Questions — AQA Further AS Paper 2 Discrete (60 questions)

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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- 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 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
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.
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
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
  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
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.
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
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
  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
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
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
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
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
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
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
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
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
  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
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
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}
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
  1. Complete the Cayley table for \(S\) under matrix multiplication.
    ABCD
    AAD
    BB
    CC
    DD
    6
  2. Using the Cayley table above, explain why \(\mathbf { B }\) is the identity element of \(S\) under matrix multiplication.
    [0pt] [1 mark] 6
  3. 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\(\mathbf { L } _ { \mathbf { 1 } }\)\(\mathbf { L } _ { \mathbf { 2 } }\)\(\mathbf { L } _ { \mathbf { 3 } }\)
\(\mathrm { Kez } \quad \mathbf { K } _ { \mathbf { 1 } }\)41- 2
\(\mathbf { K } _ { \mathbf { 2 } }\)- 4- 20
\(\mathbf { K } _ { \mathbf { 3 } }\)- 2- 12
7
  1. State, with a reason, why Kez should never play strategy \(\mathbf { K } _ { \mathbf { 2 } }\) 7
  2. \(\quad\) 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
1 The graph \(G\) has 8 vertices and 13 edges as shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{372edcfa-c3cd-4c83-89e9-2bb5fd9825f1-03_494_392_482_806} Graph \(H\) is a simple-connected subgraph of graph \(G\) Which of the following diagrams could represent graph \(H\) ? Tick ( ✓ ) one box.
\includegraphics[max width=\textwidth, alt={}, center]{372edcfa-c3cd-4c83-89e9-2bb5fd9825f1-03_312_310_1354_351}
\includegraphics[max width=\textwidth, alt={}, center]{372edcfa-c3cd-4c83-89e9-2bb5fd9825f1-03_321_310_1676_351}
\includegraphics[max width=\textwidth, alt={}, center]{372edcfa-c3cd-4c83-89e9-2bb5fd9825f1-03_117_115_1448_822}
\includegraphics[max width=\textwidth, alt={}, center]{372edcfa-c3cd-4c83-89e9-2bb5fd9825f1-03_312_310_2014_351}
\includegraphics[max width=\textwidth, alt={}, center]{372edcfa-c3cd-4c83-89e9-2bb5fd9825f1-03_122_117_1777_822}
\includegraphics[max width=\textwidth, alt={}, center]{372edcfa-c3cd-4c83-89e9-2bb5fd9825f1-03_314_314_2343_349}
\includegraphics[max width=\textwidth, alt={}, center]{372edcfa-c3cd-4c83-89e9-2bb5fd9825f1-03_120_115_2108_822}
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
  2. (ii) State the minimum completion time for the community project.
    4
  3. Write down the critical activities of the network.
    4
  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
  6. (ii) Describe a situation in which Glyn's claim would be true.
AQA Further AS Paper 2 Discrete 2023 June Q5
5
  1. 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
  3. (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
  4. \(\quad\) 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
    \(a\)\(a\)cb
    \(b\)\(b\)\(а\)
    cc
    \end{table} 5
    1. Complete the Cayley table in Figure 1 5
  6. (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\(\mathbf { Y } _ { \mathbf { 1 } }\)\(\mathbf { Y } _ { \mathbf { 2 } }\)\(\mathbf { Y } _ { \mathbf { 3 } }\)
\(\mathbf { X } _ { \mathbf { 1 } }\)- 41- 3
\(\mathbf { X } _ { \mathbf { 2 } }\)4- 3- 3
\(\mathbf { X } _ { \mathbf { 3 } }\)- 11- 2
\end{table} 6
  1. Show that the game has a stable solution.
    6
  2. State the play-safe strategy for each player. Play-safe strategy for Xander is \(\_\_\_\_\)
    Play-safe strategy for Yvonne is \(\_\_\_\_\) 6
  3. 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.