4e5n2009emprelimp2 (andss)

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ANDSS 4E5N Prelim 2009 Math (4016/02) [Turn over

ANDERSON SECONDARY SCHOOL

2009 Preliminary Examination

Secondary Four Express / Four Normal / Five Normal

CANDIDATE

NAME

CENTRENUMBER

SINDEXNUMBER

MATHEMATICS 4016/02Paper 2 03 September 2009

2 hours 30 minutesAdditional Materials: Writing paper (10 sheets)

Graph paper (2 sheets)

Geometrical instruments

READ THESE INSTRUCTIONS FIRST

Write your name, centre number and index number on all the work you hand in.Write in dark blue or black pen 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.

Answer all the questions.

If working is needed for any question it must be neatly and clearly shown in thespace below the question.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 decimalplace.

For , use either your calculator value or 3.142, unless the question requires theanswer 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 partquestion.The total of the marks for this paper is 100.


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Mathematical Formulae

Compound Interest

Total amount =

nr

P

+ 1001

Mensuration

Curved surface area of a cone = rl

Surface area of a sphere = 24 r

Volume of a cone = hr2

3

1

Volume of a sphere = 3

3

4r

Area of triangleABC= Cab sin2

1

Arc length = r , where is in radians

Sector area = 22

1r , where is in radians

Trigonometry

C

c

B

b

A

a

sinsinsin==

Abccba cos2222 +=

Statistics

Mean =f

fx

Standard deviation =

22

f

fx

f

fx



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1 (a) (i) Factorise completely. [2]23 9182 aaa +

(ii) Hence simplify352

91822

23

+

+

aa

aaa. [2]

(b) Solve the equation 3

1

1

2

1

322

=

+

xx

x

. [3]

(c) Express in the form , where a, b, and c are constants.2241 xx + 2)( cxba ++

Hence, solve the equation , leaving your answers to0241 2 =+ xx2 decimal places. . [3]

2

Diagram 1 Diagram 2 Diagram 3 Diagram 4

In each diagram, unit equilateral triangles shaped are arranged inside a largerequilateral shaped triangle. The unit triangles are shaded if they have one or two oftheir vertices on the edges of the larger triangle. Those with none or three of their

vertices on the larger triangle are not shaded.

The table below shows some of the patterns.

Diagram Number 1 2 3 4 5 n

Number of shaded unit triangles 0 3 6 9 a p

Total number of unit triangles 4 9 16 25 b q

Number of unshaded unit triangles 4 6 10 16 c r

(a) By considering the number patterns in the above table,

(i) state the value ofa, ofb and ofc, [3]

(ii) find, in terms ofn, an expression forp and for q. [2]

(b) By considering the relation amongp, q and r, show that r . [2]42 += nn

(c) Form an equation relatingp and q. [1]



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3 In the triangle KLM, andLMKL = NLMLKM = .


(a) Prove that . [2]NMLN =

(b) Prove that KLMand LNMare similar. [2]

(c) Given that KL = 8 cm and KM= 12 cm, find

(i) the length ofLN,

(ii) the ratioKLM

KLN

ofarea

ofarea. [4]

4

ABCD is a rectangle. Points P and Q lying on the linesAB andBCrespectivelysuch that ,cmxADAP == BQPB = and cmyAB = .

(a) Find an expression, in terms ofx and/ory, for the area of

(i) APD,

(ii) BPQ. [2]

(b) If the area ofAPD : area ofBPQ = 4 : 1, form an equation inx andy, and

show that it can be simplified to 3 . [2]048 22 =+ yxyx

22 =+ yxyx

(c) Solve this equation 3 , expressingx in terms ofy. [2]048

(d) Show that the perimeter of the quadrilateral PQCD is x2

234 +cm. [3]

L

N

K

A

M

P B

Q

CD


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5

North

T

B

A

C

35

52

A,B and Care points on horizontal ground whereB is due south ofA,AB = 35 m,BC= 52 m and area of triangleABCis 468.7 m2.

(a) TriangleABCis represented on a map with a scale of .250:1Find the area on the map, giving your answer correct to nearest whole. [2]

(b) Show that the bearing ofCfromB is 329. [3]

(c) Find the length ofAC. [2]

A vertical mast TA stands atA and the angle of elevation ofTfromB is 22.

Find

(d) the height ofTA, [2]

(e) the angle of depression ofCfrom T. [1]



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6 (a) The exchange rate between Singapore dollars (S$) and United State dollars(US$) was .65.0$US1$S =For every transaction, the bank charges a 1.5 % commission.

A traveller bought US$3500 from the bank.Calculate the total amount, in Singapore dollars (S$), the traveller paid to

the bank. [2]

(b) A bank offers its customers different saving and investment schemes to growtheir money:

Option A: Deposit a sum of money in a special saving account to earnan interest of 0.4 % per annum compounded monthly.

Option B: Deposit a sum of money in a fixed deposit account at a simpleinterest ofr% per annum.

Option C: Invest in a financial investment product which guaranteesyearly 3.2 % interest return plus a special bonus of $200 forinvesting for a full 5 years.

For Option C, the following charges also apply:

An initial sales charge of 3% on the amount of money invested and

a yearly administrative fee of 1.5%.

A retiree has $ 30 000 to invest for a duration of 5 years.

(i) Calculate the amount of money he will receive at the end of 5 years ifhe takes up Option A. [2]

(ii) Find the value ofr, to three decimal places, if the interest receivedfrom Option B is three times more than that obtained from Option Aat the end of 5 years. [2]

(iii) If Option C is chosen, find the percentage return to his investmentat the end of 5 years. [2]



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7

Figure 1 Figure 2


A goldsmith made a solid gold pendant which comprises of a right cone anda hemisphere of radius 0.6 cm. The height of the pendant is 1.4 cm.

(a) Show that the length of the slant edge of the cone is 1 cm. [1]

(b) Find, in terms of, the total surface area of the pendant. [2]

(c) Show that the volume of pendant is 0.24 cm3. [2]

The goldsmith decided to make another solid pendant with the same amount ofmaterial. The new pendant was in the shape of prism of length 2x cm andthe cross-section of the prism is a regular hexagon with sidex cm.

(d) Find

(i) the area of hexagon, in terms ofx,

(ii) the value ofx. [4]

8

The coordinates of the pointsA andB are (1, 3) and (2, 1) respectively.

(a) The pointA is reflected in the linex =2

1.

State the coordinates ofC, the image ofA under this reflection. [1]

(b) Find the value of . [1]2AB

(c) Find the area of triangleABC. [1]

(d) Hence, find the shortest distance from the point Cto the lineAB. [1]

A(1, 3)

B (2, 1)

x

y

1.4

0.6


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9 (a)

In the diagram,BD is a diameter of a circle centre O.= 22APD and = 35ADB .

Find, stating the circle properties as you use them,

(i) BDC,

(ii) BAC,

(iii) AOB. [4]

(b)

ABCD is a cyclic quadrilateral.The tangents to the circle, centre O, atA andB meets at the pointE.

BD bisects andCBO = 18CAD .

Stating the circle properties as you use them, prove that

(i) the pointsA,E,B and O lie on the circumference of a circle. [2]

(ii) CB is parallel toDO. [3]



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

A parts manufacturer makes a profit ofy thousand dollars forx thousand pieces of

a certain component produced and .25 2 = xxy

The table below gives some of the corresponding values ofx and y.

x 0 0.5 1.0 1.5 2.0 2.5 3.0 4.0

y

2.0 0.25 2.0 3.25 4.0 4.25 4.0 2.0

(a) Using a scale of 4 cm to represent one unit on the horizontal axis anda scale of 2 cm to represent one unit on the vertical axis, draw the graph of

for25 2 = xxy 40 x by joining the points with a smooth curve. [3]

(b) Use your graph to find

(i) the number of pieces of the component the company must producein order to obtain the maximum profit, [1]

(ii) the minimum number of pieces the company must produce in orderto cover the cost of production, [1]

(iii) the range of values ofx for which the profit is more than $2850. [1]

(c) (i) On the same axes, draw the graph of 1=

x

yfor 0.30 x . [1]

(ii) Write down thex-coordinate of the point where the two graphsintersect. [1]

(iii) State briefly what the value of thisx-coordinate represents. [1]

(iv) The value ofx in (c)(ii) is the solution of the equation

. Find the value ofA and ofB. [2]02 =++ BAxx



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11 In an international beauty pageant, each contestants height was measured andtheir results were displayed on the cumulative frequency curve.


1.5 1.6 1.7 1.8 1.9 2.0 2.1Height of contestant (in metres)

60

50

40

30

20

10

0

CumulativeFrequency

(a) Using the above cumulative frequency curve, find

(i) the median height,

(ii) the interquartile range,

(iii) the minimum height of the top 10 tallest contestants. [3]

(b) On a sheet of graph paper, draw a box-and-whisker diagram to illustratethe above information. [2]


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(c) The information from the cumulative curve above is tabulated in the frequencydistribution table below.

Height(x m)

6.15.1


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ANDERSON SECONDARY SCHOOLSecondary Four Express / Five Normal / Four NormalPreliminary Examination 2009MATHEMATICS Paper 2 4016/02

1 (a) (i) )3)(3)(12( ++ aaa 9 (a) (i) = 33BDC

(ii)12

)3)(12(

+

a

aa (ii) = 33BAC

(b) 2=x (iii) = 70AOB (c) 22.0or22.2 =x (to 2 d.p.) 10 (a)

2 (a) (i) , ,12=a 36=b 24=c (c)(i)

(ii) )1(3 = np 2)1( += nq

(c)

2

2

3

+=

pq

3 (c) (i)3

16=LN cm

(ii)9

5

ofarea

ofarea=

KLM

KLN

4 (a) (i)22 cm

2

1x

(b) (i) 2500 pieces(ii) 450 pieces(iii) 7.33.1


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(d)5

12units (g) Standard deviation decreases.

ANDSS 4E5N Prelim 2009 Math (4016/02) Answer Key

4e5n2009emprelimp2 (andss)

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