2009 ESpring Prelim Sec 4Ex EM P2
Transcript of 2009 ESpring Prelim Sec 4Ex EM P2
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Name : __________________________ ______________________ _____ ( )
S 4/5____
MathematicsPaper 2
Tuesday, 25th
August 2009
Additional materials:Writing paper.Graph paper
INSTRUCTIONS TO CANDIDATES :
Write your name, class and register number on all the work you hand in.Write in dark blue or black pen.You may use pencil for any diagrams or graphs.Do not use staples, paper clips, highlighters, glue or correction fluid.
Answerall questions.If working is needed for any question it must be shown with the answer.Omission of essential working will result in loss of marks.
Calculators should be used where appropriate.If the degree of accuracy is not specified in the question, and if th e answer isnot exact, give the answer to three significant figures. Give answers indegrees to one decimal place.ForT, use either your calculator value or 3.142, unless the question requiresthe answer in terms of T.
At the end of the examination, fasten all your work securely together.The number of marks is given in brackets [ ] at the end of each question orpart question.
The total number of marks for this paper is 100.
This question paper consists of 11 printed pages including the cover page.
4016/2
East Spring Secondary SchoolTowards Excellence and Success
Preliminary Examinations 2009Secondary 4 Express
2 hour 30 minutes10:15 am 12:45 pm
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Mathematical Formulae
Mathematical Formulae
Compound interest
Total amount = 1100
n
rP
Mensuration
Curved surface area of a cone = rlT
Surface area of a sphere = 24 rT
Volume of a cone = 21
3
r hT
Volume of a sphere = 34
3rT
Area of triangle ABC=1
sin2
ab C
Arc length = rU , where U is in radians.
Sector area = 21
2rU , where U is in radians.
Trigonometry
sin sin sin
a b c
A B
! !
2 2 2 2 cosa b c bc A!
Statistics
Mean =fx
f
Standard deviation =
22fx fx
f f
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1. Johnny walked from pointA to point B at an average speed ofx km/h. Edwin
jogged along the same route from point B to point A at an average speed of
)5( x km/h. They met at the midpoint between pointsA and B. The distance
between pointA and B is 4 km.
(a) Write down an expression, in terms ofx, for the number of hours taken
by
(i) Johnny when they met. [1]
(ii) Edwin when they met. [1]
(b) Given that Edwin started 18 minutes later than Johnny, [2]
form an equation in x and show that it reduces to
0100153 2 ! xx
(c) FindE
dwins average speed, giving your answer in one decimal place.[3]
2. In 2008, Mr Sim received the following tax relief.
Type of Tax relief AmountEarned Income $1500Wife Relief $2300Qualifying Child Relief $4000Parents $5000NS men $1200CPF Contributions $14,400
(a) Calculate his total tax relief. [1]
(b) It is given that
Chargeable income = total income for the year total tax relief
If Mr Sim earns $6000 per month, calculate his chargeable inco me. [2]
(c) In 2008 Mr Sim paid tax according to rule NO TAX on the first $25000 of
his chargeable income, 5% on the next $10,000 and 8% on the
remainderof his chargeable income. Calculate the tax that Mr Sim paidin 2008. [3]
(d) Calculate Mr Sims CPF Contributions as a percentage of his total
income for the year. Given that CPF deduct 20% of employees total
income, conclude did Mr Sim calculate his CPF contribution correctly?
(Show your working clearly) [2]
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3. A model consists of a solid cone and a solid hemisphere attached to a solid
cylinder as shown in the figure below.
(a) Calculate the volume of the model. [2]
(b) Given that the model is made from material of density 0.35 g/cm3,
calculate its mass in grams. [2]
(c) Calculate
(i) the slant length of the cone, [2]
(ii) the total surface area of the mode l, correct to the nearest whole
number. [3]
(d) If the model is to be plated with material costing $28 per cm2, find the
cost of the plating. [1]
4. In a group of 40 people, 10 can drive lorries, 25 can drive cars and 12 canride motorcycles. Given also that all those who drive lorries can drive cars as
well. There are 4 people without any driving license. There is nobody who
can ride a motorcycle and drive a lorry at the same time. It is given that
I = {people in the group}
L = {those who can drive lorries}
C= {those who can drive cars}
M= {those who can ride motorcycles}
(a) Using Venn diagram to illustrate the given i nformation, find the number
of people who can both ride motorcycles and drive cars. [3]
(b) Express in set notation the statement people who can drive a car [2]
only.
(c) Express in words the set notation 'MC . [2]
8 cm 15 cm 4 cm
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5.
In the diagram,AB is the diameter of the circle AFGBCD, centre O. Eis the
point onAB produced where DB = BEand r! xAED . The straight line ED
cuts the circle at C.
(a) Find in terms ofx,
(i) CFB , [2]
(ii) DBE . [2]
(iii) AB . [2]
(b) radians4.0!FOB and OF= 7 cm,
(i) Calculate the area of sectorFOBG. [2]
(ii) Hence find the area of the shaded segment FGB [2]
C
D
EB
A
F
O
G
rx
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6.
In the diagram, R, SandT
are three points on horizontal ground. RS = 10 mand ST= 13.5 m. A vertical flag pole RQ stands at Rand the angle of
depression ofSfrom Q is r35 . Given that Sis due south ofRand the bearing
ofT from Sis r052 , find
(a) the length ofRT, [2]
(b) RTS , [2]
(c) the bearing ofT from R, [2]
(d) the height of flag Pole RQ, [2]
(e) angle of elevation ofQfrom T, [2](f) A man walks from Sto T. Find his distance from Sat which the angle
of elevation ofQ from the man is the largest. [2]
R
T
r35
r52
10
N
13.5
Q
S
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7. A survey was conducted in a housing block on the heights of 160 residents.The cumulative frequency curve below represents the results of the survey.
(a) Use the graph to find
(i) the median height, [2]
(ii) the inter-quartile range, [2]
(iii) the percentage of residents who are shorter than 125 cm. [2]
(iv) what is the probability that a randomly selected resident is taller
than 125cm [2]
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(b) Using the cumulative frequency curve for the height of the 160residents, copy and complete the frequency table below. [2]
Height (cm)Number ofresidents
110100 ex 120110 ex 20130120 ex 140130 ex 40150140 ex 160150 ex 15170160 ex
8.
In the diagram, aOA ! and bOB ! . Mand Nare midpoints ofAB and OBrespectively.
C is a point on OM such that OMO
3
2! .
(a) Express the following vectors in terms of a and b,
(i) AM [2]
(ii) OM [2]
(iii) OC [2]
(iv) AN [2]
(b) OM is produced to D such that ahBD ! and OMkOD ! . [2]
Show that h = 1 and k= 2.
O A
N
B
C
M
D
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9. Vouchers to the Ministry of Food Carnival were sold by students during the
June holiday. The carnival had 4 food stalls. The number of vouchers sold
for Friday and Saturday are summarized in the table below:
Stall A Stall B Stall C Stall D
Friday 55 100 123 75
Saturday 60 72 111 82
The price per voucher is $20 for Stall A, $15 for Stall B, $18 for Stall Cand
$12 for Stall D.
(a) Write down a 42 v matrix T to represent the number of vouchers sold
by the students for the two days. [1]
(b) Write down a column matrix S to represent the price per voucher . [1]
(c) Find TSand interpret the elements in the matrix. [3]
(d) The principal decided to hold the carnival for one more day. The price
of each voucher for the food in Stall A was increased while the
vouchers prices for the other stalls remained unchanged. A total of
$7 950 was raised and that 109 StallA, 90 Stall B, 125 Stall Cand 90
StallD
vouchers were sold.Using a 41v matrix for the number of vouchers of each stall and a
14 v matrix for the price of each voucher, find the new price of each
voucher in StallA [3]
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10. BoxA contains 4 blue and 5 white marbles. Box B contains 5 blue and 3
white marbles. Syasya draws a marble at random from BoxA followed by
another marble from Box B.
(a) Copy and complete the probability tree di agram. [2]
(b) Calculate the probability that
(i) the two marbles drawn are of the same colours, [2]
(ii) the second marble drawn is white. [2]
(iii) Sally then draws another marble from Box A again without
replacement. Find the probability that all three marbles drawn
are blue. [2]
blue
blue
white
white
blue
white8
3
9
4
( )
Box A Box B
( )
( )
( )
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11 Answer the whole of this question on a sheet of graph paper.
The variablesxand yare connected by the equation xy 21 ! . Some
corresponding values ofxand yare given in the following table.
x 3 2 1 0 1 2 3
y 1.1 1.25 1.5 2 3 5 9
(a) Taking 2 cm to represent 1 unit on each axis, draw the graph of
xy 21 ! for values ofxin the range 33 ee x . [3]
(b) Use your graph to solve the equation 2x
+1= 4. [2]
(c) By drawing another line on your g raph, find the solutions of the [2]
equation
122 ! xx .
(d) By drawing a tangent, find the gradient of the curve xy 21 ! at the
point wherex= 2. [3]
~End of Paper~
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1ai)
aii)
b)
c)
x
2
5
2
x
)(0100153
10
3
)5(
2102
103
522
2 shownxx
xx
xx
xx
@!
!
!
)3(2
)100)(3(41515 2 s!x
=3.792 or -8.792 (NA)hkm /8.3}
5x = 8.8 km/h
[B1]
[B1]
[M1]
[A1]
[M1]
[A1]
[A1]
2a)
b)
c)
d)
$28,400
$6000(12)-$28,400=$43,600
$43,600 - $25 000 = $18 600 [ecf]
Total income tax=
1188$)8600)(08.0()10000)(05.0( !
Percentage of CPF: %20%100000,72
14400!v
YES
[B1]
[A2]
[M1]
[A2]
[B1]
[B1]
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3a)
b)
ci)
ii)
d)
3
3
322
cm022.21
cm197.1022
)4(3
2)15)(4()8)(4(
3
1modeltheo
olume
!
!
! TTT
358g
357.77g
1022.1970.35modeltheoass v
[ecf]
height = unit8048 22 !
2
2
22
cm590
cm589.9956cm)4(2)15)(4(2)80)(4(areasur
aceTotal
!
!! TTT
[ecf]
$16,520=
28)$(590=platingtheofCost v [ecf]
[M1]
[A1]
[M1]
[A1]
[M1][A1]
[M1]
[M1]
[A1]
[B1]
4a) I
L
4
14
2612
26
12
2510
!
!
!
!
!
c
c
mxc
xm
xc
1!@x
[B2 for all
correct
answer]
[B1 for 1
error]
[B1]
C
14
0
10
x M 11
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bi)
ii)
5ai)
ii)
iii)
bi)
bii)
'')( MCL
those that can drive cars or lorries or both
but not ride a motorcycle.
)(
2180
)(
)(
(
!
!!
(
!!
sumof
xDBE
segmentsamein
xCFBCDB
isos
xCEBCDB
x
xABC
xADC
!
!
!
0
00
90
)90(180
90
Area of sector = 22 8.9
2
1cmr !.
Area of segment =
cm
cm
26.0
259.0
5.0sin)7)(7(2
18.9
!
!
[B2]
[B2]
[M1]
[A1]
[M1][A1]
[M1]
[A1]
[M1][A1]
[M1]
[A1]
6a)
b)
Using Cos Rule,
RT= m77.1052cos)5.13)(10(25.1310 22 !
Using Sine Rule,
[B2]
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c)
d)
e)
f)
047
10
sin
77.10
52sin
!
!
RTS
RTS
[ecf]
Bearing = 0994752 !
mh
H
7
1035tan 0
!
!
033
77.10
7tan
!
!
U
U
mx
x
16.6
1052cos 0
!
!
[B2]
[B2]
[M1]
[A1]
[M1]
[A1]
[M1]
[A1]
7ai)
ii)
iii)
iv)
b)
132 1s
143-123=20 ( 2s )
%4.29%100160
47!v
100% - 29.4% = 70.6% [ecf]
P(resident is taller than 125 cm) = 0.706
Height (cm)Number ofresidents
110100 ex 10
120110e
x 20130120 ex 40140130 ex 40150140 ex 30160150 ex 15170160 ex 5
[B2]
[B2]
[M1][A1]
[M1]
[A1]
[B2 for all
correctanswer]
[B1 for 1
error]
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8ai)
ii)
iii)
iv)
b)
)(2
1
2
1abABAM
abOAOBAB
!!
!!
)(2
1)(
2
1baaba
AMOAOM
!!
!
)(3
1
3
2baOMOC !!
abOAONAN !!
2
1
sho nkh
haa
habab
habakb
ODBOBD
21
)2
1
2
1(2
)2
1
2
1(
!!@
!!
!!
!!
!
[M1]
[A1]
[M1]
[A1]
[M1][A1]
[M1][A1]
[M1]
[A1]
9a)
b)
c)
821117260
7512310055
12
18
15
20
TS=
821117260
7512310055x
12
18
15
20
=
5262
5714
Total amount of money earn for each respective
day
[B1]
[B1]
[M1][A1]
[B1]
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d)
307950$108022501350109
7950$
12
18
159012590109
!@!!
!
xx
x
[M1]
[M1]
[A1]
10
a)
bi)
ii)
iii)
72
35
8
3
9
5
8
5
9
4!vv [ecf]
8
3
8
3
9
5
8
3
9
4!vv [ecf]
48
5
8
3
8
5
9
4!vv [ecf]
[B2 for all
correct
answer]
[B1 for 1error]
[M1][A1]
[M1][A1]
[M1][A1]
B
B
W
W
B
W83
94
(
1st 2nd
(
(
5
(
9
5
8
5
8
3
8
5
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11a)
b)
c)
d)
e)
a= 1.25;b= 2
Scale
Shape
Smoothness
From graph,
1.05.1 s!x
22
2212
122
!
!
!
xy
x
x
x
x
Plot y=2x+2
From graph,
1.07.21.00 ss! orx
Gradient = 2.03.23.2
3.5s!
Tangent line draw at x =2
[B1][B1]
[B1]
[B1]
[B1]
[B2]
[M1]
[A1]
[M1][A1]
[B1]