ACSIEM2

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ACS(Independent)Math Dept/Y4EXP_GEP/EM2/2008/Prelim

Answer all the questions

1 (a) Given thatu

vuw

+

=

1, express u in terms ofw and v. [2]

(b) Solve the equation 094

314

32

12 =

+

x

x

x

x. [3]

(c) Solve the equation3

3

5

3 842 ++ =

x

x . [3]

2 Tickets to the Carnival of Arts fundraising concert were sold by students in a particular

school. The seats were divided into 4 blocks. The number of tickets sold for Friday and

Saturday are summarized in the table below:

The price per ticket is $25 for Block A, $18 for Block B, $15 for Block C and

$12 for Block D.

(a) Write down a 2 4 matrix T to represent the number of tickets sold by the studentsfor the two days. [1]

(b) Write down a column matrix S to represent the price per seat. [1]

(c) Find TS and interpret the elements in the matrix. [3]

(d) The principal decided to hold the concert for one more day. The price of each ticket

for the seats in Block A was increased while the ticket prices for the other blocks

remained unchanged. What was the new price of each ticket in Block A if $6 950

was raised given that 70 Block A, 90 Block B, 120 Block C and 90 Block D tickets

were sold? [3]

Block A Block B Block C Block D

Friday 54 95 120 77

Saturday 60 75 118 80

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3 In the diagram below,A,B, C,D andEare points on the circle with centerO.DCis parallel

to the diameterEB of the circle with centerO.

(a) Given that OAC= 45, ACB = 30,DEFis a straight line andFAHis a tangent to

the circle, calculate,

(i) AOE, [2]

(ii) BAC, [2]

(iii) AEF. [2]

(b) Prove that AEFis similar to CAE. [2]

C

B

A

D

E

F

O

30

45

G

H

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4 In the diagram,BC= 4BD andDA = 5DX. Mis the midpoint ofAC. BD = a and CM= 2b.

(a) Express, as simply as possible, in terms ofa and/orb,

(i) DC,

(ii) DA ,

(iii) DX . [3]

(b) Show that5

4=BX (2a + b). [1]

(c) Express BM as simply as possible, in terms ofa and b. [1]

(d) Find

(i)BM

BX, [1]

(ii)AMXofarea

ABXofarea

, [1]

(iii) ABCofarea

ABXofarea

. [2]

B

X

D C

M

a

2b

A

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5 Peter drove from Singapore to Kuala Lumpur and then returned to Singapore. The distance between Singapore and Kuala Lumpur is approximately 340 km via the North-SouthHighway. During the journey from Singapore to Kuala Lumpur, he drove at an averagespeed ofx km/h.

(a) Write down the time, in hours, taken for the journey from Singapore to KualaLumpur. [1]

(b) On the return journey, his average speed was reduced by 15 km/h. Write down thetime, in hours, taken for his return journey. [1]

(c) Given that the difference in time for the two journeys was 50 minutes, form an

equation in x and show that it reduces to 06120152 = xx . Hence, solve theequation, giving your answer correct to one decimal place. [4]

(d) Peter paid a total toll of RM $71.40 for the two journeys. Calculate the total toll hepaid in Singapore dollars given that the exchange rate was S $50 = RM $117.50.Give your answer correct to the nearest cent. [2]

(e) Peter set off at 22 15 hrs on Saturday for the Singapore-Kuala Lumpur journey.Find the day on which he arrived in Kuala Lumpur and also give the time of arrivalcorrect to the nearest minute. [2]

6 (a) The utilities bill of a household consists of three components: water, gas andelectricity. In a certain month, Sams household used 18.5 m3 of water, 78 kWh ofgas and 510 kWh of electricity. The tariff rate for water, gas and electricity is$1.24 per m3, $0.18 per kWh and $0.24 per kWh respectively.

(i) Find the amount payable for water usage. [1]

(ii) Given that the rate of GST on the utilities bill is 7%, find the GST Sam has to pay.

(b) In the following month, the consumption of gas decreased to 70 kWh, that ofelectricity increased by 20%, while the consumption of water remained unchanged.Calculate

(i) the percentage decrease in the consumption of gas, [1]

(ii) the total utilities bill inclusive of GST to 2 decimal places, [2]

(iii) the percentage change in Sams utilities bill inclusive of GST for the two

months, stating whether it is a decrease or increase. [2]

(c) On 1st January 2004, Sam deposited $4 500 in a Platinum-saver account with abank. Subsequently on 1st January of each following year, he deposited $4 500 intothe same account. If the bank pays compound interest at 4% per annum and Samwithdrew all the money in the account on 31st December 2005, how much moneydid he get? [3]

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7 (a) 40 students from class Y were weighed and the results, recorded to the nearest

kilogram were as follows.

(i) Show that the values ofp and q are 3 and 6 respectively given that the mean

is 68 kg. [3]

(ii) Find the standard deviation of the weights of the students in the class. [2]

(iii) Given that the mean and standard deviation of another class Z is 60 kg and

7 kg respectively, comment on the distribution of the weights of the students

in both classes. [2]

(b) To achieve the ideal weight, a student must be 70 kg and below.

(i) Calculate the probability that if two students in class Y are chosen at random,

one achieved an ideal weight but the other did not. [2]

(ii) Calculate the probability that if three students in class Y are chosen at random,

at least one student achieved the ideal weight. [2]

Weight (x kilograms) Frequency

54

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8 Figure I shows an open trough, constructed by taking a slice of a cylinder of radius 1.2 m

and length 2.8 m. The cross-section ACB of the trough is a segment of a circle. O is the

center of this circle and AOB = 90.

Water is poured into the trough by using a conical bucket shown in Figure II, which has

radius 0.5 m and height 1.6 m. Find

(a) the area of the segmentACB, [2]

(b) the volume of the trough, [1]

(c) the total internal surface area of the trough, [3]

(d) the volume of the bucket. [2]

The trough is to be completely filled with water.

(e) What is the minimum number of buckets that must be used? [1]

(f) What is the height of the water in the last bucket used in (e) if all earlier buckets

used had been full? Give your answer correct to 1 decimal place. [2]

0.5 m

1.6 m

Figure II (bucket)

A

Figure I (trough)

B

C

1.2 m

2.8 m

O

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9 In a friendly street soccer match, some of the players are positioned in a formation as shown

in the figure below, which is not drawn to scale. W, X, Yand Zare four players on a level

ground. Given that X is on a bearing of 153 from W, = 51WYZ , = 85WZY ,

Yis due south ofZ, WZ= 7.8 m and WX= 5.6 m.

(a) Calculate

(i) the bearing ofYfrom W, [2]

(ii) WY, [2]

(iii) area of WXY. [3]

(b) Another playerP was standing in the line between W and Y such that he was

nearest toX. Find this distancePX. [2]

(c) A bird was flying at a point 5 m above X. Find the angle of elevation of the bird

from Y, when it was directly aboveX. [3]

W

Y

Z

N

N

7.8 m

X

85

153

51

5.6 m

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Yr 4 Prelim Exams EM P2 Answer Key

1a))1)(1(

2

ww

vwu

+

+=

b) 1.08 or5.08

c) 0.5 or 3

2a)

=

801187560

771209554T

b)

=

12

15

18

25

S

c)

=

5580

5784TS

The elements represent the total amount collected on Friday and Saturday from the saleof the tickets.

d) $35

3ai) 120

aii) 15

aiii) 75

4ai) aDC 3=

aii) baDA 43 +=

aiii) )43(5

1baDX +=

c) )2(2 baBM +=

di)5

2

ii)3

2

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iii)5

1

5a) hx

time340

=

b) hx

time15

340

=

c) 86.1

d) S$30.38

e) 02 12 hrs, Sunday

6ai) $22.94

aii) $11.16

bi) 10.3%

bii) $195.19

biii) 14.5% (increase)

c) $9547.20

7aii) 5.73kg

aiii)

]1[.lub

,

]1[.

,,

AYclassofthatthangreaterisZcofstudents

ofweightsofspreadthethatshowsthisYSDthangreaterisSDSince

AZclassofthatthangreaterisYclass

ofstudentsofweightaveragethethatshowsthisxthangreaterisxSince

Z

ZY

bi)260

133

bii) 520

469

8a) 0.411m2

b) 1.15m2

c) 6.10 m2

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d) 0.419m3

e) 3

f) 1.5m

9ai) 129

aii) 10.0m

aiii) 11.4 m2

b) 2.28m

c) 28.4

10ai) p = 8.5, q = -0.5

e) 0.6 or 5.9

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