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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 ./!
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T#*s +%est*$n paper ,$ns*sts $- 5 pr*nted pa.es
[Turn ovr
Mathematical Formulae
1! ALGE"RAQuadratic Equation
For the equation ax2 + bx + c = 0,
x =a
acbb
2
42−±−
.
Binomial Theorem
(a + b)n = an +
1
nan − 1b +
2
n an − 2b2 + . . . +
r
nan − rbr + . . . + bn,
where n is a positive integer and r n
= !)!(
!
r r n
n
− =
!)1).......(1(
r r nnn +−−
2! TRIGONOMETRY Identities
sin2 A + cos2 A = 1.
sec2 A = 1 + tan2 A.
cosec2 A = 1 + cot2 A.
sin ( A ± B) = sin A cos B ± cos A sin B
cos ( A ± B) = cos A cos B ∓ sin A sin B
tan ( A ± B) = B A
B A
tantan1
tantan
±
sin 2 A = 2 sin A cos A.
cos 2 A = cos2 A − sin2 A = 2 cos2 A − 1 = 1 − 2 sin2 A
tan 2 A = A
A2
tan1
tan2
−
sin A + sin B = 2 sin 21 ( A + B) cos 2
1 ( A − B)
sin A −
sin B = 2 cos 2
1
( A + B) sin 2
1
( A −
B)
cos A + cos B = 2 cos 21 ( A + B) cos 2
1 ( A − B)
cos A − cos B = −2 sin 21 ( A + B) sin 2
1 ( A − B)
Formulae for ABC
sinsin"sin
cba== .
a2
= b2
+ c2
−
2bc cos A.
=2
1bc sin A.
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1! iven that T =
−−
-4
/, ind T1 and hence sove the si*utaneous equations
12/ =− y x ,-4- −= x y . 43
2! #he ine px y =+ - intersects the curve yqx xy −=− 2 at the points S (2, -)
and T (a, b). Find
#$% 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
#an$ong %atong &econdar' &choo rei* "dd athe*atics1
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#$%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
.
#$% &tate the a*pitude and the period o . 23
#&% &>etch the graph o 1)(0 += x y . 3
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#.% 5n the diagra* drawn in part #&%/ s>etch the graph oπ /
-1
x y +=
or-
/0
π ≤≤ x . 13
#% &tate the nu*6er o soutions, or -
/0
π ≤≤ x
, o the equation
π <
-
2
-sin2
x x= . 23
1'! ?n the 6ino*ia e7pansion o ,2 n
x x
+ where n is a positive integer, the
coeicient o the third ter* is 420.
#$% &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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