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1 Class Register Number Name

CHIJ SECONDARY (TOA PAYOH)

PRELIMINARY EXAMINATION 2010 SECONDARY FOUR (SPECIAL/EXPRESS)

MATHEMATICS 4016/02 Paper 2 31 August 2010

2 hour 30 minutes Additional Materials: Answer Paper

Graph paper (1 sheet)

READ THESE INSTRUCTIONS FIRST Write your name, register number and class on all the work 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 the answer to three significant figures. Give answers in degrees to one decimal place. For , use either 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 the brackets [ ] at the end of each question or part question. The total number of marks for this paper is 100.

This document consists of 9 printed pages. [Turn over


2

Mathematical Formulae Compound interest

Total amount = 1100

nrP +

Mensuration

Curved surface area of a cone = rl

Surface area of a sphere = 24 r

Volume of a cone = 213r h

Volume of a sphere = 343r

Area of triangle ABC = 1 sin2ab C

Arc length = r , where is in radians

Sector area = 212r , where is in radians

Trigonometry

Cc

Bb

Aa

sinsinsin==

Abccba cos2222 += .

Statistics

Mean = fxf

Standard deviation = 22fx fx

f f


3

1 (a) Express 24132

)12(25

xx

x

as a single fraction in the simplest form. [3]

(b) Given that sst

+

=342 , express s in terms of t. [3]

(c) Solve the equation xx 37)32( 2 = . [2]

(d) Factorise completely m3 3m2 9m + 27. [2]

2 Mr Chan wishes to buy a car that runs economically on petrol.

(a) The petrol consumption of Car A is x km per litre.

Write down an expression for the number of litres of petrol, in terms of x, used by Car

A to travel 250 km. [1]

(b) The petrol consumption of Car B is )2( +x km per litre.

Write down an expression for the number of litres of petrol, in terms of x, used by Car

B to travel 250 km. [1]

(c) Car A uses 3 litres of petrol more than Car B for the 250 km journey. Use this information to form an equation, in terms of x, and show that it simplifies to

050063 2 =+ xx . [2]

(d) Solve the equation 050063 2 =+ xx , giving your answers correct to 2 decimal places.

[3]

(e) Explain which value of x from your answer in part (d) is valid. [1]

(f) Hence find the number of litres of petrol used by Car B for every kilometre travelled.

Give your answer correct to 3 significant figures. [1]

(g) The price of petrol in Singapore is S$1.78 per litre while the same grade of petrol costs

RM 2.09 per litre in Malaysia. Calculate the amount that Mr Chan can save, in S$ when

he bought 19.5 litres of petrol in Malaysia. The exchange rate was S$1=RM 2.31. [2]

__________________________________________________________________________


4

3 The diagram shows the positions P, Q, R and S of four locations on a flat land. P, S and R are on a straight line. QS = 1.4 km, RS = 1.9 km, angle SPQ = 25, angle PQS = 35 and Q is due east of P. Calculate

(a) the distance QR, [3]

(b) angle SRQ, [2]

(c) the bearing of Q from R. [2]

A vertical tower T of height 165 m stands at the point S. A man walks from the point Q towards P. Calculate

(d) the greatest angle of elevation of the top of the tower from the man, [3]

(e) the smallest angle of depression of the man from the top of the tower as he walks from Q to P, leaving your answer correct to one decimal place. [2]

4 There are 20 boys and 20 girls in class A. In class B there are 22 boys and 18 girls. A pupil is

selected at random from class A and moved to class B, then a pupil from class B is selected at random and moved to class A. Using a probability tree diagram or otherwise, find the probability that

(a) class A has exactly 19 boys and 21 girls, [1]

(b) class B has exactly 22 boys and 18 girls, [2]

(c) class A has exactly 18 boys and 22 girls, [1]

(d) class A has more boys than girls. [2]

R

P

N S

T

Q

1.9 km

1.4 km

25 35


5 5 The rows, R1, R2, R3, R4 , of a sequence of even numbers in the number triangle are given

as follows:

Row Number Triangle No of even

numbers (N) Sum of Row (S)

Average of Row

(A) R1 2 1 2 2 R2 4 6 2 10 5 R3 8 10 12 3 30 10 R4 14 16 18 20 4 68 17 R5 22 24 26 28 30 5 130 26 R6 32 34 36 38 40 42 6 p q RN N S A

(a) Find the values of p and q. [1]

(b) Write down a formula connecting A and N. [2]

(c) Write down a formula connecting S and N. [2]

(d) Write down the value of S and of A for the 20th row. [2]

(e) Explain why the number 900 could not appear in the A column. [1]

6 Mr Chan has $50000 for investment.

He decides to invest in an antique Chinese vase with a selling price of $12500. (a) If he pays cash, he will get a discount and needs to pay only $10250 for the vase.

Calculate the percentage discount. [2] (b) Given that the S10250 price tag is inclusive of the 7% Goods and Services Tax (GST),

calculate the amount of GST paid. [2] After investing $10250 on the vase, he invests $20000 into a fixed deposit in Bank A that pays 2.25% per annum simple interest and the remaining money into an investment product in Bank B that pays 2.1% compound interest per annum compounded every 3 months. (c) Calculate the amount that he will get from Bank A at the end of 5 years. [1] (d) Calculate the amount that he will get from Bank B at the end of 5 years. [2] At the end of 5 years Mr Chan sold his antique vase at an auction for $13800. (e) Calculate the percentage gain for keeping the vase for 5 years. [1]

(f) Calculate the overall percentage gain for Mr Chans investment of $50000 over the 5 year period. [2] ___________________________________________________________________________


6 7 Answer the whole of this question on a sheet of graph paper.

It is given that 924 +=x

xy . Corresponding values of x and y are given in the following

table.

x 2 2.3 2.6 3 4 5 6 7 8 y 5 3.7 a 2 1 0.8 1 1.4 2

(a) Calculate the value of a correct to 1 decimal place where necessary. [1] (b) Using a scale of 2 cm to represent 1 unit on the x-axis and 4 cm to represent 1 unit

on the y-axis, draw the graph of 924 +=x

xy for 82 x . [3]

(c) By drawing a suitable straight line on your graph, find the solutions of the equation .072364 2 =+ xx [3]

(d) Find the range of values of x for which .6

45.524 xx

x ++ [2]

(e) By drawing a suitable tangent to your curve, find the x-coordinate of the point on the curve at which the tangent is equal to 5.0 . [2]

(f) Given that hy = , where h is a constant, is a tangent to the curve in the given range,

find the approximate value of h. [1]

____________________________________________________________________________

8 In the diagram, DC is a diameter of the circle, centre O. The tangent to the circle at point A

meets DC produced at B. The length of OA is 18 cm and =AOC 1.08 radians.

(a) Calculate the length of BC. [2]

(b) Calculate the area of the shaded region. [2]

(c) Calculate the perimeter of the shaded region. [3]

_________________________________________________________________________


7

9 A group of 70 pupils from an all boys school sat for a Math quiz and the following cumulative graph shows the distribution of the marks scored.

(a) Find the value of h and of k in the following frequency table. [1]

Marks (x) 6560


8 10

In the diagram, the points A, B, C, D and E are on the circle with centre O. AD and CE meet at the

point X. AXC = 98o, EAD =(x+7)o, ADC = 4yo and AOC = (5y + x)o .

(a) Explain clearly why yx 3= . [2]

(b) Explain clearly why .)91(4 xy = [2]

(c) Find the values of x and y and hence find the value of the obtuse angle ABC. [2]

(d) Given that AX=CX, find the value of the acute angle OCE. [2]

(5y + x)

98

4y

(x + 7)

A

B

C

DE

O

X


9 11

The figure shows a solid made up of a cylinder of radius r cm, and a cone of radius 1.2 cm sitting at the centre of the cylinder. The slant height of the cone when produced will meet at the end of the cylinder as shown in the diagram.

The height of the cylinder is 6.8 cm and the height of the cone is 4.6 cm.

(a) Find the slant height of the cone. [1]

(b) Find the value of r, giving your answer correct to 3 decimal places. [2]

(c) Calculate, giving your answers correct to 2 decimal places,

(i) the volume of the solid, [2]

(ii) the total surface area of the solid. [3] (d) Given that the solid is to be painted with a coat of paint of thickness 0.005 cm,

calculate the number of 5-litre tins of paint that must be purchased to paint 38500 of such solids. [2]

********** THE END **********

r cm

4.6 cm

6.8 cm

1.2 cm


10 Qn. # Answer Qn. # Answer 1a )12)(12(2

94+

+

xxx 5a p= 222 and q= 37

1b 2

2

41)31(4

tts

+

=

5b 12 += NA

1c 41

=x or 2=x 5c NNNNS +=+= 32 )1(

1d ( )( )( )333 + mmm 5d 8020,20 == SN 401=N 2a

lx250 5e 900 is a perfect square while the numbers in the A column are not

2b l

x 2250+

6a 18%

2c

x250

2250+x

= 3, 050063 2 =+ xx

6b 7% GST = $670.56

2d 95.11=x or 95.13=x (2 dec pl)

6c Amount from Bank A= $22250

2e 95.11=x as negative x is not valid 6d Amount in bank B= $21930.51 2f 0717.0 l 6e % gain = 34.6% 2g S 07.17$ 6f Total Return = $7980.51 (M1)

% Gain = 16.0% 3a 71.1 km 8a BC = 20.2 cm 3b 3.45 8b Area of shaded region = 128 cm 2 3c 7.199 8c Perimeter = 20.19 + 19.44 + 33.6819

(M2) = 73.3 cm

3d 6.11 9a h = 28 k = 23 3e 5.0 (1 dec pl) 9b Mean = 70 SD = 5.26 4a

419 9c The width of the graph will be narrower.

The graph will look more like an elongated S The gradient will be steeper.

4b 4121

4c 0 4d

4111


11 Qn. # Answer Qn. # Answer 7a 8.2=a ( 1 dec pl ) 10a AOC = 2ADC

( at centre = 2 at circumference)

yxyxy

3)4(25

=

=+

7b Draw axes and plot all given points

Draw smooth curve through all plots 10b AEC = y4 ( in same segment)

98 yx 4)7( ++= ( )int soppsumofext =

xy = 914

3

2.5

2

1.5

1

0.5

-0.5

-1

1 2 3 4 5 6 7

f x( ) = x+5

x( )-4

(1, 2)

10c =13y

== 39133x ABC = 180 y4 = 180 )13(4 = 128 ( s in opp seg are supplementary)

7c

6,3

313

331924

0241234

072364

072364 2

orxphfromthegra

xdrawy

xx

x

xx

xx

xx

=

=

+=+

=+

=+

=+

10 d OCE= = 33841

7d 5.36

45.35.52 +=+ xxxx

Draw 21

6+=xy

From graph 8.78.3 x

11a cml 754.4=

7e x = 4 11b 974.2=r cm (3 dec pl) 7f =h ( RangeAcceptable : 7.0 to )8.0 11ci Vol of solid = 195.87 cm 3 ( 2 dec pl )

11cii Total surface area =196 .03 cm 2 ( 2 dec pl )

11d 8 tins of 5 l paint is needed.