■ ■
Further Mathematics
Advanced
PAPER 4D: Decision Mathematics 2
Mathematical Formulae and Statistical Tables (Green), calculator, Decision
Mathematics 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
Use black ink or ball‑point pen.
• Write your answers for this paper in the Decision Mathematics answer book
provided.
Fill in the boxes at the top of the answer book with your name, centre number
and candidate number.
Do not return the question paper with the answer book.
• 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.
A booklet ‘Mathematical Formulae and Statistical Tables’ is provided.
• There are 8 questions in this question paper. The total mark for this paper is 75.
– use this as a guide as to how much time to spend on each question.
• Read each question carefully before you start to answer it.
• Check your answers if you have time at the end. Turn over
,1.
C1
A 2 2 E
15 8 17
x 13 11
13 13
12 12 B 19 7 D 5 5
S T
4
7 10 4
15 5 y
10 12 4 16
16
C 11 8 F
C1
Figure 1
Figure 1 shows a capacitated, directed network of pipes. The numbers in circles
represent an initial flow from S to T. The other number on each arc represents the
capacity, in litres per second, of the corresponding pipe.
(a) (i) State the value of x
(ii) State the value of y
(2)
(b) State the value of the initial flow.
(1)
(c) State the capacity of cut C1
(1)
(d) Find, by inspection, a flow‑augmenting route to increase the flow by four units.
You must state your route.
(1)
The flow‑augmenting route from (d) is used to increase the flow from S to T.
(e) Prove that the flow is now maximal.
(3)
A vertex restriction is now applied so that no more than 12 litres per second can flow
through E.
(f) (i) Complete Diagram 1 in the answer book to show this restriction.
(ii) State the value of the maximum flow through the network with this restriction.
(2)
(Total for Question 1 is 10 marks)
,2. The general solution of the first order recurrence relation
un1 aun bn2 cn d n 0
is given by
un A(3)n 5n2 1
where A is an arbitrary non‑zero constant.
By considering expressions for un+1 and un , find the values of the constants a, b, c and d.
(Total for Question 2 is 3 marks)
, 3. The table below shows the cost, in pounds, of transporting one unit of stock from each
of four supply points, E, F, G and H, to three sales points, A, B and C. It also shows the
stock held at each supply point and the amount required at each sales point.
A minimum cost solution is required.
A B C Supply
E 23 28 22 21
F 26 19 29 32
G 29 24 20 29
H 24 26 19 23
Demand 45 19 23
(a) Explain why it is necessary to add a dummy demand point.
(1)
(b) On Table 1 in the answer book, insert appropriate values in the dummy demand
column, D.
(1)
After finding an initial feasible solution and applying one iteration of the stepping‑stone
method, the table becomes
A B C D
E 21
F 19 13
G 6 23
H 5 18
(c) Starting with GD as the next entering cell, perform two further iterations of the
stepping‑stone method to obtain an improved solution. You must make your method
clear by showing your routes and stating the
• shadow costs
• improvement indices
• entering and exiting cells
(6)
(d) State the cost of the solution found in (c).
(1)
(e) Determine whether the solution obtained in (c) is optimal, giving a reason for
your answer.
(3)
(Total for Question 3 is 12 marks)
Further Mathematics
Advanced
PAPER 4D: Decision Mathematics 2
Mathematical Formulae and Statistical Tables (Green), calculator, Decision
Mathematics 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
Use black ink or ball‑point pen.
• Write your answers for this paper in the Decision Mathematics answer book
provided.
Fill in the boxes at the top of the answer book with your name, centre number
and candidate number.
Do not return the question paper with the answer book.
• 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.
A booklet ‘Mathematical Formulae and Statistical Tables’ is provided.
• There are 8 questions in this question paper. The total mark for this paper is 75.
– use this as a guide as to how much time to spend on each question.
• Read each question carefully before you start to answer it.
• Check your answers if you have time at the end. Turn over
,1.
C1
A 2 2 E
15 8 17
x 13 11
13 13
12 12 B 19 7 D 5 5
S T
4
7 10 4
15 5 y
10 12 4 16
16
C 11 8 F
C1
Figure 1
Figure 1 shows a capacitated, directed network of pipes. The numbers in circles
represent an initial flow from S to T. The other number on each arc represents the
capacity, in litres per second, of the corresponding pipe.
(a) (i) State the value of x
(ii) State the value of y
(2)
(b) State the value of the initial flow.
(1)
(c) State the capacity of cut C1
(1)
(d) Find, by inspection, a flow‑augmenting route to increase the flow by four units.
You must state your route.
(1)
The flow‑augmenting route from (d) is used to increase the flow from S to T.
(e) Prove that the flow is now maximal.
(3)
A vertex restriction is now applied so that no more than 12 litres per second can flow
through E.
(f) (i) Complete Diagram 1 in the answer book to show this restriction.
(ii) State the value of the maximum flow through the network with this restriction.
(2)
(Total for Question 1 is 10 marks)
,2. The general solution of the first order recurrence relation
un1 aun bn2 cn d n 0
is given by
un A(3)n 5n2 1
where A is an arbitrary non‑zero constant.
By considering expressions for un+1 and un , find the values of the constants a, b, c and d.
(Total for Question 2 is 3 marks)
, 3. The table below shows the cost, in pounds, of transporting one unit of stock from each
of four supply points, E, F, G and H, to three sales points, A, B and C. It also shows the
stock held at each supply point and the amount required at each sales point.
A minimum cost solution is required.
A B C Supply
E 23 28 22 21
F 26 19 29 32
G 29 24 20 29
H 24 26 19 23
Demand 45 19 23
(a) Explain why it is necessary to add a dummy demand point.
(1)
(b) On Table 1 in the answer book, insert appropriate values in the dummy demand
column, D.
(1)
After finding an initial feasible solution and applying one iteration of the stepping‑stone
method, the table becomes
A B C D
E 21
F 19 13
G 6 23
H 5 18
(c) Starting with GD as the next entering cell, perform two further iterations of the
stepping‑stone method to obtain an improved solution. You must make your method
clear by showing your routes and stating the
• shadow costs
• improvement indices
• entering and exiting cells
(6)
(d) State the cost of the solution found in (c).
(1)
(e) Determine whether the solution obtained in (c) is optimal, giving a reason for
your answer.
(3)
(Total for Question 3 is 12 marks)
