SPEC/5/MATME/SP2/ENG/TZ0/XX/M
MARKSCHEME
SPECIMEN
MATHEMATICS
Standard Level
Paper 2
15 pages
±5±
SPEC/5/MATME/SP2/ENG/TZ0/XX/M
SE C T I O N A
1.
(a)
attempt to substitute into sum formula for AP
20
20
e.g. S20
2(7) 19d , (7 u20 )
2
2
M1
setting up correct equation using sum formula
A1
e.g.
d
(b)
20
2(7) 19d 620
2
4
correct substitution 7 77(4)
u78 301
A1
A1
A1
N2
[3 marks]
N2
[2 marks]
Total [5 marks]
±6±
2.
(a)
SPEC/5/MATME/SP2/ENG/TZ0/XX/M
M1
intercepts when f ( x) 0
A1A1
(0.827, 0) (4.78, 0) (accept x 0.827 x 4.78)
N3
[3 marks]
(b)
A1A1A1
Note: Award A1 for maximum point in circle, A1 for x-intercepts in circles,
A1 for correct shape (y approximately greater than 3.14 ).
(c)
gradient is 1.28
N3
[3 marks]
A1
N1
[1 mark]
Total [7 marks]
±7±
3.
SPEC/5/MATME/SP2/ENG/TZ0/XX/M
(a)
A1A1A1
N3
Note: Award A1 for 450 , A1 for a to the right of the mean, A1 for area 0.27 .
[3 marks]
(b)
valid approach
e.g. P(X a) 1 P(X ! a), 0.73
M1
a 462.256}
A1
a 462
A1
N3
[3 marks]
Total [6 marks]
4.
(a)
L1
A1
N1
[1 mark]
(b)
§1·
¨ ¸
¨ 2¸
¨ 3¸
© ¹
A1
N1
[1 mark]
(c)
choosing correct direction vectors
§ 3 · § 1 ·
e.g. ¨¨ 2 ¸¸ , ¨¨ 2 ¸¸
¨ 1 ¸ ¨ a ¸
© ¹ © ¹
A1A1
recognizing that a b 0
correct substitution
e.g. 3 4 a 0
M1
A1
a 7
A1
N3
[5 marks]
Total [7 marks]
±8±
5.
(a)
(i)
(ii)
(b)
SPEC/5/MATME/SP2/ENG/TZ0/XX/M
recognizing binomial with n 600, p 0.4
E( X ) 240
M1
A1
correct substitution into formula for variance or standard deviation
e.g. 144, 600 u 0.4 u 0.6
A1
sd 12
A1
attempt to find range of values
e.g. 240 r 12 228 X 252
M1
evidence of correct approach
e.g. P( X d 251) P( X d 228)
A1
P 228 X 252 0.662
A1
N2
N1
[4 marks]
N2
[3 marks]
Total [7 marks]
±9±
6.
SPEC/5/MATME/SP2/ENG/TZ0/XX/M
METHOD 1
appropriate approach
e.g. completed diagram
M1
attempt at set up
e.g. correct placement of one angle
A1
h
h 1.6
, tan 35
x
x
attempt to set up equation
e.g. isolate x
A1A1
tan 30
M1
correct equation
h
h 1.6
e.g.
tan 30 tan 35
A1
h 7.52
A1
N3
[7 marks]
METHOD 2
sin 30
h
l
in triangle ATB, A 5 , T
choosing sine rule
correct substitution
h / sin 30 1.6
e.g.
sin 55
sin 5
1.6 u sin 30 u sin 55
sin 5
h 7.52
h
A1
55
A1A1
M1
A1
A1
A1
N3
[7 marks]
± 10 ±
7.
(a)
(b)
SPEC/5/MATME/SP2/ENG/TZ0/XX/M
substitution into formula for area of triangle
1
e.g.
r u r sin T
2
A1
evidence of subtraction
M1
correct expression
1
1
1
e.g. r 2T r 2 sin T , r 2 (T sin T )
2
2
2
A1
evidence of recognizing that shaded area is
e.g.
N2
[3 marks]
1
of area of circle
8
M1
1
seen anywhere
8
A1
setting up correct equation
1
1 2
e.g. r 2 (T sin T )
Sr
2
8
M1
eliminating 1 variable
1
1
e.g. (T sin T )
S , T sin T
2
8
S
4
M1
attempt to solve
e.g. a sketch, writing sin x x T 1.77 (do not accept degrees)
S
4
0
A1
N1
[5 marks]
Total [8 marks]
± 11 ±
SPEC/5/MATME/SP2/ENG/TZ0/XX/M
SE C T I O N B
8.
A1A1
N2
[2 marks]
additional cost per box (unit cost)
A1
N1
fixed costs
A1
N1
[2 marks]
(a)
y 10.7 x 121
(b)
(i)
(ii)
(c)
(d)
attempt to substitute into regression equation
e.g. y 10.7 u 60 121, y 760.12}
M1
cost
A1
$760 (accept $763 from 3 s.f. values)
setting up inequality (accept equation)
e.g. 19.99 x ! 10.7 x 121
M1
x ! 12.94}
A1
13 boxes (accept 14 from x ! 13.02, using 3 s.f. values)
A1
Note: Exception to the F T rule: if working shown, award the final A1 for a
correct integer solution for their value of x.
(e)
N2
[2 marks]
N2
[3 marks]
(i)
this would be extrapolation, not appropriate
R1R1
N2
(ii)
this regression line cannot predict x from y, not appropriate
R1R1
N2
[4 marks]
Total [13 marks]
± 12 ±
9.
(a)
y
2x 1
x 1
interchanging x and y (seen anywhere)
2 y 1
e.g. x
y 1
M1
correct working
e.g . xy x 2 y 1
A1
collecting terms
e.g. x 1 2 y xy , x 1 y (2 x)
A1
x 1
2 x
h1 ( x)
(b)
SPEC/5/MATME/SP2/ENG/TZ0/XX/M
A1
N2
[4 marks]
(i)
A1A1A1A1
N4
A1A1
N2
A1
N1
Note: Award A1 for approximately correct intercepts,
A1 for correct shape, A1 for asymptotes,
A1 for approximately correct domain and range.
(ii)
x 1, y 2
(iii)
1
2
[7 marks]
FRQWLQXHG«
± 13 ±
SPEC/5/MATME/SP2/ENG/TZ0/XX/M
Question 9 continued
(c)
A2
(i)
area 2.06
(ii)
attempt to substitute into volume formula do not accept S³ y 2dx
a
M1
2
§ 2x 1 ·
volume S³ 1 ¨
¸ dx
2 © x 1 ¹
3
b
N2
A2
N3
[5 marks]
Total [16 marks]
± 14 ±
10.
(a)
(i)
106 m
(ii)
substitute t
h 44.9 m
(iii)
(b)
4.5
SPEC/5/MATME/SP2/ENG/TZ0/XX/M
A1
N1
M1
A1
N2
set up suitable equation
e.g. f (t ) 30
M1
t
A1
4.91
recognising that height is 0
set up suitable equation
e.g. g (t ) 0
A1
M1
t 5.39secs
A1
N1
[5 marks]
N2
[3 marks]
FRQWLQXHG«
± 15 ±
SPEC/5/MATME/SP2/ENG/TZ0/XX/M
Question 10 continued
(c)
(i)
A1A2
N3
Note: Award A1 for correct scales on axes,
A2 for 5 correct points, A1 for 3 or 4 correct points.
(ii)
-DQH¶VIXQFWLRQZLWK2 valid reasons
A1R1R1
e.g. -DQH¶VSDVVHVYHU\close to all the points, .HYLQ¶VKDVWKHUock
clearly going up initially ± not possible if rock falls
Note: $OWKRXJK-DQH¶VDOVRJRHVXSLQLWLDOO\LWRQO\JRHVXSYHU\VOLJKWO\DQG
so is the better model.
N3
[6 marks]
Total [14 marks]