Questions D2 - A-Level Maths

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Edexcel D2 2004 June Q9

7 marks Standard +0.3

\includegraphics{figure_5} 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 \(C_1\), \(C_2\) and \(C_3\) are shown on The diagram above.

Find the capacity of each of the three cuts. [3]
Verify that the flow of 26 is maximal. [1]

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.

By considering both options, explain which one meets the government's aim [3]

Edexcel D2 2004 June Q10

18 marks Moderate -0.5

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 £50, £80 and £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.

Formulate the above situation as a linear programming problem, listing clearly the constraints as inequalities in their simplest form, and stating the objective function. [4]

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.

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	\(-\frac{1}{2}\)	0	10
\(y\)	\(\frac{1}{2}\)	1	\(\frac{1}{2}\)	0	\(\frac{1}{2}\)	0	20
\(t\)	2	0	0	0	\(-1\)	1	10
P	\(-10\)	0	\(-20\)	0	40	0	1600

[4]

You may not need to use all of the tableaux.

Basic variable	\(x\)	\(y\)	\(z\)	\(r\)	\(s\)	\(t\)	Value	Row operations
\(r\)
\(s\)
\(t\)
P

Basic variable	\(x\)	\(y\)	\(z\)	\(r\)	\(s\)	\(t\)	Value	Row operations

Explain the practical meaning of the value 10 in the top row. [2]
1. Perform one further complete iteration of the Simplex algorithm.
  Basic variable \(x\) \(y\) \(z\) \(r\) \(s\) \(t\) Value Row operations
  
  Basic variable \(x\) \(y\) \(z\) \(r\) \(s\) \(t\) Value Row 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]

(Total 18 marks)

Edexcel D2 2006 June Q1

4 marks Easy -1.8

State Bellman's principle of optimality. [1]
Explain what is meant by a minimax route. [1]
Describe a practical problem that would require a minimax route as its solution. [2]

(Total 4 marks)

Edexcel D2 2006 June Q2

Moderate -0.8

Three workers, \(P\), \(Q\) and \(R\), are to be assigned to three tasks, 1, 2 and 3. Each worker is to be assigned to one task and each task must be assigned to one worker. The cost, in hundreds of pounds, of using each worker for each task is given in the table below. The cost is to be minimised.

Cost (in £100s)	Task 1	Task 2	Task 3
Worker \(P\)	8	7	3
Worker \(Q\)	9	5	6
Worker \(R\)	10	4	4

Formulate the above situation as a linear programming problem, defining the decision variables and making the objective and constraints clear. (Total 7 marks)

Edexcel D2 2006 June Q3

11 marks Moderate -0.5

A college wants to offer five full-day activities with a different activity each day from Monday to Friday. The sports hall will be used for these activities. Each evening the caretaker will prepare the hall by putting away the equipment from the previous activity and setting up the hall for the activity next day. On Friday evening he will put away the equipment used that day and set up the hall for the following Monday. The 5 activities offered are Badminton (\(B\)), Cricket nets (\(C\)), Dancing (\(D\)), Football coaching (\(F\)) and Tennis (\(T\)). Each will be on the same day from week to week. The college decides to offer the activities in the order that minimises the total time the caretaker has to spend preparing the hall each week. The hall is initially set up for Badminton on Monday. The table below shows the time, in minutes, it will take the caretaker to put away the equipment from one activity and set up the hall for the next.

	To
\cline{2-6} \multicolumn{1}{c\|}{Time}	\(B\)	\(C\)	\(D\)	\(F\)	\(T\)
\(B\)	--	108	150	64	100
\(C\)	108	--	54	104	60
From \(D\)	150	54	--	150	102
\(F\)	64	104	150	--	68
\(T\)	100	60	102	68	--

Explain why this problem is equivalent to the travelling salesman problem. [2]
Find the total time taken by the caretaker each week using this ordering. A possible ordering of activities is
Monday Tuesday Wednesday Thursday Friday
\(B\) \(C\) \(D\) \(F\) \(T\)
[2]
Starting with Badminton on Monday, use a suitable algorithm to find an ordering that reduces the total time spent each week to less than 7 hours. [3]
By deleting \(B\), use a suitable algorithm to find a lower bound for the time taken each week. Make your method clear. [4]

(Total 11 marks)

Edexcel D2 2006 June Q4

11 marks Standard +0.3

During the school holidays four building tasks, rebuilding a wall (\(W\)), repairing the roof (\(R\)), repainting the hall (\(H\)) and relaying the playground (\(P\)), need to be carried out at a Junior School. Four builders, \(A\), \(B\), \(C\) and \(D\) will be hired for these tasks. Each builder must be assigned to one task. Builder \(B\) is not able to rebuild the wall and therefore cannot be assigned to this task. The cost, in thousands of pounds, of using each builder for each task is given in the table below.

Cost	\(H\)	\(P\)	\(R\)	\(W\)
\(A\)	3	5	11	9
\(B\)	3	7	8	--
\(C\)	2	5	10	7
\(D\)	8	3	7	6

Use the Hungarian algorithm, reducing rows first, to obtain an allocation that minimises the total cost. State the allocation and its total cost. You must make your method clear and show the table after each stage. [9]
State, with a reason, whether this allocation is unique. [2]

(Total 11 marks)

Edexcel D2 2006 June Q5

Moderate -0.5

Victor owns some kiosks selling ice cream, hot dogs and soft drinks. The network below shows the choices of action and the profits, in thousands of pounds, they generate over the next four years. The negative numbers indicate losses due to the purchases of new kiosks. \includegraphics{figure_5} Use a suitable algorithm to determine the sequence of actions so that the profit over the four years is maximised and state this maximum profit. (Total 12 marks)

Edexcel D2 2006 June Q6

14 marks Moderate -0.5

Explain briefly the circumstances under which a degenerate feasible solution may occur to a transportation problem. [2]
Explain why a dummy location may be needed when solving a transportation problem. [1]

The table below shows the cost of transporting one unit of stock from each of three supply points \(A\), \(B\) and \(C\) to each of two demand points 1 and 2. It also shows the stock held at each supply point and the stock required at each demand point.

	1	2	Supply
\(A\)	62	47	15
\(B\)	61	48	12
\(C\)	68	58	17
Demand	16	11

Complete the table below to show a possible initial feasible solution generated by the north-west corner method.
1 2 3
\(A\)
\(B\) 0
\(C\)
[1]
Use the stepping-stone method to obtain an optimal solution and state its cost. You should make your method clear by stating shadow costs, improvement indices, stepping-stone route, and the entering and exiting squares at each stage. [10]

(Total 14 marks)

Edexcel D2 2006 June Q7

16 marks Standard +0.8

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	7	2
\(A\) plays 2	3	8	4
\(A\) plays 3	6	4	9

Formulate the game as a linear programming problem for player \(A\), writing the constraints as equalities and clearly defining your variables. [5]
Explain why it is necessary to use the simplex algorithm to solve this game theory problem. [1]
Write down an initial simplex tableau making your variables clear. [2]
Perform two complete iterations of the simplex algorithm, indicating your pivots and stating the row operations that you use. [8]

(Total 16 marks)

Edexcel D2 2006 June Q8

16 marks Standard +0.3

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

Basic variable	\(x\)	\(y\)	\(z\)	\(r\)	\(s\)	\(t\)	Value
\(r\)	7	10	10	1	0	0	3600
\(s\)	6	9	12	0	1	0	3600
\(t\)	2	3	4	0	0	1	2400
\(P\)	-35	-55	-60	0	0	0	0

Write down the four equations represented in the initial tableau above. [4]
Taking the most negative number in the profit row to indicate the pivot column at each stage, solve this linear programming problem. State the row operations that you use. [9]
State the values of the objective function and each variable. [3]

(Total 16 marks)

Edexcel D2 2006 June Q9

14 marks Moderate -0.3

\includegraphics{figure_9} The figure above shows a capacitated, directed network. The capacity of each arc is shown on each arc. The numbers in circles represent an initial flow from \(S\) to \(T\). Two cuts \(C_1\) and \(C_2\) are shown on the figure.

Write down the capacity of each of the two cuts and the value of the initial flow. [3]
Complete the initialisation of the labelling procedure on the diagram below by entering values along arcs \(AC\), \(CD\), \(DE\) and \(DT\). \includegraphics{figure_9b} [2]
Hence use the labelling procedure to find a maximal flow through the network. You must list each flow-augmenting path you use, together with its flow. [5]
Show your maximal flow pattern on the diagram below. \includegraphics{figure_9d} [2]
Prove that your flow is maximal. [2]

(Total 14 marks)

OCR MEI D2 Q1

16 marks Easy -1.2

The switching circuit in Fig. 1.1 shows switches, s for a car's sidelights, h for its dipped headlights and f for its high-intensity rear foglights. It also shows the three sets of lights. \includegraphics{figure_1} (Note: s and h are each "ganged" switches. A ganged switch consists of two connected switches sharing a single switch control, so that both are either on or off together.)

1. Describe in words the conditions under which the foglights will come on. [2]
Fig. 1.2 shows a combinatorial circuit. \includegraphics{figure_2}
1. Write the output in terms of a Boolean expression involving s, h and f. [2]
2. Use a truth table to prove that \(s \wedge h \wedge f = \sim (\sim s \vee \sim h) \wedge f\). [3]
A car's first gear can be engaged (g) if either both the road speed is low (r) and the clutch is depressed (d), or if both the road speed is low (r) and the engine speed is the correct multiple of the road speed (m).
1. Draw a switching circuit to represent the conditions under which first gear can be engaged. Use two ganged switches to represent r, and single switches to represent each of d, m and g. [2]
2. Draw a combinatorial circuit to represent the Boolean expression \(r \wedge (d \vee m) \wedge g\). [4]
3. Use Boolean algebra to prove that \(r \wedge (d \vee m) \wedge g = ((r \wedge d) \vee (r \wedge m)) \wedge g\). [2]
4. Draw another switching circuit to represent the conditions under which first gear can be selected, but without using a ganged switch. [1]

OCR MEI D2 Q2

16 marks Moderate -0.8

Karl is considering investing in a villa in Greece. It will cost him 56000 euros (€ 56000). His alternative is to invest his money, £35000, in the United Kingdom. He is concerned with what will happen over the next 5 years. He estimates that there is a 60% chance that a house currently worth € 56000 will appreciate to be worth € 75000 in that time, but that there is a 40% chance that it will be worth only € 55000. If he invests in the United Kingdom then there is a 50% chance that there will be 20% growth over the 5 years, and a 50% chance that there will be 10% growth.

Given that £1 is worth € 1.60, draw a decision tree for Karl, and advise him what to do, using the EMV of his investment (in thousands of euros) as his criterion. [4]

In fact the £/€ exchange rate is not fixed. It is estimated that at the end of 5 years, if there has been 20% growth in the UK then there is a 70% chance that the exchange rate will stand at 1.70 euros per pound, and a 30% chance that it will be 1.50. If growth has been 10% then there is a 40% chance that the exchange rate will stand at 1.70 and a 60% chance that it will be 1.50.

Produce a revised decision tree incorporating this information, and give appropriate advice. [3]

A financial analyst asks Karl a number of questions to determine his utility function. He estimates that for x in cash (in thousands of euros) Karl's utility is \(x^{0.5}\), and that for y in property (in thousands of euros), Karl's utility is \(y^{0.75}\).

Repeat your computations from part (ii) using utility instead of the EMV of his investment. Does this change your advice? [3]
Using EMVs, find the exchange rate (number of euros per pound) which will make Karl indifferent between investing in the UK and investing in a villa in Greece. [2]
Show that, using Karl's utility function, the exchange rate would have to drop to 1.277 euros per pound to make Karl indifferent between investing in the UK and investing in a villa in Greece. [4]

OCR MEI D2 Q3

20 marks Standard +0.3

The distance and route matrices shown in Fig. 3.1 are the result of applying Floyd's algorithm to the incomplete network on 4 vertices shown in Fig. 3.2. \includegraphics{figure_3} \includegraphics{figure_4}

Draw the complete network of shortest distances. [2]
Explain how to use the route matrix to find the shortest route from vertex 4 to vertex 1 in the original incomplete network. [2]

A new vertex, vertex 5, is added to the original network. Its distances from vertices to which it is connected are shown in Fig. 3.3. \includegraphics{figure_5}

Draw the extended network and the complete 5-node network of shortest distances. (You are not required to use an algorithm to find the shortest distances.) [3]
Produce the shortest distance matrix and the route matrix for the extended 5-node network. [3]
Apply the nearest neighbour algorithm to your \(5 \times 5\) distance matrix, starting at vertex 1. Give the length of the cycle produced, together with the actual cycle in the original 5-node network. [3]
By deleting vertex 1 and its arcs, and by using Prim's algorithm on the reduced distance matrix, produce a lower bound for the solution to the practical travelling salesperson problem in the original 5-node network. Show clearly your use of the matrix form of Prim's algorithm. [4]
In the original 5-node network find a shortest route starting at vertex 1 and using each of the 6 arcs at least once. Give the length of your route. [3]

OCR MEI D2 Q4

20 marks Standard +0.8

Kassi and Theo are discussing how much oil and how much vinegar to use to dress their salad. They agree to use between 5 and 10ml of oil and between 3 and 6ml of vinegar and that the amount of oil should not exceed twice the amount of vinegar. Theo prefers to have more oil than vinegar. He formulates the following problem to maximise the proportion of oil: Maximise \(\frac{x}{x + y}\) subject to \(0 \leq x \leq 10\), \(0 \leq y \leq 6\), \(x - 2y \leq 0\).

Explain why this problem is not an LP. [1]
Use the simplex method to solve the following LP. Maximise \(x - y\) subject to \(0 \leq x \leq 10\), \(0 \leq y \leq 6\), \(x - 2y \leq 0\). [7]
Kassi prefers to have more vinegar than oil. She formulates the following LP. Maximise \(y - x\) subject to \(5 \leq x \leq 10\), \(3 \leq y \leq 6\), \(x - 2y \leq 0\). Draw separate graphs to show the feasible regions for this problem and for the problem in part (ii). [5]
Explain why the formulation in part (ii) produced a solution for Theo's problem, and why it is more difficult to use the simplex method to solve Kassi's problem in part (iii). [2]
Produce an initial tableau for using the two-stage simplex method to solve Kassi's problem. Explain briefly how to proceed. [5]

Edexcel D2 Q1

6 marks Moderate -0.8

This question should be answered on the sheet provided. The table below shows the distances in miles between five villages. Jane lives in village A and is about to take her daughter's friends home to villages B, C, D and E. She will begin and end her journey at A and wishes to travel the minimum distance possible.

	A	B	C	D	E
A	\(-\)	4	7	8	2
B	4	\(-\)	1	5	6
C	7	1	\(-\)	2	7
D	8	5	2	\(-\)	3
E	2	6	7	3	\(-\)

Obtain a minimum spanning tree for the network and hence find an upper bound for the length of Jane's journey. [4 marks]
Using a shortcut, improve this upper bound to find an upper bound of less than 15 miles. [2 marks]

Edexcel D2 Q2

8 marks Standard +0.3

The payoff matrix for player A in a two-person zero-sum game with value V is shown below.

	B
	I	II	III
\multirow{3}{*}{A}	I	6	\(-4\)	\(-1\)
	II	\(-2\)	5	3
	III	5	1	\(-3\)

Formulate this information as a linear programming problem, the solution to which will give the optimal strategy for player B.

Rewrite the matrix as necessary and state the new value of the game, v, in terms of V. [2 marks]
Define your decision variables. [2 marks]
Write down the objective function in terms of your decision variables. [2 marks]
Write down the constraints. [2 marks]

Edexcel D2 Q3

9 marks Moderate -0.3

This question should be answered on the sheet provided. The table below gives distances, in miles, for a network relating to a travelling salesman problem.

	A	B	C	D	E	F	G
A	\(-\)	83	57	68	103	91	120
B	83	\(-\)	78	63	41	82	52
C	57	78	\(-\)	37	59	63	74
D	68	63	37	\(-\)	60	52	62
E	103	41	59	60	\(-\)	48	51
F	91	82	63	52	48	\(-\)	77
G	120	52	74	62	51	77	\(-\)

Use the nearest neighbour algorithm, starting at A, to find an upper bound for the length of a tour beginning and ending at A and state the tour. [4 marks]
By deleting A, obtain a lower bound for the length of a tour. [4 marks]
Hence, write down an inequality which must by satisfied by d, the minimum distance travelled in miles. [1 mark]

Edexcel D2 Q4

10 marks Challenging +1.8

This question should be answered on the sheet provided. A rally consisting of four stages is being planned. The first stage will begin at A and the last stage will end at L. Various routes are being considered, with the end of one stage being the start of the next. The organisers want the shortest stage to be as long as possible. The table below shows the length, in miles, of each of the possible stages.

			Finishing point
		C	D	E	F	G	H	I
\multirow{3}{*}{Starting point}	A	1	4.5	13	10	8	11	4
	B					5	10.5
	C					9	6
	D					12	7	15
	E
	F							5
	G							8
	H							10
	I
	J
	K

Finishing point
J	K	L

		2
9	2	3
2	9
		5
		6
		10

Use dynamic programming to find the route which satisfies the wish of the organisers. State the length of the shortest stage on this route. [10 marks]

Edexcel D2 Q5

11 marks Standard +0.3

Four athletes are put forward for selection for a mixed stage relay race at a local competition. They may each be selected for a maximum of one stage and only one athlete can be entered for each stage. The average time, in seconds, for each athlete to complete each stage is given below, based on past performances.

	Stage
	1	2	3
Alex	19	69	168
Darren	22	64	157
Leroy	20	72	166
Suraj	23	66	171

Use the Hungarian algorithm to find an optimal allocation which will minimise the team's total time. Your answer should show clearly how you have applied the algorithm. [11 marks]

Edexcel D2 Q6

13 marks Moderate -0.3

The payoff matrix for player X in a two-person zero-sum game is shown below.

	Y
	\(Y_1\)	\(Y_2\)
\multirow{2}{*}{X}	\(X_1\)	\(-2\)	4
	\(X_2\)	6	\(-1\)

Explain why the game does not have a saddle point. [3 marks]
Find the optimal strategy for
1. player X, [8 marks]
2. player Y.
Find the value of the game. [2 marks]

Edexcel D2 Q7

18 marks Standard +0.3

A transportation problem has costs, in pounds, and supply and demand, in appropriate units, as given in the transportation tableau below.

	D	E	F	Supply
A	13	11	14	20
B	10	9	12	15
C	15	6	8	25
Demand	30	5	25

Find the initial solution given by the north-west corner rule and state why it is degenerate. [3 marks]
Use the stepping-stone method to obtain an optimal solution minimising total cost. State the resulting transportation pattern and its total cost. [15 marks]