Questions Further Paper 3 Discrete

Complete the activity network for the building project. 3
Deva Construction Ltd is able to reduce the duration of a single activity to 1 hour by using specialist equipment. State, with a reason, which activity should have its duration reduced to 1 hour in order to minimise the completion time for the building project.
3
State one limitation in the building project used by Deva Construction Ltd. Explain how this limitation affects the project.
[0pt] [2 marks]

AQA Further Paper 3 Discrete Specimen Q4

3 marks

4 Optical fibre broadband cables are being installed between 5 neighbouring villages. The distance between each pair of villages in metres is shown in the table.

	Alvanley	Dunham	Elton	Helsby	Ince
Alvanley	-	2000	4000	750	5500
Dunham	2000	-	2500	2250	4000
Elton	4000	2500	-	3000	1250
Helsby	750	2250	3000	-	4250
Ince	5500	4000	1250	4250	-

The company installing the optical fibre broadband cables wishes to create a network connecting each of the 5 villages using the minimum possible length of cable. Find the minimum length of cable required.
[0pt] [3 marks]

AQA Further Paper 3 Discrete Specimen Q5

3 marks

5 The binary operation * is defined as $$a * b = a + b + 4 ( \bmod 6 )$$ where $a , b \in \mathbb { Z }$. 5

Show that the set $\{ 0,1,2,3,4,5 \}$ forms a group $G$ under *.
5
Find the proper subgroups of the group $G$ in part (a).
5
Determine whether or not the group $G$ in part (a) is isomorphic to the group $K = \left( \langle 3 \rangle , \times _ { 14 } \right)$
[0pt] [3 marks]

AQA Further Paper 3 Discrete Specimen Q6

10 marks

6
The network shows a system of pipes, where $S$ is the source and $T$ is the sink.
The lower and upper capacities, in litres per second, of each pipe are shown on each arc.
\includegraphics[max width=\textwidth, alt={}, center]{88669bc0-9d3f-431a-8939-8aef2682412b-09_649_1399_580_424} 6

There is a feasible flow from $S$ to $T$. 6
1. Explain why arc $A D$ must be at its lower capacity.
  [0pt] [1 mark] 6
(ii) Explain why arc $B E$ must be at its upper capacity.
[0pt] [1 mark] 6
Explain why a flow of 11 litres per second through the network is impossible.
[0pt] [1 mark] 6
The network in Figure 2 shows a second system of pipes, where $S$ is the source and $T$ is the sink. The lower and upper capacities, in litres per second, of each pipe are shown on each edge. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{88669bc0-9d3f-431a-8939-8aef2682412b-10_760_1372_680_470}
\end{figure} Figure 3 shows a feasible flow of 17 litres per second through the system of pipes. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{88669bc0-9d3f-431a-8939-8aef2682412b-10_750_1371_1811_466}
\end{figure} 6
1. Using Figures 2 and 3, indicate on Figure 4 potential increases and decreases in the flow along each arc.
  [0pt] [2 marks] \begin{figure}[h]
  \captionsetup{labelformat=empty} \caption{Figure 4} \includegraphics[alt={},max width=\textwidth]{88669bc0-9d3f-431a-8939-8aef2682412b-11_749_1384_457_426}
  \end{figure} 6
(ii) Use flow augmentation on Figure 4 to find the maximum flow from $S$ to $T$.
You should indicate any flow augmenting paths clearly in the table below and modify the potential increases and decreases of the flow on Figure 4.
[0pt] [3 marks]
Augmenting Path Flow
6
(iii) Prove the flow found in part (d) (ii) is maximum.
6
(iv) Due to maintenance work, the flow through node $E$ is restricted to 9 litres per second.
[0pt] Interpret the impact of this restriction on the maximum flow through the system of pipes. [2 marks]

AQA Further Paper 3 Discrete Specimen Q7

4 marks

7 A company repairs and sells computer hardware, including monitors, hard drives and keyboards. Each monitor takes 3 hours to repair and the cost of components is $\pounds 40$. Each hard drive takes 2 hours to repair and the cost of components is $\pounds 20$. Each keyboard takes 1 hour to repair and the cost of components is $\pounds 5$. Each month, the business has 360 hours available for repairs and $\pounds 2500$ available to buy components. Each month, the company sells all of its repaired hardware to a local computer shop. Each monitor, hard drive and keyboard sold gives the company a profit of $\pounds 80 , \pounds 35$ and $\pounds 15$ respectively. The company repairs and sells $x$ monitors, $y$ hard drives and $z$ keyboards each month. The company wishes to maximise its total profit. 7

Find five inequalities involving $x , y$ and $z$ for the company's problem.
[0pt] [3 marks]
7
1. Find how many of each type of computer hardware the company should repair and sell each month.
  7
(ii) Explain how you know that you had reached the optimal solution in part (b) (i).
7
(iii) The local computer shop complains that they are not receiving one of the types of computer hardware that the company repairs and sells. Using your answer to part (b) (i), suggest a way in which the company's problem can be modified to address the complaint.
[0pt] [1 mark]

AQA Further Paper 3 Discrete Specimen Q8

8 John and Danielle play a zero-sum game which does not have a stable solution. The game is represented by the following pay-off matrix for John.

\multirow{2}{*}{}		Danielle
	Strategy	$\boldsymbol { X }$	$Y$	$\boldsymbol { Z }$
\multirow{3}{*}{John}	$A$	2	1	-1
	B	-3	-2	2
	$\boldsymbol { C }$	-3	-4	1

Find the optimal mixed strategy for John.

AQA Further Paper 3 Discrete 2019 June Q1

1 Deanna and Will play a zero-sum game.
The game is represented by the following pay-off matrix for Deanna.

\multirow{6}{*}{Deanna}	Will
	Strategy	X	Y	Z
	A	-1	0	2
	B	-2	-1	3
	C	5	-2	-3
	D	6	-2	0

Which strategy is Deanna's play-safe strategy?
Circle your answer.
A
B
C
D

AQA Further Paper 3 Discrete 2019 June Q2

2 The graph $D$ is shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{22f11ce2-8d07-4f51-9326-b578d1e454f9-02_115_150_1756_945} Which of the graphs below is a subdivision of $D$ ?
Circle your answer.
\includegraphics[max width=\textwidth, alt={}, center]{22f11ce2-8d07-4f51-9326-b578d1e454f9-02_218_154_2124_534}
\includegraphics[max width=\textwidth, alt={}, center]{22f11ce2-8d07-4f51-9326-b578d1e454f9-02_208_150_2129_808}
\includegraphics[max width=\textwidth, alt={}, center]{22f11ce2-8d07-4f51-9326-b578d1e454f9-02_208_154_2129_1082}
\includegraphics[max width=\textwidth, alt={}, center]{22f11ce2-8d07-4f51-9326-b578d1e454f9-02_208_149_2129_1361}

AQA Further Paper 3 Discrete 2019 June Q3

3 The Simplex tableau below is optimal.

$\boldsymbol { P }$	$\boldsymbol { x }$	$\boldsymbol { y }$	$\boldsymbol { z }$	$\boldsymbol { r }$	$\boldsymbol { s }$	value
1	$k ^ { 2 } + k - 6$	0	0	$k - 1$	1	20
0	0	0	1	1.5	0	6
0	0	1	0	0	0.5	86

Deduce the range of values that $k$ must satisfy.
3
Write down the value of the variable $s$ which corresponds to the optimal value of $P$.

AQA Further Paper 3 Discrete 2019 June Q4

4 The connected planar graph $P$ has the adjacency matrix

\cline { 2 - 6 } \multicolumn{1}{c\|}{}	$A$	$B$	$C$	$D$	$E$
$A$	0	1	1	0	1
$B$	1	0	1	0	1
$C$	1	1	0	1	1
$D$	0	0	1	0	1
$E$	1	1	1	1	0

Draw $P$ 4
Using Euler's formula for connected planar graphs, show that $P$ has exactly 5 faces. 4
Ore's theorem states that a simple graph with $n$ vertices is Hamiltonian if, for every pair of vertices $X$ and $Y$ which are not adjacent, $$\text { degree of } X + \text { degree of } Y \geq n$$ where $n \geq 3$
Using Ore's theorem, prove that the graph $P$ is Hamiltonian.
Fully justify your answer.

AQA Further Paper 3 Discrete 2019 June Q5

5 The set $S$ is defined as $$S = \{ A , B , C , D \}$$ where
$A = \left[ \begin{array} { l l } 1 & 0
0 & 1 \end{array} \right] \quad B = \left[ \begin{array} { c c } 0 & - 1
1 & 0 \end{array} \right] \quad C = \left[ \begin{array} { c c } - 1 & 0
0 & - 1 \end{array} \right] \quad D = \left[ \begin{array} { c c } 0 & 1
- 1 & 0 \end{array} \right]$ The group $G$ is formed by $S$ under matrix multiplication.
The group $H$ is defined as $H = ( \langle \mathrm { i } \rangle , \times )$, where $\mathrm { i } ^ { 2 } = - 1$
5

1. Prove that $B$ is a generator of $G$.
  Fully justify your answer.
  5
(ii) Show that $G \cong H$.
Fully justify your answer.
5
1. Explain why $H$ has no subgroups of order 3
  Fully justify your answer.
  5
(ii) Find all of the subgroups of $H$.

AQA Further Paper 3 Discrete 2019 June Q6

6 A council wants to monitor how long cars are being parked for in short-stay parking bays in a town centre. They employ a traffic warden to walk along the streets in the town centre and issue fines to drivers who park for longer than the stated time. The network below shows streets in the town centre which have short-stay parking bays. Each node represents a street corner and the weight of each arc represents the length, in metres, of the street. The short-stay parking bays are positioned along only one side of each street.
\includegraphics[max width=\textwidth, alt={}, center]{22f11ce2-8d07-4f51-9326-b578d1e454f9-08_483_1108_733_466} The council assumes that the traffic warden will walk at an average speed of 4.8 kilometres per hour when not issuing fines. 6

To monitor all of the parking bays, the traffic warden needs to walk along every street in the town centre at least once, starting and finishing at the same street corner. Find the shortest possible time, to the nearest minute, it can take the traffic warden to monitor all of the parking bays. Fully justify your answer.
6
Explain why the actual time for the traffic warden to walk along every street in the town centre at least once may be different to the value found in part (a).

AQA Further Paper 3 Discrete 2019 June Q7

7 Figure 1 shows a system of water pipes in a manufacturing complex. 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 38 litres of water per second. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{22f11ce2-8d07-4f51-9326-b578d1e454f9-10_874_1360_609_338}

\end{figure} 7

1. Calculate the value of the cut $\{ S , A , B , C \} \{ D , E , F , G , H , T \}$. 7
(ii) Explain, in the context of the question, what can be deduced from your answer to part (a)(i). 7
1. Using the initial feasible flow shown in Figure 1, indicate on Figure 2 potential increases and decreases in the flow along each arc. \begin{figure}[h]
  \captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{22f11ce2-8d07-4f51-9326-b578d1e454f9-11_997_1554_475_242}
  \end{figure} 7
(ii) Use flow augmentation on Figure 2 to find the maximum flow through the manufacturing complex. You must indicate any flow augmenting paths clearly in the table and modify the potential increases and decreases of the flow on Figure 2.
Augmenting Path Flow
Maximum Flow $=$ $\_\_\_\_$ 7
The management of the manufacturing complex want to increase the maximum amount of water which can flow through the system of pipes. To do this they decide to upgrade one of the water pipes by replacing it with a larger capacity pipe. Explain which pipe should be upgraded.
Deduce what effect this upgrade will have on the maximum amount of water which can flow through the system of pipes.
\includegraphics[max width=\textwidth, alt={}, center]{22f11ce2-8d07-4f51-9326-b578d1e454f9-13_2488_1716_219_153}

AQA Further Paper 3 Discrete 2019 June Q8

8 A motor racing team is undertaking a project to build next season's racing car. The project is broken down into 12 separate activities $A , B , \ldots , L$, as shown in the precedence table below. Each activity requires one member of the racing team.

Activity	Duration (days)	Immediate Predecessors
$A$	7	-
B	6	-
C	15	-
D	9	$A , B$
$E$	8	D
$F$	6	C, D
G	7	C
H	14	$E$
$I$	17	$F , G$
$J$	9	H, I
K	8	$I$
L	12	J, K

1. Complete the activity network for the project on Figure 3. 8
(ii) Find the earliest start time and the latest finish time for each activity and show these values on Figure 3. 8
Write down the critical path(s).
\section*{Figure 3} Figure 3
\includegraphics[max width=\textwidth, alt={}, center]{22f11ce2-8d07-4f51-9326-b578d1e454f9-15_469_1360_356_338} 8
1. Using Figure 4, draw a resource histogram for the project to show how the project can be completed in the shortest possible time. Assume that each activity is to start as early as possible. \begin{figure}[h]
  \captionsetup{labelformat=empty} \caption{Figure 4} \includegraphics[alt={},max width=\textwidth]{22f11ce2-8d07-4f51-9326-b578d1e454f9-16_698_1534_541_251}
  \end{figure} 8
(ii) The racing team's boss assigns two members of the racing team to work on the project. Explain the effect this has on the minimum completion time for the project.
You may use Figure 5 in your answer. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 5} \includegraphics[alt={},max width=\textwidth]{22f11ce2-8d07-4f51-9326-b578d1e454f9-16_704_1539_1695_248}
\end{figure}

AQA Further Paper 3 Discrete 2020 June Q1

1 The diagram below shows a network of pipes with their capacities.
\includegraphics[max width=\textwidth, alt={}, center]{c297a67f-65fd-47e0-a60c-d38fd86c6081-02_734_1275_630_386} A supersource and a supersink will be added to the network.
To which nodes should the supersource and supersink be connected?
Tick $( \checkmark )$ one box.

Supersource	Supersink
$P , Q$	$U , V , W$
$U , V , W$	$P , Q$
$V , X$	$U , W$
$U , W$	$V , X$

□
□
□
\includegraphics[max width=\textwidth, alt={}, center]{c297a67f-65fd-47e0-a60c-d38fd86c6081-02_118_113_2261_1324}

AQA Further Paper 3 Discrete 2020 June Q2

2 Which of the following statements is true about the operation of matrix multiplication on the set of all $2 \times 2$ real matrices? Tick ( $\checkmark$ ) one box. Matrix multiplication is associative and commutative.
\includegraphics[max width=\textwidth, alt={}, center]{c297a67f-65fd-47e0-a60c-d38fd86c6081-03_109_112_552_1599} Matrix multiplication is associative but not commutative. □ Matrix multiplication is commutative but not associative. □ Matrix multiplication is not commutative and not associative. □

AQA Further Paper 3 Discrete 2020 June Q3

4 marks

3 A company is installing an internal telephone network between the offices in a council building. Each office is required to be connected with telephone cables, either directly or indirectly, to every other office in the building. The lengths of cable, in metres, needed to connect the offices are shown in the table below.

	Education	Housing	Refuse Collection	Payroll	Social Care	Transport
Education	-	27	13	35	16	24
Housing	27	-	29	30	22	24
Refuse Collection	13	29	-	26	23	17
Payroll	35	30	26	-	20	40
Social Care	16	22	23	20	-	21
Transport	24	24	17	40	21	-

The council wants the total length of cable that is used to be as small as possible.
The cost to the council to install one metre of cable is $\pounds 8$
3

1. Find the minimum total cost to the council to install the cable required for the internal telephone network.
  [0pt] [4 marks]
  3
(ii) Suggest a reason why the total cost to the council for installing the internal telephone network is likely to be different from your answer to part (a)(i). 3
Before the company starts installing the cable, it is told that the Education office cannot be connected directly to the Transport office due to issues with the building. Explain the possible impact of this on your answer to part (a)(i).

AQA Further Paper 3 Discrete 2020 June Q4

4 Joe, a courier, is required to deliver parcels to six different locations, $A , B , C , D , E$ and $F$. Joe needs to start and finish his journey at the depot.
The distances, in miles, between the depot and the six different locations are shown in the table below.

	Depot	$\boldsymbol { A }$	$\boldsymbol { B }$	C	$\boldsymbol { D }$	$E$	$F$
Depot	-	18	17	15	16	19	30
$\boldsymbol { A }$	18	-	29	20	25	35	21
B	17	29	-	26	30	16	14
C	15	20	26	-	28	31	27
D	16	25	30	28	-	34	24
E	19	35	16	31	34	-	28
F	30	21	14	27	24	28	-

The minimum total distance that Joe can travel in order to make all six deliveries, starting and finishing at the depot, is $L$ miles. 4

Using the nearest neighbour algorithm starting from the depot, find an upper bound for $L$.
4
By deleting the depot, find a lower bound for $L$.
4
Joe starts from the depot, delivers parcels to all six different locations and arrives back at the depot, covering 134 miles in the process. Joe claims that this is the minimum total distance that is possible for the journey. Comment on Joe's claim.

AQA Further Paper 3 Discrete 2020 June Q5

5 The planar graph $P$ is shown below.
\includegraphics[max width=\textwidth, alt={}, center]{c297a67f-65fd-47e0-a60c-d38fd86c6081-08_410_406_360_817} 5

Determine the number of faces of $P$.
5
Akwasi claims that $P$ is semi-Eulerian as it is connected and it has exactly two vertices with even degree. Comment on the validity of Akwasi's claim.
\includegraphics[max width=\textwidth, alt={}, center]{c297a67f-65fd-47e0-a60c-d38fd86c6081-09_2488_1716_219_153}

AQA Further Paper 3 Discrete 2020 June Q6

1. State the order of $G$. 6
(ii) Prove that the inverse of $q r$ is $q r$.
6
Complete the Cayley table for elements of $G$. 6

Complete the Cayley table for elements of $G$.

A	$e$	$r$	$r ^ { 2 }$	$q$	$q r$	$q r ^ { 2 }$
$e$	$e$	$r$	$r ^ { 2 }$	$q$	$q r$	$q r ^ { 2 }$
$r$	$r$	$r ^ { 2 }$	$e$
$r ^ { 2 }$	$r ^ { 2 }$	$e$	$r$
$q$	$q$	$q r$	$q r ^ { 2 }$	$e$
$q r$	$q r$	$q r ^ { 2 }$	$q$	$r ^ { 2 }$
$q r ^ { 2 }$	$q r ^ { 2 }$	$q$	$q r$	$r$	$r ^ { 2 }$	$e$

State the name of a group which is isomorphic to $G$.

AQA Further Paper 3 Discrete 2020 June Q7

7 An engineering company makes brake kits and clutch kits to sell to motorsport teams. The table below summarises the time taken and costs involved in making the two different types of kit.

Type of kit	Time taken to make a kit (hours)	Cost to engineering company per kit (£)	Profit to engineering company per kit (£)
Brake kit	5	500	2000
Clutch kit	3	200	1000

The workers at the engineering company have a combined 2500 hours available to make the kits every month. The engineering company has $\pounds 200000$ available to cover the costs of making the kits every month. To meet the minimum demands of the motorsport teams, the engineering company must make at least 100 of each type of kit every month. 7

Using a graphical method on the grid opposite, find the number of each type of kit that the engineering company should make every month, in order to maximise its total monthly profit. Show clearly how you obtain your answer.
\includegraphics[max width=\textwidth, alt={}, center]{c297a67f-65fd-47e0-a60c-d38fd86c6081-13_2486_1709_221_153} Do not write outside the box 7
Give a reason why the engineering company may not be able to make the number of each kit that you found in part (a). 7
During one particular month the engineering company removes the need to make at least 100 of each type of kit. Explain whether or not this has an effect on your answer to part (a).

AQA Further Paper 3 Discrete 2020 June Q8

8 Daryl and Clare play a zero-sum game. The game is represented by the following pay-off matrix for Daryl. Clare

AQA Further Paper 3 Discrete 2021 June Q1

1 Which of the following statements about critical path analysis is always true? Tick ( $\checkmark$ ) one box. All activity networks have exactly one critical path. □ All critical activities have a non-zero float. □ The first activity in a critical path has an earliest start time of zero. □ A delay on a critical activity may not delay the project. □

\(\boldsymbol { P }\)	\(\boldsymbol { x }\)	\(\boldsymbol { y }\)	\(\boldsymbol { z }\)	\(\boldsymbol { r }\)	\(\boldsymbol { s }\)	value
1	\(k ^ { 2 } + k - 6\)	0	0	\(k - 1\)	1	20
0	0	0	1	1.5	0	6
0	0	1	0	0	0.5	86

\cline { 2 - 6 } \multicolumn{1}{c\|}{}	\(A\)	\(B\)	\(C\)	\(D\)	\(E\)
\(A\)	0	1	1	0	1
\(B\)	1	0	1	0	1
\(C\)	1	1	0	1	1
\(D\)	0	0	1	0	1
\(E\)	1	1	1	1	0

Supersource	Supersink
\(P , Q\)	\(U , V , W\)
\(U , V , W\)	\(P , Q\)
\(V , X\)	\(U , W\)
\(U , W\)	\(V , X\)

A	\(e\)	\(r\)	\(r ^ { 2 }\)	\(q\)	\(q r\)	\(q r ^ { 2 }\)
\(e\)	\(e\)	\(r\)	\(r ^ { 2 }\)	\(q\)	\(q r\)	\(q r ^ { 2 }\)
\(r\)	\(r\)	\(r ^ { 2 }\)	\(e\)
\(r ^ { 2 }\)	\(r ^ { 2 }\)	\(e\)	\(r\)
\(q\)	\(q\)	\(q r\)	\(q r ^ { 2 }\)	\(e\)
\(q r\)	\(q r\)	\(q r ^ { 2 }\)	\(q\)	\(r ^ { 2 }\)
\(q r ^ { 2 }\)	\(q r ^ { 2 }\)	\(q\)	\(q r\)	\(r\)	\(r ^ { 2 }\)	\(e\)

	Depot	\(\boldsymbol { A }\)	\(\boldsymbol { B }\)	C	\(\boldsymbol { D }\)	\(E\)	\(F\)
Depot	-	18	17	15	16	19	30
\(\boldsymbol { A }\)	18	-	29	20	25	35	21
B	17	29	-	26	30	16	14
C	15	20	26	-	28	31	27
D	16	25	30	28	-	34	24
E	19	35	16	31	34	-	28
F	30	21	14	27	24	28	-

Augmenting Path	Flow

Augmenting Path	Flow

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