Pearson Edexcel Level 3 GCE
reference9FM0/4D
Paper
Time 1 hour 30 minutes
Further Mathematics
Advanced
PAPER 4D: Decision Mathematics 2
You must have:
Mathematical Formulae and Statistical Tables (Green),
calculator, Decision Mathematics Answer Book (enclosed)
A-Level Pearson Edexcel GCE
AL Further Mathematics (9FM0)
Advanced
Paper 4D Decision Mathematics 2
summer Exam Question Paper
(AUTHENTIC MARKING SCHEME
ATTACHED)
1/1/1/ *P72118A*
P72118A
,1. Four workers, A, B, C and D, are to be assigned to four tasks, 1, 2, 3
and 4. Each task must be assigned to just one worker and each
worker must do only one task.
The cost of assigning each worker to each task is shown in
the table below. The total cost is to be minimised.
1 2 3 4
A 32 45 34 48
B 37 39 50 46
C 46 44 40 42
D 43 45 48 52
(a) Reducing rows first, use the Hungarian algorithm to obtain an
allocation that minimises the total cost. You must make your
method clear and show the table after each stage.
(5)
(b) State the minimum total cost.
(1)
(Total for Question 1 is 6 marks)
2. The general solution of the second order recurrence relation
un + 2 + k1un + 1 + k2un = 0 n 0
is given by
un = (A + Bn)(–3)n
where A and B are arbitrary non-zero constants.
(a) Find the value of k1 and the value of k2
(2)
Given that u0 = u1 = 1
(b) find the value of A and the value of B.
(2)
(Total for Question 2 is 4 marks)
2 P72118A
,3. The table below shows the transport options, usual travel times, possible delay times
and corresponding probabilities of delay for a journey. All times are in minutes.
Transport option Usual travel time Possible delay time Probability of delay
10 0.10
Car 52
25 0.02
15 0.05
Train 45
25 0.03
5 0.05
Coach 55
15 0.01
(a) Draw a decision tree to model the transport options and the possible outcomes.
(5)
(b) State the minimum expected travel time and the corresponding transport option
indicated by the decision tree.
(2)
(Total for Question 3 is 7 marks)
P72118A 3
Turn over
, 4.
C1
A 20 20 E
C2
33 9 22 42
29 1
4 22 53
41 41 30 30
B
S G
10
30 25 12 14
6 14 F 72
C 6 75
5 8 24
C1 17 15 19
C2 T
D 4 4
Figure 1
Figure 1 shows a capacitated, directed network of pipes. The
uncircled number on each arc represents the capacity of the
corresponding pipe. The numbers in circles represent an initial flow.
(a) List the saturated arcs.
(1)
(b) State the value of the initial flow.
(1)
(c) Explain why arc FT cannot be full to capacity.
(1)
(d) State the capacity of cut C1 and the capacity of cut C2
(2)
(e) By inspection find one flow-augmenting route to increase the
flow by three units. You must state your route.
(1)
(f) Prove that, once the flow-augmenting route found in part (e) has
been applied, the flow is maximal.
(3)
(Total for Question 4 is 9 marks)
4 P72118A
reference9FM0/4D
Paper
Time 1 hour 30 minutes
Further Mathematics
Advanced
PAPER 4D: Decision Mathematics 2
You must have:
Mathematical Formulae and Statistical Tables (Green),
calculator, Decision Mathematics Answer Book (enclosed)
A-Level Pearson Edexcel GCE
AL Further Mathematics (9FM0)
Advanced
Paper 4D Decision Mathematics 2
summer Exam Question Paper
(AUTHENTIC MARKING SCHEME
ATTACHED)
1/1/1/ *P72118A*
P72118A
,1. Four workers, A, B, C and D, are to be assigned to four tasks, 1, 2, 3
and 4. Each task must be assigned to just one worker and each
worker must do only one task.
The cost of assigning each worker to each task is shown in
the table below. The total cost is to be minimised.
1 2 3 4
A 32 45 34 48
B 37 39 50 46
C 46 44 40 42
D 43 45 48 52
(a) Reducing rows first, use the Hungarian algorithm to obtain an
allocation that minimises the total cost. You must make your
method clear and show the table after each stage.
(5)
(b) State the minimum total cost.
(1)
(Total for Question 1 is 6 marks)
2. The general solution of the second order recurrence relation
un + 2 + k1un + 1 + k2un = 0 n 0
is given by
un = (A + Bn)(–3)n
where A and B are arbitrary non-zero constants.
(a) Find the value of k1 and the value of k2
(2)
Given that u0 = u1 = 1
(b) find the value of A and the value of B.
(2)
(Total for Question 2 is 4 marks)
2 P72118A
,3. The table below shows the transport options, usual travel times, possible delay times
and corresponding probabilities of delay for a journey. All times are in minutes.
Transport option Usual travel time Possible delay time Probability of delay
10 0.10
Car 52
25 0.02
15 0.05
Train 45
25 0.03
5 0.05
Coach 55
15 0.01
(a) Draw a decision tree to model the transport options and the possible outcomes.
(5)
(b) State the minimum expected travel time and the corresponding transport option
indicated by the decision tree.
(2)
(Total for Question 3 is 7 marks)
P72118A 3
Turn over
, 4.
C1
A 20 20 E
C2
33 9 22 42
29 1
4 22 53
41 41 30 30
B
S G
10
30 25 12 14
6 14 F 72
C 6 75
5 8 24
C1 17 15 19
C2 T
D 4 4
Figure 1
Figure 1 shows a capacitated, directed network of pipes. The
uncircled number on each arc represents the capacity of the
corresponding pipe. The numbers in circles represent an initial flow.
(a) List the saturated arcs.
(1)
(b) State the value of the initial flow.
(1)
(c) Explain why arc FT cannot be full to capacity.
(1)
(d) State the capacity of cut C1 and the capacity of cut C2
(2)
(e) By inspection find one flow-augmenting route to increase the
flow by three units. You must state your route.
(1)
(f) Prove that, once the flow-augmenting route found in part (e) has
been applied, the flow is maximal.
(3)
(Total for Question 4 is 9 marks)
4 P72118A
