Questions - OCR MEI D1 - A-Level Maths

Draw a tree to show all of the possibilities for the player's first three moves.
Show how a player can win in 3 turns.
List all squares which it is possible for a counter to occupy after 3 turns.
Show that a game can continue indefinitely.

OCR MEI D1 Q3

12 marks

3 The following algorithm finds the highest common factor of two positive integers. ("int (x)" stands for the integer part of x, e.g. int (7.8) = 7.) \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{5d8d35b7-e4ba-4bc0-93a1-0cae58093a02-004_888_693_422_717} \captionsetup{labelformat=empty} \caption{Fig. 3.1}

\end{figure}

Run the algorithm with $\mathrm { A } = 84$ and $\mathrm { B } = 660$, showing all of your calculations.
Run the algorithm with $\mathrm { A } = 660$ and $\mathrm { B } = 84$, showing as many calculations as are necessary.
The algorithm is run with $\mathrm { A } = 30$ and $\mathrm { B } = 42$, and the result is shown in Table 3.2 below. \begin{table}[h]
A B Q R 1 R 2
30 42 1 12
12 30 2 6
6
6 12 2 0
\captionsetup{labelformat=empty} \caption{Print 6}
\end{table} Table 3.2 The first line of the table shows that $12 = 42 - 1 \times 30$.
Use the second line to obtain a similar expression for 6 in terms of 30 and 12.
Hence express 6 in the form $\mathrm { m } \times 30 - \mathrm { n } \times 42$, where m and n are integers.

OCR MEI D1 2005 January Q1

\includegraphics[alt={},max width=\textwidth]{b9ee9306-18ca-42b3-9f2e-b23849374b5e-2_723_1287_569_425} \captionsetup{labelformat=empty} \caption{Fig. 1}

\end{figure}

Draw a tree to show all of the possibilities for the player's first three moves.
Show how a player can win in 3 turns.
List all squares which it is possible for a counter to occupy after 3 turns.
Show that a game can continue indefinitely.

OCR MEI D1 2005 January Q3

3 The following algorithm finds the highest common factor of two positive integers. ("int (x)" stands for the integer part of x, e.g. int (7.8) = 7.) \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{b9ee9306-18ca-42b3-9f2e-b23849374b5e-4_888_693_422_717} \captionsetup{labelformat=empty} \caption{Fig. 3.1}

\end{figure}

Run the algorithm with $\mathrm { A } = 84$ and $\mathrm { B } = 660$, showing all of your calculations.
Run the algorithm with $\mathrm { A } = 660$ and $\mathrm { B } = 84$, showing as many calculations as are necessary.
The algorithm is run with $\mathrm { A } = 30$ and $\mathrm { B } = 42$, and the result is shown in Table 3.2 below. \begin{table}[h]
A B Q R 1 R 2
30 42 1 12
12 30 2 6
6
6 12 2 0
\captionsetup{labelformat=empty} \caption{Print 6}
\end{table} Table 3.2 The first line of the table shows that $12 = 42 - 1 \times 30$.
Use the second line to obtain a similar expression for 6 in terms of 30 and 12.
Hence express 6 in the form $\mathrm { m } \times 30 - \mathrm { n } \times 42$, where m and n are integers.

OCR MEI D1 2005 January Q5

5 There is an insert for use in parts (iii) and (iv) of this question.
This question concerns the simulation of cars passing through two sets of pedestrian controlled traffic lights. The time intervals between cars arriving at the first set of lights are distributed according to Table 5.1. \begin{table}[h]

Time interval (seconds)	2	5	15	25
Probability	$\frac { 3 } { 7 }$	$\frac { 2 } { 7 }$	$\frac { 1 } { 7 }$	$\frac { 1 } { 7 }$

\captionsetup{labelformat=empty} \caption{Table 5.1}

\end{table}

Give an efficient rule for using two-digit random numbers to simulate arrival intervals.
Use two-digit random numbers from the list below to simulate the arrival times of five cars at the first lights. The first car arrives at the time given by the first arrival interval. Random numbers: $24,01,99,89,77,19,58,42$ The two sets of traffic lights are 23 seconds driving time apart. Moving cars are always at least 2 seconds apart. If there is a queue at a set of lights, then when the red light ends the first car in the queue moves off immediately, the second car 2 seconds later, the third 2 seconds after that, etc. In this simple model there is to be no consideration of accelerations or decelerations, and the lights are either red or green. Table 5.2 shows the times when the lights are red. \begin{table}[h]
\multirow{2}{*}{
first set
of lights
} red start time 14 50 105 155
\cline { 2 - 6 } red end time 29 65 120 170
\multirow{2}{*}{
second set
of lights
} red start time 10 55 105 150
\cline { 2 - 6 } red end time 25 70 120 165
\captionsetup{labelformat=empty} \caption{Table 5.2}
\end{table}
Complete the table in the insert to simulate the passage of 10 cars through both sets of traffic lights. Use the arrival times given there.
Find the mean delay experienced by these cars in passing through each set of lights.
How could the output from this simulation model be made more reliable?

OCR MEI D1 2005 January Q6

6 A recipe for jam states that the weight of sugar used must be between the weight of fruit used and four thirds of the weight of fruit used. Georgia has 10 kg of fruit available and 11 kg of sugar.

Define two variables and formulate inequalities in those variables to model this information.
Draw a graph to represent your inequalities.
Find the vertices of your feasible region and identify the points which would represent the best mix of ingredients under each of the following circumstances.
(A) There is to be as much jam as possible, given that the weight of jam produced is the sum of the weights of the fruit and the sugar.
(B) There is to be as much jam as possible, given that it is to have the lowest possible proportion of sugar.
(C) There is to be as much jam as possible, given that it is to have the highest possible proportion of sugar.
(D) Fruit costs $\pounds 1$ per kg, sugar costs 50 p per kg and the objective is to produce as much jam as possible within a budget of $\pounds 15$.

OCR MEI D1 2006 January Q1

1 Table 1 shows a precedence table for a project. \begin{table}[h]

Activity	Immediate predecessors	Duration (days)
A	-	5
B	-	3
C	A	3
D	A, B	4
E	A, B	5

\captionsetup{labelformat=empty} \caption{Table 1}

\end{table}

Draw an activity-on-arc network to represent the precedences.
Find the early event time and late event time for each vertex of your network, and list the critical activities.
Extra resources become available which enable the durations of three activities to be reduced, each by up to two days. Which three activities should have their durations reduced so as to minimise the completion time of the project? What will be the new minimum project completion time?

OCR MEI D1 2006 January Q3

3 Fig. 3 shows a graph representing the seven bus journeys run each day between four rural towns. Each directed arc represents a single bus journey. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{ee39642f-f323-4614-a02a-4500199626de-4_317_515_392_772} \captionsetup{labelformat=empty} \caption{Fig. 3}

\end{figure}

Show that if there is only one bus, which is in service at all times, then it must start at one town and end at a different town. Give the start town and the end town.
Show that there is only one Hamilton cycle in the graph. Show that, if an extra journey is added from your end town to your start town, then there is still only one Hamilton cycle.
A tourist is staying in town B. Give a route for her to visit every town by bus, visiting each town only once and returning to B . Section B (48 marks)

OCR MEI D1 2006 January Q4

4 Table 4 shows the butter and sugar content in two recipes. The first recipe is for 1 kg of toffee and the second is for 1 kg of fudge. \begin{table}[h] \section*{Table 6.1} (ii) Specify an efficient rule for using one-digit random numbers to simulate the time taken at the till by customers purchasing fuel. Table 6.2 shows the distribution of time taken at the till by customers who are not buying fuel.

Time taken (mins)	1	1.5	2	2.5	3
Probability	$\frac { 1 } { 7 }$	$\frac { 2 } { 7 }$	$\frac { 2 } { 7 }$	$\frac { 1 } { 7 }$	$\frac { 1 } { 7 }$

\section*{Table 6.2} (iii) Specify an efficient rule for using two-digit random numbers to simulate the time taken at the till by customers not buying fuel. What is the advantage in using two-digit random numbers instead of one-digit random numbers in this part of the question? The table in the insert shows a partially completed simulation study of 10 customers arriving at the till.
(iv) Complete the table using the random numbers which are provided.
(v) Calculate the mean total time spent queuing and paying.

OCR MEI D1 2008 January Q1

3 marks

1 The graph shows routes that are available to an international lorry driver. The solid arcs represent motorways and the broken arcs represent ferry crossings.
\includegraphics[max width=\textwidth, alt={}, center]{dfe6db33-33d0-4dff-95f7-fbf097e3963e-2_668_1131_587_466}

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 $\pounds 100$ ?

Time taken (mins)	1	1.5	2	2.5	3
Probability	\(\frac { 1 } { 7 }\)	\(\frac { 2 } { 7 }\)	\(\frac { 2 } { 7 }\)	\(\frac { 1 } { 7 }\)	\(\frac { 1 } { 7 }\)

Time interval (seconds)	2	5	15	25
Probability	\(\frac { 3 } { 7 }\)	\(\frac { 2 } { 7 }\)	\(\frac { 1 } { 7 }\)	\(\frac { 1 } { 7 }\)

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

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