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OCR MEI D1 2013 January Q1
8 marks Moderate -0.3
1 The weights on the arcs in the network represent times in minutes to travel between vertices. \includegraphics[max width=\textwidth, alt={}, center]{d1e0f047-6484-435b-a685-2cbbf81a6b02-2_606_782_402_628}
  1. Use Dijkstra's algorithm to find the fastest route from A to F . Give the route and the time.
  2. Use an algorithm to find the minimum connector for the network, showing your working. Find the minimum time to travel from A to F using only arcs in the minimum connector.
OCR MEI D1 2013 January Q2
8 marks Moderate -0.5
2 A small party is held in a country house. There are 10 men and 10 women, and there are 10 dances. For each dance a number of pairings, each of one man and one woman, are formed. The same pairing can appear in more than one dance. A graph is to be drawn showing who danced with whom during the evening, ignoring repetitions.
  1. Name the type of graph which is appropriate.
  2. What is the maximum possible number of arcs in the graph? Dashing Mr Darcy dances with every woman except Elizabeth, who will have nothing to do with him. She dances with eight different men. Prince Charming only dances with Cinderella. Cinderella only dances with Prince Charming and with Mr Darcy. The three ugly sisters only have one dance each.
  3. Add arcs to the graph in your answer book to show this information.
  4. What is the maximum possible number of arcs in the graph?
OCR MEI D1 2013 January Q3
8 marks Moderate -0.3
3 The following algorithm computes an estimate of the square root of a number which is between 0 and 2.
Step 1 Subtract 1 from the number and call the result \(x\) Step 2 Set oldr = 1
Step 3 Set \(i = 1\) Step 4 Set \(j = 0.5\) Step 5 Set \(k = 0.5\) Step 6 Set change \(= x ^ { i } \times k\) Step 7 Set newr \(=\) oldr + change
Step 8 If \(- 0.005 <\) change < 0.005 then go to Step 17
Step 9 Set oldr = newr
Step 10 Set \(i = i + 1\) Step 11 Set \(j = j - 1\) Step 12 Set \(k = k \times j \div i\) Step 13 Set change \(= x ^ { i } \times k\) Step 14 Set newr \(=\) oldr + change
Step 15 If \(- 0.005 <\) change < 0.005 then go to Step 17
Step 16 Go to Step 9
Step 17 Print out newr
  1. Use the algorithm to find an estimate of the square root of 1.44 , showing all of the steps.
  2. Consider what happens if the algorithm is applied to 0.56 , and then use your four values of change from part (i) to calculate an estimate of the square root of 0.56 .
OCR MEI D1 2013 January Q4
16 marks Moderate -0.3
4 A room has two windows which have the same height but different widths. Each window is to have one curtain. The table lists the tasks involved in making the two curtains, their durations, and their immediate predecessors. The durations assume that only one person is working on the activity.
TaskDuration (minutes)Immediate predecessor(s)
Ameasure windows5-
Bcalculate material required5A
Cchoose material15-
Dbuy material15B, C
Ecut material5D
Fstitch sides of wide curtain30E
Gstitch top of wide curtain30F
Hstitch sides of narrow curtain30E
Istitch top of narrow curtain15H
Jhang curtains and pin hems20G, I
Khem wide curtain30J
Lhem narrow curtain15J
Mfit curtains10K, L
  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. Kate and Pete have two rooms to curtain, each identical to that above. Tasks A, B, C and D only need to be completed once each. All other tasks will have two versions, one for room 1 and one for room 2, eg E1 and E2. Kate and Pete share the tasks between them so that each task is completed by only one person.
  3. Complete the diagram to show how the tasks can be shared between them, and scheduled, so that the project can be completed in the least possible time. Give that least possible time.
  4. How much extra help would be needed to curtain both rooms in the minimum completion time from part (ii)? Explain your answer.
OCR MEI D1 2013 January Q5
16 marks Easy -1.2
5 A chairlift for a ski slope has 160 4-person chairs. At any one time half of the chairs are going up and half are coming down empty. An observer watches the loading of the chairs during a moderately busy period, and concludes that the number of occupants per 'up' chair has the following probability distribution.
number of occupants01234
probability0.10.20.30.20.2
  1. Give a rule for using 1-digit random numbers to simulate the number of occupants of an up chair in a moderately busy period.
  2. Use the 10 random digits provided to simulate the number of occupants in 10 up chairs. The observer estimates that, at all times, on average \(20 \%\) of chairlift users are children.
  3. Give an efficient rule for using 1-digit random numbers to simulate whether an occupant of an up chair is a child or an adult.
  4. Use the random digits provided to simulate how many of the occupants of the 10 up chairs are children, and how many are adults. There are more random digits than you will need.
  5. Use your results from part (iv) to estimate how many children and how many adults are on the chairlift (ie on the 80 up chairs) at any instant during a moderately busy period. In a very busy period the number of occupants of an up chair has the following probability distribution.
    number of occupants01234
    probability\(\frac { 1 } { 13 }\)\(\frac { 1 } { 13 }\)\(\frac { 3 } { 13 }\)\(\frac { 3 } { 13 }\)\(\frac { 5 } { 13 }\)
  6. Give an efficient rule for using 2-digit random numbers to simulate the number of occupants of an up chair in a very busy period.
  7. Use the 2-digit random numbers provided to simulate the number of occupants in 5 up chairs. There are more random numbers provided than you will need.
  8. Simulate how many of the occupants of the 5 up chairs are children and how many are adults, and thus estimate how many children and how many adults are on the chairlift at any instant during a very busy period.
  9. Discuss the relative merits of simulating using a sample of 10 chairs as against simulating using a sample of 5 chairs.
OCR MEI D1 2013 January Q6
16 marks Moderate -0.3
6 Jean knits items for charity. Each month the charity provides her with 75 balls of wool.
She knits hats and scarves. Hats require 1.5 balls of wool each and scarves require 3 balls each. Jean has 100 hours available each month for knitting. Hats require 4 hours each to make, and scarves require 2.5 hours each. The charity sells the hats for \(\pounds 7\) each and the scarves for \(\pounds 10\) each, and wants to gain as much income as possible. Jean prefers to knit hats but the charity wants no more than 20 per month. She refuses to knit more than 20 scarves each month.
  1. Define appropriate variables, construct inequality constraints, and draw a graph representing the feasible region for this decision problem.
  2. Give the objective function and find the integer solution which will give Jean's maximum monthly income.
  3. If the charity drops the price of hats in a sale to \(\pounds 4\) each, what would be an optimal number of hats and scarves for Jean to knit? Assuming that all hats and scarves are sold, by how much would the monthly income drop?
OCR MEI D1 2005 June Q3
8 marks Moderate -0.8
3 Table 3 gives the durations and immediate predecessors for the five activities of a project. \begin{table}[h]
ActivityDuration (hours)Immediate predecessor(s)
A3-
B2-
C5-
D2A
E1A, B
\captionsetup{labelformat=empty} \caption{Table 3}
\end{table}
  1. Draw an activity-on-arc network to represent the precedences.
  2. Find the early and late event times for the vertices of your network, and list the critical activities.
  3. Give the total and independent float for each activity which is not critical.
OCR MEI D1 2005 June Q5
16 marks Moderate -0.8
5 A computer store has a stock of 10 laptops to lend to customers while their machines are being repaired. On any particular day the number of laptop loans requested follows the distribution given in Table 5.1. \begin{table}[h]
Number requested01234
Probability0.200.300.200.150.15
\captionsetup{labelformat=empty} \caption{Table 5.1}
\end{table}
  1. Give an efficient rule for using two-digit random numbers to simulate the daily number of requests for laptop loans.
  2. Use two-digit random numbers from the list below to simulate the number of loans requested on each of ten successive days. Random numbers: \(23,02,57,80,31,72,92,78,04,07\) The number of laptops returned from loan each day is modelled by the distribution given in Table 5.2, independently of the number on loan (which is always at least 5 ). \begin{table}[h]
    Number returned0123
    Probability\(\frac { 1 } { 6 }\)\(\frac { 1 } { 4 }\)\(\frac { 1 } { 4 }\)\(\frac { 1 } { 3 }\)
    \captionsetup{labelformat=empty} \caption{Table 5.2}
    \end{table}
  3. Give an efficient rule for using two-digit random numbers to simulate the daily number of laptop returns.
  4. Use two-digit random numbers from the list below to simulate the number of returns on each of ten successive days. Random numbers: \(32,98,01,32,14,21,32,71,82,54,47\) At the end of day 0 there are 7 laptops out on loan and 3 in stock. Each day returns are made in the morning and loans go out in the afternoon. If there is no laptop available the customer is disappointed and never gets a loaned laptop.
  5. Use your simulated numbers of requests and returns to simulate what happens over the next 10 days. For each day record the day number, the number of laptops in stock at the end of the day, and the number of customers that have to be disappointed.
    [0pt] [3] To try to avoid disappointing customers, if the number of laptops in stock at the end of a day is 2 or fewer, the store sends out e-mails to customers with loaned laptops asking for early return if possible. This changes the return distribution for the next day to that given in Table 5.3. \begin{table}[h]
    Number returned01234
    Probability0.10.10.40.20.2
    \captionsetup{labelformat=empty} \caption{Table 5.3}
    \end{table}
  6. Simulate the 10 days again, but using this new policy. Use the requests you produced in part (ii). Use the random numbers given in part (iv) to simulate returns, but use either the distribution given in Table 5.2 or that given in Table 5.3, depending on the number of laptops in stock at the end of the previous day. Is the new policy better?
OCR MEI D1 2005 June Q6
16 marks Moderate -0.8
6 A company manufactures two types of potting compost, Flowerbase and Growmuch. The weekly amounts produced of each are constrained by the supplies of fibre and of nutrient mix. Each litre of Flowerbase requires 0.75 litres of fibre and 1 kg of nutrient mix. Each litre of Growmuch requires 0.5 litres of fibre and 2 kg of nutrient mix. There are 12000 litres of fibre supplied each week, and 25000 kg of nutrient mix. The profit on Flowerbase is 9 p per litre. The profit on Growmuch is 20 p per litre.
  1. Formulate an LP to maximise the weekly profit subject to the constraints on fibre and nutrient mix.
  2. Solve your LP using a graphical approach.
  3. Consider each of the following separate circumstances.
    (A) There is a reduction in the weekly supply of fibre from 12000 litres to 10000 litres. What effect does this have on profit?
    (B) The price of fibre is increased. Will this affect the optimal production plan? Justify your answer.
    [0pt] (C) The supply of nutrient mix is increased to 30000 kg per week. What is the new profit? [1]
OCR MEI D1 2006 June Q2
8 marks Moderate -0.8
2 Fig. 2.1 represents the two floors of a house. There are 5 rooms shown, plus a hall and a landing, which are to be regarded as separate rooms. Each " × " represents an internal doorway connecting two rooms. The " ⊗ " represents the staircase, connecting the hall and the landing. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c429bfed-9241-409a-9cd5-9553bf16c9df-3_401_1287_447_388} \captionsetup{labelformat=empty} \caption{Fig. 2.1}
\end{figure}
  1. Draw a graph representing this information, with vertices representing rooms, and arcs representing internal connections (doorways and the stairs). What is the name of the type of graph of which this is an example?
  2. A larger house has 12 rooms on two floors, plus a hall and a landing. Each ground floor room has a single door, which leads to the hall. Each first floor room has a single door, which leads to the landing. There is a single staircase connecting the hall and the landing. How many arcs are there in the graph of this house?
  3. Another house has 12 rooms on three floors, plus a hall, a first floor landing and a second floor landing. Again, each room has a single door on to the hall or a landing. There is one staircase from the hall to the first floor landing, and another staircase joining the two landings. How many arcs are there in the graph of this house?
  4. Fig. 2.2 shows the graph of another two-floor house. It has 8 rooms plus a hall and a landing. There is a single staircase. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{c429bfed-9241-409a-9cd5-9553bf16c9df-3_208_666_1896_694} \captionsetup{labelformat=empty} \caption{Fig. 2.2}
    \end{figure} Draw a possible floor plan, showing internal connections.
OCR MEI D1 2006 June Q3
8 marks Moderate -0.8
3 An incomplete algorithm is specified in Fig. 3. \(\mathrm { f } ( \mathrm { x } ) = \mathrm { x } ^ { 2 } - 2\) Initial values: \(\mathrm { L } = 0 , \mathrm { R } = 2\).
Step 1 Compute \(\mathrm { M } = \frac { \mathrm { L } + \mathrm { R } } { 2 }\).
Step 2 Compute \(\mathrm { f } ( \mathrm { M } )\).
Step 3 If \(\mathrm { f } ( \mathrm { M } ) < 0\) change the value of L to that of M .
Otherwise change the value of \(R\) to that of \(M\).
Step 4 Go to Step 1. \section*{Fig. 3}
  1. Apply two iterations of the algorithm.
  2. After 10 iterations \(\mathrm { L } = 1.414063 , \mathrm { R } = 1.416016 , \mathrm { M } = 1.416016\) and \(\mathrm { f } ( \mathrm { M } ) = 0.005100\). Say what the algorithm achieves.
  3. Say what is needed to complete the algorithm.
OCR MEI D1 2006 June Q4
16 marks Moderate -0.8
4 Table 4.1 shows some of the activities involved in preparing for a meeting. \begin{table}[h]
ActivityDuration (hours)Immediate predecessors
AAgree date1-
BConstruct agenda0.5-
CBook venue0.25A
DOrder refreshments0.25C
EInform participants0.5B, C
\captionsetup{labelformat=empty} \caption{Table 4.1}
\end{table}
  1. Draw an activity-on-arc network to represent the precedences.
  2. Find the early event time and the late event time for each vertex of your network, and list the critical activities.
  3. Assuming that each activity requires one person and that each activity starts at its earliest start time, draw a resource histogram.
  4. In fact although activity A has duration 1 hour, it actually involves only 0.5 hours work, since 0.5 hours involves waiting for replies. Given this information, and the fact that there is only one person available to do the work, what is the shortest time needed to prepare for the meeting? Fig. 4.2 shows an activity network for the tasks which have to be completed after the meeting. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{c429bfed-9241-409a-9cd5-9553bf16c9df-5_533_844_1688_294} \captionsetup{labelformat=empty} \caption{Fig. 4.2}
    \end{figure} P: Clean room
    Q: Prepare draft minutes
    R: Allocate action tasks
    S: Circulate draft minutes
    T: Approve task allocations
    U: Obtain budgets for tasks
    V: Post minutes
    W: Pay refreshments bill
  5. Draw a precedence table for these activities.
OCR MEI D1 2006 June Q5
16 marks Moderate -0.3
5 John is reviewing his lifestyle, and in particular his leisure commitments. He enjoys badminton and squash, but is not sure whether he should persist with one or both. Both cost money and both take time. Playing badminton costs \(\pounds 3\) per hour and playing squash costs \(\pounds 4\) per hour. John has \(\pounds 11\) per week to spend on these activities. John takes 0.5 hours to recover from every hour of badminton and 0.75 hours to recover from every hour of squash. He has 5 hours in total available per week to play and recover.
  1. Define appropriate variables and formulate two inequalities to model John's constraints.
  2. Draw a graph to represent your inequalities. Give the coordinates of the vertices of your feasible region.
  3. John is not sure how to define an objective function for his problem, but he says that he likes squash "twice as much" as badminton. Letting every hour of badminton be worth one "satisfaction point" define an objective function for John's problem, taking into account his "twice as much" statement.
  4. Solve the resulting LP problem.
  5. Given that badminton and squash courts are charged by the hour, explain why the solution to the LP is not a feasible solution to John's practical problem. Give the best feasible solution.
  6. If instead John had said that he liked badminton more than squash, what would have been his best feasible solution?
OCR MEI D1 2007 June Q1
8 marks Easy -1.8
1 Bus routes connect towns A and B and towns A and C .
Train lines connect towns B and D, towns C and D, and towns A and C.
John represents this information in a graph with four nodes, one for each town, in which an arc is drawn for each connection, giving five arcs in all.
  1. Draw John's graph.
  2. Is John's graph simple? Justify your answer. Jamil represents the information in a graph with five nodes. He uses one node for each of towns A, B and D. Because in town C the bus station and train station are some distance apart, he uses a node labelled C (bus) and a node labelled C (train). Again there are 5 arcs, each representing a connection.
  3. Draw Jamil's graph.
  4. Is Jamil's graph a tree? Justify your answer.
OCR MEI D1 2007 June Q2
8 marks Easy -1.2
2 Two hikers each have a 25 litre rucksack to pack. The items to be packed have volumes of 14, 6, 11, 9 and 6 litres.
  1. Apply the first fit algorithm to the items in the order given and comment on the outcome.
  2. Write the five items in descending order of volume. Apply the first fit decreasing algorithm to find a packing for the rucksacks.
  3. The hikers argue that the first fit decreasing algorithm does not produce a fair allocation of volumes to rucksacks. Produce a packing which gives a fairer allocation of volumes between the two rucksacks. Explain why the hikers might not want to use this packing.
OCR MEI D1 2007 June Q3
8 marks Moderate -0.8
3 Use a graphical approach to solve the following LP. $$\begin{aligned} & \text { Maximise } \quad 2 x + 3 y \\ & \text { subject to } \quad x + 5 y \leqslant 14 \\ & \quad x + 2 y \leqslant 8 \\ & \quad 3 x + y \leqslant 21 \\ & \quad x \geqslant 0 \\ & y \geqslant 0 \end{aligned}$$ Section B (48 marks)
OCR MEI D1 2007 June Q4
16 marks Moderate -0.3
4 Colin is setting off for a day's sailing. The table and the activity network show the major activities that are involved, their durations and their precedences.
ARig foresail
BLower sprayhood
CStart engine
DPump out bilges
ERig mainsail
FCast off mooring ropes
GMotor out of harbour
HRaise foresail
IRaise mainsail
JStop engine and start sailing
\includegraphics[max width=\textwidth, alt={}, center]{21ab732d-435e-4f0b-bc88-21ddc2a398c9-3_480_912_555_925}
  1. Complete the table in your answer book showing the immediate predecessors for each activity.
  2. Find the early time and the late time for each event. Give the project duration and list the critical activities. When he sails on his own Colin can only do one thing at a time with the exception of activity G, motoring out of the harbour.
  3. Use the activity network to determine which activities Colin can perform whilst motoring out of the harbour.
  4. Find the minimum time to complete the activities when Colin sails on his own, and give a schedule for him to achieve this.
  5. Find the minimum time to complete the activities when Colin sails with one other crew member, and give a schedule for them to achieve this.
OCR MEI D1 2007 June Q5
16 marks Challenging +1.2
5 The table shows the weights on the arcs of a network.
ABCDEFG
A-11--1035
B11-85--14
C-8-2-7-
D-52-6--
E10--6-6-
F3-7-6--
G514-----
  1. Draw the network.
  2. Apply Dijkstra's algorithm to find the least weight route from G to D. (Do this on the network you drew for part (i).) Give your route and its total weight.
  3. Find by inspection the route from \(G\) to \(D\) such that the minimum of the weights for arcs on the route is as large as possible. Give your route and its minimum arc weight. Give an application in which this might be needed.
  4. Consider how Dijkstra's algorithm could be modified to solve the problem in part (iii). Explain how to update working values. Explain how to select the next vertex to be permanently labelled.
OCR MEI D1 2007 June Q6
16 marks Moderate -0.3
6 In winter in Metland the weather each day can be classified as dry, wet or snowy. The table shows the probabilities for the next day's weather given the current day's weather.
\cline { 3 - 5 } \multicolumn{2}{c|}{}next day's weather
\cline { 3 - 5 } \multicolumn{2}{c|}{}drywetsnowy
\multirow{3}{*}{
current
day's
weather
}		dry\(\frac { 4 } { 10 }\)\(\frac { 3 } { 10 }\)\(\frac { 3 } { 10 }\)
\cline { 2 - 5 }wet\(\frac { 2 } { 10 }\)\(\frac { 5 } { 10 }\)\(\frac { 3 } { 10 }\)
\cline { 2 - 5 }snowy\(\frac { 2 } { 7 }\)\(\frac { 2 } { 7 }\)\(\frac { 3 } { 7 }\)
You are to use two-digit random numbers to simulate the winter weather in Metland.
  1. Give an efficient rule for using two-digit random numbers to simulate tomorrow's weather if today is
    (A) dry,
    (B) wet,
    (C) snowy.
  2. Today is a dry winter's day in Metland. Use the following two-digit random numbers to simulate the next 7 days' weather in Metland. $$\begin{array} { l l l l l l l l l l } 23 & 85 & 98 & 99 & 56 & 47 & 82 & 14 & 03 & 12 \end{array}$$
  3. Use your simulation from part (ii) to estimate the proportion of dry days in a Metland winter.
  4. Explain how you could use simulation to produce an improved estimate of the proportion of dry days in a Metland winter.
  5. Give two criticisms of this model of weather.
OCR MEI D1 2008 June Q1
8 marks Easy -1.2
1 Consider the following LP.
Maximise \(x + y\) subject to \(2 x + y < 44\) \(2 x + 3 y < 60\) \(10 x + 11 y < 244\) Part of a graphical solution is produced below and in your answer book.
Complete this graphical solution in your answer book. \includegraphics[max width=\textwidth, alt={}, center]{8eba759f-38bc-4d14-ac65-9a0ee6c79741-2_1316_1346_916_356}
OCR MEI D1 2008 June Q2
8 marks Easy -1.2
2 The following algorithm acts on a list of three or more numbers.
Step 1: Set both X and Y equal to the first number on the list.
Step 2: If there is no next number then go to Step 5.
Step 3: If the next number on the list is bigger than X then set X equal to it. If it is less than Y then set Y equal to it. Step 4: Go to Step 2.
Step 5: Delete a number equal to X from the list and delete a number equal to Y from the list.
Step 6: If there is one number left then record it as the answer and stop.
Step 7: If there are two numbers left then record their mean as the answer and stop.
Step 8: Go to Step 1.
  1. Apply the algorithm to the list \(5,14,153,6,24,2,14,15\), counting the number of comparisons which you have to make.
  2. Apply the algorithm to the list \(5,14,153,6,24,2,14\), counting the number of comparisons which you have to make.
  3. Say what the algorithm is finding.
  4. The order of the algorithm is quadratic. Explain what this means when it is applied to long lists.
OCR MEI D1 2008 June Q3
8 marks Moderate -0.8
3 The graph represents four towns together with (two-way) roads connecting them. \includegraphics[max width=\textwidth, alt={}, center]{8eba759f-38bc-4d14-ac65-9a0ee6c79741-4_191_286_319_886} A path is a set of connected arcs linking one vertex to another. A path contains no repeated vertex. \(\mathrm { T } _ { 1 } \rightarrow \mathrm {~T} _ { 2 }\) and \(\mathrm { T } _ { 1 } \rightarrow \mathrm {~T} _ { 3 } \rightarrow \mathrm {~T} _ { 2 }\) are paths.
  1. There are six paths from \(\mathrm { T } _ { 1 }\). List them.
  2. List the paths from \(\mathrm { T } _ { 4 }\).
  3. How many paths are there altogether? For this question a route is defined to be a path in which the direction of the arcs is not relevant. Thus \(\mathrm { T } _ { 1 } \rightarrow \mathrm {~T} _ { 2 }\) and \(\mathrm { T } _ { 2 } \rightarrow \mathrm {~T} _ { 1 }\) are the same route. Similarly \(\mathrm { T } _ { 1 } \rightarrow \mathrm {~T} _ { 3 } \rightarrow \mathrm {~T} _ { 2 }\) and \(\mathrm { T } _ { 2 } \rightarrow \mathrm {~T} _ { 3 } \rightarrow \mathrm {~T} _ { 1 }\) are the same route (but note that \(\mathrm { T } _ { 1 } \rightarrow \mathrm {~T} _ { 2 } \rightarrow \mathrm {~T} _ { 3 }\) is different).
  4. How many routes are there altogether?
OCR MEI D1 2008 June Q4
16 marks Moderate -0.8
4 Joe is to catch a plane to go on holiday. He has arranged to leave his car at a car park near to the airport. There is a bus service from the car park to the airport, and the bus leaves when there are at least 15 passengers on board. Joe is delayed getting to the car park and arrives needing the bus to leave within 15 minutes if he is to catch his plane. He is the \(10 ^ { \text {th } }\) passenger to board the bus, so he has to wait for another 5 passengers to arrive. The distribution of the time intervals between car arrivals and the distribution of the number of passengers per car are given below.
Time interval between cars (minutes)12345
Probability\(\frac { 1 } { 10 }\)\(\frac { 3 } { 10 }\)\(\frac { 2 } { 5 }\)\(\frac { 1 } { 10 }\)\(\frac { 1 } { 10 }\)
Number of passengers per car123456
Probability\(\frac { 1 } { 6 }\)\(\frac { 1 } { 3 }\)\(\frac { 1 } { 12 }\)\(\frac { 1 } { 4 }\)\(\frac { 1 } { 12 }\)\(\frac { 1 } { 12 }\)
  1. Give an efficient rule for using 2-digit random numbers to simulate the intervals between car arrivals.
  2. Give an efficient rule for using 2-digit random numbers to simulate the number of passengers in a car.
  3. The incomplete table in your answer book shows the results of nine simulations of the situation. Complete the table, showing in each case whether or not Joe catches his plane.
  4. Use the random numbers provided in your answer book to run a tenth simulation.
    [0pt]
  5. Estimate the probability of Joe catching his plane. State how you could improve your estimate. [2]
OCR MEI D1 2008 June Q5
16 marks Moderate -0.5
5
  1. The graphs below illustrate the precedences involved in running two projects, each consisting of the same activities \(\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D }\) and E . \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Project 1} \includegraphics[alt={},max width=\textwidth]{8eba759f-38bc-4d14-ac65-9a0ee6c79741-6_280_385_429_495}
    \end{figure} \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Project 2} \includegraphics[alt={},max width=\textwidth]{8eba759f-38bc-4d14-ac65-9a0ee6c79741-6_255_392_429_1187}
    \end{figure}
    1. For one activity the precedences in the two projects are different. State which activity and describe the difference.
    2. The table below shows the durations of the five activities.
      ActivityABCDE
      Duration21\(x\)32
      Give the total time for project 1 for all possible values of \(x\).
      Give the total time for project 2 for all possible values of \(x\).
  2. The durations and precedences for the activities in a project are shown in the table.
    ActivityDurationImmediate predecessors
    R2-
    S1-
    T5-
    w3R, S
    X2R, S, T
    Y3R
    Z1W, Y
    1. Draw an activity on arc network to represent this information.
    2. Find the early time and the late time for each event. Give the project duration and list the critical activities.
OCR MEI D1 2008 June Q6
16 marks
6 The matrix gives the lengths of the arcs of a network.
ABCDEF
A-107-95
B10--1-4
C7---3-
D-1--2-
E9-32--
F54----
  1. Using the copy of the matrix in your answer book, apply the tabular form of Prim's algorithm to find a minimum connector for the network. Start by choosing vertex A and show the order in which you include vertices.
    List the arcs in your connector and give its total length. Serena takes a different approach to find a minimum connector. She first uses Dijkstra's algorithm to find shortest paths from A to each of the other vertices. She then uses the arcs in those paths to construct a connector.
  2. Draw the network using the vertices printed in your answer book.
  3. Apply Serena's method to produce a connector. List the arcs in the connector and give its total length. Serena adapts her method by starting from each vertex in turn, producing six connectors, from which she chooses the best.
  4. Serena's approach will not find the minimum connector in all networks, but it is an algorithm. What is its algorithmic complexity? Justify your answer.