Paper
� �
Further Mathematics
Advanced Subsidiary
Further Mathematics options
28: Decision Mathematics 2
(Part of option K only)
D2 Answer Book (enclosed)
Candidates may use any calculator allowed by Pearson regulations. Calculators
must not have the facility for symbolic algebra manipulation, differentiation
and integration, or have retrievable mathematical formulae stored in them.
Instructions
• If pencil is used for diagrams/sketches/graphs it must be dark (HB or B).
centre number and candidate number.
Answer all questions and ensure that your answers to parts of questions are
clearly labelled.
Answer the questions in the answer book provided
– there may be more space than you need.
You should show sufficient working to make your methods clear. Answers without
working may not gain full credit.
Inexact answers should be given to three significant figures unless
otherwise stated.
Do not return the question paper with the D2 Answer Book.
Information
• The total mark for this part of the examination is 40. There are 4 questions.
– use this as a guide as to how much time to spend on each question.
Advice
Read each question carefully before you start to answer it.
• Check your answers if you have time at the end.
Turn over
,1. Five workers, A, B, C, D and E, are each to be assigned to one of five tasks, J, K, L,
M and N. Each task must be assigned to exactly one worker and each worker must do
exactly one task.
Worker C cannot do task L and worker D cannot do task K.
The profit, in pounds, that each worker will make while assigned to each task is shown
in the table below.
J K L M N
A 38 33 40 35 32
B 26 24 27 25 23
C 33 29 – 30 27
D 36 – 41 37 33
E 32 27 31 29 25
The Hungarian algorithm is to be used to find the maximum total profit that can be
earned by the five workers.
(a) Explain how the contents of the table must be modified to allow the algorithm to be
used.
(2)
(b) Reducing rows first, use the Hungarian algorithm to obtain the maximum total
profit. You should explain how any initial row and column reductions are made and
also how you determine if the table is optimal at each stage.
(7)
(Total for Question 1 is 9 marks)
2
■■■■
,2. C1
C
31 C2
A 31 G
0 31 39
8
70 20
32 20 21
63 72
45 21 80
T
D
S F
32 36 14 14
20 25
20
21
44
48 6 3
6
6
B
H C1
C2 30
24 24 18
E
Figure 1
Figure 1 shows a capacitated, directed network of pipes. The number on each arc
represents the capacity of the corresponding pipe. The numbers in circles represent a
feasible flow from S to T.
(a) State the two conditions satisfied by a feasible flow.
(2)
(b) List the seven saturated arcs in Figure 1.
(1)
(c) Find the capacity of
(i) cut C1
(ii) cut C2
(2)
(d) Write down a flow-augmenting route that increases the flow by four units.
(1)
(e) Use the answer to part (d) to draw the resulting flow pattern on Diagram 1 in the
answer book.
(2)
(f) Prove that the answer to part (e) is a maximum flow.
(3)
(Total for Question 2 is 11 marks)
3
■■■■ Turn over
, 3. Layla and Mohsin play a zero-sum game represented by the following pay-off matrix
for Layla.
Mohsin plays X Mohsin plays Y Mohsin plays Z
Layla plays P 1 −2 2
Layla plays Q −4 3 −5
Layla plays R −1 1 −3
(a) (i) Find the play-safe strategies for each player.
(ii) State, giving a reason, whether there is a stable solution to this game.
(3)
(b) Option R is now removed from Layla’s choices.
(i) For each of Mohsin’s three options, find the expected pay-off to Layla when she
plays options P and Q equally often.
(1)
(ii) Use a graphical method to determine Layla’s optimal mixed strategy. You should
define any variables you use.
(7)
(Total for Question 3 is 11 marks)
4. A sequence{un}, where n , satisfies the recurrence relation
4un1 2un 3n 5
Given that u1 2
(a) solve the recurrence relation, giving un in terms of n
(6)
(b) hence determine the number of negative terms in the sequence {un}. You must
justify your answer.
(3)
(Total for Question 4 is 9 marks)
TOTAL FOR DECISION MATHEMATICS 2 IS 40 MARKS
4
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� �
Further Mathematics
Advanced Subsidiary
Further Mathematics options
28: Decision Mathematics 2
(Part of option K only)
D2 Answer Book (enclosed)
Candidates may use any calculator allowed by Pearson regulations. Calculators
must not have the facility for symbolic algebra manipulation, differentiation
and integration, or have retrievable mathematical formulae stored in them.
Instructions
• If pencil is used for diagrams/sketches/graphs it must be dark (HB or B).
centre number and candidate number.
Answer all questions and ensure that your answers to parts of questions are
clearly labelled.
Answer the questions in the answer book provided
– there may be more space than you need.
You should show sufficient working to make your methods clear. Answers without
working may not gain full credit.
Inexact answers should be given to three significant figures unless
otherwise stated.
Do not return the question paper with the D2 Answer Book.
Information
• The total mark for this part of the examination is 40. There are 4 questions.
– use this as a guide as to how much time to spend on each question.
Advice
Read each question carefully before you start to answer it.
• Check your answers if you have time at the end.
Turn over
,1. Five workers, A, B, C, D and E, are each to be assigned to one of five tasks, J, K, L,
M and N. Each task must be assigned to exactly one worker and each worker must do
exactly one task.
Worker C cannot do task L and worker D cannot do task K.
The profit, in pounds, that each worker will make while assigned to each task is shown
in the table below.
J K L M N
A 38 33 40 35 32
B 26 24 27 25 23
C 33 29 – 30 27
D 36 – 41 37 33
E 32 27 31 29 25
The Hungarian algorithm is to be used to find the maximum total profit that can be
earned by the five workers.
(a) Explain how the contents of the table must be modified to allow the algorithm to be
used.
(2)
(b) Reducing rows first, use the Hungarian algorithm to obtain the maximum total
profit. You should explain how any initial row and column reductions are made and
also how you determine if the table is optimal at each stage.
(7)
(Total for Question 1 is 9 marks)
2
■■■■
,2. C1
C
31 C2
A 31 G
0 31 39
8
70 20
32 20 21
63 72
45 21 80
T
D
S F
32 36 14 14
20 25
20
21
44
48 6 3
6
6
B
H C1
C2 30
24 24 18
E
Figure 1
Figure 1 shows a capacitated, directed network of pipes. The number on each arc
represents the capacity of the corresponding pipe. The numbers in circles represent a
feasible flow from S to T.
(a) State the two conditions satisfied by a feasible flow.
(2)
(b) List the seven saturated arcs in Figure 1.
(1)
(c) Find the capacity of
(i) cut C1
(ii) cut C2
(2)
(d) Write down a flow-augmenting route that increases the flow by four units.
(1)
(e) Use the answer to part (d) to draw the resulting flow pattern on Diagram 1 in the
answer book.
(2)
(f) Prove that the answer to part (e) is a maximum flow.
(3)
(Total for Question 2 is 11 marks)
3
■■■■ Turn over
, 3. Layla and Mohsin play a zero-sum game represented by the following pay-off matrix
for Layla.
Mohsin plays X Mohsin plays Y Mohsin plays Z
Layla plays P 1 −2 2
Layla plays Q −4 3 −5
Layla plays R −1 1 −3
(a) (i) Find the play-safe strategies for each player.
(ii) State, giving a reason, whether there is a stable solution to this game.
(3)
(b) Option R is now removed from Layla’s choices.
(i) For each of Mohsin’s three options, find the expected pay-off to Layla when she
plays options P and Q equally often.
(1)
(ii) Use a graphical method to determine Layla’s optimal mixed strategy. You should
define any variables you use.
(7)
(Total for Question 3 is 11 marks)
4. A sequence{un}, where n , satisfies the recurrence relation
4un1 2un 3n 5
Given that u1 2
(a) solve the recurrence relation, giving un in terms of n
(6)
(b) hence determine the number of negative terms in the sequence {un}. You must
justify your answer.
(3)
(Total for Question 4 is 9 marks)
TOTAL FOR DECISION MATHEMATICS 2 IS 40 MARKS
4
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