Decision Mathematics - Antony Males3.antonymale.co.uk/Exam Papers and Mark Schemes... · Implement the algorithm given by the flow chart above and ... draw a cascade (Gantt) chart

6689/01

Edexcel GCE

Decision Mathematics

Unit D1 Mock paper

Advanced Subsidiary / Advanced

Time: 1 hour 30 minutes

Materials required for the examination Mathematical Formulae

Items included with these question papers None

Candidates may use any calculator EXCEPT those with a facility for symbolic algebra, differentiation and/or integration. Thus candidates may NOT use calculators such as Texas TI 89, TI 92, Casio CFX 9970G, Hewlett Packard HP 48G. Instructions to Candidates

In the boxes on the Answer Book provided, write the name of the Examining Body (Edexcel), your Centre Number, Candidate Number, the Unit Title (Decision Mathematics D1), the Paper Reference (6689), your surname, other names and signature.

Information for Candidates

Full marks may be obtained for answers to ALL questions. This paper has 7 questions. There are no blank pages.

Advice to Candidates

You must ensure that your answers to parts of questions are clearly labelled. You must show sufficient working to make your methods clear to the Examiner. Answers without working will gain no credit.

UA009018 – AS/Advanced GCE Mathematics: Mock Papers with Mark Schemes (Part 1) Decision Mathematics D1

© 2000 Edexcel Foundation This publication may only be reproduced in accordance with Edexcel copyright policy. Edexcel Foundation is a Registered charity.


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1. This question should be answered on the page opposite.

Start A = 1 A = A + 1

B = A60

No

Is B an integer?

Yes

Print A

Is A = 60?

No

Yes End

Implement the algorithm given by the flow chart above and state what the algorithm actually produces. (5 marks)

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Sheet for use in answering question 1

Is B = A60 an integer?

A Yes or No A Yes or No 1 31 2 32 3 33 4 34 5 35 6 36 7 37 8 38 9 39 10 40 11 41 12 42 13 43 14 44 15 45 16 46 17 47 18 48 19 49 20 50 21 51 22 52 23 53 24 54 25 55 26 56 27 57 28 58 29 59 30 60

What the algorithm produces:

…………………………………………………………………………………………………

…………………………………………………………………………………………………

…………………………………………………………………………………………………


4


A college wishes to staff classes in French (F), German (G), Italian (I), Russian (R) and

Spanish (S). Five language teachers are available – Mr Ahmed (A), Mrs Brown (B), Ms Corrie (C), Dr Donald (D) and Miss Evans (E). The languages they can teach are shown in the table.

Mr Ahmed (A) French and German Mrs Brown (B) French and Italian Ms Corrie (C) Russian Dr Donald (D) Russian and Spanish Miss Evans (E) French and Spanish

The first three teachers were allowed to choose their preferences. Their choices were:

Mr Ahmed (A) – French (F) Dr Donald (D) – Russian (R) Miss Evans (E) – Spanish (S)

(a) Using the dots on the answer sheet, draw a bipartite graph to show the information in the table. Indicate the above choices in a distinctive way. (1 mark) (b) Using your answer to part (a) as the initial matching, apply the maximum matching algorithm to obtain a complete matching. Alternating paths and the final matching should be stated. (5 marks)

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(a) A • • French (F) B • • German (G) C • • Italian (I) D • • Russian (R) E • • Spanish (S) (b) Alternating paths...…………………………………………………………………………

…………………………………………………………………………………………………

…………………………………………………………………………………………………

…………………………………………………………………………………………………

…………………………………………………………………………………………………

(c) Complete matching………………………………………………………………………...

…………………………………………………………………………………………………


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3. C 10 15 A E 35 28 25 30 S D T 20 45 14 45 B F 28

Fig. 1

Figure 1 shows a capacitated network. The number on each arc indicates the capacity of the arc. (a) State the maximum flow along SADET…………………………………………………….. ………………………………………………………………………………………...(1 mark)

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C A E S D T

B F

Fig. 2

Figure 2 shows a feasible flow of value 72 through the same network. (b) Explaining your reasoning carefully, find the value of the flows x, y z and t. ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………(5 marks) (c) Explain why 72 is not the maximum flow………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ……………………..…………………………………………………………………(2 marks)

28

10 10

t 20 z

42 45 yx

30


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4. In the course of an investigation the following linear programming problem arose.

Maximise P = x – y + 3z

subject to 3x + 5y + z ≤ 6,

4x + y + z ≤ 5,

x – y + z ≤ 3,

x ≥ 0, y ≥ 0, z ≥ 0 .

The Simplex algorithm is to be used to solve the problem. (a) Explaining the purpose of r, s and t, show that the initial tableau can be written as Basic Variable x y z r s t Value

r 3 5 1 1 0 0 6 s 4 1 1 0 1 0 5 t 1 −1 1 0 0 1 3 P −1 1 −3 0 0 0 0

(3 marks)

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(b) Solve the problem using the Simplex algorithm. Start by increasing z for your first iteration and continue by increasing y. (9 marks)


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5. B 34 A 30 6 9 60 D 7

E 25 35 C

Fig. 3

Figure 3 shows the paths in Bill’s garden. The length of each path is given in metres. Each morning Bill likes to inspect the garden. He starts and finishes at A and traverses each path at least once. (a) Use the route inspection algorithm to find the minimum distance he must walk. State which paths must be traversed more than once and state a route of minimum length.

(6 marks)

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Caroline, a friend of Bill’s, wishes to look at the garden covering each path at least once. However, she wants to start at B and finish at a vertex other than B. (b) In order for Caroline to walk a minimum distance, determine where she should end her walk and the distance she will cover. Explain your method carefully and give a possible route. (7 marks)


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B

20 12 10 15 14 36 28 10 30 30

A

F

K

D

C 815

E

Fig. 4

Figure 4 shows a number of satellite towns A, B, C, D, E and F surrounding a city K. The number on each edge give the length of the road in km. (a) Use Dijkstra’s algorithm to find the shortest route from A to E in the network. Show your working in the boxes provided on the answer sheet. (8 marks) It is planned to link all the sites A, B, C, D, E and F and K by telephone lines laid alongside the roads. (b) Use Kruskal’s algorithm to find a minimum spanning tree for the network and hence obtain the minimum total length of cable required. Draw your tree. (6 marks)

UA009018 – AS/Advance

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(a) 20

A

B

15 8C

Length of Shortes Determination of …..……………… ……..…………… …..……………… ……..…………… (b) Minimum span Total weight of m

T

d GCE Mathematics: Mock Papers with Mark Schemes (Part 1) Decision Ma

12 10 15 14

36 28

10 30 30

KEY:

E

D

F

K

t Path: …………………………………………………………………

shortest route: …………………………………………………………

………………………………………………………………………

………………………………………………………………………

………………………………………………………………………

………………………………………………………………………

ning tree

inimum spanning tree: ………………………………………………

Vertex Permanent Label

Temporary Labels

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thematics D1

……

……

………

………

………

………

………


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7. This question should be answered on the pages opposite and overleaf.

E (10) D (18) H (12) F (18) A (30) 36 C (20) I (10) B (52) G (15)

7

3

5

62

4

1

Fig. 5

A building project is modelled by the activity network shown in Fig. 5. The activities involved in the project are represented by the arcs. The numbers in brackets on each arc gives the time, in days, taken to complete the activity. (a) Complete the boxes on the answer sheet by calculating the earliest and latest event times. (4 marks) (b) Hence write down the critical activities and the length of the critical path. (2 marks) (c) Obtain the total float for each non-critical activity. (2 marks) (d) On the grid on the answer sheet, draw a cascade (Gantt) chart showing the information found in parts (b) and (c). (3 marks) Given that each activity requires one worker, (e) draw up a schedule to determine the minimum number of workers needed to complete the project in the critical time. (3 marks) Due to unforeseen circumstances, activity C takes 30 days rather than 20 days. (f ) Determine how this affects the length of the critical path and state the critical activities now. (3 marks)

UA009018 – AS/Advanced GCE Mathematics: M

15Sheet for use in answering question 7 D (18) A (30) 36

C B (52)

1 2

(b) Length of critical path: ……… Critical activities: ……………… (c) Floats on critical activities:…… …………………………………… ……………………………………

ock Papers with Mark Schemes (Part 1)

E (10)

H (12) F (18)

(20) I (10) G (15)

KEY:

4

6

5

3

7

EE

………………………………

………………………………

………………………………

………………………………

………………………………

arliest Latest

vent Time Event Time

Decisi

…………………

………………

…………………

…………………

…………………

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on Mathematics D1

…………

…………..

…………

…………

…………..


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(d)

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90

(e)

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90

(f ) ………………………………………………………………………………………………

…………………………………………………………………………………………………..

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END

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Decision Mathematics - Antony Males3.antonymale.co.uk/Exam Papers and Mark Schemes... · Implement the algorithm given by the flow chart above and ... draw a cascade (Gantt) chart