Post on 27-May-2015
description
|CS-GATE-2012 Paper| www.gateforum.com
Correspondence Course All India Test Series Classroom Coaching eTutor-Online Classes © All rights reserved by GATE Forum Educational Services Pvt. Ltd. No part of this booklet may be reproduced or utilized in any form without the written permission.
!!
"!!
#!! $%!!"!# $&! $'!"!# $"!#
% ($
)( '*+, - .!!/! 0
" #
$%!" $&!"# $'!# $!"#% ($'
)( ! !=
12!121"2
111"11
111"1
1"111
" 3#! 4 ''
$%# $&5 $' $&#
% ($'
)( *'!6 4#&"
78 4
# 9
$%1:;3 $&3 $';'3 $<3
% ($'
)( ) 1:;3!9 ;'3
9
= '4> 97=
;*$2 >
$%77= $&7
$'7= $7=75=
|CS-GATE-2012 Paper| www.gateforum.com
Correspondence Course All India Test Series Classroom Coaching eTutor-Online Classes © All rights reserved by GATE Forum Educational Services Pvt. Ltd. No part of this booklet may be reproduced or utilized in any form without the written permission.
% ($'
)( ;
( ) ( )( ) ( ) ( )( ) ( ) ( ) ( )
* 3 ?
* 3 ? 3 ? 7=
* 3 ? 3 ? 3 ? 7= 7=
= ≤
− = ≤ − = = − =
+ = ≤ + = = − + = + = + =
@ ,
$%
$&
$'46
$
% ($'
)( ,%,A%6
5 ;,B:#
$%@# $&C $'D $D
% ($
)( *# # # C =@× = =
B C
1,B: C C D× =
C $%$9!
6
$% ( ) ( )( )% = Ω $& ( ) ( )( )% = Θ
$' ( ) ( )( )% B = $ ( ) ( )( )% =
% ($'
)( &$!%$$!
39&$2B$%$!%$2
B$$
E -7 =-
! 9-
F
$%" $&# $'= $@
% ($
)( G )>*2!9 !
*25B5!@
7 ;
$% ( ) ( ); ; = − + $& ( ) ( ); ; = − +
$' ( ) ( ); ; A = + $ ( ) ( ); ; = − +
|CS-GATE-2012 Paper| www.gateforum.com
Correspondence Course All India Test Series Classroom Coaching eTutor-Online Classes © All rights reserved by GATE Forum Educational Services Pvt. Ltd. No part of this booklet may be reproduced or utilized in any form without the written permission. !
% ($
)( %!&'!%'
&
; F9
H%&; %
%'
" H&'9 1!;$2;$ >
1IF9
3(%1IF9J%GK-
-,B<3&L
I(%1IF9J%GK-9-,B<3&L
,(%-,B<3&L1))';
1(K-,B<3&L
1))';
$%3, $&31 $'I, $I1
% ($%
)( J%GK- -,B<3&L!J%GK-
9 !
94.1!3I
*-,B<3&L
1))';-,B<3&L
1!,1
- ), !
"
$%%9
$&%9
$' !
$ !9
K<
% ($'
)( !
" 9K*%
% ./ Σ = ε 9
$%∅ $& ε $'0 $ ! ε
% ($&
ε
ε
|CS-GATE-2012 Paper| www.gateforum.com
Correspondence Course All India Test Series Classroom Coaching eTutor-Online Classes © All rights reserved by GATE Forum Educational Services Pvt. Ltd. No part of this booklet may be reproduced or utilized in any form without the written permission. #
)( 9K*%>!.M/
#
N1O
$% ( ) ( )( ) ∃ $& ( ) ( )( ) ∀ →
$' ( ) ( )( ) ∃ ∧ $ ( ) ( )( ) ∃ →
% ($'
)( B%(;
B&(%
B'(;
B(;
= % 222>2 ;
2 ;%E
$%7# 7# $&7# 7# −
$'# # − $= = −
% ($
)( = = " 7% % %
7 "
= = −
# 7 =
= = #
# 7 C C% % %
7 # C C
# # = =% % %
# # = =
= = −
= = − −
( )( )( ) ( )
( )( )
E
= =%
= =
= =7
= =
= = 7
=
=
∴ = −
− λ=
− + λ
− − λ − =
λ = ×
λ = ±
@ ;$3<94
$%1 $& $': $*
% ($'
)( ;3<49!K 49!;9%9
!!9
|CS-GATE-2012 Paper| www.gateforum.com
Correspondence Course All India Test Series Classroom Coaching eTutor-Online Classes © All rights reserved by GATE Forum Educational Services Pvt. Ltd. No part of this booklet may be reproduced or utilized in any form without the written permission.
5 '$2$ A#!5 A#∈ π π ;
$
$%B! Aπ $&B!" Aπ
$'; ! Aπ " Aπ $; ! A#" Aπ π
% ($&
)( ;6 ( ) = π
1π!
P
π"
; ( ) =
5
!# #
π π ∴
!π= "
C %
4$8
4$8
4$8
;$%
$%" $&# $'5 $C
% ($'
)( 4 !
1!
E ;7=)))
$%777Q7777
$&777Q777
$'77Q7777
$
% ($&
)( ( ) ( )
7 7= 7 −= ×
1!2 777 777
|CS-GATE-2012 Paper| www.gateforum.com
Correspondence Course All India Test Series Classroom Coaching eTutor-Online Classes © All rights reserved by GATE Forum Educational Services Pvt. Ltd. No part of this booklet may be reproduced or utilized in any form without the written permission. &
7 ;
' ()'*+
7 7 7
7 7
7
&
$%? $&?>L $'? L⊕ $ L
% ($%
)( ( )?LR?L ? LRL ? ?+ = + = =
; 9
$% ( )Θ $& ( )Θ $' ( )Θ $ ( )Θ
% ($'
)( ; &1;S+1!2
!$ 2>$2>2T$
%3 K3!≠
$%K3 2K3 $&K3 3∩ = ∅
$'K3 2K3 $32K3
% ($&
)( 3 U2K3! K3 '
9 3K3'V
" '
''2S%+8
$'.
S%+($N'%WO8
S&+(
S'+($N'&O8
S+(
S)+(
($NK'O8/
|CS-GATE-2012 Paper| www.gateforum.com
Correspondence Course All India Test Series Classroom Coaching eTutor-Online Classes © All rights reserved by GATE Forum Educational Services Pvt. Ltd. No part of this booklet may be reproduced or utilized in any form without the written permission. ,
$%K
$&'%
$''%
'&K
$3
% ($'
)( 1 S4+ ! F
N%O
# -
$%)9"K*&'K*
$&% , "K* 9 , 9 9
949,
$')9&'K*"K*
$K&'K*"K*
% ($'
)( B%&'K*"K*
B&K*
B'!&'K*"K*!9&'K*
"K*
= '
( 4 9
;4 9
"(.;
( 4 9
"(.;4 9
$%&
$&
$'
$&
% ($&
|CS-GATE-2012 Paper| www.gateforum.com
Correspondence Course All India Test Series Classroom Coaching eTutor-Online Classes © All rights reserved by GATE Forum Educational Services Pvt. Ltd. No part of this booklet may be reproduced or utilized in any form without the written permission. /
)(
( )
( , X' X,GX''
YYYYYYX,
YYYYYY
( , ' X,X'X, X,
YYYYYYX,'YYYYYY
→ ≈
→ ≈
( )
& 01
@ '.7!/ !
6*!777777 !
7777 % " %9*%
;*%
$% $&
22 2 2 3
22 7
2
2 7
7
22 2 2 3
22 7
2
2 7
7
ε
7
7
7
77
7
77
7
F 7!
|CS-GATE-2012 Paper| www.gateforum.com
Correspondence Course All India Test Series Classroom Coaching eTutor-Online Classes © All rights reserved by GATE Forum Educational Services Pvt. Ltd. No part of this booklet may be reproduced or utilized in any form without the written permission. 4
$' $
% ($
)( ;*%
5 ;
; 4$
9
$
.$22K< 8
( ) K<→ ==
( ) K< 78→ ==
& 8 AA&
.2$→ 8
( ) ( ) K< 8→ == +
. $ 8= →
& 8 AA&
/
/
/
22 2 2 3
22 7
2
2 7
7
22 2 2 3
22 7
2
2 7
7
ε
77
7
77
7
7 7
7
F 7!
77
7
|CS-GATE-2012 Paper| www.gateforum.com
Correspondence Course All India Test Series Classroom Coaching eTutor-Online Classes © All rights reserved by GATE Forum Educational Services Pvt. Ltd. No part of this booklet may be reproduced or utilized in any form without the written permission. 2
; &&
$% ( )( )&( + → $& ( )( )&( →
( )( )& ( !+ ( )( )& ( !+
$' ( )&( →
( )& ( ! $ ( )( )&( + →
( )& ( !
% ($%
)(
$
. $22P8→ A0 ! 0A
( ) K< A0 RR0A→ == →
( ) K< B8→ ==
A0 Z → RR! 0A
$> ( ) → 8
→ A0 !!
;
> ( ) → 0A
. → A0S+ 0A
2 ( ) 8→
→ A0**S+0A
( ) K<→ == ( ) 8+
→ A0
S+!2$ → >
2> ( ) → 2> 0A
.2 ( ) → 8
→ A0 !
S+ 0A
( )( ) !+ 8
A0→ ;S+2
>:$1!,
2>:$! 0A
/
/
|CS-GATE-2012 Paper| www.gateforum.com
Correspondence Course All India Test Series Classroom Coaching eTutor-Online Classes © All rights reserved by GATE Forum Educational Services Pvt. Ltd. No part of this booklet may be reproduced or utilized in any form without the written permission.
C ' ;'3+% : $%:
6 :11
C:11%
* 6
$%C:11 $&#:11 $'5:11 $:11
% ($'
)(
-!2C
;2!! 62$
;2! 62#$4
;2"! 62C
;2#! 62E!$9%
;2=!7:11! 627
J2$' 6A27 =
=
;2@!
;25!#
;2C!=$=
;2E!@
;27!5
∴ 7!5
6 5:11
"
#
=
@
5
C
E
7
" # = @ 5 C E 7 B
;
; →
6
↑
|CS-GATE-2012 Paper| www.gateforum.com
Correspondence Course All India Test Series Classroom Coaching eTutor-Online Classes © All rights reserved by GATE Forum Educational Services Pvt. Ltd. No part of this booklet may be reproduced or utilized in any form without the written permission.
E ' $1 6 7@
$ 4 $,,
4$!!"1,8,,8"
,4774%
7C%4
4:4 77746777
*9 1
$%77= $&77 $'"777 $"77"
% ($%
)(
;941!,, 4
394!"4
;994
4
;4@
;45
19 9777477=
"7 1, ( )%! & , ( )'!
&49',
9 9 !
$% ( ) ( )& ' Π − Π = ∅
$& ( ) ( )' & Π − Π = ∅
$' ( ) ( )& ' Π = Π
$ ( ) ( )& ' Π − Π ≠ ∅
% ($%
)( 1&49'! ( ) ( )& ' Π ⊆ Π ( ) ( )& ' Π − Π = ∅
" '
!!"!!#!!"!!#!
999
96" 99,<!**B
B3;:% 9;
$%B3;:%[,<[**B $&B3;:%[**B[,<
$'B3;:%2,< $B3;:%2**B
1 ,
,
"
|CS-GATE-2012 Paper| www.gateforum.com
Correspondence Course All India Test Series Classroom Coaching eTutor-Online Classes © All rights reserved by GATE Forum Educational Services Pvt. Ltd. No part of this booklet may be reproduced or utilized in any form without the written permission. !
% ($&
)( **B
# # #
" " "
→ $@
B
# #
" "
→ $=
,<
# # #
" " "
" " # #
→ $E
B[**B[,<
" % 9 "77 -&9 4 C 4
!494;6
4 4 C &9 6 4 4 C
&9;69
$%"D&9
$&"=D&9
$'C7D&9
$64
% ($&
)( )462C&9
442C&9
∴ )42C
@C
=
16C4(CC
164(@C
1694(@@C
1694(@@C
1!6(
C C @ C @ @ C 7# 7#C "5@C "=C#7 &9 "=4&9= × + × + × × = + + = =
"" ' ;
1 ; 9
V4+ % ! 9 !
9 9
|CS-GATE-2012 Paper| www.gateforum.com
Correspondence Course All India Test Series Classroom Coaching eTutor-Online Classes © All rights reserved by GATE Forum Educational Services Pvt. Ltd. No part of this booklet may be reproduced or utilized in any form without the written permission. #
$%1; $&1&; $'1%'; $1%');
% ($
)( \]S1+
9\1]27!\%]2 ∞ ! & ! ;= ∞ − − − − = ∞ %3\] S1+S+
3\]2 9S+
→ I
→
→ 19!I2 1!%!&!'!!)!*!-! ; ! = φ
,I9
.
2I \]
I I = −
"V
#2<./
/
\1]27!\%]2 ∞ !\&]2 ∞ !QQQ!\;]2 ∞
"55$
1(21
1( I %!&!'!!)!*!-! ;=
1"(V
1 7! % #! & "! ' ! 5! ) ! ;= = = = ∞ = = ∞ − −− = ∞
3 % 1! 3 & 1!3 ' !3 1!3 ) !3 ; = = = − = = − − −− = −
1#( 1=
& "
1
#
%
' )
#
"
= =
*
"
-
;
"
5
" #
|CS-GATE-2012 Paper| www.gateforum.com
Correspondence Course All India Test Series Classroom Coaching eTutor-Online Classes © All rights reserved by GATE Forum Educational Services Pvt. Ltd. No part of this booklet may be reproduced or utilized in any form without the written permission.
"55$.
1 ( &
1 (I %!'!!)!*!-! ;
" ( V
1 7! % #! & "! ' ! 5! ) ! ;
3 % 1! 3 & 1!3 ' !3 1!3 ) !3 ;
1 # ( 1!&
==
= = = = ∞ = = ∞ − −− = ∞
= = = − = = − − −− = −
=
"55$!.
1 ( %
1 (I '!!)!*!-! ;
" ( V
1 7! % #! & "! ' =! 5! ) ! ;
3 % 1! 3 & 1!3 ' %!3 1!3 ) !3 ;
1 # ( 1!&!%
==
= = = = = = ∞ − −− = ∞
= = = = = − − −− = −
=
"55$#.
1 ( '
1 (I !)!*!-! ;
" ( V
1 7! % #! & "! ' =! 5! ) @! ! ;
3 % 1! 3 & 1!3 ' %!3 1!3 ) '! !3 ;
1# ( 1!&!%!'
==
= = = = = = − − − = ∞
= = = = = − − − = −
=
"55$.
1 ( )
1 (I !*!-! ;
" ( V
1 7! % #! & "! ' =! 5! ) @! * ! - C! ; 7
3 % 1! 3 & 1!3 ' %!3 1!3 ) '!3 * !3 - )
==
= = = = = = = ∞ = =
= = = = = = − =
!3 ; )
1 # ( 1!&!%!'!)
=
=
%=! 3\;]2)!3\)]2'!3\']2%!3\%]21!
11;1%');
"# %!! ;
$% ( )B $& ( )B
$' ( )B + $ ( )B
% ($&
)( ;
!
|CS-GATE-2012 Paper| www.gateforum.com
Correspondence Course All India Test Series Classroom Coaching eTutor-Online Classes © All rights reserved by GATE Forum Educational Services Pvt. Ltd. No part of this booklet may be reproduced or utilized in any form without the written permission. &
F9 !1
( )B
"= -@-!9#-F
$%= $&"7 $'E7 $"@7
% ($'
)( #@@# 9 9
#9$# U!#
9@#"U2E7
"@ J 9$V ( ) ≥
$% $& $' $$
% ($'
)( ;! 9
9!
"5 ' ! 4
38
G
3%8
GQ
'%8
)%
3%8
G
3%8
G
'%8
)%
'%8
)%
'%8
)
'(:→ % → %→ %→ %
; 49
|CS-GATE-2012 Paper| www.gateforum.com
Correspondence Course All India Test Series Classroom Coaching eTutor-Online Classes © All rights reserved by GATE Forum Educational Services Pvt. Ltd. No part of this booklet may be reproduced or utilized in any form without the written permission. ,
% ($
)( % 4 4 9
4 ! 4 9 % !% %!%
"C 1F9$P
9 %
,)%,*,BK;9!9
9!,)%,2*,BK;27;FF9
$%$,)%,>22*,BK;$&$,)%,>22*,BK;
,)%,22*,BK; $*,BK;>22,)%,
$',)%,22*,BK; $$*,BK;>22,)%,
$,)%,>22*,BK;,)%,22*,BK;
:
%
%
%
% *,%:)
3BK;), %'')11
KD1
$% :
%
%
%
% *,%:)
3BK;), %'')11
KD1
$&
:
%
%
%
*,%:)
3BK;),
%'')11
KD1
$'
:
%
%
%
% *,%:)
3BK;), %'')11
KD
$
|CS-GATE-2012 Paper| www.gateforum.com
Correspondence Course All India Test Series Classroom Coaching eTutor-Online Classes © All rights reserved by GATE Forum Educational Services Pvt. Ltd. No part of this booklet may be reproduced or utilized in any form without the written permission. /
% ($%
)(
; (,)%,2*,BK;
9 I9,)%,2*,BK;279
; ($*,BK;>2,)%,
9 I9,)%,2*,BK;272"!
,)%,2*,BK;27!
!
I!
; ($,)%,>2*,BK;
9 I9,)%,2*,BK;272!
,)%,2*,BK;27!
!F
9
; ($*,BK;>2,)%,
9 I9,)%,2*,BK;272!
,)%,2*,BK;27!
!F
9
"E %13$13 4', 3
(#=#CC7A7;13
4B6%!FB6&!
%&
$%#=#C"@7A#=#CC7A
$&#=#CC7A#=#CC7A
$'#=#C"7A#=#C"7A
$#=#C"@7A##=#C"7A
% ($%
)( 1#7E@6%!
46%!
%#=#C"@7A
K 6&!S7+ 97#C
!"S7+ 4
6&&#=#CC7A
77 777777 − − − − − − − − − − − −
K 4
"
|CS-GATE-2012 Paper| www.gateforum.com
Correspondence Course All India Test Series Classroom Coaching eTutor-Online Classes © All rights reserved by GATE Forum Educational Services Pvt. Ltd. No part of this booklet may be reproduced or utilized in any form without the written permission. 4
#7 1 !!"!
9
@
$%7A $&=A $'A" $A@
% ($&
)(
" # = =,F 9
@ @ @ @ @ @ @ "@ ∴ = × + × + × + = =
# *Y%Y%$?!, :9
9 ?! 9 !
? 9
4 6 7 ; 7
4! 9 6
4
%F4$.
$*Y%Y%$!
28
/
,4$.
278
/
;
$%
$&4 6 49
$' 499
$ 49
Toss a coin
#
@
"
=
@
=
#
@
=
#
@
"
=
@
@
@
@
@
9
@
=
"9
@
=
#9
@
=
@
|CS-GATE-2012 Paper| www.gateforum.com
Correspondence Course All India Test Series Classroom Coaching eTutor-Online Classes © All rights reserved by GATE Forum Educational Services Pvt. Ltd. No part of this booklet may be reproduced or utilized in any form without the written permission. 2
% ($&
)(
%F4$.
$*Y%Y%$!
" 2
/
#,4$.
= 278
@/
3I 99
3!!"I3#!=!@27
I"9
6!9
# '"!3!33"
611 $7%
5$8
$8$51
3$
3 7 =
3 5
3" " #
;"*'*1,,$ '3<F
$%99.3!3!3".3!3!3"
$&99.3!3"!3.3!3"!3
$'99.3!3!3".3!3"!3
$99.3!3"!3.3!3!3"
% ($'
)( ;
%2
3 9F!9F (
%2"
3"9 F
%2#!
F3
3
↑
3
↑
"3
|CS-GATE-2012 Paper| www.gateforum.com
Correspondence Course All India Test Series Classroom Coaching eTutor-Online Classes © All rights reserved by GATE Forum Educational Services Pvt. Ltd. No part of this booklet may be reproduced or utilized in any form without the written permission.
F39FZ
4,9F!3
,9F(
1!3!3"!3
#" D %?+
$% $& +
$' + $ + +
% ($&
)(
RRRR+
## - -+
9F -;;+
--+9! +
$%;+2; +2
$&;+2; +[
$';+ ≠ ; +2 $K
77 7 7
77
7
7
? ?
?
?
77 7 7
77
7
7
? ?
?
?
"3
↑
3
|CS-GATE-2012 Paper| www.gateforum.com
Correspondence Course All India Test Series Classroom Coaching eTutor-Online Classes © All rights reserved by GATE Forum Educational Services Pvt. Ltd. No part of this booklet may be reproduced or utilized in any form without the written permission.
% ($
)(
--$&$'--
$%
#= ;6$2# "
P#\!E];YYYYYY
$% $&" $'= $5
% ($&
)(
( )
( ) ( )( )
# "
7
#
E 7 E 7 =
= 7 !=
= − − −
+< > ∴ = =
> ∴
( )
( )
= "
" 7 !"
"
7
+= =
> ∴
+= =
= ∴
#@
$% $&
$' $
2
G
2 3
G1
|CS-GATE-2012 Paper| www.gateforum.com
Correspondence Course All India Test Series Classroom Coaching eTutor-Online Classes © All rights reserved by GATE Forum Educational Services Pvt. Ltd. No part of this booklet may be reproduced or utilized in any form without the written permission. !
% ($&
)( ;$%"9
"9
;$' #
#
;$#9
#9
#5 ' 3I66(
($38
$I8
327I(2I>8
$I
($I8
$3
I273(23>8
$3
%9 ;;
$%6
$&6
$'6
$
% ($&
)( 1 ! 9;
;;;$3
; $3!1;
;! 1 ;$$I
;$ $I
!1
;;!1 9
9 ;
6
9 ;
6
|CS-GATE-2012 Paper| www.gateforum.com
Correspondence Course All India Test Series Classroom Coaching eTutor-Online Classes © All rights reserved by GATE Forum Educational Services Pvt. Ltd. No part of this booklet may be reproduced or utilized in any form without the written permission. #
88:515$1.#/;#4
' %!&'(
%
" 8 <
% @7
= 19 #
EE ,
&
" 8 <
= 19 #
= J #7
EC , 7
EE ,
'
" 6
7 7 7
EE 77 7
#C J 9 1IF9
%
9%
=%% $&%
9&
J),)&K2S%+
$%# $&" $'7 $
% ($&
)( % F9 9 ! S%+ ;! +% %
#E J 9
%% &∪ %
( ) % #7 ' =% & '> <∪
$%5 $&# $'= $E
|CS-GATE-2012 Paper| www.gateforum.com
Correspondence Course All India Test Series Classroom Coaching eTutor-Online Classes © All rights reserved by GATE Forum Educational Services Pvt. Ltd. No part of this booklet may be reproduced or utilized in any form without the written permission.
% ($%
)( ;
>" 8 < " 6
% @7 7 77 7
= 19 # 7 77 7
= J #7 7 77 7
EC , 7 7 77 7
EC , 7 EE 77 7
EE , 7 77 7
EE , EE 77 7
88:515$1.2;
' '(
!!278
*$8
$
.28 A00A
*$8
>28
*$
$NW__N!!8
/
G*$
. 28 A00A
28
>2>>8
$NW__N!!8
/
=7 9(
95$5?8
9<$15$5?8
$% $& $' $
|CS-GATE-2012 Paper| www.gateforum.com
Correspondence Course All India Test Series Classroom Coaching eTutor-Online Classes © All rights reserved by GATE Forum Educational Services Pvt. Ltd. No part of this booklet may be reproduced or utilized in any form without the written permission. &
"
#
#
#
@
@
#
@
7
#
#
7
% ($
@A.55$%%7$%1.1A4
585$%%7$%1.
199
<$15 7$%1. 1
J94
= 9
$% $& $' $
"
#
#
#
@
@
#
@
7
"
=
=
% ($'
)(
7
7
7
( ) #
( ) ( # → ( )
( !7 ( )
# @
( ) 4
( ) ( @! → #
@
7
7
7
7
( ) #
( ) ( # → ( )
( )→ ( ! 7 "
#
#
( ) ( # → ( )
|CS-GATE-2012 Paper| www.gateforum.com
Correspondence Course All India Test Series Classroom Coaching eTutor-Online Classes © All rights reserved by GATE Forum Educational Services Pvt. Ltd. No part of this booklet may be reproduced or utilized in any form without the written permission. ,
$ 1015$1.5 01
5585($ 1015$.;!
%=@D&9!# 9! 4
4 6 " &9 ; "
)99!!
!
= ;
$% $&# $'@ $5
% ($'
)( K42C
"
=
=@D& 4
"&9 = =
%#P 9!2"
=
=" ;69
$%@7D $&"@D $'#7D $"D
% ($%
)( ;%-@>#2794
J!692 × "7 2@7 D
5585($ 1015$.#;
* ! $
) ** ! ε 9 !` ! a
1 %& % &→ ε
% 1→
& 1→
`
1 1 → ε
% % 1→ % 1→
& & 1→ & 1→ !
"
;%- 1);
&9
= @
|CS-GATE-2012 Paper| www.gateforum.com
Correspondence Course All India Test Series Classroom Coaching eTutor-Online Classes © All rights reserved by GATE Forum Educational Services Pvt. Ltd. No part of this booklet may be reproduced or utilized in any form without the written permission. /
=# ;** %&
$%*,1;$%2.!! ε /2*,1;$&
*BB$%2.!/
*BB$&2.!!`/
$&*,1;$%2.!!`/
*,1;$&2.!! ε /
*BB$%2.!/
*BB$&2.`/
$'*,1;$%2.!! ε /2*,1;$&
*,1;$%2.!/
*BB$&2∅
$*,1;$%2.!!/2*,1;$&
*,1;$%2.!/
*BB$&2.!/
% ($%
)( ( ) ( ) ( ) ( ) ( )* % * 1 * %& * %& *= = ∪ ∪ ∈
( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
!!
* & * 1 !!
* % * & * & !
* & * 1 ` * % `! !
= ∪ ∪ ∈ = ∈= = ∈
= ∪ == = ∪ =
== ;)!)!)"
$%)( 1 %&! % 1→ → $&)( 1 %&! 1→ → ε
)( 1 %&! & 1→ → )( 1 %&! 1→ → ε
)"( & 1→ )"( 1 → ε
$')( 1 %&! 1→ → ε $)( % 1! 1→ → ε
)( 1 %&! 1→ → ε )( & 1! 1→ → ε
)"( & 1→ )"( & 1→
% ($'
)(
( ) ( )
( ) ( )
* 1 !! ! * 1 `!! `
1 1 %& 1 %& 1
1 1
* & !! ! * & `!! `
& & 1 & 1 & 1
= ∈ =
→ → →∈→∈ →∈
= ∈ =
→ → →
|CS-GATE-2012 Paper| www.gateforum.com
Correspondence Course All India Test Series Classroom Coaching eTutor-Online Classes © All rights reserved by GATE Forum Educational Services Pvt. Ltd. No part of this booklet may be reproduced or utilized in any form without the written permission. 4
&B&2
=@ ;9 =F ! F
;4,=7
; 9 6
$%= $&7 $'= $=
% ($%
)( 3 =7F =F= −
=7 7F8 7
F F
=7 7F 7 ! F =
= − <
∴ − = =
)4
=5 '
(
1C1 <$15DDDDDDDD015$5158A
$% $& $' $
% ($%
=C '9"(
$%;9 9 4 ;J
$&;9R&
$'; 9 4 *9 4 !
$;9R9
% ($
=E
$5$<5
$% $& $' $
% ($%
|CS-GATE-2012 Paper| www.gateforum.com
Correspondence Course All India Test Series Classroom Coaching eTutor-Online Classes © All rights reserved by GATE Forum Educational Services Pvt. Ltd. No part of this booklet may be reproduced or utilized in any form without the written permission. !2
@7 '
(
:1A$5 17% DDDDDDDDDD 5 8$11$ 1 $ $51 558A5 5
1%75(%$5
$% $&4 $' $
% ($&
&B& 01
@ ;9! 3 *
,F(
J 1 ! 9! ) 19 4;!
$%- 9 $&?
$'K4
$K 9
% ($'
)( -$&9
@ -F!%'-*Db3!
$%BG $&B $'3G $3
% ($%
@" 'B,,)';
3( %595
I( %595
,( 9
1( 9
$%3!I $&I!, $'3!, $,!1
% ($'
)( 3, 9
).!!"!#/49
% ! '- ! *D ! b3 ! BG
+ "+ #+ =+
"+ #+ =+ @+
|CS-GATE-2012 Paper| www.gateforum.com
Correspondence Course All India Test Series Classroom Coaching eTutor-Online Classes © All rights reserved by GATE Forum Educational Services Pvt. Ltd. No part of this booklet may be reproduced or utilized in any form without the written permission. !
@# %94 ?
L?@7_L#7_4%4
V F9 ; F9
B?R4!E@_BLR4!5_
; 9 9 4 !
!9L
$%7CC $&7""# $'7@@5 $757
% ($&
)( 9
19 @7_ #7_
, E@_ 5_
B 7=5@ 7CC
( ) 7CC3 7""#
7=5@ 7CC∴ = =
+
@= %9
; F 9 7= −
9;
$%C $&7 $' $#
% ($&
)( 9 7= −
9 7
97 9 7 7
7
9 7 7 7
= −
< ∴ − =
=∴ = − =