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Class Index Number Name
ANG MO KIO SECONDARY SCHOOL PRELIMINARY EXAMINATION 2009 SECONDARY FOUR EXPRESS / FIVE NORMAL ACADEMIC
MATHEMATICS Paper 1
4016/01 Wednesday 2 Sep 2009 2 hours Name of Setter: Mr Chio Kah Leong Candidates answer on the Question Paper. Additional Materials: Geometrical Instruments
READ THESE INSTRUCTIONS FIRST Write your name, class and index number in the spaces on the top of this page. Write in dark blue or black pen in the spaces provided on the Question Paper. You may use a pencil for any diagrams or graphs. Do not use staple, paper clips, highlighters, glue or correction fluid. Answer all questions. If working is needed for any question it must be shown in the space below that question. You are expected to use a scientific calculator to evaluate explicit numerical expressions. If the degree of accuracy is not specified in the question, and if the answer is not exact, give the answer to three significant figures. Give answers in degrees to one decimal place. For , use either your calculator value or 3.142, unless the question requires the answer in terms of . The number of marks is given in brackets [ ] at the end of each question or part question. The total of the marks for this paper is 80. For Examiner's Use
This document consists of 20 printed pages
80
[ Turn Over
Mathematical Formulae
Compound interest
Total amount = nrP
1001
Mensuration
Curved surface area of a cone = rl
Surface area of a sphere = 4r2
Volume of a cone = hr 2
31
Volume of a sphere = 3
34 r
Area of triangle ABC = Cabsin21
Arc length = rθ, where θ is in radians
Sector area = 221 r , where θ is in radians
Trigonometry
Cc
Bb
Aa
sinsinsin
Abccba cos2222
Statistics
Mean = ffx
Standard deviation = 22
ffx
ffx
3
AMKSS 4E / 5N E Math Prelim Exam 2009, 4016/01 [ Turn Over
1 (a) The mass of a bacteria cell 111002.1 g can be written as k nanograms. Find k.
(b) Find the ratio of 2103.1 to 1105.6 . Give your answer in its simplest form.
Answer (a) k = ....………….………………… [1]
(b) …....………….. : .………….… [1]
2 The numbers 756 and 2352, written as a product of their prime factors, are
732756 32 and 24 7322352 .
Use these results to find
(a) the largest integer which is a factor of 756 and 2352,
(b) the smallest positive integer k for which 756k is a multiple of 2352,
(c) the smallest positive integer value of n for which 3 756n is a whole number.
Answer (a) ….....………….………………… [1]
(b) …....………….……………….… [1]
(c) ………………………………….. [1]
For Examiner's
Use
4
AMKSS 4E / 5N E Math Prelim Exam 2009, 4016/01 [ Turn Over
3 Solve the following equations
(a) x23
1
64
7xx + 3 = 0,
(b) yy
1255
2553
2
.
Answer (a) x = ……….……………………. [2]
(b) y = .……….……………………. [2]
5
AMKSS 4E / 5N E Math Prelim Exam 2009, 4016/01 [ Turn Over
4 Given that Y varies inversely as 42 x and that Y = 25 when x = 2, (a) express Y in terms of x,
(b) find the percentage increase in Y when x decreases by 25% from 2.
Answer (a) ….....………….………………… [2]
(b) …....………….………………. % [2]
6
AMKSS 4E / 5N E Math Prelim Exam 2009, 4016/01 [ Turn Over
5 Kelvin bought a crystal display for $180. He still made a percentage profit of 65%
despite offering a 40% discount to his customer. Calculate the selling price of the crystal
display before discount.
Answer $ ..………….…………………… [2] 6 The actual area of 2 km2 is represented on a map by an area of 50 cm2. Calculate
(a) the actual distance, in kilometres, represented by a length of 10.4 cm,
(b) the scale of the map, in the form n:1 .
Answer (a) …....………………………. km [2]
(b) 1 : …....……………………… [1]
For Examiner's
Use
7
AMKSS 4E / 5N E Math Prelim Exam 2009, 4016/01 [ Turn Over
7 In the diagram, a toy-car wheel of radius 6 cm is in contact with the horizontal ground at
P and touching the stair at R. Find in terms of ,
(a) the length of PQ,
(b) the height of the stair QR.
If = 3 , calculate
(c) the area of the sector OPR,
(d) the shaded area PQR.
Answer (a) ….....……………….……… cm [1]
(b) …....………………………. cm [1]
(c) …....……………………… cm2 [1]
(d) …....……………………… cm2 [2]
For Examiner's
Use
6 cm
P Q
R
O
8
AMKSS 4E / 5N E Math Prelim Exam 2009, 4016/01 [ Turn Over
8 In the diagram, the ratio of tan is 45 .
(a) Write down the coordinates of the point A.
(b) Find the equation of the line AB.
(c) Find the ratio of
(i) CABsin ,
(ii) CABcos .
Answer (a) (…...……..…. , …..……..….) [1]
(b) …....…………………………… [2]
(c)(i) …………………...…..……… [1]
(ii) …………………........……… [1]
A
B (2, 5)
y
x O
C
9
AMKSS 4E / 5N E Math Prelim Exam 2009, 4016/01 [ Turn Over
9 In the diagram (not drawn to scale), CEDCDB , CD = DE = 8 cm and
CE = 13 cm.
(a) Write down a second pair of equal lengths.
(b) Calculate the ratio of
(i) CD : CB ,
(ii) area of CBD : area of CDE .
Answer (a) …..…………...………..……….. [1]
(b)(i) …………………...…..……… [1]
(ii) …………………........……… [1]
B
C
E
13 cm
8 cm D
10
AMKSS 4E / 5N E Math Prelim Exam 2009, 4016/01 [ Turn Over
10 In the diagram, EC is a tangent to the circle center O. Given that 63CED and ABD
is a straight line.
(a) Find, giving your reasons,
(i) CBA ,
(ii) reflex AOC ,
(iii) ACO ,
(iv) ECA .
(b) Is DE parallel to CA? State briefly a reason for your answer.
Answer (a)(i) …..…………...………..……..…° [1]
(ii) …..…………...………..……..…° [1]
(iii) …..…………...………..……..…° [1]
(iv) …..…………...………..……..…° [1]
Answer (b) ……………………………………………………………………………….
……………………………………………………………………………….. [1]
For Examiner's
Use
E
D
B A
O
C
63°
11
AMKSS 4E / 5N E Math Prelim Exam 2009, 4016/01 [ Turn Over
11 During a supermarket sale, a fruiterer sells two types of packages consisting of kiwi,
mango and strawberry.
Kiwi Mango Strawberry Package A 4 3 8 Package B 6 7 5
The cost of each kiwi, mango and strawberry is $0.90, $1.50 and $1.20 respectively.
A total of 50 Package A and 30 Package B were sold.
Given that
58
73
64
P ,
2.15.19.0
C and 3050T ,
(a) find PC.
(b) hence find TPC and explain what the answer represents.
Answer (a) PC = …….………..……………. [1]
(b) TPC = ………...…...…..……… [1]
Answer (b) ……………………………………………………………………………….
……………………………………………………………………………….. [1]
For Examiner's
Use
12
AMKSS 4E / 5N E Math Prelim Exam 2009, 4016/01 [ Turn Over
12 In the Venn diagrams below,
(a) write down the set notation that represents the shaded region,
Answer (a) …..…………...………..……….. [1]
(c) shade the region representing (B′ ∩ A)′.
Answer (b)
[1]
For Examiner's
Use
ε
A B
ε
A B
13
AMKSS 4E / 5N E Math Prelim Exam 2009, 4016/01 [ Turn Over
13 Two toy boats are geometrically similar and one is 3½ times longer than the other.
(a) Given that the height of the mast of the smaller boat is 12 cm, calculate the height
of the mast of the larger boat.
(b) The mass of the larger boat is 6.517 kg. What is the mass of the small boat in
grams?
Answer (a) ..…………...………..………...cm [2]
(b) .…………………...…..……......g [2]
14
AMKSS 4E / 5N E Math Prelim Exam 2009, 4016/01 [ Turn Over
14 Find the smallest integer value of x such that x + 3 15 4x 39.
Answer x = …..………...………..……….. [2]
15 The number of siblings that students in a class have is shown in the following table.
(a) If there are 40 students in the class, calculate the mean number of siblings.
(b) If the mode is 2, write down the range of values of x.
(c) Find the largest value of x if the median is 2.
Answer (a) …..…………...………..………… [2]
(b) ……………………...…..……….. [1]
(c) ……………………...…..……….. [1]
Number of siblings 0 1 2 3 4
Number of students 11 8 15 x 3
For Examiner's
Use
15
AMKSS 4E / 5N E Math Prelim Exam 2009, 4016/01 [ Turn Over
16 (a) Solve the equation 2231 tt .
(b) Factorise 4422 aba completely.
Answer (a) t = .…....…..………, .…..……..………… [2]
(b) …..……..….…….…...………………...... [2]
For Examiner's
Use
16
AMKSS 4E / 5N E Math Prelim Exam 2009, 4016/01 [ Turn Over
17 (a) Factorise 2542 2 xxy by using the completing the square method.
(b) Hence sketch the curve 2542 2 xxy in the axes below.
Label the y-intercept, x-intercepts and turning point clearly.
Answer (a) y = …………………...……… [2] Answer (b)
[2]
For Examiner's
Use
y
x 0
17
AMKSS 4E / 5N E Math Prelim Exam 2009, 4016/01 [ Turn Over
18 Solve the simultaneous equations
0242
32712
yx
yx
Answer x = ...…………...………..…………
y = ..…………………...…..………. [3]
19 The diagram shows two containers A and B of the same volume and height.
Water is poured into the containers at a constant rate until they are completely filled.
Given that the two containers are filled up completely at the same time, sketch on the
given axes, the height of water in the 2 containers against time as they are being filled.
Label your sketch clearly.
Answer
[2] time
height of water, h Maximum level of water
0
Container A Container B
h
18
AMKSS 4E / 5N E Math Prelim Exam 2009, 4016/01 [ Turn Over
20 (a) In the diagram, OPRQ is a parallelogram. OP p, OQ q and OT 3p.
(i) Express QT in terms of p and q.
(ii) Find the value of PTSofareaQRSofarea .
(iii) Hence, find the value of OPRQofarea
PSTofarea .
(b) Given a
013
, b
h9
, c
4h
and d
41
.
(i) Evaluate 3d a .
(ii) Given that b is parallel to c, calculate the values of h .
Answer (a)(i) …..…………...………..………… [1]
(ii) …..…………...………..………… [1]
(iii) .....…………...………..………… [1]
(b)(i) …..…………...………..………… [2]
(ii) h = …...……...………..………… [1]
For Examiner's
Use
T
S
Q
R
P
O p
q
19
AMKSS 4E / 5N E Math Prelim Exam 2009, 4016/01 [ Turn Over
21 A car travels at a constant speed of v m/s for 10 seconds and then accelerates uniformly
to a speed of 70 m/s before coming to a stop at 35 seconds.
(a) Calculate the value of v if the distance travelled in the first 25 seconds is 1470 m.
(b) Calculate the speed of the car, in m/s, when time is 20seconds.
(c) Calculate the acceleration of the car in the last 10 seconds.
(d) On the axes in the answer space below, sketch the acceleration-time graph of the
car for the 35 seconds. Label the vertical axis clearly.
Answer (a) …..…………...………..……...m/s [2]
(b) ……………………...…..…….m/s [2]
(c) ……………………...…..……m/s2 [1] (d)
[1]
For Examiner's
Use
Speed in m/s
Time (t) in seconds 10 25 35
v
70
0
Acceleration in m/s2
Time in seconds 10 25 35 0
20
AMKSS 4E / 5N E Math Prelim Exam 2009, 4016/01 [ Turn Over
22 The diagram shows a radio station A in a Town. Using a scale of 1 cm to 10 km, draw and label
(a) housing estate B located 90 km apart, at a bearing of 320 from radio station A,
(b) industrial estate C located 120 km west of radio station A.
Construct
(c) the angle bisector of BAC,
(d) the perpendicular bisector of AC. [4]
For Examiner's
Use END OF PAPER
N
A
2
AMKSS 4E / 5N E Math Prelim Exam 2009, 4016/01 [ Turn Over
Sec 4E / 5N Preliminary Exam 2009 E-Maths Paper 1
Answer Marking Scheme
1(a)
1(b)
21002.10102.0 or 1 : 50
B1
B1
2(a) 2(b) 2(c)
847322 HCF
2872732732
7327567322352756
2
23432
234
234
kk
kandofLCM
9872756
,756
2
3
nnumbercubeabemustn
numberwholeaisnIf
B1
B1
B1
3(a) 3(b)
4520
01218720232372
03232
723
1
0364
723
1
xx
xxxx
xx
x
xx
x
321
33122
55
55 3
31
22
y
yy
yy
M1
A1
M1
A1
4(a)
4200200825
,25,2
.tan,4
2
2
xY
kYxWhen
tconsaiskwherex
kY
M1
A1
3
AMKSS 4E / 5N E Math Prelim Exam 2009, 4016/01 [ Turn Over
4(b)
%28
%10025
2532%
3245.1
2005.1275.0
%,25
2
Yinincrease
Ynew
xnewbydecreasesxWhen
M1
A1
5
Selling price after discount 297$65.1180$
Selling price before discount
495$
%10060297$
M1
A1
6(a) 6(b)
kmcm
kmcmkmcmkmcm
08.25
4.104.10
1:51:252:50
22
22
00020:100020:1
51:1
cmcm
kmcm
M1
A1
B1
7(a) 7(b) 7(c) 7(d)
sin6 cos66
2
2
2
2
8.186
36
2121
sec
cmcm
r
OPRtorofArea
B1 B1
B1
4
AMKSS 4E / 5N E Math Prelim Exam 2009, 4016/01 [ Turn Over
2
2
53.4
62
327
63
sin63
cos66621
621
cm
cm
PQRQOP
PQRareaShaded
M1
A1
8(a) 8(b) 8(c)(i) 8(c)(ii)
0,22
45
205
Ax
x
105425
45
2455
xyor
xy
xy
415sinsin CAB
414coscos CAB
B1
M1
A1
B1
B1
9(a) 9(b)(i) 9(b)(ii)
DB = BC
8:13:8
13.8
.
CBCDCDCE
CBCD
cmDECDCEDtosimilarisCDB
169:64..169642
eiCDCB
CDEofAreaCBDofArea
B1
B1
B1
10(a)(i) 10(a)(ii)
117.63180
63linestraonsadjCBA
segmentsameinsDECDBC
.22342117
circumatcentreatAOCreflex
B1
B1
5
AMKSS 4E / 5N E Math Prelim Exam 2009, 4016/01 [ Turn Over
10(a)(iii) 10(a)(iv) 10(b)
isosofsbase
AOC
272
234360180
63tan2790 radiusECA
Yes. Since ECADEC , they form alternating angles i.e. DE // AC.
B1
B1
B1
11(a) 11(b)
90.2170.17
PC
1542TPC
It represents the total amt. of money collected from the sale of 50 package A and 30 package B.
B1
B1
B1
12(a) 12(b)
''''''
,
BAorABBAorAB
followingtheAccept
B1
B1
13(a) 13(b)
cmh
h
4215.3
12
gkgmass
mass
smaller
smaller
152152.0
3438
72
5.31
517.6
33
M1
A1
M1
A1
ε
A B
6
AMKSS 4E / 5N E Math Prelim Exam 2009, 4016/01 [ Turn Over
14
.5int4.26
64.2244125
394154153
isxegerSmallestx
xandxxandx
xandxx
M1
M1
A1
15(a) 15(b) 15(c)
If total = 40, x = 3.
.48.1475.148.1
475.140
343315281110
tooffroundingbeforeshowstudentsifacceptOnly
Mean
150140 xorx
30
M1
A1
B1
B1
16(a) 16(b)
134
0143043
22312
tort
tttt
tt
baba
ba
aba
222
4422
22
M1
A1
M1
A1
17(a)
2712
22712
225
22
2222
22522
2
2
222
2
x
x
xx
xxy
M1
A1
7
AMKSS 4E / 5N E Math Prelim Exam 2009, 4016/01 [ Turn Over
17(b)
B1 – Correct shape of curve (minimum) B1 – Label the y-intercept, x-intercepts and turning point.
18 15,6 yx M2 + A1
19
B2
20(a)(i) 20(a)(ii) 20(a)(iii)
qp 3
41
32
B1
B1
B1
-2.67
-25
4.67
(1, -27)
time
height of water, h Maximum level of water
0
A
B
8
AMKSS 4E / 5N E Math Prelim Exam 2009, 4016/01 [ Turn Over
20(b)(i) 20(b)(ii)
2012163
1216
3
22
ad
ad
64
9
h
hh
M1
A1
B1
21(a) 21(b) 21(c) 21(d)
smv
vv
/54
147015702110
sms
s
/3264
155470
1054
- 7 m/s2
M1
A1
M1
A1
B1
B1
Acceleration in m/s2
Time in seconds 10 25 35 0 1511
-7