Tampines Prelim 2009 Am 2
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Transcript of 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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TPSS/PRELIM AM P2 4E 2009 2
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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TPSS/PRELIM AM P2 4E 2009 3
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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TPSS/PRELIM AM P2 4E 2009 4
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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TPSS/PRELIM AM P2 4E 2009 5
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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TPSS/PRELIM AM P2 4E 2009 6
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