Questions - OCR MEI D1 - A-Level Maths

	Activity	Duration (hours)	Immediate predecessors
A	Agree date	1	-
B	Construct agenda	0.5	-
C	Book venue	0.25	A
D	Order refreshments	0.25	C
E	Inform participants	0.5	B, C

\captionsetup{labelformat=empty} \caption{Table 4.1}

\end{table}

Draw an activity-on-arc network to represent the precedences.
Find the early event time and the late event time for each vertex of your network, and list the critical activities.
Assuming that each activity requires one person and that each activity starts at its earliest start time, draw a resource histogram.
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
Draw a precedence table for these activities.

OCR MEI D1 2006 June Q5

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.

Define appropriate variables and formulate two inequalities to model John's constraints.
Draw a graph to represent your inequalities. Give the coordinates of the vertices of your feasible region.
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.
Solve the resulting LP problem.
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.
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

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.

Draw John's graph.
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.
Draw Jamil's graph.
Is Jamil's graph a tree? Justify your answer.

OCR MEI D1 2007 June Q2

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.

Apply the first fit algorithm to the items in the order given and comment on the outcome.
Write the five items in descending order of volume. Apply the first fit decreasing algorithm to find a packing for the rucksacks.
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

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

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.

A	Rig foresail
B	Lower sprayhood
C	Start engine
D	Pump out bilges
E	Rig mainsail
F	Cast off mooring ropes
G	Motor out of harbour
H	Raise foresail
I	Raise mainsail
J	Stop engine and start sailing

\includegraphics[max width=\textwidth, alt={}, center]{21ab732d-435e-4f0b-bc88-21ddc2a398c9-3_480_912_555_925}

Complete the table in your answer book showing the immediate predecessors for each activity.
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.
Use the activity network to determine which activities Colin can perform whilst motoring out of the harbour.
Find the minimum time to complete the activities when Colin sails on his own, and give a schedule for him to achieve this.
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

5 The table shows the weights on the arcs of a network.

	A	B	C	D	E	F	G
A	-	11	-	-	10	3	5
B	11	-	8	5	-	-	14
C	-	8	-	2	-	7	-
D	-	5	2	-	6	-	-
E	10	-	-	6	-	6	-
F	3	-	7	-	6	-	-
G	5	14	-	-	-	-	-

Draw the network.
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.
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.
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

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|}{}

You are to use two-digit random numbers to simulate the winter weather in Metland.

Give an efficient rule for using two-digit random numbers to simulate tomorrow's weather if today is
(A) dry,
(B) wet,
(C) snowy.
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. $$\begin{array} { l l l l l l l l l l } 23 & 85 & 98 & 99 & 56 & 47 & 82 & 14 & 03 & 12 \end{array}$$
Use your simulation from part (ii) to estimate the proportion of dry days in a Metland winter.
Explain how you could use simulation to produce an improved estimate of the proportion of dry days in a Metland winter.
Give two criticisms of this model of weather.

OCR MEI D1 2008 June Q1

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

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.

Apply the algorithm to the list $5,14,153,6,24,2,14,15$, counting the number of comparisons which you have to make.
Apply the algorithm to the list $5,14,153,6,24,2,14$, counting the number of comparisons which you have to make.
Say what the algorithm is finding.
The order of the algorithm is quadratic. Explain what this means when it is applied to long lists.

OCR MEI D1 2008 June Q3

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.

There are six paths from $\mathrm { T } _ { 1 }$. List them.
List the paths from $\mathrm { T } _ { 4 }$.
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).
How many routes are there altogether?

OCR MEI D1 2008 June Q4

2 marks

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)	1	2	3	4	5
Probability	$\frac { 1 } { 10 }$	$\frac { 3 } { 10 }$	$\frac { 2 } { 5 }$	$\frac { 1 } { 10 }$	$\frac { 1 } { 10 }$

Number of passengers per car	1	2	3	4	5	6
Probability	$\frac { 1 } { 6 }$	$\frac { 1 } { 3 }$	$\frac { 1 } { 12 }$	$\frac { 1 } { 4 }$	$\frac { 1 } { 12 }$	$\frac { 1 } { 12 }$

Give an efficient rule for using 2-digit random numbers to simulate the intervals between car arrivals.
Give an efficient rule for using 2-digit random numbers to simulate the number of passengers in a car.
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.
Use the random numbers provided in your answer book to run a tenth simulation.
[0pt]
Estimate the probability of Joe catching his plane. State how you could improve your estimate. [2]

OCR MEI D1 2008 June Q5

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.
  Activity A B C D E
  Duration 2 1 $x$ 3 2
  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 $x$.
The durations and precedences for the activities in a project are shown in the table.
Activity Duration Immediate predecessors
R 2 -
S 1 -
T 5 -
w 3 R, S
X 2 R, S, T
Y 3 R
Z 1 W, 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

6 The matrix gives the lengths of the arcs of a network.

	A	B	C	D	E	F
A	-	10	7	-	9	5
B	10	-	-	1	-	4
C	7	-	-	-	3	-
D	-	1	-	-	2	-
E	9	-	3	2	-	-
F	5	4	-	-	-	-

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.
Draw the network using the vertices printed in your answer book.
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.
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.

OCR MEI D1 2009 June Q1

1 The numbers on opposite faces of the die shown (a standard die) add up to 7. The adjacency graph for the die is a graph which has vertices representing faces. In the adjacency graph two vertices are joined with an arc if they share an edge of the die. For example, vertices 2 and 6 are joined by an arc because they share an edge of the die.
\includegraphics[max width=\textwidth, alt={}, center]{dab87ac5-eda4-433f-b07a-0a609aca2f65-2_246_213_488_1608}

List the pairs of numbers which are opposite each other.
Draw the adjacency graph.
Identify and sketch a solid which has the following adjacency graph.
\includegraphics[max width=\textwidth, alt={}, center]{dab87ac5-eda4-433f-b07a-0a609aca2f65-2_287_307_1027_879}

OCR MEI D1 2009 June Q2

2 In this question INT( $m$ ) means the integer part of $m$. Thus INT(3.5) $= 3$ and INT(4) $= 4$.
A game for two players starts with a number, $n$, of counters. Players alternately pick up a number of counters, at least 1 and not more than half of those left. The player forced to pick up the last counter is the loser. Arif programs his computer to play the game, using the rule "pick up INT(half of the remaining counters), or the last counter if forced".

You are to play against Arif's computer with $n = 5$ and with Arif's computer going first. What happens at each turn?
You are to play against Arif's computer with $n = 6$ and with Arif's computer going first. What happens at each turn?
Now play against Arif's computer with $n = 7$ and with Arif's computer going first. Describe what happens.

OCR MEI D1 2009 June Q3

3 Consider the following linear programming problem:
Maximise $\quad 3 x + 4 y$
subject to $\quad 2 x + 5 y \leqslant 60$
$x + 2 y \leqslant 25$
$x + y \leqslant 18$

Graph the inequalities and hence solve the LP.
The right-hand side of the second inequality is increased from 25 . At what new value will this inequality become redundant?

OCR MEI D1 2009 June Q4

4 The diagram represents a very simple maze with two vertices, A and B. At each vertex a rat either exits the maze or runs to the other vertex, each with probability 0.5 . The rat starts at vertex A .
\includegraphics[max width=\textwidth, alt={}, center]{dab87ac5-eda4-433f-b07a-0a609aca2f65-4_79_930_534_571}

Describe how to use 1-digit random numbers to simulate this situation.
Use the random digits provided in your answer book to run 10 simulations, each starting at vertex A. Hence estimate the probability of the rat exiting at each vertex, and calculate the mean number of times it runs between vertices before exiting. The second diagram represents a maze with three vertices, A, B and C. At each vertex there are three possibilities, and the rat chooses one, each with probability $1 / 3$. The rat starts at vertex A.
\includegraphics[max width=\textwidth, alt={}, center]{dab87ac5-eda4-433f-b07a-0a609aca2f65-4_566_889_1082_589}
Describe how to use 1-digit random numbers to simulate this situation.
Use the random digits provided in your answer book to run 10 simulations, each starting at vertex A. Hence estimate the probability of the rat exiting at each vertex.

OCR MEI D1 2009 June Q5

5 The diagram represents canals connecting five cities. Canal lengths (shown on the arcs) are in km.
\includegraphics[max width=\textwidth, alt={}, center]{dab87ac5-eda4-433f-b07a-0a609aca2f65-5_410_990_306_539}

Draw a network in your answer book with nodes representing the five cities and arcs representing direct canal connections, i.e. canal connections which do not involve passing through another city. The company operating the canal system wishes to close some canals to save money, whilst preserving the connectivity.
Starting at A, use Prim's algorithm on your answer to part (i) to find a minimum connector for the network. Give the order in which you include arcs. Draw your minimum connector and give its total length. Consider the original network together with an extra vertex, X , at the junction of four arcs.
\includegraphics[max width=\textwidth, alt={}, center]{dab87ac5-eda4-433f-b07a-0a609aca2f65-5_405_987_1361_539}
Draw the minimum connector which results from applying Prim's algorithm, starting at A , to this network. Give the length of that minimum connector. Hence advise the company on which canals to close.
Give a reason why the company might face objections to such closures.

OCR MEI D1 2009 June Q6

6 Joan and Keith have to clear and tidy their garden. The table shows the jobs that have to be completed, their durations and their precedences.

Jobs		Duration (mins)	Immediate predecessors
A	prune bushes	100	-
B	weed borders	60	A
C	cut hedges	150	-
D	hoe vegetable patch	60	-
E	mow lawns	40	B
F	edge lawns	20	D, E
G	clean up cuttings	30	B, C
H	clean tools	10	F, G

Draw an activity on arc network for these activities.
Mark on your diagram the early time and the late time for each event. Give the minimum completion time and the critical activities.
Each job is to be done by one person only. Joan and Keith are equally able to do all jobs. Draw a cascade chart indicating how to organise the jobs so that Joan and Keith can complete the project in the least time. Give that least time and explain why the minimum project completion time is shorter.

OCR MEI D1 2010 June Q1

Use Dijkstra's algorithm to find the shortest distances and corresponding routes from A to each of the other vertices in the given network.
\includegraphics[max width=\textwidth, alt={}, center]{839adc96-1bea-44ef-917e-f03e396a3061-2_588_792_632_632}
If the shortest distances and routes between every pair of vertices are required how many applications of Dijkstra will be needed?

OCR MEI D1 2010 June Q2

2 The following steps define an algorithm which acts on two numbers.
STEP 1 Write down the two numbers side by side on the same row.
STEP 2 Beneath the left-hand number write down double that number. Beneath the right-hand number write down half of that number, ignoring any remainder. STEP 3 Repeat STEP 2 until the right-hand number is 1.
STEP 4 Delete those rows where the right-hand number is even. Add up the remaining left-hand numbers. This is the result.

Apply the algorithm to the left-hand number 3 and the right-hand number 8.
Apply the algorithm to the left-hand number 26 and the right-hand number 42.
Use your results from parts (i) and (ii), together with any other examples you may choose, to write down what the algorithm achieves.

OCR MEI D1 2010 June Q3

3 Traffic flows in and out of a junction of three roads as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{839adc96-1bea-44ef-917e-f03e396a3061-3_296_337_333_863} Assuming that no traffic leaves the junction by the same road as it entered, then the digraph shows traffic flows across the junction.
\includegraphics[max width=\textwidth, alt={}, center]{839adc96-1bea-44ef-917e-f03e396a3061-3_362_511_852_776}

Redraw the digraph to show that it is planar.
Draw a digraph to show the traffic flow across the junction of 4 roads, assuming that no traffic leaves the junction by the same road as it entered.
\includegraphics[max width=\textwidth, alt={}, center]{839adc96-1bea-44ef-917e-f03e396a3061-3_366_366_1512_854}
(Note that the resulting digraph is also planar, but you are not required to show this.)
The digraphs showing flows across the junctions omit an important aspect in their modelling of the road junctions. What is it that they omit?

OCR MEI D1 2010 June Q4

4 A wall 4 metres long and 3 metres high is to be tiled. Two sizes of tile are available, 10 cm by 10 cm and 30 cm by 30 cm .

If $x$ is the number of boxes of ten small tiles used, and $y$ is the number of large tiles used, explain why $10 x + 9 y \geqslant 1200$. There are only 100 of the large tiles available.
The tiler advises that the area tiled with the small tiles should not exceed the area tiled with the large tiles.
Express these two constraints in terms of $x$ and $y$. The smaller tiles cost 15 p each and the larger tiles cost $\pounds 2$ each.
Given that the objective is to minimise the cost of tiling the wall, state the objective function. Use the graphical approach to solve the problem.
Give two other factors which would have to be taken into account in deciding how many of each tile to purchase.

OCR MEI D1 2010 June Q5

5 The diagram shows the progress of a drunkard towards his home on one particular night. For every step which he takes towards his home, he staggers one step diagonally to his left or one step diagonally to his right, randomly and with equal probability. There is a canal three steps to the right of his starting point, and no constraint to the left. On this particular occasion he falls into the canal after 5 steps.
\includegraphics[max width=\textwidth, alt={}, center]{839adc96-1bea-44ef-917e-f03e396a3061-5_723_622_488_724}

Explain how you would simulate the drunkard's walk, making efficient use of one-digit random numbers.
Using the random digits in the Printed Answer Book simulate the drunkard's walk and show his progress on the grid. Stop your simulation either when he falls into the canal or when he has staggered 6 steps, whichever happens first.
How could you estimate the probability of him falling into the canal within 6 steps? On another occasion the drunkard sets off carrying a briefcase in his right hand. This changes the probabilities of him staggering to the right to $\frac { 2 } { 3 }$, and to the left to $\frac { 1 } { 3 }$.
Explain how you would now simulate this situation.
Simulate the drunkard's walk (with briefcase) 10 times, and hence estimate the probability of him falling into the canal within 6 steps. (In your simulations you are not required to show his progress on a grid. You only need to record his steps to the right or left.)

Time interval between cars (minutes)	1	2	3	4	5
Probability	\(\frac { 1 } { 10 }\)	\(\frac { 3 } { 10 }\)	\(\frac { 2 } { 5 }\)	\(\frac { 1 } { 10 }\)	\(\frac { 1 } { 10 }\)

Number of passengers per car	1	2	3	4	5	6
Probability	\(\frac { 1 } { 6 }\)	\(\frac { 1 } { 3 }\)	\(\frac { 1 } { 12 }\)	\(\frac { 1 } { 4 }\)	\(\frac { 1 } { 12 }\)	\(\frac { 1 } { 12 }\)

	A	B	C	D	E	F	G
A	-	11	-	-	10	3	5
B	11	-	8	5	-	-	14
C	-	8	-	2	-	7	-
D	-	5	2	-	6	-	-
E	10	-	-	6	-	6	-
F	3	-	7	-	6	-	-
G	5	14	-	-	-	-	-

Activity	Duration	Immediate predecessors
R	2	-
S	1	-
T	5	-
w	3	R, S
X	2	R, S, T
Y	3	R
Z	1	W, Y

	A	B	C	D	E	F	G
A	-	11	-	-	10	3	5
B	11	-	8	5	-	-	14
C	-	8	-	2	-	7	-
D	-	5	2	-	6	-	-
E	10	-	-	6	-	6	-
F	3	-	7	-	6	-	-
G	5	14	-	-	-	-	-

Questions — OCR MEI D1 (128 questions)

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	A	B	C	D	E	F	G
A	-	11	-	-	10	3	5
B	11	-	8	5	-	-	14
C	-	8	-	2	-	7	-
D	-	5	2	-	6	-	-
E	10	-	-	6	-	6	-
F	3	-	7	-	6	-	-
G	5	14	-	-	-	-	-