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OCR MEI D1 2014 June Q2
8 marks Easy -1.2
2 Honor either has coffee or tea at breakfast. On one third of days she chooses coffee, otherwise she has tea. She can never remember what she had the day before.
  1. Construct a simulation rule, using one-digit random numbers, to model Honor's choices of breakfast drink.
  2. Using the one-digit random numbers in your answer book, simulate Honor's choice of breakfast drink for 10 days. Honor also has either coffee or tea at the end of her evening meal, but she does remember what she had for breakfast, and her choice depends on it. If she had coffee at breakfast then the probability of her having coffee again is 0.55 . If she had tea for breakfast, then the probability of her having tea again is 0.15 .
  3. Construct a simulation rule, using two-digit random numbers, to model Honor's choice of evening drink given that she had coffee at breakfast. Construct a simulation rule, using two-digit random numbers, to model Honor's choice of evening drink given that she had tea at breakfast.
  4. Using your breakfast simulation from part (ii), and the two-digit random numbers in your answer book, simulate Honor's choice of evening drink for 10 days.
  5. Use your results from parts (ii) and (iv) to estimate the proportion of Honor's drinks, breakfast and evening meal combined, which are coffee. \section*{Question 3 begins on page 4}
OCR MEI D1 2014 June Q3
8 marks Easy -1.3
3 Six remote villages are linked by a set of roads. Two villages are connected directly if there is a road between them which does not pass through another village. The table gives the lengths in miles of all direct connections.
ABCDEF
A67123
B6108
C7102
D12298
E89
F38
  1. Why might it be thought surprising that the direct distance between A and D is as long as 12 miles? Give a possible reason why the distance is longer than might have been expected.
  2. Use the tabular form of Prim's algorithm, starting at A , to find a minimum connector for these villages. Draw your connector and give its total length.
OCR MEI D1 2014 June Q4
16 marks Moderate -0.3
4 The table lists tasks which are involved in adding a back door to a garage. The table also lists the duration and immediate predecessor(s) for each task. Each task is undertaken by one person.
TaskDuration (hours)Immediate predecessor(s)
Ameasure0.5-
Bmanufacture frame and door5A
Ccut hole in wall2A
Dfit lintel and marble step1.5C
Efit frame1B, C
Ffit door1E
Grepair plaster around door1E
  1. Draw an activity on arc network for these activities.
  2. Mark on your diagram the early time and the late time for each event. Give the minimum completion time and the critical activities.
  3. Produce a schedule to show how two people can complete the project in the minimum time. Soon after starting activity D , the marble step breaks. Getting a replacement step adds 4 hours to the duration of activity D.
  4. How does this delay affect the minimum completion time, the critical activities and the minimum time needed for two people to complete the project? \section*{Question 5 begins on page 6}
OCR MEI D1 2014 June Q5
16 marks Easy -1.2
5
  1. The following instructions operate on positive integers greater than 4.
    Step 10 Choose any positive integer greater than 4, and call it \(n\).
    Step 15 Write down \(n\).
    Step 20 If \(n\) is even then let \(n = \frac { n } { 2 }\) and write down the result.
    Step 30 If \(n\) is odd then let \(n = 3 n + 1\) and write down the result.
    Step 40 Go to Step 20.
    1. Apply the instructions with 6 as the chosen integer, stopping when a sequence repeats itself.
    2. Apply the instructions with 256 as the chosen integer, stopping when a sequence repeats itself.
    3. Add an instruction to stop the process when \(n\) becomes 1 .
    4. It is not known if, when modified to stop cycling through \(4,2,1\), the instructions form an algorithm. What would need to be known for it to be an algorithm?
  2. Six items with weights given in the table are to be packed into boxes each of which has a capacity of 10 kg .
    ItemABCDEF
    Weight \(( \mathrm { kg } )\)216335
    The first-fit algorithm is as follows. \includegraphics[max width=\textwidth, alt={}, center]{aac29742-fee8-48a9-896c-e96696742251-7_809_1280_660_356}
    1. Use the first-fit algorithm to pack the items in the order given, and state how many boxes are needed.
    2. Place the items in increasing order of weight, and then apply the first-fit algorithm.
    3. Place the items in decreasing order of weight, and then apply the first-fit algorithm. An optimal solution is one which uses the least number of boxes.
    4. Find a set of weights for which placing in decreasing order of weight, and then applying the firstfit algorithm, does not give an optimal solution. Show both the results of first-fit decreasing and an optimal solution.
    5. First-fit decreasing has quadratic complexity. If it takes a person 30 seconds to apply first-fit decreasing to 6 items, about how long would it take that person to apply it to 60 items?
OCR MEI D1 2014 June Q6
16 marks Moderate -0.3
6 Ian the chef is to make vegetable stew and vegetable soup for distribution to a small chain of vegetarian restaurants. The recipes for both of these require carrots, beans and tomatoes. 10 litres of stew requires 1.5 kg of carrots, 1 kg of beans and 1.5 kg of tomatoes.
10 litres of soup requires 1 kg of carrots, 0.75 kg of beans and 1.5 kg of tomatoes. Ian has available 100 kg of carrots, 70 kg of beans and 110 kg of tomatoes.
  1. Identify appropriate variables and write down three inequalities corresponding to the availabilities of carrots, beans and tomatoes.
  2. Graph your inequalities and identify the region corresponding to feasible production plans. The profit on a litre of stew is \(\pounds 5\), and the profit on a litre of soup is \(\pounds 4\).
  3. Find the most profitable production plan, showing your working. Give the maximum profit. Ian can buy in extra tomatoes at \(\pounds 2.50\) per kg .
  4. What extra quantity of tomatoes should Ian buy? How much extra profit would be generated by the extra expenditure? \section*{END OF QUESTION PAPER} \section*{OCR}
OCR MEI D1 2015 June Q1
8 marks Easy -1.8
1 The directed bipartite graph represents links between chairlifts and ski runs in one part of a ski resort. Chairlifts are represented by capital letters, and ski runs are represented by numbers. For example, chairlift A takes skiers to the tops of ski runs 1 and 2, whereas ski run 2 takes skiers to the bottom of chairlift B . \includegraphics[max width=\textwidth, alt={}, center]{a27c868b-4fc4-4e82-b27f-d367b15b42c2-2_551_333_493_849}
  1. The incomplete map in your answer book represents the three chairlifts and ski run 2 . Complete the map by drawing in the other 4 ski runs. Angus wants to ski all 5 ski runs, starting and finishing at the bottom of chairlift A .
  2. Which chairlifts does Angus have to repeat, and why?
  3. Which ski runs does Angus have to repeat, and why? The chairlifts and ski runs shown above form only part of the resort. In fact, chairlift C also takes skiers to the bottom of chairlift \(D\).
  4. Why can this information not be represented in a bipartite graph?
OCR MEI D1 2015 June Q2
8 marks Easy -1.8
2 The following algorithm operates on the equations of 3 straight lines, each in the form \(y = m _ { i } x + c _ { i }\).
Step 1Set \(i = 1\)
Step 2Input \(m _ { i }\) and \(c _ { i }\)
Step 3If \(i = 3\) then go to Step 6
Step 4Set \(i = i + 1\)
Step 5Go to Step 2
Step 6Set \(j = 1\)
Step 7Set \(a = j + 1\)
Step 8If \(a > 3\) then set \(a = a - 3\)
Step 9Set \(b = j + 2\)
Step 10If \(b > 3\) then set \(b = b - 3\)
Step 11Set \(d _ { j } = m _ { b } - m _ { a }\)
Step 12If \(d _ { j } = 0\) then go to Step 20
Step 13Set \(x _ { j } = \frac { c _ { a } - c _ { b } } { d _ { j } }\)
Step 14Set \(y _ { j } = m _ { a } \times x _ { j } + c _ { a }\)
Step 15Record \(\left( x _ { j } , y _ { j } \right)\) in the print area
Step 16If \(j = 3\) then go to Step 19
Step 17Set \(j = j + 1\)
Step 18Go to Step 7
Step 19Stop
Step 20Record "parallel" in the print area
Step 21Go to Step 16
  1. Run the algorithm for the straight lines \(y = 2 x + 8 , y = 2 x + 5\) and \(y = 4 x + 3\) using the table given in your answer book. The first five steps have been completed, so you should continue from Step 6. [7]
  2. Describe what the algorithm achieves.
OCR MEI D1 2015 June Q3
8 marks Moderate -0.8
3 Mary takes over a small café. She will sell two types of hot drink: tea and coffee.
A coffee filter costs her \(\pounds 0.10\), and makes one cup of coffee. A tea bag costs her \(\pounds 0.05\) and makes one cup of tea. She has a total weekly budget of \(\pounds 50\) to spend on coffee filters and tea bags. She anticipates selling at least 500 cups of hot drink per week. She estimates that between \(50 \%\) and \(75 \%\) of her sales of cups of hot drink will be for cups of coffee. Mary needs help to decide how many coffee filters and how many tea bags to buy per week.
  1. Explain why the number of tea bags which she buys should be no more than the number of coffee filters, and why it should be no less than one third of the number of coffee filters.
  2. Allocate appropriate variables, and draw a graph showing the feasible region for Mary's problem. Mary's partner suggests that she buys 375 coffee filters and 250 tea bags.
  3. How does this suggestion relate to the estimated demand for coffee and tea?
OCR MEI D1 2015 June Q4
16 marks Standard +0.3
4 The table defines a network on 6 nodes, the numbers representing distances between those nodes.
ABCDEF
A32783
B345
C246
D75
E862
F32
  1. Use Dijkstra's algorithm to find the shortest routes from A to each of the other vertices. Give those routes and their lengths.
  2. Jack wants to find a minimum spanning tree for the network.
    1. Apply Prim's algorithm to the network, draw the minimum spanning tree and give its length. Jill suggests the following algorithm is easier.
      Step 1 Remove an arc of longest length which does not disconnect the network
      Step 2 If there is an arc which can be removed without disconnecting the network then go to Step 1
      Step 3 Stop
    2. Show the order in which arcs are removed when Jill's algorithm is applied to the network.
    3. Explain why Jill's algorithm always produces a minimum spanning tree for a connected network.
    4. In a complete network on \(n\) vertices there are \(\frac { n ( n - 1 ) } { 2 }\) arcs. There are \(n - 1\) arcs to include when using Prim's algorithm. How many arcs are there to remove using Jill's algorithm? For what values of \(n\) does Jill have more arcs to remove than Prim has to include?
OCR MEI D1 2015 June Q5
16 marks Moderate -0.3
5 The table lists activities which are involved in framing a picture. The table also lists their durations and their immediate predecessors. Except for activities C and H, each activity is undertaken by one person. Activities C and H require no people.
ActivityDuration (mins)Immediate predecessor(s)
Aselect mounting5-
Bglue picture to mounting5A
Callow mounting glue to dry20B
Dmeasure for frame5A
Eselect type of frame10A
Fcut four frame pieces5D, E
Gpin and glue frame pieces together5F
Hallow frame glue to dry20G
Icut and bevel glass30D
Jfit glass to frame5H, I
Kfit mounted picture to frame5C, J
  1. Draw an activity on arc network for these activities.
  2. Mark on your diagram the early time and the late time for each event. Give the minimum completion time and the critical activities. A picture is to be framed as quickly as possible. Two people are available to do the job.
  3. Produce a schedule to show how two people can complete the picture framing in the minimum time. To reduce the completion time an instant glue is to be used. This will reduce the time for activities C and H to 0 minutes.
  4. Produce a schedule for two people to complete the framing in the new minimum completion time, and give that time.
OCR MEI D1 2015 June Q6
16 marks Moderate -0.8
6 Adrian and Kleo like to go out for meals, sometimes to a French restaurant, and sometimes to a Greek restaurant. If their last meal out was at the French restaurant, then the probability of their next meal out being at the Greek restaurant is 0.7 , whilst the probability of it being at the French restaurant is 0.3 . If their last meal out was at the Greek restaurant, then the probability of their next meal out being at the French restaurant is 0.6 , whilst the probability of it being at the Greek restaurant is 0.4 .
  1. Construct two simulation rules, each using single-digit random numbers, to model their choices of where to eat.
  2. Their last meal out was at the Greek restaurant. Use the random digits printed in your answer book to simulate their choices for the next 10 of their meals out. Hence estimate the proportion of their meals out which are at the French restaurant, and the proportion which are at the Greek restaurant. Adrian and Kleo find a Hungarian restaurant which they like. The probabilities of where they eat next are now given in the following table.
    \backslashbox{last meal out}{next meal out}FrenchGreekHungarian
    French\(\frac { 1 } { 5 }\)\(\frac { 3 } { 5 }\)\(\frac { 1 } { 5 }\)
    Greek\(\frac { 1 } { 2 }\)\(\frac { 3 } { 10 }\)\(\frac { 1 } { 5 }\)
    Hungarian\(\frac { 1 } { 3 }\)\(\frac { 1 } { 3 }\)\(\frac { 1 } { 3 }\)
  3. Construct simulation rules, each using single-digit random numbers, to model this new situation.
  4. Their last meal out was at the Greek restaurant. Use the random digits printed in your answer book to simulate their choices for the next 10 of their meals out. Hence estimate the proportion of their meals out which are at each restaurant. \section*{END OF QUESTION PAPER}
OCR MEI D1 2016 June Q1
8 marks Moderate -0.8
1 Pierre knows that, if he gambles, he will lose money in the long run. Nicolas tries to convince him that this is not the case. Pierre stakes a sum of money in a casino game. If he wins then he gets back his stake plus the same amount again. If he loses then he loses his stake. Nicolas says that Pierre can guarantee to win by repeatedly playing the game, even though the probability of winning an individual game is less than 0.5 . His idea is that Pierre should bet in the first game with a stake of \(\pounds 100\). If he wins then he stops, as he will have won \(\pounds 100\). If he loses then he plays again with a stake of \(\pounds 200\). If he wins then he has lost \(\pounds 100\) and won \(\pounds 200\). This gives a total gain of \(\pounds 100\), and he stops. If he loses then he plays again with a stake of \(\pounds 400\). If he wins this time he has lost \(\pounds 100\) and \(\pounds 200\) and won \(\pounds 400\). This gives a total gain of \(\pounds 100\), and he stops. Nicolas's advice is that Pierre simply has to continue in this way, doubling his stake every time that he loses, until he eventually wins. Nicolas says that this guarantees that Pierre will win \(\pounds 100\). You are to simulate what might happen if Pierre tries this strategy in a casino game in which the probability of him winning an individual game is 0.4 , and in which he has \(\pounds 1000\) available.
  1. Give an efficient rule for using 1-digit random numbers to simulate the outcomes of individual games, given that the probability of Pierre winning an individual game is 0.4 .
  2. Explain why at most three random digits are needed for one simulation of Nicolas's strategy, given that Pierre is starting with \(\pounds 1000\).
  3. Simulate five applications of Nicolas's strategy, using the five sets of three 1-digit random numbers in your answer book.
  4. Summarise the results of your simulations, giving your mean result.
OCR MEI D1 2016 June Q2
8 marks Moderate -0.8
2 A bag contains 26 cards. A different letter of the alphabet is written on each one. A card is chosen at random and its letter is written down. The card is returned to the bag. The bag is shaken and the process is repeated several times. Tania wants to investigate the probability of a letter appearing twice. She wants to know how many cards need to be chosen for this probability to exceed 0.5. Tania uses the following algorithm. Step 1 Set \(n = 1\) Step 2 Set \(p = 1\) Step 3 Set \(n = n + 1\) Step 4 Set \(p = p \times \left( \frac { 27 - n } { 26 } \right)\) Step 5 If \(p < 0.5\) then stop
Step 6 Go to Step 3
  1. Run the algorithm.
  2. Interpret your results. A well-known problem asks how many randomly-chosen people need to be assembled in a room before the probability of at least two of them sharing a birthday exceeds 0.5 (ignoring anyone born on 29 February).
  3. Modify Tania's algorithm to answer the birthday problem. (Do not attempt to run your modified algorithm.)
  4. Why have 29 February birthdays been excluded?
OCR MEI D1 2016 June Q3
8 marks Moderate -0.3
3 The adjacency graph of a cube \includegraphics[max width=\textwidth, alt={}, center]{e88abde1-8769-4a3c-b115-031cea08d9a6-4_106_108_214_735}
is shown. Vertices on the graph represent faces of the cube. Two vertices are connected by an arc if the corresponding faces of the cube share an edge. \includegraphics[max width=\textwidth, alt={}, center]{e88abde1-8769-4a3c-b115-031cea08d9a6-4_401_464_246_1334} The second graph is the complement of the adjacency graph, i.e. the graph that consists of the same vertices together with the arcs that are not in the adjacency graph. \includegraphics[max width=\textwidth, alt={}, center]{e88abde1-8769-4a3c-b115-031cea08d9a6-4_403_464_737_1334} Throughout this question we wish to colour solids so that two faces that share an edge have different colours. The second graph shows that the minimum number of colours required for a cube is three, one colour for the top and base, one for the front and back, and one for the left and right.
  1. Draw the adjacency graph for a square-based pyramid, and draw its complement. Hence find the minimum number of colours needed to colour a square-based pyramid. \includegraphics[max width=\textwidth, alt={}, center]{e88abde1-8769-4a3c-b115-031cea08d9a6-4_161_202_1434_1548}
  2. (A) Draw the adjacency graph for an octahedron, and draw its complement.
    (B) An octahedron can be coloured using just two colours. Explain how this relates to the complement of the adjacency graph. \includegraphics[max width=\textwidth, alt={}, center]{e88abde1-8769-4a3c-b115-031cea08d9a6-4_227_205_1731_1548}
OCR MEI D1 2016 June Q4
16 marks Moderate -0.3
4 Two products are to be made from material that is supplied in a single roll, 20 m long and 1 m wide. The two products require widths of 47 cm and 32 cm respectively. Two ways of cutting lengths of material are shown in the plans below. \includegraphics[max width=\textwidth, alt={}, center]{e88abde1-8769-4a3c-b115-031cea08d9a6-5_408_1538_520_269} \includegraphics[max width=\textwidth, alt={}, center]{e88abde1-8769-4a3c-b115-031cea08d9a6-5_403_1533_952_274}
  1. Given that there should be no unnecessary waste, draw one other cutting plan that might be used for a cut of length \(z\) metres.
  2. Write down an expression for the total area that is wasted in terms of \(x , y\) and \(z\). All of the roll is to be cut, so \(x + y + z = 20\).
    There needs to be a total length of at least 20 metres of the material for the first product, the one requiring width 47 cm .
  3. Write this as a linear constraint on the variables. There needs to be a total length of at least 24 metres of the material for the second product, the one requiring width 32 cm .
  4. Write this as a linear constraint on the variables.
  5. Formulate an LP in terms of \(x\) and \(y\) to minimise the area that is wasted. You will need to use the relationship \(x + y + z = 20\), together with your answers to parts (ii), (iii) and (iv).
  6. Solve your LP graphically, and interpret the solution.
OCR MEI D1 2016 June Q5
16 marks Moderate -0.8
5 A village amateur dramatic society is planning its annual pantomime. Three rooms in the village hall have been booked for one evening per week for 12 weeks. The following activities must take place. Their durations are shown.
ActivityDuration (weeks)
Achoose lead actors1
Bchoose rest of actors1
Cchoose dancers1
Drehearse lead actors8
Erehearse rest of actors6
Frehearse dancers6
Gprepare scenery6
Hinstall scenery1
Iprepare music2
Jmake costumes4
Kdress rehearsals2
Each activity needs a room except for activities G, I and J.
Choosing actors and dancers can be done in the same week. Rehearsals can begin after this. Rehearsing the dancers cannot begin until the music has been prepared. The scenery must be installed after rehearsals, but before dress rehearsals.
Making the costumes cannot start until after the actors and dancers have been chosen. Everything must be ready for the dress rehearsals in the final two weeks of the 12-week preparation period.
  1. Complete the table in your answer book by showing the immediate predecessors for each activity.
  2. Draw an activity on arc network for these activities.
  3. Mark on your network the early time and the late time for each event. Give the critical activities. It is discovered that there is a double booking and that one of the rooms will not be available after week 6.
  4. Using the space provided, produce a schedule showing how the pantomime can be ready in time for its first performance.
OCR MEI D1 2016 June Q6
16 marks Standard +0.3
6 A mountain ridge separates two populated areas. Networks representing roads connecting the villages in each area are shown below. The numbers on the arcs represent distances in kilometres. \includegraphics[max width=\textwidth, alt={}, center]{e88abde1-8769-4a3c-b115-031cea08d9a6-7_524_1429_340_319} There is also a mountain road of length 15 kilometres connecting C to Z .
  1. A national bus company needs a route from A to X .
    1. Use Dijkstra's algorithm on the complete network, including CZ, to find the shortest route from A to X . Give the route and its corresponding distance.
    2. Would it need fewer computations to use Dijkstra's algorithm on the network for villages A to F to find the shortest route from A to C, and then use Dijkstra's algorithm on the network for villages U to Z to find the shortest route from Z to X? Give a brief justification for your answer.
  2. The local council needs to discover which roads it should keep clear of snow during the winter to keep all the villages connected, and the corresponding total length of road.
    1. Use Kruskal's algorithm on the network for villages A to F to find a minimum connector for \(\{ \mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D } , \mathrm { E } , \mathrm { F } \}\). Show your use of the algorithm. Draw your minimum connector.
    2. Use Prim's algorithm on the network for villages U to Z to find a minimum connector for \(\{ \mathrm { U } , \mathrm { V } , \mathrm { W } , \mathrm { X } , \mathrm { Y } , \mathrm { Z } \}\), starting at U . Show your use of the algorithm. Draw your minimum connector.
    3. What is the total length of road that the council must keep clear of snow?
OCR MEI D2 2005 June Q1
16 marks Moderate -0.5
1 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. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ab28be76-9329-41c8-90fe-ff1bdb28f788-2_284_917_404_580} \captionsetup{labelformat=empty} \caption{Fig. 1.1}
\end{figure} (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. Fig. 1.2 shows a combinatorial circuit. \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{ab28be76-9329-41c8-90fe-ff1bdb28f788-2_367_1235_1183_431} \captionsetup{labelformat=empty} \caption{Fig. 1.2}
      \end{figure}
    2. Write the output in terms of a Boolean expression involving \(s , h\) and \(f\).
    3. Use a truth table to prove that \(\mathrm { s } \wedge \mathrm { h } \wedge \mathrm { f } = \sim ( \sim \mathrm { s } \vee \sim \mathrm { h } ) \wedge \mathrm { f }\).
  2. 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 \(\mathrm { d } , \mathrm { m }\) and g .
    2. Draw a combinatorial circuit to represent the Boolean expression \(\mathrm { r } \wedge ( \mathrm { d } \vee \mathrm { m } ) \wedge \mathrm { g }\).
    3. Use Boolean algebra to prove that \(\mathrm { r } \wedge ( \mathrm { d } \vee \mathrm { m } ) \wedge \mathrm { g } = ( ( \mathrm { r } \wedge \mathrm { d } ) \vee ( \mathrm { r } \wedge \mathrm { m } ) ) \wedge \mathrm { g }\).
    4. Draw another switching circuit to represent the conditions under which first gear can be selected, but without using a ganged switch.
OCR MEI D2 2005 June Q2
16 marks Moderate -0.5
2 Karl is considering investing in a villa in Greece. It will cost him 56000 euros ( € 56000 ). His alternative is to invest his money, \(\pounds 35000\), 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.
  1. Given that \(\pounds 1\) 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. In fact the \(\pounds / €\) 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 .
  2. Produce a revised decision tree incorporating this information, and give appropriate advice. 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 \(x ^ { 0.8 }\), and that for \(y\) in property (in thousands of euros), Karl's utility is \(y ^ { 0.75 }\).
  3. Repeat your computations from part (ii) using utility instead of the EMV of his investment. Does this change your advice?
  4. 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.
  5. 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.
OCR MEI D2 2005 June Q3
20 marks Standard +0.8
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. Distance Matrix \begin{center} \begin{tabular}{ | c | c | c | c | c | } \multicolumn{1}{l}{} & \(\mathbf { 1 }\) & \(\mathbf { 2 }\) & \(\mathbf { 3 }\) & \(\mathbf { 4 }\) \hline \(\mathbf { 1 }\) & 4 & 2 & 3 & 9
\hline \(\mathbf { 2 }\) & 2 & 2 & \(\mathbf { 1 }\) & 7
\hline \(\mathbf { 3 }\) & 3 & \(\mathbf { 1 }\) & 2 & 6
\hline
OCR MEI D2 2005 June Q4
20 marks Standard +0.8
\(\mathbf { 4 }\) & 9 & 7 & 6 & 12
\hline \end{tabular} \end{center} Route Matrix
\(\mathbf { 1 }\)\(\mathbf { 2 }\)\(\mathbf { 3 }\)\(\mathbf { 4 }\)
\(\mathbf { 1 }\)2222
\(\mathbf { 2 }\)\(\mathbf { 1 }\)333
\(\mathbf { 3 }\)2224
\(\mathbf { 4 }\)3333
Fig. 3.1 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ab28be76-9329-41c8-90fe-ff1bdb28f788-4_296_310_918_904} \captionsetup{labelformat=empty} \caption{Fig. 3.2}
\end{figure}
  1. 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. 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. \begin{table}[h]
    \cline { 2 - 5 } \multicolumn{1}{c|}{}1234
    5-3-1
    \captionsetup{labelformat=empty} \caption{Fig. 3.3}
    \end{table}
  3. 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.)
  4. Produce the shortest distance matrix and the route matrix for the extended 5-node network.
  5. 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.
  6. 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.
  7. In the original 5-node network find a shortest route starting at vertex \(\mathbf { 1 }\) and using each of the 6 arcs at least once. Give the length of your route. 4 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 10 ml of oil and between 3 and 6 ml 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: $$\begin{array} { l c } \text { Maximise } & \frac { x } { x + y } \\ \text { subject to } & 0 \leqslant x \leqslant 10 , \\ & 0 \leqslant y \leqslant 6 , \\ & x - 2 y \leqslant 0 . \end{array}$$
  8. Explain why this problem is not an LP.
  9. Use the simplex method to solve the following LP. $$\begin{array} { l c } \text { Maximise } & x - y \\ \text { subject to } & 0 \leqslant x \leqslant 10 \\ & 0 \leqslant y \leqslant 6 \\ & x - 2 y \leqslant 0 \end{array}$$
  10. Kassi prefers to have more vinegar than oil. She formulates the following LP. $$\begin{array} { l l } \text { Maximise } & y - x \\ \text { subject to } & 5 \leqslant x \leqslant 10 , \\ & 3 \leqslant y \leqslant 6 , \\ & x - 2 y \leqslant 0 . \end{array}$$ Draw separate graphs to show the feasible regions for this problem and for the problem in part (ii).
  11. 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).
  12. Produce an initial tableau for using the two-stage simplex method to solve Kassi's problem. Explain briefly how to proceed.
OCR MEI D2 2006 June Q1
16 marks Moderate -0.5
1
  1. Use a truth table to prove \(\sim ( \sim \mathrm { T } \Rightarrow \sim \mathrm { S } ) \Leftrightarrow ( \sim \mathrm { T } \wedge \mathrm { S } )\).
  2. Prove that \(( \mathrm { A } \Rightarrow \mathrm { B } ) \Leftrightarrow ( \sim \mathrm { A } \vee \mathrm { B } )\) and hence use Boolean algebra to prove that $$\sim ( \sim \mathrm { T } \Rightarrow \sim \mathrm {~S} ) \Leftrightarrow ( \sim \mathrm { T } \wedge \mathrm {~S} ) .$$
  3. A teacher wrote on a report "It is not the case that if Joanna doesn't try then she won't succeed." He meant to say that if Joanna were to try then she would have a chance of success. By letting T be "Joanna will try" and S be "Joanna will succeed", find the real meaning of what the teacher wrote.
OCR MEI D2 2006 June Q3
20 marks Moderate -0.8
3 Emma has won a holiday worth \(\pounds 1000\). She is wondering whether or not to take out an insurance policy which will pay out \(\pounds 1000\) if she should fall ill and be unable to go on the holiday. The insurance company tells her that this happens to 1 in 200 people. The insurance policy costs \(\pounds 10\). Thus Emma's monetary value if she buys the insurance and does not fall ill is \(\pounds 990\).
  1. Draw a decision tree for Emma's problem. Use the EMV criterion in your calculations.
  2. Interpret your tree and say what the maximum cost of the insurance would have to be for Emma to consider buying it if she uses the EMV criterion. Suppose that Emma's utility function is given by utility \(= \sqrt [ 3 ] { \text { monetary value } }\).
  3. Using expected utility as the criterion, should Emma purchase the insurance? Under this criterion what is the cost at which she will be indifferent to buying or not buying it? Emma could pay for a blood pressure check to help her to make her decision. Statistics show that \(75 \%\) of checks are positive, and that when a check is positive the chance of missing a holiday through ill heath is 0.001 . However, when a check is negative the chance of cancellation through ill health is 0.017.
  4. Draw a decision tree to help Emma decide whether or not to pay for the check. Use EMV, not expected utility, in your calculations and assume that the insurance policy costs \(\pounds 10\). What is the maximum amount that she should pay for the blood pressure check?
OCR MEI D2 2006 June Q4
20 marks Standard +0.3
4 The "Cuddly Friends Company" produces soft toys. For one day's production run it has available \(11 \mathrm {~m} ^ { 2 }\) of furry material, \(24 \mathrm {~m} ^ { 2 }\) of woolly material and 30 glass eyes. It has three soft toys which it can produce: The "Cuddly Aardvark", each of which requires \(0.5 \mathrm {~m} ^ { 2 }\) of furry material, \(2 \mathrm {~m} ^ { 2 }\) of woolly material and two eyes. Each sells at a profit of \(\pounds 3\). The "Cuddly Bear", each of which requires \(1 \mathrm {~m} ^ { 2 }\) of furry material, \(1.5 \mathrm {~m} ^ { 2 }\) of woolly material and two eyes. Each sells at a profit of \(\pounds 5\). The "Cuddly Cat", each of which requires \(1 \mathrm {~m} ^ { 2 }\) of furry material, \(1 \mathrm {~m} ^ { 2 }\) of woolly material and two eyes. Each sells at a profit of \(\pounds 2\). An analyst formulates the following LP to find the production plan which maximises profit. $$\begin{array} { l l } \text { Maximise } & 3 a + 5 b + 2 c \\ \text { subject to } & 0.5 a + b + c \leqslant 11 , \\ & 2 a + 1.5 b + c \leqslant 24 , \\ & 2 a + 2 b + 2 c \leqslant 30 . \end{array}$$
  1. Explain how this formulation models the problem, and say why the analyst has not simplified the last inequality to \(a + b + c \leqslant 15\).
  2. The final constraint is different from the others in that the resource is integer valued. Explain why that does not impose an additional difficulty for this problem.
  3. Solve this problem using the simplex algorithm. Interpret your solution and say what resources are left over. On a particular day an order is received for two Cuddly Cats, and the extra constraint \(c \geqslant 2\) is added to the formulation.
  4. Set up an initial simplex tableau to deal with the modified problem using either the big-M approach or two-phase simplex. Do not perform any iterations on your tableau.
  5. Show that the solution given by \(a = 8 , b = 2\) and \(c = 5\) uses all of the resources, but that \(a = 6 , b = 6\) and \(c = 2\) gives more profit. What resources are left over from the latter solution?
OCR MEI D2 2007 June Q1
16 marks Moderate -0.5
1
  1. A joke has it that army recruits used to be instructed: "If it moves, salute it. If it doesn't move, paint it." Assume that this instruction has been carried out completely in the local universe, so that everything that doesn't move has been painted.
    1. A recruit encounters something which is not painted. What should he do, and why?
    2. A recruit encounters something which is painted. Do we know what he or she should do? Justify your answer.
  2. Use a truth table to prove \(( ( ( m \Rightarrow s ) \wedge ( \sim m \Rightarrow p ) ) \wedge \sim p ) \Rightarrow s\).
  3. You are given the following two rules. $$\begin{aligned} & 1 \quad ( a \Rightarrow b ) \Leftrightarrow ( \sim b \Rightarrow \sim a ) \\ & 2 \quad ( x \wedge ( x \Rightarrow y ) ) \Rightarrow y \end{aligned}$$ Use Boolean algebra to prove that \(( ( ( m \Rightarrow s ) \wedge ( \sim m \Rightarrow p ) ) \wedge \sim p ) \Rightarrow s\).