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COMMONWEALTH SECONDARY SCHOOLPRELIMINARY EXAMINATION 2009

SECONDARY FOUR EXPRESS/FIVE NORMAL

MATHEMATICS 4016/02

Paper 2 27 August 2009

10 45 13 15 2 hours 30 minutes

Addi tional Materials: Writing PaperGraph Paper (1 sheet)

NAME: _____________________________ ( ) CLASS: ________

READ THESE INSTRUCTIONS FIRST

Write your name, index number and class on all the work you hand in.

Write in dark blue or black pen on both sides of the paper.

You may use a pencil for any diagrams or graphs.

Do not use staples, paper clips, highlighters, glue or correction fluid.

Answerallquestions.

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 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 .

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 of the marks for this paper is 100.

This question paper consists of 11 printed pages including the cover page.


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Mathematical Formulae

Compound Interest

Total amount = 1 100

n

r

P

+

Mensuration

Curved surface area of a cone = rl

Surface area of a sphere = 24 r

Volume of a cone = 21

3r h

Volume of a sphere = 34

3r

Area of triangle =ABC1

sin2

ab C

Arc length = r , where is in radians

Sector area = 21

2r , where is in radians

Trigonometry

sin sin sin

a b c

A B= =

C

A

2 2 22 cosa b c bc= +

Statistics

Mean =fx

f

Standard deviation =

22

fx fx

f f

CSS/Prelim2009/MATH/SEC4E5N/P2/AGS/Page 2 of 18


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Answerall the questions.

1 (a) National Petroleum Company (NPC) provides 3 different gradesof petrol. The price per litre of each grade of petrol is as follows:

Petro l Grade Price per Lit re ($)Grade 92 1.687Grade 95 1.767Grade 98 1.870

(i) Mr Soh pumped 42 litres of Grade 98 petrol for his car.Calculate the amount of money he paid for the petrol. [1]

(ii) Mr Sohs car has a petrol consumption rate of 12.5 kmper litre. Calculate the distance his car can travel with$50 worth of Grade 98 petrol. [2]

(iii) During a promotion month, the cost per litre of Grade 95

petrol was reduced by 15% but an instant rebate of $5was given to car owners who pumped Grade 92 petrol.What is the maximum volume of Grade 92 petrol to bepumped before the total cost becomes more than thecost of pumping Grade 95 petrol? Give your answer inlitres correct to 1 decimal place. [3]

(b) A shopkeeper sells two types of luxury handbags, Elegant andConvenient. Elegant handbags cost $7500 a piece andConvenient handbags cost $240 less.

(i) Write down, in its simplest form, the ratio of the cost of

Elegant handbags to Convenient handbags. [1]

(ii) Given that the shopkeeper sold an Elegant handbag at adiscount of 15% and a Convenient handbag at adiscount of $50, calculate the total percentage discountgiven on the sale of the handbags. [2]



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2 Each diagram in the sequence below is made up of a number of dots.

Diagram 1 Diagram 2 Diagram 3 Diagram 4

(a) Draw the next diagram in the sequence. [1]

(b) The table shows the number of dots in each diagram.Diagram 1 2 3 4 5 6Number of dots 1 6 13 22 p q

Write down the values ofp and of .q [2]

(c) The formula for finding the number of dots in the n th diagram is2An Bn C+ + , where A , B and C are constants. Find the

values ofA , B and ofC. [3] (d) Find the number of dots in Diagram 10. [1]

(e) Which diagram has 253 dots? [2]

3 (a) (i) Simplify2 2 2 26 9 5 45

6 3 2 6 3

a ab b a b

ac ad ac ad bc bd

+

+ .

[3]

(ii) Solve

( ) ( )3 5

42 1 3 1x x

=

.[2]

(b) A box contains several red discs and green discs. A disc is

randomly chosen and then placed back into the box and theprocess is repeated several times. The probability of choosing ared disc is p .

(i) Write down, in terms ofp , the probability of choosing a

green disc. [1]

(ii) The process was repeated 8 times. Find the probabilitythat(a) a red disc was chosen every time, [1]

(b) at least one green disc was chosen. [1]



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4 In the diagram, ,= 90ACB = 51ABC , = 35BEC ,, cm, = 103ACD = 4CD = 4.6BC cm and = 7.3CE cm.

Calculate

(a) CBE, [2]

(b) the length ofCA , [1]

(c) the length ofAD , [3]

(d) the area of triangle BCE, [2] (e) the shortest distance from E to CB produced. [2]



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5 An airplane is scheduled to fly to its destination 3500 km away. Thespeed of the airplane in still air is 600 km/h and the speed of wind,which is constant throughout, is x km/h. Due to a haze, the speed ofthe airplane in still air is reduced by 10%.

(a) Write down an expression, in terms ofx, for the time taken by

the airplane, in hours, if it is flying in the direction of the wind. [1]

(b) Write down an expression, in terms ofx, for the time taken bythe airplane, in hours, if it is flying against the wind. [1]

(c) The difference in arrival time is 1 hour and 10 minutes. Writedown an equation in terms ofx, and show that it reduces to

= .2 6000 291600 0x x+ [3]

(d) Solve the equation 2 6000 291600 0x x+ = . [3]

(e) Hence, find the time taken by the airplane, in hours andminutes, if it is flying in the direction of the wind when there isno haze. [2]



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6 In the diagram, O is the centre of the circle and points , S , T andlie on the circumference of the circle. The tangent at meets

produced at Q . ,

P

PR RT= PSTS =TQ SQ and = 36TRP .

(a) Find

(i) reflex angle POT, [2]

(ii) ,PTS [2]

(iii) .PQS [3]

(b) Show that PS bisects QPT. [3]



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7 (a) The diagram shows the cross-section of a swing in a childrensplayground. The seat is suspended on a 1.8 m long rope. Tooscillate the swing, the seat is pulled back to pointA andreleased to swing an angle of 62 to pointA . The seat makesone complete oscillation when it moves from pointA to pointA

and back to pointA again.

(i) Calculate the distance moved by the swing seat frompointA to pointA. [2]

(ii) Assuming that the swing oscillates regularly from pointAto pointA, find the speed of the swing, in metres perminute, if it makes 5 complete oscillations in 2 minutes. [2]

(b) The diagram shows the swing and a bench, 4 m away, in thechildrens playground. Both the bench seat and swing seat areat the same height above the ground.

(i) Calculate the angle of depression of the edge of thebench seat from the top of the swing. [2]

(ii) A bird flies from the edge of the bench seat to the top ofthe swing. Calculate the distance the bird flies. [2]

AA

1.8 m62

1.8 m

4 m



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8 (a) A factory manufactures small decorative ornaments. Eachdecorative ornament is made up of two parts: a solidhemisphere with radius 7 cm and a solid cone with a height 10cm, as shown in Diagram I.

Diagram I

(i) Calculate the volume of the hemisphere. [2]

(ii) The volume of the hemisphere is 3 times the volume ofthe cone. Find the base radius of the cone. [2]

(iii) Given that the solid cone is made with a light plasticmaterial with a density of 0.9 g/cm3, find the mass of thematerial used for the cone. [2]

The two pieces are joined together to form thedecorative ornament as shown in Diagram II.(iv) Calculate the total external surface area of

the ornament.

Diagram II [4]

(b) Given that the area of the major sector is98 cm2, find the value of and hencecalculate the perimeter of the majorsector.

[2]

10 cm7 cm

6 cm rad.



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Daily Wages ($)10 20 30 40 50 60 70

9 The cumulative frequency curve below represents the daily wages of80 male employees in a company.

Use the graph to estimate

(a) the median daily wage, [1]

(b) the interquartile range, [2]

(c) the value of z such that 77.5% of the male employees have adaily wage more than $ z. [2]

The box-and-whisker diagramrepresents the daily wages of60 female employees in thesame company.

(d) Find the median daily wage of the female employees and theinterquartile range. [3]

(e) Compare and comment briefly on the daily wages of the maleand female employees in the company. [2]

(f) Find the probability that an employee chosen at random from allthe employees has a daily wage less than or equal to $38. [2]

Daily Wages ($)

15 20 25 30 35 40 45 50 55 60 65

10

20

30

40

50

60

70

80

0

CumulativeFrequency



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

The following table gives the corresponding values ofxand y, which

are connected by the equation5

2y xx

9= + , correct to 1 decimal

place.x 1 2 3 4 5 6 7 8

12 7.5 4.7 2.3 0 -2.2y p -6.4

(a) Calculate the value ofp correct to 1 decimal place. [1]

(b) Using a scale of 2 cm for 1 unit on the x-axis and 1 cm for 1

unit on the y-axis, draw the graph of5

2 9y xx

= + for the

values ofx in the range 1 8x . [3]

(c) Use your graph to find the value of y when 2.5x= . [1]

(d) Use your graph to solve the equation =

52 1x

x.

[2]

(e) Find the coordinates of the point on the graph for which thegradient of the curve is -4. [2]

(f) By drawing a suitable straight line, solve the equation

0 for1 8x24 6 5x x = . [3]

END OF PAPER



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COMMONWEALTH SECONDARY SCHOOLPRELIMINARY EXAMINATION 2009

SECONDARY FOUR EXPRESS/FIVE NORMALMATHEMATICS 4016/02

1 (c) (i) Amt. of money paid = $1.870 42= $78.54 [B1]

(ii) Amt. of petrol =50

1.870

26.73797 litresDist. = 26.73797 12.5

334 km

[M1]

[A1]

(iii) Let the max. volume be V litres.

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