Modul Matematik Tambahan Form 4 Jpnp

8/12/2019 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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4

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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1"

#. 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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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'

21

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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3%

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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3$

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


66/162

66

% 3 - 1 '

% / 4H '

3. % SPM 03, P1, Q5 'G*= - ; 2 T ? ; 4 \ @ 3,


67/162

% 39 2/ 1 '

% < @ 1 '

11. % SPM 06, P1, Q '

G*= - ; 2 xy @ 2 3; 2 x ; 2 y ,


68/162

< @ ?3 J

< @ " J1 . % SPM 0", P1, Q6 'G*= - 3 -?3 $ 2 - @ 243 , >-8 =(; -.

- @ 2 J

1!. % SPM 0", P1, Q 'G*= - ( ; ! ; 2 H @ 0,


69/162

1. T %/ * , %&*'/6,


70/162

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


71/162

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


73/162

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


74/162

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


75/162


76/162

77

y

D*( 9(

y

D*( 9( x

y

"


77/162

. S 87, '88 * , f 88 %/g

x "

2

x "

3 D*( 9(

4

7$


78/162

:!#/% %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#


79/162

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

$"


80/162

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


81/162

% 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


82/162

@ 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


93/162

3.3 F 9 (+ ;; *- , >-8 (- ; F *- 9(8*(- *= - =(; ) & (-8 .

E


94/162

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


95/162

'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


96/162

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 )


97/162

. ( 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+;


98/162

/ - /* % " , / *- ;* ) - " (-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

##


99/162

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


100/162

(!) 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


101/162

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


102/162

( ) ( @%(' (- , 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 >


103/162

%(' * )*- (- *= ,+(;+ ;( 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* - (;

(


104/162

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


108/162

)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


109/162

( )( ) ( )

( )( )

( ) [ ]( )

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


110/162

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< )*-+ %


111/162

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 * )


112/162

8 9 *-9 (+ *) (

(==dx

yd

( > F'8 & f ' /6*% /


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 *;; (


114/162

%4' T 8*( 9( ; ) ) ) ; - ( *9 . # < * (9 + (- ;(9 () x 3 x (-8 * h .

G*= - (; ; - *9 *)34! + ,


115/162

%(' 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


116/162

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, *


117/162

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$


118/162

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 ()


119/162

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


120/162

"$"

)"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


121/162

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


122/162

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 < ', +(;+ ;(

(' =(; < - *) ( (


123/162

124

9(8* - - 9 (; ( *)14

. F*-8 + 98*-( ) 9 ) x . 3 m!&k J


124/162

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,

% '


125/162

/ @ 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


126/162

9 @ 4 J

SPM QUESTIONS PAPER 21. ( SPM 05 P2 2 ) A + 9= () ( 9(8* - -+ * - /< 2 4


127/162

@ 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$


128/162

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#


129/162

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"


130/162

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


131/162

!.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


132/162

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


133/162

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


134/162

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


135/162

% ' 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.


136/162

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 *)(+ .


142/162

- (-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


145/162

146


146/162

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

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