St Gabriels Prelim 2009 Em P2

2

Name: …………………………………………(

)

Class:

Sec ………………..

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

: : : : : :

Mathematics 4016 / 02 4E / 5N / 4N1 2 hour 30 minutes 3 September 2009 Mrs Chang / Mrs Olsen / Ms Liu

READ THESE INSTRUCTIONS FIRST Write your answers and working on the separate answer paper provided. 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 answer. Omission of essential working will result in loss of marks. Calculator should be used where appropriate. You are expected to use an electronic calculator to evaluate explicit numerical expressions. 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.

Target Set:

100 This question paper consists of 12 printed pages including this cover page.

2 Mathematical Formulae

Compound interest

r   Total amount = P 1   100 

n

Mensuration Curved Surface area of a cone = πrl Surface area of a sphere = 4πr 2 Volume of a cone =

1 2 πr h 3 4 3 πr 3

Volume of a sphere = Area of triangle ABC =

1 ab sin 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

3 Answer all questions In the diagram, ABC represents a horizontal triangular field and AP represents a vertical flagpole. B is 50 m from A on a bearing 018° and C is 125 m from A. The length of BC is 110 m and the height of the flagpole is 20 m.

1

B North

P

50 m

20 m

110 m

18° A 125 m

C

(a)

(b)

Calculate (i)

the bearing of A from B ,

[1]

(ii)

angle ABC.

[3]

A man walks along CB. Calculate (i)

the shortest distance the man is from A as he walks along CB produced,

(ii)

the greatest angle of elevation of the top of the flagpole when viewed by the man as he walks along CB produced.

[2]

[2]

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4 2

(a)

(b)

(c)

3

Factorise completely (i)

4y − x 2y + 4z − x 2z,

[2]

(ii)

4t 2 − 9. Hence, find the prime factors of 391.

[3]

Solve the equation

Given that q 

8 x + 2 = 16 9 −3x .

2 p  6rs , 3 ps  2r

(i)

find the value of q when p = 5.4, r = 3 and s = 2.5,

[1]

(ii)

express r in terms of p, q and s.

[3]

A car travels from P to Q, by a 300 km route, at an average speed of x kilometres per hour. Write down an expression for the time taken, in hours, for the journey. (a)

(b)

(c)

[2]

[1]

A van travels from P to Q, by a different route, which is 8 km longer. The average speed of the van is 4 km/h less than that of the car. Write down the expression for (i)

the average speed, in km/h for the van,

[1]

(ii)

the time taken, in hours, for the journey made by the van.

[1]

Given that the van takes 30 minutes longer than the car to travel from P to Q , write down an equation which x must satisfy and show that it simplifies to x2 − 20x − 2400 = 0 .

[2]

(d)

Solve the equation x2 − 20x − 2400 = 0.

[2]

(e)

If the van had set off from P at 2230, at what time would it reach Q ?

[2]

 SGSS 2009

4016/02 Prelim/4E5N/09

5 4

(a)

ε = {1 ≤ x ≤ 9, x is an integer} P = {odd number} Q = {prime number} (i)

Draw a Venn Diagram to illustrate this information.

[1]

(ii)

List the elements contained in the set (P  Q)'.

[1]

(iii)

Write down n( P  Q ' ).

[1]

(iv)

If one of the elements of

ε

is chosen at random, what is the probability [1]

that it is a member of (P  Q) ?

(b)

A florist prepares 3 different types of bridal bouquets each containing 4 types of flowers. The number of each type of flower is tabulated below.

“Love” Bouquet “Passion” Bouquet “Devotion” Bouquet

(i)

Carnation 2 5 7

Roses 2 5 7

Lilies 3 6 8

Baby Breath 4 6 8

Write down a 3 by 4 matrix showing the number of each type of flower in each type of bouquet.

[1]

The cost price per flower of each type is shown in the table below. Flowers Carnations Roses Lilies Baby Breath

Cost Price $0.70 $1.10 $2.30 $0.50

(ii)

Write down a column matrix showing the cost price per flower of each type. [1]

(iii)

By using matrix multiplication, find the cost price of each type of bouquet.

[2]

(iv)

The florist decides to make a profit of 40% from the “Love” Bouquet, 50% from the “Passion” Bouquet and 20% from the “Devotion” Bouquet. By using matrix multiplication, find the selling price of each type of bouquet.

[2]

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6 5

(a)

Rooney works as a sales representative for a greeting card company. He is paid a basic salary of $850 a month and commission on the sales he achieves for the month. His commission is 10% of the first $2000 in sales, 12% of the next $4000, and 20% of all sales over $6000. If his sales for the month of June were $9650, how much did Rooney earn at the end of June. [3]

(b)

Jackson Limited pays its 12 employees 8.2% of total profits in a profit sharing plan. (i)

(ii)

(c)

 SGSS 2009

If the total profits were $475,000 last year and have decreased by 26% this year due to recession, find the amount of profit each employee is going to receive this year.

[2]

Sonia is one of the 12 employees of Jackson Limited. She would like to send the amount of profit that she receives this year to her family in Philippines. The rate of exchange between Singapore dollar (S$) and Philippines peso was S$4.50 = 100 peso. Calculate the amount of Philippines peso Sonia’s family received, giving your answer correct to the nearest peso.

[2]

In one state, property is taxed at a rate of 3.21% per annum on 40% of its market value. In a second state, property is taxed at a rate of 5.02% per annum on 24% of its market value. Gerry wants to invest in a warehouse of market value worth $295, 000. (i)

Which state would charge a lower property tax?

[2]

(ii)

Find the percentage difference in taxes between the two states.

[2]

4016/02 Prelim/4E5N/09

7 6 B B O C 4 cm 26 E

A

6 cm

D

A, B and C are points on the circumference of a circle centre O. BC is the diameter of the circle. EAD is a tangent to the circle at A such that it meets the line BC produced at D and angle BDA = 26. (a)

(i)

State angle AOC.

[1]

(ii)

Calculate angle BAE.

[2]

(b)

Show that triangle ACD is similar to triangle BAD .

[3]

(c)

It is given that CD = 4 cm and AD = 6 cm. Using similar triangles, show that the radius of the circle is 2.5 cm.

[2]

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8 7

F

C D

8

20

57 A

E

B

10

The diagram shows a triangular prism in which three of the faces are rectangular. BE = 20 cm, AB = 10 cm, AC = 8 cm and BAC = 57. (a)

(b)

 SGSS 2009

Find (i)

the volume of the prism,

[3]

(ii)

the total surface area of the prism.

[4]

Calculate angle BME, where M is the midpoint of CF.

4016/02 Prelim/4E5N/09

[3]

9 8

y

A B x

O (a)

 6 15  In the diagram, OA    and OB     6 1 Find (i) OA ,

[1]

(ii)

BA ,

[1]

(iii)

the coordinates of D where OADB forms a parallelogram.

[2]

A

E F B

a

D

b C

 In the diagram, BC = 4BD and AD = 5FD. E is the mid-point of AC. BD  a and  CE  b . (b)

(c)

Express, as simply as possible, in terms of a and/or b, (i)

DC ,

[1]

(ii)

DA ,

[2]

(iii)

DF .

[1]

Write down the value of area of triangle ABE area of triangle ABC .

[1]

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10 9

Answer the whole of this question on a sheet of graph paper. A building stands on horizontal ground. A stone is thrown from a window of the building. At any instant, the horizontal distance of the stone from the building is x metres and the height of the stone above the ground is y metres, given by the formula y  15  4 x  x 2 . The table below shows some values of x and the corresponding values of y. x y

0 15

1 18

2 a

3 18

4 15

5 b

6 3

(a)

Calculate the value of a and of b.

[1]

(b)

Using a scale of 2 cm to represent 1 metre on the x-axis and 2 cm to represent 2 metre on the y-axis, draw the graph of y  15  4 x  x 2 for 0  x  8 .

[3]

(c)

Write down the height of the window above the ground.

[1]

(d)

From the graph, estimate (i)

(ii)

the horizontal distance of the stone from the tower when it land on the ground.

[1]

the height of the stone from the ground when the stone is 4.6 m away from the building.

[1]

(e)

By drawing a tangent to the graph, find the gradient of the graph at x = 3.5 .

[2]

(f)

Use your graph to find (i) a solution of 9  4 x  x 2  0

[1]

(ii)

 SGSS 2009

a solution of  1  4 x  x 2  0

4016/02 Prelim/4E5N/09

[2]

11 10

Number of pellets 600

500

400

300

200

100

0

18

22

20

24

26

28

30

Time taken to dissolve pellet (s) The time taken for 600 pellets to dissolve in water was measured. The cumulative frequency curve below shows the distribution of time taken to dissolve pellets. (a)

Use the information provided to find the value of a, of b and of c in the box–and–whisker plot shown below. (You do not need to copy the plot onto your working.)

18

(b)

a

b

c

[2]

30

Using the graph, estimate (i)

the number of pellets taking 25 seconds or less to dissolve,

[1]

(ii)

the percentage of pellets taking more than 27 seconds to dissolve.

[1]

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12 The number of cars passing the CTE ERP gantry in 20 consecutive five-minute intervals on a certain day was recorded below.

(c)

21

30

22

46

35

55

27

42

37

41

21

23

35

29

21

34

42

47

24

21

(i)

Copy and complete the stem-and-leaf diagram for this data in ascending order.

[2]

2 3 4 5 (ii)

(d)

Write down the mode.

[1]

David played the game shown below at his school’s annual carnival. Throw a dart for $1 and be a winner. Strike the Green zone and win a teddy bear. Strke the Orange zone and win a can of coke. The dart board is shown below. 20 cm

Orange

20 cm

150 cm Diameter 10 cm Green

250 cm Find the probability that David

 SGSS 2009

(i)

wins a teddy bear,

[2]

(ii)

wins a can of coke,

[2]

(iii)

does not win any prizes.

[1]

4016/02 Prelim/4E5N/09

13 2009 ‘O’ Level Preliminary Examination Elementary Mathematics Paper 2 Answer Key

1

(a) (i)

198

(ii)

95.3

(b) (i) (ii) 2

(a) (i) (ii)

4

(b) (i)

49.8 m

(ii)

3

(a)

q = –0.735 r=

5

p (2  3qs) 2(q  3s)

300 hours x

(x – 4) km/h

(ii)

308 hours x4

6

7

38.7 42 

$2401.92

(ii)

53 376 peso

(c) (i)

Second state 6.57%

(a) (i)

64

(ii)

58

(a) (i)

671 cm3

(ii)

602 cm2

(e) 0400

(b) 97.5

(a) (i)

 Q

P

8

(a) (i)

3

1

(ii)

2

5

9 4

6

8

(b) (i)

{4, 6, 8}

(iii) 2 2 3

8.49 units  9   5  

(iii) D(21, 7)

7

(iv)

 12.5  (iii)  25.8   35   

(a) $2260

(ii)

(b) (i)

(ii)

17.5

(b) (i)

(d) x = 60 or –40

4

(iv)

(2t + 3)(2t – 3) prime factors are 17 and 23

(b) x = 2 (c) (i)

(ii)

 0.7   1.1     2.3     0.5 

21.9 (2 + x)(2 – x)(y + z)

 2 2 3 4  5 5 6 6   7 7 8 8  

(c)

3a

(ii)

3a + 2b

(iii)

1 (3a + 2b) 5

1 2

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14 9

(a) a = 19, b = 10 (c) 15 m (d) (i)

6.4  0.1 m

(ii)

12  0.2 m

(e) 3  0.3 5.55  0.1

(f) (i)

0.3  0.1 < x < 3.75  0.1

(ii)

10 (a) a = 20.4, b = 21.4, c = 22.6 (b) (i)

550 pellets

(ii)

5%

(c) (i) 2 3 4 5 (ii) (d) (i) (ii)

1 1 1 1 2 3 4 7 9 0 4 5 5 7 1 2 2 6 7 5 21 0.00628 0.0257

(iii) 0.968

 SGSS 2009

4016/02 Prelim/4E5N/09