$ Circle your answer. 4
8
28
32 Turn over for the next question

AQA Further Paper 3 Discrete 2022 June Q4

4
8
28
32 Turn over for the next question 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 $A D G J L$ is critical. Number of workers
\includegraphics[max width=\textwidth, alt={}, center]{bcb1dd40-4e54-4ac7-a623-3a4b46e3ea9d-04_504_1145_719_548} 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. Number of workers
\includegraphics[max width=\textwidth, alt={}, center]{bcb1dd40-4e54-4ac7-a623-3a4b46e3ea9d-05_510_1145_502_459} Number of workers
\includegraphics[max width=\textwidth, alt={}, center]{bcb1dd40-4e54-4ac7-a623-3a4b46e3ea9d-05_515_1145_1059_459} Number of workers
\includegraphics[max width=\textwidth, alt={}, center]{bcb1dd40-4e54-4ac7-a623-3a4b46e3ea9d-05_517_1147_1619_459} Number of workers
\includegraphics[max width=\textwidth, alt={}, center]{bcb1dd40-4e54-4ac7-a623-3a4b46e3ea9d-05_517_1149_2179_458} 4 Ben and Jadzia play a zero-sum game. The game is represented by the following pay-off matrix for Ben.

\multirow{6}{*}{Ben}	Jadzia
	Strategy	X	Y	Z
	A	-3	2	3
	B	6	0	-4
	C	7	-1	1
	D	6	-2	1

State, with a reason, which strategy Ben should never play.
4
Determine whether or not the game has a stable solution.
Fully justify your answer.
4
Ben knows that Jadzia will always play her play-safe strategy. Explain how Ben can maximise his expected pay-off.

AQA Further Paper 3 Discrete 2022 June Q5

4 marks

5 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[max width=\textwidth, alt={}, center]{bcb1dd40-4e54-4ac7-a623-3a4b46e3ea9d-08_1036_1514_616_264} 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. 5

Find the value of $x$
Fully justify your answer.
[0pt] [4 marks] 5
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.

AQA Further Paper 3 Discrete 2022 June Q6

1. Find the earliest start time and the latest finish time for each activity and show these values on the activity network above. 6
(ii) Identify all of the critical activities. 6
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.

AQA Further Paper 3 Discrete 2022 June Q8

8
28
32 Turn over for the next question 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 $A D G J L$ is critical. Number of workers
\includegraphics[max width=\textwidth, alt={}, center]{bcb1dd40-4e54-4ac7-a623-3a4b46e3ea9d-04_504_1145_719_548} 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. Number of workers
\includegraphics[max width=\textwidth, alt={}, center]{bcb1dd40-4e54-4ac7-a623-3a4b46e3ea9d-05_510_1145_502_459} Number of workers
\includegraphics[max width=\textwidth, alt={}, center]{bcb1dd40-4e54-4ac7-a623-3a4b46e3ea9d-05_515_1145_1059_459} Number of workers
\includegraphics[max width=\textwidth, alt={}, center]{bcb1dd40-4e54-4ac7-a623-3a4b46e3ea9d-05_517_1147_1619_459} Number of workers
\includegraphics[max width=\textwidth, alt={}, center]{bcb1dd40-4e54-4ac7-a623-3a4b46e3ea9d-05_517_1149_2179_458} 4 Ben and Jadzia play a zero-sum game. The game is represented by the following pay-off matrix for Ben.

\multirow{6}{*}{Ben}	Jadzia
	Strategy	X	Y	Z
	A	-3	2	3
	B	6	0	-4
	C	7	-1	1
	D	6	-2	1

State, with a reason, which strategy Ben should never play.
4
Determine whether or not the game has a stable solution.
Fully justify your answer.
4
Ben knows that Jadzia will always play her play-safe strategy. Explain how Ben can maximise his expected pay-off.

AQA Further Paper 3 Discrete 2022 June Q9

9 The binary operation ⊕ acts on the positive integers $x$ and $y$ such that $$x \oplus y = x + y + 8 \quad \left( \bmod k ^ { 2 } - 16 k + 74 \right)$$ where $k$ is a positive integer.
9

1. Show that ⊕ is commutative.
  9
(ii) Determine whether or not ⊕ is associative.
Fully justify your answer.
9
Find the values of $k$ for which 3 is an identity element for the set of positive integers under

AQA Further Paper 3 Discrete 2022 June Q10

10 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. $$\begin{array} { l l } \text { Maximise } & v
\text { subject to } & 7 p _ { 1 } + p _ { 2 } + 8 p _ { 3 } \geq v
& 3 p _ { 1 } + 7 p _ { 2 } + 2 p _ { 3 } \geq v
& 9 p _ { 1 } + 2 p _ { 2 } + 4 p _ { 3 } \geq v \end{array}$$ and $$\begin{array} { r } p _ { 1 } + p _ { 2 } + p _ { 3 } \leq 1
p _ { 1 } , p _ { 2 } , p _ { 3 } \geq 0 \end{array}$$ 10

1. Explain why the condition $p _ { 1 } + p _ { 2 } + p _ { 3 } \leq 1$ is necessary in Kira's linear programming problem. 10
(ii) Explain why the condition $p _ { 1 } , p _ { 2 } , p _ { 3 } \geq 0$ is necessary in Kira's linear programming problem. 10

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.

	Julian
\cline { 2 - 5 }	Strategy	$\mathbf { J } _ { 1 }$	$\mathbf { J } _ { 2 }$	$\mathbf { J } _ { 3 }$
\multirow{3}{*}{Kira}	$\mathbf { K } _ { 1 }$	7
\cline { 2 - 5 }	$\mathbf { K } _ { 2 }$
\cline { 2 - 5 }	$\mathbf { K } _ { 3 }$

AQA Further Paper 3 Discrete 2023 June Q1

1 The simple-connected graph $G$ is shown below.
\includegraphics[max width=\textwidth, alt={}, center]{5ff6e3bb-6392-49cf-b64d-23bc595cd92e-02_271_515_632_762} The graph $G$ has $n$ faces. State the value of $n$ Circle your answer. 2345

AQA Further Paper 3 Discrete 2023 June Q2

1 marks

2 Jonathan and Hoshi play a zero-sum game.
The game is represented by the following pay-off matrix for Jonathan.

\multirow{6}{*}{Jonathan}	Hoshi
	Strategy	$\mathbf { H } _ { \mathbf { 1 } }$	$\mathbf { H } _ { \mathbf { 2 } }$	$\mathbf { H } _ { \mathbf { 3 } }$
	$\mathbf { J } _ { \mathbf { 1 } }$	-2	3	2
	$\mathbf { J } _ { \mathbf { 2 } }$	3	2	0
	$\mathbf { J } _ { \mathbf { 3 } }$	4	-1	3
	$\mathbf { J } _ { \mathbf { 4 } }$	3	1	0

The game does not have a stable solution.
Which strategy should Jonathan never play?
Circle your answer.
[0pt] [1 mark]
$\mathbf { J } _ { \mathbf { 1 } }$
$\mathbf { J } _ { \mathbf { 2 } }$
$\mathbf { J } _ { \mathbf { 3 } }$
$\mathbf { J } _ { \mathbf { 4 } }$

AQA Further Paper 3 Discrete 2023 June Q3

3 A student is solving a maximising linear programming problem. The graph below shows the constraints, feasible region and objective line for the student's linear programming problem.
\includegraphics[max width=\textwidth, alt={}, center]{5ff6e3bb-6392-49cf-b64d-23bc595cd92e-03_1248_1184_502_427} Which vertex is the optimal vertex? Circle your answer.
$A$
B
C
D

AQA Further Paper 3 Discrete 2023 June Q4

4 The network below represents a system of water pipes in a geothermal power station. The numbers on each arc represent the lower and upper capacity for each pipe in gallons per second.
\includegraphics[max width=\textwidth, alt={}, center]{5ff6e3bb-6392-49cf-b64d-23bc595cd92e-04_837_1413_493_312} The water is taken from a nearby river at node $A$
The water is then pumped through the system of pipes and passes through one of three treatment facilities at nodes $H , I$ and $J$ before returning to the river. 4

The senior management at the power station want all of the water to undergo a final quality control check at a new facility before it returns to the river. Using the language of networks, explain how the network above could be modified to include the new facility. 4
Find the value of the cut $\{ A , B , C , D , E \} \{ F , G , H , I , J \}$ 4
Tim, a trainee engineer at the power station, correctly calculates the value of the cut $\{ A , B , C , D , E , F \} \{ G , H , I , J \}$ to be 106 gallons per second. Tim then claims that the maximum flow through the network of pipes is 106 gallons per second. Comment on the validity of Tim's claim.

AQA Further Paper 3 Discrete 2023 June Q5

2 marks

5 A student is solving the following linear programming problem. $$\begin{array} { l r } \text { Minimise } & Q = - 4 x - 3 y
\text { subject to } & x + y \leq 520
& 2 x - 3 y \leq 570
\text { and } & x \geq 0 , y \geq 0 \end{array}$$ 5

The student wants to use the simplex algorithm to solve the linear programming problem. They modify the linear programming problem by introducing the objective function $$P = 4 x + 3 y$$ and the slack variables $r$ and $s$ State one further modification that must be made to the linear programming problem so that it can be solved using the simplex algorithm. 5
1. Complete the initial simplex tableau for the modified linear programming problem.
  [0pt] [2 marks]
  $P$ $x$ $y$ $r$ $S$ value
  5
(ii) Hence, perform one iteration of the simplex algorithm.
$P$ $x$ $y$ $r$ $s$ value
5
The student performs one further iteration of the simplex algorithm, which results in the following correct simplex tableau.
$P$ $x$ $y$ $r$ $s$ value
1 0 0 $\frac { 18 } { 5 }$ $\frac { 1 } { 5 }$ 1986
0 0 1 $\frac { 2 } { 5 }$ $- \frac { 1 } { 5 }$ 94
0 1 0 $\frac { 3 } { 5 }$ $\frac { 1 } { 5 }$ 426
5
1. Explain how the student can tell that the optimal solution to the modified linear programming problem can be determined from the above simplex tableau.
  5
(ii) Find the optimal solution of the original linear programming problem.

AQA Further Paper 3 Discrete 2023 June Q6

8 marks

6 A council wants to grit all of the roads on a housing estate. The network shows the roads on a housing estate. Each node represents a junction between two or more roads and the weight of each arc represents the length, in metres, of the road.
\includegraphics[max width=\textwidth, alt={}, center]{5ff6e3bb-6392-49cf-b64d-23bc595cd92e-08_1145_1458_539_292} The total length of all of the roads on the housing estate is 9175 metres.
In order to grit all of the roads, the council requires a gritter truck to travel along each road at least once. The gritter truck starts and finishes at the same junction. 6

The gritter truck starts gritting the roads at 7:00 pm and moves with an average speed of 5 metres per second during its journey. Find the earliest time for the gritter truck to have gritted each road at least once and arrived back at the junction it started from, giving your answer to the nearest minute. Fully justify your answer.
[0pt] [6 marks]

Explain how a refinement to the council's requirement, that the gritter truck must start and finish at the same junction, could reduce the time taken to grit all of the roads at least once.

[2 marks] $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$

The planning involves producing an activity network for the project, which is shown in Figure 1 below. The duration of each activity is given in weeks. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{5ff6e3bb-6392-49cf-b64d-23bc595cd92e-10_965_1600_559_221}

\end{figure}

AQA Further Paper 3 Discrete 2023 June Q7

1. Find the earliest start time and the latest finish time for each activity and write these values on the activity network in Figure 1 7
(ii) Write down the critical path. 7
On Figure 2 below, draw a cascade diagram (Gantt chart) for the planned building project, assuming that each activity starts as early as possible. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{5ff6e3bb-6392-49cf-b64d-23bc595cd92e-11_1127_1641_539_201}
\end{figure} 7
During further planning of the building project, Nova Merit Construction find that activity $F$ is not necessary and they remove it from the project. Explain the effect removing activity $F$ has on the minimum completion time of the project.

AQA Further Paper 3 Discrete 2023 June Q8

8 The graph $G$ is shown below.
\includegraphics[max width=\textwidth, alt={}, center]{5ff6e3bb-6392-49cf-b64d-23bc595cd92e-12_301_688_351_676} 8

1. State, with a reason, whether or not $G$ is simple. 8
(ii) A student states that $G$ is Eulerian.
Explain why the student is correct. 8
The graph $H$ has 8 vertices with degrees 2, 2, 4, 4, 4, 4, 4 and 4 Comment on whether $H$ is isomorphic to $G$
8
The formula $v - e + f = 2$, where
$v =$ number of vertices
$e =$ number of edges
$f =$ number of faces
can be used with graphs which satisfy certain conditions. Prove that $G$ does not satisfy the conditions for the above formula to apply.

AQA Further Paper 3 Discrete 2023 June Q9

2 marks

9 The group $\left( C , + _ { 4 } \right)$ contains the elements $0,1,2$ and 3 9

1. Show that $C$ is a cyclic group.
  9
(ii) State the group of symmetries of a regular polygon that is isomorphic to $C$
9
The group ( $V , \otimes$ ) contains the elements (1, 1), (1, -1), (-1, 1) and (-1, -1) The binary operation $\otimes$ between elements of $V$ is defined by $$( a , b ) \otimes ( c , d ) = ( a \times c , b \times d )$$ 9
1. Find the element in $V$ that is the inverse of $( - 1,1 )$
  Fully justify your answer.
  [0pt] [2 marks]
  9
(ii) Determine, with a reason, whether or not $C \cong V$
$\mathbf { 9 }$ (c) The group $G$ has order 16
Rachel claims that as $1,2,4,8$ and 16 are the only factors of 16 then, by Lagrange's theorem, the group $G$ will have exactly 5 distinct subgroups, including the trivial subgroup and $G$ itself. Comment on the validity of Rachel's claim.
\includegraphics[max width=\textwidth, alt={}, center]{5ff6e3bb-6392-49cf-b64d-23bc595cd92e-16_2493_1721_214_150}

AQA Further Paper 3 Discrete 2024 June Q1

1 Which one of the following sets forms a group under the given binary operation?
Tick ( ✓ ) one box.

Set	Binary Operation
\{1, 2, 3\}	Addition modulo 4	□
\{1, 2, 3\}	Multiplication modulo 4	□
\{0, 1, 2, 3\}	Addition modulo 4	□
\{0, 1, 2, 3\}	Multiplication modulo 4	□

AQA Further Paper 3 Discrete 2024 June Q2

2 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 ( ✓ ) one box.

Best Lower Bound	Best Upper Bound
15	48	□
15	51	□
19	48	□
19	51	□

The simple-connected graph $G$ has the adjacency matrix

\cline { 2 - 5 } \multicolumn{1}{c\|}{}	$A$	$B$	$C$	$D$
$A$	0	1	1	1
$B$	1	0	1	0
$C$	1	1	0	1
$D$	1	0	1	0

Which one of the following statements about $G$ is true?
Tick ( ✓ ) one box.
$G$ is a tree □
$G$ is complete □
$G$ is Eulerian □ G is planar □

\multirow{6}{*}{Jonathan}	Hoshi
	Strategy	\(\mathbf { H } _ { \mathbf { 1 } }\)	\(\mathbf { H } _ { \mathbf { 2 } }\)	\(\mathbf { H } _ { \mathbf { 3 } }\)
	\(\mathbf { J } _ { \mathbf { 1 } }\)	-2	3	2
	\(\mathbf { J } _ { \mathbf { 2 } }\)	3	2	0
	\(\mathbf { J } _ { \mathbf { 3 } }\)	4	-1	3
	\(\mathbf { J } _ { \mathbf { 4 } }\)	3	1	0

\(P\)	\(x\)	\(y\)	\(r\)	\(S\)	value

\(P\)	\(x\)	\(y\)	\(r\)	\(s\)	value

\(P\)	\(x\)	\(y\)	\(r\)	\(s\)	value
1	0	0	\(\frac { 18 } { 5 }\)	\(\frac { 1 } { 5 }\)	1986
0	0	1	\(\frac { 2 } { 5 }\)	\(- \frac { 1 } { 5 }\)	94
0	1	0	\(\frac { 3 } { 5 }\)	\(\frac { 1 } { 5 }\)	426

\cline { 2 - 5 } \multicolumn{1}{c\|}{}	\(A\)	\(B\)	\(C\)	\(D\)
\(A\)	0	1	1	1
\(B\)	1	0	1	0
\(C\)	1	1	0	1
\(D\)	1	0	1	0

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