Tampines Prelim 2009 Am 2

8/14/2019 Tampines Prelim 2009 Am 2

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TPSS/PRELIM AM P2 4E 2009 1

TAMPINES SECONDARY SCHOOL

PRELIMINARY EXAMINATION 2009

SECONDARY FOUR EXPRESS

ADDITIONAL MATHEMATICS

PAPER 2 2 hours 30 min

15 September 2009

Additional Materials: Writing Paper

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.

You may use a pencil for any diagrams or graphs.Do not use staples, paper clips, highlighters, glue or correction fluid.

Write your calculator model on the top right hand corner of your answer script.

Answer all questions.

Write your answers on the separate writing papers provided.

Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place inthe case of angles in degrees, unless a different level of accuracy is specified in thequestion.

The use of a scientific calculator is expected, where appropriate.You are reminded of the need for clear presentation in your answers.

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 number of marks for this paper is 100.

This question paper consists of 6printed pages.

Setter: Mdm Loh M W


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Mathematical Formulae

1. ALGEBRA

Quadratic Equation

For the equation 02

cbxax , . a

acbb

x 2

42

Binomial expansion

nrrnnnnn bbar

nba

nba

naba

......

21

221 ,

where n is a positive integer and !

)1)...(1(

!!

!

r

rnnn

rrn

n

r

n

.

2. TRIGONOMETRY

Identities

Formulae forABC

Abc

Abccba

C

c

B

b

A

a

sin2

1

cos2

sin

sin

sin222

)(21sin)(

21sin2coscos

)(2

1cos)(

2

1cos2coscos

)(2

1sin)(

2

1cos2sinsin

)(2

1cos)(

2

1sin2sinsin

tan1

tan22tan

sin211cos2sincos2cos

cossin22sin

tantan1

tantan)tan(

sinsincoscos)cos(

sincoscossin)sin(

cot1cosec

tan1sec

1cossin

2

2222

22

22

22

BABABA

BABABA

BABABA

BABABA

A

AA

AAAAA

AAA

BA

BABA

BABABA

BABABA

AA

AA

AA


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1. The massM, in grams, of a radioactive substance present tyears after first being

observed is given by the formula

tM

03472.0)

2

1(24

(i) Find the initial mass of the substance at the beginning of the observation. [1]

(ii) Calculate the mass of the substance after 45 years. [2]

(iii) How long will it take for the substance to reduce to 10 grams? [3]

2. The roots of the equation 0423 2 xx are and . Find the quadratic equationwhose roots are 2 + and + 2. [7]

3. (i) Prove the identity AAA

AA

AA

AA2sec2

sincos

sincos

sincos

sincos

. [4]

(ii) Find all the angles between 0o

and 360o

which satisfy the equation

5sincos

sincos

sincos

sincos

AA

AA

AA

AA[4]

4. (a) Given thatx m is a factor of the expression 84)2( 22 mmxmx ,

calculate the possible values ofm. [4]

(b) Given that DxCxxBAxxxx )1()2)(1)((3643 23 for all real

values ofx, find the values of A, B, C and D.

Hence deduce the remainder when 3643 23 xxx is divided by 22 xx . [5]

5. (a) A rectangle has sides of length )343( metres and )3

65( metres.

Express, in the form 3ba , where a and b are integers, the area of therectangle. [3]

(b) Given that 2)2(log)53(log 44 yx , expressy in terms ofx. [3]

(c) Solve the equation 7log28log 2 xx . [5]


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6.

In the diagram,B, C, D, Eand Fare points on the circle.AB is a tangent to the circleatB. CFA, DFNandDEA are straight lines.BA is parallel to CE.

(a)(i) Prove that trianglesNFA andNAD are similar. [4]

(ii) Hence, show that NDNFNA 2 . [1]

(b) Given thatNis the mid-point ofAB, prove that ACAFNB 4

12 . [3]

7.

The diagram shows part of the curve 2)3( xxy . The curve has a maximum point

at P and a minimum point at Q. Find

(a) the coordinates ofP and Q. [4]

(b) the area of the shaded region. [5]

A

N

B

C

D

E

F

y=x(x-3)2

y

x0

P

Q


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8. In the diagram, OD = 3m, OC= 7m and DOC= DAO = CBO = 90o.It is given that DOA is a variable angle, where 0o < < 90o.The pointEis on the lineBCsuch thatED is parallel toBA.

(i) Show that .cos3sin7 AB [3]

(ii) ExpressAB in the form )sin( R whereR is positive and is acute. [3]

(iii) Find the value of for whichAB = 6.5 m. [3]

9. The functionfis defined by f(x) = 3 sin 2x 1.

(i) State the amplitude off. [1]

(ii) State the period off. [1]

The equation of a curve is y = 3 sin 2x 1 for 0ox 180

o.

(iii) Find the coordinates of the maximum and minimum points of the curve. [2]

(iv) Sketch the graph of y = 3 sin 2x 1 for 0ox 180

o. [2]

(v) Sketch, separately from (iv), the graph of 12sin3 xy for 0ox 180o.

State the range ofy. [3]

(vi) Using (v), state the number of solutions of the equation 112sin3 x in the

range 0ox 180

o. [1]

AD

C EB

O

3

7


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10. A curve is such that22

11

xdx

dy and P is a point on the curve. Thex-coordinate

ofP is positive and the equation of the tangent at P isy = 3x + 1.

(i) Find the coordinates ofP. [3]

(ii) Find the equation of the curve. [3]

(iii) Find the equation of the normal at P. [2]

A point (x, y) moves along the curve in such a way that the y-coordinate increases at

a constant rate of 0.15 units per second.

(iv) Find the corresponding rate of change of thex-coordinate at point P. [3]

11.

In the diagram, the pointsA (8, - 6) andD ( - 4, 10) are at the opposite ends of a

diameter of a circle with centre C. The diameterBEis perpendicular toAD.

(i) Find the equation of the circle. [4]

(ii) Find the coordinates of the pointsB andE. [8]

EA

x

BD

C

O

y

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Tampines Prelim 2009 Am 2