2009 ESpring Prelim Sec 4Ex EM P2

8/8/2019 2009 ESpring Prelim Sec 4Ex EM P2

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49229296.doc1/11

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


2/18

East Spring Secondary SchoolMathematics DepartmentDo It Right. Always

49229296.docTowards Excellence and Success page 2/11

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


3/18



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]


4/18



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


5/18



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


6/18



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


7/18



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]


8/18



(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]


10/18



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~


12/18



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


14/18



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]


16/18



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]


17/18



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


18/18



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]

Documents

2009 ESpring Prelim Sec 4Ex EM P2