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!

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( ) ( )% % & = ω − φ

φ (

%

−φ %

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1

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*+#$566"

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1

1

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%

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< <

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A & ( ) ( )B'9'? %'1'4'8 '=

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( ) ( ) ( )

%

7

7 8/&2 8/

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7 7 /; // & .. =

$//// //& // // //& ////

// &%//// //& &

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/

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/

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% % @ @ 1

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π π + +

%

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= = × =

1 I " ' +

( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 8 * * 1 & & 1 & 8 = ω + ω = ω − φ + ω − φ + ω

" !

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1

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*+#$ ( ) + D'+- $=+

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%

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πθ

π −π

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8 %/

8/ %/ / /8

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α "

%

α − π

%

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απ

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D D%5 /'5 4/= =

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%

<

<

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= =∂ − −−∂

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%1 " ( )4 :1− Ω

( )8 // // π +

44%! 8/! >%8! %8!

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% %

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−

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%

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% %

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∴ =

%> %%/7'8;!'/// " /%8Ω '

>0 %%

+( >//"" 8%0 0

00O 00O

1>1>O 1>1>O

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% %

*3

3 *

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φ

%2 /%8 =& > %% >80=& 8%0 0 8= = − = = − =

%

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%

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%A & "'" //;@"

8/OD ∆ >;$ $;&/8' ;

>>

8/

80

4%

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% %

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%0 /8 " $

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8

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+

8 5

8

×= =

5 /8=

, , + , +5 5 0/ 1/ = 0/ 0/ 8/= δ = δ = − =

( ) / %0/ +− δ = δ π − +

, /8 8/ A.>1

− π = π − + =

P

* 8

+

==

= µ

/8

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2< 1& 7 = θ

1> /48& %%4

8

×= =

5 7& 1> %%4 /48= θ = × × A8/ !≅

%%

%

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%. I ""' % +"

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K

%

/% /%>0 /1% /4

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< %= + = + =

( ) ( ) ( )

%

% % %

%

% % %

5" $ $%7 7

- = /8B /8 /8 1/= / 7 / = 1/Q=7 1/−

×∴ θ − θ = θ − θ θ − θ =

θ − θ = = θ = = ∴ θ = − = −

( ) ( )

%%

% D%

% K % %

7 7 / 1/& :/%00

% :/8 1/= K 1/ :/%>0 /%>0 %/

72 - 7 & 7 & /%>0 ./ /%>0

− − −= = = −

= − = − − − = − = × × + =

1/ "

D

5 =

5% =

%&

@ :/8=

%7 1/−=

7 /=

© All rights reserved by Gateforum Educational Services Pvt. Ltd. No part of this booklet may be reproduced or utilized in any form without the written permission.

( )1

%

<" " )

2 2 =

+

1

" )

2 =

1

<" " )

2 =

( )1

%

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+

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= / '

% %

= = =

( ) ( ) ( ) ( ) ( )

S ∞

=−∞

= = −

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

$ = / / / /

4 %

/ /% %

= × + × + ×

= = + =

1% "

( ) / ( )

/

( )

( )

)%

)4

( ) −

/−

/ %

%−

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( ) −

/−

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© All rights reserved by Gateforum Educational Services Pvt. Ltd. No part of this booklet may be reproduced or utilized in any form without the written permission.

"#$

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( )5 K 3+

/ / /

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11 ""

%//≈

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& %>@ C//; %%

@ C @ TT %% = 4% 4% @ C2 %% /;

/

+ − − = = µ = β = =

∴ = = Ω = β = Ω ∴ = =

= = Ω = = + + + ≈

K /= K =

/=

/=

= =

/;

//;

−

//; %;

7

© All rights reserved by Gateforum Educational Services Pvt. Ltd. No part of this booklet may be reproduced or utilized in any form without the written permission.

14 & 7 7 >7' 7 7− = −

E87 %7 17 >7

"#$

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% %7 7 &-1

−= = =

−

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18 + ( ) 1 %+ + .+ %4+ 8= − + + F'>G

% %8 4 4>

"#$

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( )( ) ( ) ( )

( )

%

1 %

C+ / 1+ 0+ %4 / +-%'4

U+ >+ 0=U% % 0 /=U4 %4 0 /

+ + +-%

+% . % %4 % 8 %8 " + '

= − + = = − = − < = − >

∴ − × + × + =

( ) 1 %

>

> > . > %4 > 8 4

∴ = − × + × + =

1> D 18 1 / & '

% / /

− − = =

8 %&+ . 1/&+ A 8&+ A %&+

"#$

−+/7

2%Ω2

2

2

+ −

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7

2

2

Ω

%

2

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7

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1≡

© All rights reserved by Gateforum Educational Services Pvt. Ltd. No part of this booklet may be reproduced or utilized in any form without the written permission.

*+#$$8 $1

D # - =% /

( ) ( )( ) ( )

%

% 1 % %

1

1

8 1 ( $& / /

% /8 > / 8 > /

8 > 8 > 8 8 > > 8 >

%8 > 1/ . 1/* + ( . 1/&

− − λ −λ = =

− λ − − λ −λ + = λ + λ + = λ = − λ − λ = − λ − λ = − − λ − − λ λ = − λ − λ = λ − λ + = λ +

∴ = +

1A /;7'8/@ " ;7""/8

88!' ('

" %

;

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& %;7 8/@

' "' '

/A'AA0! /A'88>!

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///88

& //88=///

6 @ '& & & /8 //88 /4.>.2 /// %///

3" 2 = %% % 3 % 3

φ

= × = =

= =

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φ =

( )

%

%

%

%%

& 3 & 3 %& 3= %

2 2 22

& %& %& % /4.>. /A/%2

φ φ φ

φ φ φ

= = φ = φ =

∴ = = = × =

10 ; 1$' 8;!'4$'%1/7'8/

@ 8/3 @

' + ; " " %1>7'8A@

808 48/ 48A 88>

"#$

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% %% % % %% %

* * * *& -

@ B <2 B= = =

ω+

%% %

% %

* 8/ %1/ 8A& = & 48

& 8/ %1>∝ = × =

© All rights reserved by Gateforum Educational Services Pvt. Ltd. No part of this booklet may be reproduced or utilized in any form without the written permission.

1. + !

%/; 'Ω 44/7$" 0/;Ω 18%7I 4/;Ω'

1A7 1017 1.47 4/>7

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2 240/

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+

= × =+

4/ + V ( ) ( ) ( )

+ 1 −∞

= τ τ τ

$ $

$ $

"#$

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= τ τ τ

+ " ' $

< + ;

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= τ τ < ∞

4 ; "" %)"

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+

−7

%/;Ω

2;Ω

+

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44/7

+

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2

+

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18%7

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*+#$ ( ) ( ) ( ND /=

( ) ( ) ( )1 %

1 %

; /= ; % ; /

%

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1

%

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;

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( )

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% %

; /=-:= = %

; /= ; =4-;N=I ';-%'-/A8

+ + = ω = −ω ω =

− ω + + = ω = +

4% I ( ) ( ) ( ) ( ) : % % )ω = ω ω ω

/

)4 )% %

"#$

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41 <&

( )

% % %

1 1 1

%

1

+ / / + /

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+ / / +

+

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+

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=

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% 1 /' /' /= = =

% 1 /' /' /≠ ≠ =

"#$

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%

% %

% %

1

/ / / /

K =- / / =- / =- = /

/ / / /

= =

( ) C %

=

% %−

%ωω

: :

%%

% ω − ω

ω ω ω

ω + ω

( ) ( ) ( ) C C = − + +

1− 1

%

© All rights reserved by Gateforum Educational Services Pvt. Ltd. No part of this booklet may be reproduced or utilized in any form without the written permission.

%

%

%

1

/ /

K / / =&;K 1 +' K

/ /

/

/ K /

/

= = =

≠ ≠ ≠=

44 ' 2 "+"

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"#$

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A > /2 4% A/25" -7&-

2N% 2 % 2 %

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%

4

/ 1 A A2 > /2& =7-1N&2-1N

% 2 % 2 %N2 % 2

2 % A/ 4% A/2 % 2 %5/

2 2 %

− + = = = + + + + − + +

= =+

( ) ( ) ( ) ( ) ( )%A/ 2 % 4% A/2 % 2 % =8 2N% % 1 8282 / > /2=4-82=2-/0

+ = + + = ++ = + Ω

48 (

( ) ( ) ( ) ( ) ( )

%

% / /

% " % /

−=

=

+ + = δ = − =

/

+=

E% E /

+

−17/7

+

−

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:− Ω

/7

+

−

%

+

−

7

2

+

−17

&

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© All rights reserved by Gateforum Educational Services Pvt. Ltd. No part of this booklet may be reproduced or utilized in any form without the written permission.

"#$

*+#$( ) ( ) ( ) ( )

%

%

%

+ + = δ

( ) ( ) ( ) ( ) ( ) ( )( )

( ) ( )

%

%

%

'

/ C/ % /

% % 4

1 %

%

− − − + − + =

+ + + + =

− −=+ +

I

( ) ( )( )

( )

/

%

%

%

+

− −

− − −

=

= − −

= + −

= − =

4> " ' " % % % % ; " + @= = + + ; "

/∇ =

E% % /

"#$

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!;" '

∂∇ =∂

( )%%

3"'

∂∇ =∂

( ) ( )

( )( )

%

% %

;; %

; %

I' /' % / %

+

+

∂= = +∂

= +

∴ ∇ = + = = −

4A

( '

)1 )% %)1 1)4

"#$

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1 8 % 4

%

% % % % % %

4 %

% 4 4 % % 1 1

4

= + + + = + + +

= + + + = = × =

−

© All rights reserved by Gateforum Educational Services Pvt. Ltd. No part of this booklet may be reproduced or utilized in any form without the written permission.

!"#"$%&

! /7 "$ ";""' ""#

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40 I ";'" >7 ' Ω "A)1&07 '

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0/ $ "

8/

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%26 -7 7

1=

<

7 % %26 - 7 1// 44%7

1 11= = × =

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8 & 7 1//7'= "1$

8;! %/;! %8;! 1/;!

"#$

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7 44%5 1 1 1///!

2 %/= ⋅ = × =

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17 1>7=

8% "

/48 /8/ /88 />/

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%

%%/ %% 1> /48

% % %% 1>

− − − −θ = = =× ×

81 &<2 8 '= Ω + "

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% %1< <

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@ = = = = = × =

*"" "+'$()#"$%&

( )

D

+=+

84 ( )D

' %= = 1' %= = 1' = − = − 1' = =

© All rights reserved by Gateforum Educational Services Pvt. Ltd. No part of this booklet may be reproduced or utilized in any form without the written permission.

"#$

*+#$

− −ω ωφ = −β

φ +

− −ω ω >

β

<

" @

2$

,

88 +

% ) 1 ) > ) ) 1 )

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+

= -'-%' %)∴ ω = =

8> ! " ""X

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6

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8A "

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,+""$(-($"("-$(./////////" ""#""0-

; ;

; " ;

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,2#"("""#-(03$4"($4$3"""( $"(+"

(

"

"#$

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8. " " " #

5$4"$#+"$"#"$"" ("+.$//$ 6$4

:

"#$

>/ & ( ) ( ) ( )%8. %/>% 11%// 18% // A08' //= = =

%%1 4%1 1A %A>4

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%8. %/>%

11% %8. %/>%

+ 18%+ A08

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