Commonwealth Em 2 Paper

COMMONWEALTH SECONDARY SCHOOL PRELIMINARY EXAMINATION 2009 SECONDARY FOUR EXPRESS/FIVE NORMAL MATHEMATICS

4016/02

Paper 2

27 August 2009

10 45 – 13 15 Additional Materials:

2 hours 30 minutes Writing Paper Graph Paper (1 sheet)

NAME: _____________________________ (

)

CLASS: ________

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

 , use either your calculator value or 3.142, unless the question requires the answer in terms of  . For

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.

This question paper consists of 11 printed pages including the cover page.

2 Mathematical Formulae

Compound Interest r   Total amount = P 1    100 

n

Mensuration Curved surface area of a cone =  rl Surface area of a sphere = 4 r 2 Volume of a cone =

1 2 r h 3

Volume of a sphere = Area of triangle ABC =

4 3 r 3

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

CSS/Prelim2009/MATH/SEC4E5N/P2/AGS/Page 2 of 11

3 Answer all the questions. 1

(a)

National Petroleum Company (NPC) provides 3 different grades of petrol. The price per litre of each grade of petrol is as follows: Petrol Grade Price per Litre ($) Grade 92 1.687 Grade 95 1.767 Grade 98 1.870 (i) (ii)

(iii)

(b)

Mr Soh pumped 42 litres of Grade 98 petrol for his car. Calculate the amount of money he paid for the petrol.

[1]

Mr Soh’s car has a petrol consumption rate of 12.5 km per litre. Calculate the distance his car can travel with $50 worth of Grade 98 petrol.

[2]

During a promotion month, the cost per litre of Grade 95 petrol was reduced by 15% but an instant rebate of $5 was given to car owners who pumped Grade 92 petrol. What is the maximum volume of Grade 92 petrol to be pumped before the total cost becomes more than the cost of pumping Grade 95 petrol? Give your answer in litres correct to 1 decimal place.

[3]

A shopkeeper sells two types of luxury handbags, Elegant and Convenient. Elegant handbags cost $7500 a piece and Convenient handbags cost $240 less. (i) (ii)

Write down, in its simplest form, the ratio of the cost of Elegant handbags to Convenient handbags.

[1]

Given that the shopkeeper sold an Elegant handbag at a discount of 15% and a Convenient handbag at a discount of $50, calculate the total percentage discount given on the sale of the handbags.

[2]

CSS/Prelim2009/MATH/SEC4E5N/P2/AGS/Page 3 of 11

4 2

Each diagram in the sequence below is made up of a number of dots. ● ● ● ● ● ●

●

Diagram 1

● ● ● ● ● ● ● ● ● ● ● ● ●

Diagram 3

● ● ● ● ● ● ● ● ●

Diagram 4

(a)

Draw the next diagram in the sequence.

(b)

The table shows the number of dots in each diagram. Diagram 1 2 3 4 5 p Number of dots 1 6 13 22 Write down the values of p and of q .

[1] 6 q [2]

The formula for finding the number of dots in the n th diagram is An 2  Bn  C , where A , B and C are constants. Find the values of A , B and of C .

[3]

(d)

Find the number of dots in Diagram 10.

[1]

(e)

Which diagram has 253 dots?

[2]

(a)

(i)

Simplify

(ii)

Solve

(c)

3

Diagram 2

● ● ● ● ● ● ● ● ● ● ● ● ●

(b)

a2  6ab  9b 2 5a2  45b 2  . 6ac  3ad 2ac  ad  6bc  3bd

3 5 4 . 2  x  1 3  x  1

[3]

[2]

A box contains several red discs and green discs. A disc is randomly chosen and then placed back into the box and the process is repeated several times. The probability of choosing a red disc is p . (i) (ii)

Write down, in terms of p , the probability of choosing a green disc.

[1]

The process was repeated 8 times. Find the probability that (a) a red disc was chosen every time,

[1]

(b)

[1]

at least one green disc was chosen.

CSS/Prelim2009/MATH/SEC4E5N/P2/AGS/Page 4 of 11

5 4

In the diagram, ACB  90 , ABC  51 , BEC  35 , ACD  103 , CD  4 cm, BC  4.6 cm and CE  7.3 cm.

Calculate (a)

CBE ,

[2]

(b)

the length of CA ,

[1]

(c)

the length of AD ,

[3]

(d)

the area of triangle BCE ,

[2]

(e)

the shortest distance from E to CB produced.

[2]

CSS/Prelim2009/MATH/SEC4E5N/P2/AGS/Page 5 of 11

6 5

An airplane is scheduled to fly to its destination 3500 km away. The speed of the airplane in still air is 600 km/h and the speed of wind, which is constant throughout, is x km/h. Due to a haze, the speed of the airplane in still air is reduced by 10%. Write down an expression, in terms of x , for the time taken by the airplane, in hours, if it is flying in the direction of the wind.

[1]

Write down an expression, in terms of x , for the time taken by the airplane, in hours, if it is flying against the wind.

[1]

The difference in arrival time is 1 hour and 10 minutes. Write down an equation in terms of x , and show that it reduces to x 2  6000 x  291600  0 .

[3]

(d)

Solve the equation x 2  6000 x  291600  0 .

[3]

(e)

Hence, find the time taken by the airplane, in hours and minutes, if it is flying in the direction of the wind when there is no haze.

[2]

(a) (b) (c)

CSS/Prelim2009/MATH/SEC4E5N/P2/AGS/Page 6 of 11

7 6

In the diagram, O is the centre of the circle and points P , S , T and R lie on the circumference of the circle. The tangent at P meets RT produced at Q . TS  PS , TQ  SQ and TRP  36 . (a) Find

(b)

(i)

reflex angle POT ,

[2]

(ii)

PTS ,

[2]

(iii)

PQS .

[3]

Show that PS bisects QPT .

[3]

CSS/Prelim2009/MATH/SEC4E5N/P2/AGS/Page 7 of 11

8 7

(a)

The diagram shows the cross-section of a swing in a children’s playground. The seat is suspended on a 1.8 m long rope. To oscillate the swing, the seat is pulled back to point A and released to swing an angle of 62° to point A ‘. The seat makes one complete oscillation when it moves from point A to point A’ and back to point A again.

62°

1.8 m

A’

(i) (ii)

(b)

A

Calculate the distance moved by the swing seat from point A to point A’.

[2]

Assuming that the swing oscillates regularly from point A to point A’, find the speed of the swing, in metres per minute, if it makes 5 complete oscillations in 2 minutes.

[2]

The diagram shows the swing and a bench, 4 m away, in the children’s playground. Both the bench seat and swing seat are at the same height above the ground.

1.8 m

4m

(i) (ii)

Calculate the angle of depression of the edge of the bench seat from the top of the swing. A bird flies from the edge of the bench seat to the top of the swing. Calculate the distance the bird flies.

[2] [2]

CSS/Prelim2009/MATH/SEC4E5N/P2/AGS/Page 8 of 11

9 8

(a)

A factory manufactures small decorative ornaments. Each decorative ornament is made up of two parts: a solid hemisphere with radius 7 cm and a solid cone with a height 10 cm, as shown in Diagram I.

7 cm

10 cm

Diagram I (i)

Calculate the volume of the hemisphere.

[2]

(ii)

The volume of the hemisphere is 3 times the volume of the cone. Find the base radius of the cone.

[2]

Given that the solid cone is made with a light plastic material with a density of 0.9 g/cm3, find the mass of the material used for the cone.

[2]

(iii)

The two pieces are joined together to form the decorative ornament as shown in Diagram II. (iv) Calculate the total external surface area of the ornament.

Diagram II (b)

Given that the area of the major sector is 98 cm2, find the value of  and hence calculate the perimeter of the major sector.

6 cm

[4]

 rad.

[2]

CSS/Prelim2009/MATH/SEC4E5N/P2/AGS/Page 9 of 11

10 9

The cumulative frequency curve below represents the daily wages of 80 male employees in a company.

80

Cumulative Frequency

70 60 50 40 30 20 10 0

15

20

25

30

35 40 45 Daily Wages ($)

50

55

60

65

Use the graph to estimate (a)

the median daily wage,

[1]

(b)

the interquartile range,

[2]

(c)

the value of z such that 77.5% of the male employees have a daily wage more than $ z .

[2]

The box-and-whisker diagram represents the daily wages of 60 female employees in the same company.

(d) (e) (f)

10

20

30 40 50 Daily Wages ($)

60

70

Find the median daily wage of the female employees and the interquartile range.

[3]

Compare and comment briefly on the daily wages of the male and female employees in the company.

[2]

Find the probability that an employee chosen at random from all the employees has a daily wage less than or equal to $38.

[2]

CSS/Prelim2009/MATH/SEC4E5N/P2/AGS/Page 10 of 11

11 10

Answer the whole of this question on a sheet of graph paper. The following table gives the corresponding values of x and y , which 5 are connected by the equation y   2x  9 , correct to 1 decimal x place. 1 2 3 4 5 6 7 8 x y p 12 7.5 4.7 2.3 0 -2.2 -6.4 (a)

Calculate the value of p correct to 1 decimal place.

[1]

(b)

Using a scale of 2 cm for 1 unit on the x -axis and 1 cm for 1 5 unit on the y -axis, draw the graph of y   2x  9 for the x values of x in the range 1  x  8 .

[3]

(c)

Use your graph to find the value of y when x  2.5 .

[1]

(d)

Use your graph to solve the equation

(e)

Find the coordinates of the point on the graph for which the gradient of the curve is -4.

[2]

By drawing a suitable straight line, solve the equation 4 x 2  6 x  5  0 for 1  x  8 .

[3]

(f)

5  2 x  1. x

[2]

END OF PAPER

CSS/Prelim2009/MATH/SEC4E5N/P2/AGS/Page 11 of 11