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OCR Further Discrete AS 2019 June Q2
10 marks Moderate -0.3
2 Two graphs are shown below. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8473d9aa-a4db-4001-ac71-e5fbbaee530c-2_396_353_1343_479} \captionsetup{labelformat=empty} \caption{Graph G1}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8473d9aa-a4db-4001-ac71-e5fbbaee530c-2_399_328_1340_1233} \captionsetup{labelformat=empty} \caption{Graph G2}
\end{figure}
  1. List the vertex degrees for each graph.
  2. Prove that the graphs are non-isomorphic. The two graphs are joined together by adding an arc connecting J and T .
    1. Explain how you know that the resulting graph is not Eulerian.
    2. Describe how the graph can be made Eulerian by adding one more arc. The vertices of the graph \(K _ { 3 }\) are connected to the vertices of the graph \(K _ { 4 }\) to form the graph \(K _ { 7 }\).
  4. Explain why 12 arcs are needed connecting \(K _ { 3 }\) to \(K _ { 4 }\).
OCR Further Discrete AS 2019 June Q3
11 marks Moderate -0.5
3
  1. Give an example of a standard sorting algorithm that can be used when some of the values are not known until after the sorting has been started. Becky needs to sort a list of numbers into increasing order.
    She uses the following algorithm:
    STEP 1: Let \(L\) be the first value in the input list.
    Write this as the first value in the output list and delete it from the input list.
    STEP 2: If the input list is empty go to STEP 7.
    Otherwise let \(N\) be the new first value in the input list and delete this value from the input list. STEP 3: \(\quad\) Compare \(N\) with \(L\).
    STEP 4: If \(N\) is less than or equal to \(L\)
    • write the value of \(N\) immediately before \(L\) in the output list,
    • replace \(L\) with the first value in the new output list,
    • then go to STEP 2.
    STEP 5: If \(N\) is greater than \(L\)
    • if \(L\) is the value of the last number in the output list, go to STEP 6;
    • otherwise, replace \(L\) with the next value in the output list and then go to STEP 3.
    STEP 6: \(\quad\) Write the value of \(N\) immediately after \(L\) in the output list. Let \(L\) be the first value in the new output list and then go to STEP 2. STEP 7: Print the output list and STOP.
  2. Trace through Becky's algorithm when the input list is $$\begin{array} { l l l l l l } 6 & 9 & 5 & 7 & 6 & 4 \end{array}$$ Complete the table in the Printed Answer Booklet, starting a new row each time that STEP 3 or STEP 7 is used.
    You should not need all the lines in the Answer Booklet. Becky measures the efficiency of her sort by counting using the number of times that STEP 3 is used.
    1. How many times did Becky use STEP 3 in sorting the list from part (b)?
    2. What is the greatest number of times that STEP 3 could be used in sorting a list of 6 values? A computer takes 15 seconds to sort a list of 60 numbers using Becky's algorithm.
  4. Approximately how long would you expect it to take the computer to sort a list of 300 numbers using the algorithm?
OCR Further Discrete AS 2019 June Q4
10 marks Standard +0.3
4 The table shows the activities involved in a project, their durations in hours and their immediate predecessors. The activities can be represented as an activity network.
ActivityABCDEFGH
Duration24543324
Immediate predecessors-A-A, CB, CB, DD, EF, G
  1. Use standard algorithms to find the activities that form
    • the longest path(s)
    • the shortest path(s)
      through the activity network.
    You must show working to demonstrate the use of the algorithms. Only one of the paths from part (a) has a practical interpretation.
  2. What is the practical interpretation of the total weight of that path? The duration of activity E can be changed. No other durations change.
  3. What is the smallest increase to the duration of E that will make activity E become part of a longest path through the network?
OCR Further Discrete AS 2019 June Q5
12 marks Moderate -0.3
5 Corey is training for a race that starts in 18 hours time. He splits his training between gym work, running and swimming.
  • At most 8 hours can be spent on gym work.
  • At least 4 hours must be spent running.
  • The total time spent on gym work and swimming must not exceed the time spent running.
Corey thinks that time spent on gym work is worth 3 times the same time spent running or 2 times the same time spent swimming. Corey wants to maximise the worth of the training using this model.
  1. Formulate a linear programming problem to represent Corey's problem. Your formulation must include defining the variables that you are using. Suppose that Corey spends the maximum of 8 hours on gym work.
    1. Use a graphical method to determine how long Corey should spend running and how long he should spend swimming.
    2. Describe why this solution is not practical.
    3. Describe how Corey could refine the LP model to make the solution more realistic.
OCR Further Discrete AS 2019 June Q6
12 marks Challenging +1.2
6 Drew and Emma play a game in which they each choose a strategy and then use the tables below to determine the pay-off that each receives.
Drew's pay-offEmma
XYZ
\cline { 2 - 5 } \multirow{2}{*}{Drew}P31411
Q1247
R1146
Emma's pay-offEmma
XYZ
\cline { 2 - 5 } \multirow{3}{*}{Drew}P1325
Q4129
R51210
  1. Convert the game into a zero-sum game, giving the pay-off matrix for Drew.
  2. Determine the optimal mixed strategy for Drew.
  3. Determine the optimal mixed strategy for Emma.
OCR Further Discrete AS 2022 June Q1
4 marks Easy -1.2
1 The flowchart below has positive inputs \(X , Y\) and \(M\). \includegraphics[max width=\textwidth, alt={}, center]{74b6f747-7045-4902-8b21-0b59c007f7f6-2_1274_643_392_242}
  1. Trace through the flowchart above using the inputs \(X = 1 , Y = 2\) and \(M = 2\). You only need to record values when they change.
  2. Explain why the process in the flowchart is finite.
OCR Further Discrete AS 2022 June Q2
7 marks Standard +0.3
2 The activities involved in a project and their durations, in hours, are represented in the activity network below. \includegraphics[max width=\textwidth, alt={}, center]{74b6f747-7045-4902-8b21-0b59c007f7f6-3_446_1139_338_230}
  1. Carry out a forward pass and a backward pass through the network.
  2. Calculate the float for each activity. A delay means that activity B cannot finish until \(t\) hours have elapsed from the start of the project.
  3. Determine the maximum value of \(t\) for which the project can be completed in 16 hours.
OCR Further Discrete AS 2022 June Q3
10 marks Easy -1.8
3
  1. The list below is to be sorted into increasing order using bubble sort. \(\begin{array} { l l l l l l l l l l } 52 & 38 & 15 & 61 & 27 & 49 & 10 & 33 & 96 & 74 \end{array}\)
    1. Determine the list that results at the end of the first, second and third passes. You do not need to show the individual swaps in each pass.
    2. Write down the number of comparisons and the number of swaps used in each of these passes.
  2. The list below is to be sorted into increasing order using shuttle sort. \(\begin{array} { l l l l l l l l l l } 52 & 38 & 15 & 61 & 27 & 49 & 10 & 33 & 96 & 74 \end{array}\)
    1. Determine the list that results at the end of the first, second and third passes. You do not need to show the individual swaps in each pass.
    2. Write down the number of comparisons and the number of swaps used in each of these passes.
  3. Use the results from parts (a) and (b) to compare the efficiency of bubble sort with the efficiency of shuttle sort for the first three passes of this list. You do not need to consider what happens after these three passes.
OCR Further Discrete AS 2022 June Q4
10 marks Standard +0.3
4 Kareem and Sam play a game in which each holds a hand of three cards.
  • Kareem's cards are numbered 1, 2 and 5.
  • Sam's cards are numbered 3, 4 and 6 .
In each round Kareem and Sam simultaneously choose a card from their hand, they show their chosen card to the other player and then return the card to their own hand.
  • If the sum of the numbers on the cards shown is even then the number of points that Kareem scores is \(2 k\), where \(k\) is the number on Kareem's card.
  • If the sum of the numbers on the cards shown is odd then the number of points that Kareem scores is \(4 - s\), where \(s\) is the number on Sam's card.
    1. Complete the pay-off matrix for this game, to show the points scored by Kareem.
    2. Write down which card Kareem should play to maximise the number of points that he scores for each of Sam’s choices.
    3. Determine the play-safe strategy for Kareem.
    4. Explain why Kareem should never play the card numbered 1.
Sam chooses a card at random, so each of Sam’s three cards is equally likely.
  • Calculate Kareem's expected score for each of his remaining choices.
    •
    OCR Further Discrete AS 2022 June Q5
    13 marks Standard +0.8
    5 A baker makes three types of jam-and-custard doughnuts.
    • Each batch of type X uses 6 units of jam and 4 units of custard.
    • Each batch of type Y uses 7 units of jam and 3 units of custard.
    • Each batch of type Z uses 8 units of jam and 2 units of custard.
    The baker has 360 units of jam and 180 units of custard available. The baker has plenty of doughnut batter, so this does not restrict the number of batches made. From past experience the baker knows that they must make at most 30 batches of type X and at least twice as many batches of type Y as batches of type Z . Let \(x =\) number of batches of type X made \(y =\) number of batches of type Y made \(z =\) number of batches of type Z made.
    1. Set up an LP formulation for the problem of maximising the total number of batches of doughnuts made. The baker finds that type Z doughnuts are not popular and decides to make zero batches of type Z .
    2. Use a graphical method to find how many batches of each type the baker should make to maximise the total number of batches of doughnuts made.
    3. Give a reason why this solution may not be practical. The baker finds that some of the jam has been used so there are only \(k\) units of jam (where \(k < 360\) ).
      There are still 180 units of custard available and the baker still makes zero batches of type Z .
    4. Find the values of \(k\) if exactly one of the other (non-trivial) constraints is redundant. Express your answer using inequalities.
    OCR Further Discrete AS 2022 June Q6
    9 marks Moderate -0.8
    6 Eight footpaths connect six villages. The lengths of these footpaths, in km , are given in the table.
    Villages connectedA BA DB EB FC DC ED EE F
    Length of footpath, km32465731
    1. The shortest route from B to C using these footpaths has length 10 km . Without using an algorithm, write down this shortest route from B to C.
    2. Use an appropriate algorithm to find the shortest route from A to F .
    3. Write down all the pairs of villages for which the shortest route between them uses at least one footpath that is not in the minimum spanning tree for the six villages.
    OCR Further Discrete AS 2022 June Q7
    7 marks Standard +0.8
    7
    1. List the 15 partitions of the set \(\{ \mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D } , \mathrm { E } \}\) in which A and E are in the same subset.
    2. By considering the number of subsets in each of the partitions in part (a), or otherwise, explain why there are 8 partitions of the set \(\{ \mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D } , \mathrm { E } \}\) into two subsets with A and E in different subsets. Ali says that each of the 15 partitions from part (a) can be used to give two partitions in which A and E are in different subsets by moving E into a subset on its own or by moving E into another subset.
      [0pt]
      1. By considering the partition from part (a) into just one subset, show that Ali is wrong. [1]
      2. By considering a partition from part (a) into more than two subsets, show that Ali is wrong.
    OCR Further Discrete AS 2023 June Q1
    3 marks Moderate -0.8
    1 Jane wants to travel from home to the local town. Jane can do this by train, by bus or by both train and bus.
    1. Give an example of a problem that Jane could be answering that would give a construction problem. A website gives Jane all the possible buses and trains that she could use.
      Jane finds 7 possible ways to make the journey.
      • 2 of the 7 journeys involve travelling by train for at least part of the journey
      • 6 of the 7 journeys involve travelling by bus for at least part of the journey
      • Use the inclusion-exclusion principle to find how many of the 7 journeys involve travelling by both train and bus.
    OCR Further Discrete AS 2023 June Q2
    6 marks Moderate -0.5
    2 A network is shown below. \includegraphics[max width=\textwidth, alt={}, center]{c9fb5dad-1069-46cd-b167-80b77016f03c-2_533_869_1160_244}
    1. Use an appropriate algorithm to find the least weight (shortest) path from A to D.
    2. Use Kruskal's algorithm to find a minimum spanning tree for the network.
    OCR Further Discrete AS 2023 June Q3
    12 marks Easy -1.2
    3 The list of numbers below is to be sorted into increasing order. \(\begin{array} { l l l l l l l l } 23 & 10 & 18 & 7 & 62 & 54 & 31 & 82 \end{array}\)
    1. Sort the list using bubble sort. You do not need to show intermediate working.
      1. Record the list that results at the end of each pass.
      2. Record the number of swaps used in each pass.
    2. Now sort the original list using shuttle sort. You do not need to show intermediate working.
      1. Record the list that results at the end of each pass.
      2. Record the number of swaps used in each pass.
    3. Using the total number of comparisons plus the total number of swaps as a measure of efficiency, explain why shuttle sort is more efficient than bubble sort for sorting this particular list. Bubble sort and shuttle sort are both \(\mathrm { O } \left( n ^ { 2 } \right)\).
    4. Explain what this means for the run-time of the algorithms when the length of the list being sorted changes from 1000 to 3000.
    OCR Further Discrete AS 2023 June Q4
    10 marks Standard +0.3
    4 Graph G is a simply connected Eulerian graph with 4 vertices.
      1. Explain why graph G cannot be a complete graph.
      2. Determine the number of arcs in graph G, explaining your reasoning.
      3. Show that graph G is a bipartite graph. Graph H is a digraph with 4 vertices and no undirected arcs. The adjacency matrix below shows the number of arcs that directly connect each pair of vertices in digraph H . From \begin{table}[h]
        \captionsetup{labelformat=empty} \caption{To}
        ABCD
        A0101
        B0020
        C2101
        D0110
        \end{table}
      1. Write down a feature of the adjacency matrix that shows that H has no loops.
      2. Find the number of \(\operatorname { arcs }\) in H .
      3. Draw a possible digraph H .
      4. Show that digraph H is semi-Eulerian by writing down a suitable trail.
    OCR Further Discrete AS 2023 June Q5
    11 marks Moderate -0.5
    5 Hiro has been asked to organise a quiz.
    The table below shows the activities involved, together with the immediate predecessors and the duration of each activity in hours.
    ActivityImmediate predecessorsDuration (hours)
    AChoose the topics-0.5
    BFind questions for round 1A2
    CCheck answers for round 1B2.5
    DFind questions for round 2A2
    ECheck answers for round 2D2.5
    FChoose pictures for picture roundA1
    GGet permission to use picturesF1.5
    HChoose music for music roundA2
    IGet permission to use musicH1.5
    JProduce answer sheetsG0.5
    1. A sketch of the activity network is provided in the Printed Answer Booklet. Apply a forward pass to determine the minimum project completion time.
    2. Use a backward pass to determine the critical activities. You can show your working on the activity network from part (a).
    3. Give the total float for each non-critical activity. Hiro decides that there should be a final check of the answers which he will include as activity \(L\). Activity L needs to be done after checking the answers for rounds 1 and 2 and also after getting permission to use the pictures and music but before producing the answer sheets.
      1. Complete the activity network provided in the Printed Answer Booklet to show the new precedences, with the final check of the answers included as activity \(L\).
      2. As a result of including L , the minimum project completion time found in part (a) increases by 2.5 hours. Determine the duration of L .
    OCR Further Discrete AS 2023 June Q6
    6 marks Challenging +1.2
    6 Ryan and Casey are playing a card game in which they each have four cards.
    • Ryan's cards have the letters A, B, C and D.
    • Casey's cards have the letters W, X, Y and Z.
    Each player chooses one of their four cards and they simultaneously reveal their choices. The table shows the number of points won by Ryan for each combination of strategies. \begin{table}[h]
    \captionsetup{labelformat=empty} \caption{Casey}
    WXYZ
    \cline { 2 - 6 } RyanA4021
    B02- 34
    C14- 45
    D6- 150
    \end{table} For example, if Ryan chooses A and Casey chooses W then Ryan wins 4 points (and Casey loses 4 points). Both Ryan and Casey are trying to win as many points as possible.
    1. Use dominance to reduce the \(4 \times 4\) table for the zero-sum game above to a \(4 \times 2\) table.
    2. Determine an optimal mixed strategy for Casey.
    OCR Further Discrete AS 2023 June Q7
    12 marks Challenging +1.2
    7 A linear programming problem is
    Maximise \(P = 4 x + y\) subject to $$\begin{aligned} 3 x - y & \leqslant 30 \\ x + y & \leqslant 15 \\ x - 3 y & \leqslant 6 \end{aligned}$$ and \(x \geqslant 0 , y \geqslant 0\)
    1. Use a graphical method to find the optimal value of \(P\), and the corresponding values of \(x\) and \(y\). An additional constraint is introduced.
      This constraint means that the value of \(y\) must be at least \(k\) times the value of \(x\), where \(k\) is a positive constant.
      1. Determine the set of values of \(k\) for which the optimal value of \(P\) found in part (a) is unchanged.
      2. Determine, in terms of \(k\), the values of \(x , y\) and \(P\) in the cases when the optimal solution is different from that found in part (a).
    OCR Further Discrete AS 2024 June Q1
    8 marks Moderate -0.3
    1 There are six non-isomorphic trees with exactly six vertices.
    A student has drawn the diagram below showing six trees each with exactly six vertices. However, two of the trees that the student has drawn are isomorphic. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{6f64abca-108c-4b81-8ccf-124dfd9cc2f6-2_222_243_388_246} \captionsetup{labelformat=empty} \caption{A}
    \end{figure} \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{6f64abca-108c-4b81-8ccf-124dfd9cc2f6-2_214_240_392_635} \captionsetup{labelformat=empty} \caption{B}
    \end{figure} \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{6f64abca-108c-4b81-8ccf-124dfd9cc2f6-2_214_241_392_1023} \captionsetup{labelformat=empty} \caption{C}
    \end{figure} \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{6f64abca-108c-4b81-8ccf-124dfd9cc2f6-2_218_246_758_246} \captionsetup{labelformat=empty} \caption{D}
    \end{figure} \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{6f64abca-108c-4b81-8ccf-124dfd9cc2f6-2_216_243_760_632} \captionsetup{labelformat=empty} \caption{E}
    \end{figure} \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{6f64abca-108c-4b81-8ccf-124dfd9cc2f6-2_214_241_762_1023} \captionsetup{labelformat=empty} \caption{F}
    \end{figure}
    1. Identify which two of these trees are isomorphic.
    2. Draw an example of the missing tree. Graph G has exactly six vertices.
      The degree sequence of G is \(1,1,1,3,3,3\).
    3. Without using a sketch, calculate the number of edges in graph G.
    4. Explain how the result from part (c) shows that graph G is not a tree. In a simple graph with six vertices each vertex degree must be one of the values \(0,1,2,3,4\) or 5 .
    5. Use the pigeonhole principle to show that in a simple graph with six vertices there must be at least two vertices with the same vertex degree.
    OCR Further Discrete AS 2024 June Q2
    5 marks Moderate -0.5
    2 In a game two players are each dealt five cards from a set of ten different cards.
    Player 1 is dealt cards A, B, F, G and J.
    Player 2 is dealt cards C, D, E, H and I. Each player chooses a card to play.
    The players reveal their choices simultaneously. The pay-off matrix below shows the points scored by player 1 for each combination of cards. Pay-off for player 1 \begin{table}[h]
    \captionsetup{labelformat=empty} \caption{Player 2}
    C
    \cline { 3 - 7 } \multirow{4}{*}{Alayer 1}DEHI
    \cline { 3 - 7 }A41322
    \cline { 3 - 7 }B02121
    \cline { 3 - 7 }F01123
    \cline { 2 - 7 }G20333
    \cline { 3 - 7 }J12302
    \cline { 3 - 7 }
    \cline { 3 - 7 }
    \end{table}
    1. Determine the play-safe strategy for player 1, ignoring any effect on player 2. The pay-off matrix below shows the points scored by player 2 for each combination of cards.
      Pay-off for player 2 Player 1 \begin{table}[h]
      \captionsetup{labelformat=empty} \caption{Player 2}
      CDEH
      \cline { 2 - 6 } A2201I
      \cline { 2 - 6 } B31212
      \cline { 2 - 6 } F32210
      \cline { 2 - 6 } G13000
      \cline { 2 - 6 } J21031
      \cline { 2 - 6 }
      \cline { 2 - 6 }
      \end{table}
    2. Use a dominance argument to delete two columns from the pay-off matrix. You must show all relevant comparisons.
    3. Explain, with reference to specific combinations of cards, why the game cannot be converted to a zero-sum game.
    OCR Further Discrete AS 2024 June Q3
    6 marks Standard +0.3
    3 Heidi has a pack of cards.
    Each card has a single digit on one side and is blank on the other side.
    Each of the digits from 1 to 9 appears on exactly four cards.
    Apart from the numerical values of the digits, the cards are indistinguishable from each other.
    Heidi draws four cards from the pack, at random and without replacement. She places the four cards in a row to make a four-digit number. Determine how many different four-digit numbers Heidi could have made in each of the following cases.
    1. The four digits are all different.
    2. Two of the digits are the same and the other two digits are different.
    3. There is no restriction on whether any of the digits are the same or not.
    OCR Further Discrete AS 2024 June Q4
    9 marks Standard +0.8
    4 A project is represented by the activity network below. The activity durations are given in minutes. \includegraphics[max width=\textwidth, alt={}, center]{6f64abca-108c-4b81-8ccf-124dfd9cc2f6-5_447_1020_392_246}
    1. Give the reason for the dummy activity from event (3) to event (4).
    2. Complete a forward pass to determine the minimum project completion time.
    3. By completing a backward pass, calculate the float for each activity.
    4. Determine the effect on the minimum project completion time if the duration of activity A changes from 2 minutes to 3 minutes. The duration of activity C changes to \(m\) minutes, where \(m\) need not be an integer. This reduces the minimum project completion time.
    5. By considering the range of possible values of \(m\), determine the minimum project completion time, in terms of \(m\) where necessary.
    OCR Further Discrete AS 2024 June Q5
    10 marks Moderate -0.8
    5 A garden centre sells mixed packs of flower bulbs.
    Each pack contains bulbs which produce flowers of two different colours. The cost, in \(\pounds\), of a pack of each colour combination is shown in the table below.
    Pack typeABCDE
    ColoursRed, orangeRed, yellowOrange, yellowRed, pinkOrange, pink
    Cost \(( \pounds )\)1.503.004.005.256.50
    1. Represent the information in the table as a network in which the vertices are the colours and the arcs are the packs available, weighted using the costs.
    2. Construct a minimum spanning tree for the network. Taylor wants to buy at most four different packs of bulbs and to ensure that the packs include bulbs capable of producing all four flower colours. Taylor wants to minimise the total cost of these packs.
    3. Determine whether or not buying the packs represented by the solution to part (b) solves Taylor's problem.
    4. Represent Taylor's problem as an LP formulation, in which the variables are the number of packs of each type.
    OCR Further Discrete AS 2024 June Q6
    9 marks Moderate -0.3
    6 Beth wants to buy some tokens for use in a game.
    Each token is either a silver token or a gold token.
    Silver tokens and gold tokens have different points values in the game.
    Silver tokens have a value of 1.5 points each.
    Gold tokens have a value of 4 points each.
    Beth already has 2 silver tokens and 1 gold token.
    She also has \(\pounds 10\) that can be spent on buying more tokens.
    Silver tokens can be bought for \(\pounds 2\) each.
    Gold tokens can be bought for \(\pounds 6\) each.
    After buying some tokens, Beth has \(x\) silver tokens and \(y\) gold tokens.
    She now has a total of at least 5 tokens and no more than 8 tokens.
    1. Set up an LP formulation in \(x\) and \(y\) for the problem of maximising the points value of tokens that she finishes with.
    2. Use a graphical method to determine how many tokens of each type Beth should buy to maximise the points value of her tokens.