2009 RV EM P2

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2009 RVHS Preliminary Examination Page 1 of 10 O Level Mathematics (4016) P2

RIVER VALLEY HIGH SCHOOL

2009 PRELIMINARY EXAMINATION

SECONDARY FOUR

CANDIDATE

NAME

CLASS 4 INDEX NUMBER

___________________________________________________________________________

MATHEMATICS 4016/02

Paper 2 15 September 2009

2 hours 30 minutes

Additonal Materials: Answer Paper

Graph paper (1 sheet)___________________________________________________________________________

READ THESE INSTRUCTIONS FIRST

Write your class, index number and name 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 100.

________________________________________________________________________

This document consists of 10 printed pages including this page

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2

Mathematical Formulae

Compound interest

Total amount = P( 1 +100

r)n

Measuration

Curved surface area of a cone = rl

Surface area of a sphere = 4r2

Volume of a cone =3

1r2h

Volume of a sphere =3

4r3

Area of triangle ABC =21 ab sin C

Arc length = r, where is in radians

Sector area =2

1r

2, where is in radians

Trigonometry

C

c

B

b

A

a

sin=

sin=

sin

a2

= b2

+ c2 2bc cos A

Statistics

Mean =fx

f

Standard deviation =

22fx fx

f f

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Answer all the questions.

1 (a) Given that 1and3,2 cba , find the value of22

5

ab ac

c

. [2]

(b)By applying factorisation, simplify the expression2

2

2 8

2 6

y

y y

. [3]

(c) Solve the equation1 3

1 4 6

x

x x

. [3]

___________________________________________________________________________

2

In the diagram,A,B, CandD are four points on a horizontal field. The bearing ofB from

A is 320,AB=AD = 455 m,AC= 765 m, BAC= 25 and CDA = 70.

(a) Calculate

(i)BC, [2]

(ii)ACD, [2]

(iii)the bearing of

Dfrom

A,[2]

(iv) area of triangleABC. [2]

(b) A tower of height 99 m is situated at the pointB. Find the angle of elevation of

the top of the tower from the pointA. [2]

___________________________________________________________________________

A

B

C

N

785 25

D

455 m

765 m

70

B

455 m

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3 (a) John inherits $120,000 from his grandparents. He decides to deposit the whole sum

in the bank to earn interest. The bank offers him two plans. Plan A offers a compound

interest of 3.5% per annum. Plan B offers a simple interest rate of 3.7% per annum. If

John only intends to leave the money in the bank for 4 years, which plan would be a

better choice? Explain your reasons. [4]

(b) A digital camera can be purchased online at a price of US$234. A Singaporean

shopkeeper decides to sell the same type of digital camera in his shop at a discounted

price (in S$) that is equivalent to what a buyer can buy online. If the discount given

is 15%, calculate the original selling price (in S$) of the digital camera in the

Singaporean shop. [US$1 = S$1.70] [3]

(c) The value of a motorcycle bought in Jan 2007 decreased by 12% by the end of 2007.

However, by end of 2008, its value increased by 15% from its value at the end of

2007. Express the eventual value of the motorcycle at the end of 2008 as a

percentage of its original value in Jan 2007. [2]___________________________________________________________________________

4 The first four terms in a pattern of numbers, T1, T2, T3, T4, ., are given below.

T1 = 12 + 10 = 1

T2 = 32 + 21 = 11

T3 = 52

+ 32 = 31

T4 = 72

+ 43 = 61

(a) Write down an expression for T5 and show that T5 = 101. [1]

(b) Write down an expression for T6 and evaluate it. [1]

(c) Find an expression, in terms ofn, for the nth term, Tn, of the pattern. [3]

(d) Evaluate T15. [1]

(e) (i) Simplify 2 2(2 1) (2 3)n n [1]

(ii) Hence or otherwise, find and simplify, an expression, in terms ofn,

for TnTn1. [2]

__________________________________________________________________________

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F

5

5 Mr Lim bought some superior quality cooking oil for $800 for his mini-mart. For this,

he has to pay $x for each litre of cooking oil.

(a) Write down an expression, in terms ofx, for the number of litres he bought. [1]

(b)Due to leak in some containers , he lost 3 litres of cooking oil. He then sold each litreof the reminder of the cooking oil at a price of $2 more than what he has originally

paid for each litre. Write down an expression, in terms ofx, for the money he

received from the sale of all the reminder of the cooking oil. [2]

(c)Given further that he made a profit of $100. Write down an equation to represent all

the above information, and show that it simplifies to

23 106 1600 0x x . [2]

(d)Solve the equation 23 106 1600 0x x . [3]

(e)Find, correct to the nearest whole number, the litres of cooking oil sold. [2]

___________________________________________________________________________

6

O

In the above diagram, the points A, B, C, D and E lie on the circumference of the two

circles such that AEDC forms a cyclic quadrilateral and BCD is a straight line. The

centre of the smaller circle is marked as O. The lineAD and CEintersect at the point F.

Given thatAB =BC, AED = 65 and ACE= 70.

(a)Explain, with geometrical reason, why ECD = 45. [2]

(b)Calculate (i)ACB, [1]

(ii)ABC, [1]

(iii)AOC. [2]

(c) Find ADEand hence show that AFCis similar to EFD. [2]

___________________________________________________________________________

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7 (a)

The diagram above shows part of a circle with centre O and radius 17 cm and the

chord PR = 16 cm.

(i) Show that angle POR = 0.980 radians. [2]

(ii) Find the arc length PQR [1]

(iii) Calculate the area of the shaded segment. [3]

(b)

The diagram above shows a solid pyramid with a rectangular base ABCD and

a vertical height VN. Given thatAB = 8 cm,AD = 6 cm and VN= 12 cm,

(i) Show that VB = 13 cm, [2]

(ii) Hence, determine angleDVB. [2]

___________________________________________________________________________

A

V

C

D

N

B

21

156 cm

8 cm

O

RP

Q

16 cm

17 cm

56

17 cm

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8

A cone, a cylinder and a hemisphere are joined together to form a container as shown in

the above diagram. The height and radius of the cone is 18 cm and 6 cm respectively and

the height of the cylinder is 8 cm.

(a) Calculate the volume of the whole container. [4]

(b) The container is then filled with water up to a height level of 20 cm from the base

tip of the hemisphere. The figure below shows the water level in the cone portion of

the container in this situation with CA = 6 cm :

(i) Find the slant height VB of the cone. [2]

(ii) By applying similar triangle property, find the length CD and VD. [2]

(iii) Hence, find the totalsurface area of container that is in contact with water

in this situation. [4]

___________________________________________________________________________

18cm

6 cm8 cm

6 cm

V

C D

A B6 cm

18 cm

6 cm

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10 (a) The cumulative frequency curve below illustrates the marks obtained, out of 60, by

120 students in a Mathematics test.

(i) Use the graph to determine

(a) the median mark, [1]

(b) the upper quartile, [1]

(c) the passing mark if1

6of the sudent failed the test. [1]

(ii) The following is the grouped frequency table of marks of the 120 students.

Marks (x) 0

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10 (b) A bag contains six identical balls numbered 1, 2, 3, 4, 5 and 6.

Two balls are drawn at random, one after the other, from the bag without

replacement.

(i) Copy and complete the following possibility diagram and use to find the

probability that the sum of the numbers drawn is a prime number.

[2]

(ii) Copy and complete the following probability tree diagram and use it to find

the probability that 1 even and 1 odd numbered ball are drawn.

[3]

___________________________________________________________________________

1st drawn number

+ 1 2 3 4 5 6

1 3 4 5

2 3

3 4 5

4 5

52nd

drawnnumber

6

even

even

odd

odd

even

odd2

5

3

6

( )

1st draw 2nd draw

( )

( )

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