Questions D1 - A-Level Maths

Give a route from Milan to Chania involving exactly two ferry crossings. How many such routes are there?
Give a route from Milan to Chania involving exactly three ferry crossings. How many such routes are there?
Give a route from Milan to Chania using as many ferry crossings as possible, without repeating any arc.
[0pt]
Give a route leaving Piraeus and finishing elsewhere which uses every arc once and only once.[3]

OCR MEI D1 2008 January Q2

2 Consider the following linear programming problem.
Maximise $$\mathrm { P } = 6 x + 7 y$$ subject to $$\begin{aligned} & 2 x + 3 y \leqslant 9
& 3 x + 2 y \leqslant 12
& x \geqslant 0
& y \geqslant 0 \end{aligned}$$

Use a graphical approach to solve the problem.
Give the optimal values of $x , y$ and P when $x$ and $y$ are integers.

OCR MEI D1 2008 January Q3

3 The following algorithm (J. M. Oudin, 1940) claims to compute the date of Easter Sunday in the Gregorian calendar system.
The algorithm uses the year, y, to give the month, m, and day, d, of Easter Sunday.
All variables are integers and all remainders from division are dropped. For example, 7 divided by 3 is 2 remainder 1 . The remainder is dropped, giving the answer 2. $$\begin{aligned} & c = y / 100
& n = y - 19 \times ( y / 19 )
& k = ( c - 17 ) / 25
& i = c - ( c / 4 ) - ( c - k ) / 3 + ( 19 \times n ) + 15
& i = i - 30 \times ( i / 30 )
& i = i - ( i / 28 ) \times ( 1 - ( i / 28 ) \times ( 29 / ( i + 1 ) ) \times ( ( 21 - n ) / 11 ) )
& j = y + ( y / 4 ) + i + 2 - c + ( c / 4 )
& j = j - 7 \times ( j / 7 )
& p = i - j
& m = 3 + ( p + 40 ) / 44
& d = p + 28 - 31 \times ( m / 4 ) \end{aligned}$$ For example, for 2008:
$\mathrm { y } = 2008$
$\mathrm { c } = 2008 / 100 = 20$
$n = 2008 - 19 \times ( 2008 / 19 ) = 2008 - 19 \times ( 105 ) = 13$, etc.
Complete the calculation for 2008.

OCR MEI D1 2008 January Q4

4 In a population colonizing an island 40\% of the first generation (parents) have brown eyes, $40 \%$ have blue eyes and $20 \%$ have green eyes. Offspring eye colour is determined according to the following rules. \section*{Eye colours of parents Eye colour of offspring} (1) both brown
(2) one brown and one blue $50 \%$ brown and $50 \%$ blue
(3) one brown and one green blue
(4) both blue $25 \%$ brown, $50 \%$ blue and $25 \%$ green
(5) one blue and one green 50\% blue and $50 \%$ green
(6) both green green

Give an efficient rule for using 1-digit random numbers to simulate the eye colour of a parent randomly selected from the colonizing population.
Give an efficient rule for using 1-digit random numbers to simulate the eye colour of offspring born of parents both of whom have blue eyes. The table in your answer book shows an incomplete simulation in which parent eye colours have been randomly selected, but in which offspring eye colours remain to be determined or simulated.
Complete the table using the given random numbers where needed. (You will need your own rules for cases $( 2 )$ and 5 .)
Each time you use a random number, explain how you decide which eye colour for the offspring. $\square$

OCR MEI D1 2008 January Q5

5 The table shows some of the activities involved in building a block of flats. The table gives their durations and their immediate predecessors.

Activity		Duration (weeks)	Immediate Predecessors
A	Survey sites	8	-
B	Purchase land	22	A
C	Supply materials	10	-
D	Supply machinery	4	-
E	Excavate foundations	9	B, D
F	Lay drains	11	B, C, D
G	Build walls	9	E, F
H	Lay floor	10	E, F
I	Install roof	3	G
J	Install electrics	5	G

Draw an activity on arc network for these activities.
Mark on your diagram the early and late times for each event. Give the minimum completion time and the critical activities. Each of the tasks E, F, H and J can be speeded up at extra cost. The maximum number of weeks by which each task can be shortened, and the extra cost for each week that is saved, are shown in the table below.
Task E F H J
Maximum number of weeks by
which task may be shortened
3 3 1 3
Cost per week of shortening task
(in thousands of pounds)
30 15 6 20
Find the new shortest time for the flats to be completed.
List the activities which will need to be speeded up to achieve the shortest time found in part (iii), and the times by which each must be shortened.
Find the total extra cost needed to achieve the new shortest time.

OCR MEI D1 2008 January Q6

6 The diagram shows routes between points in a town. The distances are in kilometres.
\includegraphics[max width=\textwidth, alt={}, center]{dfe6db33-33d0-4dff-95f7-fbf097e3963e-6_817_1219_319_422}

Use an appropriate algorithm to find a set of connecting arcs of minimum total length. Indicate your connecting arcs on the copy of the diagram in your answer book, and give their total length.
Give the name of the algorithm you have used, and describe it briefly.
Using the second diagram in your answer book, apply Dijkstra's algorithm to find the shortest distances from A to each of the other points. List the connections that are used, and give their total length.

OCR MEI D1 2009 January Q1

1 Alfred, Ben, Charles and David meet, and some handshaking takes place.

Alfred shakes hands with David.
Ben shakes hands with Charles and David.
Charles shakes hands with Ben and David.
1. Complete the bipartite graph in your answer book showing A (Alfred), B (Ben), C (Charles) and D (David), and the number of people each shakes hands with.
2. Explain why, whatever handshaking takes place, the resulting bipartite graph cannot contain both an arc terminating at 0 and another arc terminating at 3 .
3. Explain why, whatever number of people meet, and whatever handshaking takes place, there must always be two people who shake hands with the same number of people.

OCR MEI D1 2009 January Q2

2 The following algorithm computes the number of comparisons made when Prim’s algorithm is applied to a complete network on $n$ vertices ( $n > 2$ ). Step 1 Input the value of $n$
Step 2 Let $i = 1$
Step 3 Let $j = n - 2$
Step 4 Let $k = j$
Step 5 Let $i = i + 1$
Step 6 Let $j = j - 1$
Step 7 Let $k = k + ( i \times j ) + ( i - 1 )$
Step 8 If $j > 0$ then go to Step 5
Step 9 Print $k$
Step 10 Stop

Apply the algorithm when $n = 5$, showing the intermediate values of $i , j$ and $k$. The function $\mathrm { f } ( n ) = \frac { 1 } { 6 } n ^ { 3 } - \frac { 7 } { 6 } n + 1$ gives the same output as the algorithm.
Showing your working, check that $\mathrm { f } ( 5 )$ is the same value as your answer to part (i).
What does the structure of $\mathrm { f } ( n )$ tell you about Prim's algorithm?

OCR MEI D1 2009 January Q3

3 Whilst waiting for her meal to be served, Alice tries to construct a network to represent the meals offered in the restaurant.
\includegraphics[max width=\textwidth, alt={}, center]{9bb2d545-3764-4930-a8a8-9dc5e25d0836-3_684_1310_365_379}

Use Dijkstra's algorithm to find the cheapest route through the undirected network from "start" to "end". Give the cost and describe the route. Use the lettering given on the network in your answer book.
Criticise the model and suggest how it might be improved.

OCR MEI D1 2009 January Q4

4 A ski-lift gondola can carry 4 people. The weight restriction sign in the gondola says "4 people - 325 kg ". The table models the distribution of weights of people using the gondola.

\cline { 2 - 4 } \multicolumn{1}{c\|}{}	Men	Women	Children
Weight $( \mathrm { kg } )$	90	80	40
Probability	$\frac { 1 } { 2 }$	$\frac { 1 } { 3 }$	$\frac { 1 } { 6 }$

Give an efficient rule for using 2-digit random numbers to simulate the weight of a person entering the gondola.
Give a reason for using 2-digit rather than 1-digit random numbers in these circumstances.
Using the random numbers given in your answer book, simulate the weights of four people entering the gondola, and hence give its simulated load.
Using the random numbers given in your answer book, repeat your simulation 9 further times. Hence estimate the probability of the load of a fully-laden gondola exceeding 325 kg .
What in reality might affect the pattern of loading of a gondola which is not modelled by your simulation?

OCR MEI D1 2009 January Q5

5 The tasks involved in turning around an "AirGB" aircraft for its return flight are listed in the table. The table gives the durations of the tasks and their immediate predecessors.

Activity	Duration (mins)	Immediate Predecessors
A Refuel	30	-
B Clean cabin	25	-
C Pre-flight technical check	15	A
D Load luggage	20	-
E Load passengers	25	A, B
F Safety demonstration	5	E
G Obtain air traffic clearance	10	C
H Taxi to runway	5	G, D

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. Because of delays on the outbound flight the aircraft has to be turned around within 50 minutes, so as not to lose its air traffic slot for the return journey. There are four tasks on which time can be saved. These, together with associated costs, are listed below.
Task A B D E
New time (mins) 20 20 15 15
Extra cost 250 50 50 100
List the activities which need to be speeded up in order to turn the aircraft around within 50 minutes at minimum extra cost. Give the extra cost and the new set of critical activities.

OCR MEI D1 2009 January Q6

6 A company is planning its production of "MPowder" for the next three months.

Over the next 3 months 20 tonnes must be produced.
Production quantities must not be decreasing. The amount produced in month 2 cannot be less than the amount produced in month 1 , and the amount produced in month 3 cannot be less than the amount produced in month 2.
No more than 12 tonnes can be produced in total in months 1 and 2.
Production costs are $\pounds 2000$ per tonne in month $1 , \pounds 2200$ per tonne in month 2 and $\pounds 2500$ per tonne in month 3.

The company planner starts to formulate an LP to find a production plan which minimises the cost of production: $$\begin{array} { l l } \text { Minimise } & 2000 x _ { 1 } + 2200 x _ { 2 } + 2500 x _ { 3 }
\text { subject to } & x _ { 1 } \geq 0 x _ { 2 } \geq 0 x _ { 3 } \geq 0
& x _ { 1 } + x _ { 2 } + x _ { 3 } = 20
& x _ { 1 } \leq x _ { 2 }
& \bullet \cdot \cdot \end{array}$$

Explain what the variables $x _ { 1 } , x _ { 2 }$ and $x _ { 3 }$ represent, and write down two more constraints to complete the formulation.
Explain how the LP can be reformulated to: $$\begin{array} { l l } \text { Maximise } & 500 x _ { 1 } + 300 x _ { 2 }
\text { subject to } & x _ { 1 } \geq 0 x _ { 2 } \geq 0
& x _ { 1 } \leq x _ { 2 }
& x _ { 1 } + 2 x _ { 2 } \leq 20
& x _ { 1 } + x _ { 2 } \leq 12 \end{array}$$
Use a graphical approach to solve the LP in part (ii). Interpret your solution in terms of the company's production plan, and give the minimum cost.

OCR MEI D1 2010 January Q1

1 The table shows the activities involved in a project, their durations and their precedences.

Activity	Duration (mins)	Immediate predecessors
A	3	-
B	2	-
C	3	A
D	5	A, B
E	1	C

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 critical activities.

OCR MEI D1 2010 January Q2

2 The vertices of a graph are to be coloured using the following rules:

all vertices are to be coloured
no two vertices joined by an edge are to have the same colour.

The following graph has been coloured with four colours.
\includegraphics[max width=\textwidth, alt={}, center]{71ca9c4e-573b-43b7-910d-4bd610af6b27-2_357_883_1683_591} Kempe's rule allows for colours to be swapped. The rule is:

choose two colours
draw the subgraph consisting of the vertices coloured with these two colours, together with the edges that connect them
in any connected part of this subgraph consisting of two or more vertices, the two colours can be swapped.
1. Use Kempe's rule, choosing the colours blue and red.

Show that the graph can then be coloured with two colours.

Why does Kempe's rule not constitute an algorithm for colouring graphs?

OCR MEI D1 2010 January Q3

3 Consider the following graph in which the arcs are straight lines.
\includegraphics[max width=\textwidth, alt={}, center]{71ca9c4e-573b-43b7-910d-4bd610af6b27-3_928_938_317_566}

Explain how you know that the graph is simple.
Explain how you know that the graph is not connected.
On the copy of the graph in your answer book, add as many arcs as you can whilst keeping it both simple and not connected. Give the number of arcs which you have added.
Imagine that a new graph is produced in which two vertices are connected if there is no connection between them, direct or indirect, on the original graph. How many arcs would this new graph have?

OCR MEI D1 2010 January Q4

4 An air charter company has the following rules for selling seats on a flight.

The total number of seats sold must not exceed 120.
There must be at least 100 seats sold, or the flight will be cancelled.
For every child seat sold there must be a seat sold for a supervising adult.
1. Define two variables so that the three constraints can be formulated in terms of your variables. Formulate the three constraints in terms of your variables.
2. Graph your three inequalities from part (i).
The price for a child seat is $\pounds 50$ and the price for an adult seat is $\pounds 100$.
Find the maximum income available from the flight, and mark and label the corresponding point on your graph.
Find the minimum income available from a full plane, and mark and label the corresponding point on your graph.
Find the minimum income available from the flight, and mark and label the corresponding point on your graph.
At $\pounds 100$ for an adult seat and $\pounds 50$ for a child seat the company would prefer to sell 100 adult seats and no child seats rather than have a full plane with 60 adults and 60 children. What would be the minimum price for a child's seat for that not to be the case, given that the adult seat price remains at $\pounds 100$ ?

OCR MEI D1 2010 January Q5

5 The matrix shows the distances in miles between towns where direct routes exist.

	A	B	C	D	E	F
A	-	22	-	12	10	-
B	22	-	-	-	-	13
C	-	-	-	6	5	11
D	12	-	6	-	-	-
E	10	-	5	-	-	26
F	-	13	11	-	26	-

Draw the network.
Use Dijkstra's algorithm to find the shortest route from A to F . Give the route and its length.
Use Kruskal's algorithm to find a minimum connector for the network, showing your working. Draw your connector and give its total length.
How much shorter would AD have to be if it were to be included in
(A) a shortest route from A to F ,
(B) a minimum connector?

OCR MEI D1 2010 January Q6

6 An apple tree has 6 apples left on it. Each day each remaining apple has a probability of $\frac { 1 } { 3 }$ of falling off the tree during the day.

Give a rule for using one-digit random numbers to simulate whether or not a particular apple falls off the tree during a given day.
Use the random digits given in your answer book to simulate how many apples fall off the tree during day 1 . Give the total number of apples that fall during day 1 .
Continue your simulation from the end of day 1 , which you simulated in part (ii), for successive days until there are no apples left on the tree. Use the same list of random digits, continuing from where you left off in part (ii). During which day does the last apple fall from the tree? Now suppose that at the start of each day the gardener picks one apple from the tree and eats it.
Repeat your simulation with the gardener picking the lowest numbered apple remaining on the tree at the start of each day. Give the day during which the last apple falls or is picked. Use the same string of random digits, a copy of which is provided for your use in this part of the question.
How could your results be made more reliable?

OCR MEI D1 2011 January Q1

1 The diagram shows an electrical circuit with wires and switches and with five components, labelled A, B, C, D and E.
\includegraphics[max width=\textwidth, alt={}, center]{11c2d98f-1f72-4f1b-b971-5521bee09358-2_328_730_609_319}
\includegraphics[max width=\textwidth, alt={}, center]{11c2d98f-1f72-4f1b-b971-5521bee09358-2_261_519_621_1224}

Draw a graph showing which vertices are connected together, either directly or indirectly, when the two switches remain open.
How many arcs need to be added to your graph when both switches are closed? The graph below shows which components are connected to each other, either directly or indirectly, for a second electrical circuit.
\includegraphics[max width=\textwidth, alt={}, center]{11c2d98f-1f72-4f1b-b971-5521bee09358-2_410_494_1356_788}
Find the minimum number of arcs which need to be deleted to create two disconnected sets of vertices, and write down your two separate sets.
Explain why, in the second electrical circuit, it might be possible to split the components into two disconnected sets by cutting fewer wires than the number of arcs which were deleted in part (iii).

OCR MEI D1 2011 January Q2

2 King Elyias has been presented with eight flagons of fine wine. Intelligence reports indicate that at least one of the eight flagons has been poisoned. King Elyias will have the wine tasted by the royal wine tasters to establish which flagons are poisoned. Samples for testing are made by using wine from one or more flagons. If a royal wine taster tastes a sample of wine which includes wine from a poisoned flagon, the taster will die. The king has to make a very generous payment for each sample tasted. To minimise payments, the royal mathematicians have devised the following scheme:
Test a sample made by mixing wine from flagons $1,2,3$ and 4.
If the taster dies, then test a sample made by mixing wine from flagons $5,6,7$ and 8 .
If the taster lives, then there is no poison in flagons $1,2,3$ or 4 . So there is poison in at least one of flagons 5, 6, 7 and 8, and there is no need to test a sample made by mixing wine from all four of them. If the sample from flagons $1,2,3$ and 4 contains poison, then test a fresh sample made by mixing wine from flagons 1 and 2, and proceed similarly, testing a sample from flagons 3 and 4 only if the taster of the sample from flagons 1 and 2 dies. Continue, testing new samples made from wine drawn from half of the flagons corresponding to a poisoned sample, and testing only when necessary.

Record what happens using the mathematicians' scheme when flagon number 7 is poisoned, and no others.
Record what happens using the mathematicians' scheme when two flagons, numbers 3 and 7, are poisoned.

OCR MEI D1 2011 January Q3

1 marks

3 The network shows distances between vertices where direct connections exist.
\includegraphics[max width=\textwidth, alt={}, center]{11c2d98f-1f72-4f1b-b971-5521bee09358-3_518_691_1742_687}

Use Dijkstra's algorithm to find the shortest distance and route from A to F .
Explain why your solution to part (i) also provides the shortest distances and routes from A to each of the other vertices.
[0pt]
Explain why your solution to part (i) also provides the shortest distance and route from B to F. [1]

OCR MEI D1 2011 January Q4

4 The table shows the tasks involved in preparing breakfast, and their durations.

Task	Description	Duration (mins)
A	Fill kettle and switch on	0.5
B	Boil kettle	1.5
C	Cut bread and put in toaster	0.5
D	Toast bread	2
E	Put eggs in pan of water and light gas	1
F	Boil eggs	5
G	Put tablecloth, cutlery and crockery on table	2.5
H	Make tea and put on table	0.5
I	Collect toast and put on table	0.5
J	Put eggs in cups and put on table	1

Show the immediate predecessors for each of these tasks.
Draw an activity on arc network modelling your precedences.
Perform a forward pass and a backward pass to find the early time and the late time for each event.
Give the critical activities, the project duration, and the total float for each activity.
Given that only one person is available to do these tasks, and noting that tasks B, D and F do not require that person's attention, produce a cascade chart showing how breakfast can be prepared in the least possible time.

OCR MEI D1 2011 January Q5

5 Viola and Orsino are arguing about which striker to include in their fantasy football team. Viola prefers Rocinate, who creates lots of goal chances, but is less good at converting them into goals. Orsino prefers Quince, who is not so good at creating goal chances, but who is better at converting them into goals. The information for Rocinate and Quince is shown in the tables.

\multirow{2}{*}{}	Number of chances created per match
	Rocinate				Quince
Number	6	7	8	9	5	6	7	8
Probability	$\frac { 1 } { 20 }$	$\frac { 1 } { 4 }$	$\frac { 1 } { 2 }$	$\frac { 1 } { 5 }$	$\frac { 1 } { 3 }$	$\frac { 1 } { 3 }$	$\frac { 1 } { 6 }$	$\frac { 1 } { 6 }$

Probability of converting a chance into a goal
Rocinate	Quince
0.1	0.12

Give an efficient rule for using 2-digit random numbers to simulate the number of chances created by Rocinate in a match.
Give a rule for using 2-digit random numbers to simulate the conversion of chances into goals by Rocinate.
Your Printed Answer Book shows the result of simulating the number of goals scored by Rocinate in nine matches. Use the random numbers given to complete the tenth simulation, showing which of your simulated chances are converted into goals.
Give an efficient rule for using 2-digit random numbers to simulate the number of chances created by Quince in a match.
Your Printed Answer Book shows the result of simulating the number of goals scored by Quince in nine matches. Use the random numbers given to complete the tenth simulation, showing which of your simulated chances are converted into goals.
Which striker, if any, is favoured by the simulation? Justify your answer.
How could the reliability of the simulation be improved?

OCR MEI D1 2011 January Q6

6 A manufacturing company holds stocks of two liquid chemicals. The company needs to update its stock levels. The company has 2000 litres of chemical A and 4000 litres of chemical B currently in stock. Its storage facility allows for no more than a combined total of 12000 litres of the two chemicals. Chemical A is valued at $\pounds 5$ per litre and chemical B is valued at $\pounds 6$ per litre. The company intends to hold stocks of these two chemicals with a total value of at least $\pounds 61000$. Let $a$ be the increase in the stock level of A, in thousands of litres ( $a$ can be negative).
Let $b$ be the increase in the stock level of B , in thousands of litres ( $b$ can be negative).

Explain why $a \geqslant - 2$, and produce a similar inequality for $b$.
Explain why the value constraint can be written as $5 a + 6 b \geqslant 27$, and produce, in similar form, the storage constraint.
Illustrate all four inequalities graphically.
Find the policy which will give a stock value of exactly $\pounds 61000$, and will use all 12000 litres of available storage space.
Interpret your solution in terms of stock levels, and verify that the new stock levels do satisfy both the value constraint and the storage constraint.

OCR MEI D1 2012 January Q1

1 A graph is obtained from a solid by producing a vertex for each exterior face. Vertices in the graph are connected if their corresponding faces in the original solid share an edge. The diagram shows a solid followed by its graph. The solid is made up of two cubes stacked one on top of the other. This solid has 10 exterior faces, which correspond to the 10 vertices in the graph. (Note that in this question it is the exterior faces of the cubes that are being counted.)
\includegraphics[max width=\textwidth, alt={}, center]{3239d012-5699-4789-ba64-f1295f4b4642-2_455_309_571_653}
\includegraphics[max width=\textwidth, alt={}, center]{3239d012-5699-4789-ba64-f1295f4b4642-2_444_286_573_1135}

Draw the graph for a cube.
Obtain the number of vertices and the number of edges for the graph of three cubes stacked on top of each other.
\includegraphics[max width=\textwidth, alt={}, center]{3239d012-5699-4789-ba64-f1295f4b4642-2_643_305_1302_881}

\multirow{2}{*}{}	Number of chances created per match
	Rocinate				Quince
Number	6	7	8	9	5	6	7	8
Probability	\(\frac { 1 } { 20 }\)	\(\frac { 1 } { 4 }\)	\(\frac { 1 } { 2 }\)	\(\frac { 1 } { 5 }\)	\(\frac { 1 } { 3 }\)	\(\frac { 1 } { 3 }\)	\(\frac { 1 } { 6 }\)	\(\frac { 1 } { 6 }\)

	A	B	C	D	E	F
A	-	22	-	12	10	-
B	22	-	-	-	-	13
C	-	-	-	6	5	11
D	12	-	6	-	-	-
E	10	-	5	-	-	26
F	-	13	11	-	26	-

Task	A	B	D	E
New time (mins)	20	20	15	15
Extra cost	250	50	50	100

	A	B	C	D	E	F
A	-	22	-	12	10	-
B	22	-	-	-	-	13
C	-	-	-	6	5	11
D	12	-	6	-	-	-
E	10	-	5	-	-	26
F	-	13	11	-	26	-

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	A	B	C	D	E	F
A	-	22	-	12	10	-
B	22	-	-	-	-	13
C	-	-	-	6	5	11
D	12	-	6	-	-	-
E	10	-	5	-	-	26
F	-	13	11	-	26	-