Other School 3 EM P2

7/29/2019 Other School 3 EM P2

1/23

Class Registration No.Name:

MATHEMATICS

PAPER 2

O LEVEL PRELIMINARY EXAMINATION

SECONDARY 4 EXPRESS /

Duration: 2 hours 30 minutes

Additional Materials: Answer PaperGraph paper (1 sheet)

READ THESE INSTRUCTIONS FIRST

Write your name, class and registration number in the spaces at the top of this page and on all thework you hand in.Write in dark blue or black pen.

You may use a pencil for any diagrams or graphs.Do not use staples, paper clips, highlighters, glue or correction fluid.

Answer all 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 the answer is not exact, give theanswer 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 termsof .

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 or part question.

The total number of marks for this paper is 100.

This question paper consists of12printed pages (including this cover page).[Turn over


2/23

Mathematical Formulae

Compound interest

Total amount =

n

100

r1P

+

Mensuration

Curved surface area of a cone = rl

Surface area of a sphere = 2r4

Volume of a cone = hr2

3

1

Volume of a sphere = 3

3

4r

Area of triangle ABC= Cabsin2

1

Arc length = r , where is in radians

Sector area = 2

2

1r , where is in radians

Trigonometry

Asin

a=

Bsin

b=

Csin

c

Abccba cos2222 +=

Statistics

Mean =f

fx

Standard deviation =

22

f

fx

f

fx

2


3/23

Answer all the questions.

1 A rectangular water tank whose base measures 45 cm by 12 cm, is9

7filled with water to a

height of 14 cm.

3[Turn over

14 cm

12 cm

45 cm

(a) Find the capacity of the tank in cm3. [2]

(b) Ah Heng has some identical lead balls, each of radius 5 cm. He gently puts theballs one at a time into the tank until the water overflows.

(i) Find the volume of each lead ball. [1]

(ii) How many lead balls has he put into the tank? [2]

(iii) What is the volume of water that overflows from the tank? Correct your answerto two decimal places. [2]

2 The diagram shows a straight linePQR. The coordinates ofP,QandRare (0, 2), (h, k) and(7, 8) respectively.

y

P(0, 2)

Q(h, k)

R(7, 8)

x

(a) Find the gradient ofPR. [1]

(b) Find the equation of the linePR. [1]

(c) S is a point ( )b,7 such thatS,P andRare collinear. Find the value of [2].b(d) IfPQ=3QR, find the values ofhandk. [3]


4/23

3

C B

4

O

M

OABC is a parallelogram. M is the midpoint ofOBandQ is the midpoint ofOM.

P is the point onOA such thatOP =2PA. =3aand =c.

OA

OC

(a) Express, in terms ofaandc,

(i) , [1]

OB

(ii) , [1]

OQ

(iii) , [1]

CQ

(iv)

MP . [1]

(b) Explain why is parallel to

CQ

MP . [1]

(c) Find the value of the ratio of

(i) area ofMOP : area ofQBC, [1]

(ii) area ofMPQ : area ofQCM. [2]

A

Q

3a

c

P


5/23

4Cumulative Frequency

Time (hour5 10 15 20 25 30 35 40

10

20

30

40

50

60

70

80

Time (hours)

CumulativeFrequency

The cumulative frequency curve shows the number of hours a group of 72 students spend

playing computer games each week.

(a) Use the graph to estimate

(i) the range, [1]

(ii) the median number of hours, [1]

(iii) the interquartile range of the data, [2]

(iv) the number of students who spend at least 20 hours playing each week. [1]

(b) The box-and-whisker plot for the same data is given below.

1x 2x 5x3x 4xTime (hours)

(i) Find the values of , , , and . [2]1x 2x 3x 4x 5x(ii) What does the value of 24 xx represent? [1]

5[Turn over


6/23

5 X,Y andZare three points on horizontal ground.XY=100 m,XZ=80 m, ZXY =140and the bearing ofX fromY is 295.

(a) Calculate

(i) the bearing ofX fromZ, [1]

(ii) YZ, [2]

(iii) the area of triangleXYZ. [2]

(b) A vertical pole of height 60 m stands atX. A man walks alongYZ. Calculate

(i) the shortest distance fromX to the lineYZ, [2]

(ii) the greatest angle of elevation of the top of the pole from the man. [2]

6


7/23

6 (a) In the diagram,O is the centre of the circle,TDandTE are the tangents to the circleatD andE respectively, = 34BEC and = 52EAD .

Find(i) , [1]ECD

(ii) , [1]EOD

(iii) , [1]ODE

(iv) , [1]ETD

(v) . [1]ESD

7[Turn over


8/23

6 (b) BagA contains 5 blue balls and 8 red balls. BagBcontains 2 blue balls and 4 redballs. A ball is taken at random from BagAand placed in BagB. A ball is thenchosen from BagB. The tree diagram of the sample space is shown below.

8

B

R

B

R

B

R

BagA BagB

13

5

7

3

p

q

r

s

Outcome

BB

BR

RB

RR

Find(i) the values ofp,q, r ands, [2]

(ii) the probability that

(a) both the balls taken are blue, [1]

(b) at least one of the balls taken is red. [1]


9/23

7 The diagram below shows the speed-time graph of a truck and a car travelling on the samestretch of road. The car started its journey from rest, 5 seconds after the truck andtravelled at an acceleration equal to that of the truck in the first 6 seconds. The carreachedvm/s when .15=t

0 6 155

car

12

16

v

Time (tseconds)

Speed (m/s)

truck

(a) Calculate the acceleration of the truck in the first 6 seconds. [1]

(b) Calculate the speed of the truck when 13=t . [2]

(c) Find the value ofv. [2]

(d) Given that the car and truck started from the same point, find the time taken forthe car to overtake the truck. [4]

9[Turn over


10/23

8 (a) The table shows the income tax rates for the year of assessment 2001 for both Cherryand Shawn.

Chargeable Income ($) Rate (%) Gross Tax Payable ($)On the firstOn the next

20 00010 000

04

0400

On the firstOn the next

30 00010 000 6

400600


40 00020 000 9

10001800


60 00040 000 12

28004800

On the first

On the next

100 000

100 000 18

7600

18 000On the firstAbove

200 000200 000 21

25 600

(i) Cherrys chargeable income in the year 2000 was $45000. Calculate herincome tax payable in the year 2001. [2]

(ii) If Cherry were to invest half of her chargeable income in year 2000 in afinancial scheme for 10 months which promised 8% simple interest returnper annum, how much interest would she get? [2]

(iii) Shawns tax payable was $9796 in the year 2001. Find his chargeableincome in the year 2000. [2]

(b) Shawn bought a car by hire purchase. The cash price of the car was $90000. Hepaid a deposit of $9000 and the rest by monthly instalments over a period of

2

17 years. The bank charges an interest rate of 3% per annum on the amount

loaned.

(i) Calculate the total interest he had to pay. [2]

(ii) If the first instalment was $900, find the monthly instalment for theremaining months. (Assume that the monthly instalments for each of therest of the months are equal.) [2]

10


11/23

9 (a) Express as a single fraction in its simplest form

13

11

32

8

+

xx. [2]

(b) Simplify22

22

23

818

nmnm

nm

. [3]

(c) Given that (p+q)2 =8 andpq=3, find the value of

2

2

qp. [2]

(d) If ,3

5

32

23=

+

xy

yxfind the value of .

y[3]

10 Water flows out at different rates from two tapsAandB, when turned on to its fullest.Water flows out from TapAat litres per minute and from tapBat 5 litres per minutefaster than tapA.

A rectangular tank with dimensions of 300 cm by 250 cm by 120 cm is to be filled withwater. It takes 5 hours longer to fill the tank with water using tapAalone than by usingtapBalone.

(a) Find the volume of the tank in litres. [1]

(b) Write down an expression, in terms ofx, the time taken to fill the tank by usingonly(i) tapA, [1](ii) tapB. [1]

(c) Form an equation inxand show that it reduces to . [2]015052 =+ xx

(d) Solve the equation . [2]

015052 =+ xx

(e) State the rates of water flow from tapAand tapB. [2]

(f) Find the time taken, in hours, to fill the rectangular tank if both tapsAandBare turned on together. [2]

11[Turn over


12/23

Answer the whole of this question on a sheet of graph paper.11

412

The variablesxandyare connected by the equationy=6

+x

. The table below2x

ofxandy. The values ofyare given correct to one

here appropriate.

shows some corresponding values

decimal place w

x 1 1.5 2 3 4 5 6 7 8y 8.2 4.4 2.7 1.5 1.7 2.6 4 k 8.2

(a) Calculate the value ofk, correct to one decimal place. [1]

(b) c axis, draw a horizontal x-axis for 0x 8

[3]

(c) Using your graph, find the value ofywhenx=6.5. [1]

(d) By drawing a tangent, find the gradient of the curve at the point (1.5, 4.4). [2]

(e)

Using a scale of 2 cm to 1 unit on ea h

and a vertical y-axis for 0y 10.

On your axes, plot the points given in the table and join them with a smooth curve.

Use your graph to find the values ofxfor

(i) 2412

6

2

=+x

, [2]

(ii) 01624

3

2

+ xx

. [3]

END OF PAPER

12


13/23

13[Turn over

1

)(aHeight of the tank = 1814

9=

7

Vol =45 x 12 x 18 = 9720 cm3(b) (i)

of each ball =Vol

3

53

4

=523.5987756 =524 cm

3

Or 66666667.5235142.33

4 3 =

Acc 2ept 5 4 (3 sf)

ro3

2523

(ii)

No. of balls =1571

1964

3

2523

2160

3

2523

9

29720

==

=5 balls

r(O 124498759

7.

= =.4

523

2160

7.523

9

29720

=

5 balls)

(iii)5 = (2 d.p.)33.458

9

29720

3

2523

2(a)

radient of PR =G7

6

07

28=

(b)quation of PR:E 2

7

6+= xy

(c) Gradient of =gradient ofPR SP

07

2

7

6

=b

b= 4

OR 2

7

6+= xy

277

6+= b

b= 4


14/23

2 (d)

(d)

4

3

28

2=

k

2

16=k

4

3

7=

h

4

15=h

3

14


15/23

15[Turn over

3)(a (i) OB = +c3a

(ii)c

Accept4

1=

4

1(3a+c) =

4

3a+

4

1OQ

(3a +c)

(iii) cAccept

4

3(

=c+43a +

41c =

43a -

43+OQCQ=CO

a c)

(iv) OM +MP =OP

MP =2a -2

1OB =2a -

2

1=OP - OM (3a+c)

=2

1c

Accept2

1(

a-2

1a c)

a c)(b)

SinceCQ = 4

3a - 4

3c= 4

3(

MP=

2

1(a c)

CQ =4

3=

2

3MP x 2 MP

CQ is parallel to MP

(c) )(i SinceMOP is similar toMOP

aarea ofMOP : are ofQBC

=2

2

: 3

2

=4 : 9 = .9

4

(ii)

3

2

2

12

1

==

=

CQ

MP

hCQ

hMP

QCMofArea

MPQofArea

4Cumulative Frequency

80


16/23

) =35(a (i) Range =40 5(ii) Median =21 h Accept 20.5 21.0(iii) ccept 25 25.3Upper Quartile =25 A

Lower Quartile =15Interquartile Range =25 15 =10 h ccept 10 10.2A

(iv) 2 32 =407

(b) (i) =15, =21, =25 to 25.3 (part (a)(iii)),

=40

1x = 5, 2x 3x 4x

5x(ii) represents inter-quartile range24 xx

16


17/23

5

(a) (i) 140 - (360 - 295) =75Bearing ofX fromZ=075

(ii) YZ2 =802 +1002 2(80)(100) cos 140YZ=169.28 169 m

(iii)Area of XYZ = 140sin)100)(80(

2

1 257015.2571 = m2

(b) (i) Lethbe the shortest distance fromX toYZ

15.257128.1692

1= h

h=30.4 m(ii) Let be the angle required

4.30

60tan =

=63.1

17[Turn over


18/23

6 (a)

(a) (i) ECD =52(ii) EOD =2 x 52 =104(iii) ODE =180 - 90 - 52 =38 Accept 90 - 52 =38(iv) ETD =180 - 52 - 52 =76(v) ESD =180 - 52 =128

(b)

B

R

B

RB

R

BagA BagB

5

3

r

s

Outcome

BB

BRRB

RR

(b) (i)

13

8=p ,

7

4=q ,

7

2=r ,

7

5=s

(ii) (a)

91

15

7

3

13

5= (OR 0.165)

(b)

91

76

91

151 =

Accept

91

76

7

5

13

8

7

2

13

8

7

4

13

5=++

18


19/23

7

19[Turn over

0 6 155

car

12

16

v

Time (tseconds)

Speed (m/s)

truck

(a)Acceleration = 2

6

12= m/s2

(b) using similar triangles method

Speed = )613(9

121612 =

+

9

115 m/s

OR cmxy +=

cxy +=9

4, ( )12,6

c+= 69

412 ,

3

28=c ,

3

28

9

4+= xy

When 13=x , 1.159

115

9

136

3

2813

9

4===+=y m/s

(c)Acceleration =

515

v

20

102

=

=

v

v


20/23

7 (d)

7(d) Lettbe the time in seconds when the car overtook the truck from the start of thetrucks journey.Speed of the car attseconds = )5(2 t m/s

For overtaking, Distance travelled by truck =Distance travelled by car

1.2188.4

2

)103)(1(42626

010326

251024016162

)5(2)5(2

1)15(169)1612(

2

1126

2

1

2

2

2

or

t

tt

ttt

ttt

=

=

=+

+=+

=+++

Since ,5>t 1.21=t

Time taken for car to overtake truck = 1.1651.21=

s

20


21/23

8

(a)(i) 1000 + 5000100

9 =$1450

(ii)

10012

108

2

45000

=$1500

(iii) Let Shawns chargeable income be $(100 000 +x)Income tax for first $100 000 =$7600Income tax for $x=9796 7600 =$2196

2196100

18= x

x=$12 200100 000 +12 200 =$112 200

(b) (i) Amount remaining =$90000 - $9000 =$81000Total interest

=100

5.7381000 =$18225

(ii) Amount paid by instalment =$81000 +$18225 =$99225

Monthly instalment for other months =190

90099225

=$1104.78

9

(a))13)(32(

412)13)(32(

)32(11)13(8+

+=+

+xx

xxxxx

(b)

)(

)23(2

))(23(

)23)(23(2

))(23(

)49(2 22

nm

nm

nmnm

nmnm

nmnm

nm

=

+

+=

+

(c)5

4

)3(48

4

22)(

4

2 222=

=

+=

+ pqpqqppqqp

(d)35

3223 =+ xyyx

( ) ( yxxy 233325 =+ )yxxy 691510 =+

yx 166 =

yx 83 =

8

3=

y

21[Turn over


22/23

10

(a) Volume of tank = lcm 9000109120250300 36 ==

(b)(i)

Time taken by tapAonly = x

9000mins (OR hx

150)

(ii)Time taken by tapBonly =

5

9000

+xmins (OR h

x 5

150

+)

(c)

01505

)5(3015030

1)5(

30)5(30

15

3030

3005

90009000

2 =+

+=+

=+

+

=+

=+

xx

xxxx

xx

xx

xx

xx

10(c)(OR 5

5

150150=

+xx

)

(d)

10or15

0)10)(15(

015052

==

=+

=+

xx

xx

xx

(e) tapA : 10 l/mintapB : 10 +5 =15l/min

(f) Speed of tapAandB=10 +15 =25l/min

Time taken to fill the tank = 36025

9000= mins =6 hours

22


23/23

11 Answer the whole of this question on a sheet of graph paper.

(a) 9.5=k (1 d.p.) B1(c) 9.4,5.6 == yx (4.7 to 5 acceptable) A1

(d) m = 4.8 (4.6 to 5.0 acceptable) Tangent 1 mark;Gradient 1 mark

(e) (i) Draw 2=y M1

4.2=x or 4.4 A1(ii)

01624

3

2

+ xx

02

812

6

2

+xx

M1

02

4412

6

2

+xx

Draw the straight line2

4 xy += M1

9.74.1 x A1Accept Range )1.0(

(b)

Scales, points, smooth curve, A3

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