CHIJ Toa Payoh EM P2 2011 Prelim

Class Register Number Name

CHIJ SECONDARY (TOA PAYOH)

PRELIMINARY EXAMINATION 2011

SECONDARY FOUR (EXPRESS/THROUGH-TRAIN)

& SECONDARY FIVE (NORMAL ACADEMIC)

MATHEMATICS 4016/02

Paper 2 3 August 2011

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 10 printed pages. [Turn over

chijsectp.4E/4TT/5N.prelim.emath2.2011

2

Mathematical Formulae

Compound interest

Total amount = 1100

nr

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 ABC = 1

sin2

ab C

Arc length = r , where is in radians

Sector area = 21

2r , where is in radians

Trigonometry

C

c

B

b

A

a

sinsinsin

Abccba cos2222

.

Statistics

Mean = fx

f

Standard deviation =

22fx fx

f f


3

1 (a) Solve the equation 127

42 xx x3

3 = 2. Give your answers correct to 2 decimal

places. [3]

(b) Solve the simultaneous equations

,2

1153 xy

.523 yx [3]

(c) Express 23

32

2

3222

xx

x

xx

x as a single fraction in its simplest form. [3]

(d) If y

yx

21

43

4

, express y in terms of x. [2]

__________________________________________________________________________________

2 The theatre at Marina Bay Sands has 1680 seats. In June 2011, tickets are priced at $180 per

adult and $50 per child.

(a) One evening, 80% of the seats in the cinema are occupied. Twenty of the people present

are children. Calculate the total amount of money collected from the sale of tickets for

the evening. [2]

(b) The price of the adult ticket in June 2011 is 4% cheaper than that in December 2010.

Calculate the price of the adult ticket in December 2010. [2]

(c) The amount of money collected from the sale of tickets for one day in June 2011 is

$108 000. This sum is divided among cost of operation, wages and profit in the ratio

2 : 3 : 7.

(i) Calculate the compound interest earned if the profit of the day is invested at a rate of 5% per annum compounded every 3 months for a period of 2 years. [2]

(ii) The cost of operation per day in June 2011 is $2000 more than that in December

2010. Find the percentage increase of the cost. [2]

__________________________________________________________________________________


4

3

In the diagram, O is the centre of the circle and B, C, D, E and F lie on the circle. AB is a

tangent to the circle at B. AFC is a straight line, FD is parallel to BC, AF = 5 cm.

34DCE and 28BAC .

(a) (i) Find ABO . Give a reason. [1]

(ii) Calculate the value of the radius of the circle. [3]

(b) Find the following angles, stating your reasons clearly,

(i) OBC , [2]

(ii) BEC , [1]

(iii) FDE . [1]

__________________________________________________________________________________

D

C A

B

F O

5 28o

E

34o

R


5

4 A universal set A and B are given by

20} 10 andinteger an is :{ xxx ,

A = { x : x is a prime number },

B = { x : x is an integer that is a perfect square}.

(a) Draw a Venn diagram showing , A and B and place each of the members in the

appropriate part of the diagram. [2]

(b) Find

(i) 'BA , [1]

(ii) n ( )'BA . [1]

__________________________________________________________________________________

5 In the diagram, 4OP p and OQ = 5q.

X is the point on PQ such that PX = PQ3

1.

The line OX when produced, meets PY at Z.

(a) Express as simply as possible, in terms of p and q,

(i) QX , [1]

(ii) OX . [1]

(b) Y is the point such that PY = OQ5

8. Find QY in terms of p and q. [1]

(c) Calculate

(i) OQX

PXZ

of area

of area, [1]

(ii) the area of OQX , given that the area of OPX is 8 square units. [1]

_________________________________________________________________________________

O

Q

P

X

Y

5q

4p


6

6 For Teachers Day, the quantities and types of flowers sold by two shops, X and Y are given in

the table below.

Roses Carnation Orchid

Shop X 500 350 800

Shop Y 900 620 200

The above information about the quantities and types of flowers sold can be represented by a

matrix A =

200620900

800350500 .

The selling price and cost price of each Rose, Carnation and Orchid are given below.

Rose Carnation Orchid

Selling price ($) 2.00 1.80 1.50

Cost price ($) 0.60 0.45 0.50

The information on the selling price and cost price is represented by a matrix B =

50.050.1

45.080.1

60.000.2

.

(a) Calculate P = AB. [2]

(b) Describe what is represented by the elements of P. [2]

(c) Given that Q = 1

1

and R = PQ, find R and describe what is represented by its

elements. [3]

_________________________________________________________________________


7

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

The following table gives the corresponding values of x and y which are connected by the

equation 133 xxy .

x 1.75 1.5 1 0.5 0 0.5 1 1.5 2 2.25

y 1.1 0.1 1 0.4 1 2.4 3 a 1 3.6

(a) Find the value of a, giving your answer correct to 1 decimal place. [1]

(b) Using a scale of 4 cm to represent 1 unit on the x-axis and 2 cm to represent 1 unit on

the y-axis, draw the graph of y against x for values of x in the range 25.275.1 x .

[3]

(c) Find the range of values of x for which 133 xx . [2]

(d) Use your graph to solve the equation

(i) 05.033 xx . [2]

(ii) 152 3 xx . [2] (e) By drawing a suitable straight line find the gradient of the curve when x = 1.5. [2]

__________________________________________________________________________________

8 The figure shows a regular hexagon of side 12 cm. Each of the arcs is drawn using each vertex

of the hexagon as the centre, and having a radius equal to the length of each side of the hexagon.

Find (a) the perimeter of the shaded region, [2]

(b) the area of the un-shaded region. [3]

_____________________________________________________________________________

9 The masses, in grams, of 50 oranges are recorded in the table below.

Mass ( x g) 10050 x 150100 x 200150 x 250200 x

No. of oranges 5 12 26 7

(a) Calculate an estimate of the mean mass of the 50 oranges. [2]

(b) Calculate an estimate of the standard deviation. [2]

_________________________________________________________________________________


8

10 P, Q, R and S are four points on level ground. P is due west of Q and the bearing

of R from P is 054. PQ = 19 m, QR = 45 m, QS = 72 m and RS = 90 m.

(a) Calculate

(i) PRQ , [2]

(ii) SQR , [2]

(iii) the bearing of R from Q, [2]

(iv) the area of SQR. [1]

(b) A tower of height h metres stands at Q and the angle of elevation of the top of the

tower from S is 38. Calculate

(i) the value of h, [2]

(ii) the shortest distance of Q from SR, [2]

(c) A man walks from S to R and he reaches a point K where the angle of elevation of the

top of the tower from K is at its greatest. Calculate the distance of SK. [2]

________________________________________________________________________________

19 m

45 m 72 m

90 m

P Q

R

54

S


9

11 The diagram shows a right pyramid on a horizontal rectangular base ABCD.

Given that AB = 6 cm, BC = 8 cm and VA = 13 cm, find

(a) the length of AC and VX, [2]

(b) the value of AVC , [2]

(c) VCW in degrees, where W is the midpoint of VX. [2]

__________________________________________________________________________________

12 In the diagram, the circle with centre at B has a radius (2r + 1) cm. The semicircle with centre

at A and the semicircle with centre at C are identical and each has a radius )612( r cm. O is

the centre of the largest semicircle.

(a) Write down an expression, in terms of r, for

(i) BC, [1]

(ii) BO. [1]

(b) By using Pythagoras Theorem, form an equation in r and show that it reduces to

0425718 2 rr . [2]

(c) Solve the equation to find the possible value of r. [2]

(d) Find the area of the un-shaded region. [Take = 3.142] [2]

2r + 1

A C O

12 r6

B


10

13 A bag contains 21 coins, of which 9 are 50 cents coins, 7 are 20 cents coins and 5 are 10 cents

coins. Two coins are removed one after another from the bag at random and without

replacement.

(a) Copy and complete the following probability tree diagram. [2]

(b) Showing your methods clearly, find the probability that

(i) the value of the sum of the two coins is 30 cents, [2]

(ii) the sum of the value of both coins is more than 40 cents, [2]

(iii) the value of the difference of the two coins is less than 25 cents. [2]

(c) If a third coin is picked from the bag, what is the probability that the sum of the value

of the three coins is more than 35 cents. [1]

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

Second coin

10

First coin

50

cent

s

20

cent

s

7

3

3

1

21

5

50

20

10

50

20

10

50

20

10


11

CHIJ Sec Toa payoh Sec 4 Elementary Mathematics Preliminary Examination 2011

Answer Key

Level: Sec 4E Paper 2 (100 marks)

Qn. # Answer Qn. # Answer

1a

64.4x or 86.0x 6a

9193216

50.8572830ABP

1b

3

2x ;

2

13y

6b The total selling price and total cost price of

the shops A and B respectively

1c

)2)(1(

32

xxx

x

6c

2297

50.1972PQR ; The profit made by the

shops X and Y respectively

1d

482

642

2

x

x

or )24(2

)8)(8(2

x

xx

7a 1.21)5.1(35.1 3 a

2a $239320 7b Refer to graph below

2b $187.50

2ci $6582.62

2cii 12.5% 7c 35.055.1 x or x > 1.85

3ai 90ABO ; Tangent perpendicular to the radius

7di Draw 5.0y

65.1x or 15.0 or 1.8

3aii 42.4x (2 dec pl) 7dii Draw 5.05.0 xy

2.0or 45.1 x or 1.65

3bi 31 7e Gradient = (3.6 to 3.9)

3bii 59 8ai 75.4 cm

3biii 25 8aii 295.86 cm 2

4a

9i

9ii

160 g

41.5

4bi 'BA ={11,13,17,19} 10ai 4.14

4bii n( BA ) = 4 10aii 9.97

5ai qp 54

3

2

10aiii 6.039

5aii

qp

3

5

3

8

10aiv 1604.6 m 2

5b 4p+3q 10bi 56.3 m

5ci

4

1

10bii 35.7 m

A

B

11 13 17

19

16

12 14 15 18


12

Qn. # Answer Qn. # Answer

11a AC = 10 cm; VX = 12 cm 13a Diagram

11b 2.45 13bi

6

1

11c 2.17 13bii

35

24

12ai r413 13biii

35

17

12aii r1423 13c

133

132

12b 0425718 2 rr

12c

6

11r

.).(2 anr

12d 43.64cm 2

7(b)

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