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Edexcel D2 2002 June Q10
6 marks Moderate -0.8
10. While solving a maximizing linear programming problem, the following tableau was obtained.
Basic variable\(x\)\(y\)\(z\)\(r\)\(s\)\(t\)Value
\(r\)00\(1 \frac { 2 } { 3 }\)10\(- \frac { 1 } { 6 }\)\(\frac { 2 } { 3 }\)
\(y\)01\(3 \frac { 1 } { 3 }\)01\(- \frac { 1 } { 3 }\)\(\frac { 1 } { 3 }\)
\(x\)10-30-1\(\frac { 1 } { 2 }\)1
P00101111
  1. Explain why this is an optimal tableau.
  2. Write down the optimal solution of this problem, stating the value of every variable.
  3. Write down the profit equation from the tableau. Use it to explain why changing the value of any of the non-basic variables will decrease the value of \(P\).
Edexcel D2 2002 June Q11
11 marks Moderate -0.5
11. A company wishes to transport its products from 3 factories \(F _ { 1 } , F _ { 2 }\) and \(F _ { 3 }\) to a single retail outlet \(R\). The capacities of the possible routes, in van loads per day, are shown in Fig. 5. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 5} \includegraphics[alt={},max width=\textwidth]{c4c64221-0373-4be9-abe3-5ff281922cdb-10_723_1172_476_337}
\end{figure}
  1. On Diagram 1 in the answer booklet add a supersource \(S\) to obtain a capacitated network with a single source and a single sink. State the minimum capacity of each arc you have added.
    1. State the maximum flow along \(S F _ { 1 } A B R\) and \(S F _ { 3 } C R\).
    2. Show these maximum flows on Diagram 2 in the answer booklet, using numbers in circles. Taking your answer to part (b)(ii) as the initial flow pattern,
    1. use the labelling procedure to find a maximum flow from \(S\) to \(R\). Your working should be shown on Diagram 3. List each flow-augmenting route you find together with its flow.
    2. Prove that your final flow is maximal.
Edexcel D2 2002 June Q12
11 marks Moderate -0.3
12. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{c4c64221-0373-4be9-abe3-5ff281922cdb-11_618_1211_253_253}
\end{figure} A company has 3 warehouses \(W _ { 1 } , W _ { 2 }\), and \(W _ { 3 }\). It needs to transport the goods stored there to 2 retail outlets \(R _ { 1 }\) and \(R _ { 2 }\). The capacities of the possible routes, in van loads per day, are shown in Fig 2. Warehouses \(W _ { 1 } , W _ { 2 }\) and \(W _ { 3 }\) have 14, 12 and 14 van loads respectively available per day and retail outlets \(R _ { 1 }\) and \(R _ { 2 }\) can accept 6 and 25 van loads respectively per day.
  1. On Diagram 1 on the answer sheet add a supersource \(W\), a supersink \(R\) and the appropriate directed arcs to obtain a single-source, single-sink capacitated network. State the minimum capacity of each arc you have added.
  2. State the maximum flow along
    1. \(W \quad W _ { 1 } \quad A \quad R _ { 1 } \quad R\),
    2. \(W W _ { 3 } \quad C \quad R _ { 2 } \quad R\).
  3. Taking your answers to part (b) as the initial flow pattern, use the labelling procedure to obtain a maximum flow through the network from \(W\) to \(R\). Show your working on Diagram 2. List each flowaugmenting route you use, together with its flow.
  4. From your final flow pattern, determine the number of van loads passing through \(B\) each day.
Edexcel D2 2003 June Q1
6 marks Challenging +1.2
  1. A two person zero-sum game is represented by the following pay-off matrix for player \(A\).
\cline { 2 - 4 } \multicolumn{1}{c|}{}\(B\) plays I\(B\) plays II\(B\) plays III
\(A\) plays I- 325
\(A\) plays II4- 1- 4
  1. Write down the pay off matrix for player \(B\).
  2. Formulate the game as a linear programming problem for player \(B\), writing the constraints as equalities and stating your variables clearly.
Edexcel D2 2003 June Q2
10 marks Moderate -0.3
2. (a) Explain the difference between the classical and practical travelling salesman problems. \includegraphics[max width=\textwidth, alt={}, center]{eabe577b-80d9-45f8-a8e8-0c3b139b96a8-1_691_1297_1014_383} The network in the diagram above shows the distances, in kilometres, between eight McBurger restaurants. An inspector from head office wishes to visit each restaurant. His route should start and finish at \(A\), visit each restaurant at least once and cover a minimum distance.
(b) Obtain a minimum spanning tree for the network using Kruskal's algorithm. You should draw your tree and state the order in which the arcs were added.
(c) Use your answer to part (b) to determine an initial upper bound for the length of the route.
(d) Starting from your initial upper bound and using an appropriate method, find an upper bound which is less than 135 km . State your tour.
Edexcel D2 2003 June Q3
13 marks Moderate -0.8
3. Talkalot College holds an induction meeting for new students. The meeting consists of four talks: I (Welcome), II (Options and Facilities), III (Study Tips) and IV (Planning for Success). The four department heads, Clive, Julie, Nicky and Steve, deliver one of these talks each. The talks are delivered consecutively and there are no breaks between talks. The meeting starts at 10 a.m. and ends when all four talks have been delivered. The time, in minutes, each department head takes to deliver each talk is given in the table below.
\cline { 2 - 5 } \multicolumn{1}{c|}{}Talk ITalk IITalk IIITalk IV
Clive12342816
Julie13323612
Nicky15323214
Steve11333610
  1. Use the Hungarian algorithm to find the earliest time that the meeting could end. You must make your method clear and show
    1. the state of the table after each stage in the algorithm,
    2. the final allocation.
  2. Modify the table so it could be used to find the latest time that the meeting could end.
Edexcel D2 2003 June Q4
14 marks Moderate -0.5
4. A two person zero-sum game is represented by the following pay-off matrix for player \(A\).
\cline { 2 - 4 } \multicolumn{1}{c|}{}\(B\) plays I\(B\) plays II\(B\) plays III
\(A\) plays I2- 13
\(A\) plays II130
\(A\) plays III01- 3
  1. Identify the play safe strategies for each player.
  2. Verify that there is no stable solution to this game.
  3. Explain why the pay-off matrix above may be reduced to
    \cline { 2 - 4 } \multicolumn{1}{c|}{}\(B\) plays I\(B\) plays II\(B\) plays III
    \(A\) plays I2- 13
    \(A\) plays II130
  4. Find the best strategy for player \(A\), and the value of the game.
Edexcel D2 2003 June Q5
14 marks Moderate -0.8
5. The manager of a car hire firm has to arrange to move cars from three garages \(A , B\) and \(C\) to three airports \(D , E\) and \(F\) so that customers can collect them. The table below shows the transportation cost of moving one car from each garage to each airport. It also shows the number of cars available in each garage and the number of cars required at each airport. The total number of cars available is equal to the total number required.
Airport \(D\)Airport \(E\)Airport \(F\)Cars available
Garage \(A\)£20£40£106
Garage \(B\)£20£30£405
Garage C£10£20£308
Cars required694
  1. Use the North-West corner rule to obtain a possible pattern of distribution and find its cost.
    (3)
  2. Calculate shadow costs for this pattern and hence obtain improvement indices for each route.
  3. Use the stepping-stone method to obtain an optimal solution and state its cost.
Edexcel D2 2003 June Q6
18 marks Standard +0.3
6. Kris produces custom made racing cycles. She can produce up to four cycles each month, but if she wishes to produce more than three in any one month she has to hire additional help at a cost of \(\pounds 350\) for that month. In any month when cycles are produced, the overhead costs are \(\pounds 200\). A maximum of 3 cycles can be held in stock in any one month, at a cost of \(\pounds 40\) per cycle per month. Cycles must be delivered at the end of the month. The order book for cycles is
MonthAugustSeptemberOctoberNovember
Number of cycles required3352
Disregarding the cost of parts and Kris' time,
  1. determine the total cost of storing 2 cycles and producing 4 cycles in a given month, making your calculations clear. There is no stock at the beginning of August and Kris plans to have no stock after the November delivery.
  2. Use dynamic programming to determine the production schedule which minimises the costs, showing your working in the table below.
    StageDemandStateActionDestinationValue
    \multirow[t]{3}{*}{1 (Nov)}\multirow[t]{3}{*}{2}0 (in stock)(make) 20200
    1 (in stock)(make) 10240
    2 (in stock)(make) 0080
    \multirow[t]{2}{*}{2 (Oct)}\multirow[t]{2}{*}{5}140\(590 + 200 = 790\)
    230
    The fixed cost of parts is \(\pounds 600\) per cycle and of Kris' time is \(\pounds 500\) per month. She sells the cycles for \(\pounds 2000\) each.
  3. Determine her total profit for the four month period.
    (3)
    (Total 18 marks)
Edexcel D2 2003 June Q7
18 marks Moderate -0.8
7. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{eabe577b-80d9-45f8-a8e8-0c3b139b96a8-4_759_1529_715_267}
\end{figure} Figure 1 shows a capacitated, directed network. The unbracketed number on each arc indicates the capacity of that arc, and the numbers in brackets show a feasible flow of value 68 through the network.
  1. Add a supersource and a supersink, and arcs of appropriate capacity, to Diagram 1 below. \section*{Diagram 1} \includegraphics[max width=\textwidth, alt={}, center]{eabe577b-80d9-45f8-a8e8-0c3b139b96a8-4_684_1531_1816_267}
  2. Find the values of \(x\) and \(y\), explaining your method briefly.
  3. Find the value of cuts \(C _ { 1 }\) and \(C _ { 2 }\). Starting with the given feasible flow of 68,
  4. use the labelling procedure on Diagram 2 to find a maximal flow through this network. List each flow-augmenting route you use, together with its flow. \section*{Diagram 2} \includegraphics[max width=\textwidth, alt={}, center]{eabe577b-80d9-45f8-a8e8-0c3b139b96a8-5_647_1506_612_283}
  5. Show your maximal flow on Diagram 3 and state its value. \section*{Diagram 3} \includegraphics[max width=\textwidth, alt={}, center]{eabe577b-80d9-45f8-a8e8-0c3b139b96a8-5_654_1511_1567_278}
  6. Prove that your flow is maximal.
Edexcel D2 2003 June Q8
14 marks Challenging +1.2
8. The tableau below is the initial tableau for a maximising linear programming problem.
Basic
variable
\(x\)\(y\)\(z\)\(r\)\(s\)Value
\(r\)234108
\(s\)3310110
\(P\)- 8- 9- 5000
  1. For this problem \(x \geq 0 , y \geq 0 , z \geq 0\). Write down the other two inequalities and the objective function.
  2. Solve this linear programming problem. You may not need to use all of these tableaux.
    b.v.\(x\)\(y\)\(z\)\(r\)\(s\)Value
    \(P\)
    b.v.\(x\)\(y\)\(z\)\(r\)\(s\)Value
    \(P\)
    b.v.\(x\)\(y\)\(z\)\(r\)\(s\)Value
    \(P\)
    b.v.\(x\)\(y\)\(z\)\(r\)\(s\)Value
    \(P\)
  3. State the final value of \(P\), the objective function, and of each of the variables.
Edexcel D2 2004 June Q1
4 marks Easy -1.2
  1. In game theory explain what is meant by
    1. zero-sum game,
    2. saddle point.
      (Total 4 marks)
    3. In a quiz there are four individual rounds, Art, Literature, Music and Science. A team consists of four people, Donna, Hannah, Kerwin and Thomas. Each of four rounds must be answered by a different team member. The table shows the number of points that each team member is likely to get on each individual round.
    \cline { 2 - 5 } \multicolumn{1}{c|}{}ArtLiteratureMusicScience
    Donna31243235
    Hannah16101922
    Kerwin19142021
    Thomas18152123
    Use the Hungarian algorithm, reducing rows first, to obtain an allocation which maximises the total points likely to be scored in the four rounds. You must make your method clear and show the table after each stage.
    (Total 9 marks)
Edexcel D2 2004 June Q3
12 marks Moderate -0.5
3. The table shows the least distances, in km, between five towns, \(A , B , C , D\) and \(E\). Nassim wishes to find an interval which contains the solution to the travelling salesman problem for this network.
  1. Making your method clear, find an initial upper bound starting at \(A\) and using
    1. the minimum spanning tree method,
    2. the nearest neighbour algorithm.
      \(A\)\(B\)\(C\)\(D\)\(E\)
      \(A\)-15398124115
      \(B\)153-74131149
      \(C\)9874-82103
      \(D\)12413182-134
      \(E\)115149103134-
  2. By deleting \(E\), find a lower bound.
  3. Using your answers to parts (a) and (b), state the smallest interval that Nassim could correctly write down.
    (Total 12 marks)
Edexcel D2 2004 June Q4
14 marks Standard +0.8
4. Emma and Freddie play a zero-sum game. This game is represented by the following pay-off matrix for Emma. \(\left( \begin{array} { r r r } - 4 & - 1 & 3 \\ 2 & 1 & - 2 \end{array} \right)\)
  1. Show that there is no stable solution.
  2. Find the best strategy for Emma and the value of the game to her.
  3. Write down the value of the game to Freddie and his pay-off matrix.
Edexcel D2 2004 June Q5
16 marks Standard +0.3
5. (a) Describe a practical problem that could be solved using the transportation algorithm. A problem is to be solved using the transportation problem. The costs are shown in the table. The supply is from \(A , B\) and \(C\) and the demand is at \(d\) and \(e\).
\cline { 2 - 4 } \multicolumn{1}{c|}{}\(d\)\(e\)Supply
\(A\)5345
\(B\)4635
\(C\)2440
Demand5060
(b) Explain why it is necessary to add a third demand \(f\).
(c) Use the north-west corner rule to obtain a possible pattern of distribution and find its cost.
\cline { 2 - 5 } \multicolumn{1}{c|}{}\(d\)\(e\)\(f\)Supply
\(A\)5345
\(B\)4635
\(C\)2440
Demand5060
(d) Calculate shadow costs and improvement indices for this pattern.
(e) Use the stepping-stone method once to obtain an improved solution and its cost.
(Total 16 marks)
Edexcel D2 2004 June Q6
18 marks Standard +0.3
6. Joan sells ice cream. She needs to decide which three shows to visit over a three-week period in the summer. She starts the three-week period at home and finishes at home. She will spend one week at each of the three shows she chooses travelling directly from one show to the next. Table 1 gives the week in which each show is held. Table 2 gives the expected profits from visiting each show. Table 3 gives the cost of travel between shows. \begin{table}[h]
\captionsetup{labelformat=empty} \caption{Table 1}
Week123
ShowsA, B, CD, EF, G, H
\end{table} Table 2 \begin{table}[h]
\captionsetup{labelformat=empty} \caption{Table 1}
ShowA\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)
Expected Profit \(( \pounds )\)900800100015001300500700600
\end{table} \begin{table}[h]
\captionsetup{labelformat=empty} \caption{Table 3}
Travel costs (£)ABCDEFGH
Home7080150809070
A180150
B140120
C200210
D200160120
E170100110
\end{table} It is decided to use dynamic programming to find a schedule that maximises the total expected profit, taking into account the travel costs.
  1. Define suitable stage, state and action variables.
  2. Determine the schedule that maximises the total profit. Show your working in a table.
  3. Advise Joan on the shows that she should visit and state her total expected profit.
    (3) (Total 18 marks)
Edexcel D2 2004 June Q7
13 marks Moderate -0.3
7. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{343fdcef-660e-4e8c-bd9c-a7f929dc668e-3_526_903_1455_575}
\end{figure} Figure 1 shows a capacitated directed network. The number on each arc is its capacity. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{343fdcef-660e-4e8c-bd9c-a7f929dc668e-3_499_967_2231_548}
\end{figure} Figure 2 shows a feasible initial flow through the same network.
  1. Write down the values of the flow \(x\) and the flow \(y\).
  2. Obtain the value of the initial flow through the network, and explain how you know it is not maximal.
  3. Use this initial flow and the labelling procedure on Diagram 1 below to find a maximum flow through the network. You must list each flow-augmenting route you use, together with its flow. \section*{Diagram 1} \includegraphics[max width=\textwidth, alt={}, center]{343fdcef-660e-4e8c-bd9c-a7f929dc668e-4_1920_1175_742_450}
    d) Show your maximal flow pattern on Diagram 2. \section*{Diagram 2} \includegraphics[max width=\textwidth, alt={}, center]{343fdcef-660e-4e8c-bd9c-a7f929dc668e-5_679_1102_315_479}
    (2)
  4. Prove that your flow is maximal.
    (2)
    (Total 13 marks)
Edexcel D2 2004 June Q8
6 marks Moderate -0.5
8. A three-variable linear programming problem in \(x , y\) and \(z\) is to be solved. The objective is to maximise the profit P . The following tableau was obtained.
Basic variable\(x\)\(y\)\(Z\)\(r\)\(s\)\(t\)Value
S30201\(- \frac { 2 } { 3 }\)\(\frac { 2 } { 3 }\)
\(r\)40\(\frac { 7 } { 2 }\)108\(\frac { 9 } { 2 }\)
\(y\)5170037
P30200863
  1. State, giving your reason, whether this tableau represents the optimal solution.
  2. State the values of every variable.
  3. Calculate the profit made on each unit of \(y\).
Edexcel D2 2004 June Q9
7 marks Standard +0.3
9. \includegraphics[max width=\textwidth, alt={}, center]{343fdcef-660e-4e8c-bd9c-a7f929dc668e-6_1088_1509_219_285} The diagram above shows a network of roads represented by arcs. The capacity of the road represented by that arc is shown on each arc. The numbers in circles represent a possible flow of 26 from \(B\) to \(L\). Three cuts \(\mathrm { C } _ { 1 } , \mathrm { C } _ { 2 }\) and \(\mathrm { C } _ { 3 }\) are shown on The diagram above.
  1. Find the capacity of each of the three cuts.
  2. Verify that the flow of 26 is maximal. The government aims to maximise the possible flow from \(B\) to \(L\) by using one of two options.
    Option 1: Build a new road from \(E\) to \(J\) with capacity 5.
    or
    Option 2: Build a new road from \(F\) to \(H\) with capacity 3.
  3. By considering both options, explain which one meets the government's aim
Edexcel D2 2004 June Q10
18 marks Moderate -0.5
10. Flatland UK Ltd makes three types of carpet, the Lincoln, the Norfolk and the Suffolk. The carpets all require units of black, green and red wool. For each roll of carpet, the Lincoln requires 1 unit of black, 1 of green and 3 of red, the Norfolk requires 1 unit of black, 2 of green and 2 of red, and the Suffolk requires 2 units of black, 1 of green and 1 of red. There are up to 30 units of black, 40 units of green and 50 units of red available each day.
Profits of \(\pounds 50 , \pounds 80\) and \(\pounds 60\) are made on each roll of Lincoln, Norfolk and Suffolk respectively. Flatland UK Ltd wishes to maximise its profit. Let the number of rolls of the Lincoln, Norfolk and Suffolk made daily be \(x , y\) and \(z\) respectively.
  1. Formulate the above situation as a linear programming problem, listing clearly the constraints as inequalities in their simplest form, and stating the objective function. This problem is to be solved using the Simplex algorithm. The most negative number in the profit row is taken to indicate the pivot column at each stage.
  2. Stating your row operations, show that after one complete iteration the tableau becomes
    Basic variable\(x\)\(y\)\(z\)\(r\)\(s\)\(t\)Value
    \(r\)\(\frac { 1 } { 2 }\)0\(1 \frac { 1 } { 2 }\)1\(- \frac { 1 } { 2 }\)010
    \(y\)\(\frac { 1 } { 2 }\)1\(\frac { 1 } { 2 }\)0\(\frac { 1 } { 2 }\)020
    \(t\)2000-1110
    \(P\)-100-2004001600
    You may not need to use all of the tableaux.
    Basic variable\(x\)\(y\)\(z\)\(r\)\(s\)\(t\)ValueRow operations
    \(r\)
    \(S\)
    \(t\)
    \(P\)
    Basic variable\(x\)\(y\)\(z\)\(r\)\(s\)\(t\)ValueRow operations
  3. Explain the practical meaning of the value 10 in the top row.
    1. Perform one further complete iteration of the Simplex algorithm.
      Basic variable\(x\)\(y\)\(z\)\(r\)\(s\)\(t\)ValueRow operations
      Basic variable\(x\)\(y\)\(z\)\(r\)\(s\)\(t\)ValueRow operations
    2. State whether your current answer to part (d)(i) is optimal. Give a reason for your answer.
    3. Interpret your current tableau, giving the value of each variable.
      (8)
Edexcel D2 2005 June Q1
11 marks Moderate -0.5
  1. Freezy Co. has three factories \(A , B\) and \(C\). It supplies freezers to three shops \(D , E\) and \(F\). The table shows the transportation cost in pounds of moving one freezer from each factory to each outlet. It also shows the number of freezers available for delivery at each factory and the number of freezers required at each shop. The total number of freezers required is equal to the total number of freezers available.
\cline { 2 - 5 } \multicolumn{1}{c|}{}\(D\)\(E\)\(F\)Available
\(A\)21241624
\(B\)18231732
\(C\)15192514
Required203020
\cline { 1 - 4 }
\cline { 1 - 4 }
  1. Use the north-west corner rule to find an initial solution.
  2. Obtain improvement indices for each unused route.
  3. Use the stepping-stone method once to obtain a better solution and state its cost.
Edexcel D2 2005 June Q2
11 marks Standard +0.3
2. \includegraphics[max width=\textwidth, alt={}, center]{be329a47-a709-4719-abe6-41d388a6c631-1_613_1269_1318_392} The network in the diagram shows the distances, in km , of the cables between seven electricity relay stations \(A , B , C , D , E , F\) and \(G\). An inspector needs to visit each relay station. He wishes to travel a minimum distance, and his route must start and finish at the same station. By deleting C, a lower bound for the length of the route is found to be 129 km .
  1. Find another lower bound for the length of the route by deleting \(F\). State which is the better lower bound of the two.
  2. By inspection, complete the table of least distances. The table can now be taken to represent a complete network.
  3. Using the nearest-neighbour algorithm, starting at \(F\), obtain an upper bound to the length of the route. State your route.
    (4) (Total 11 marks)
Edexcel D2 2005 June Q3
7 marks Moderate -0.3
3. Three warehouses \(W , X\) and \(Y\) supply televisions to three supermarkets \(J , K\) and \(L\). The table gives the cost, in pounds, of transporting a television from each warehouse to each supermarket. The warehouses have stocks of 34, 57 and 25 televisions respectively, and the supermarkets require 20, 56 and 40 televisions respectively. The total cost of transporting the televisions is to be minimised.
\(J\)\(K\)\(L\)
\(W\)363
\(X\)584
\(Y\)257
Formulate this transportation problem as a linear programming problem. Make clear your decision variables, objective function and constraints.
(Total 7 marks)
Edexcel D2 2005 June Q4
14 marks Standard +0.3
4. (a) Explain what is meant by a maximin route in dynamic programming, and give an example of a situation that would require a maximin solution.
(3) \includegraphics[max width=\textwidth, alt={}, center]{be329a47-a709-4719-abe6-41d388a6c631-2_700_1392_1069_338} A maximin route is to be found through the network shown in the diagram.
(b) Complete the table in the answer book, and hence find a maximin route.
(9)
(c) List all other maximin routes through the network.
(Total 14 marks)
Edexcel D2 2005 June Q5
15 marks Moderate -0.3
5. Four salesperson \(A , B , C\) and \(D\) are to be sent to visit four companies \(1,2,3\) and 4 . Each salesperson will visit exactly one company, and all companies will be visited. Previous sales figures show that each salesperson will make sales of different values, depending on the company that they visit. These values (in \(\pounds 10000\) s) are shown in the table below.
\cline { 2 - 5 } \multicolumn{1}{c|}{}1234
Ann26303030
Brenda30232629
Connor30252724
Dave30272521
  1. Use the Hungarian algorithm to obtain an allocation that maximises the sales. You must make your method clear and show the table after each stage.
  2. State the value of the maximum sales.
  3. Show that there is a second allocation that maximises the sales.
    (Total 15 marks)