Modul Matematik Tambahan Form 4 Jpnp
Transcript of Modul Matematik Tambahan Form 4 Jpnp
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PANITIA MATEMATIK
TAMBAHAN SMK MAT KILAU, 25150 KUANTAN NAMA :..
KELAS : .................. .................. ............ DISEMBER 2010
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CONTENT
TOPIC PAGE1. FUNCTIONS 232. QUADRATIC EQUATIONS 223. QUADRATIC FUNCTIONS 35
4. SIMULTANEOUS EQUATIONS 515. INDICES AND LOGARITMA 536. COORDINATES GEOMETRY 66. CIRCULAR MEASURES !5!. STATISTICS "". DIFFERENTIATION 10010.SOLUTIONS OF TRIANGLES 12111.NUM#ER INDE$S 13"
FUNCTIONS % C&APTER 1 '
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1. #()*+ C -+ / OF F -+ * -1
A relation from P into Q is defined by the set ofordered pairs { (1, 4), (1, 6), (2, 6), (3, 7) }S (
%(' * ( 1,% ' 7 + 4,%+' 8 (*-,%8' + 8 (*-,% ' 9(- ,% ' / 9 ;( * -
2
A relation from P into Q is defined by the setof ordered pairs
{ (2, 2), (4, 2), (6, 7) }S (
%(' * ( 4,% ' 7 + ) 2,%+' 8 (*-,%8' + 8 (*-,% ' 9(- ,% ' / 9 ;( * -
3
S PS Q
%(' * ( % ' 7 + ) -8
%(' %0'% ' %3'%+' %?2'%8' %4'
% ' %?3'
2 . G*= - -+ * - x x x g 3! 2 , >-8
(' %0'' %2'+' %4'8' %?3'' %?1'
3. G*= - -+ * -
x xh
436
! , >-8(' %0'' %1'
+' %2
1'
8' %?3'
' %41
'
P @ 1, 2, 3B
P @ 2, 4, 6B
8
<
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3. Problem of function involving simple equations
1. G*= - ( 34! x x f , >-8 =(; x * f % x ' @ 1 .
< @5
2. G*= - ( 73! + x x f , >-8 =(; x * f % x ' @ 4.
< @ ?1
3. G*= - ( 23! x x g , >-8 =(; y * % ' @ 2 4.
@ 6
4. G*= - ( 1"2! x x f ,24! + x x g , >-8 =(; < *
%-8 =(; < *
%-8 =(; ) < * f %-8 =(; ) / * % p ' = - 5p .
/ @ 1 2 , ?3
". G*= - ( 63! 2 x x g , >-8 =(; ) * % ' @ 6 .
@ ?2 , 2
10. G*= - -+ * -) 11. G*= - -+ * -) 12. G*= - -+ * -) 2! x x f
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%
%2! x x f (-82! + x x g , >-8
(' =(; < * %
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4. Finding the value of composite functions
1. G*= - ( 12! + x x f (-8 x x g 3! ,
>-8 %1' .
2. G*= - ( #2! x x f (-8 x x g 31! + , >-8 %3'.
G*= - ( x x f 23! (-82! x x g , >-8 %4'.
4. G*= - -+ * -) 3! + x x f (-8
14! x x g , >-8(' %1' % ' %1'
5. G*= - -+ * -)73! x x f (-8
x x g 24! , >-8(' %0' % ' %0'
6 G*= - ( x x f 3! (-822! x x g , >-8
%(' %?2' % ' %?2'
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5. Finding the value of composite functions f 2 , g 2
1. G*= - ( 32! + x x f , >-8 2%2' .
2. G*= - ( %! x x g , =(; (%3' .
3. G*= - ( 12! x x g ,=(; ( 2%3'.
4. G*= - ( 43! + x x f , >-8 %0' .
5. G*= - ( x x g %2! , >-8 2%1'.
6. G*= - ( 13! 2 x x f , >-8 %1' .
. G*= - ( x x g 4! , =(; (%?2'.
!. D* 9* 342! 2 + x x x f . N*;(* (-
f 2%0' .
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6. Solve all the follo ing
1. G*= - 32! + x x f ,f(x) = 2x + 3
) %(' %2' @
% ' %(' @
%+' %/' @
%8' %2 ' @
% ' %
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#
4. G*= - ( 43! + x x f (-8 x x g 2%! , >-8
%(' % '
5. G*= - ( x x f 3! (-822! x x g , >-8 + / )*
-+ * -)%(' % '
6. G*= - ( x x f %2! (-821! x x g + , >-8 + / )*
-+ * -%(' % '
. G*= - ( x x f %2! , >-8 -+ * - f 2 .
!. G*= - ( x x g 21
! , >-8 g 2% x '. ". G*= - (2
2!
x xh , >-8
-+ * - f 2.
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#. Finding the $nverse Functions
1. G*= - ( : < 4 !< , >-8 1.
2. G*= - ( : < 10 2< , >-81 .
3. G*= - ( : < 4 3< , >-8 1.
4. G*= - ( : < 15 6< , >-81 .
5. G*= - ( : < 10 !< , >-8 1 . 6. G*= - ( : < 3 2
1< ,
>-8 1 .
. G*= - ( : < 5 2< , >-8 1 . !. G*= - ( : < 3 4
3< , >-8
1 .
". G*= - ( %
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%. Finding the value of inverse function hen & given
1. G*= - ( : < < 2, >-8 =(; 1 %3'.
2. G*= - ( : < < , >-8 =(; 1 %2'.
3. G*= - ( : < 3< , >-8 =(; 1 %?5'.
4. G*= - ( : < 5 2< , >-8 =(;
1 %5'.
5. G*= - ( : < 3< ?2, >-8 =(;
1 %10'.
6. G*= - ( : < < , >-8 =(;
1 %2'.
. G*= - ( : < 21
< , >-8 =(; 1 %4'.
!. G*= - ( : < 34
x , >-8 =(;
1 % ?1 '.
". D* 9* : < x23
, +(9* -*;(*?1%?1'.
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'(. Finding the other related component function hen given a composite function and one of its component function ) *ethod '+
1. G*= - -+ * -) : < 2< 3(-8 : < 2< 3, >-8 -+ * - . A-) 9 :
2. G*= - -+ * -) : < < 3 (-8 : < 2< , >-8 -+ * - .
A-) 9 :
3. G*= - -+ * -) : < 3< 4(-8 : < 6< 1, >-8 -+ * - . A-) 9 :
4. G*= - -+ * -) : < 2< ? 1(-8 : < 6< , >-8 -+ * - . A-) 9 :
5. G*= - -+ * -) : < 2 -8 -+ * - ,. A-) 9 :
6. G*= - -+ * -) / : < 2< (-8 /H : < 4 2< , >-8 -+ * - H. A-) 9 :
. G*= - -+ * -) : < 2 4-8 -+ * - . A-) 9 :
!. G*= - -+ * -) : < 1 -8 -+ * - .
A-) 9 :
". G*= - -+ * -) : < 3< 1(-8 : < "< 2 , >-8 -+ * - .
A-) 9 :
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1. G*= - -+ * -) : < 2< ? 3(-8 : < 2< , >-8 -+ * - ,. A-) 9 :
2. G*= - -+ * -) : < < 3(-8 : < 2< 3 , >-8 -+ * - . A-) 9 :
3. G*= - -+ * -) : < 3< 4(-8 : < 6< , >-8 -+ * - ,. A-) 9 :
4. G*= - -+ * -) : < 2< ? 1(-8 : < 6< 1 , >-8 -+ * -. A-) 9 :
5. G*= - -+ * -) : < 2 -8 -+ * - ,. A-) 9 :
6. G*= - -+ * -) : < 2-8 -+ * - . A-) 9 :
. G*= - -+ * -) : < 3-8 -+ * -
!. G*= - -+ * -) : < 2 4-8
MET&OD 2
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,. A-) 9 :
-+ * - ,. A-) 9 :
QUESTIONS #ASED ON SPM FORMAT
1. T ;; *- *- 9 ( * - 9 9 ) A (-8 ) #.
T 9 ;( * -) - ) A (-8 ) # *) 8 >- 8 ) 98 9 8
/(*9 % ?3 , " ' , % ?2 , 4 ' , % 2 , 4 ' ,% 3 , " '(' S ( / 9 ;( * -)' U)*- -+ * -) - ( * -, 9* ( 9 ;( * - -
) A (-8 ) #
2. G*= - ( -+ * -) : < 2< 6 (-8 ?1 : < < / , 9 (-8 / (9 + -) (- ), >-8 =(; (-8 /.
K @
2
1 /
@ ?33. G*= - -+ * -) : < < 4 (-8 : < 2< 3 ,>-8 (' %-8 =(; (' %4'
' * ?1%2 ' @21
S A @ ?3 ,? 2 , 2 , 3 B
S # @ 4, " B
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1%
%
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@ / @ 23 4 J ' % '
". T -+ * - ?1%
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< @ 5 9, ?2 J
(' 2 ' 4 J
15. G*= - ( ?1 : < < 1 (-8 : < 3< 2 2 , >-8
(' -+ * - %
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1$
S P S QN ( ( (-
(' ;( - (- * ,' -*) - (- *
D* 9* - )* : < 4< 8(- ?1 : < 2 < $
%, 8 - (-
(8(- 8(- (8(;( / (;(9, +(9* (- -*;(* 8(- -*;(* .
@ ?5 2 @ 1 !
5. % SPM 04, P1 , Q3 '
D* 9* - )* %
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1#
T- (-(' ?1%5'' %2'
.
C(9* (- ?1 %
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2"
%(' 23
x + % '
1%1%+ x %+'10x 4 m = 4
11. %SPM 0 , P1, Q1' D*( 9( ) ;*- (9 -+ * - .
%(' S ( =(; % ' U)*- -+ * - - ( * -,
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%(' =(; # ,% ' 9(- f(x) + 99 )/ -8*- *= - 8 (*-.
%(' 12
x =
% ' " ( ) # f x
(!) 4$5 %&0.8 (b) 25x 2 10x - 1 '
15. %SPM 200!, P1 , Q3'G*= - -+ * -) f(x) = x 1 (-8 g(x) = kx + 2 ,>-8
%(' f(5) ,% ' =(; ) + ( gf(k) @ 14.
%(' 4 ' 4, ?3
16. %SPM 200", P1 , Q1'D*( 9( ) ) 9 ;( * - - ) $ (-8 ) Y *- 9(/ 9
S (
(' 7 + ) H' + 8 (*- 9 ;( * -
%(' 2 9 6 % ' /, H , 9 , ) B
1 . %SPM 200", P1 , Q2'G*= - -+ * -) : < 2< 3 (-8 : < 4-8
(' %
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%-8 T =(; %10'
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J
!ADRATI" E !ATI#NS ( "HAPTER 2 )
'. he roots of -uadratic quation 1 . D 9 *- * 3 *) ( 9 H ( * - 2< 2 < 15 @ 0.
2 . D 9 *- * $ %& ' * f * ,, '*% / x 2 + 2x 2 = 0.
3. D 9 *- * 3 *) ( 9 H ( * - x 2 2x + 3 = 0.
S Y
)9H
/
1 2 3
S $
4
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4. I 3 *) ( 9 H ( * - x 2 2 x + 12 = 0 4/ .
5. If -2 '/ p ' , *& f * ,' '*%6 , '*% / 2x 2 + 3x + =
0 4/ * , 7'8 , f '/ p.
9. If -1 ' , *& f * ,' '*%6 , '*% / px 2 x + 3p = 0 4/ p.
2. Finding the roots of -uadratic quation ) b/ factorisation +
L1 . S 87, x2
x 5 = 0 .
A/& : 1 5
L2 . S 87, x ( 1 + x) = 9.
A/& : 3 2
S ;= H (89( *+ H ( * - 2< %< 1' @ 6.
< @
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23 9 < @ 2
S 87, (x 3) 2 = 1.
A/& : 2
L . S 87, 1 + 2x 2 = 5x + .
A/& : 1 3;2
S ;= %2< 1' 2 @ 2< 1 .
A-) : , 1S ; )(* (- %< 3'%< 3' @ 16.
A-) : 5 , 5
L4. S ;= 5< 2 45 @ 0.
A-) : 3 , 3
L!. S ;= 2%< 2 "' @ 5
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2%
< @ ?5 9 < @ 1(A/& : 1 3 )
A-) : .31 , 0.6!34S 87, x 2 2x = 0 p8,*%/g * , & ' , g%7, '/& , & 6 ,6* * &%g/%46'/*4g ,&.
A/& : 2.212 .192
L4. S ;= < 2 10< 5 @ 0 , *=9 (-) 9) + 99 + 4 )* -*>+(-
> 9 ).
A-) : 0.52 ", ".4 2
S ;= < 2 < ! @ 0. G*= 9(-) 9 + 99 + 4 )* -*>+(- > 9 ).
%A-) : 2.3 2, ?3.3 2'
L5. S ;= < 2 5< 4 @ 0. G*=
9 (-) 9 + 99 + 4 )* -*>+(-> 9 ).
L . S ;= < 2 < 1 @ 0 , G*=
9 (-) 9 + 99 + 4 )* -*>+(-> 9 ).
L10. S ;= 2< 2 ? 12< 5 @ 0
+ 99 + 8 +* (; /;(+ ).
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%A-) : 0. 016, ?5. 02' %A-) : ?0.145", ?6.!541'%A-) : 5.55 , 0 .45 '
L11. S ;= ?2< 2 10< " @ 0+ 99 + 8 +* (; /;(+ ).
%A-) : ?0. ! , 5. ! '
S ;= ? < 2 < 3 @ 0 + /; *- )H (9 .
%A-) : ? .405, 0.4051'
L13. S ;= +(- > 9 ).
L2. # )*- 9 ;(, ) ;= 3 < 2 @ ? 3%4< 3' + 99 + 8 +* (; /;(+ ).
S ;=
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%A-) : 5.550, 0.4505'%A-): 0.52 , 11.4! '
%A-) : 1.2!, ?0. !'
S ;= < 2 4< @ 2 )*- 9 ;(.G*= 9 (-) 9 + 99 + 4)* -*>+(- > 9 ).
%A-) : 4.44" , ?0.44"5'
L6. S ;= H (89( *+ H ( * -
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A-) : < 2 2< 15 @ 0
A-) : < 2 3< @ 0 A-) : 2-8 =(; ) /.
/ @ 2 9 ?2 J
5. G*= - H (89( *+ H ( * - 3< % < 2 ' @ 3< % < 2 ', >-8 (' ) 9 )
' /9 8 + 9 ).
6. I H (89( *+ H ( * - < 2 2< 3 @ ? () - 9 ),>-8 9(- =(; ) .
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?4 3 , 2 3 J
X ?2 J
. S ;= H (89( *+ H ( * - 2< 2 6< @
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?! < ! J
/ @ 6 , < @ ?3 , " J
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SPM QUESTIONS
1. % SPM 03 , P1, Q3'S ; )(* (- / 9)( ((- (89( * 2< % < 4 ' @ % 1 < ' % < 2 '. T ;*) (- 7( (/(- (-8( ; /(8( /( (- (9 9 *.
% 2.5"1 , ?0.25 3 '
2. % SPM 03, P1, Q4'P 9)( ((- (89( * < % < 1 ' @ /< 4 / - (* 8 (/ -+( 9 (. C(9* (- 7 ;( -*;(* /.
% / ?3 9 / X 5 '
3. % SPM 04, P1, Q4 '# - (- / 9)( ((- (89( * (- / - (* / -+(?/ -+( ?3 8(- .# 9* (- 7( (/(- (-8( 8(;( - (< 2 < + @ 0,8 - (- (8((- (, , 8(- + (8(;( / (;(9.
% 2
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% / @ ! 9 ?4 '
% < ?4 9 < X 1 '
". %SPM 0 , P1, Q4 ' (' S ;= ;; *- H (89( *+ H ( * - : 3< 2 5< 2@ 0
' T H (89( *+ H ( * - < 2 < 3 @ 0 , 9 (-8 (9 + -) (- ), () H (; 9 ). E-8 =(; ) /.
/ @?3 J
12. %SPM 0" , P1, Q4 ' T H (89( *+ H ( * - < 2 < @ 2/< / 2 , 9 / *)+ -) (- , () 8*[ 9 - 9 ). F*-8 9(- =(; ) /.
/ J
13. % SPM 0", P2 , Q2 ' T H (89( *+ H ( * - < 2 5< 6 @ 0 () 9 ) (-8 ,
9 X .(' F*-8
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*' =(; (-8 **' 9(- < * < 2 5< 6 X 0
5 (9 ) J' U)*- =(; ) (-8 9 13 %(' , 9
H (89( *+ H ( * - *+ () 9 ) 2 (-8 3 2 . 2 (9 ) J
@ 3 , @ 2 , < 2 9 < X 3 J
QUADRATIC FUNCTIONS %C&APTER 3 F4 '
1. T ) ( M%/%> > 7'8 , f ' ' '*%6 /6*% / f(x) = 'x 2 + 6 ' C 0
N . /6*% /M%/%> > 7'8 , f " ,&p / %/g
7'8 , f x
M%/%> >
P %/* )S etched+ 0raph
1. %
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4. %
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3#
3. (x) = (x 1) 2 - 3
4. = (x 2)2
5. %
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!. % > 7'8 , f " ,&p / %/g
7'8 , f x
M'x%> >
P %/*)S etched+ 0raph
1.
%
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4. % > F'8 , f ' ' '*%6 /6*% / %/ * , f >. f(x) = ' (x + p) 2 + ' 0
N . /6*% /M'x%> > 7'8 , f " ,&p / %/g
7'8 , f x
M'x%> >
P %/*)S etched+ 0raph
1. f(x) = (x 1) 2 + 2 2 % x 1 ' 2 @ 0 x = 1
(1 2)
2. g(x) = - (x- 2) 2 +
x &
y
x &
y
x &
y
x &
y
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3. (x) = - (x 1) 2 - 3
4. f(x) = - 3(x + 2) 2 - 1
5. f(x) = 2 - (x + 1) 2
5. T ) + G9(/ ) Q (89( *+ F-+ * -) *- 9 %
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2.
f(x) = (x + 2) 2 + 1 M%/. P %/*. :A
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2.
%
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4%
%< ' 2 ? 25 4 %< ' 2 ? 25 4 %< 3 2' 2 ? . @ 2< 2 ? 6< 3
2%< 3 2' 2 ? 3 2
!. %
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C() 2 2 4(+ @ 0 . G9(/ @ %
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/ ?! , / X 0
.
?5 11 @ ?3 , @ 2
. %(' G*= - %
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4$
/ @ ?2 , H @ ? 1 2 (x+1) 2 - 5
@ 1 9 @ 4
4. E
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4#
p = = 25 13 2 %< 2' 2
%A-) : / @ ? , H @ 2 2 (
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%"
% 9 @ 3 9 ?1 @ 4 9 0 '
C(9* (-(' -*;(* ,' / 9)( ((- /( )* )* 9*
+' K 98*-( * * ( )*
3. % SPM 05, P1, Q6 'R(7( 2 - -7 (- 9( (89( * %
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%1
. %SPM 0! , P1, Q5 ' T H (89( *+ -+ * - %
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%2
(' S ( + 98*-( A. 1 (9 ) J
' # )*- 8 + /; *- )H (9 , >-8 =(; (-8 /.
4 (9 ) J+' D 9 *- 9(- =(; ) < , * %
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%3
/ X 1 10 J
@1 J
SIMULTANEOUS EQUATIONS % C&APTER 4 F4 '
1. % SPM 00, P2, Q1 'S ;= )* ; (- ) H ( * -)
3< 5 @ 2
%< ' @
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%4
% < @ 3 @ 2 ' % < @
1, @ 4 < @ 4, @ 1 '
3. % SPM 03, P2, Q1 'S ;= )* ; (- ) H ( * -) :
4< @ !
' / =
' > + /' > ' / =
' > /(' > ) / =
' >/' 0 =
1m
m
aa
1= (' =
' > < >
1. S* /;* (;; ;; *-
(. 5 3 $ 5 4 @. 2 x $ 2 3< @
+' 2 3 $ 5 3 @8' 4
122
aa @
'
4
3
22
@
' % 3 ' ?2 @
' %33 ' ?2 $ 3 2 @ ' 11 $2 ++ nn
*' 133 %% ++ nn
7' 1312 22 nn
' ( ) nn + 231 77
;' ( )ba
baab3
423
'
nn
n
+
+
112
273# -' 133 ++ n ' 122 nn /' 11 %%% + + nnn
2. S ;= (;; ;; *-
%7
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%7
!) 3 x @ !1 b) 2 x @ 32
c) 4 x @ 64
) ! x @ 16
*) 4 x @ 32
f) 2 x @ "
g) ! < @ 16 < 3 h) 4 < 2 @ 32 < ? 1
/) 2 1 x @ " 2 x
) 2 _ ! 2< @ 16 < 3
k) 16 _ 4 2< 3 @ 32 2 <
) 25 1 3< @ 5 _125 %6 >
1. 10 2 @ 100 ; 10 100 @ 2 11. ; 3 x @
2. 2 3 @ ! 12. ; ( @ 2
3. / H @ 9 13. ; 2 32 @ 5
4. 10 4 @ 10000 14. ; 3 %< ' @ 2
5. ( 3 @ 15. ; 10< @ 3
6. !1 @ 3 4 16. ; 10 100 @ /
. @ / 1 . ; / @
!. 2 x @ 1!. L 5 125 @ 3
". = @ 10 .
1. ; ( 3/9 @ ; ( 3 ; ( /
; ( 9
2. ; ( 2 - @ 3. ; ( 3(H @
6"
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6
4. ; 10 10 @ 5. ; 10 1000< @ 6. ; 2 4 - @
. ; ( q p
@ ; ( / ; ( H !. ; ( r p2
". ; 2 m4
@
10.; 10 kx1" @ 11. ; 10 1""
xy @ 12 (e) lo a ma3 -
13. ; ( x 3 @ 3 ; ( x 14. ; 2 % 4 xy ' @ 15. ; ( 21
x
61
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61
16. ; 2 % 44 y ' @
17. ; 2 $
4 y @ 18. ; 2 x
y 4
@
19. ; 10 10000 x 5 @
20. ; 2 3
2
4 x
k @
21. ; 2 %$
4 xy ' @
22. ; / % %$ p ' @
23. ; 4 64
3 y @
24. ; 10 100 x 3 @
9. EQUATIONS IN INDICES % *+ *-= ;= ) ) LOGARITM'
1. 3 x @ 1! ; 10 3 x @ ; 10 1! x ; 10 3 @ ; 10 1!
3lo1$lo
1"
1"= x x @ 2.631
2. 5 x+2 @ 16; 10 5 x+2 @ ; 10 16% x+2) ; 10 3 @ ; 10 1!
%lo16lo
21"
1"=+ x x+2 @
3. 2 x @ "
4. x @ 20
62
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62
x @
5. 4 x+1 @ 2!
6. 3 x-2 @ !
7. 2 < 3 @ 200
8. 1? x @ 2.!
9. 6 3x-2 = 66
10. 2 3< 1 @ 5 11. 3 ; 2 < @ 5 12. 7 = 9 0g3 x
!. "hange of ase of ogarithms . .orm*la ! log a x = a x
b
b
lolo
1. ; 4 ! @4lo$lo
2
2
@23
2. ; 4 32 @
4lo32lo
2
2
3. ; 16 ! @ 4. ; ! 2 @
63
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63
5. ; " 2 @ 6. ; 25 625 . ; " 3 @ !. ; 4 2 @
% * ( +(;+ ;( 9' C (- () 101.
; 4 " @ 4lo#lo
1"
1"
@ 1.5!5
%(' ; 5 20 @%lo
2"lo
1"
1"
@
% ' ; 4 0.! @
%+' ; 2 @ %8' ; " @ % ' ; 3 ".6 @
% ' ; 6 2.5 @ % ' ; 5 2000 @ % ' ; 12 6 @
!. A/;*+( * - L( ) L (9* T S ;= S* /; EH ( * -) *-= ;= - ; (9* )
S ;= H ( * - ; 2 %< 1' @ 3. *& : ; 2 %< 1' @ 3 < 1 @ 2 3
< 1 @ ! < @
1. S ;= H ( * - ; 2 %< 3 ' @2.
< @
2.S ;= H ( * - ; 10 %3< 2'@ 1 .
< @ 0.3. S ;= H ( * - ; 5 %4< 1 '@ 1 .
4. S ;= H ( * - ; 3 %< 6' @2.
5. S ;= H ( * - ; 10 %1 3
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64
A-) : < @ 0.3 A-) : < @ 15 A-) : < @ 33
6.S ;= H ( * - ; 3 %2< 1'
; 2 4 @ 5 .
A-) : < @ 14
. S ;= H ( * - : ; 4 %< 2' 3; 2 ! @ 10.
A-) : < @ 6
!.S ;= H ( * - ; 2 %< 5' @ ; 2 %< 2' 3.
A-) : < @ 3". S ;= H ( * - ; 5 %4< ' @ ; 5 %< 2' 1.
A-) : < @ 3
10.S ;= ; 3 3%2< 3' @ 4
A-) : < @ 12
11 . S ;= ; 2 !% 3
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6%
%8' ; 2 10
% ' ; 4 5 % ' ; 5 2
L1. G*= - ; 3 5 @ 1.465 , ; 3 @1. 1 . * )*- +(;+ ;( 9,=(; (
%(' ; 3 35 @
% ' ; 3 4" @ %+' ; 3 1.4 @
%8' ; 3 21 @ % ' ; " 21 @ % ' ; 5 3 @
SPM QUESTIONS
1. % SPM 01, P1, Q 'G*= - ; 3 x @ (-8 ; 2 x @ -. F*-8 ; x 24 *- 9 ) (-8 -.
2. % SPM "!, P1, Q 'G*= - ; 3 x @ / (-8 ; " y @ H. F*-8 ; 3 xy 2 *- 9 ) / (-8H.
66
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66
% 3 - 1 '
% / 4H '
3. % SPM 03, P1, Q5 'G*= - ; 2 T ? ; 4 \ @ 3,
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% 39 2/ 1 '
% < @ 1 '
11. % SPM 06, P1, Q '
G*= - ; 2 xy @ 2 3; 2 x ; 2 y ,
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< @ ?3 J
< @ " J1 . % SPM 0", P1, Q6 'G*= - 3 -?3 $ 2 - @ 243 , >-8 =(; -.
- @ 2 J
1!. % SPM 0", P1, Q 'G*= - ( ; ! ; 2 H @ 0,
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1. T %/ * , %&*'/6,
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E4. T 8*) (-+ - / *- )K%?1, /' (-8 L%/, "' *) %" . F*-8 /.
/ @ 0, 6
E5. T 8*) (-+ - / *- ) U%4, ?5' (-8 \%2, ' *) 2" .F*-8 / ))* ; =(; ) .
@?", ?1
E6. I 8*) (-+ - O%0, 0'(-8 P% , 2 ' *) )( () 8*) (-+ - / *- ) A%?4,3' (-8 #%1, ? ', >-8 / ))* ;=(; ) .
k = 5, -5
D*=*)* - ( L*- S -
2. T >-8 *8?/ *- T G*= - P *- ). F 9 ;( : M*8/ *- M @
++
2,
22121 y y x x
E . P%3, 2' (-8 Q%5, '
M*8/ *- , M @
++
2
72,
2
%3
@ %4 ,2# '
E1 P%?4, 6' (-8 Q%!, 0'
%2, 3'
E2 P%6, 3' (-8 Q%2, ?1'
%4, 1'
E3 P%0,?1', (-8 Q%?1, ?5'
%? , ?3'
E4 P%?3,!' (-8 Q%?11, 0'
% , 4'
E5 P%?2,? 5' (-8 Q%?10, ?5'
%?6 , ?5'
71
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3. Q 8*=*8 ) ;*- ) - PR *- 9( * @ : -. P% x , y ', R% x , y '
Q (x,y) @
++
++
nm
myny
nm
mxnx 2121 ,
E 1. T / *- P *- 9-(;; 8*=*8 ) ;*- ) - 7 *-*- / *- M%3, '(-8 N%6,2' *- 9( * 2 : 1. F*-8 + 98*-( ) / *- P.
P @
++
++
12)2(2)7(1
,12
)6(2)3(1
@
311
,3
1%
@ 11%,3
E1. T / *- P *- 9-(;; 8*=*8 ) ;*- ) - 7 *-*- / *- M
%4,5' (-8 N%?!,?5' *- 9( * 1 : 3.F*-8 + 98*-( ) / *- P.
%1,
2
E2. P 8*=*8 ) A# *- 9( * 3 : 2.F*-8 + 98*-( ) P *
(' A%2, ?3' (-8 #% ?!, '
' A%? , 5' (-8 #%!, ?5'
%(' %?4, ?3' % ' %2 , ?1'
E3. M *) ( / *- ( ;* ) - E4. P *) ( / *- ( ;* ) - E5. T / *- ) P%3, /', #%?1, 2' (-8
-
P%-8 + 98*-( ) / *- M.
3RM @ MS
MS
RM @
3
1, RM : MS @ 1 : 3
A-) :
2%
,1
) 9(* ;*- TU ) + ( 3TP @2PU. I + 98*-( ) / *- )
T (-8 U (9 %?", ' (-8 %1,?3'9 )/ + *= ; , >-8 + 98*-( ) / *- P.
%?5, 3'
C%", ' ;* - ( ) 9(* ;*- . I P8*=*8 ) #C *- 9-(;; *- 9( * :- , >-8 %(' : - , % ' =(; /.
%(' 2 : 3 % ' / @ 4
. T %/ A ,'& f P 8 g /& . A9 ( ( / ; - @21
1 2 3 11 2 3 1
x x x x
y y y y
E1. P%0, 1', Q%1, 3' (-8 R%2, '
A9 ( a PQR @21 " 1 2 "
1 3 7 1
1 -* 2
1. P%2,3', Q%5,6' (-8 R%?4,4'
A9 ( a PQR @
17
2 -* 2
2. T + 98*-( ) 9*(- ;A#C (9 %5, 10', %2,1' (-8 %!, '9 )/ + *= ; . F*-8 / ))* ;=(; ) , *= - ( (9 ( 9*(- ; A#C *) 24 -* ) 2.
@ 3 , 35
73
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3. T + 98*-( ) 9*(- ; RST(9 %4, 3', %?1, 1' (-8 % , ?3'9 )/ + *= ; . F*-8 / ))* ; =(; ) , *= - ( (9 ( 9*(- ;
RST *) 11 -* ) 2.
@ 0 , ?22
4. P%1,5', Q%4, ', R%6,6' (-8 S%3,1'.A9 ( PQRS @
! -* 2
5. P%2, ?1', Q%3,3', R%?1, 5' (-8 S%?4,?1'.
2 J
5. ' th( ar(a i) *(ro$ th(+ th( ,oi+t) ar( -o i+(ar.
1. G*= - ( / *- ) P%5, ', Q%4, 3'(-8 R%?5, ' (9 + ;;*- (9, >-8 =(; .
@ 33
2. S ( / *- ) K%4, !', L%2, 2'(-8 M%1, ?1' (9 + ;;*- (9.
3. G*= - ( / *- ) A%2,/', #%?1,?2'(-8 C%5,6' (9 + ;;*- (9, >-8 =(; .
/@2
74
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9. E '*% /& f S* '%g * L%/,&
(. I *= - 9(8* - (-8 -/ *- :
1 y y @ )( 1 x xm
E1. F*-8 H ( * - ( ) 9(* ;*-( /()) ) 9 / *- %5,2'
(-8 () ( 9(8* - ?2.
@ ?2< 12
E2. F*-8 H ( * - ( ) 9(* ;*-( /()) ) 9 / *- %?!,3'
(-8 () ( 9(8* - 4
3.
4 @ 3< 36
. I / *- ) (9 *= - :N : Y ( >-8 9(8* - >9) ,
- ) * 9 %(' y = mx + cO9 % ) y y 1 = m( x x 1 )
O9
%+'1
1
x x
y y
@12
12
x x y y
E1. F*-8 H ( * - ( ) 9(* ;*-( /()) ) 9 / *- ) %2, ?1'
(-8 %3,0'
@ < ? 3
E2. F*-8 H ( * - ( ) 9(* ;*-( /()) ) 9 / *- ) %?4,3'
(-8 %2,?5'
4< 3 @ 0+ % Th( x/i+t(r-(,t a+ th( y/i+t(r-(,t E1. T
-
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-
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77
y
D*( 9(
y
D*( 9( x
y
"
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. S 87, '88 * , f 88 %/g
x "
2
x "
3 D*( 9(
4
7$
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:!#/% %f ! %c:
". T F*-8 H ( * - ; + ) =*- / *- P ) + ( * ) 8*) (-+ ) P 9 / *- ) Q (-8 R (9H (;.
E1. Q%2,5' (-8 R%4,2'
4x 6y+9 =0
E2. Q%?3, 0' (-8 R%6, 4'
18x + 8y = 43
E3. Q%2, ?3' (-8 R%?4, 5'
3x 4y + 3 = 0E4. Q%6, ?2' (-8 R%0, 2'
3x 2y 9 = 0
E5. G*= - / *- ) A%3, 2' (-8 #% ,?4'. F*-8 H ( * - / 9/ -8*+ ;(9 *) + 9 A#.
3y =2x - 13
E6. G*= - / *- ) P%4, 10' (-8 Q#%?6, 0'. F*-8 H ( * - / 9/ -8*+ ;(9 *) + 9 PQ.
x + y = 4
7#
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10. : T 4/ * , , '*% / f * , 8 6 & f * , > 7%/g p %/* P & 6 * '* %*& %&*'/6,& f > * , p %/*& A '/ B ' ,%/ * , '*% > : /
E1. A%1, 5', #%4, 2' (-8 : - @ 2: 1
x 2+y 2 10x 2y + 19 = 0
E2. A%?3, 2', #%3, 2' (-8 : - @2 : 1
x 2+y 2 10x 6y + 13 = 0
E3. A%1, 3', #%?2, 6' (-8 : - @ 1 :2
x 2+ y 2 3x 3y = 0E4. A%5, ?2', #%?4, 1' (-8 : - @1 : 2
E5. P%?1, 3', Q% 4, ?2' (-8 : - @ 2 :3
E6. A%1, 5', #%?4, ?5' (-8 : - @3 : 2
$"
-
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x 2+ y 2 16x + 6y + 33 = 0 x 2+y 2+ 10x 14y + 2 = 0 x 2+y 2 + 16x +26y + 53 = 0SPM QUESTIONS PAPER 1
1. % SPM 03, P1, Q" ' T / *- ) A%2 , ' , #% / , ' (-8 C %2/, 3 ' (9 - ( ) 9(*;*- . # 8*=*8 ) AC *- 9-(;; *- 9( * 2 : 3 . E
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% 2 33
y x= + )%5-8 =(; ) /.
x
y
O
A#0$ 4%
C
D*( 9(5
B
> y = px + k ? y = (k 2)x + p
$2
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@ 2 , @ 1 3 J
/ @ ?" 9 21 J
". % SPM 0! , P1, Q13 'D*( 9( 13 ) ) ( ) 9(* ;*- /())*- 9 S%3,0'
(-8 T%0,4'.
(' 9* 8 - H ( * - ) 9(* ;*- ST *-
9 1=+b y
a x
' A / *- P%< , ' = ) ) + ( PS @ PT. F*-8 H ( * - ; + ) P.
143
=+ y x , 6< ! @ 0 J
10. %SPM 0!, P1, Q14 ' T / *- %0 ,3', % 2 , ' (-8 % ?2 , ?1 ' (9 = 9 *+ ) (
9*(- ; ). G*= - ( (9 ( 9*(- ; *) 4 -*2
, >-8 =(; ) .
@ 11 9 3 J
11. %SPM 0" , P1, Q15 'D*( 9( ) ) ( ) 9(* ;*- AC.
T / *- # ;* ) - AC ) + ( A# : #C @ 3 : 1. F*-8 + 98*-( ) #.
12. %SPM 10 , P1 , Q13 'A ) 9(* ;*- /()) ) 9 A % ?2, ?5 ' (-8 # % 6, '.(' G*= - C% , 10 ' ;* ) - ) 9(* ;*- A# , >-8 =(;
.' P *- D 8*=*8 ) ;*- ) - A# *- 9( * 1 : 3. F*-8
+ 98*-( ) D.
13. % SPM 10, P1, Q14 'P *- P = ) ) + ( * ) 8*) (-+ *) (; ( ) 5 -* ) 9 Q%?3,4'. F*-8 H ( * - ; + ) P.
A%?2,3'
T%0,4'
S %3,0 '
8 ( F * 9(8*( * ( ) &( 8
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3.3 F 9 (+ ;; *- , >-8 (- ; F *- 9(8*(- *= - =(; ) & (-8 .
E
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4 *&/m*#*& ! &*! %f *gm* # %f ! @/&c *
F 9 (+ ;; *- 8*( 9( ), >-8
%*' / 9* 9 (-8%**' (9 ( ) (8 8 ) - .
%('
P 9* 9 @ 4." 4 + , A9 ( @ 0.25 " + 2J
% '
P 9* 9 @ ".4 2 + , A9 ( @ 1.4!" + 2J
%+'
P 9* 9 @ 10.1" + , A9 ( @ 1.145 + 2JSPM QUESTIONS PAPER 1
'. ) SP* (4, P', -'% + 2. ) SP* (5 , P', -'#+
#%
'h di h / i l /ith t O 0i th t th l th f th 'h di h / i l /ith t O 'h l th f th i AB i 16
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'he dia ram sho/s a ir le /ith entre O 0i en that the len th of thema or ar AB is 4% %1 m, find the len th, in m, of the radi*s( se - 3 142)
" 3%4 rad
A
B
O
% .6 5+ 2 '
'he dia ram sho/s a ir le /ith entre O 'he len th of the minor ar AB is 16m and the an le of the ma or se tor AOB is 2#" o
sin - 3 142, find(a) the al*e of , in radians,
(0i e yo*r ans/er orre t to fo*r si nifi ant fi *res )(b) the len th, in m, of the radi*s of the ir le
(!) 1.222 &! (b) 13.09 cm
3. ) SP* (6 P' - '6 +
'he dia ram sho/s se tor OAB /ith entre O andse tor AXY /ith entre A 0i en that OB - 1" m, AY - 4 m,5 POQ - 1 1 radians and the len th of the ar AB - 7 m, al *late(a) the al*e of , in radian,(b) the area, in m 2, of the shaded re ion
(!) 0.7 &! (b) 26.2 cm 2
4. ) SP* (! P' - '# +
'he dia ram sho/s a se tor BOC of a ir le /ith entre Ot is i en that ( AD - $ m) 5 AOD - 1 $% radians and
BA - AO - OD - DC - % m .ind(a) the len th, in m, of the ar BC ,(b) the area, in m 2, of the shaded re ion
(!) 18.5 cm (b) 80.5 cm 2
5. ) SP* (# P' - '# +'he dia ram sho/s a ir le /ith entre O and radi*s 1" m 0i en that P , Q
6. ) SP* (% P' - '2 +
#6
and R are points s* h that &P PQ and 5 &P8 #" " find 'he dia ram sho/s a se tor BOC of a ir le /ith entre O
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and R are points s* h that &P - PQ and 5 &P8 - #" , findthat OP - PQ and 5 OPR - #" o, find9 se - 3 142:
(a) 5 QOR , in radians,(b) the area, in m 2, of the shaded re ion
(!) 1.047 (b) 30.72cm 2
'he dia ram sho/s a se tor BOC of a ir le /ith entre Ot is i en that 5 ; OC - 1 42 radians , A< - $ m and&A - AB - OD - DC - % m .ind(a) the len th, in m, of the ar BC ,(b) the area, in m 2, of the shaded re ion
(!) 14.2 cm (b) 59cm 2
!. ) SP* '( P' - '! +
'he dia ram sho/s a se tor &PQ of a ir le /ith entre O and se tor =8> of a ir le /ith entre =
t is i en that 5 P&Q - 1 % radians and 5 8=> - " % radian, find the area ,in m 2 of the shaded re ion
SPM QUESTIONS PAPER 2
P
QO
N
S
R
! +10 +
#7
( SPM 03 P2 ) 2 ( SPM 0 P2 )
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. ( SPM 03 P2 ) T 8*( 9( ) ) ( ) + 9 " * + - 9 " (-8 9(8* )10 + . P *- ;* ) - ) 9(* ;*- " * + -8* * - (" : " @ 3 : 5. C(;+ ;(
%(' =(; e, *- 9(8,% ' (9 (, *- + 2 ) (8 8 9 * -.
(!) 0.9274 &! (b) 22.37 cm 2
2. ( SPM 0 P2 ) T 8*( 9( ) ) ( +*9+; ? * + - 9 " (-8 9(8* ) 5 + . > *) (- - +*9+; ( .T ) 9(* ;*- ) " (-8 >" ,*- 9) + * +*9+;( (-8 9 )/ + *= ;. " *) ( 9 ). > *) (- (9+ +*9+; * + - 9 " . C(;+ ;(
%(' (- ; f, *- 9 ) ,% ' ; - , *- + , (9+ > ,%+' (9 (, *- + 2, ) (8 8 9 * -.
(!) 32
(b) 20.95 cm (c) 61.43 cm 2
3. ( SPM 05 P2 10 ) T 8*( 9( ) ) ( ) + 9 " ( +*9+; , + - 9 " . T
. ( SPM 09 P2 10 ) T 8*( 9( ) ) /;(- ( (98 -. @ *) ( ) *+*9+;
#$
/ *- * + - 9 " (-8 () ( 9(8* ) ! *) ) + 9 ( +*9+;
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/ - /* % " , / *- ;* ) - " (-8 *) / 9/ -8*+ ;(9
" . T ; - " @ ! + (-8 " @6
9(8*(-. I *)
*= - ( " : " @ 4 : .%U) c @ 3.142' C(;+ ;(
%(' ; - , *- + , , 1 J
% ' / 9* 9, *- + , ) (8 8 9 * -, 5 J
%+' (9 (, *- + 2, ) (8 8 9 * -. 4 J
(!) 6 cm (b) 24.403 cm (c) 37.463 cm 2
+ - 9 (-8 () ( 9(8 ) ! . ) ) + 9 ( + 9+;*+ - 9 (-8 () ( 9(8* ) 14 . S + 9 @" *) ( ;( -. T) (8 89 * - *) ( g 9 8 (-8 () -+ 8. I *) *= - ( @ @ ! (-8 " @ 1."56 9(8*(-).
%U) c @ 3.142' C(;+ ;(
%(' (9 (, *- 2, ;( -, 2 J% ' ; - , *- , -+ 9 H *9 8 9 -+*-
g 9 8, 4 J%+' (9 (, *- 2, g 9 8. 4 J
(!) 62.592 m 2 (b) 38.252 m (c) 14.71 m 2
##
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5. ( SPM 0 P2 ) T 8*( 9( ) ) ( +*9+; , + - 9 " (-8 9(8* ) 10 +*-)+9* 8 *- ( ) + 9 ( +*9+; , + - 9 . T ) 9(*;*- ),
(-8 (9 (- - ) +*9+; ( / *- (-8 ,9 )/ + *= ; .
U) c @ 3.142J . C(;+ ;(%(' ; - , *- + , (9+ , 5 J
% ' (9 (, *- + 2, ) (8 8 9 * -. 5 J
9. ( SPM 0 P2 ) T 8*( 9( ) ) +*9+; ). T ;(9 9 +*9+; () + - 9 G (-8 9(8* ) 12 + . T ) (;; 9 +*9+; () + - 9 < (-8 9(8* ) !+ .
T +*9+; ) + ( / *- . T ) 9(* ;*- *) ( + -(- -
+*9+; ) ( / *- (-8 / *- .U) c @ 3.142JG*= - ( G @ f 9(8*(-,
%(' ) ( f @ 1.3 % 8 +* (; /;(+ )', 2 J% ' +(;+ ;( ; - , *- + , *- 9 (9+ , 3 J%+' +(;+ ;( (9 (, *- + 2, ) (8 8 9 * -. 5 J
1""
(!) 14.176 cm (b) 40.616 cm 2
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(!) 31.41 cm (b) 88.6 cm 2 ( ) ( )
. ( SPM 0 P2 10 )I- 8*( 9( ; , POQ *) ( ) + 9 ( +*9+; * + - 9 O(-8 9(8* ) 26 + . SRT *) ( H (89(- ( +*9+; R (-8 9(8* )5 + .
I *) *= - ( S *) *8/ *- OP. U) c @ 3.142 (-8*= (-) 9) + 99 + 8 +* (; /;(+ ) . C(;+ ;(
%(' POQ, *- 9(8*(- 2 (9 ) J% ' / 9* 9 *- + + ; 9 8 9 * - . 4 (9 ) J %+' (9 ( *- + 2, + ; 9 8 9 * - . 4 (9 ) J
. ( SPM 10 P2 11 )D*( 9( ; ) ) ( +*9+; * + - 9 O. D *) *8/ *- AC (-8 DOC @ 60 .
I *) *= - ( (9 ( (7 9 ) + 9 OAC *) 20"1%7
+ 2.
U) c @ 3.142 F*-8%(' =(; *- 9(8*(-
1 (9 ) J% ' 9(8* ) , *- + , +*9+; ,
2 (9 ) J %+' / 9* 9 , *- + , ) (8 8 9 * - , 4 (9 ) J %8' (9 ( , *- + 2 , ) - A#C 3 (9 ) J
Q
P
S
O R T
P
AC
#
D
60
O
1"1
-
8/12/2019 Modul Matematik Tambahan Form 4 Jpnp
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O 0.3 ' 0 6> 2.1 6> 2
O1.0 21.15 .1
STATISTICS % C&APTER ! F4 '
1. ( SPM 05 P1 23 ) T (- 9 - 9) *) m . T ) )H (9 )
- 9) *) 100 (-8 ) (-8(98 8 =*( * - *) 3 k . E= - 9). T ) - 9) *) 60 (-8
) )H (9 ) - 9) *) !00. F*-8, 9 >= - 9)
%(' (-,% ' ) (-8(98 8 =*( * -. 3 J
(!) 12 (b) 4
. ( SPM 0 P1 22 )A ) ) = - - 9) () ( (- ".
%(' F*-8 h x .% ' - ( - 9 k *) (88 8 *) ) , - (- *)
!.5.F*-8 =(; k .
(!) 63 (b) 5
5. ( SPM 0 P1 2 )A ) 12 - 9) ,,,, 1221 x x x () ( =(9*(-+
1"2
40 (-8 * *) *= - ( h x 2 @ 10!0 . F*-8
-
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( ) ( @%(' (- , x .% ' =(; h < .
(!) 7.0711 (b) 84.853
1. ( SPM 03 P2 5 )A ) ) (9 ): x 1, x 2, x 3, x 4, x 5, x 6 () ( (- =(; 5(-8 ) (-8(98 8 =*( * - 1.5.
%(' F*-8 %*' ) (9 ), h x ,
%**' ) )H (9 8 (9 ),h x 2. 3 J
% ' E(+ (9 *) ; */;* 8 2 (-8 - 3 *) (88 8.F 9 ) - (9 ), >-8 %*' (-,
%**' =(9*(-+ . 4 J
(!) (/) 30(//) 163.50 (b) (/) 13 (//) 3
2. ( SPM 0 P2 )A ) 8( ( + - (*-) 10 - 9). T ) ) - 9)*) 150(-8 ) *9 )H (9 ) *) 24 2.
%(' F*-8 (- (-8 =(9*(-+ 9 ) 10 - 9).% ' A- 9 - 9 *) (88 8 *) ) 8( ( (-8
(- *-+9 () 1.F*-8
%*' =(; *) - 9,%**' ) (-8(98 8 =*( * - 9 ) 11 - 9).
4 J
(!) m*! = 15, B!&/! c* = 22.2(b) (/) 26 (//) 5.494
3. ( SPM 05 P2 ) T 8*( 9( ) ) ( *) 9( *+ 9 /9 ) - ) 8*) 9* * - (9 ) (*- 8 40 / /*;) *- ( ) .
. ( SPM 09 P2 9 ) T ( ; )*8 ) ) 9 H -+ 8*) 9* * - )+ 9 ) ( 9 / / /*;) *- ( ( .
S6 ,N >
-
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%(' * )*- (- *= ,+(;+ ;( 8*(- (9 . 3 J
% ' C(;+ ;( ) (-8(98 8 =*( * - 8*) 9* * -. 4 J
(!) 24.07 (b) 11.74
%(' I *) *= - ( 8*(- )+ 9 8*) 9* * - *)42.
C(;+ ;( =(; k . (b )H * #h* g&!ph p!p*& p&%B/ * by #h* / B/g/ !#%& #%! *& #h/
:* #/% .U)*- ( )+(; 2 + 10 )+ 9 ) - 9* - (;
(
-
8/12/2019 Modul Matematik Tambahan Form 4 Jpnp
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4, 6, 10, 8, 4 (b)3
x = 13, y = 5, m% ! c ! = 20 - 29
. ( SPM 10 P2 9 ) T ( ; ) ) 9 H -+ 8*) 9* * - (9 ) (
9 / ) 8 - .
(' U)*- ( )+(; 2+ 10 (9 ) - 9* - (; (%/, * , 4 &* , %7'*%7, f * , f /6*% / = 'x / &%/g f > 8'.
@ p &%*, f /6*% / &%/g 6 '%/ 8,.
E
-
8/12/2019 Modul Matematik Tambahan Form 4 Jpnp
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)23(6
3)23(2
+=
+
x
xdx
!%2< 3' 3J
20%4< 2' 4J
15%5< !' 2J3 2)2(2 += x y
)2(4
1)2(22 1
+=
+=
x
xdxdy
( 4)2(% += x y
20%< 2' 3J
%)24(3 += x y
60%4< 2' 4J
+ 3)$2(2 += x y
12%2< !' 3J4
2)2(
2
+=
x y
3
3
3
2
)2(
4
)2(4
)2(22
)2(2
+
=
+=
+=
+=
x
x
xdxdy
x y
(4)2(
%
+=
x y
%2"(?@2)9+ :
%)2(
3
+=
x y
61%(?@2)9+ :
+4)43(2
%
= x
y
%1"(3?+4)9 :
5. P9 ; S + -8 D 9*=( *= )
Ex'>p8, :G*= - ( ( )%223)( x x f = , >-8 f I(x) .& -+ , 8 9 *- =(; f I(1)S 8 *% /:
%1' G*= - (3642 23 ++= x x x y , >-8
2
2
dx yd
)( x f y =
)(A x f dxdy=
)(AA22
x f dx yd
=
% &*, %7'*%7,
S + -88 9*=( *=
D%Q, ,/*%'*, * , 4 &**%>,
D%Q, ,/*%'*, * ,&,6 / *%>,
1"#
( )23)( %2= x x f
-
8/12/2019 Modul Matematik Tambahan Form 4 Jpnp
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( )( ) ( )
( )( )
( ) [ ]( )
3""
6"(1)(%) (1)B
1)+(6?)2?+6"(3
31$)2?+2"(3
)23(162?+32"
)23(32")23(2"
):4(234)92"(:2"9)23()(B
2?+3+2"?
423%)(
)(
232
232
2232
32242
3242
42
42
===
== +=
+==
=
f
x
x x
x x x
x x x x x f
x x x f
12 x !
%2' G*= - ( ( ) x x x f 34"2)( 2 = , >-8fI(x) .
160 36%/, * , G ' %,/* f ' T'/g,/* '/ ' N >'8 '* ' p %/* / ' " 7,.
Ex'>p8, 1 : F*-8 9(8* - (- - + 9=
%732 23 += x x x y ( / *- %?2,5'
Ex'>p8, : G*= - )32()( = x x x f (-8 9(8* - (- - ( / *- P - + 9= @ %-8 9(8* - (- - + 9= ( / *- %?1,?3'
11"
S 8 *% / : %732 23 += x x x y S 8 *% / : )( x f y =2
-
8/12/2019 Modul Matematik Tambahan Form 4 Jpnp
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766 2 += x xdxdy
A / *- %?2,5', x= ?2 & -+ , 9(8* - (- - ( / *- %?2,5'
%
7)2(6)2(6
2/hen766
2
2
2
=+=
=+=
==
x x x
x#hendxdy
m $
@ 2< 2 3< )*-+ %
-
8/12/2019 Modul Matematik Tambahan Form 4 Jpnp
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126(2)d?dy
2,/hen ?
6
%3 2
===
=
=
xdxdy
x y
G9(8* - (- - , $ @12EH ( * - (- - *)
( )11 x xm y y $ =
( )
"1712?y 241272127
=+==
x y x y
46(1)2d?dy
,1/hen ?
62
324 2
=+==
+=
+=
xdxdy
x x y
G9(8* - - 9 (;,41= % m
- < @ 1 , @ 4 2%1' 3%1' 2 @ 5 EH ( * - - 9 (; *) ( )11 x xm y y % =
( )
"21+y4?
)1(12"4
141
%
=+=
=
x y
x y
2112 += x y
%2' F*-8 H ( * - (- - + 9= ( )( )112 += x x y ( / *- 9 * )
-
8/12/2019 Modul Matematik Tambahan Form 4 Jpnp
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8 9 *-9 (+ *) (
(==dx
yd
( > F'8 & f ' /6*% /
-
8/12/2019 Modul Matematik Tambahan Form 4 Jpnp
113/162
. P & f M'x%> > '/ M%/%> > F'8 ,& f ' /6*% /.
%1' G*= - )22%(2 x x & = , >-8 =(; x 9 *+ *) (-8
=(; x 9 *+ y *) *-* .
& -+ , 8 9 *- *-*=(; y .
3%
min34 , == y x
x @ f ( )( *) * ) "=dxdy
*;; *-* * y * =(; "2
2
>dx
yd ( x @ f
x @ j ( )( *) * ) "=dx
dy *;; (
-
8/12/2019 Modul Matematik Tambahan Form 4 Jpnp
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%4' T 8*( 9( ; ) ) ) ; - ( *9 . # < * (9 + (- ;(9 () x 3 x (-8 * h .
G*= - (; ; - *9 *)34! + ,
-
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%(' h @ ! 4 x % ' %+' ma?' @1!2"1. 5 + 3
% ' 9 @ 40 , d @ 2 9(8 %+' 16""ma? = A %(' h @r
r
21%" % ' %+' 3##or 1"" %"
. P9 ; ) R( ) C (-
%1' G*= - ( x x y 23 2 = (-8 x *)*-+9 ()*- ( (
+ -) (- 9( 2 -* / 9) + -8, >-8 9( + (-
y - x @ 4 -* .
1
2
44
)2)(22(
22
2)4(6
4
26
23
2
==
=
=
=
=
==
=
"!n(t
dt dx
dxdy
dt dy
dxdy
x)hen
xdxdy
x x ydt dx
%2' G*= - ( x x y = 24 (-8 x *)*-+9 ()*- ( ( + -) (- 9( 4 -*/ 9 ) + -8, >-8 9( + (-
y - x @ 0.5 -* .
%3' G*= - ( x
x1
# = (-8 x *)*-+9 ()*- ( (+ -) (- 9( 3 -* / 9 ) + -8, >-8 9( + (- B - x @ 1
-* .
I y = ' ( x ) (-8 x = g ( t ) , - )*- + (*- 9 ;dt dx
dxdy
dt dy = ,
9dt dy *) 9( + (- y (-8
dt dx *) 9(
116
-
8/12/2019 Modul Matematik Tambahan Form 4 Jpnp
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12 -* ) 1
30 -* ) 1
%4' T (9 ( ( +*9+; 9(8* ) & +*-+9 () ) ( ( + -) (- 9( 10+ 2 / 9 ) + -8. F*-8 9( + (- & - & @ 2 + .% U) k @ 3.142 '
A-) 9 :
1
2
7#%7"
41"
4
)2(2
2
2
1"
=
=
=
==
=
=
=
=
"cmdt dr
dt dr
dt dr
dr dA
dt dA
dr dA
cmr )hen
r dr dA
r Adt dA
%5' T (9 ( ( +*9+; 9(8* ) & +*-+9 () ) ( ( + -) (- 9( 16 + 2 / 9 ) + -8. F*-8 9( + (- & - & @ 3 + . % U)k @ 3.142 '
0.!4! + ) 1
%6' T = ; ( )/ 9 9(8* ) & + *-+9 () ) ( ( + -) (- 9( 20+ 3 / 9 ) + -8. F*-8 9( + (-
& - & @ 1 + . % U) k @ 3.142 '
1.5"1 + ) 1
% ' T = ; ( 9 , D + l, *- (
+ - (*- 9 *) *= - ,$31 3
hh' += 9 h + *) *
( 9 *- + - (*- 9. ( 9 *)/ 9 8 *- + - (*- 9 ( 9(
sm1" +13 F*-8 9( + (- * ( 9, *-
,sm +13 ( *-) (- - * )
%!' A )/ 9*+(; (*9 ; *) 9 8 ( () ( / -8. - ;= ) ) 9 (+ ( 9, *
-
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6
% + ) 1 $" + 3 ) 1 21 + 2 ) 1
%10' T 9(8* ) ( )/ 9*+(;(;; - *-+9 () ) ( 9( 0.5+ ) 1 . F*-8 9( + (- *- = ; - 9(8* ) *) 15 + .
%11' T 8 ( + *) 8 +9 ()*-( 9( 3 + ) 1 . F*-8 9( + (- *- = ; - = ; *) 64 + 3.
%12' D%'g '> 1 ) ) ( + -*+(;+ - (*- 9 * (8*( 9 60 + (-8 * 40
+ . ( 9 *)/ 9 8 *- + - (*- 9 ( ( + -) (-
9( 1 000 +13 sm .
C(;+ ;( 9( + (- 9(8* ) ( 9; = ; ( *-) (- - 9(8* )
( 9 *) 6 + .%U) c @ 3.142 = ; + -
hr 231 = '
60 +
( 940 +
:iagram 1
11$
-
8/12/2019 Modul Matematik Tambahan Form 4 Jpnp
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4%" + 3 ) 1144 + 3 ) 1
6.631+ 3 ) 1
%13' O*; *) / 9 8 *- (- *-= 9 89* +*9+ ;(9 + - () 9(8* ) 6+ (-8 * 1! + ( 9( 2 +13 sm . F*-8 9( *-+9 ()
* ( 9 ; = ; - ( 9 ; = ; *) 6 + * . % U) k @3.142 '
%14'
T ( = > 9 ) ) (- *-= 9 8+ - * (
* 20 + (-8 ( () ?9(8* ) 4+ . ( 9 *) / 9 8 *- + - (
9( 5 + l ) 1 ( )(* , ( 9 *) 89*//*- 9 + - 8 ( ; ( ( ( 9( 1+ l ) 1 .
%(' I * (-8 = ; ( 9 ( * t ) (9 h + (-8 ; + l9 )/ + *= ; , ) (
7%1 3h' =
% ' F*-8 9( *-+9 () ( 9 ; = ; *-
%15'
T ( = 8*( 9( ) ) ( ) ;*8 *++ -)*) )
( + *8 * ( )H (9 () )*86 x + ,) 9 - 8 ( / 9( *8 * 4 x
+ . T= ; + *8 *) 5!32 + l.
%(' S ( (; ) 9 (+ (9 ( ) ;*8,
+ m, *) *= - 3$$$
#6 2 x
x A +=
% ' I =(; x *-+9 ()*- ( 9(
4 +
20 +
h
+
#
&
F G
C
A
D
4 x +
6 x +
6 x +
\
E
11#
+ - ( - ( 9; = ; *) 12 + . % U) k @ 3.142 '
0.0! ,sm +1 >-8 9( *-+9 ()
-
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0.15"1 + ) 1
; ; ) ) @
% ' 0.2210 + ) 1
+9 () (; ) 9 (+ (9 ( ) ;*8 (
*-) (- x @ 4.
42 2 ) 1
11. P & f S>'88 " '/g,& '/ App x%>'*% /& .
%1' G*= - ( x x y 42 += , >-8 ) (;;
+ (- *- y - x *-+9 () ) 92 2.01.
%2' G*= - (9 ( ( 9 + (- ; , x x A 23 2 += ,
9 < *) *8 , >-8 ) (;;+ (- *- (9 ( - *88 +9 () ) 9 3 + 2."!+ .
A-) 9 :
%3' G*= - ( %4
x y = , +(;+ ;(
=(; dxdy
* x @ 2. & -+ , ) * ( =(; )
"324
%
So tio+
Sma
Cha+g()
A,,roximat(
;a (
12"
42 +=dy
x x y 2 23 +=dA
x x A4
4 %% ==
x y
-
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"$"
)"1")($(
$24)2(2
"1"
2"12
"122
42
==+=
==
=
+=
y
y
xdxdy
y
x#hendx
dy
x
x
xdxdy
12sm4"
)"2")(2"(
2"
32)3(6
"2"
#$23
#$23
26
=
=+=
===
+=
A
A
xdxdA
A
x#hendxdA
x
x
xdxdA
16
%2",2Dhen
2"2"
6
66
%
===
==
xdx
dy x
x x
dxdy
xy
x
dxdy
y
y y ya
*r(g(na+
*r(g(na+
+
+=
)( ne/
( )
116%2%" 32"
3324
2"3216%
2
42
4
22
4,
2
4
"32
4
%
%
%%%
+
+
==+=
xdxdy
xand y#here y
%4' G*= - ( x x y = 22 , >-8 ) (;;
+ (- *- y - x 8 +9 () ) 9! ."!.
%5' A + *8 * )H (9 () () (
(; ) 9 (+ (9 (, x x A 432 = , 9
< *) ; - )*8 () .F*-8 ) (;; + (- *- (;) 9 (+ (9 ( - ; - )*8 () 8 +9 () ) 9 5 + 4."" + .
%6' G*= - 327 x
y = , >-8 =(; dxdy
- x @ 3. & -+ , ) * ( =(;
"33
273
121
-
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0.620.26 + 2
1=dxdy 0."
% ' G*= - ( x y 4= , >-8 dxdy
.& -+ , >-8 ) (;; + (- *- y - x *-+9 () ) 9 4 4.02.
"2"C2 == y xdxdy
%!' T = ; , D + 3 , ( + *8 *9 + (- ;(9 () *) *= -
x x x' 32 23 += , 9 x + *) *8 () . F*-8 ) (;;+ (- *- = ; - *8*-+9 () ) 9 4 + 4.05 + .
1. 5 + 3
%"' G*= - 432
x y = , >-8 dxdy
. & -+ ,
) * ( =(; ##132
4
%12$ xdx
dy = 2.04
%10' G*= - ( x
y= 22"
, >-8
(//9 -8 =(;
dxdy - x @ 4.& -+ , ) * ( =(;
%(' ( )3"24%
% '
( )3##3%
122
-
8/12/2019 Modul Matematik Tambahan Form 4 Jpnp
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16"1
.2 + 2
2%61%=dxdy %(' 0.0 "30 % ' 0.0 ! 1
SPM QUESTIONS PAPER 1 ^ 2
1. ( SPM 03 P1 15 )G*= - ( @ 4< % 5 < ', +(;+ ;(
(' =(; < - *) ( (
-
8/12/2019 Modul Matematik Tambahan Form 4 Jpnp
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124
9(8* - - 9 (; ( *)14
. F*-8 + 98*-( ) 9 ) x . 3 m!&k J
-
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4. 3 m!&k J
P % , 4 ' J
14 %3< 5 ' 6 J
. ( SPM 09 P1 1 ) G*= - ( y = 3x 2 + x 4 ,
%(' >-8 =(; dydx
- x @ 1,
% '
-
8/12/2019 Modul Matematik Tambahan Form 4 Jpnp
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/ @ 16 J h2
J
13. ( SPM 0 P1 20 ) T - 9 (; + 9= @ < 2 5< ( / *- P /(9(;; ; ) 9(* ;*- @ ?< 12 . F*-8 H ( * - - 9 (; + 9= ( / *- P
3 (9 ) J
@ ?< 3 J
1 . ( SPM 0 P1 1 )
T 9(8* - -+ * - ( + 9= *)dydx
@ < 6 , 9 *)
( + -) (- . I *) *= - ( + 9= () ( 9-*- / *- ( % 2,
1 ' . F*-8
(' =(;
@ 3 J
15. ( SPM 0 P1 20 )A ; + *+ *- 9 ( + * )*8 ) < + , ; )( ( 9( ". 2 + 3 / 9 *- ). F*-8 9( + (- ) < ( *-) (- - < @ 12 3 (9 ) J
? 0.0225 J
19. ( SPM 10 P1 20 )G*= - @ 2< % < 6 ' , >-8
('dxdy
,
' =(; < - *) *-* ,+' *-* =(;
3 (9 ) J
4< 12, < @ 3 , @ ?1! J
1 . ( SPM 10 P1 21 ) T = ; ( )/ 9 *) *-+9 ()*- ( ( + -) (- 9( 12.! + 3) ?1 . F*-8 9(8* ) )/ 9 ( *-) (-
- 9(8* ) *) *-+9 ()*- ( ( 9( 0.2 + ) ?1.
\; )/ 9 @ 334
r 3
(9 ) J
126
-
8/12/2019 Modul Matematik Tambahan Form 4 Jpnp
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9 @ 4 J
SPM QUESTIONS PAPER 21. ( SPM 05 P2 2 ) A + 9= () ( 9(8* - -+ * - /< 2 4
-
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@ 2 (-8 @ 6 J
(-8 - - - )*8 ) :
C
c
B
b
A
a
sinsinsin ==
T >-8 - - - (- ; ) :
c
C
b
B
a
A sinsinsin
==
D%'g '> 1
D%' '> 2
D
E900
35 0 19@
156>
D%'g '>3
12$
-
8/12/2019 Modul Matematik Tambahan Form 4 Jpnp
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12.32
!. "4 +
10.00 +
%8' D*( 9( 4 ) ) 9*(- ;KLM.
C(;+ ;( ; - KM.
11.26 + J
% ' D*( 9( 5 ) ) 9*(- ; A#C.
C(;+ ;( ; - AC.
11.56 + J
% ' D*( 9( 6 ) ) 9*(- ; PQR
C(;+ ;( ; - PR.
6.52 + J
2. F*-8 - - - )*8 ) ( 9*(- ; - * ) (- ; ) (-8 )*8 - + 99 )/ -8*- (- ; ) (9 - -.%(' D*( 9( 10 ) ) 9*(- ;A#C.
% ' D*( 9( 11 ) ) 9*(- ; PQR. %+' D*( 9( 12 ) ) 9*(- ; DEF.
LK
M
2 093 0
156>D%'g '>
D%'g '> 5
D%' '> 9
RP 25 0
2 0
.26>
D%'g '>11
D
E2
051 0
5.96>D%'g '>
12#
-
8/12/2019 Modul Matematik Tambahan Form 4 Jpnp
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C(;+ ;( ; - AC.
4.51 + J
C(;+ ;( ; - PQ.
3.!10 + J
C(;+ ;( ; - DE.
5.1!" + J
3. F*-8 - - - (- ; ) ( 9*(- ; - )*8 ) (-8 ( - -?*-+; 8 8 (- ; (9 *= -.%(' D*( 9( 1 ) ) 9*(- ;A#C.
F*-8 AC#.
35.2 J
% ' D*( 9( 2 ) ) 9*(- ; KLM
F*-8 KLM
2 .36 0 J
%+' D*( 9( 3 ) ) 9*(- ; DEF.
F*-8 DFE.
11.0" 0 J
%8' D*( 9( 4 ) ) 9*(- ;PQR.
% ' D*( 9( 5 ) ) 9*(- ; A#C. % ' D*( 9( 6 ) ) 9*(- ; KLM.
A
B "900
156>
D%'g '> 1
106>
LK
M
6> 50 0
156>
D%'g '>2
D
E
3.56>
3 0 2 @
12.56>
D%'g '>3
RP
106>13
0 0
136>
D%'g '>
B
A
"
110 0
16>
D%'g '> 5
6>
LK
M
.26>
25 02.6>D%'g '
> 9
13"
-
8/12/2019 Modul Matematik Tambahan Form 4 Jpnp
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F*-8 QPR.
36.11 0 J
F*-8 A#C.
32.!4 J
F*-8 KLM.
13!.64 0 J
4. F*-8 - - - (- ; ) ( 9*(- ; - )*8 ) (-8 ( - -?*-+; 8 8 (- ; (9 *= -.%(' D*( 9( 1 ) ) 9*(- ;A#C.
G*= - ( AC# *) (- ) (- ; ,>-8
; - AC.
% ' D*( 9( 2 ) ) 9*(- ; KLM
G*= - ( KLM *) (- ) (- ; ,>-8
; - ML.
%+' D*( 9( 3 ) ) 9*(- ; DEF.
G*= - ( =(; EDF *) 9 ( 9(-"0 0, >-8 ; - DE.
B
A
"
30
16>
D%'g '> 1
6>
LK
M
6>
00
6>
D%'g '>2
D
E 116>
2 0
6>
D%'g '>3
131
-
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!.01! + 2."52 + J 5.040 + J
%8' D*( 9( 4 ) ) 9*(- ;PQR.
G*= - ( PQR *) (- (- ; *- ) + -8
H (89(- +(9 )*(- /;(- , >-8
; - QR.
% ' D*( 9( 5 ) ) 9*(- ; KLM.
G*= - ( KLM *) (- (- ; *- ) + -8
H (89(- +(9 )*(- /;(- , >-8
; - KL.
RP
9.6>
9 0.56>
D%'g '>
LK
M
1 .36>
230
.26>D%'g '>
5
132
-
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2. 0 + ".6!6 + J
U) C )*- R ; >-8 - - - )*8 ) 9 (- ; ) ( 9*(- ; .
1. F*-8 - - - )*8 ( 9*(- ; - )*8 ) (-8 (- *-+; 8 8 (- ; (9 *= -.%(' D*( 9( 1 ) ) 9*(- ;PQR ) +
( PR @12.3 + , QR @16.4+ (-8
PRQ @ 6 .
% ' D*( 9( 2 ) ) 9*(- ; PQR) + (
PQ @ + , QR @5 + (-8 PQR @5 .
%+' D*( 9( 3 ) ) ( 9*(- ; *)*8 ) 5 + , 13 + (-8 (- *-+; 8 8 (- ; 43 .
bcacb A
2os
222 +=
ac
bca B
2os
222 +=
ab
cbaC
2os
222 +=
5+
x6>
6>
P
R
50
56>
D%'g '>
x6>
E 136>
3 0
5 6>
D%'g '>
133
12.3
P
-
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F*-8 =(; x .
< @ 16.21 J
F*-8 =(; x .
.4 5 J
F*-8 =(; x .
"."46 J
%8' D*( 9( 4 ) ) 9*(- ;PQR.
F*-8 ; - #C.
5."6 + J
% ' D*( 9( 5 ) ) 9*(- ; KLM.
F*-8 ; - LM.
4.312 + J
% ' D*( 9( 6 ) ) 9*(- ; PQR.
F*-8 ; - PR.
5.302 + J
D%'g '>1
.36>
19.6>
x 6>
R90
"A
B
9.36>
53 0 6>
D%'g '>
LK
M
6>
0
5.6>
D%'g '>5
5 0 31@
2.236>
5. 06>
P
R
D%'g '>9
134
2. F*-8 - - - (- ; ( 9*(- ; - 9 )*8 ) (9 *= -.%(' I- D*( 9( 1, A#C *) ( 9*(- ;
9% ' D*( 9( 2 ) ) ( 9*(- ; A#C
9%+' D*( 9( 3 ) ) ( 9*(- ; A#C
9
-
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9A# @ 13 + , AC @ 14 + (-8
#C@ 15 + .
F*-8 BAC .
6 .3! J
9A# @ 11 + , AC @ 13 + (-8 #C@
16 + .
F*-8 BAC .
!3.1 J
9A# @ 13 + , AC @ 16 + (-8 #C @
1 .5 + .
C(;+ ;( BAC
3.41 J%8' D*( 9( 4 ) ) ( 9*(- ; A#C
9A# @ 12.6 + , AC @ 16. ! +
(-8#C@ 1"." + .
C(;+ ;( BCA
% ' I- D*( 9( 5, PQR *) ( 9*(- ; ) +(
PR @ 6.45 + , RQ @ 2.23 + (-8PQ @ 5.40 + .
F*-8 RQP .
% ' I- D*( 9( 6, PQR *) ( 9*(- ; ) +(
PR @ 23.5 + , RQ @ 12.5 + (-8PQ@ 1!. + .
C(;+ ;( ) (;; ) (- ; *- 9*(- ; .
C
A
;
13+14 +
15 +D*( 9( 1
C
A
;
11+ 13 +
16 +D*( 9( 2
C
A
;
13+ 16 +
1 .5 +D*( 9( 3
A
;
12.6 + 16. ! +
1"." +
D*( 9( 4
C
2.23 +6.45 +
5.40 +P
Q
R
D*( 9( 5
12.5+
23.5+
1!. +P Q
R
D*( 9( 6
13%
3".1 J10!.0 J
31."6 J
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% ' F 9 9*(- ; A#C *- D*( 9( , A#@ !.56
+ , AC @ 11.23 + (-8 #[email protected] + .
C(;+ ;( ;(9 ) (- ; *- 9*(- ; .
"3.33 J
% ' F 9 9*(- ; A#C *- D*( 9( !, A#@ 13
+ , AC @ 16 + (-8 #C@ 1 .5 + .
C(;+ ;( ) + -8 ;(9 ) (- ; *-
9*(- ; .
61.1" J
U) 9 ;( C ab sin21
9 * ) H *=(; - >-8 (9 ( ( 9*(- ; .
1. F*-8 (9 ( ( 9*(- ; ) *= - *- (+ ;; *- .%(' I- D*( 9( 1, A#C *) ( 9*(- ;*
A#@ 6 + , AC @ " + (-8= %3 BAC .
% ' I- D*( 9( 2, A#C *) ( 9*(- ; *AC@ 6 + , #C @ 5 + (-8
= 7$ ACB .
%+' I- D*( 9( 3, A#C *) ( 9*(- ; *AC@ 6 + , #C @ ! + (-8
=12" ACB .
! +3
6 +3
120 C
#
A
D*( 9( 3
C
A
;
!.56+ 11.23+
14.5 1+
D*( 9(
C
A
;
13+ 16 +
1 .5 +D*( 9( !
Area of = C ab sin21
= Abc sin21 =
" +
6 +53
#
A
CD*( 9(1
136
F*-8 (9 ( A#C A F*-8 (9 ( A#C.
-
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12.5 +3
6 +3
250 C
#
A
21.56 J
5 +
6 +
!
C#
D*( 9( 2F*-8 (9 ( A#C.
14.6 + 2 J
20. ! + 2 J
%8' I- D*( 9( 4, A#C *) ( 9*(- ;*
AC@ 6 + , #C @ 12.5 + (-8 9 g JC .
%(' 12."2 + % ' 31. 3 0 %+' 141.65 + 2 J
D*( 9( 5
F*-8 %(' , % ' ; - , %+' (9 ( 9*(- ; .
%(' 54.31 0 % ' 4.15 + %+' 4. 5 + 2 J
D*( 9( 6
C(;+ ;( %(' ; - , % ' @ A , %+' (9 ( 9*(- ;
@A .
%(' 6.6 ! + % '!4. 4 0 %+' 13.1 + 2 J
SPM QUESTIONS PAPER 2
142
1. ( SPM 03 P2 15 )D*( 9( ; ) ) ( - \A#C *- ) (/ ( / 9( *8* 9*(- ; A#C () 9* - (; () . \ *) = 9 <
2. ( SPM 0 P2 13 )D*( 9( ; ) ) ( H (89*;( 9(; @A ) + ( ABC *)(+ .
-
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- (-8 (- ; - *-+;*- 8 /;(- \#C (-8 () *) %"
%2. 00, 3.14", 2.!2" '
(
% ' A 9*(- ; Ab#bCb () )( () 9 - ) () )*= - 9 9*(- ; A#C, ( *), [email protected] + , Cb#b@".5+ (-8
4" % B A C = , *+ *) 8*[ 9 - *- ) (/ 9*(- ; A#C.
%*') + 9*(- ; Ab#bCb
%**' S ( )* A ; E 2 m!&k J
G*= - ( \# @\C @2.2 (-8 A# @AC @2.6 , +(;+ ;(
%(' ; - #C * (9 ( () *) 3 2. 3 (9 ) J
% ' ; - A\ * (- ; - A\ (-8 () *)25 0. 3 m!&k J
%+' (9 ( 9*(- ; \A# 4 m!&k J
%('C(;+ ;(
%*' ABC
%**' ADC
%***' (9 (, *- + 2, H (89*;( 9(; @A
143
% 5 .23, 106.0 , !0."6, 122. '
5. ( SPM 05 P2 12 )D*( 9( ; ) ) 9*(- ; A#C
9. ( SPM 0 P2 )D*( 9( ; ) ) ( H (89*;( 9(; A#CD
-
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D ( 9( ; ) ) 9 (- ; A#C
1".2 , 50. 3, 24.!", 2"0.1 J
D ( 9( ; ) ) ( H (89 ;( 9(; A#CD.
% 60.0 , 5.5 3, 116.55 35.43 '
%(' C(;+ ;( ; - , *- + , AC. 2m!&k J
% ' A H (89*;( 9(; A#CD *) - 9 8 ) (AC *) ( 8*( -(;, 4" ACD = (-8 AD @16 + .C(;+ ;( / ))* ; =(; ) ADC 2 m!&k J
% +' # )*- (+ ADC 9 % ' , +(;+ ;(
%*' ; - , *- + , CD 2
T (9 ( 9*(- ; #CD *) 13 + 2 (-8 BCD *) (+ . C(;+ ;(
%(' BCD , 2
m!&k J% ' ; - , *- + , #D, 2m!&k J
%+' , ABD 3 m!&k J
144
5. ( SPM 0 P2 15 ) 9. ( SPM 0 P2 1 )I- 8*( 9( ; , @ *) ( 9*(- ; . AK , @ ! L@ (9
-
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% 13.36, 23.!! 13.!0'
) 9(* ;*- ). T ) 9(* ;*- KL *) / 9/ -8*+ ;(9 @.
I *) *= - ( A @ 1" + , A @16 + , @ 14 + ,$"=
DA, (-84%=
-B.
%(' +(;+ ;( ; - , *- + , %*' A%**' @
5 m!&k J% ' T (9 ( 9*(- ; A *) *+ (9 ( 9*(- ;
K L,C(;+ ;( ; - , *- + , L 4 m!&k J
%+' S + 9*(- ; N N@N *+ () ( 8*[ 9 - ) (/ 99*(- ; @ 1 m!&k J
%1".344, 16.213, 10.502'
D*( 9( ) ) H (89*;( 9(; A#CD.
%(' C(;+ ;(
%*' ; - , *- + , AC
%**' ACB 4 m!&k J
% ' P *- Ab ;* ) - AC ) + ( Ab# @A#.
%*' S + A A BC
%**' C(;+ ;( (9 ( , *- + 2 , A BC 6(9 ) J
-
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146
-
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LN @ 23.23 , MN @ 21. 61 , LMN @"! 12b, (9 ( LMN @ 60.30! J
147
NUM#ER INDE$S % C&APTER 11 F4 '
1. C(;+ ;( I-8 < - 9
-
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(' T ) 8 - / / ;( * - )+ ; VAW *-+9 () 8 96! ) 8 - ) *- (9 1""" "60 ) 8 - ) *- (9
2002. C(;+ ;( *-8
-
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*=' A /(9 *+ ;(9 / - + ) RM0.!0 *- (9 1"" (-8 RM0."0 *- (9 1""". C(;+ ;( /9*+ *-8 < / - *- (9 1""" () 8 -
(9 1"" .
112.5J
=' G*= -
I P9*+*-
1""0
P9*+*-
1""5
P9*+*-8 ++(;+ ;( 9 *- (9 1""! (-8 (92000 (9 130 (-8 145 9 )/ + *= ; -
(9 1"!5 *) ) 8 () () (9.F*-8 /9*+ )+* - *>+ +(;+ ;( 9*- (9 1""! * /9*+ *- (92000 *) RM5!.00
RM52J
4. C(;+ ;( + / )* *-8 p &%*, %/ ,x f * , 6 &* f >' %/g * ,
6' , %/ * , ,' 2005
-
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5. ( SPM 0 P2 13 )T'
-
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9. ( SPM 0 P2 13 ) T( ; ; ) ) /9*+ ) (-8 /9*+ *-8*+ ) 9*- 9 8* - ), >) , g 9, )(; (-8 ) (9, ) 8 ( ( /
>) +9(+ 9.I- 9 8* - ) P9*+ %RM' / 9 9
(9P9*+ *-8