Pearson Edexcel International Advanced Level

Monday 20 January 2020

Afternoon (Time: 1 hour 30 minutes) Paper Reference WDM11/01

Mathematics
International Advanced Subsidiary/Advanced Level
Decision Mathematics D1

You must have:

Decision Mathematics Answer Book (enclosed), calculator

Candidates may use any calculator permitted 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
•• Use black ink or ball-point pen.
If pencil is used for diagrams/sketches/graphs it must be dark (HB or B).
• Write your answers for this paper in the Decision Mathematics answer book
provided.
• centre
Fill in the boxes at the top of the answer book with your name,
number and candidate number.
•• Answerreturn
Do not the question paper with the answer book.
all questions and ensure that your answers to parts of questions are
clearly labelled.
• Answer the questions in the spaces 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.
Information
•• There are 7 questions in this question paper. The total mark for this paper is 75.
The marks for each question are shown in brackets
– 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.
Try to answer every question.
•• Check your answers if you have time at the end.
If you change your mind about an answer, cross it out and put your new answer
and any working underneath.
Turn over

P60708A
©2020 Pearson Education Ltd.

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 Write your answers in the D1 answer book for this paper.

1. The table below shows the distances, in km, between six data collection points, A, B, C, D, E and F.

A B C D E F
A – 35 42 55 48 50
B 35 – 40 49 52 31
C 42 40 – 47 53 49
D 55 49 47 – 39 44
E 48 52 53 39 – 52
F 50 31 49 44 52 –

Ferhana must visit each data collection point. She will start and finish at A and wishes to minimise
the total distance she travels.

(a) Starting at A, use the nearest neighbour algorithm to obtain an upper bound for the distance
Ferhana must travel. Make your method clear.
(2)

(b) Starting by deleting B, and all of its arcs, find a lower bound for the distance Ferhana must
travel. Make your calculation clear.
(3)

(Total 5 marks)

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 2. C
B 19
A 32

25 34 29
30
22 E 15 F
D 30

32 13 17 11
23

G 18 H 20 J

Figure 1

(a) Define the terms

(i) tree,

(ii) minimum spanning tree.

(3)

(b) Use Kruskal’s algorithm to find the minimum spanning tree for the network shown in Figure 1.
You must clearly show the order in which you consider the edges. For each edge, state whether
or not you are including it in the minimum spanning tree.
(3)

(c) Draw the minimum spanning tree using the vertices given in Diagram 1 in the answer book and
state the weight of the minimum spanning tree.
(2)

(Total 8 marks)

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 3.

4 D(x) 24

10 24

A(4) E(7) J(9)

I(7)

0 7 F(5) z K(10) 33

0 B(5) y 17 33
L(11)

C(7) G(9)
28
7 17

H(10) M(9) 33
7 17

Figure 2

The network in Figure 2 shows the activities that need to be undertaken by a company to complete
a project. Each activity is represented by an arc and the duration, in days, is shown in brackets.
Each activity requires one worker. The early event times and late event times are shown at each
vertex.

The total float on activity D is twice the total float on activity E.

(a) Find the values of x, y and z.

(3)

(b) Draw a cascade chart for this project on Grid 1 in the answer book.
(4)

(c) Use your cascade chart to determine a lower bound for the minimum number of workers
needed to complete the project in the shortest possible time. You must make specific reference
to time and activities. (You do not need to provide a schedule of the activities.)
(2)

(Total 9 marks)

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 4.
35 17 10 7 28 23 41 15 20 29

(a) Use the first-fit bin packing algorithm to determine how the numbers listed above can be
packed into bins of size 60
(3)

(b) The list of numbers is to be sorted into descending order. Use a quick sort to obtain the sorted
list. You should show the result of each pass and identify your pivots clearly.
(4)

(c) Use the first-fit decreasing bin packing algorithm on your ordered list to pack the numbers into
bins of size 60
(3)

The ten distinct numbers below are to be sorted into descending order.

20 24 17 26 8 15 x y 19 12

A bubble sort, starting at the left-hand end of the list, is to be used to obtain the sorted list.

After the second complete pass the list is

24 26 20 17 15 y 19 12 x 8

(d) Find the constraints on the values of x and y.

(3)
(Total 13 marks)

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 5.

Activity Immediately preceding activities

A –
B –
C –
D A
E C
F A, B, C
G A, B, C
H D, F, G
I A, B, C
J D, F, G
K H
L D, E, F, G, I

(a) Draw the activity network described in the precedence table above, using activity on arc.
Your activity network must contain only the minimum number of dummies.
(5)

Given that all critical paths for the network include activity H,

(b) state which activities cannot be critical.

(2)

(Total 7 marks)

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 6. A 8 E

5 20
14 2
C
7 H
28
B
42
17
8 21 2
4 F 20
D
11
19

G 10 J
31
[The total weight of the network is 269]

Figure 3

Figure 3 models a network of roads. The number on each edge gives the time taken, in minutes, to
travel along the corresponding road.

(a) Use Dijkstra’s algorithm to find the shortest time needed to travel from A to J. State the
quickest route.
(6)

Alan needs to travel along all the roads to check that they are in good repair. He wishes to complete
his route as quickly as possible and will start at his home, H, and finish at his workplace, D.

(b) By considering the pairings of all relevant nodes, find the arcs that will need to be traversed
twice in Alan’s inspection route from H to D. You must make your method and working clear.
(5)
For Alan’s inspection route from H to D

(c) (i) state the number of times vertex C will appear,

(ii) state the number of times vertex D will appear.

(2)

(d) Determine whether it would be quicker for Alan to start and finish his inspection route at H,
instead of starting at H and finishing at D. You must explain your reasoning and show all your
working.
(2)

(Total 15 marks)

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 7. A school is planning to run several training days next year for its new teachers, middle leaders and
senior leaders.

Next year, the school will need to run

• at least 20 training days in total,
• at most twice as many training days for new teachers when compared to the total number
of training days required for both middle and senior leaders,
• at most 25% of the training days for senior leaders.

The costs of running a training day for new teachers, middle leaders and senior leaders are £400,
£550 and £750 respectively.

The school wants to minimise the total cost of running the training days.

Let x be the number of training days required for new teachers.

Let y be the number of training days required for middle leaders.
Let z be the number of training days required for senior leaders.

(a) Formulate this information as a linear programming problem. State the objective and list the
constraints as simplified inequalities with integer coefficients.
(6)
The school decides that the number of training days for middle leaders and the number of training
days for senior leaders should be in the ratio 5 : 3
This reduces two of the constraints to 5x - 16y and 4y - 5x.

(b) (i) Express the constraint representing the requirement for a total of at least 20 training days
as a simplified inequality with integer coefficients in terms of x and y only.

(ii) Express the objective in the form ax + by where a and b are integers.
(3)

(c) Represent the constraints in x and y only on Diagram 1 in the answer book. Hence determine,
and label, the feasible region, R.
(4)

(d) Use the objective line method to locate the optimal vertex, V, of the feasible region. You must
make your objective line clear and label the optimal vertex, V.
(3)

(e) Hence, determine

(i) the total cost of running the training days,

(ii) the number of training days required for senior leaders.

(2)

(Total 18 marks)
TOTAL FOR PAPER: 75 MARKS
END

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 Please check the examination details below before entering your candidate information
Candidate surname Other names

Centre Number Candidate Number

Pearson Edexcel
International
Advanced Level

Monday 20 January 2020

Afternoon (Time: 1 hour 30 minutes) Paper Reference WDM11/01

Mathematics
International Advanced Subsidiary/Advanced Level
Decision Mathematics D1

Answer Book Total Marks

Do not return the question paper with the answer book.

Turn over

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1.
A B C D E F
A – 35 42 55 48 50
B 35 – 40 49 52 31
C 42 40 – 47 53 49
D 55 49 47 – 39 44
E 48 52 53 39 – 52
F 50 31 49 44 52 –

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A B C D E F
A – 35 42 55 48 50
B 35 – 40 49 52 31
C 42 40 – 47 53 49
D 55 49 47 – 39 44
E 48 52 53 39 – 52
F 50 31 49 44 52 –

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2.
C
B 19
A 32

25 34 29
30
22 E 15 F
D 30

32 13 17 11
23

G 18 H 20 J

Figure 1

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B C
A

E F
D

G H J

Diagram 1

Weight of the minimum spanning tree ____________________________________________ Q2

(Total 8 marks)

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3.
4 D(x) 24

10 24

A(4) E(7) J(9)

I(7)

0 7 F(5) z K(10) 33

0 B(5) y 17 33
L(11)

C(7) G(9)
Key:
28
7 17
Early
event M(9) 33
7 H(10) 17
time
Late
event Figure 2
time

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0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34

Grid 1
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(Total 9 marks)

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35 17 10 7 28 23 41 15 20 29
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(Total 13 marks)

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(Total 7 marks)

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 6.

14
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A 1 0 8 E

14 2 20
5
B 7 C 28 H
42

8 17
21
2
F
D 4
20
Key:

11 Order of Final
Vertex
19 labelling value
G J
Working values
10
31

*P60708A01420*
Shortest time: _______________________________________________________________

Quickest route: ______________________________________________________________

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(Total 15 marks)

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y
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14

12

10

0
x
0 5 10 15 20 25 30

Diagram 1

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(Total 18 marks)
TOTAL FOR PAPER: 75 MARKS
END
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