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AQA Further AS Paper 2 Discrete 2022 June Q6

5 marks Standard +0.3

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 marks Challenging +1.2

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.

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

7 marks Standard +0.3

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 marks Moderate -0.5

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

7 marks Standard +0.3

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

7 marks Moderate -0.5

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 marks Easy -1.8

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

1 marks Moderate -0.5

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 Standard +0.8

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

6 marks Moderate -0.5

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

8 marks Moderate -0.8

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

11 marks Standard +0.3

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

4 marks Moderate -0.5

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}

AQA Further AS Paper 2 Discrete Specimen Q8

8 marks Moderate -0.3

8 A family business makes and sells two kinds of kitchen table.
Each pine table takes 6 hours to make and the cost of materials is $\pounds 30$.
Each oak table takes 10 hours to make and the cost of materials is $\pounds 70$.
Each month, the business has 360 hours available for making the tables and $\pounds 2100$ available for the materials.
Each month, the business sells all of its tables to a wholesaler.
The wholesaler specifies that it requires at least 10 oak tables per month and at least as many pine tables as oak tables. Each pine table sold gives the business a profit of $\pounds 40$ and each oak table sold gives the business a profit of $\pounds 75$. Use a graphical method to find the number of each type of table the business should make each month, in order to maximise its total profit. Show clearly how you obtain your answer.
[0pt] [8 marks]

T...,

\includegraphics[max width=\textwidth, alt={}]{ba9e9840-ce27-4ca7-ab05-50461d135445-13_24_64_421_1323}

T.....

\multirow{2}{*}{}

Toloolo

\includegraphics[max width=\textwidth, alt={}]{ba9e9840-ce27-4ca7-ab05-50461d135445-13_43_351_660_197}

\includegraphics[max width=\textwidth, alt={}]{ba9e9840-ce27-4ca7-ab05-50461d135445-13_52_208_752_200}

\includegraphics[max width=\textwidth, alt={}]{ba9e9840-ce27-4ca7-ab05-50461d135445-13_27_77_934_1310}

- -

\includegraphics[max width=\textwidth, alt={}]{ba9e9840-ce27-4ca7-ab05-50461d135445-13_47_169_898_1567}

\includegraphics[max width=\textwidth, alt={}]{ba9e9840-ce27-4ca7-ab05-50461d135445-13_24_147_950_738}

\includegraphics[max width=\textwidth, alt={}]{ba9e9840-ce27-4ca7-ab05-50461d135445-13_45_261_946_1475}

"
"
"

- - -

\multirow{4}{*}{}

\includegraphics[max width=\textwidth, alt={}]{ba9e9840-ce27-4ca7-ab05-50461d135445-13_29_150_1450_1310}

\includegraphics[max width=\textwidth, alt={}]{ba9e9840-ce27-4ca7-ab05-50461d135445-13_24_169_1448_1567}

\includegraphics[max width=\textwidth, alt={}]{ba9e9840-ce27-4ca7-ab05-50461d135445-13_44_183_1637_536}

- -

$\%$
- 1
- 1

- - -

\includegraphics[max width=\textwidth, alt={}]{ba9e9840-ce27-4ca7-ab05-50461d135445-13_35_171_2116_1319}

\includegraphics[max width=\textwidth, alt={}]{ba9e9840-ce27-4ca7-ab05-50461d135445-13_41_250_2104_1485}

- - - -

\includegraphics[max width=\textwidth, alt={}]{ba9e9840-ce27-4ca7-ab05-50461d135445-13_38_158_2143_729}

\includegraphics[max width=\textwidth, alt={}]{ba9e9840-ce27-4ca7-ab05-50461d135445-13_45_183_2281_536}

- -

AQA Further Paper 1 2019 June Q1

1 marks Easy -1.2

1 Which one of these functions has the set $\{ x : | x | < 1 \}$ as its greatest possible domain? Circle your answer. $$\cosh x \quad \cosh ^ { - 1 } x \quad \tanh x \quad \tanh ^ { - 1 } x$$

AQA Further Paper 1 2019 June Q2

1 marks Standard +0.3

2 The first two non-zero terms of the Maclaurin series expansion of $\mathrm { f } ( x )$ are $x$ and $- \frac { 1 } { 2 } x ^ { 3 }$ Which one of the following could be $\mathrm { f } ( x )$ ?
Circle your answer. $x \mathrm { e } ^ { \frac { 1 } { 2 } x ^ { 2 } }$ $\frac { 1 } { 2 } \sin 2 x$ $x \cos x$ $\left( 1 + x ^ { 3 } \right) ^ { - \frac { 1 } { 2 } }$

AQA Further Paper 1 2019 June Q3

1 marks Moderate -0.8

3 The function $\mathrm { f } ( x ) = x ^ { 2 } - 1$ Find the mean value of $\mathrm { f } ( x )$ from $x = - 0.5$ to $x = 1.7$ Give your answer to three significant figures.
Circle your answer.

AQA Further Paper 1 2019 June Q4

4 marks Moderate -0.5

4 Solve the equation $2 z - 5 \mathrm { i } z ^ { * } = 12$

AQA Further Paper 1 2019 June Q5

3 marks Standard +0.3

5 A plane has equation r. $\left[ \begin{array} { l } 1 \\ 1 \\ 1 \end{array} \right] = 7$ A line has equation $\mathbf { r } = \left[ \begin{array} { l } 2 \\ 0 \\ 1 \end{array} \right] + \mu \left[ \begin{array} { l } 1 \\ 0 \\ 1 \end{array} \right]$ Calculate the acute angle between the line and the plane.
Give your answer to the nearest $0.1 ^ { \circ }$ \includegraphics[max width=\textwidth, alt={}, center]{68359582-cd8b-4807-9127-eaf8fd339746-05_2491_1716_219_153}

AQA Further Paper 1 2019 June Q6

8 marks Standard +0.3

Show that $$\cosh ^ { 3 } x + \sinh ^ { 3 } x = \frac { 1 } { 4 } \mathrm { e } ^ { m x } + \frac { 3 } { 4 } \mathrm { e } ^ { n x }$$ where $m$ and $n$ are integers.
6
Hence find $\cosh ^ { 6 } x - \sinh ^ { 6 } x$ in the form $$\frac { a \cosh ( k x ) + b } { 8 }$$ where $a , b$ and $k$ are integers.

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-	\(a\)	\(b\)	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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AQA Further AS Paper 2 Discrete 2022 June Q6

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