Pearson Edexcel GCE In A Level Further Mathematics (9FM0) Paper 4D
Pure Mathematics QP & MS JUNE 2024
, ■ ■
Further Mathematics
Advanced
PAPER 4D: Decision Mathematics 2
Mathematical Formulae and Statistical Tables (Green), calculator, Decision
Mathematics Answer Book (enclosed)
•. Turn over
P72803A
©2024 Pearson Education Ltd.
F:1/1/1/1/1/
,1.
C1
A 2 2 E
x 15 8 17
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 P72803A
■■■■
, 2. The general solution of the first order recurrence relation
un+1 + 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)
P72803A 3
■■■■ Turn over
Pure Mathematics QP & MS JUNE 2024
, ■ ■
Further Mathematics
Advanced
PAPER 4D: Decision Mathematics 2
Mathematical Formulae and Statistical Tables (Green), calculator, Decision
Mathematics Answer Book (enclosed)
•. Turn over
P72803A
©2024 Pearson Education Ltd.
F:1/1/1/1/1/
,1.
C1
A 2 2 E
x 15 8 17
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 P72803A
■■■■
, 2. The general solution of the first order recurrence relation
un+1 + 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)
P72803A 3
■■■■ Turn over
