1. !" #$ %" &' (" #) *" #
! + *"
, + , = -'&
− −
. / / = = =
# #
/
# &0
= + +) = = #+ &0+ ) × $ = &
$ $ $
1 ≈ #
#
# /
#
!" 2 3 %" 3 (" 0 3 *" 3
! + !"
# −
, + − 4 (0 =
) − 4 = −4 = -< 3
#
# $ )
$ + + + + +
#5 $5 )5
!" %" (" *" + #
! + %"
, +
6
4 7# 8 0 #
=09
) 4
!" %"
(" *"
! + %"
−
, + µ= =4
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written permission.

2. +
1 8
11 8
111 8
1: 8
;!< ,
!" 1 %" 11 (" 111 *" 1:
! + ("
=
, + =: ( )
= ( )
#
#
=ψ ( )
& 1
/
!" %" (" *" #
#
! + !"
, + σ = σ = σ> = 4
τ = ( )
σ −σ
/ τ = #θ − τ #θ = 0
#
#
σ −σ
τ = + τ# =
#
0 ! 1 ) &+# / < * /
!" %" + # (" *" + #
, , , ,
! + *"
, + 6
; 1 ) &+# 9
' ?
@
!" < %" <
(" < *" <
! + ("
, + < / A /
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written permission.

3. - 1 /
!" %" ( (" ; *"
! + !"
A
/
/ 8
!" %" (" # *" )
! + *"
)π#,1 π#,1
, + ; / 4( = B; / 4(# =
<# <#
4(
C / =)
4(#
/ $'
!" 0' %" & & (" )'' *" )$'
! + !"
, + τ=(+σ φ
(= →
φ= $' = 0'
τ
/ τ = σ× 0' = 0'
σ
#
? /
1
11
(DEE,(
!" D 1 E., %" D 11 E.,
(" % 1 11 E., *" F 1 11 E.,
! + %"
C
, + / D=( /
<
=
G
D=( D=(# 3
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written permission.

4. $ ! 1 /
8 ) 3 #'3
!" (C %" (1 (" (< *" (< =<
! + %"
, + << 7 ) 3 41 7 #'3
(C
(1 !
! 1 (D*,/
! / 41 = 0 (<< − #
$ )= ) & < (#' ) (<
# 3 < << = ) 3 < 3
# $
/ (1
) ! 1
A A A ;
/
= ( H
!" 4
!
%" F " ! !
(" D 8
*" D 8
! + %"
! A A
=# AFI $ " 7#
A " !
4
0/ " A
!" 0 %" (" ) *" #
! + !"
& ! > )
> $ #
(= = #) "
$
!" # %" $ (" $ *" #
! + *"
, + =)+ +)= '
C
/ != + ' × $ = )# #
#
$ $
= $# + )# = )
4 / 4 = #× + =#
! )#
C E /E = = =#
4 #
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written permission.

5. $
0 ! I
; ' "
!" ) %" (" & *" #
! + !"
, + θ= $
I
:
%= B =
; = '
: θ
= ' = ' =)
%
' A
I" J
EK
=)
4 L
: =& I : = I
E
=#
: =@
!" $ %" ) (" *" &
! + %"
, + . 8 (! ) = (! )
E
+ (! ) L
π π π
× )# × & = × ## × : + × )# × :=) I
) ) )
-
!" ) %" & - (" *" #
! + ("
# ! / /
(D*"
;! "
!" = %" 4
(" *"
! + %"
# + ! /
+
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written permission.

6. (DEE,( +
!" % " " " "
%" % " " " "
(" % " "
*" " "
! + !"
## H
J
MK J
GK M G
!" # %" $$ (" *" 0
! + !"
, + F 7& "
F 7 #) "
&
/ = = #
#)
#$ J
MK J
GK ) ( & (/
( +
1 : J
MK J
GK
11 4 J
MK J
GK
(DEE,(
!" % 1 11 E., %" 1 ;!< , 11 E.,
(" % 1 11 ;!< , *" 1 E., 11 ;!< ,
! + ("
, + < / #
#) E
!" % A %" %
(" * *" ;
! + %"
# ?
@
1 ,
11 ,
!" % 1 11 %" 1 (" 11 *" F 1 11
! + !"
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written permission.

7. !"
#& ( ) 7 $
=
( + )− ( − )/
( ) ≈
#
#× −$
=
( ) -0' /
7 #
−) −) −) −)
!" $ × %" $ × (" ) × *" - ×
! + !"
#0 1 / 8 A
# $ &
!" %" (" *"
$# $# $# $#
! + *"
, + 4 = 8=
#
)
− &
4 = (48 = ( + ( =
# # # # $#
-
#'
'
!" −# )# & '& %" $ )' $ $
(" ) 0 & '& *" & '& -
! + %"
-− λ
, +
'−λ
= ( -− λ ) ( ' − λ ) = # λ = $ )'/ $ $
#- ; D4LE A / D4 = N+ N
H DE = N+ N
H
L
!" −
E
%" O
(" O 4
*" P D
! + !"
H A
, + ! / ! = D4 × DE = Q ( − ) D4 × DE =
! = −
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written permission.

8. $ 8 +# =
/ 7 7
−# −# −# −#
!" = %" =# (" = - *" = $& -
! + *"
, + R / +# = = =
(* + # ) = * = −# ∴ =( −#
−# #
F / = =( = ( = $& -
−#
∴ 8 = $& -
$ ! H
,1
) ) ) )
!" %" (" *"
& )) ,1 0&' ,1 $') ,1 - ,1
#
! + %"
L
$# A
1 H 4 =
" E I I# 4
!" ! 8 4 =
%" ! 8 4I#
I#
(" ! 8 4 > 8 =I
*" ! 8 4I
# > 8 =I E
! + ("
$$ ! A
&# AF
) AI
F
0
)
( −
0$
( )
!
#
FI "
!" #- %" ( #-
(" ) $# *" ( ) $#
© 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. #

9. ! + %"
, + ! = $0 × $ #
#
1
= >= = )&'0 × $ $
&
#
=* = . =
*< = #- 0 ×
- & &
F
'
−4 4 =
/ − =
!
− + * = −# - FI
> #
#
( )
$) ! 4LE 4L LE
S
LT H / 4 E > 4E 1
? LT L
? ? ?
!" %" (" *" U
# ) '
! + *"
$ > H /
A A #
# =4 ) =4
A (γ = # ) ! < =
1 ' +# 0/
" 4 8 #0 AF
!" %" (" *"
! + %"
$& H
# 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. $

10. ! + !"
Q!
, + ; / =
# ∂<
!
= A
α= =
# ∂<
/E = = # = A
#
E# = = A
= ( ) = (E ) = (
A
# ) =$
#
$0 ! / A H
# A4
+ 7# 0/ 7 ) 3/
7 ) / 7
A4 ! >
/ "
!" # %" - (" ) *" &
! + *"
$' A L/ #
A > 4/ L/
> E
L A
$ $
'AFI -' AFI
A "
4 L
4
#
L
E
!" ) )# %" '- (" & $' *" ' $)
! + %"
, + ! 4/ = B E/ = # &B L/ =#
/ θ = σ = σ − υ = G × C − Gω ×
= ' × # − -' × # = '-AFI #
$- )
/
# ' 0 I/ & I ) I
. /
$
I"
!" ) %" # (" *" '
© 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. %

11. ! + ("
, + !
: +: &+ )
: = # '
= = I
# #
/ L = :×! = × ×)× = $
I
#
) R 1 R 11 I
& ' )
( & ' )
( )
4 !
L , # 4 T
E 1 $ C T
? ) (
(DEE,( R 1 R 11
!" 4 − / L − #/E − $/ −) %" 4 − )/ L − $/E − #/ −
(" 4 − )/ L − #/E − $/ − *" 4 − / L − $/E − #/ −)
! + ("
) ? 8 #
! & 3
"
$
I
!" $ '# %" '' (" #$ ) *" #$ )
! + !"
)# ! C -#
−-# −00
!" I< %" I<
(" $ # I
< *" $ # I
<
! + ("
$
)$ ! 8 )# I
; !
# I
$ 8
#
"
!" # %" $ (" 0#' *" #
! + %"
$
L L )# I
, + D = != = −$
=$ #
! DE & × # × I
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written permission.

12. )) !
1E(+$0
+ 7 /
7) I / 7 '3
!" $ %" # #) (" $ && *" ) )
! + %"
$& ! ( + ) −
, + != × :*;
( + ') −
× &
= $& × ) × :*; :*; = # #)
) >
+ 7 # / 7 $
70 1 *" ' /
A " 8
!" # ) %" ) (" 0 *" 0-&
! + %"
)& ! '
I . H /
;
I / I "
A
!" # %" ' (" &0 *" 0
! + ("
)0 > 4 L
4 L $
> / 4 L
> / 4/ L/
4 L F
T / "
!" & - #
0 %" # $'0 (" # ) *" #' 0)
! + %"
)( )
D
, + ( $+ )=( − D
+ D
$D
4
D
D
= ##' 0 '
C = ##' 0 ' ( $D + D
)
= # $'0
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written permission.

13. ** + ' , #- $
A
+ 7 - I "/ 7# 0
7 '
$
$
)' "
!" $$ %" $' (" )$ *" )
! + %"
)-
!" $ %" # -$ (" # & *" $-
! + ("
** + ' , %-
! A " 8
? + = $
I /
%D* 7 I
</ %D* 7 I !
< A +
7 ' / 7 #) /
$
7) / 8 7# I
<
1 ! ,
A (
E
?
;I
=" A %D* A "
A
!" %" # (" ## *" #)
! + ("
; $
I × I< −$
, + = = = -$0 × A I = ## A I
= × # I ×)
<
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written permission.

14. AI "
!" #) %" (" ' *" $$
! + !"
, + = =# × × $
AI = × '
× #) A I
<
= #) A I
. / " ' , !"
0 * 1 . / " ' , -
A !
=# ; )
#
!" ) !
( > )
0
&
!
$#
# " < = 1 ) &+#
!" # %" ##$ $ (" #$& $ *" #0 &
$
! + ("
, + $& A + ) A ; *; = '0 !
$& × # × $# × ) × ) ×# ×( − $# )× = '0× ) ×)
= #$& $
$ A "
F / <
= 1 ) &+#
!" )0 # %" 0 0 (" 0 & ) *" 0&#
! + %"
0 * 1 . / " ' , -
A #
φ I
$
I D 7 "
A
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written permission.

15. $
) A I I "
!" %" ' # (" $0 $ *" ' #
! + %"
, + ×%×C = ! ×( )
#
× × $& ×C = × &
× C= ' # $
I
#
; / A
( $
I )
!" %" '# & (" - &
# *" #&
! + ("
, + E =E −φ= − =
4 A C =4 A C ×
$
I
= ' # × = ' # $
I
%
& (
+
+ ( 2 34 4 4 4 4
4 4 4 4 4 * ' // *(
32 1
3
!" %" A (" *"
! + %"
#
0 8/ 8
A E
>
!" %" (" *" #
! + !"
, + = 8#
=E
/ = 8
4=4 = 4 − (4 = 8− 8#
> 4/ = − 8= 8=
8
' (
+
0' 5/ 6 4 4 4 4 "
4 4 4 4 ' *( /
!" %" (" *" -
! + !"
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written permission.

16. - ( ) ! +
!" A C
%" %
(" A ; < A /
*"
! + *"
& ?
@
6
!" * %" * (" * *" *
! + !"
!"
& !
8 =# − #
!" ' %" (" # *" )
! + %"
, + =# − #
=#− #
#
#
< ∴ #− # =
=
∴ =# − =
&# ? /4 ; 1
E 8 + C
/ * / , A
/
? @
!" R
%" M
(" F A
*" F
! + ("
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written permission.

17. &$ R 8 / !* (R ;Q V4/
!" D: %" D? (" 4: *" 4?
! + !" )
# $
, +
! * ( R ; Q V 4 D:
# $ )
&) ? (DEE,( @
4+ ! 0 0
L+ ! 0 0
E+ *
+ *
!" 4/ L %" L/ E (" 4/ E *" E/
! + ("
, + 4" ( /
+#+$+)
< / #/ $/ )B / = =#
)
! 0
'/ -/ /
' + -+ +
= = =- =# + (0)
)
E" * /#/ )/ &/ '
#+)+&+'
= = = = #×#
)
& ! A M G
M & 3 G ) 3 A ! A
H 8 8
D M A /-&3 D G A /0#3
A / /
G
!" #'' %" $$) (" &&0 *" 0#
! + %"
, +
& 3 ) 3
E -&3 0#3
D 0& #''
#''
∴ 4( )= = $$)
0 + #''
&
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written permission.