General Certificate of Education
January 2007
Advanced Subsidiary Examination
MATHEMATICS
Unit Decision 1
Tuesday 16 January 2007
MD01
9.00 am to 10.30 am
For this paper you must have:
*
an 8-page answer book
*
the blue AQA booklet of formulae and statistical tables
*
an insert for use in Questions 6 and 7 (enclosed).
You may use a graphics calculator.
Time allowed: 1 hour 30 minutes
Instructions
Use blue or black ink or ball-point pen. Pencil or coloured pencil should only be used for
drawing.
Write the information required on the front of your answer book. The Examining Body for this
paper is AQA. The Paper Reference is MD01.
Answer all questions.
Show all necessary working; otherwise marks for method may be lost.
The final answer to questions requiring the use of calculators should be given to three significant
figures, unless stated otherwise.
Fill in the boxes at the top of the insert.
*
*
*
*
*
*
Information
The maximum mark for this paper is 75.
The marks for questions are shown in brackets.
*
*
P91441/Jan07/MD01 6/6/6/
MD01
2
Answer all questions.
1 The following network shows the lengths, in miles, of roads connecting nine villages.
C
E
14
8
B
9
12
D
13
12
F
5.5
12
11
A
15
17
21
9
G
I
16.5
H
(a) Use Prim’s algorithm, starting from A , to find a minimum spanning tree for the
network.
(5 marks)
(b) Find the length of your minimum spanning tree.
(c) Draw your minimum spanning tree.
(1 mark)
(3 marks)
(d) State the number of other spanning trees that are of the same length as your answer in
part (a).
(1 mark)
P91441/Jan07/MD01
3
2 Five people A , B , C , D and E are to be matched to five tasks R , S , T , U and V .
The table shows the tasks that each person is able to undertake.
Person
Tasks
A
R, V
B
R, T
C
T, V
D
U, V
E
S, U
(a) Show this information on a bipartite graph.
(2 marks)
(b) Initially, A is matched to task V , B to task R , C to task T, and E to task U .
Demonstrate, by using an alternating path from this initial matching, how each person
can be matched to a task.
(4 marks)
3 Mark is driving around the one-way system in Leicester. The following table shows the
times, in minutes, for Mark to drive between four places: A , B , C and D . Mark decides to
start from A , drive to the other three places and then return to A .
Mark wants to keep his driving time to a minimum.
To
A
B
C
D
A
–
8
6
11
B
14
–
13
25
C
14
9
–
17
D
26
10
18
–
From
(a) Find the length of the tour ABCDA .
(2 marks)
(b) Find the length of the tour ADCBA .
(1 mark)
(c) Find the length of the tour using the nearest neighbour algorithm starting from A .
(4 marks)
(d) Write down which of your answers to parts (a), (b) and (c) gives the best upper bound
for Mark’s driving time.
(1 mark)
P91441/Jan07/MD01
s
Turn over
4
4
(a) A student is using a bubble sort to rearrange seven numbers into ascending order.
Her correct solution is as follows:
Initial list
18
17
13
26
10
14
24
After 1st pass
17
13
18
10
14
24
26
After 2nd pass
13
17
10
14
18
24
26
After 3rd pass
13
10
14
17
18
24
26
After 4th pass
10
13
14
17
18
24
26
After 5th pass
10
13
14
17
18
24
26
Write down the number of comparisons and swaps on each of the five passes.
(6 marks)
(b) Find the maximum number of comparisons and the maximum number of swaps that
might be needed in a bubble sort to rearrange seven numbers into ascending order.
(2 marks)
P91441/Jan07/MD01
5
5 A student is using the following algorithm with different values of A and B .
(a)
Line 10
Input A, B
Line 20
Let C ¼ 0 and let D ¼ 0
Line 30
Let C ¼ C þ A
Line 40
Let D ¼ D þ B
Line 50
If C ¼ D then go to Line 110
Line 60
If C > D then go to Line 90
Line 70
Let C ¼ C þ A
Line 80
Go to Line 50
Line 90
Let D ¼ D þ B
Line 100
Go to Line 50
Line 110
Print C
Line 120
End
(i) Trace the algorithm in the case where A ¼ 2 and B ¼ 3 .
(3 marks)
(ii) Trace the algorithm in the case where A ¼ 6 and B ¼ 8 .
(3 marks)
(b) State the purpose of the algorithm.
(1 mark)
(c) Write down the final value of C in the case where A ¼ 200 and B ¼ 300 .
(1 mark)
Turn over for the next question
P91441/Jan07/MD01
s
Turn over
6
6 [Figure 1, printed on the insert, is provided for use in this question.]
Dino is to have a rectangular swimming pool at his villa.
He wants its width to be at least 2 metres and its length to be at least 5 metres.
He wants its length to be at least twice its width.
He wants its length to be no more than three times its width.
Each metre of the width of the pool costs £1000 and each metre of the length of the pool
costs £500.
He has £9000 available.
Let the width of the pool be x metres and the length of the pool be y metres.
(a) Show that one of the constraints leads to the inequality
2x þ y 4 18
(1 mark)
(b) Find four further inequalities.
(3 marks)
(c) On Figure 1, draw a suitable diagram to show the feasible region.
(6 marks)
(d) Use your diagram to find the maximum width of the pool. State the corresponding
length of the pool.
(3 marks)
P91441/Jan07/MD01
7
7 [Figure 2, printed on the insert, is provided for use in this question.]
The network shows the times, in seconds, taken by Craig to walk along walkways connecting
ten hotels in Las Vegas.
130
Circus (C)
Venetian (V)
75
300
150
Aladin (A)
220
60
Imperial (I)
Stardust (S)
80
100
100
Bellagio (B)
120
95
MGM (M)
100
New York (N)
250
150
150
80
Oriental (O)
Luxor (L)
120
The total of all the times in the diagram is 2280 seconds.
(a)
(i) Craig is staying at the Circus (C ) and has to visit the Oriental (O).
Use Dijkstra’s algorithm on Figure 2 to find the minimum time to walk from
C to O .
(6 marks)
(b)
(ii) Write down the corresponding route.
(1 mark)
(i) Find, by inspection, the shortest time to walk from A to M .
(1 mark)
(ii) Craig intends to walk along all the walkways. Find the minimum time for Craig
to walk along every walkway and return to his starting point.
(6 marks)
Turn over for the next question
P91441/Jan07/MD01
s
Turn over
8
8
(a) The diagram shows a graph G with 9 vertices and 9 edges.
(i) State the minimum number of edges that need to be added to G to make a
connected graph. Draw an example of such a graph.
(2 marks)
(ii) State the minimum number of edges that need to be added to G to make the
graph Hamiltonian. Draw an example of such a graph.
(2 marks)
(iii) State the minimum number of edges that need to be added to G to make the
graph Eulerian. Draw an example of such a graph.
(2 marks)
(b) A complete graph has n vertices and is Eulerian.
(i) State the condition that n must satisfy.
(1 mark)
(ii) In addition, the number of edges in a Hamiltonian cycle for the graph is the same
as the number of edges in an Eulerian trail. State the value of n .
(1 mark)
END OF QUESTIONS
Copyright Ó 2007 AQA and its licensors. All rights reserved.
P91441/Jan07/MD01
2
Figure 1 (for use in Question 6)
y m
20 –
15 –
10 –
5–
5
10
m
–
P91441/Jan07/MD01
–
–
0–
0
x
3
Figure 2 (for use in Question 7)
130
Circus (C)
Venetian (V)
75
300
150
Aladin (A)
220
60
Imperial (I)
Stardust (S)
80
100
100
Bellagio (B)
120
95
MGM (M)
100
250
New York (N)
150
150
80
Oriental (O)
120
Luxor (L)
P91441/Jan07/MD01
MD01 - AQA GCE Mark Scheme 2007 January series
Key to mark scheme and abbreviations used in marking
M
m or dM
A
B
E
or ft or F
CAO
CSO
AWFW
AWRT
ACF
AG
SC
OE
A2,1
–x EE
NMS
PI
SCA
mark is for method
mark is dependent on one or more M marks and is for method
mark is dependent on M or m marks and is for accuracy
mark is independent of M or m marks and is for method and accuracy
mark is for explanation
follow through from previous
incorrect result
correct answer only
correct solution only
anything which falls within
anything which rounds to
any correct form
answer given
special case
or equivalent
2 or 1 (or 0) accuracy marks
deduct x marks for each error
no method shown
possibly implied
substantially correct approach
MC
MR
RA
FW
ISW
FIW
BOD
WR
FB
NOS
G
c
sf
dp
mis-copy
mis-read
required accuracy
further work
ignore subsequent work
from incorrect work
given benefit of doubt
work replaced by candidate
formulae book
not on scheme
graph
candidate
significant figure(s)
decimal place(s)
No Method Shown
Where the question specifically requires a particular method to be used, we must usually see evidence of use of this
method for any marks to be awarded. However, there are situations in some units where part marks would be
appropriate, particularly when similar techniques are involved. Your Principal Examiner will alert you to these and
details will be provided on the mark scheme.
Where the answer can be reasonably obtained without showing working and it is very unlikely that the correct
answer can be obtained by using an incorrect method, we must award full marks. However, the obvious penalty to
candidates showing no working is that incorrect answers, however close, earn no marks.
Where a question asks the candidate to state or write down a result, no method need be shown for full marks.
Where the permitted calculator has functions which reasonably allow the solution of the question directly, the correct
answer without working earns full marks, unless it is given to less than the degree of accuracy accepted in the mark
scheme, when it gains no marks.
Otherwise we require evidence of a correct method for any marks to be awarded.
Jan 07
3
MD01 - AQA GCE Mark Scheme 2007 January series
MD01
Q
Solution
5.5
1(a) AB
BC
8
AI
9
BD
13
DE
9
DG
11
DF, EF, GF 12
IH
16.5
(b) 84
Marks
B1
M1
A1
A1
Total
A1
5
B1
1
Comments
8 edges
SCA
AI 3rd
BD 4th
All correct
(c)
(d) 2
M1
B1
A1
3
B1
1
Total
10
4
Minimum spanning tree
8 edges
All correct including labelling
(or including DF or GF instead of EF)
MD01 - AQA GCE Mark Scheme 2007 January series
MD01 (cont)
Q
2(a)
Solution
Marks
Total
Bipartite graph
M1
A1
(b) Start with D (or S)
D − U −/ E − S
or
D − V −/ A − R −/ B − T −/ C
−V −/ D − U −/ E − S
B1
B C D A
8 13 17 26
= 64
D
11
(c) A
C
18
C
6
9
B
9
4
Must be 5 pairs
6
4 numbers (either part)
M1
2
= 52
A1
1
A
M1
M1
A1
B1
A
14
D
25
For attempt at any path
A1
B
All correct
A1
Total
(b) A
2
B1
M1
Match:
AV, BR, CT, DU, ES
or
AR, BT, CV, DU, ES
3(a) A
Comments
26
= 66}
Tour
Visits every vertex
Correct order
4
Alternative if matrix used:
M1 3 numbers
all different rows
M1 4th number
and columns
A1 correct numbers
B1 66
(d) 52 (their lowest of (a), (b), (c))
B1F
Total
5
1
8
Allow “ part (b) ”
MD01 - AQA GCE Mark Scheme 2007 January series
MD01 (cont)
Q
4(a)
Solution
Comparisons
6
5
4
3
2
Swaps
5
3
2
1
0
Marks
Total
B1B1
B1B1
B1
B1
6
B1
B1
(b) 21
21
Total
5(a)(i)
(A )
2
(B )
3
C
0
2
D
0
3
4
6
--------
--------
(A )
6
(B )
8
(ii)
6
--------
C
0
6
Comments
Other 3 comparisons
Other 3 swaps. Ignore 6th pass
2
8
M1
SCA: as far as D = 3
A1
For 4
A1
3
All correct
--------
D
0
8
12
M1
SCA: as far as D = 8
A1
For 12
16
18
24
---------
---------
24
---------
A1
3
All correct
B1
1
Allow lowest common denominator
B1
1
8
---------
(b) Find LCM
(c) 600
Total
6
MD01 - AQA GCE Mark Scheme 2007 January series
MD01 (cont)
Q
Solution
6(a) 1000 x + 500 y ≤ 9000
Marks
B1
Total
1
Comments
( 2 x + y ≤ 18)
(b)
x ≥ 2, y ≥ 5
y ≥ 2x
y ≤ 3x
B1
B1
B1
3
−1 for strict inequalities
−1 for ‘w’s and ‘l’s
(c)
B1
x = 2, y = 5
B1
2 x + y = 18
M1
Line y = mx
A1
y = 2x
A1
y = 3x
B1
(d) Considering an extreme point on their f.r.
x = 4.5
y=9
Total
M1
A1
A1
7
6
Feasible region
Extreme point - vertex
3
13
MD01 - AQA GCE Mark Scheme 2007 January series
MD01 (cont)
Q
7(a)(i)
Solution
Marks
(ii) CASINO
(b)(i)
A → M = 255
(ii) Odds ( C, A, S, M )
CA + SM = 270
CS + AM = 390
CM + AS = 390
Min 2280 + 270
= 2550
Total
Comments
M1
SCA
M1
4 values at I
M1
2 values at M
M1
2 values at O
A1
All correct
B1
6
B1
1
B1
1
465 at O
Or ONISAC
M1
PI
A3
M1
A1
(–1 EE)
2280 + their best pairing
SC 2/6 for answer 2550 with no working
Total
8
6
14
MD01 - AQA GCE Mark Scheme 2007 January series
MD01 (cont)
Q
8(a)(i) 2
Solution
(ii) 3
Marks
B1
Total
Comments
B1
2
OE
2
OE
2
OE
B1
B1
(iii) 3
B1
B1
SC
4
OE
B1(must have number and diagram)
(b)(i) n is odd
B1
1
(ii) 3 (only)
B1
1
8
75
Total
TOTAL
9
AQA January Examinations 2007
Scaled Mark Component Grade Boundaries (GCE Specifications)
Generally, scaled mark boundaries are the same as raw mark boundaries as there is no scaling of marks.
However, they may be different if a unit of assessment consists of more than one component.
Component
Code
LAT1
LAT2
LAT3
LAT4
LAT5
LAT6
LAW1
LAW2
LAW3
LAW4
LAW5
MD01
MD02
MFP1
MFP2
MFP3
MFP4
MM1A/W
MM1A/C
MM1B
MM2A/W
MM2A/C
Maximum
Scaled Mark
35
50
50
35
70
50
A
24
29
29
21
45
32
LAW UNIT 1
LAW UNIT 2
LAW UNIT 3
LAW UNIT 4
LAW UNIT 5
65
65
65
85
85
48
45
44
60
58
43
40
39
55
54
38
36
34
50
50
33
32
29
45
47
29
28
25
40
44
MATHEMATICS UNIT MD01
MATHEMATICS UNIT MD02
MATHEMATICS UNIT MFP1
MATHEMATICS UNIT MFP2
MATHEMATICS UNIT MFP3
MATHEMATICS UNIT MFP4
MECHANICS 1A - WRITTEN
MECHANICS 1A - COURSEWORK
MATHEMATICS UNIT MM1B
MECHANICS 2A - WRITTEN
MECHANICS 2A - COURSEWORK
75
75
75
75
75
75
75
25
75
75
25
61
60
60
61
61
61
60
20
60
61
20
53
52
53
53
53
53
51
18
51
54
18
45
44
46
45
45
45
44
15
43
46
15
38
37
39
37
38
38
36
13
35
40
13
31
30
32
29
31
31
29
10
27
34
10
Component Title
LATIN UNIT 1
LATIN UNIT 2
LATIN UNIT 3
LATIN UNIT 4
LATIN UNIT 5
LATIN UNIT 6
Page 9 of 13
Scaled Mark Grade Boundaries
B
C
D
21
18
15
26
23
20
26
23
20
18
16
14
39
33
28
29
26
23
E
12
17
18
12
23
20