Paper
� �
Further Mathematics
Advanced Subsidiary
Further Mathematics options
27: Decision Mathematics 1
(Part of options D, F, H and K)
D1 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 D1 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.
,1
16 10 25 30 13 12 28 22 23 20
The list of ten numbers above is to be sorted into ascending order.
(a) Carry out a bubble sort, starting at the left-hand end of the list, to produce the sorted
list. You should only give the state of the list after each pass.
(3)
B 22 E
13 16
12 23
10
A F
C 20
28
30 25
D
Figure 1
(b) Use Prim‟s algorithm, starting at A, to find the minimum spanning tree for the
network in Figure 1. You must list the arcs in the order in which you select them.
(2)
(c) (i) Draw the minimum spanning tree on Diagram 1 in the answer book.
(ii) Find the weight of the minimum spanning tree.
(2)
(Total for Question 1 is 7 marks)
2
■■■■
,2
Activity Immediately preceding activities
A -
B -
C -
D B
E A, B
F B
G B
H C, D
I G, H
J E
K E, F, I
L G, H
M G, H
(a) Explain how you can deduce from the precedence table that at least one dummy
will be needed when drawing the activity network. Your explanation should refer to
specific activities.
(1)
(b) Draw the activity network described in the precedence table, using activity on arc.
Your activity network must contain the minimum number of dummies only.
(5)
Each activity in the precedence table takes 3 hours to complete.
(c) State the minimum completion time.
(1)
One of the activities now needs to be chosen to be extended by an hour. When the
change is made the minimum completion time must not be affected.
(d) List the activities that could be chosen.
(1)
(Total for Question 2 is 8 marks)
3
■■■■ Turn over
, 3
12
B 3 D 6
H
15 4 8 9
9
6 8
A 10 F
C 2 4
14 10 J
E
30 3
G
Figure 2
[The total weight of the network is 153]
Figure 2 models a network of roads in a town, where the nodes represent road
junctions. The numbers on the edges are the times, in minutes, taken to walk along the
corresponding roads.
(a) State, with a reason, whether the graph in Figure 2 is Eulerian, semi-Eulerian
or neither.
(1)
(b) (i) Use Dijkstra‟s algorithm to find the quickest path from A to J.
(ii) State the shortest time needed to walk from A to J.
(6)
Ruby manages road maintenance from her office located at junction B. She needs to
walk along each road at least once, starting and finishing at her office. Road AE is
temporarily blocked so she is unable to walk along it. Ruby wishes to minimise her
journey time.
(c) (i) Use an appropriate algorithm to find the roads that Ruby needs to traverse twice.
You must make your method and working clear.
(ii) Calculate Ruby‟s journey time.
(5)
Ruby can save some time by choosing to start her route from any junction and to finish
at her home at A. Road AE is still blocked and Ruby again wishes to minimise her
journey time.
(d) (i) Write down the junction at which Ruby should start.
(ii) Calculate how much time she would save.
(2)
(Total for Question 3 is 14 marks)
4
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� �
Further Mathematics
Advanced Subsidiary
Further Mathematics options
27: Decision Mathematics 1
(Part of options D, F, H and K)
D1 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 D1 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.
,1
16 10 25 30 13 12 28 22 23 20
The list of ten numbers above is to be sorted into ascending order.
(a) Carry out a bubble sort, starting at the left-hand end of the list, to produce the sorted
list. You should only give the state of the list after each pass.
(3)
B 22 E
13 16
12 23
10
A F
C 20
28
30 25
D
Figure 1
(b) Use Prim‟s algorithm, starting at A, to find the minimum spanning tree for the
network in Figure 1. You must list the arcs in the order in which you select them.
(2)
(c) (i) Draw the minimum spanning tree on Diagram 1 in the answer book.
(ii) Find the weight of the minimum spanning tree.
(2)
(Total for Question 1 is 7 marks)
2
■■■■
,2
Activity Immediately preceding activities
A -
B -
C -
D B
E A, B
F B
G B
H C, D
I G, H
J E
K E, F, I
L G, H
M G, H
(a) Explain how you can deduce from the precedence table that at least one dummy
will be needed when drawing the activity network. Your explanation should refer to
specific activities.
(1)
(b) Draw the activity network described in the precedence table, using activity on arc.
Your activity network must contain the minimum number of dummies only.
(5)
Each activity in the precedence table takes 3 hours to complete.
(c) State the minimum completion time.
(1)
One of the activities now needs to be chosen to be extended by an hour. When the
change is made the minimum completion time must not be affected.
(d) List the activities that could be chosen.
(1)
(Total for Question 2 is 8 marks)
3
■■■■ Turn over
, 3
12
B 3 D 6
H
15 4 8 9
9
6 8
A 10 F
C 2 4
14 10 J
E
30 3
G
Figure 2
[The total weight of the network is 153]
Figure 2 models a network of roads in a town, where the nodes represent road
junctions. The numbers on the edges are the times, in minutes, taken to walk along the
corresponding roads.
(a) State, with a reason, whether the graph in Figure 2 is Eulerian, semi-Eulerian
or neither.
(1)
(b) (i) Use Dijkstra‟s algorithm to find the quickest path from A to J.
(ii) State the shortest time needed to walk from A to J.
(6)
Ruby manages road maintenance from her office located at junction B. She needs to
walk along each road at least once, starting and finishing at her office. Road AE is
temporarily blocked so she is unable to walk along it. Ruby wishes to minimise her
journey time.
(c) (i) Use an appropriate algorithm to find the roads that Ruby needs to traverse twice.
You must make your method and working clear.
(ii) Calculate Ruby‟s journey time.
(5)
Ruby can save some time by choosing to start her route from any junction and to finish
at her home at A. Road AE is still blocked and Ruby again wishes to minimise her
journey time.
(d) (i) Write down the junction at which Ruby should start.
(ii) Calculate how much time she would save.
(2)
(Total for Question 3 is 14 marks)
4
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