Questions — Edexcel (10514 questions)

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Edexcel D2 2004 June Q6
18 marks Standard +0.3
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. Table 1
Week123
ShowsA, B, CD, EF, G, H
Table 2
Show\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)
Expected Profit (£)900800100015001300500700600
Table 3
Travel costs (£)\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)
Home7080150809070
\(A\)180150
\(B\)140120
\(C\)200210
\(D\)200160120
\(E\)170100110
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. [3]
  2. Determine the schedule that maximises the total profit. Show your working in a table. [12]
  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 -0.5
\includegraphics{figure_1} Figure 1 shows a capacitated directed network. The number on each arc is its capacity. \includegraphics{figure_2} 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]
  2. Obtain the value of the initial flow through the network, and explain how you know it is not maximal. [2]
  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. Diagram 1 \includegraphics{figure_3} [5]
  4. Show your maximal flow pattern on Diagram 2. Diagram 2 \includegraphics{figure_4} [2]
  5. Prove that your flow is maximal. [2]
(Total 13 marks)
Edexcel D2 2004 June Q8
6 marks Moderate -0.3
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
\(s\)30201\(-\frac{2}{3}\)\(\frac{2}{3}\)
\(r\)40\(\frac{7}{2}\)108\(\frac{9}{2}\)
\(y\)5170037
P30200863
  1. State, giving your reason, whether this tableau represents the optimal solution. [1]
  2. State the values of every variable. [3]
  3. Calculate the profit made on each unit of \(y\). [2]
(Total 6 marks)
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.
  1. Find the capacity of each of the three cuts. [3]
  2. 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.
  1. 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.
  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. [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.
  1. 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\(-1\)110
    P\(-10\)0\(-20\)04001600
    [4]
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
  1. 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\(x\)\(y\)\(z\)\(r\)\(s\)\(t\)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]
(Total 18 marks)
Edexcel D2 2006 June Q1
4 marks Easy -1.8
  1. State Bellman's principle of optimality. [1]
  2. Explain what is meant by a minimax route. [1]
  3. 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 1Task 2Task 3
Worker \(P\)873
Worker \(Q\)956
Worker \(R\)1044
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}\(B\)\(C\)\(D\)\(F\)\(T\)
\(B\)--10815064100
\(C\)108--5410460
From \(D\)15054--150102
\(F\)64104150--68
\(T\)1006010268--
  1. Explain why this problem is equivalent to the travelling salesman problem. [2]
  2. Find the total time taken by the caretaker each week using this ordering. A possible ordering of activities is
    MondayTuesdayWednesdayThursdayFriday
    \(B\)\(C\)\(D\)\(F\)\(T\)
    [2]
  3. 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]
  4. 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\(H\)\(P\)\(R\)\(W\)
\(A\)35119
\(B\)378--
\(C\)25107
\(D\)8376
  1. 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]
  2. 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
  1. Explain briefly the circumstances under which a degenerate feasible solution may occur to a transportation problem. [2]
  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.
12Supply
\(A\)624715
\(B\)614812
\(C\)685817
Demand1611
  1. Complete the table below to show a possible initial feasible solution generated by the north-west corner method.
    123
    \(A\)
    \(B\)0
    \(C\)
    [1]
  2. 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 1572
\(A\) plays 2384
\(A\) plays 3649
  1. Formulate the game as a linear programming problem for player \(A\), writing the constraints as equalities and clearly defining your variables. [5]
  2. Explain why it is necessary to use the simplex algorithm to solve this game theory problem. [1]
  3. Write down an initial simplex tableau making your variables clear. [2]
  4. 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
\(r\)710101003600
\(s\)69120103600
\(t\)2340012400
\(P\)-35-55-600000
  1. Write down the four equations represented in the initial tableau above. [4]
  2. 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]
  3. 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.
  1. Write down the capacity of each of the two cuts and the value of the initial flow. [3]
  2. 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]
  3. 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]
  4. Show your maximal flow pattern on the diagram below. \includegraphics{figure_9d} [2]
  5. Prove that your flow is maximal. [2]
(Total 14 marks)
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.
ABCDE
A\(-\)4782
B4\(-\)156
C71\(-\)27
D852\(-\)3
E2673\(-\)
  1. Obtain a minimum spanning tree for the network and hence find an upper bound for the length of Jane's journey. [4 marks]
  2. 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
IIIIII
\multirow{3}{*}{A}I6\(-4\)\(-1\)
II\(-2\)53
III51\(-3\)
Formulate this information as a linear programming problem, the solution to which will give the optimal strategy for player B.
  1. Rewrite the matrix as necessary and state the new value of the game, v, in terms of V. [2 marks]
  2. Define your decision variables. [2 marks]
  3. Write down the objective function in terms of your decision variables. [2 marks]
  4. 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.
ABCDEFG
A\(-\)83576810391120
B83\(-\)7863418252
C5778\(-\)37596374
D686337\(-\)605262
E103415960\(-\)4851
F9182635248\(-\)77
G1205274625177\(-\)
  1. 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]
  2. By deleting A, obtain a lower bound for the length of a tour. [4 marks]
  3. 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
CDEFGHI
\multirow{3}{*}{Starting point}A14.513108114
B510.5
C96
D12715
E
F5
G8
H10
I
J
K
Finishing point
JKL
2
923
29
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
123
Alex1969168
Darren2264157
Leroy2072166
Suraj2366171
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\(-1\)
  1. Explain why the game does not have a saddle point. [3 marks]
  2. Find the optimal strategy for
    1. player X, [8 marks]
    2. player Y.
  3. 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.
DEFSupply
A13111420
B1091215
C156825
Demand30525
  1. Find the initial solution given by the north-west corner rule and state why it is degenerate. [3 marks]
  2. Use the stepping-stone method to obtain an optimal solution minimising total cost. State the resulting transportation pattern and its total cost. [15 marks]
Edexcel AEA 2002 June Q1
8 marks Challenging +1.8
Solve the following equation, for \(0 \leq x \leq \pi\), giving your answers in terms of \(\pi\). $$\sin 5x - \cos 5x = \cos x - \sin x.$$ [8]
Edexcel AEA 2002 June Q2
9 marks Challenging +1.8
In the binomial expansion of $$(1 - 4x)^p, \quad |x| < \frac{1}{4},$$ the coefficient of \(x^2\) is equal to the coefficient of \(x^4\) and the coefficient of \(x^3\) is positive. Find the value of \(p\). [9]
Edexcel AEA 2002 June Q3
11 marks Challenging +1.8
The curve \(C\) has parametric equations $$x = 15t - t^3, \quad y = 3 - 2t^2.$$ Find the values of \(t\) at the points where the normal to \(C\) at \((14, 1)\) cuts \(C\) again. [11]
Edexcel AEA 2002 June Q4
14 marks Hard +2.3
Find the coordinates of the stationary points of the curve with equation $$x^3 + y^3 - 3xy = 48$$ and determine their nature. [14]