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ST JOSEPH'S INSTITUTION

SECONDARY 4 PRELIMINARY EXAMINATION

MATHEMATICS 4016 /2

THURSDAY 26 AUGUST 2010 2 HOURS 30 MINUTES

(0800 1030 h)

INSTRUCTIONS TO CANDIDATES

1. Answer ALL questions.

2. Write your answers on the separate writing papers provided.

3. Hand in your answers to Questions 1 to 6 and Questions 7 to 11 separately.

4. If you use more than one sheet of paper, fasten the sheets together.

5. 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.

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

7. You are reminded of the need for clear presentation in your answers.

INFORMATION FOR CANDIDATES

1. The number of marks is given in brackets [ ] at the end of each question or part

question.

2. The total of the marks for this paper is 100.

3. You are expected to use an electronic calculator to evaluate explicit numerical expressions.

This question paper consists of 10 printed pages including the Cover Sheet.

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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

sin sin sin

a b c

A B C

2 2 2 2 cosa b c bc A

Statistics

Mean = fx

f

Standard deviation =

22fx fx

f f

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Answer ALL questions.

1a) Simplify

aab

aab

105

4530 2

. [1]

b) Simplify

xx

x

3

1

182

42

. [3]

2a) Paul wanted to buy a new car. He decided to take up the hire purchase scheme offered by the

car dealer. According to the scheme, Paul needed to pay a downpayment of $25 000 and

settle the rest of the payment by instalments. He was given 2 instalment plans to choose from.

Plan A required Paul to pay $800 every month for 4 years.

i) Calculate the total amount Paul paid for the car according to Plan A. [2]

Plan B charges a simple interest rate of 4 % per annum on the outstanding amount. Given

that the cash price of the car is $59 000,

ii) calculate the total amount of money Paul paid for the car if the repayment period

is also 4 years.

[2]

b) Susan wanted to go Hong Kong for her holiday with her parents. Before leaving, she needs to

help her parents exchange some money. The foreign exchange rate of Hong Kong Dollars

(HKD) and Singapore Dollars (SGD) is 1 SGD = 5.82 HKD on a particular day.

i) Calculate, in HKD, the amount she received for SGD 500. [1]

ii) In Hong Kong, Susan ran out of Hong Kong dollars. She needed to exchange

additional Hong Kong dollars with the Singapore dollars she had brought along.

The exchange rate at a particular bank in Hong Kong is 1 HKD = 0.1790 SGD on

a particular day. Calculate how much she would receive for 100 SGD, giving your

answer to the nearest dollar.

[1]

iii) Upon her return to Singapore, Susan had leftover Hong Kong dollars. But she was

going on a school trip to Malaysia so she decided to exchange her remaining

Hong Kong dollars into Malaysian Ringgit (MYR). The exchange rates at a

particular bank is 0.1739 SGD = 1 HKD and 0.4250 SGD = 1 MYR. If she had

320 HKD remaining, calculate how much Malaysian Ringgit she would receive,

giving your answer to the nearest dollar.

[2]

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3a) A water tank consists of a right cylindrical container of height 30 cm and radius 12 cm, and a

cone of slant length 15 cm that is fully filled with water. After a spherical object is immersed

into the tank, some water is displaced. The diameter of the sphere is the same as the diameter

of the cylinder.

The diagram below illustrates the sphere being immersed into the tank. Find, leaving your

answers in terms of ,

i) the volume of water remaining in the container, [4]

ii) the total external surface area of the figure as shown. [4]

b) All the water in the tank is now drained into another container which stands upright. The

container is a prism whose cross-section is right-angled triangle. The dimensions of the

triangle are 0.25 m and 0.6 m as shown in the figure below. Calculate the minimum integer,

in centimetres, the height of the container if it were to contain all the water from the tank.

[3]

30 cm

15 cm

12 cm

0.25 m

0.6 m

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4 In the figure, B, C, D and E are points on the circle with centre O. BD and CE intersect at F

and the tangent to the circle at B meets DE produced at A. Given that 34DEC and

26ECB . Find, with reasons stated clearly,

i) reflex DOC , [2]

ii) OCF , [2]

iii) DAB , [1]

iv) EBF . [1]

5 In the diagram, ABCD is a park and BD and AE are paths running through it. D is due north

of B. CD = 6.5 km, ED = 3.5 km, AE = 4 km, 63AED , 50CBD .

i) Find the length of AD. [2]

ii) If the area of ABE is 8 km2, find the length of BE. [2]

iii) Find BCD . [2]

iv) Find the bearing of B from C. [2]

v) A vertical tower ET of height 800 m stands at E where T is the top of the tower.

Find the angle of depression from the top of the tower to A.

[2]

B Diagram is not drawn to scale.

O

C

A

D

E

34

26 F

N

A B

C

D

E 3.5 km

6.5 km

4 km 50

63

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6 Mr Li can travel to work by taxi or express bus. The distance from his house to work is 25

km, and the speed of the taxi is v km/h. The taxi travels 20 km/h faster than the bus.

a) Write down an expression, in terms of v, for

i) the time taken when Mr Li travelled by taxi, [1]

ii) the time taken when Mr Li took the express bus. [1]

b) Given also that the bus journey is 15 minutes more than by taxi, form an equation in v and

show that it reduces to

02000202 vv .

[3]

c) Solve the equation 02000202 vv , giving both answers correct to 2 decimal places. [3]

d) The taxi charges a peak-hour surcharge of 35% of the metered fare. The taxis hop-in fare is

$3 and charges $0.20 for every 45 seconds travelled or less.

Calculate the amount Mr Li has to pay if he takes a taxi to work during the peak-hour on a

particular day. Leave your answer to the nearest five-cent.

[3]

Begin Question 7 on a fresh sheet of paper.

7 In the diagram, ED // AC // OB , where OAE is a straight line . P is the point OB produced

meets EC produced. AE = 3a , OA = 2a, OB = 6b and OP = k OB , where k is a scalar.

a) Express in terms of a and/or b, and k,

i) AC, [1]

ii) CP. [1]

b) If

4

5k , find the position vector of Q such that

[2]

OAQP is a parallelogram.

c) If the area of triangle ACE is 5 cm

2, and k =

4

5 , find the area of

i) triangle OEP. [1]

ii) triangle CDE. [2]

iii) OACB. [2]

B P O

C

D E

A

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8 The graph shows the cumulative frequency curve of the Mathematics weekly test scores of

200 students in a particular week.

a) Use the graph to find the

i) 60th

percentile, [1]

ii) median, [1]

iii) interquartile range, [2]

iv) probability that for any two students selected randomly, one will have a score of more than 86 marks and the other with a score of at most 70 marks.

[2]

Test

scores 0

50

100

150

200

0 20 40 60 80 100

Cumulative frequency

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b) The box-and-whiskers diagram below shows the scores of the same group of students taking

a Science test in the same week.

i) Using the data above, find the median and interquartile range. [2]

ii) Compare and comment on the weekly test scores of the two tests taken by the same group of students in two different ways.

[2]

9a) A survey was conducted on the place that students study in order to prepare for their exam.

The number of students surveyed is 400.

In the Venn Diagram shown below, is the set of students involved in the survey.

H is the set of students who study at home. L is the set of students who study at the library.

i) Copy the above Venn Diagram and shade 'LH . [1]

ii) Describe, as simply as possible, in words, the set 'LH . [1]

iii) It is given that n() = 400, n(H) = 250 and n(L) = 180. Find the largest possible

value of )'( LHn .

[1]

iv) Express the following statement in set notation: All students who study at home

also study at the library.

[1]

Scores

20 40 60 80 100

L H

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b) Three types of material nylon, cotton and denim are required to manufacture shirts and

trousers. The mass, in grams, of each type of material required to make one shirt and one pair

of trousers, and the cost of each type of material, in dollars per gram, are shown in the table

below.

i) The manufacturer produces 80 shirts and 60 pairs of trousers of each material

daily. Represent the above information as a row matrix P.

[1]

ii) The matrix M is given as

803020

402010. The product of P and M is denoted by T.

Find T and describe what T represents.

[3]

iii) Find the total cost of manufacturing 80 shirts and 60 pairs of trousers using matrix

multiplication once.

[2]

iv) Find Q such that

100

1PQ represents the number of shirts and trousers

manufactured on a particular day, such that there is a decrease of 20% and 25%

in the manufacture of shirts and trousers respectively.

[1]

v) Hence evaluate

100

1PQ. [1]

10 The diagram shows a circle, centre O, and another smaller circle with centre M. RN is a

tangent to both circles. OM = 6 cm and the length of minor arc PN = 3

10 cm.

a) i) Show that NOQ =

18

5 radians,

ii) Find the major arc length of QN. iii) Find the area of sector QON.

[1]

[1]

[2]

b) Calculate the area of the shaded region. [4]

Nylon Cotton Denim

Mass of material for 1

shirt

10 20 40

Mass of material for 1

pair of trousers

20 30 80

Cost (dollars per gram)

0.30

0.45

0.58

O

M

N

P

Q

R

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11 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 612

6

2

x

xy , correct to 1 decimal place.

x 1 1.5 2 3 4 5 6 7

y 6.2 2.4 0.7 5.0 p 0.6 2 3.9

a) Calculate the value of p. [1]

b) Using a scale of 2 cm to represent 1 unit on both axes, join the points in the table with

a smooth curve for 612

6

2

x

xy , 71 x .

[3]

c) Use your graph to solve 8

12

6

2

x

x. [2]

d) Find the x - coordinate of the point on the graph for which the gradient of the curve

is 2 . [2]

e) i) On the same axes, draw the graph of 82 yx . Write down the x coordinate(s) of

the point(s) where the two graphs intersect. [2]

ii) This value of x is a solution of the equation 023 CBxAxx . Find the value

of A, of B and of C. [1]

The End