Questions D2 - A-Level Maths

	Greg plays 1	Greg plays 2	Greg plays 3
Rani plays 1	- 3	1	2
Rani plays 2	0	2	1
Rani plays 3	2	4	- 5

Explain what the term 'zero-sum game' means.
State the number of points that Greg scores if he plays his strategy 3 and Rani plays her strategy 3.
Verify that there is no stable solution to this game.
Reduce the game so that Greg has only two possible strategies. Write down the reduced pay-off matrix for Greg.
Find the best strategy for Greg and the value of the game to him.

Edexcel D2 2015 June Q3

	A	B	C	D	E	F	G
A	-	\(x\)	41	43	38	21	30
B	\(x\)	-	27	38	19	29	51
C	41	27	-	24	37	35	40
D	43	38	24	-	44	52	25
E	38	19	37	44	-	20	28
F	21	29	35	52	20	-	49
G	30	51	40	25	28	49	-

The network represented by the table shows the least distances, in km, between seven theatres, A, \(\mathrm { B } , \mathrm { C } , \mathrm { D } , \mathrm { E } , \mathrm { F }\) and G . Jasmine needs to visit each theatre at least once starting and finishing at A. She wishes to minimise the total distance she travels. The least distance between A and B, is \(x \mathrm {~km}\), where \(21 < x < 27\)

Using Prim's algorithm, starting at A , obtain a minimum spanning tree for the network. You should list the arcs in the order in which you consider them.
Use your answer to (a) to determine an initial upper bound for the length of Jasmine's route.
Use the nearest neighbour algorithm, starting at A , to find a second upper bound for the length of the route. The nearest neighbour algorithm starting at F gives a route of \(\mathrm { F } - \mathrm { E } - \mathrm { B } - \mathrm { A } - \mathrm { G } - \mathrm { D } - \mathrm { C } - \mathrm { F }\).
State which of these two nearest neighbour routes gives the better upper bound. Give a reason for your answer. Starting by deleting A, and all of its arcs, a lower bound of 159 km for the length of the route is found.
Find \(x\), making your method clear.
Write down the smallest interval that you can be confident contains the optimal length of Jasmine's route. Give your answer as an inequality.

Edexcel D2 2015 June Q4

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{57c75bde-811a-421c-899a-3689bdba6724-5_837_1217_233_424} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows a capacitated network. The capacity of each arc is shown on the arc. Two cuts \(\mathrm { C } _ { 1 }\) and \(\mathrm { C } _ { 2 }\) are shown.

Find the capacity of each of the two cuts. Given that one of these two cuts is a minimum cut,
write down the maximum possible flow through the network. Given that the network now has a maximal flow from S to T ,
determine the flow along arc SB.
Explain why arcs GF and GT cannot both be saturated. Given that arcs EC, AD and DF are saturated and that there is no flow along arc GF,
determine a maximum flow pattern for this network and draw it on Diagram 1 in the answer book. You do not need to use the labelling procedure to determine the maximum flow.

Edexcel D2 2015 June Q5

5. The table shows the cost, in pounds, of transporting one unit of stock from each of four supply points, \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) and D , to each of three sales points, \(\mathrm { P } , \mathrm { Q }\) and R . It also shows the stock held at each supply point and the amount required at each sales point. A minimum cost solution is required.

	P	Q	R	Supply
A	20	5	13	74
B	7	15	8	58
C	9	14	21	63
D	22	16	10	85
Demand	145	57	78

The north-west corner method gives the following initial solution.

	P	Q	R	Supply
A	74			74
B	58			58
C	13	50		63
D		7	78	85
Demand	145	57	78

Taking AQ as the entering cell, use the stepping stone method to find an improved solution. Make your route clear.
Perform one further iteration of the stepping stone method to obtain an improved solution. You must make your method clear by stating your shadow costs, improvement indices, route, entering cell and exiting cell.
Determine whether your current solution is optimal. Justify your answer.
State the cost of the solution you found in (b).
Formulate this problem as a linear programming problem. You must define your decision variables and make the objective function and constraints clear.

Edexcel D2 2015 June Q6

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{57c75bde-811a-421c-899a-3689bdba6724-7_614_1264_239_402} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} The staged, directed network in Figure 2 represents a series of roads connecting 11 towns, \(\mathrm { S } , \mathrm { A }\), B, C, D, E, F, G, H, J and T. The number on each arc shows the weight limit, in tonnes, for the corresponding road. Janet needs to drive a truck from S to T, passing through exactly three other towns. She needs to find the maximum weight of the truck that she can use.

Write down the type of dynamic programming problem that Janet needs to solve.
Use dynamic programming to complete the table in the answer book.
Hence find the maximum weight of the truck Janet can use.
Write down the route that Janet should take. Janet intends to ask for the weight limit to be increased on one of the three roads leading directly into T. Janet wishes to maximise the weight of her truck.
1. Determine which of the three roads she should choose and its new minimum weight limit.
2. Write down the maximum weight of the truck she would be able to use and the new route she would take.

Edexcel D2 2016 June Q1

(a) Explain the difference between the classical travelling salesperson problem and the practical travelling salesperson problem.

	A	B	C	D	E	F	G
A	-	31	15	12	24	17	22
B	31	-	20	25	14	25	50
C	15	20	-	16	24	19	21
D	12	25	16	-	21	32	17
E	24	14	24	21	-	28	41
F	17	25	19	32	28	-	25
G	22	50	21	17	41	25	-

The table above shows the least direct distances, in miles, between seven towns, \(\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D } , \mathrm { E } , \mathrm { F }\) and G. Yiyi needs to visit each town, starting and finishing at A, and wishes to minimise the total distance she will travel.
(b) Show that there are two nearest neighbour routes that start from A. State these routes and their lengths.
(c) Starting by deleting A, and all of its arcs, find a lower bound for the length of Yiyi's route.
(d) Use your results to write down the smallest interval which you can be confident contains the optimal length of Yiyi's route.

Edexcel D2 2016 June Q2

2. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{a12f9520-85c0-43f6-a104-2502011d5657-3_780_1155_246_461} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows a capacitated, directed network of pipes. The number on each arc represents the capacity of the corresponding pipe. The numbers in circles represent an initial flow.

List the saturated arcs.
State the value of the initial flow.
State the capacities of the cuts \(C _ { 1 }\) and \(C _ { 2 }\)
By inspection, find a flow-augmenting route to increase the flow by three units. You must state your route.
Prove that the new flow is maximal.

Edexcel D2 2016 June Q3

3. Four pupils, Alexa, Ewan, Faith and Zak, are to be allocated to four rounds, 1, 2, 3 and 4, in a mathematics competition. Each pupil is to be allocated to exactly one round and each round must be allocated to exactly one pupil. Each pupil has been given a score, based on previous performance, to show how suitable they are for each round. The higher the score the more suitable the pupil is for that round. The scores for each pupil are shown in the table below.

	1	2	3	4
Alexa	61	50	47	23
Ewan	71	62	20	61
Faith	70	49	48	49
Zak	72	68	67	67

Reducing rows first, use the Hungarian algorithm to obtain an allocation that maximises the total score. You must make your method clear and show the table after each stage.
(8)
State the maximum total score.

Edexcel D2 2016 June Q4

4. 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 is obtained after the first iteration.

Basic Variable	\(x\)	\(y\)	\(z\)	\(r\)	\(s\)	\(t\)	Value
r	0	5	2	1	-3	0	10
\(x\)	1	2	3	0	1	0	18
\(t\)	0	1	-1	0	4	1	3
\(P\)	0	3	-4	0	1	0	7

State which variable was increased first, giving a reason for your answer.
Perform one complete iteration of the simplex algorithm, to obtain a new tableau, T. Make your method clear by stating the row operations you use.
Write down the profit equation given by T .
State whether T is optimal. You must use your answer to (c) to justify your answer.

Edexcel D2 2016 June Q5

5. The table below shows the cost of transporting one unit of stock from each of four supply points, 1 , 2,3 and 4, to each of three demand points, A, B and C. It also shows the stock held at each supply point and the stock required at each demand point. A minimal cost solution is required.

	A	B	C	Supply
1	18	23	20	15
2	22	17	25	36
3	24	21	19	28
4	21	22	17	20
Demand	40	20	25

Explain why it is necessary to add a dummy demand point.
Add a dummy demand point and appropriate values to Table 1 in the answer book.
Use the north-west corner method to obtain a possible solution. After one iteration of the stepping-stone method the table becomes
A B C D
1 15
2 19 17
3 3 25
4 6 14
Taking D3 as the entering cell, use the stepping-stone method twice to obtain an improved solution. You must make your method clear by stating your shadow costs, improvement indices, routes, entering cells and exiting cells.
Determine whether your solution from (d) is optimal. Justify your answer.

Edexcel D2 2016 June Q6

6. A two-person zero-sum game is represented by the following pay-off matrix for player A.

	B plays 1	B plays 2	B plays 3
A plays 1	5	- 3	1
A plays 2	2	5	0
A plays 3	- 4	- 1	4

Verify that there is no stable solution to this game.
Formulate the game as a linear programming problem for player A. Define your variables clearly. Write the constraints as equations.
Write down an initial simplex tableau, making your variables clear.

Edexcel D2 2016 June Q7

7. Remy builds canoes. He can build up to five canoes each month, but if he wishes to build more than three canoes in any one month he has to hire an additional worker at a cost of \(\pounds 400\) for that month. In any month when canoes are built, the overhead costs are \(\pounds 150\) A maximum of three canoes can be held in stock in any one month, at a cost of \(\pounds 25\) per canoe per month. Canoes must be delivered at the end of the month. The order book for canoes is

Month	January	February	March	April	May
Number ordered	2	2	5	6	4

There is no stock at the beginning of January and Remy plans to have no stock after the May delivery.

Use dynamic programming to determine the production schedule that minimises the costs given above. Show your working in the table provided in the answer book and state the minimum cost.
(13) The cost of materials is \(\pounds 200\) per canoe and the cost of Remy’s time is \(\pounds 450\) per month. Remy sells the canoes for \(\pounds 700\) each.
Determine Remy's total profit for the five-month period.
(2)
(Total 15 marks)

Edexcel D2 Q1

A company has five machines \(A , B , C , D\) and \(E\), which it can assign to four tasks \(1,2,3\) and 4 . Each task must be assigned to just one machine and each machine may only be assigned to just one task.

The profit, in \(\pounds 100\) s, of using each machine to do each task is given in the table below.

	1	2	3	4
\(A\)	14	12	11	17
\(B\)	14	13	15	16
\(C\)	17	16	10	12
\(D\)	16	14	13	12
\(E\)	13	15	13	15

Explain why it is necessary to add a dummy column to the table.
(2)
Use the Hungarian algorithm to allocate machines to tasks in order to maximise the total profit. You must make your method clear and show the state of the table after each iteration.
(7)
(Total 9 marks)

Edexcel D2 Q2

2. The following transportation problem is to be solved.

	\(P\)	\(Q\)	\(R\)	Supply
\(A\)	7	5	7	12
\(B\)	5	6	5	7
\(C\)	14	12	9	11
Demand	10	9	11

A possible north-west corner solution is:

	\(P\)	\(Q\)	\(R\)
\(A\)	10	2
\(B\)		7	0
\(C\)			11

Use the stepping-stone method once to obtain an improved solution. You must make your shadow costs, improvement indices, entering cell, exiting cell and stepping-stone route clear.
Demonstrate that your solution is optimal.
(3)

Edexcel D2 Q3

3. A two-person zero-sum game is represented by the following pay-off matrix for player A.

	B plays 1	B plays 2	B plays 3
A plays 1	- 2	4	3
A plays 2	4	- 1	2

Find the best strategy for player A and the value of the game.
(Total 7 marks)

Edexcel D2 Q4

4. The table shows the least distances, in km, between six towns \(A , B , C , D , E\) and \(F\).

	\(A\)	\(B\)	\(C\)	\(D\)	\(E\)	\(F\)
\(A\)	-	98	123	68	96	71
\(B\)	98	-	74	129	47	120
\(C\)	123	74	-	102	111	63
\(D\)	68	129	102	-	85	59
\(E\)	96	47	111	85	-	115
\(F\)	71	120	63	59	115	-

Starting at \(A\), and making your method clear, find an upper bound for the travelling salesman problem using the nearest neighbour algorithm.
By deleting \(A\), and all of its arcs, find a lower bound for the travelling salesman problem.
Write down an inequality about the length of the optimal route.

Edexcel D2 Q5

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{e80fcab6-7c7d-4a0c-84e0-c23f5a969a75-4_924_1646_221_207} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows a capacitated, directed, network. The capacity of each arc is shown on that arc and the numbers in circles represent an initial flow from S to T . Two cuts, \(\mathrm { C } _ { 1 }\) and \(\mathrm { C } _ { 2 }\) are shown on Figure 1.

Find the capacity of each of the two cuts and the value of the initial flow.
(3)
Complete the initialisation of the labelling procedure on Figure 1 in the answer book, by entering values along \(\mathrm { SB } , \mathrm { AB } , \mathrm { BE }\) and BG .
(2)
Hence use the labelling procedure to find a maximum flow of 85 through the network. You must list each flow-augmenting path you use, together with its flow.
(5)
Show your flow pattern on Figure 2.
(2)
Prove that your flow is maximal.
(2)

Edexcel D2 Q6

6. The tableau below is the initial tableau for a maximising linear programming problem.

Basic variable	\(x\)	\(y\)	\(z\)	\(r\)	\(s\)	\(t\)	Value
\(r\)	16	2	4	1	0	0	350
\(s\)	18	- 2	6	0	1	0	480
\(t\)	5	0	5	0	0	1	360
\(P\)	- 18	- 7	- 20	0	0	0	0

Write down the four equations represented in the initial tableau.
Taking the most negative number in the profit row to indicate the pivot column at each stage, perform two complete iterations of the Simplex algorithm. State the row operations that you use.
State whether or not your last tableau is optimal. Give a reason for your answer.

Edexcel D2 Q7

7. D2 make industrial robots. They can make up to four in any one month, but if they make more than three they need to hire additional labour at a cost of \(\pounds 300\) per month. They can store up to three robots at a cost of \(\pounds 100\) per robot per month. The overhead costs are \(\pounds 500\) in any month in which work is done. The robots are delivered to buyers at the end of each month. There are no robots in stock at the beginning of January and there should be none in stock at the end of May. The order book for January to May is:

Month	January	February	March	April	May
Number of robots required	3	2	2	5	4

Use dynamic programming to determine the production schedule that minimises the costs, showing your working in the table provided in the answer book. State the minimum cost.
(Total 14 marks)

Edexcel D2 Specimen Q1

1. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{899a26d1-7599-4051-b1cf-596542624997-2_730_1534_285_264} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows a directed, capacitated network where the number on each arc is its capacity. A possible flow is shown from \(S\) to \(T\) and the value in brackets on each arc is the flow in that arc.

Find the values of \(x , y\), and \(z\).
(3)
Find, by inspection, the maximal flow from \(S\) to \(T\) and verify that it is maximal.
(2)

Edexcel D2 Specimen Q2

2. 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 initial tableau was obtained.

Basic variable	\(x\)	\(y\)	\(z\)	\(r\)	\(s\)	Value
\(r\)	2	0	4	1	0	80
\(s\)	1	4	2	0	1	160
\(P\)	- 2	- 8	- 20	0	0	0

Taking the most negative number in the profit row to indicate the pivot column, perform one complete iteration of the simplex algorithm, to obtain tableau \(T\). State the row operations that you use.
Write down the profit equation shown in tableau \(T\).
State whether tableau \(T\) is optimal. Give a reason for your answer.

Edexcel D2 Specimen Q3

3. 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
\(A\)	21	24	16	24
\(B\)	18	23	17	32
\(C\)	15	19	25	14
Required	20	30	20
\cline { 1 - 4 }
\cline { 1 - 4 }

Use the north-west corner rule to find an initial solution.
Obtain improvement indices for each unused route.
Use the stepping-stone method once to obtain a better solution and state its cost.

Edexcel D2 Specimen Q4

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{899a26d1-7599-4051-b1cf-596542624997-5_602_1255_196_406} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} The network in Figure 2 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 .

Find another lower bound for the length of the route by deleting \(F\). State which is the best lower bound of the two.
By inspection, complete the table of least distances. The table can now be taken to represent a complete network.
Using the nearest-neighbour algorithm, starting at \(F\), obtain an upper bound to the length of the route. State your route.

Edexcel D2 Specimen Q5

5. 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.

	\(J\)	\(K\)	\(L\)
\(W\)	3	6	3
\(X\)	5	8	4
\(Y\)	2	5	7

Formulate this transportation problem as a linear programming problem. Make clear your decision variables, objective function and constraints.

Edexcel D2 Specimen Q6

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{899a26d1-7599-4051-b1cf-596542624997-6_705_1424_1034_338} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} A maximin route is to be found through the network shown in Figure 3.
Complete the table in the answer book, and hence find a maximin route.

	A	B	C	D	E	F	G
A	-	31	15	12	24	17	22
B	31	-	20	25	14	25	50
C	15	20	-	16	24	19	21
D	12	25	16	-	21	32	17
E	24	14	24	21	-	28	41
F	17	25	19	32	28	-	25
G	22	50	21	17	41	25	-

	A	B	C	D	E	F	G
A	-	31	15	12	24	17	22
B	31	-	20	25	14	25	50
C	15	20	-	16	24	19	21
D	12	25	16	-	21	32	17
E	24	14	24	21	-	28	41
F	17	25	19	32	28	-	25
G	22	50	21	17	41	25	-

Questions D2 (547 questions)

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Edexcel D2 2015 June Q2

Edexcel D2 2015 June Q3

Edexcel D2 2015 June Q4

Edexcel D2 2015 June Q5

Edexcel D2 2015 June Q6

Edexcel D2 2016 June Q1

Edexcel D2 2016 June Q2

Edexcel D2 2016 June Q3

Edexcel D2 2016 June Q4

Edexcel D2 2016 June Q5

Edexcel D2 2016 June Q6

Edexcel D2 2016 June Q7

Edexcel D2 Q1

Edexcel D2 Q2

Edexcel D2 Q3

Edexcel D2 Q4

Edexcel D2 Q5

Edexcel D2 Q6

Edexcel D2 Q7

Edexcel D2 Specimen Q1

Edexcel D2 Specimen Q2

Edexcel D2 Specimen Q3

Edexcel D2 Specimen Q4

Edexcel D2 Specimen Q5

Edexcel D2 Specimen Q6

	A	B	C	D	E	F	G
A	-	31	15	12	24	17	22
B	31	-	20	25	14	25	50
C	15	20	-	16	24	19	21
D	12	25	16	-	21	32	17
E	24	14	24	21	-	28	41
F	17	25	19	32	28	-	25
G	22	50	21	17	41	25	-