Questions Further AS Paper 2 Discrete

Complete the Cayley Table shown below for $S$ under the binary operation multiplication modulo 5
$\times _ { 5 }$ 1 2 3 4
1
2
3
4
4
State the identity element for $S$ under multiplication modulo 5 4
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

Draw, in the space below, the spanning tree of minimum total length for this road network. 5
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

Determine if $\nabla$ is commutative on $\mathbb { R }$ Fully justify your answer. 7
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

By considering the flows at the source and the sink, explain why $x = 7$
8
8
Prove that the maximum flow through the network is at most $27 \mathrm {~m} ^ { 3 } \mathrm {~s} ^ { - 1 }$
$\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$
[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:

Activity	Immediate Predecessor(s)	Duration (Days)
A	None	1
B	None	1
C	A	10
D	A	2
E	B, D	5
F	$E$	6
G	$E$	1
H	$F$	1
$I$	$F$	2
J	C, G, H, I	4

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
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	- 1	1
\cline { 2 - 5 }	$\mathbf { B } _ { \mathbf { 2 } }$	4	- 3	1
\cline { 2 - 5 }	$\mathbf { B } _ { \mathbf { 3 } }$	- 1	$x$	0

The game has a stable solution. 10

Show that there is only one possible value for $x$
Fully justify your answer.
13 10
State the value of the game for Bilal.

AQA Further AS Paper 2 Discrete 2018 June Q1

1 The table shows some of the outcomes of performing a modular arithmetic operation.

\cline { 2 - 3 } \multicolumn{1}{c\|}{}	2	3
2		1
3	1

Which pair are operations that could each be represented by the table?
Tick ( ✓ ) one box.
\includegraphics[max width=\textwidth, alt={}, center]{5a826f8b-4751-4589-ad0a-109fc5c821f2-02_109_111_1338_497} Addition $\bmod 6$ and multiplication $\bmod 5$
\includegraphics[max width=\textwidth, alt={}, center]{5a826f8b-4751-4589-ad0a-109fc5c821f2-02_108_109_1471_497} Addition mod 6 and multiplication $\bmod 6$
\includegraphics[max width=\textwidth, alt={}, center]{5a826f8b-4751-4589-ad0a-109fc5c821f2-02_113_109_1603_497} Addition mod 4 and multiplication $\bmod 5$
\includegraphics[max width=\textwidth, alt={}, center]{5a826f8b-4751-4589-ad0a-109fc5c821f2-02_107_109_1742_497} Addition mod 4 and multiplication mod 6

AQA Further AS Paper 2 Discrete 2018 June Q2

1 marks

2 The binary operation ⊗ is given by
$a \otimes b = 3 a ( 5 + b ) ( \bmod 8 )$
where $a , b \in \mathbb { Z }$
Given that $2 \otimes x = 6$, which of the integers below is a possible value of $x$ ?
Circle your answer.
[0pt] [1 mark]
0123

AQA Further AS Paper 2 Discrete 2018 June Q3

3 Alex and Sam are playing a zero-sum game. The game is represented by the pay-off matrix for Alex.

Sam
\cline { 2 - 5 }	Strategy
\cline { 2 - 5 }	$\mathbf { S } _ { \mathbf { 1 } }$	$\mathbf { S } _ { \mathbf { 2 } }$	$\mathbf { S } _ { \mathbf { 3 } }$
$\mathbf { A } _ { \mathbf { 1 } }$	2	2	3
\cline { 2 - 5 }	$\mathbf { A } _ { \mathbf { 2 } }$	0	3	5
$\mathbf { A } _ { \mathbf { 3 } }$	- 1	2	- 2

Explain why the value of the game is 2
3
Identify the play-safe strategy for each player.
Each pipe is labelled with its upper capacity in $\mathrm { cm } ^ { 3 } \mathrm {~s} ^ { - 1 }$
\includegraphics[max width=\textwidth, alt={}, center]{5a826f8b-4751-4589-ad0a-109fc5c821f2-04_620_940_450_550}

AQA Further AS Paper 2 Discrete 2018 June Q4

1. Find the value of the cut given by $\{ A , B , C , D , F , J \} \{ E , G , H \}$.
  4
(ii) State what can be deduced about the maximum flow through the network.
4
1. List the nodes which are sources of the network. 4
(ii) Add a supersource $S$ to the network. 4
1. List the nodes which are sinks of the network. 4
(ii) Add a supersink $T$ to the network.

AQA Further AS Paper 2 Discrete 2018 June Q5

2 marks

5 A group of friends want to prepare a meal. They start preparing the meal at 6:30 pm Activities to prepare the meal are shown in Figure 1 below. \begin{table}[h]

\captionsetup{labelformat=empty} \caption{Figure 1}

Label	Activity	Duration (mins)	Immediate predecessors
A	Weigh rice	1	-
$B$	Cook rice	18	$A$
C	Drain rice	1	B
D	Chop vegetables	10	-
$E$	Fry vegetables	12
$F$	Combine fried vegetables and drained rice	1
G	Prepare sauce ingredients	4	-
$H$	Boil sauce	12
$I$	Serve meal on plates	2

\end{table} 5

1. Use Figure 2 shown below to complete Figure 1 above. 5
(ii) Complete Figure 2 showing the earliest start time and latest finishing time for each activity. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{5a826f8b-4751-4589-ad0a-109fc5c821f2-06_700_1650_1781_194}
\end{figure} 5
1. State the activity which must be started first so that the meal is served in the shortest possible time. Fully justify your answer.
  5
(ii) Determine the earliest possible time at which the preparation of the meal can be completed.
Question 5 continues on the next page 5
The group of friends want to cook spring rolls so that they are served at the same time as the rest of the meal. This requires the additional activities shown in Figure 3. \begin{table}[h]
\captionsetup{labelformat=empty} \caption{Figure 3}
Label Activity Duration Immediate predecessors
J Switch on and heat oven -
K Put spring rolls in oven and cook
$L$ Transfer spring rolls to serving dish
\end{table} It takes 15 seconds to switch on the oven. The oven must be allowed to heat up for 10 minutes before the spring rolls are put in the oven. It takes 15 seconds to put the spring rolls in the oven.
The spring rolls must cook in the hot oven for 8 minutes.
It takes 30 seconds to transfer the spring rolls to a serving dish.
5
1. Complete Figure 3 above. 5
(ii) Determine the latest time at which the oven can be switched on in order for the spring rolls to be served at the same time as the rest of the meal.
[0pt] [2 marks]
\includegraphics[max width=\textwidth, alt={}, center]{5a826f8b-4751-4589-ad0a-109fc5c821f2-09_2488_1716_219_153}

AQA Further AS Paper 2 Discrete 2018 June Q6

6 An animal sanctuary has a rainwater collection site. The manager of the sanctuary is installing a pipe system to connect the rainwater collection site to five other sites in the sanctuary. Each site does not need to be connected directly to the rainwater collection site. There are nine possible routes between the sites that are suitable for water pipes. The distances, in metres, of the nine possible routes are given in the table below.

From/To	Henhouse (H)	Goatshed (G)	Kennels (K)	Cattery (C)
Rainwater collection site (R)	840	810	520	370
Cattery (C)	-	680	610	\multirow{3}{*}{}
Duckpond (D)	480	310
Goatshed (G)	150

Water pipe costs 60 pence per metre. Find the minimum cost of connecting all the sites to the rainwater collection site. Fully justify your answer.
$7 \quad$ A linear programming problem has the constraints $$\begin{aligned} 1 \leq x & \leq 6
1 \leq y & \leq 6
y & \geq x
x + y & \leq 11 \end{aligned}$$

AQA Further AS Paper 2 Discrete 2018 June Q7

1. Complete Figure 4 to identify the feasible region for the problem. \begin{figure}[h]
  \captionsetup{labelformat=empty} \caption{Figure 4} \includegraphics[alt={},max width=\textwidth]{5a826f8b-4751-4589-ad0a-109fc5c821f2-12_922_940_849_552}
  \end{figure} 7
(ii) Determine the maximum value of $5 x + 4 y$ subject to the constraints.
7
The simple-connected graph $G$ has seven vertices. The vertices of $G$ have degree $1,2,3 , v , w , x$ and $y$
7
1. Explain why $x \geq 1$ and $y \geq 1$
  7
(ii) Explain why $x \leq 6$ and $y \leq 6$
7
(iii) Explain why $x + y \leq 11$
7
(iv) State an additional constraint that applies to the values of $x$ and $y$ in this context.
7
The graph $G$ also has eight edges. The inequalities used in part (a)(i) apply to the graph $G$. 7
1. Given that $v + w = 4$, find all the feasible values of $x$ and $y$.
  7
(ii) It is also given that the graph $G$ is semi-Eulerian. On Figure 5, draw $G$. Figure 5

AQA Further AS Paper 2 Discrete 2019 June Q1

1 marks

1 The network represents a system of pipes.
The number on each arc represents the upper capacity for each pipe in $\mathrm { cm } ^ { 3 } \mathrm {~s} ^ { - 1 }$
\includegraphics[max width=\textwidth, alt={}, center]{dcf97b92-d067-41d4-89a6-ea5bab9ea4ff-03_691_1067_721_482} The value of the cut $\{ S , A , B \} \{ C , D , E , T \}$ is $V \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }$
Find $V$. Circle your answer.
[0pt] [1 mark]
25303137

AQA Further AS Paper 2 Discrete 2019 June Q2

2 Part of an activity network is shown in the diagram below.
$A B C$ is part of the critical path of the activity network.
\includegraphics[max width=\textwidth, alt={}, center]{dcf97b92-d067-41d4-89a6-ea5bab9ea4ff-04_264_908_447_566} The duration of activity $B$ is $d$.
Which of the following statements about $d$ is correct? Circle your answer. $$0 < d < 10 \quad d = 10 \quad 10 < d < 20 \quad d = 20$$

AQA Further AS Paper 2 Discrete 2019 June Q3

3 Manon makes apple cakes and banana cakes. Each apple cake is made with 3 eggs and 100 grams of flour. Each banana cake is made with 2 eggs and 150 grams of flour. Manon has 36 eggs and 1500 grams of flour.
Manon wants to make as many cakes as possible.
Formulate Manon's situation as a linear programming problem, clearly defining any variables you introduce.

AQA Further AS Paper 2 Discrete 2019 June Q4

2 marks

State the definition of a bipartite graph. 4
A jazz quintet has five musical instruments: bassoon, clarinet, flute, oboe and violin. Jay, Kay, Lee, Mel and Nish are musicians and each plays a musical instrument in the jazz quintet. Jay knows how to play the bassoon and the clarinet.
Kay knows how to play the bassoon, the oboe and the violin.
Lee knows how to play the clarinet and the flute.
Mel knows how to play the clarinet, the oboe and the violin.
Nish knows how to play the flute, the oboe and the violin. 4
1. Draw a graph to show which musicians know how to play which instruments. 4
(ii) Nish arrives late to a jazz quintet rehearsal. Each of the other four musicians is already playing an instrument: \begin{displayquote} Jay is playing the clarinet
Kay is playing the oboe
Lee is playing the flute
Mel is playing the violin. \end{displayquote} Explain how the graph in part (b)(i) shows that there is no instrument available that Nish knows how to play. 4
(iii) When Nish arrives the rehearsal stops. When they restart the rehearsal, Nish is playing the flute. Draw all possible subgraphs of the graph in part (b)(i) that show how Jay, Kay, Lee and Mel can each be assigned a unique musical instrument they know how to play.
[0pt] [2 marks]

AQA Further AS Paper 2 Discrete 2019 June Q5

Complete the Cayley table in Figure 1 for multiplication modulo 4 \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{dcf97b92-d067-41d4-89a6-ea5bab9ea4ff-08_761_1017_434_493}
\end{figure} 5
The set $S$ is defined as $$S = \{ a , b , c , d \}$$ Figure 2 shows an incomplete Cayley table for $S$ under the commutative binary operation • \begin{table}[h]
\captionsetup{labelformat=empty} \caption{Figure 2}
• $a$ $b$ $c$ $d$
$a$ $b$ $a$ $a$ $c$
$b$ $c$ $c$
$c$ $d$ $d$
$d$ $d$ $d$
\end{table} 5
1. Complete the Cayley table in Figure 2. 5
(ii) Determine whether the binary operation • is associative when acting on the elements of $S$. Fully justify your answer.

AQA Further AS Paper 2 Discrete 2019 June Q6

4 marks

6 The diagram shows a nature reserve which has its entrance at $A$, eight information signs at $B , C , \ldots , I$, and fifteen grass paths. The length of each grass path is given in metres.
The total length of the grass paths is 1465 metres.
\includegraphics[max width=\textwidth, alt={}, center]{dcf97b92-d067-41d4-89a6-ea5bab9ea4ff-10_812_1192_584_424} To cut the grass, Ashley starts at the entrance and drives a mower along every grass path in the nature reserve. The mower moves at 7 kilometres per hour. 6

Find the least possible time that it takes for Ashley to cut the grass on all fifteen paths in the nature reserve and return to the entrance. Fully justify your answer.
6
Brook visits every information sign in the nature reserve to update them, starting and finishing at the entrance. For the eight information signs, the minimum connecting distance of the grass paths is 510 metres. 6
1. Determine a lower bound for the distance Brook walks to visit every information sign.
  Fully justify your answer.
  [0pt] [2 marks]
  6
(ii) Using the nearest neighbour algorithm starting from the entrance, determine an upper bound for the distance Brook walks to visit every information sign.
[0pt] [2 marks]
6
Brook takes one minute to update the information at one information sign. Brook walks on the grass paths at an average speed of 5 kilometres per hour. Ashley and Brook start from the entrance at the same time. 6
1. Use your answers from parts (a) and (b) to show that Ashley and Brook will return to the entrance at approximately the same time. Fully justify your answer.
  6

(ii) State an assumption that you have used in part (c)(i).
\includegraphics[max width=\textwidth, alt={}, center]{dcf97b92-d067-41d4-89a6-ea5bab9ea4ff-13_2488_1716_219_153}
$7 \quad$ Ali and Bex play a zero-sum game. The game is represented by the following pay-off matrix for Ali.

\multirow{2}{*}{}	Bex
	Strategy	$\mathbf { B } _ { \mathbf { 1 } }$	$\mathbf { B } _ { \mathbf { 2 } }$	$\mathbf { B } _ { \mathbf { 3 } }$
\multirow{4}{*}{Ali}	$\mathbf { A } _ { \mathbf { 1 } }$	2	-1	3
	$\mathbf { A } _ { \mathbf { 2 } }$	-4	-2	2
	$\mathbf { A } _ { \mathbf { 3 } }$	0	1	1
	$\mathrm { A } _ { 4 }$	-3	2	-2

AQA Further AS Paper 2 Discrete 2019 June Q7

1. Write down the pay-off matrix for Bex. 7
(ii) Explain why the pay-off matrix for Bex can be written as

AQA Further AS Paper 2 Discrete 2020 June Q1

1 The network represents a system of pipes.
The number on each arc represents the upper capacity for each pipe in $\mathrm { cm } ^ { 3 } \mathrm {~s} ^ { - 1 }$
\includegraphics[max width=\textwidth, alt={}, center]{21ed3b4e-a089-4607-b5d6-69d8aac03f31-02_793_1255_731_395} The value of the cut $\{ S , A , B \} \{ C , D , E , F , T \}$ is $60 \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }$
The maximum flow through the system is $M \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }$
What does the value of the cut imply about $M$ ? Circle your answer.
$M < 60 \quad M \leq 60 \quad M \geq 60 \quad M > 60$

AQA Further AS Paper 2 Discrete 2020 June Q2

2 The graph $G$ has 5 vertices and 6 edges, as shown below.
\includegraphics[max width=\textwidth, alt={}, center]{21ed3b4e-a089-4607-b5d6-69d8aac03f31-03_547_547_360_749} Which of the following statements describes the properties of $G$ ?
Tick ( $\checkmark$ ) one box.
$G$ is Eulerian and Hamiltonian. □
$G$ is Eulerian but not Hamiltonian. □
$G$ is semi-Eulerian and Hamiltonian. □
$G$ is semi-Eulerian but not Hamiltonian. □

•	\(a\)	\(b\)	\(c\)	\(d\)
\(a\)	\(b\)	\(a\)	\(a\)	\(c\)
\(b\)		\(c\)		\(c\)
\(c\)		\(d\)	\(d\)
\(d\)			\(d\)	\(d\)

Label	Activity	Duration	Immediate predecessors
J	Switch on and heat oven		-
K	Put spring rolls in oven and cook
\(L\)	Transfer spring rolls to serving dish

\multirow{2}{*}{}	Bex
	Strategy	\(\mathbf { B } _ { \mathbf { 1 } }\)	\(\mathbf { B } _ { \mathbf { 2 } }\)	\(\mathbf { B } _ { \mathbf { 3 } }\)
\multirow{4}{*}{Ali}	\(\mathbf { A } _ { \mathbf { 1 } }\)	2	-1	3
	\(\mathbf { A } _ { \mathbf { 2 } }\)	-4	-2	2
	\(\mathbf { A } _ { \mathbf { 3 } }\)	0	1	1
	\(\mathrm { A } _ { 4 }\)	-3	2	-2

\(\times _ { 5 }\)	1	2	3	4
1
2
3
4

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