Questions - AQA - A-Level Maths

The graph $A$ is a tree with $x$ edges.
State the value of $x$ Circle your answer. 3459 1
The graph $B$ is simple-connected with $y$ edges.
Find the maximum possible value of $y$
Circle your answer. 3459

AQA Further AS Paper 2 Discrete 2022 June Q2

2 The diagram shows a network of pipes. Each pipe is labelled with its upper capacity in $\mathrm { m } ^ { 3 } \mathrm {~s} ^ { - 1 }$
\includegraphics[max width=\textwidth, alt={}, center]{ecbeedf5-148e-40ad-b8a2-a7aa3db4a115-03_424_1262_445_388} 2

Find the value of the cut $\{ A , C , D , G , H \} \{ B , E , F , I \}$ 2
Write down a cut with a value of $300 \mathrm {~m} ^ { 3 } \mathrm {~s} ^ { - 1 }$ 2
Using the values from part (a) and part (b), state what can be deduced about the maximum flow through the network. Fully justify your answer.

AQA Further AS Paper 2 Discrete 2022 June Q3

1 marks

3 A project consists of 11 activities $A , B , \ldots , K$ A completed activity network for the project is shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{ecbeedf5-148e-40ad-b8a2-a7aa3db4a115-04_972_1604_445_219} All times on the activity network are given in days.
3

Write down the critical path.
[0pt] [1 mark] 3
Due to an issue with the supply of materials, the duration of activity $G$ is doubled. Deduce the effect, if any, that this change will have on the earliest start time and latest finish time for each of the activities $I , J$ and $K$

AQA Further AS Paper 2 Discrete 2022 June Q4

4 Alun, a baker, delivers bread to community shops located in Aber, Bangor, Conwy, and E'bach. Alun starts and finishes his journey at the bakery, which is located in Deganwy.
The distances, in miles, between the five locations are given in the table below.

	Aber	Bangor	Conwy	Deganwy	E'bach
Aber	-	9.1	10.0	12.3	17.1
Bangor	9.1	-	15.5	17.8	22.7
Conwy	10.0	15.5	-	2.4	7.6
Deganwy	12.3	17.8	2.4	-	8.0
E'bach	17.1	22.7	7.6	8.0	-

The minimum total distance that Alun can travel in order to make all four deliveries, starting and finishing at the bakery in Deganwy is $x$ miles. 4

Using the nearest neighbour algorithm starting from Deganwy, find an upper bound for $x$

AQA Further AS Paper 2 Discrete 2022 June Q5

A connected planar graph has 9 vertices, 20 edges and $f$ faces. Use Euler's formula for connected planar graphs to find $f$ 5
The graph $J$, shown in Figure 1, has 9 vertices and 20 edges. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{ecbeedf5-148e-40ad-b8a2-a7aa3db4a115-09_778_760_440_641}
\end{figure} By redrawing the graph $J$ using Figure 2, show that $J$ is planar. \begin{table}[h]
\captionsetup{labelformat=empty} \caption{Figure 2}
$A$ $B$ $C$
$\bullet$ $\bullet$ $\bullet$
$D \bullet$ $E \bullet$ $\bullet F$
$\bullet$ $\stackrel { \theta } { H }$ $\bullet$
\end{table}

AQA Further AS Paper 2 Discrete 2022 June Q6

1 marks

6 The set $S$ is given by $S = \{ \mathbf { A } , \mathbf { B } , \mathbf { C } , \mathbf { D } \}$ where
$\mathbf { A } = \left[ \begin{array} { l l } 1 & 0
0 & 0 \end{array} \right]$
$\mathbf { B } = \left[ \begin{array} { l l } 1 & 0
0 & 1 \end{array} \right]$
$\mathbf { C } = \left[ \begin{array} { l l } 0 & 0
0 & 1 \end{array} \right]$
$\mathbf { D } = \left[ \begin{array} { l l } 0 & 0
0 & 0 \end{array} \right]$ 6

Complete the Cayley table for $S$ under matrix multiplication.
A B C D
A A D
B B
C C
D D
6
Using the Cayley table above, explain why $\mathbf { B }$ is the identity element of $S$ under matrix multiplication.
[0pt] [1 mark] 6
Sam states that the Cayley table in part (a) shows that matrix multiplication is commutative. Comment on the validity of Sam's statement.

AQA Further AS Paper 2 Discrete 2022 June Q7

7 Kez and Lui play a zero-sum game. The game does not have a stable solution. The game is represented by the following pay-off matrix for Kez.

	Lui
\cline { 2 - 5 }	Strategy	$\mathbf { L } _ { \mathbf { 1 } }$	$\mathbf { L } _ { \mathbf { 2 } }$	$\mathbf { L } _ { \mathbf { 3 } }$
$\mathrm { Kez } \quad \mathbf { K } _ { \mathbf { 1 } }$	4	1	- 2
$\mathbf { K } _ { \mathbf { 2 } }$	- 4	- 2	0
	$\mathbf { K } _ { \mathbf { 3 } }$	- 2	- 1	2

State, with a reason, why Kez should never play strategy $\mathbf { K } _ { \mathbf { 2 } }$ 7
$\quad$ Kez and Lui play the game 20 times.
Kez plays their optimal mixed strategy.
Find the expected number of times that Kez will play strategy $\mathbf { K } _ { \mathbf { 3 } }$
Fully justify your answer.

AQA Further AS Paper 2 Discrete 2022 June Q8

4 marks

8 Alli is planting garlic cloves and leek seedlings in a garden. The planting density is the number of plants that are planted per $\mathrm { m } ^ { 2 }$
The planting densities and costs are shown in the table below.

AQA Further AS Paper 2 Discrete 2023 June Q1

AQA Further AS Paper 2 Discrete 2023 June Q2

2 The diagram below shows a network of pipes with their capacities.
\includegraphics[max width=\textwidth, alt={}, center]{372edcfa-c3cd-4c83-89e9-2bb5fd9825f1-04_691_1155_340_424} A supersource is added to the network. Which nodes are connected to the supersource? Tick ( ✓ ) one box.
$A$ and $B$ □
$A$ and $G$ □
$G$ and $H$
\includegraphics[max width=\textwidth, alt={}, center]{372edcfa-c3cd-4c83-89e9-2bb5fd9825f1-04_104_108_1822_685}
$H$ and $I$ □

AQA Further AS Paper 2 Discrete 2023 June Q3

3 Ben is packing eggs into boxes, labelled Town Box or Country Box. Each Town Box must contain 10 chicken eggs and 2 duck eggs. Each Country Box must contain 4 chicken eggs and 8 duck eggs. Ben has 253 chicken eggs and 151 duck eggs. Ben wants to pack as many boxes as possible. Formulate Ben's situation as a linear programming problem, defining any variables you introduce.

AQA Further AS Paper 2 Discrete 2023 June Q4

4 A community project consists of 10 activities $A , B , \ldots , J$, as shown in the activity network below.
\includegraphics[max width=\textwidth, alt={}, center]{372edcfa-c3cd-4c83-89e9-2bb5fd9825f1-06_899_1083_367_466} The duration of each activity is shown in days. 4

1. Complete the activity network in the diagram above, showing the earliest start time and latest finish time for each activity. 4
(ii) State the minimum completion time for the community project.
4
Write down the critical activities of the network.
4
Glyn claims that a project's activity network can be used to determine its minimum completion time by adding together the durations of all the project's critical activities. 4
1. Show that Glyn's claim is false for this community project's activity network.
  4
(ii) Describe a situation in which Glyn's claim would be true.

AQA Further AS Paper 2 Discrete 2023 June Q5

The set $S$ is defined as $S = \{ 0,1,2,3,4,5 \}$ 5
1. State the identity element of $S$ under the operation multiplication modulo 6 5
(ii) An element $g$ of a set is said to be self-inverse under a binary operation * if $$g * g = e$$ where $e$ is the identity element of the set. Find all the self-inverse elements in $S$ under the operation multiplication modulo 6
5
$\quad$ The set $T$ is defined as $$T = \{ a , b , c \}$$ Figure 1 shows a partially completed Cayley table for $T$ under the commutative binary operation - \begin{table}[h]
\captionsetup{labelformat=empty} \caption{Figure 1}
- $a$ $b$ c
$a$ $a$ c b
$b$ $b$ $а$
c c
\end{table} 5
1. Complete the Cayley table in Figure 1 5
(ii) Prove that is not associative when acting on the elements of $T$

AQA Further AS Paper 2 Discrete 2023 June Q6

6 Xander and Yvonne are playing a zero-sum game. The game is represented by the pay-off matrix for Xander. \begin{table}[h]

\captionsetup{labelformat=empty} \caption{Yvonne} Xander

Strategy	$\mathbf { Y } _ { \mathbf { 1 } }$	$\mathbf { Y } _ { \mathbf { 2 } }$	$\mathbf { Y } _ { \mathbf { 3 } }$
$\mathbf { X } _ { \mathbf { 1 } }$	- 4	1	- 3
$\mathbf { X } _ { \mathbf { 2 } }$	4	- 3	- 3
$\mathbf { X } _ { \mathbf { 3 } }$	- 1	1	- 2

\end{table} 6

Show that the game has a stable solution.
6
State the play-safe strategy for each player. Play-safe strategy for Xander is $\_\_\_\_$
Play-safe strategy for Yvonne is $\_\_\_\_$ 6
The game that Xander and Yvonne are playing is part of a marbles challenge. The pay-off matrix values represent the number of marbles gained by Xander in each game. In the challenge, the game is repeated until one player has 24 marbles more than the other player. Explain why Xander and Yvonne must play at least 3 games to complete the challenge.

AQA Further AS Paper 2 Discrete 2023 June Q7

2 marks

7 A construction company has built eight wind turbines on a moorland site. The network below shows nodes which represent the site entrance, $E$, and the wind turbine positions, $S , T , \ldots , Z$
\includegraphics[max width=\textwidth, alt={}, center]{372edcfa-c3cd-4c83-89e9-2bb5fd9825f1-12_924_1294_479_356} Each arc represents an access track with its length given in metres.
These 17 tracks were created in order to build the wind turbines. Eight of the tracks are to be retained so that each turbine can be accessed for maintenance, directly or indirectly, from the site entrance. The other nine tracks will be removed. 7

1. To save money the construction company wants to maximise the total length of the eight tracks to be retained. Determine which tracks the construction company should retain.
  7
(ii) Find the total length of the eight tracks that are to be retained. 7
The total length of the 17 tracks is 14.6 km
The cost of removing all 17 tracks would be $\pounds 87,600$
Using your answer to part (a)(ii), calculate an estimate for the cost of removing the nine tracks that will not be retained.
[0pt] [2 marks]
7
Comment on why the modelling used in part (b) may not give an accurate estimate for the cost of removing the nine tracks.

AQA Further AS Paper 2 Discrete 2023 June Q8

The graph $G$ has 2 vertices. The sum of the degrees of all the vertices of $G$ is 6 Draw $G$ 8
The planar graph $P$ is Eulerian, with at least one vertex of degree $x$, where $x$ is a positive integer. Some of the properties of $P$ are shown in the table below. Question number Additional page, if required. Write the question numbers in the left-hand margin. Question number Additional page, if required. Write the question numbers in the left-hand margin.

AQA Further AS Paper 2 Discrete Specimen Q1

1 A graph has 5 vertices and 6 edges.
Find the sum of the degrees of the vertices. Circle your answer. 10111215

AQA Further AS Paper 2 Discrete Specimen Q2

2 A connected planar graph has $x$ vertices and $2 x - 4$ edges.
Find the number of faces of the planar graph in terms of $x$.
Circle your answer.
$x - 6$
$x - 2$
$6 - x$
$2 - x$

AQA Further AS Paper 2 Discrete Specimen Q3

2 marks

3 The function min $( a , b )$ is defined by: $$\begin{aligned} \min ( a , b ) & = a , a < b
& = b , \text { otherwise } \end{aligned}$$ For example, $\min ( 7,2 ) = 2$ and $\min ( - 4,6 ) = - 4$. Gary claims that the binary operation $\Delta$, which is defined as $$x \Delta y = \min ( x , y - 3 )$$ where $x$ and $y$ are real numbers, is associative as finding the smallest number is not affected by the order of operation. Disprove Gary's claim.
[0pt] [2 marks]

AQA Further AS Paper 2 Discrete Specimen Q4

4 marks

4 A communications company is conducting a feasibility study into the installation of underground television cables between 5 neighbouring districts. The length of the possible pathways for the television cables between each pair of districts, in miles, is shown in the table. The pathways all run alongside cycle tracks.

	Billinge	Garswood	Haydock	Orrell	Up Holland
Billinge	-	2.5	***	4.3	4.8
Garswood	2.5	-	3.1	***	5.9
Haydock	***	3.1	-	6.7	7.8
Orrell	4.3	***	6.7	-	2.1
Up Holland	4.8	5.9	7.8	2.1	-

Give a possible reason, in context, why some of the table entries are labelled as ***. 4
As part of the feasibility study, Sally, an engineer needs to assess each possible pathway between the districts. To do this, Sally decides to travel along every pathway using a bicycle, starting and finishing in the same district. From past experience, Sally knows that she can travel at an average speed of 12 miles per hour on a bicycle. Find the minimum time, in minutes, that it will take Sally to cycle along every pathway.
[0pt] [4 marks]

AQA Further AS Paper 2 Discrete Specimen Q5

3 marks

5 Charlotte is visiting a city and plans to visit its five monuments: $A , B , C , D$ and $E$.
The network shows the time, in minutes, that a typical tourist would take to walk between the monuments on a busy weekday morning.
\includegraphics[max width=\textwidth, alt={}, center]{ba9e9840-ce27-4ca7-ab05-50461d135445-06_902_1134_529_543} Charlotte intends to walk from one monument to another until she has visited them all, before returning to her starting place. 5

Use the nearest neighbour algorithm, starting from $A$, to find an upper bound for the minimum time for Charlotte's tour.
5
By deleting vertex $B$, find a lower bound for the minimum time for Charlotte's tour.
[0pt] [3 marks]
5
Charlotte wants to complete the tour in 52 minutes. Use your answers to parts (a) and (b) to comment on whether this could be possible.
5
Charlotte takes 58 minutes to complete the tour. Evaluate your answers to part (a) and part (b) given this information.
5
Explain how this model for a typical tourist's tour may not be applicable if the tourist walked between the monuments during the evening.

AQA Further AS Paper 2 Discrete Specimen Q6

5 marks

6 Victoria and Albert play a zero-sum game. The game is represented by the following pay-off matrix for Victoria.

\multirow{2}{*}{}		Albert
	Strategy	$\boldsymbol { x }$	$Y$	$z$
\multirow{3}{*}{Victoria}	$P$	3	-1	1
	$Q$	-2	0	1
	$R$	4	-1	-1

Find the play-safe strategies for each player.
6
State, with a reason, the strategy that Albert should never play.
6
1. Determine an optimal mixed strategy for Victoria.
  [0pt] [5 marks]
  6
(ii) Find the value of the game for Victoria.
6
(iii) State an assumption that must made in order that your answer for part (c)(ii) is the maximum expected pay-off that Victoria can achieve.

AQA Further AS Paper 2 Discrete Specimen Q7

3 marks

7 The network shows a system of pipes, where $S$ is the source and $T$ is the sink.
The capacity, in litres per second, of each pipe is shown on each arc.
The cut shown in the diagram can be represented as $\{ S , P , R \} , \{ Q , T \}$.
\includegraphics[max width=\textwidth, alt={}, center]{ba9e9840-ce27-4ca7-ab05-50461d135445-10_629_1168_616_557} 7

Complete the table below to give the value of each of the 8 possible cuts.

Cut		Value
\{ S \}	$\{ P , Q , R , T \}$	31
$\{ S , P \}$	$\{ Q , R , T \}$	32
$\{ S , Q \}$	$\{ P , R , T \}$
$\{ S , R \}$	$\{ P , Q , T \}$
$\{ S , P , Q \}$	$\{ R , T \}$	30
$\{ S , P , R \}$	$\{ Q , T \}$	37
$\{ S , Q , R \}$	$\{ P , T \}$	35
$\{ S , P , Q , R \}$	$\{ T \}$	30

State the value of the maximum flow through the network. Give a reason for your answer.
[0pt] [1 mark] 7
Indicate on Figure 1 a possible flow along each arc, corresponding to the maximum flow through the network.
[0pt] [2 marks] \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{ba9e9840-ce27-4ca7-ab05-50461d135445-11_618_1150_1260_557}
\end{figure}

\(A\)	\(B\)	\(C\)
\(\bullet\)	\(\bullet\)	\(\bullet\)
\(D \bullet\)	\(E \bullet\)	\(\bullet F\)
\(\bullet\)	\(\stackrel { \theta } { H }\)	\(\bullet\)

-	\(a\)	\(b\)	c
\(a\)	\(a\)	c	b
\(b\)		\(b\)	\(а\)
c			c

Cut		Value
\{ S \}	\(\{ P , Q , R , T \}\)	31
\(\{ S , P \}\)	\(\{ Q , R , T \}\)	32
\(\{ S , Q \}\)	\(\{ P , R , T \}\)
\(\{ S , R \}\)	\(\{ P , Q , T \}\)
\(\{ S , P , Q \}\)	\(\{ R , T \}\)	30
\(\{ S , P , R \}\)	\(\{ Q , T \}\)	37
\(\{ S , Q , R \}\)	\(\{ P , T \}\)	35
\(\{ S , P , Q , R \}\)	\(\{ T \}\)	30

	Lui
\cline { 2 - 5 }	Strategy	\(\mathbf { L } _ { \mathbf { 1 } }\)	\(\mathbf { L } _ { \mathbf { 2 } }\)	\(\mathbf { L } _ { \mathbf { 3 } }\)
\(\mathrm { Kez } \quad \mathbf { K } _ { \mathbf { 1 } }\)	4	1	- 2
\(\mathbf { K } _ { \mathbf { 2 } }\)	- 4	- 2	0
	\(\mathbf { K } _ { \mathbf { 3 } }\)	- 2	- 1	2

Strategy	\(\mathbf { Y } _ { \mathbf { 1 } }\)	\(\mathbf { Y } _ { \mathbf { 2 } }\)	\(\mathbf { Y } _ { \mathbf { 3 } }\)
\(\mathbf { X } _ { \mathbf { 1 } }\)	- 4	1	- 3
\(\mathbf { X } _ { \mathbf { 2 } }\)	4	- 3	- 3
\(\mathbf { X } _ { \mathbf { 3 } }\)	- 1	1	- 2

\multirow{2}{*}{}		Albert
	Strategy	\(\boldsymbol { x }\)	\(Y\)	\(z\)
\multirow{3}{*}{Victoria}	\(P\)	3	-1	1
	\(Q\)	-2	0	1
	\(R\)	4	-1	-1

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