St Gabriels Prelim 2009 Em P1

Name: …………………………………………(

)

Class: Sec ………………..

St. Gabriel’s Secondary School 2009 ‘O’ Preliminary Examination Subject Paper No Level/Stream Duration Date Setter

: : : : : :

Mathematics 4016 / 01 4E / 5N / 4N1 2 hours 02 Sep 2009 Mr Koh Keng Wee and Ms Teo Chen Nee

READ THESE INSTRUCTIONS FIRST Write your name, class and register number 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. 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 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 80.

Target Set:

80 This question paper consists of 15 printed pages including this cover page.

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Mathematical Formulae Compound interest Total amount = P(1 

r n ) 100

Mensuration Curved surface area of a cone = πrl Surface area of a sphere = 4π r 2 Volume of a cone =

1 2 πr h 3 4 3 πr 3

Volume of a sphere =

Area of triangle ABC =

1 absin C 2

Arc length = r , where  is in radians Sector area =

1 2 r  , where  is in radians 2

Trigonometry a b c   sin A sin B sin C a 2  b 2  c 2  2bc cos A

Statistics Mean =

Standard deviation =

 fx f

 fx f

2

  fx    f   

2

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Answer all the questions.

1

The cost price of a monitor is $480. It is sold at a loss of 12.5% on the cost price. Find its selling price.

Answer $ …………………………………. [2]

2

Ben can complete a piece of work in 12 days and Ali can complete it in 4 days. Find the number of days in which both of them, working together, will take to complete their work.

Answer ……………............………….. days [2]

3

Peter is x years old. He is 6 years younger than his twin sisters, Jane and Jennifer. (a)

Write down Jane’s age, in terms of x.

(b)

Find, as simply as possible, the mean age of the three children, giving your answer in terms of x.

Answer (a) …………….…..…... years old [1] (b) ……………..…..….. years old [1] ________________________________________________________________________ [Turn Over

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4

A conical flask has a surface area of 50 cm2 and a capacity of 845 cm3. Find the volume of a geometrically similar conical flask which has a surface area of 32 cm2.

Answer ……..……….……...……..…… cm3 [2]

5

(a)

Express 63 as a product of prime factors.

(b)

What is the smallest positive integer value of n , such that 63n is a multiple of 14?

Answer (a) …………….…..…................... [1] (b) ……………..…..…………….. [1]

6

Express 74 cm3 in m3, giving your answer (a)

as a decimal,

(b)

in standard form.

Answer (a) …………….…..…….…… m3 [1] (b) ……………..……..…….... m3 [1] ________________________________________________________________________

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7

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1

Simplify (2 x3 y 2 )3  4 x 2 y 2 , giving your answer in its simplest form.

Answer …………………………..………….. [2]

8

Ada invests a sum of money at 7% per year compound interest. Find the sum of money if it yields $5112 after 4 years.

Answer $ …………………………………. [2]

9

A map is drawn to a scale of 1 : 2500. (a)

Calculate the actual length, in metres, of a path represented by 9 cm on the map.

(b)

The surface area of the path is 450 m2. Calculate, in cm2, the area representing the path on the map.

Answer (a) …………….…..………….. m [1] (b) ……………...…..………. cm2 [2] ________________________________________________________________________ [Turn Over

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10

In the diagram, BCD is a straight line, AB = 8 cm , BC = 15 cm and ABC  90 . Find (a)

the length of AC,

(b)

the exact value of cos ACD ,

(c)

angle BAC .

A

8 cm

 15 cm

B

C

D

Answer (a) AC = ...…….…..………….cm [1] (b) cos ACD = ....……..……….. [1] (c) BAC = ……...………..…... ° [1]

11

(a)

Write down the next two terms in the sequence 2, 3, 7, 16, 32, 57, ____, ____.

(b)

Write down an expression, in terms of n , for the nth term of the sequence 1, 2, 4, 8, 16, ____.

Answer (a)

……………... , ..……………. [2]

(b) ………………………………. [1] ________________________________________________________________________

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  5    20  It is given that AB    and CD    .  3   6x   (a) Evaluate AB . (b)

13

For Examiner’s Use

If AB is parallel to CD, find the value of x.

Answer (a)

…..………………..…. units [1]

(b)

x = .…………………………. [2]

(a)

Factorise completely the following expression

(b)

Solve the following equation

3 xy  2 y  12 x  8 .

32 x  27 x1  0 .

Answer (a)

………….…..…………….. [2]

(b) ……...……………………. [2] ________________________________________________________________________ [Turn Over

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In the figure, PQ is parallel to BA, P is on BC such that BP : PC = 2 : 5 and R is on BA such that BR : RA = 4 : 3. If the area of BCR = 98 cm2, calculate the area of (a)

triangle ACR ,

(b)

triangle CPQ .

A R Q

B

P

C

Answer (a) ….……….…..……...… cm2 [2] (b) ………………………... cm2 [2]

15

Evaluate (a)

7  2  2  5        1     9  5  3  6

(b)

32 110  1 2 83

Answer (a) ………………………………. [2] (b) ………………………………. [2] ________________________________________________________________________

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16

Solve the equation places.

 2 x  3

2

 7 x  8 , giving your answers correct to 2 decimal

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Answer x = …….……… or ………...….. [4] ________________________________________________________________________ 17

Mr Chia has the habit of drinking coffee in the morning. The probability of him drinking coffee in the morning is k. Given that the probability of him drinking 9 coffee for two consecutive mornings is , find 25 (a)

the value of k,

(b)

the probability that for two consecutive mornings, (i)

he will not drink coffee on both mornings,

(ii)

he will drink coffee on either morning.

Answer (a) k = ……………………….…. [1] (b) (i) .………………………...… [1] (b) (ii) .…..……………….…...… [2] ________________________________________________________________________ [Turn Over

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For Examiner’s Use

In triangle ABC, AB = 7.5 cm, BC = 4.5 cm and angle BAC = 25. The side AB is drawn in the answer space below. (a)

Complete the triangle such that angle ABC is obtuse.

(b)

Construct

(c)

(i)

the perpendicular bisector of AB,

(ii)

the bisector of angle BAC.

Measure the bearing of B from C.

Answers (a) and (b)

North

A

B

[1] [2] Answer (c) ……………………………... [1] ________________________________________________________________________

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For Examiner’s Use

The graph below shows the charges of printing photographs of 4R size by Company A. Charges ($) 25

20

15

10

5

0

20

40 60 Number of Photographs

80

100 [1]

(a)

Calculate the charge of printing a photograph by Company A.

(b)

The charges of printing photographs of the same size by Company B comprise of a fixed charge of $5 together with a charge of 15 cents per photograph. Draw a line on the grid to represent the charge made by Company B.

(c)

A customer wishes to print 70 photographs. Which company would be cheaper and by how much?

Answer (a) $…………………...…….…. [1] (c) Company………...……….… [1] $.………….……….….……. [1] ________________________________________________________________________ [Turn Over

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For Examiner’s Use

In the diagram, PQ is a straight line. The x-coordinate of P is 4 and the y-coordinate of Q is 5. Given that O is the origin, OP = 5 units and OQ = 13 units, find (a)

the coordinates of P and of Q,

(b)

the equation of PQ.

y

P

4

5

O

Q

x

Answer (a) P (………… , …………) (a) Q (………… , …………)

[2]

(b) .……………………………. [3] ________________________________________________________________________

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(a)

For Examiner’s Use

Solve the simultaneous equations 2x – 3y = 12, 2y + 3x = 5.

Answer (a) x = ……………………………... (a) y = …………………………. [3] (b)

Given the inequality 32 < 5 + 3x  87, find (i)

the greatest possible value of x if x is a rational number,

(ii)

the smallest possible value of x if x is a perfect square.

Answer (b) (i) x = ………..…………...... (b) (ii) x = ……………….…...… [3] ________________________________________________________________________ [Turn Over

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22

Harry practises throwing javelin twice a week. In a particular week, the results of his practices are shown in the tables below. First practice of the week Distance in metres Number of throws

85 < x  90

90 < x  95

95 < x  100

100 < x  105

3

6

4

2

Second practice of the week Mean

92.7 m

Standard Deviation

4.57 m

(a)

(b)

For the first practice, calculate (i)

the mean distance,

(ii)

the standard deviation.

Compare the results for the two practices.

Answer (a) (i) .……………………...…m [2] (b) (ii) .…..…………….…...…m [2] Answer (b) ……………………………………………………………………………….... ………………………………………………………………………………… ………………………………………………………………………………[2] ________________________________________________________________________

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(a)

For Examiner’s Use

By expressing x 2  2 x  3 in the form (x + m)(x – n), sketch the graph of y = x 2  2 x  3 .

Answer (a) ………………...…………… [1] y

y

x

x

[2] (b)

By expressing  x 2  4 x  3 in the form a ( x  h) 2  k , sketch the graph of y =  x 2  4 x  3 .

Answer (b) ………………...…………… [2] y

y

x

x

[2] END OF PAPER

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16 St. Gabriel’s Secondary School Sec 4E/5N/4N1 EM 2009 ‘O’ Prelims - Answer Key 1

Qn

Answer $420

Marks M1, A1

16

2

3 Days

M1, A1

17

x6 x4

B1 B1

3

(a) (b)

3

4

432.64 cm

5

(a) (b)

2

3 x 3 x 7 or 3 x 7 2

B1 B1

6

(a) (b)

0.000074 m3 7.4 × 10–5 m3

B1 B1

7

2x7y

M1, A1

8

$3899.92

M1, A1

Qn

Answer 4.70 or 0.05

Marks M2, A2

(a) (b) (c)

3/5 4/25 12/25

B1 B1 M1, A1

18

(c)

200  2

B1

19

(a) (b)

$0.25 Company B, cheaper by $2.

B1 B1 B1

20

(a)

P(-4, 3) Q(12, 5) 1/8 1 7 y  x 8 2

B1 B1 M1 M1, A1

y = – 2, x = 3 1 9  x  27 3 27 + 1/3 16

M1, A2 M1

94.2 m 4.71 m Better results for 1st practice (higher mean 94.2 > 92.7) Better consistency for 2 nd practice (lower std. dev. 4.57 < 4.71)

M1, A1 M1, A1

( x  3)( x  1) Parabola through turning pt.(-1, -4) y-intercept (0, -3) roots: (-3, 0) and (1, 0)

B1

M1, A1

(b)

9

(a) (b)

225 m 0.72 cm2

B1 M1, A1

10

(a) (b) (c)

17 cm –15/17 61.9◦

B1 B1 B1

11

(a) (b)

93, 142 2 n1

B2 B1

12

(a) (b)

34 or 5.83 units –2

21

(i) (ii) 22

14

15

(a) (b)

( y  4)(3x  2) –3

(a) (b)

73.5 cm 50 cm2

(a) (b)

35/9 – 4/9

2

(a) (b)

B1

A1 A1

B1

B1

M1, A1 23

13

(a) (b)

(a)

M1, A1 M1, A1 M1, A1 M1, A1 M1, A1 M1, A1

(b)

 ( x  2) 2  1 Parabola through y-intercept: (-3, 0) turning pt. (2, 1)

B1 B1 M1, A1 B1 B1

17 18

23 (a)

y

y y

C

25° A

B

3

x

1

x

1

3 4

19 (b)

23 (b)

B1

y

y

Charges 25

20

1 15

2

x

x

x

1

3 5

0

20

40 6 80 Number of Photographs

1

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