2010AMathsPrelimP1

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Class Reg Number

Candidate Name _____________________________________

TANJONG KATONG SECONDARY SCHOOL

PRELIMINARY EXAMINATION 2010SECONDARY FOUR

ADDITIONAL MATHEMATICS 4038/01

Paper 1

Wednesday 15 Septe!er "010 " #$%rs

Additional Materials: Writing Paper

&EAD THESE INST&'CTIONS (I&ST

Write your name, class and index number in the spaces at the top of this page and onall separate riting paper used!Write in dar" blue or blac" pen!#ou may use a soft pencil for any diagram or graphs!$o not use staples, paper clips, highlighters, glue or correction fluid!

Anser a)) %uestions!Write your ansers on the riting paper pro&ided!'i&e non(exact numerical ansers correct to ) significant figures, or * decimal in the

case of angles in degree, unless a different le&el of accuracy is specified in the%uestion!+he use of a scientific calculator is expected, here appropriate!#ou are reminded of the need for clear presentation in your ansers!

At the end of the examination, fasten all your or" securely together!+he number of mar"s is gi&en in brac"ets - at the end of each %uestion or part%uestion!+he total number of mar"s for this paper is ./!



T#*s +%est*$n paper ,$ns*sts $- 5 pr*nted pa.es

[Turn ovr

Mathematical Formulae

1! ALGE"RAQuadratic Equation

For the equation ax2 + bx + c = 0,

x =a

acbb

2

42−±−

.

Binomial Theorem

(a + b)n = an +

1

nan − 1b +

2

n an − 2b2 + . . . +

r

nan − rbr + . . . + bn,

where n is a positive integer and r n

= !)!(

!

r r n

n

− =

!)1).......(1(

r r nnn +−−

2! TRIGONOMETRY Identities

sin2 A + cos2 A = 1.

sec2 A = 1 + tan2 A.

cosec2 A = 1 + cot2 A.

sin ( A ± B) = sin A cos B ± cos A sin B

cos ( A ± B) = cos A cos B ∓ sin A sin B

tan ( A ± B) = B A

B A

tantan1

tantan

±

sin 2 A = 2 sin A cos A.

cos 2 A = cos2 A − sin2 A = 2 cos2 A − 1 = 1 − 2 sin2 A

tan 2 A = A

A2

tan1

tan2

−

sin A + sin B = 2 sin 21 ( A + B) cos 2

1 ( A − B)

sin A −

sin B = 2 cos 2

1

( A + B) sin 2

1

( A −

B)

cos A + cos B = 2 cos 21 ( A + B) cos 2

1 ( A − B)

cos A − cos B = −2 sin 21 ( A + B) sin 2

1 ( A − B)

Formulae for ABC

sinsin"sin

cba== .

a2

= b2

+ c2

−

2bc cos A.

=2

1bc sin A.

#an$ong %atong &econdar' &choo rei* "dd athe*atics1

"



1! iven that T =

−−

-4

/, ind T1 and hence sove the si*utaneous equations

12/ =− y x ,-4- −= x y . 43

2! #he ine px y =+ - intersects the curve yqx xy −=− 2 at the points S (2, -)

and T (a, b). Find

#$% the vaues o p and q, 23

#&% the coordinates o T . 3

'! #$% Find the su* and the product o the roots o the equation

20)2)(2( =++ x x . 23

#&% 5ne root o the quadratic equation 0-)2(4 2 =−+−+ m xm x is the

negative o the other. Find the vaue o m. 23

(! #$% #he equation o a curve is12

2

+

+=

x

x y .

Find the equation o the nor*a o the curve where 2−= x . -3

#&% iven that

2-n

+

+=

x

x y , o6tain an e7pression or

x

y

d

d. 23

)! &ove the equation θ θ sec1tan 2 −= , where .0 <<θ 43

*! #he equation p x x p x 12−=−

+−+ has a soution x = 2. Find

#$% the vaue o p, 3

#&% the other soution o the equation. 3

+! #$% 87press x x

x

2

4

2

+

+ in partia ractions. -3

#&% 9ence, ind ∫ +

+

4

2

2

4

x x

xd x. 3

,! #he surace area, A c*

2

, o a sphere is increasing at the rate oπ

100 c*

2

:s at theinstant when its radius, r , is 4 c*. Find


3



#$%t

r

d

d, 3

#&% the rate o increase o the vou*e o the sphere at the sa*e instant. 3

-! " particuar species o snai iving in a garden is 6eing studied. "ter t 'ears, the

popuation o the snai, S , is given 6' ct eS 200 = , where c is a constant.

(a) Find the initia popuation o the snai in the pond. 13

#he popuation o the snai ater - 'ears is to 6e 1200. Find

(6) the e7pected nu*6er o snais ater 'ears, 3

(c) the nu*6er o 'ears needed or the popuation o snais to reach -000. 23

10! " po'no*ia, ( ) x g when divided 6' ( 122+ x , eaves a quadratic quotient and

no re*ainder. #he constant ter* in ( ) x g is ;. iven urther that ( ) x g eaves

re*ainders o ;< and 11 when divided 6' ( )2− x and ( )+ x respective', ind

#$% an e7pression or ( ) x g in descending powers o x , -3

#&% the nu*6er o rea roots o the equation g( x) = 0, $usti'ing 'our answer. 23

11! iven that x x y 2sin2cos22

−= , where .2

0 π << x

#$% Find an e7pression or .d

d

x

y23

#&% Find the coordinates o the stationar' point o the curve.

eter*ine the nature o this point. -3

#.% Find the range o vaues o x or which y is increasing.

&how 'our wor>ing cear'. 23

12! #he unction is deined, or the interva-

/0

π ≤≤ x , 6' ( x) =

2

-sin x

.

#$% &tate the a*pitude and the period o . 23

#&% &>etch the graph o 1)(0 += x y . 3


4



#.% 5n the diagra* drawn in part #&%/ s>etch the graph oπ /

-1

x y +=

or-

/0

π ≤≤ x . 13

#% &tate the nu*6er o soutions, or -

/0

π ≤≤ x

, o the equation

π <

-

2

-sin2

x x= . 23

1'! ?n the 6ino*ia e7pansion o ,2 n

x x

+ where n is a positive integer, the

coeicient o the third ter* is 420.

#$% &how that n = 1-. 3

#&% @sing the vaue o n ound in part #$%, ind the coeicient o A x in the

e7pansion o .1

2

−

+

x x

x x

n

3

En o P$r


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2010AMathsPrelimP1