( ) =
(
4
('
(' ('
'
*
%
$
*
."/$ !
."/0 !
$ !
0 !
Λ∇ 2 ∇ Λ
∇ /Λ
+
*
∇ 3Λ
"2 3 - ! ∂" 5 ∂ ∂-2
! 6" &$ − + ! % /
)
+ $" #$ " #- " $
'
(
! #" +!&! ,"$ &$$ #$
)
! " $" ! ) ×
(
× × ! "# "$%$& % % &$&"$ ! %"& ' ('
∩ ∩ = ∩ ∩ = ( ∩ ) =
4
*
'
! $" 7
'
6" "# . / "& %$ .%! +!" &-&$ 8
( + )
( − )
© All rights reserved by GATE Forum Educational Educational Services Pvt. Ltd. No part of this booklet may be reproduced or utilized in any form without without the written permission. Discuss this questions paper at www.gateforum.com. Page 1 of 18
! $$+. . &1 %$ .%! ! +!&! ! $$+. . !"$ # ) %$ .%!9
* ! #
:+ - . %#$ +! & 9 8 ) 3
8 ) 8
8 )
"&
! $$+. %" ' '' ' +!&! ! $$+. !! "& +"$ # & ! #
'
$-
$-
$-
#!
;!&! ! $$+. 5 "9
<%. +$$ $+- ! =>= / .
. . ! %. $+- & &-&$
! .%! # #! %1 ?")$1 $.! . " & %. $+-
$$ ! #
$-
$-
#!
;!&! ! $$+. &&9 ! &%$ &" $.". &" ! " + &" $.". &" $.". @ 1 &%$ @ #! &"$- "#$ ! @ &" ! " + &"$- "#$ $.". &"$"#$
(
;! % ! $$+. & %. A &! B 2 C#&DE &! %FG 2 C#&DE %FG 2 FG 2 H1E %CI ID %E J
© 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. Discuss this questions paper at www.gateforum.com. Page 2 of 18
& &
# #
"$ " % % # "$ " % % # 4
;! ! $$+. "& + %- . +! "$$ &!&9 > &! B &! B A +!$ B33 2 B33E J &% . &% . &. &! &!& #- &% . # &. &! &!& #- &%$
;!&! ! $$+. ," & 9 K.
K6
?-#
LM
;!&! ! $$+. $ $&) % &!9 K$ $$ "& # ,". + "& *"# ! "& ","$- ," $+ "# "& ! ! $ ," * ," "& $. "& K," $$ ," "& " # $$& # &"
'
! $ "& + + + ,"$
! "$" ," $ #$ + + = = ' + = + = ' ! $" %&$ '''
'
'
'''
:+ - .$ $" ! + + + + = ' +! &! ≥ 9 '
'
'
'
© All rights reserved by GATE Forum Educational Services Pvt. Ltd. No part of this booklet may be reproduced or utilized in any form without the written permission. Discuss this questions paper at www.gateforum.com. Page 3 of 18
! %& %%$ %& " ! %%$ . ! $- &" "& % &! ! ! %%$ . ! !+ "& $$ ! #
:+ - /# &!& & # % & '' #" $ &"& $) ". -&!" +! # # + % # 9 ''
''
4
@M
KN6
&$ "
& # " +) +$$ "$ ! & ! +) ! ! +! $$ # '
&# '1 1
% ! +) &$
:+ - ! & "&&"$$- "$"$- O!9
'''
6&%.. ! . LM
(
'
"$
;!&! ! $$+. "9 ! %&) . !"$ # ) & #- $) $- + L %&) " .. " - ) + " &!. O- "# !"$ # . * ! ML L &&$ %&$
@ < # $.". B
( B + B )
( + ( ))
< ⊂ '
∑
} % #- ! ."$ %
+!&! ! $$+. "9
⊂ <
< 2
< ∩ 2 ∅
" ! $$ ! $$+. "# 1 &%$ P 2 '' Q 2 ''''''' R 2 ' ;! "$ ''''
( )
9
'''''
'''''
''''
© All rights reserved by GATE Forum Educational Services Pvt. Ltd. No part of this booklet may be reproduced or utilized in any form without the written permission. Discuss this questions paper at www.gateforum.com. Page 4 of 18
! S" < /5
/
! "& ( ) = + " (
O$"
(
/
"
∫ +
#- <%1
((
= −
(
!
!
'(
*
! "$
4
(
' = 7
'
/5
& !+ "$ & ! ! %##$- ! & ! "# $ 7
/ 5
'
=
= = '' ! P ,"$ 7
+ ''
'' '
(
+ − −
''
− '' ('
'' − + ''
$- &) + " ( &) "# ( ! %& $" ! " ! "# ! &) + 4
4
(
∫
! " #- ! $ 2' 2 -2' -2 ! 7
© All rights reserved by GATE Forum Educational Services Pvt. Ltd. No part of this booklet may be reproduced or utilized in any form without the written permission. Discuss this questions paper at www.gateforum.com. Page 5 of 18
:+ - %$ & "$#$$ .%! %#$ +! & .9
'
'
+!&! ! $$+. "# ' ( 4 !"$ # %- #- &! . &! !.! 9 ( 4 '
' ( 4
( 4 '
' 4 (
+!&! ! $$+. "# 4 !"$ # %- #- &! . % $ ,"& 4
4
4
4
. +! +.! # ! $$+. %$ .%! . $ %. ! +! !"$ # 9
#
&
4
&
* %#$ . + $ %.
"# #- &! ! +!&! ! $$+. $ "%" ! "# 9 %
%
#!/
,"" %$ ". - ! +! ! + & &%$ #! ,"" ,"" % N N
N N
N N
N N
! " "# &$" ," &$ ! $$+. .%! 7
© All rights reserved by GATE Forum Educational Services Pvt. Ltd. No part of this booklet may be reproduced or utilized in any form without the written permission. Discuss this questions paper at www.gateforum.com. Page 6 of 18
(
! $$+. $ ( 4 !% ! +! % $ ! !% 4 (
4 (
( 4
( 4
+ " 0)1 .$ "& ! %! $.! ! $$+. ./ +.! & .%! +! ! "& '
' '
4
'
4
(
+! ! . &$" ! & +!&! ! ! %! & $T
4
( 4
( 4
( 4
( 4
;! ! $$+. & "& %$9 "# A $" 2 'E $" 2 'E "# U2 "# 8 E $"33 E " $"E J &" ! "# '1 "# &" ! "# 1 "# &%$ ! '1 1 "# * ! #
! $$+. %. "& &$$* A "& &$$* B$!$E $E "& &$$* B.!!$E JE
© 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. Discuss this questions paper at www.gateforum.com. Page 7 of 18
$" 2 'E <!. "& &$$* B% A %/V$!$ 22 *M@@ UU %/V.!!$ 22 *M@@ $"33E $ A %/V$!$ <!.%/V$!$E %/V.!!$ <!.%/V.!!$E J J ;! C$"D & ! $$ &$$ +! % ! /%- % ." ! "# ! ! "# $ ! ! "# $ ! ! !.! !
;! ! $$+. "& 9 <!. "& &$$* B% A "& &$$* B%%E % A %% 2 %/V$!$E %/V$!$ 2 %/V.!!$E %/V.!!$ 2 %%E %/V$!$ <!.%/V$!$E %/V.!!$ <!.%/V.!!$E J +% ! $ .! &!$ - +% $- ! $ .! &!$ +% $- ! $ .! &!$ $ * ! #
© 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. Discuss this questions paper at www.gateforum.com. Page 8 of 18
'
! $$+. %. 9 A W2
" E
2 8 8 8 3 B E E %CI D E J ;! ! "%" ! $ &$$ (
;! % ! $$+. & %. &! B A 2 &! B $$& ' B T&!E &%- CO''(DE J A &! B 2 CODE E %CID E J
O
O''(
<. "$
* ! #
;! % ! $$+. %. A 22 ' == 22 " E $
" 8 E
J A E 2 E %CID E J
(
'
© All rights reserved by GATE Forum Educational Services Pvt. Ltd. No part of this booklet may be reproduced or utilized in any form without the written permission. Discuss this questions paper at www.gateforum.com. Page 9 of 18
;! ! &%$- ! # "& 9 N
N$.
N
N
:+ - %#$ ! $$ " ! &&% ! $$+. $.".9 @ 2 A = #- . +! "# '1 #$ #- "# 1 #$ #- J (
! $$+. " ! &&% ! $$+. $.". ! +!&! !"$ # ) $9 @ 2 A+ = + ∈ A #JB + ! "# 1 "# #1 + ! "# #1 "# 1J # <
< #
# <
< #
U (
4
U
U
U
@ @ ! $.". . #- ! . < → '' →< = < ! -#$ ! +!&! ! $$+. "9 @ 2 ' 3 B
@ ."$ #" ' 3 B
@ & #" ."$
@ &
;!&! ! $$+. . $ !)- *$ S < → → → ∈ = # < → & → → # < → → → #
$$ ! #
$-
U $-
U $-
;! ! ,"$ S@ ! $$+. → ' ∈ '' A %$ . . + #&)+J A 2 ' ≥ 'J
© All rights reserved by GATE Forum Educational Services Pvt. Ltd. No part of this booklet may be reproduced or utilized in any form without the written permission. Discuss this questions paper at www.gateforum.com. Page 10 of 18
A ! ,"$ "# '1 1 J A 2 ' ≥ 'J
;!&! $.". ! $$+. L &&%9 6 2 A,' ,J A' J AR' PJ δ ,' R' ∅ J δ . #-
δ ,' R' 2 A,' PR'J δ,' PJ 2 A,' PPJ δ ,' ' P 2 A, PJ δ ,' ∈ R' 2 A,' ∈J δ , P 2 A, ∈J δ , ' R' 2 A,' R'J
('
@ 2 A'' +! ≥ J
@ 2 A'' +! ≥ 'J
@ 2 A ' +! ≥ J
@ 2 A ' +! ≥ 'J
M -$ / ! & % "#$ %$ & % ) #$&) T ? ) #$&) # ;! ! " %#$ $ T9 #-
(
#-
!"
X
'
X
X
X
4
&! +! 6 %!-&$ - /# "$ %& ! %. T ? +! ! %% T ! %. #$9 6
(
#-
;! ! . +. ! $$+. 0# %%
(
#-
6
6
(6
! - T " # ? ! %. T ? @SM @ S,"$- M $.! " %. %$& ! +! %. !"$ - ! ! $$+. ,"& %. & 9 ( 4 4
4
( 4
© 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. Discuss this questions paper at www.gateforum.com. Page 11 of 18
(
Y
$&) L"$
<
;! ! $" Y K 2 < 2 '9
(
Y 2 2
Y 2 ' 2
Y 2 2 '
&&" ) "# ! # ! " # ≥ ! ! &&" "%" N!+ "%" ' /%" * . /%" NK . /%" *N . % +! ! " "# . ," %$ ! &&"9
((
(
"$%$ +! P Q %" R &$ %" R 2 ' $& %" P R 2 $& %" Q ;! ! && ," $T ! /#$ $ "& 2 K + !" ". - $ :5; && K P Q R && K P ' Q R && ' P K Q R && P K Q R
(4
! $$+. &&" ! $ $" P L ' +! ! &&" $T ' K< ! $%/$% ;! $.& $" !"$ # %%$ . H'1 P9 L
P
L K<
2 ' L 2 '
K< 2 ' L 2 '
© All rights reserved by GATE Forum Educational Services Pvt. Ltd. No part of this booklet may be reproduced or utilized in any form without the written permission. Discuss this questions paper at www.gateforum.com. Page 12 of 18
2 L 2 ' (
* ! #
;! ! " " %"& % ( ) !+ ! $$+. ?".!/%9 + -T ''
+ ′
(
''
'
'
'
'
'
'
'
'
×
'
'
×
×
'
′ + ′
′ +
′ + ′
! $$+. $.& &&" +! %" "& "%" 7 ( ) ( )
( ) ( ) ! ( ) =
∑ (' )
( ) =
∑ ((4)
( ) =
∑ ( )
;! ( ) 9
4'
∑ ( )
) ∑ (
) ∑ (
∑ ((4 )
! $ $" = ' +! ! "%" L .. ! %" X ? $$ $% $% 9
Y
X
X
Y
X
Y
X
Y
X L
?
'''
?
'''
?
'
?
''
© 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. Discuss this questions paper at www.gateforum.com. Page 13 of 18
4
! $$+. %%$
( → ) ∧ ( → ) → ( ∨ ) →
4
"$.-
&&
! "$.- &&
&#$
- ! && $ $.&$ ! $$+. O- && #% .%! %$ ∀ FZ&& ∨ Z#% ∧ Z%$G ∀ FZ&& ∨ Z#% ∨ Z%$G ∀ FZ&& ∧ Z#% ∧ Z%$G ∀ FZ&& ∧ Z#% ∨ Z%$G
4
! $$+. #- $ < 2 A - = - 2 3 - ∈ A' [JJ ! -& &$" < A - = 2 - 3 - ∈ A' [JJ A - = - 2 3 - ∈ A' [JJ A - = - 2 \ - ∈ A' [JJ * ! #
4
;!&! ! $$+. " 9 T ! . ] ! ""$ . T ! FTE ]G %$$- $$- T ! . ] ! ""$ . T ! FTE ]G %$$- #" $$- M # #- 2 LM # ! &$$& $$ "# M ! FLME ⊆G % M & ! $ ! $$-
4
;! ! $$+. &"& $ $" 9 = − + +! ' =
4(
( − )
(+ − )
(+ − )
(− − )
! ' $. . #- !+ - +- & ! $. # #" . #- ! $ ' %%$ ! $ . ."!#$ 9 ''
' ('^)
''
'
© 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. Discuss this questions paper at www.gateforum.com. Page 14 of 18
44
! $$+. $ &! %. "%%$/% # <<_ <*6O LL_ N@NK
4
$ K K<_ <M< Q <*6O +! <_ ! %)- K &% + $ +!&! ! $$+. $ $ &% <_ <M< <_ Q <*6O <_ <M< <M< Q <*6O <_ <M< Q Q <*6O <_ <M< <*6O Q <M<
4
K $ +! #" ! $$+. S7 → → → → → ;! ! ,"$ "&#$ S ! # 7
'
→ → →
→ → →
→ → →
→ → →
$ *K +! #" *6O "," <KOO Q <O RL S - . RL ! 0" Q <O $ - . <KOO Q <O ! 0" RL ;! ! !.! $ #- # &%9 *S
*S
*S
*S
© All rights reserved by GATE Forum Educational Services Pvt. Ltd. No part of this booklet may be reproduced or utilized in any form without the written permission. Discuss this questions paper at www.gateforum.com. Page 15 of 18
#$ !"% " & #$ !" ' %
@ " - ! ,"" %$ ". &) () ;! ! " "# &) %
() S ! # % +! ! &%$ ,"" ,"" %
N N
N N
N N
* ! #
#$ !" * ' *%
<"%% + + -&!T + &&" %& L Y ". #%! < M + +%
+ %
L<
;7
L
P7
LM
Q7
L H1E
% H1E
L H#1E
% H#1E
><
7
>
7
>M
7
() ;! !"$ # + ; $&) !"$ &&"9
L
LM
L<
* %#$ $&) () ;! !"$ # + P Q ! # %#$9
L< L
L LM
L< LM
* ! #
© All rights reserved by GATE Forum Educational Services Pvt. Ltd. No part of this booklet may be reproduced or utilized in any form without the written permission. Discuss this questions paper at www.gateforum.com. Page 16 of 18
#$ !" ' %
R7
! $$+. #$- $.". %. !-%!&$ %& # . ! . " "& !+ & 6N>
_'
E ← '
6N>
_
E ←
6L _'
E &% +! '
XRP
E X"% P T $.
E ← 8
@K _
E @ !".! &- #- # !" ! $ $" ! &- $. 4([' &' K%&$- ! $" ! &" +$$ # (['&' 4 %&$-
X Q7
Q
E X"% Q &- $.
X6L R
E X"% R
_
E ← 3
X6L R
E X"% R
P7 () ! $ $" . ' ! $" . ! %. &" +$$ #
! "# ' #
! " $$ #
'
() ;!&! ! $$+. "& !"$ # $& P " ! ! $" . %. &" ! $ $"
*NL
@K _
_
KK _
#$ !" , ' ,%
&$ +) '''' "# $$+ ! $ ' # ! ! $$ ! "# ! . ( # "# "# () :+ - "# ! %#$ +! ! # . &!9
( '
' '
© 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. Discuss this questions paper at www.gateforum.com. Page 17 of 18
* ! #
() ;! ! . ! "#9
'' ''(
'' ''
'''' '''
* ! #
#$ !" & ' &%
! $$+. $ &! %. "1 # " $$ &" O$$$$ &" &" ;! ! %- )- "$ ! "# "%$ ! " O$$ #$ ' ' %&$- () ! "$ 0 % $$ +! ! " " "# "%$ ! %#$ 9
' '
' '
' '
' '
() ! "$ 0 % &" +! ! " " "# "%$ %#$9
' '
' '
' '
' '
© 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. Discuss this questions paper at www.gateforum.com. Page 18 of 18