JANUARY 2003
AS/A LEVEL MATHEMATICS
NUMERICAL ANSWERS
6405 Pure Mathematics P1
6406 Pure Mathematics P2
6407 Pure Mathematics P3
6408 Pure Mathematics P4
6409 Mechanics M1
6410 Mechanics M2
6411 Mechanics M3
6412 Mechanics M4
6413 Statistics T1
6414 Statistics T2
6415 Statistics T3
6416 Statistics T4
6417 Decision Mathematics D1
6671 Pure Mathematics P1
6677 Mechanics M1
6683 Statistics S1
6689 Decision Mathematics D1
Grade Boundaries January 2003
2
15
6671 Pure Mathematics P1
1
5
1.
(a) 15 x 2 (b) 7x + 4 x 2 + C
2.
(b) (0, 0.5) (150, 0) (330, 0) (c) x = 180, 300
3.
1
8
(b) y =  , y = 3; x =  , x = 4
3
3
4.
(b) 0.023 (or – 0.023) (c) Sn =
1200(1  ( 0.25) n )
1  (  0.25)
5.
(a) v = 20 (c) C = 12; cost = £30
6.
14
5
5 5
(a)  ,   (b) 2x + 3y = 1 (c) x =
; y=
13
13
 2 2
7.
8.
(c) 513 mm2 (d) 44 cm3
2
(a) A: y = 1; B: y = 4 (b) y – 1 = (x – 5) (or 5y = 2x – 5)
5
1
70
(c) x = 5 y 2 (d)
(or 23 13 , 23.3)
3
UA013196
UA013198
UA013199
UA013200
UA013202
UA013203
UA013204
UA013205
UA013206
UA013208
UA013210
Pure Mathematics P1 Question Paper January 2003
Pure Mathematics P2 Question Paper January 2003
Pure Mathematics P3 Question Paper January 2003
Pure Mathematics P4 Question Paper January 2003
Mechanics M1 Question Paper January 2003
Mechanics M2 Question Paper January 2003
Mechanics M3 Question Paper January 2003
Mechanics M4 Question Paper January 2003
Statistics S1 Question Paper January 2003
Statistics S2 Question Paper January 2003
Decision Mathematics Question Paper January 2003
Edexcel Publications
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3
14
The Uniform Mark Score (UMS):
All unit results are given in the manner of a Uniform Mark Score
(UMS). It was decided by QCA that the reporting of results on
modular examinations should be standardised among all Awarding
Bodies and all subjects, since as modular syllabuses proliferated
there were more and more scoring systems for varying numbers of
modules. All Advanced GCEs are now out of a total of 600 UMS.
AS is out of 300 UMS.
Different units in some subjects carry different weightings, but
each Edexcel AS/Advanced GCE Mathematics combination
comprises six equally weighted units with a maximum of 100 UMS
possible on each unit.
The example on the right shows the UMS score for each total
mark on a hypothetical 6671 Pure Mathematics P1 unit. The grade
A boundary is set at 57 (out of 75), grade B at 50 and grade E at 29
and these correspond to 80, 70 and 40 UMS respectively. The
notional grade boundaries are shown in bold type.
Further weighting-UMS correspondences can be found, if you
can get hold of it, in a publication called Guidance notes for
Modular Syllabuses (4th Edition October 1995), although they're
not very difficult to work out. Of course, the boundaries vary from
unit to unit and from year to year. Consequently the same
percentage score in different modules may lead to different UMS
scores.
Joint Forum, under prompting from QCA have agreed that the
stretching of UMS between the grade A boundary and the
maximum mark means it is harder for candidates to score uniform
marks once their raw mark is above the grade A threshold. An
adjustment has been made so that the maximum UMS mark is
achieved by scoring twice the mark range between the A and B
grade above that of the grade A boundary.
An N grade is still calculated at the unit level, even though this
is grade is no longer awarded at subject level.
The change to the UMS in June 2001 was communicated to
centres at the time results were issued with the following
statement:
“The UMS has been very effective in giving centres a clear
picture of progress during the course. In conjunction with the
regulatory bodies, the awarding bodies have considered ways in
which it can be made even more effective. It has been agreed
therefore that, with effect from the outcomes of the 1998 Summer
module tests, the system for converting raw marks to UMS scores
for high raw marks should be slightly modified. The modification
ensures that the ‘rate of exchange’ between raw marks and
uniform marks is closer at all grades.”
Pure Mathematics
Module P1 (6671)
(rm = raw mark)
rm
UMS
rm
75
74
73
72
71
70
69
68
67
66
65
64
63
62
61
60
59
58
57
56
55
54
53
52
51
50
49
48
47
46
45
44
43
42
41
40
39
38
37
36
100
100
100
100
100
99
97
96
94
93
91
90
89
87
86
84
83
81
80 A
79
77
76
74
73
71
70 B
69
67
66
64
63
61
60 C
59
57
56
54
53
51
50 D
35
34
33
32
31
30
29
28
27
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
UMS
49
47
46
44
43
41
40 E
39
37
36
34
33
32
30 N
29
28
26
25
23
22
21
19
18
17
15
14
12
11
10
8
7
6
4
3
1
6672 Pure Mathematics P2
1.
2( x  2)
( x  1)( x  3)
2.
n = 9, p = 2
3.
(d) x =
1
2
5.
(a)
x
0
2
4
6
8
10
y
0
6.13
7.80
7.80
6.13
0
(b) 55.7 m2 (b) 84.3 m2 (c) over-estimate
6.
x1 = 0.298280, x2 = 0.304957, x3 = 0.303731, x4 = 0.303958,
q = 0.304
7.
(a) 2 sin (x + 60) x = 40, 60, 160
8.
(c) (i) 2 (ii) 1 < f(x) < 2
(d) d = 1/585
(e) fg(x) =
3
1
x
4
13
6673 Pure Mathematics P3
Grade Boundaries for January 2003 examinations
1.
(a) A = 1, B = 2
The table below gives the lowest raw marks for the award of the stated
uniform marks (UMS):
2.
(a) a = 4, b = 5 (b) (x – 4)2 + (y – 5)2 = 25 (c) PT = 11.6
4.
(a) 1 – 6x, + 27x2 – 108x3 (b) 4 – 23x + 102x2

2
t
15
5.
(b) V = 225 + 775 e
6.
(a) r = i + 2j – 3k ± λ(4i – 5j + 3k) or r = 5i– 3j ± λ(4i – 5j + 3k)
or equivalent (c) V = 225
(c) θ = 19.5º (d) 1 unit
1
4
1
1

1
(b) π[– e  4  e  4 ] – [  ] = [1 5e  4 ]
16
4
16
16
7.
(a) x =
8.
(c) y + ln 2 = –(x – 2)
Module
UMS
Pure Mathematics P1
Pure Mathematics P2
Pure Mathematics P3
Pure Mathematics P4
Mechanics M1
Mechanics M2
Mechanics M3
Mechanics M4
Statistics S1
Statistics S2
Decision Mathematics D1
80
64
59
54
62
62
63
60
54
51
58
53
70
56
52
48
55
54
56
54
48
45
52
47
60
48
45
42
48
46
50
48
42
39
47
41
50
40
39
36
41
39
44
43
36
33
42
35
40
33
33
31
34
32
38
38
31
27
37
29
12
5
6689 Decision Mathematics D1
6674 Pure Mathematics P4
3.
c = 20x + 26y + 36z
1.
11 
 11
 i sin
12 cos
12
12 

4.
(b) BA + AE = 17 + x, BD + DE = 2x + 9, BC + CE = 21
(c) 0 < x < 6 (d) 89 minutes
2.
(b) x >
3.
(a)
4.
(a)  = 2.65 (b) x2 = 2.91
5.
(b) v = 
6.
(a) 18i (b)
7.
(a)  = 2 (c) y = cos 3x + 2x cos 3x
5.
(a) x = 31, y = 17
8.
(a)
2x + 3y + 4  8
x + 3y + z  10
p = 8x + 9y + 5z
(c) p = 28, x = 2, y =
4
, z = 0, r = 0, s = 0
3
8.
4
7
1
1

r 1 r  3
1
2
ln x  c
1 i
6
(c)
2
6
3a 2
a
2
a
 2
(a) 2
(b) A: r = ,  =
, B: r = ,  =
2
3
2
3
2
27 3a
(c) 2 14 a (d)
(e) 113.3 cm2
8
6
11
6677 Mechanics M1
6684 Statistics S2
1.
(a) v = 2.5 m s–1 (b) 15 000 Ns
2.
(a) θ = 138.2˚, (b) X = 8.94
3.
(a) (–5i + 12j) m s–2 (b) 5.2 N
4.
(a) p = 10tj; q = (6i + 12j) + (–8i + 6j)t (b) 18 km (c)
1.
(a) Continuous uniform (Rectangular), U(–0.5,0.5) (b) 0.4
(c) 0.16
3
4
5.
T = 11.0 N
6.
(b) x < 1
7.
(b) T = 4.64 s (c) F = 6390 N (d) Air resistance; variable F
8.
(a) T = 4.7 N (b) t = 0.452 s (c) t = 0.485
2.
(a) (X  2)  (X  13) (b) 0.0566
3.
(b) 0.469 (c) 0.6628
4.
(a) 0.4267 (b)
5.
(a) 0.3412 (b) 0.0022 (c) 0.0230 (d) 0.1528 (e) 0.1056
6.
(e) 0.1612 (g) 0.5553
4x
(2  x 2 ) for 0  x  1 (c) 0.057 (d) –0.812
3
10
7
6683 Statistics S1
6678 Mechanics M2
2.
(a) 0.1 (b) 0.75 (c) 0.6
1.
(b) k = –1.1
3.
(a)  = 3.90 (b) 6.18% (c)  = 55.88
2.
(a) f = 0.08 (b) d = 81 23
4.
(a) Q2 = 16, Q1 = 15; Q3 = 16.5; IQR = 1.5 (c) 16.1
(d) Almost symmetrical/Slight negative skew.
Mean (16.1)  Median (16) & Q3 – Q2 (0.5)  Q2 – Q1 (1.0)
3.
 
(b) 1 (c) 13.5
4.
(b)  =
5.
(a) v = 2t2 – 8t + 6 (b) 2 23 m
6.
(b) e =
5.
1
4
(e)
y
0
1
2
3
4
5
6
P(Y = y) 0.25 0.25 0.0625 0.25 0.125 (0) 0.0625
(f) 0.3125
6.
7.
(b) y = –1.63 + 1.33x
(c) 2653.7 + 13.3p
(d) Number sold if no money spent on advertising
p = 0 is well outside valid range – meaningless
2  13.3  27 extra cars sold
Only valid in range of data for 1990s
(c) Resistance may vary with speed
11
15
25
32
(c) Q still has velocity and will bounce back from all colliding
with stationary P.
7.
(a) 14.8 N s (b) v = 22 m s–1 (c) 48
(d) Air resistance; Wind (problem not 2 dimensional);
Rotation of ball (ball is not a particle)
8
9
6679 Mechanics M3
1.
 = 42
2.
(a) T =
3.
6680 Mechanics M4
1.
Time = 151s; bearing = 049
2.
26 km h–1; bearing = 023
(b) h = 3r
3.
v=
4.
(a) 1.3m (b) 2.6 m s–1 (c) 5.2 m s–2 (d) 0.79 s
4.
(c) Stable
5.
(b) 96.8 m
5.
(a) 3 sin 2t – 6e–t sin t (c) 1.07 (d) 
6.
(a) 6 m (b) v = 14.3 m s–1
6.
(b)
7.
5mg
4
(b) v =
(a) v2 = u2 – 3ga
(b)
6 gl
5mg
2
(c) u =
7ga
2
(d) u = 5ga

g
1 e 2kD
k
4
5
(e) 12:25
