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!!"#$%$ &’’! (’)$*+,*,-,).+’/)0’’ $ &%#!%$$#0&$12$$’’ $ & & #$’)) ’’$ &/’)/! ( ) α ( ) α ( ) ( ) α + + = ! " #$% ! ( ) $ & " ( ! )*++,-.. / 0* " 1 ,1023444. +5,0 1 6 7** " 1 6 ) $ $ π & ( ) π π - < < = < < π π π π + - - 8&$ π = " 6 . + + + + π 9 π : π π 9 ( !$$/ - + ; + 6 - http://www.satishkashyap.com/ For Video Lectures http://www.satishkashyap.com/ Not available in NPTEL FREE online GATE coaching http://www.egate.ws/ FREE online IES coaching

GATE ECE 1993

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Page 1: GATE ECE 1993

! !"#$%$&''!

( ') $*+, *,-, ) .+ '/)0 ''

$&%#!%$$#0&$12$$''$&&#$'))''$&/')/!

( )α ( )α ( ) ( ) α

+ + =

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0*"1 ,1023444. +5,016 7**"1

6 ) $ $ π&

( )

ππ

− < <= < <

'

π π ππ

∞ + − −

8&$ π=

" 6 .+ + + +

'

π

9

π

:

π

π

9 ( !$ $ /

+ ;

+

6

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Page 2: GATE ECE 1993

. ( ) " = − + +

! ! ! #$%,'#$*333333333+##$1

2$$)!%#%#)%

$1!.'/1!$/$2%#0&!

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=

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( ) &$ $ &

+ ? !$@1<<<<<<<<<

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= − + + <<<<<<<<<

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+ <<<<<<<<<

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Page 3: GATE ECE 1993

: 7 = + + # !

$

<<<<<<<<<

D + = E &

( ) ( ) λ+ = 2

η <<<<<<<<<

F$ $ ( ) & !

( ) ( ) ( ) "

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*!<<<<<<<<<

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)%#$#2$$) !%

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π π "π 'π 6π 9π .π :π

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Page 4: GATE ECE 1993

" $ ! $& !

H!!

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! &

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$ @

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( ) ( )' 9 6 6 π − − = × − ×

( & & $

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Page 5: GATE ECE 1993

"D &$ !&

, 8 B 0

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.

.

6

K- & $63+ !$"3+L

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+

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Page 6: GATE ECE 1993

"" , +- $

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&

( )

µ ω ω ( )

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( )

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Page 7: GATE ECE 1993

7

8! 8! 8!9#$%$/$'''10

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Page 8: GATE ECE 1993

'' (

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Page 9: GATE ECE 1993

': ) ! $T

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Page 10: GATE ECE 1993

6 6 6: !&H7 $ &

6 2 !! 6,

(! ! H$ $/ 5

$

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Page 11: GATE ECE 1993

6' 2 $ !9Ω.6Ω'Ω$& $ 66( J) ! $

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Page 12: GATE ECE 1993

)2$$%0''2$$1

$ 9 9 ,! U 9

6 &H, !!&$$

2$ ,8B0 ! & 2 &! $ B,P2,F#3H! & !) V)HW&$ H H &!! ! H

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Page 13: GATE ECE 1993

9' 2 ( ) ( )( ) ( ) ( )

'

#

τ

→∞ = = + +

&

'

@ R,R 6R,R

96 2 " " ' + + + = $

$

,X "R,R' R,R 6 R , R

99 $ 0 P $ E

) ! $

! $ !

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!!

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Page 14: GATE ECE 1993

9 2 $ $$

$ $

$

$

$

9 % $ $ 9 6)))22$

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9" 8 $ $ A-*+5 !$,8

! !+ ! !+ ( ) ( ) ! !+ + ( ) ( ) ! !+ +

9' B ! 2 J 2 $! ν

96 ,$!& 3O@ '$$ ! N ? $ $ ) $ $ $ $$ $ &$$ $<<<<<<

π π

J π π

J

π π

J

π π

J

J

J*

J

J

ππ

6J

J16J

"

ν = Ω

J

) )

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Page 15: GATE ECE 1993

99 2 $ !

$

H$

9. 2 $!

H$ !03, $

H$ !$$

H$ !$ ! !

!$$ $

9: ! $$ && ,<<<<<

9D ( !

( ) ( )6 π π= +

5 $ !

& -

9 ,$& ! & '66 ?O@ $ ?O@ & <<<<<<?O@

9 #H ! !$ $&H $

( ) &

( )

"

9

"

< <= ≤ <

9 #$ !$ !"(((("((O@ $ P,3 ! <<<<<<<

",

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,

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Page 16: GATE ECE 1993

9" ,& !$ &

7O@ 9σ = S1S

σσ

= &

D

"9# σ

π− = ×

9' B $ 6 F & IE6 J#(+ $ & !

I ∞ IE

96 ,$! & $ !O

$ %′=P & $&

!!

′= −

$ $= −

% %= −

′ =

. , $ & !$ B ! ,8

: ) T $ $ ! H !

5

O&

%

@

8

′′

Ω Ω

)

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Page 17: GATE ECE 1993

D B ! F2 &

( ) ( ) ( ) " $$ F2&!

( ) ( ) ( ) " $ $ $ω ω ω $) 7

( )

( )

( ) ( )

( )

"

$

&

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πωω

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=

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2 !$B1?(!? 6)B U $ B 0 H !

$ ( ) $$ !

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JBBJ

I

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Page 18: GATE ECE 1993

" ) $ ! $

$

' #,8B0 + G: $

$8 ( ) ! " + !" !+ "+= + +

6 , H,8B $A, $A!,82$A , 8 BB A ,$ $ ,,181B1

0!

7

0!0$ $

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2 &

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J

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Page 19: GATE ECE 1993

. 2 !

( ) 66

η −= & $ ! ! $ ! $ &

( ) $η= F2&*$

O F2&

: * $ ( )" ν

( ) ( ) ( ) ' ν π π= + , ( ) ( ) ( ) ν ν ν= +

8P)&!$ :O@ O@

D , !$ 8O@$&$

$ ( )% $ ( )

"

!= $

$$$!, ( )"

!$&

$ &$ $ ( )

B & ! ! $ 6 λ=

0 ! !B$ &DMH$$

3 !$ ! ) $$ &J !

3! ε ε>

3! ε ε<

3$

3$

J J J"

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8P)

∠ ° ∠ °φ φ

16λ

5

5

0 0

5 5

O1

O1

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