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

- , . /( )= ( )=
+

0 ( )≤+ =0 =
+
≤ ≤
+
0 ≤ ≤
+
=
+
$
+ $
+
# # '
= = × =
− + − + % % (

+ 1 = −
−π $+ π$+
!" &" )" *"
! , !"

( )
−
- , 2 3 / = − =

π
θ π π
4 5 =6 θ+ θ +
=6 + =
+ +

∴ () = ( π$
+
) = −π $
+

+
# 2 (7) = − 1 ) 6 6 65 7 6
7+ 7+#

7+ = (7) 7
+π8 )

!" + &" )" *" +
! , )"

7
- , (7 ) 7= 7− 7
+π ) +π ) 7+ ) 7+#
1 1+

7=− 9 6 73 # 6

7
& 6 6 : 9 1+ = 7=;
) 7+#

∴1 = 7 = /1 − 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 66 6 9 6

Ω 8 Ω
∠ ;!
∠ ;@ ∠ ;@
? ?
+ +
∠ ;!

−8 Ω Ω

+ −
!" ! &" ! )" ! *" ; !
+8 +8 +8
! , )"
- ,

Ω 8 Ω

;!
;@ ;@
;@ 8 Ω Ω
− + + −

;!
−8 Ω Ω

1< = ;× = !
+8 +8

= 6 5 9 + 9 66

5Ω ''

' 5Ω
;;Ω
+

!" =; Ω &" ;; Ω )" =>Ω *" ; 5Ω
! , !"
- ,
( 1 + '' ) 1

5
+ + +
'' '5Ω @ @+ ;;Ω @
'5 − − '' −

;;Ω

written permission.

! 6 6 9 9 6 @
@ = @+ ( ;5 ) ( − ) = ;; ( 1 + '' + ) /
;;;; = ;;1 + ;; × ;; = ;;1 + ;;;;
;; 1
−+;;;; = ;;1 =− 1= −
+;;;; +;;
−1
@ = ;; 1 + '' + = ;; 1 + ;; = =;1
+;;
@ =;1
A = = = =;Ω
1 1

( ! 6

2( )=
( +
+' )( + +)
( + )( + #) ( + %)

6 (ω ) 7

!" ω = $ &" ω = + $
)" ω = # $ *" ω = % $
! , )"
- , B

( ) = 2 ( 8ω) (ω + 2 ( 8ω ) )
() 7
2 ( 8ω ) 6 7

2 ( 8ω ) =
( −ω+
+' ) ω+ + %
ω+ + ω+ + ' ω+ + (
ω=# $ 6
2 ( 8ω ) = ; ( )=;

C 1 6 6 "3 (+ # % =) 6

!" &" D D

)" D D *" D D D D
! , !"
- ,
7
;; ; ;
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; ;

1 6 7 7 7 7
6

written permission.

E 1 3 $ FF
#" $
+" 9 6 9 6 AG)" D
D

!" < 7 <# &" < 7 <
# # +

)" < 7 <# *" < 7
+ #
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- , = ( #) −
+

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# AG)
#
< 7 <#

( $)
+ AG) 7 >
+
6 AG) < 7 <#
+

' 6 6 9
H ( θ) = 6 %
θ/ ≤ θ ≤ π $
; +

6
!" ; & &" + ( & )" = & *" E &
! , !"
H (θ)
- , * 6 =
H ( θ)

H (θ) =
+π π $+

H θ = 6 θ θ θ φ
%π ; ;
;
= +π − %
=
%π ;

* 6 = = ;

;
= ; 9;= ; &

; ! 6 6 +%
6 6 ; E' 2
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µ ; = % π × ; −C I $ ε; = H$ 6 6 6 6 6
#(π
!" ##; Ω &" ;; Ω )" %# #Ω *" %# % Ω
! ,

written permission.

- , ) 6 6 6
µ µ +%
D; = =
+π ∈ +π ∈
B D; = = # Ω

! ! "#
" $&
%
+π

! 6 6 J
9 ! 6 6 6
ε 6 6 ε !
6 9 6
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! , *"
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1 9 9 6

+ 6 6 66 9 "
6 *
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+
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−

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6 )+
*

) 0 5A 6

4 6 6 ω " 6 9 6 6 5 ;
9 5 L+ B 6 ω " L

written permission.

1 +

π
; π 6
#π +π
−
+
+

@( ) ; 6 ( )−
−

−+

# 1 66
=@

!

)

&

)

! &

!" = ! & + ) &" = ( ! + &) 6

)" (
= ! + & ) ) *" = !& + )

! , !"
- , = ! + &) = ! + & + ) = !& + )

% 4 6 " 3 ;= K
+ =

+
+
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+ + + +
! , *"

- , 2 *- + = + =

1H = = 9
= / (1H ) = ( 1H )
+
= ⋅ = +6 2
/ ( )=;= ;== +6 6= ;
+ +
+
∴ K 3 =
+ +
written permission. '

= < 6 " +
< 6
+ +
"
+ +
!" − &" −
( ) ( )
+ +
+ +
+ + + +

+ +
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( ) ( )
+ +
+ +
+ + + +

! , *"

- , 1 ( ) ↔ H( ) ( )↔− H( )

H( )= +
+ +
+ +
( )→− =
+
+ + +
+ +

( 9 6 ( % − 8#) Ω 6
=6 ( ;;π + ;; ) !
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! , &"
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0= 1 +A < = × =+ × % = =;4
+ +

C 1 9 9 ) )+ 6 6 ) 6 9 +
@ 6 B 6 3; 6 "

B =;

) )+
()

!" 7 &" 6
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6 66 6 9 9 6 9 9 6
7
6 5 6

written permission. (

E 6 6 6 66 9
υ−; C
! υ ≥ ; C@
= =;;
; ! υ < ; C@

>Ω

+ +
;@ @
− −

6 66
!" ; ! &" ' # ! )" ( (C ! *" ( + !
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=
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= =;;Ω
B 6 9 6 ; C@

Ω

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; ; ;
; ; ;
; ;
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+; ) 9 66

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O
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1 66 6
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O

)5

O
&

O = ! )<>O
= ! )<>+ O
O = ! )<>+ O
1 )<> 3 ! &3
O =
J 6
O =
1 )<> 3 !3&3;

written permission. *

O =O
J 6
O =O
6 66 66

+ 6 6 6 9 6 6 9
6 9 (
- = ; P + 8P7 ) − 8+=
K 6
7 6
!" + 2I7 66 &" % I7 66
)" + 2I7 9 66 *" % I7 9 66
! , !"
+π
- , = +=
λ
+π
λ=
+=
-7
# × ;E
= = + 2I7
+π
+=
-
< - =6 ω
π
-7 = 6 ω +
+
J 6 QR 66 7

++ ! 9 9 (
- = EP + (P − =P7 ) 8( ω + # − % )@ $

6 6 6 6 9 ≤; -
6

(
!" −EP − (P − =P7 ) 8( ω + # + % )@ $

(
&" −EP + (P − =P7 ) 8( ω + # + % )@ $

(
)" −EP − (P − =P7 ) 8( ω − # − % )@ $

(
*" −EP + (P − =P7 ) 8( ω − # − % )@ $

! , )"
- = E ( +( += 7 ) 8( ω + # − % )@ $

-

- ( 6 6 6 9 )
>; =; <;
written permission. +

- 6 6 66 6 3;
∴- =;

+ H - - 6 6
0 3;
∴- +- = ;
8( ω − # − % )
- = - = − (E +( + = 7)

+# 1 6 6 5 K 6 #=;; I7
9 9 . 9 6 C=S 6
6 9 9
6
!" C=; &" +(+= )" %;;; *" =+=;
! , )"

- , & = A B (β + )
+
AB → B
+×β
AB =
β+
β = ; C=
+ × #=;;
AB = = %;;; $ 6
+ ; C=

+% 6 6 " K 6
- +
() - () 6

π
!" (;;; $ ;

π
&" (%;; $ ;

)" (%;; $ +; $ π +
π ( )

*" (;;; $ +; $ π +
π ( )
! , &"
- , 0B* "
(
%;; δ ω + ;% ) (
%;; δ ω − ;% )
(

− − ; −'
( )
' ;
;# $ 6

written permission.

- +
() =A (;)
∞
A ( ; ) = +π B ( ω) ω
−∞

A ( τ ) ↔ B ( ω)

= × + × ;# × ( + × + × ;# × ( + %;; + %;; = (%;; $π
+π + +
B 6 0B* " 6 *) 6 "
7

+= 6 9 6 +;
6 ( ;;;; π ) ! ! #;; > π 9

!" +=; Ω &" +C = Ω )" += Ω *" ++ = Ω
! , )"

@ += @
- , π = = = +=Ω
1& !

' ,

+( 2
−= −# ;
! = 1= !#
+ ; ;
!" = ! + +1 &" ' ! + #;1 )" C ! + =1 *" C! + + 1
! , &"
= #
- , 2 , !3 /
+ ;
−= − λ −#
) 6 6 K ! ! 1λ = ; =;
+ ;−λ
( −= − λ ) ( −λ ) + ( = ;
=λ + λ+ + ( = ;
λ = −=λ − (
+
λ = −=λ+ − (λ = −= ( −=λ − ( ) − (λ
#

λ = +=λ − (λ + #; = 'λ + #;
#

- 6 6 6 K
∴ !# = '! + #;1

written permission.

# +
+C "3 L' N +% N = (
!" + &" += )" % *" %(
! , )"
- 0, 2 ( )= #
−' +
+ +% + =

: )=;
( # + − E + +% = ; 3+ %
T ) = ( − E/ T+ ) = + − E < ;/ T% ) = +% − E > ;
( ( (
I 6 ( ) 3+
∴ +# − ' × ++ + +% × + + = = +=
& (
∴ ( ( ) = (# − ' × (+ + +% × ( + = = %

+E 1 @! − @& = ( @ @) − @*

!" =@

&" +@

)" #@

*" (@
! , !"
@! − @& (
- , 1= = = #!/B 6 6 9 5
+ +
9 @) − @* 6 1 3 #!
A @! +Ω @& A

A
A
A − ;@
A Ω
+
A
−
Ω @) @*
=@
+!
Ω

@) @*
≡ @)
− + #!
@*

+! +@

@* = + + # + @) = = + @) /@) − @* = −=@

+' 9 9 !@ 66

!" ! @ ≈ +;;

&" ! @ ≈ ;;

)" ! @ ≈ +;

*" ! @ ≈ ;

written permission.

! , *" @

- ,
;;5 +5

;5
;;5
−!

>@< # C ( 1) + 1& ) +5 − ;;5 ( 1& ) − ; C = ;
1& = ' 'µ!/1) = β1& = ; '' !/1- = !
∴ =
+( !
= +(Ω/7 = β = + (5Ω/ ∴ ! =
( ;;5 F +5 ) = % +
F
1- +(
;;5 7: ++
=
7: 7 F F = ++ Ω/! = ! = ( % +)
+% + +
7 :A ++ + ;5
! ≈ ;

#; 9 966 6

! , *"
!=; =O
- , != 9
!= =O

!=;
!= ! =; !=
9 99
O=; O=

! =;
written permission.

# < 6 9 3 $+"
9 6 K 6 1 ; 3 3 $
+ 9 K
!" ; &" $+ )" *" #$+
! , !"
- ,
∞ 5
= 9 − 5)
(
5 =;
+
∞ 5
; = 9 −5 ) =
(
5 =;
+
G
9 ;) =
(
+
B 6 9 " ) K 6
9 ;) =
(
9− " 9−+" =;
∞ 5
= 9 −5
5 =;
+
;
9 + 9 ;) =
(
+ + +
9 =;

#+ 66

!" = $
# &
(A + A+ ) )

&" 9 # & = $
A )

)" # & = $
A )

*" 9 = $
# &
(A + A+ ) )
! , &"

A+
- , @; B" = − υ ( )
A +
)B
A +)B
@G ( B ) = − @ (B)
(A )B + )
6 K 6 9
A )
written permission.

## 9 6 9 9 6 6 9
9 6 6 9
I7 = # 6 (+ ;'% × ;+ ) 6 (+ ( E × ;+ ) 6 ( ( +E# × ; ;
− β7 )

6 @ 9

!" >6 &" =6 )" ; < <6 *" =;

! , *"
π π
- , I7 = IG 6 6 6 (ω − β7 )
β
2

I7 = # 6 (+ ;'% × ;+ ) 6 (+ ( E× ;
+

) 6 (( +E# × ; ;
− β7 )
) 9 K
9 3+ 3
= ; 2I7
+ + ; + +
) #× ; +
) = + = + = = 2I7
+ + # +
9 9 6 @0 3 ;

#% 9 " 5
6 " K 6 5 K 6 6 "
9 6 K 6

5 $
5 $
I7"
!" Eπ &" %π )" +π *" π
! , &"
- , 1
U 0 3> " => +

1 H K 6
+
U 0 3 +π 5 + = +π 5 × %


50
J > + = +π > × %
0 = %π
5

#= ! 66 &B)" $
E 1
6 0 3 ;" 3 '$ ;
6
!" C$
E; &" (#$
E; )" '$ ; *" $ ;
- .
- , $E
= =
C$E

C$E
=; =;
$E

( ) = (;) ( ;) + ( ) (; )
' C (# C C;
= × + ×C = + = = C
; E ; E E; E; E; E

#( 1 )UGB 6 6 6 K U
U+ K 7 6 U 9

!" @ < EC=@ &" EC@ < @ < # += @
)" @ > # +=@ *" ; < @ < =@
! , !"
- , 9 6 6 6 )UGB ,
@
U −
@GI = @**
U+ − B

U+ −
U+ − B

@
@1< @1I @**

@1< = (#@** + +@ ) = + +=@ I 6 G Q
!R
E
6 6

#C ! 6
K
!" $
# &" $
+ )" +$
# *" V
! , )"
- , 0( ) = 0 ( I) + 0 ( I) + 0 ( I) +
# =

= + + +
+ + +
+ %

= + + +
+ + +
+
% +
= + + + = = × =
+ % % + + # #
−
%

#E 6 6 ! 9 ! =5
+ + + +
= + +7 5 6 6 ∇! = ;
!" + &" + )" *" ;
! , !"
- , 4 5 ∇ != +
∂
∂
( +
! )
J ∇ != +
∂
∂
( +
! )
= + ∂
∂
(5 ) =
I+ 5
+
( + +) +

=5 ( + +) +

∴H ∇ !=; ( + +) = ; = −+

#' ) K
+
( ) ++ ( )+
+
( ) = δ( ) () = ;−
= −+ =;
= ;−

6
= ;+

!" + &" )" ; *"
! , *"
+
( )++ ( )+
- , + ( ) = δ( )
) 9 −
+
( )− ( ;) − :;) + + ( ) − (; )
( + ( )=
+
++ + ( )++ +% =


−# − +
( )=
( +
++ + )
H 6
( ) = −+ − − −
()
( ) =+ − + −
− −

() =+− =
= ;+

%; ! 9 9 6 A 6
66 ! 66 &

!" ; E Ω &" %Ω )" + Ω *" + E Ω
! , !"
C ( + ;A %+ + C;A
- , 0 66 ! 66 ! 3 @13 =
A ++ (A + + )
+
AN+

;−# C
1= =
++A ++A
CA ( + ;A
@3#N1A3#N =
+NA ++A
0 (A + + ) ( C; ) − ( %+ + C;A ) + (A + + )
+

= =;
(A + + )
%
A

+Ω A + A
1
+
+ + + +
;@ −8 Ω
−
#@ ≡ ;@ @
−
#@
− −
−

66 ! 66 &

C; (A + + ) = ( %+ + C;A ) + (A + + )
+

= (AN+ ) = + ( # + =A )
=A + ; = ( + ;A
%3=A
A3; EΩ


% 6 < 1 9

; ; ;
+ = ; ; + + + ;
# ; ; #
#

= ( ; ;) +

#

6
!" ≠ ; + =; # ≠; &" = ; + ≠; # ≠;
)" = ; + ≠; # =; *" ≠ ; + ≠; # =;
! , *"
- , 6
= & !& !+&

; ;
!= ; ; +

# ; ;

;
&= ;

; ; +
6 = ; + ;
; ;

6
≠;
+ ≠;
6
# 6 7

7

%+ 9 " I ( 8 ω ) 3 (+ 6 ω) ( +ω) $ω
(;)
!" $
% &" $
+ )" *" +
! , )"
- ,
+ω = :)
(
+
ω

−+ +

written permission. +

+ω
+6 ω
ω ( )= (
: − )+ :
( + )
8ω − 8ω +ω
+
ω
+
−# #

%# 65 6 + $

!" > 3 + 3 ; C= &" > 3 # 3 ; C=
)" > 3 % 3;= *" > 3 + 3;=
! , !"
- , + 2 ( B ) I ( B ) = B# + +
+ (+ + 5 ) + +5
#
(+ + 5 )
+
(+ + 5 )
(+ + 5 ) − (+ + 5 ) ;
( + 5)
6
(+ + 5 ) − ( + 5) = ;
+5
=
++5
+5 +5
!- +
+ ( + 5) = ; = =+ = ++5 = % 5=+

= ; C=

%% " " " 3

−∞
( τ ) 6 (#τ ) τ

!"
&"
)"
*"
! , &"

- , "= τ" 6 #τ" τ
−∞

B 6 " " 6

written permission.

< " 5

"= >6 #τ" τ < ∞
−∞

" B

%= ! 6 6 6 6 ;; Ω 6
=; Ω 6 +;; Ω 6 1 6 9 %+' UI7
2I7 9 6
!" E+ = 6 &" ;= )" =E *" C=
! , &"
- , ) 6 6 6 3 ;;Ω
6 6 3 =;Ω +;;Ω/ K 6 3 %+'UI7 2I7
( U 6
λ
9 6
9 9
9 3 λ$
%
)

=;Ω ;;Ω +;;Ω

6 9
6

λ )
%+'UI7 = = = ; C= /
% %
λ )
2I7 + = + = = ; ;C=
% %+
< 9 9 +
∴ 3 <)U + = ; =+=
I 6 ;=

%( ! &0B> 6 9 !42J 6 6
+-
J; $+ K 9 ( )= ( ω6 )
+-
+ ( )=− ( ω6 ) (; ) 1 6 6

6 6 %=G 6 6 9
9
+- - - -
!" O &" O )" O *" O
J; J; +J; %J;
! , &"
J
- , 2 , &0B> 6 !42J 6 J 6
+
-K 9
+- +-
( )= ( ω6 ) / + ( )=− ( ω6 )
6 6 6 6 %=W 6
6 9
written permission.

2 6 6 6 6 9

+
φ ( )= ω6 ; ≤ ≤ 6 6 %=W

+
φ ( )
+ %=W = ( ω6 )
+ %=W ; ≤ ≤

A 6 6 ,

( ) +
( 9 6 )

φ ( + %=W)

+- +
= ( )φ ( + %=W) = ω6 ⋅ ( ω6 + %=W)
; ;

-
= /
+
-
+ =−
+
-
5 9
+
- - -
= + J/0 + 6
= 0 ( γ < ;) = 0 +J< ; =0 J< −
+ + +

1
6 9 6 9
+

1A
- -
−
+ +

-
5 9
+
-
= + J/
+

- -
0 + 6
= 0 ( γ < ;) = 0 +J< ; =0 J< −
+ +

11 2 A @
- J;
6
+ +

1A
- -
−
+ +

written permission.

+
-
+ -
+
+ +
− +
; J ; −
- + ;
0 J< = +
= J;
/
+ −∞ J −∞ πJ;
+π ;
+

-
+ ∞ +
+ J; - − -
< = / 3 0 J< = +
=O
J; + -
+π J;
J;
+
( 9 6 ! 2 0 )
B K 0( )= 0 6
+0 + 6

+ +

- - -
∴ 0( )= O + O =O
+ J; + J; J;

%C 6 6 ( = ; ;
6 #
) 6 UGB
' #
Kµ µ 1 ; $6
6 9
C
!" ; &" ;; )" ; *" ;
! , *"
+
−#
- , = '
= ;6 / 3 ; E #
= ;− + 6 #

;
∴ 3 ; ≈;

// 0 1 )2 *

1 6 6 UGB
δ = +;
µ
8 6 ;
B B G+ 6 C #'
ε; = E ' × ; − +
H$

written permission.

%E 6 8 6 6 6 6
!" + H &" C H )" + H *" C H
! , !"
C × µ × ; +µ × E ' × ; +
- , B 6− 8 6 6 6 6 3 =+H
;

%' 9 6 6 6 6
!" ; C H &" ; C H )" ; #= H *" ; +% H
! , !"
ε! ε ( B G+ ) × ! # ' × E ' × ;− +
× µ × +;
- , )= /)9 = = =;C H

// 0 1 +2

4 ;@ 66 6 ! 6 6 5
9 ,
" Ω 6 6 & 6 #!
" + =Ω 6 6 & 6 +!

=; H 5 (@ 66 6 ! Ω 6 6 &
C$ ! 1 E@ 6
# 6 6 ! 66 9 &
!" (@ &" C@ )" E@ *" '@
! , &"

= 4 ;@ 66 6 ! 6 CΩ 6 6 &
!" #$
C! &" =$
C! )" ! *" '$
C!
! , )"
- , 9 5 6 6 K @ A
9 A 6
J 6 K
@ 3 #A N#
@ 3 +A N=
B 9 K 9 A 3+ @ 3'
J 9 K 6 5
' 3 1 × +NC × 1 13 !

written permission.

3 ! 1 ,

4 / 3 ! 1 2

+
6 6 9 2) ( )=
+
=+ 2) B" 6
!" 3 3+ &" 3# 3+
)" 3 # 3 *" 3# 3
! , !"
− ω − ω
- , φ= −
β
φ +

− ω − ω
>
β
<
!" )"
)" 7
AIB
G !" 9

=# 6
!" + $ &" # $

)" ( $ *" $# $
! , !"
- , X3 9 6 6 K 6
= +× = + $ 6

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Ec gate-2012

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