1.
(a)
A path is a (finite) sequence of edges, such that the end vertex
of one edge is the start vertex of the next and in which no vertex
appears more than once/no cycles.
B2,1,0
B2 A good, complete answer
B1 Close – mostly there. “Bad” sets B1, “route” series may be
OK
(b)
2
M1, A1, A1, A1ft
Shortest path: ABDFGI
length: 108 miles
A1, A1ft
M1 In D, F, G, H, or I working values large replaced by small
A1 A, B, C, E corrects labels in a rising sequence
6
A1
rrect
ls
ft
D,
F
labe
co

penalis
rder
bellin
ly
on
o

A1ft
, t
I
ft
labels
correc
G,
H

A1 Path c.a.o.
A1ft Length ft from I – accept 108 if a correct path
(c)
e.g.
108 – 21 = 87
87 – 15 = 72
GI
FG
72 – 21 = 51
DF
or
trace back from I
include arc XY if Y
already on the path
and if the difference
in final labels equals the
length of arc
51 – 28 = 23
BD
23 – 23 = 0
AB
B2ft1ft0
B2ft complete version of one of the two given explanations
B1ft all there bar one step “bad” sets B1 – easy mark
Ashbourne Independent School
2
1
(d)
ABEDFGI
length: 118 miles
M1 Route A to I including E
A1 c.a.o.
M1 A1
2
[12]
2.
(a)
(b)
AC + DF = 8 + 9 = 17 
AD + CF = 15 + 16 = 31
AF + CD = 13 + 7 = 20
length = 77 + 17 = 94 km
M1 3 pairs of 4 odd vertices (different) A C D F
A1 2 pairs + “total” correct
A1 all 3 pairs + totals correct 17 31 20
M1 77 + their shortest or plausible list
A1ft cao +km
M1 A1 A1
3
M1 A1ft
2
B2,1,0
2
Shortest arc is CD (7) so use A and F as end points
B2 CD identified as the smallest or arc to be repeated
and A + F stated as end points
B1 either CD identified as the smallest or arc to be
repeated
or A + F stated as end points “bad” gets B1
or picks smallest out of least 4 routes
[7]
3.
(a)
e.g.
M1 A1 A1 A1
2
5
I
A
1
E
4
D
3
C
H
J
6
K
F
M
B
4
G
7
L
1
1
8
9
1
0
N
M1 6 activities + 1 dummy, activity on arc. Condone lack of
events throughout
A1 A – F + arcs + 1 start
A1 G – K + dummy “7 – 8” (ignore label on dummy) + arrow
on dummy + other arcs (penalise once only)
A1 L – N + dummy + 1 finish “10–11” Note: dummy may be M
or N (ignore label on dummy) + arrows on dummy+ other arcs
(penalise once only)
Ashbourne Independent School
2
(b)
Reference to K, J, G and L – K depends on J and G,
but L depends on G only
Both M and N must be uniquely represented in terms of events.
B2 complete + clear K, J, G, L referred to explanation
clear + correct
B1 nearly there. “Bad” gets B1. All there but confused
explanation/vague. K J G L referred to
B1 unique representation (o.e.) start + finish at the
same events.
B2,1,0
B1
3
[7]
4.
(a)
Ashbourne Independent School
M1A1
2
M1A1
2
3
(b)
A1
(c)
Float on D = 21 – 5 – 14 = 2
Float on F = 42 – 20 – 14 = 8
(d)
Gantt Chart
(e)
Day 15:
C
Day 25:
G, H, E, F
1
B1ft
M1A1 ft
3
B4
4
B1
B2,1,0
3
[15]
Ashbourne Independent School
4