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AQA Further AS Paper 2 Discrete 2019 June Q1
1 marks Easy -1.2
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
1 marks Moderate -0.5
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
4 marks Moderate -0.8
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
6 marks Easy -1.2
4
  1. State the definition of a bipartite graph. 4
  2. 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
  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
  5. (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
5 marks Standard +0.3
5
  1. 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
  2. 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
  4. (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
13 marks Standard +0.8
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
  1. 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
  2. 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
  4. (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
  5. 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
  7. (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}\(\mathbf { A } _ { \mathbf { 1 } }\)2-13
    \(\mathbf { A } _ { \mathbf { 2 } }\)-4-22
    \(\mathbf { A } _ { \mathbf { 3 } }\)011
    \(\mathrm { A } _ { 4 }\)-32-2
AQA Further AS Paper 2 Discrete 2019 June Q7
10 marks Easy -2.5
7
    1. Write down the pay-off matrix for Bex. 7
  2. (ii) Explain why the pay-off matrix for Bex can be written as
AQA Further AS Paper 2 Discrete 2020 June Q1
1 marks Moderate -0.5
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
1 marks Moderate -0.8
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. â–¡ \(G\) is Eulerian but not Hamiltonian. â–¡ \(G\) is semi-Eulerian and Hamiltonian. â–¡ \(G\) is semi-Eulerian but not Hamiltonian. â–¡
AQA Further AS Paper 2 Discrete 2020 June Q3
5 marks Moderate -0.5
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- 40
\cline { 2 - 5 }\(\mathbf { S } _ { \mathbf { 2 } }\)- 12010
\cline { 2 - 5 }\(\mathbf { S } _ { \mathbf { 3 } }\)1046
3
  1. Show that the game has a stable solution.
    3
    1. State the value of the game for Summer. 3
  3. (ii) State the play-safe strategy for each player.
AQA Further AS Paper 2 Discrete 2020 June Q4
4 marks Challenging +1.2
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.
Number of
vertices
Number of
edges
Number of
faces
\(3 x + 6\)\(x ^ { 2 } + 8 x\)\(2 x ^ { 2 } + 2\)
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
6 marks Moderate -0.3
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.
ActivityImmediate predecessor(s)Duration (weeks)
\(A\)-10
B-5
CB12
D\(A\)8
\(E\)C, D4
\(F\)C, D3
\(G\)C, D7
\(H\)E, F8
\(I\)G6
\(J\)G15
KH, I5
\(L\)K4
5
  1. 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
  2. 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
7 marks Easy -1.2
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
  1. State why six is the minimum number of paths that need to be made wider. 6
  2. 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\(\boldsymbol { D }\)\(E\)\(F\)\(G\)
    \(\boldsymbol { A }\)-47----
    B4-623--
    C76--3-4
    \(D\)-2--45-
    \(E\)-334-37
    \(F\)---53-6
    G--4-76-
    Find the minimum number of trees that need to be removed. Fully justify your answer.
    6
  3. 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
10 marks Moderate -0.3
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.
Type of
bakery box
Number of
rolls
Number of
teacakes
Number of
croissants
Profit per
box sold
Standard1263\(\pounds 2.50\)
Luxury669\(\pounds 2.00\)
Robyn formulates a linear programming problem to find the maximum profit the bakery can make from selling the bakery boxes. 7
  1. 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
  3. (ii) Use Figure 1 to find the maximum profit that the bakery can make from selling bakery boxes.
    7
  4. 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
6 marks Standard +0.3
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
  2. (ii) State the inverse of \(b\) under the binary operation
    8
  3. Figure 3 shows a Cayley table for multiplication modulo 4 \begin{table}[h]
    \captionsetup{labelformat=empty} \caption{Figure 3}
    \(\times _ { 4 }\)0123
    00000
    10123
    20202
    30321
    \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
2 marks Easy -1.2
1
  1. 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
  2. Find the value of \(y\) Circle your answer.
    5
    7
    8
    15
AQA Further AS Paper 2 Discrete 2021 June Q2
4 marks Moderate -0.8
2 The set \(S\) is given by \(S = \{ 0,2,4,6 \}\) 2
  1. 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
  2. State the identity element for \(S\) under the binary operation addition modulo 8
AQA Further AS Paper 2 Discrete 2021 June Q3
4 marks Moderate -0.5
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
  1. Find the value of Cut \(X\) 3
  2. Find the value of Cut \(Y\) 3
  3. Add a supersink \(T\) to the network.
AQA Further AS Paper 2 Discrete 2021 June Q5
7 marks Moderate -0.8
5
7
8
12 1 (b) Find the value of \(y\) Circle your answer.
5
7
8
15 2 The set \(S\) is given by \(S = \{ 0,2,4,6 \}\) 2 (a) 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 (b) State the identity element for \(S\) under the binary operation addition modulo 8
AQA Further AS Paper 2 Discrete 2021 June Q8
5 marks Moderate -0.5
8
12 1 (b) Find the value of \(y\) Circle your answer.
5
7
8
15 2 The set \(S\) is given by \(S = \{ 0,2,4,6 \}\) 2 (a) 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 (b) State the identity element for \(S\) under the binary operation addition modulo 8
AQA Further AS Paper 2 Discrete 2022 June Q1
2 marks Easy -1.2
1 The connected graph \(G\) is shown below. \includegraphics[max width=\textwidth, alt={}, center]{ecbeedf5-148e-40ad-b8a2-a7aa3db4a115-02_542_834_630_603} The graphs \(A\) and \(B\) are subgraphs of \(G\) Both \(A\) and \(B\) have four vertices. 1
  1. The graph \(A\) is a tree with \(x\) edges.
    State the value of \(x\) Circle your answer. 3459 1
  2. 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
4 marks Moderate -0.5
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
  1. Find the value of the cut \(\{ A , C , D , G , H \} \{ B , E , F , I \}\) 2
  2. Write down a cut with a value of \(300 \mathrm {~m} ^ { 3 } \mathrm {~s} ^ { - 1 }\) 2
  3. 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
4 marks Moderate -0.5
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
  1. Write down the critical path.
    [0pt] [1 mark] 3
  2. 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
5 marks Moderate -0.8
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.
AberBangorConwyDeganwyE'bach
Aber-9.110.012.317.1
Bangor9.1-15.517.822.7
Conwy10.015.5-2.47.6
Deganwy12.317.82.4-8.0
E'bach17.122.77.68.0-
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
  1. Using the nearest neighbour algorithm starting from Deganwy, find an upper bound for \(x\)
AQA Further AS Paper 2 Discrete 2022 June Q5
3 marks Moderate -0.5
5
  1. A connected planar graph has 9 vertices, 20 edges and \(f\) faces. Use Euler's formula for connected planar graphs to find \(f\) 5
  2. 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}