Questions - AQA - A-Level Maths

\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. □

AQA Further AS Paper 2 Discrete 2020 June Q3

3 Summer and Haf play a zero-sum game. The pay-off matrix for the game is shown below. Haf

	Strategy	$\mathbf { H } _ { \mathbf { 1 } }$	$\mathbf { H } _ { \mathbf { 2 } }$	$\mathbf { H } _ { \mathbf { 3 } }$
Summer	$\mathbf { S } _ { \mathbf { 1 } }$	4	- 4	0
\cline { 2 - 5 }	$\mathbf { S } _ { \mathbf { 2 } }$	- 12	0	10
\cline { 2 - 5 }	$\mathbf { S } _ { \mathbf { 3 } }$	10	4	6

Show that the game has a stable solution.
3
1. State the value of the game for Summer. 3
(ii) State the play-safe strategy for each player.

AQA Further AS Paper 2 Discrete 2020 June Q4

4 The connected planar graph $P$ is Eulerian and has at least one vertex of degree $x$. Some of the properties of $P$ are shown in the table below.

Deduce the value of $x$.
Fully justify your answer.
\includegraphics[max width=\textwidth, alt={}, center]{21ed3b4e-a089-4607-b5d6-69d8aac03f31-07_2488_1716_219_153}

AQA Further AS Paper 2 Discrete 2020 June Q5

4 marks

5 A restoration project is divided into a number of activities. The duration and predecessor(s) of each activity are shown in the table below.

Activity	Immediate predecessor(s)	Duration (weeks)
$A$	-	10
B	-	5
C	B	12
D	$A$	8
$E$	C, D	4
$F$	C, D	3
$G$	C, D	7
$H$	E, F	8
$I$	G	6
$J$	G	15
K	H, I	5
$L$	K	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.
[0pt] [4 marks] \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{21ed3b4e-a089-4607-b5d6-69d8aac03f31-09_533_289_2124_1548} \captionsetup{labelformat=empty} \caption{Turn over -}
\end{figure} 5
Due to a change of materials during the project, the duration of activity $C$ is extended by 3 weeks. Determine the new minimum completion time of the project.
\includegraphics[max width=\textwidth, alt={}, center]{21ed3b4e-a089-4607-b5d6-69d8aac03f31-11_2488_1716_219_153}

AQA Further AS Paper 2 Discrete 2020 June Q6

2 marks

6 A garden has seven statues $A , B , C , D , E , F$ and $G$, with paths connecting each pair of statues, either directly or indirectly. To provide better access to all the statues, some of the paths are being made wider.
6

State why six is the minimum number of paths that need to be made wider. 6
The table below shows the number of trees that need to be removed to make the path between adjacent statues wider. A dash in the table means that there is no direct path between the two statues.
Statue $\boldsymbol { A }$ $\boldsymbol { B }$ C $\boldsymbol { D }$ $E$ $F$ $G$
$\boldsymbol { A }$ - 4 7 - - - -
B 4 - 6 2 3 - -
C 7 6 - - 3 - 4
$D$ - 2 - - 4 5 -
$E$ - 3 3 4 - 3 7
$F$ - - - 5 3 - 6
G - - 4 - 7 6 -
Find the minimum number of trees that need to be removed. Fully justify your answer.
6
A landscaper identifies that two new wide paths could be constructed without removing any trees. However, there are only enough resources to build one new wide path. The new wide path could be between $A$ and $D$ or between $A$ and $F$.
Explain clearly how the solution to part (b) can be adapted to find the new minimum number of trees that need to be removed.
[0pt] [2 marks]

AQA Further AS Paper 2 Discrete 2020 June Q7

4 marks

7 Robyn manages a bakery. Each day the bakery bakes 900 rolls, 600 teacakes and 450 croissants.
The bakery sells two types of bakery box which contain rolls, teacakes and croissants, as shown in the table below.

Robyn formulates a linear programming problem to find the maximum profit the bakery can make from selling the bakery boxes. 7

Part of a graphical method to solve this linear programming problem is shown on Figure 1. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{21ed3b4e-a089-4607-b5d6-69d8aac03f31-14_1283_1196_1267_424}
\end{figure} 7
1. Explain how the line shown on Figure 1 relates to the linear programming problem. Clearly define any variables that you introduce.
  [0pt] [3 marks]
  7
(ii) Use Figure 1 to find the maximum profit that the bakery can make from selling bakery boxes.
7
State an assumption that you have made in part (a)(ii).
[0pt] [1 mark]
\includegraphics[max width=\textwidth, alt={}, center]{21ed3b4e-a089-4607-b5d6-69d8aac03f31-17_2493_1732_214_139}

AQA Further AS Paper 2 Discrete 2020 June Q8

2 marks

8 The set $S$ is defined as $$S = \{ a , b , c , d \}$$ Figure 2 shows a Cayley table for $S$ under the commutative binary operation \begin{table}[h]

\captionsetup{labelformat=empty} \caption{Figure 2}

$\odot$	$a$	$b$	$c$	$d$
$a$	$a$	$a$	$a$	$a$
$b$	$a$	$d$	$b$	$c$
$c$	$a$	$b$	$c$	$d$
$d$	$a$	$c$	$d$	$a$

\end{table} 8

1. Prove that there exists an identity element for $S$ under the binary operation
  [0pt] [2 marks]
  8
(ii) State the inverse of $b$ under the binary operation
8
Figure 3 shows a Cayley table for multiplication modulo 4 \begin{table}[h]
\captionsetup{labelformat=empty} \caption{Figure 3}
$\times _ { 4 }$ 0 1 2 3
0 0 0 0 0
1 0 1 2 3
2 0 2 0 2
3 0 3 2 1
\end{table} Mali says that, by substituting suitable distinct values for $a , b , c$ and $d$, the Cayley table in Figure 2 could represent multiplication modulo 4 Use your answers to part (a) to show that Mali's statement is incorrect.
\includegraphics[max width=\textwidth, alt={}, center]{21ed3b4e-a089-4607-b5d6-69d8aac03f31-20_2491_1736_219_139}

AQA Further AS Paper 2 Discrete 2021 June Q1

A project consists of three activities $A , B$ and $C$
An activity network for the project is shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{18ce34aa-e4c3-4a84-a36d-6542d2319bf5-02_615_770_726_635} Find the value of $x$ Circle your answer.
5
7
8
12 1
Find the value of $y$
Circle your answer.
5
7
8
15

AQA Further AS Paper 2 Discrete 2021 June Q2

2 The set $S$ is given by $S = \{ 0,2,4,6 \}$ 2

Construct a Cayley table, using the grid below, for $S$ under the binary operation addition modulo 8
\includegraphics[max width=\textwidth, alt={}, center]{18ce34aa-e4c3-4a84-a36d-6542d2319bf5-03_561_563_607_831} 2
State the identity element for $S$ under the binary operation addition modulo 8

AQA Further AS Paper 2 Discrete 2021 June Q3

3 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]{18ce34aa-e4c3-4a84-a36d-6542d2319bf5-04_513_832_440_605} 3

Find the value of Cut $X$
3
Find the value of Cut $Y$
3
Add a supersink $T$ to the network.

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

Statue	\(\boldsymbol { A }\)	\(\boldsymbol { B }\)	C	\(\boldsymbol { D }\)	\(E\)	\(F\)	\(G\)
\(\boldsymbol { A }\)	-	4	7	-	-	-	-
B	4	-	6	2	3	-	-
C	7	6	-	-	3	-	4
\(D\)	-	2	-	-	4	5	-
\(E\)	-	3	3	4	-	3	7
\(F\)	-	-	-	5	3	-	6
G	-	-	4	-	7	6	-

\(\odot\)	\(a\)	\(b\)	\(c\)	\(d\)
\(a\)	\(a\)	\(a\)	\(a\)	\(a\)
\(b\)	\(a\)	\(d\)	\(b\)	\(c\)
\(c\)	\(a\)	\(b\)	\(c\)	\(d\)
\(d\)	\(a\)	\(c\)	\(d\)	\(a\)

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

	Strategy	\(\mathbf { H } _ { \mathbf { 1 } }\)	\(\mathbf { H } _ { \mathbf { 2 } }\)	\(\mathbf { H } _ { \mathbf { 3 } }\)
Summer	\(\mathbf { S } _ { \mathbf { 1 } }\)	4	- 4	0
\cline { 2 - 5 }	\(\mathbf { S } _ { \mathbf { 2 } }\)	- 12	0	10
\cline { 2 - 5 }	\(\mathbf { S } _ { \mathbf { 3 } }\)	10	4	6

\(\times _ { 4 }\)	1	2	3
0	0	0	0
1	1	2	3
2	2	0	2
3	3	2	1

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