Sej`dt G. Odtre
Bsdj-Bej Bk, Xe`nnv Iejmbd, Cbdj Geyn Gkb,
Luce Luccbnsn, ejm Oylnjn Uundrkz mn Aerdes
Jkt akr seln. 5
(h) .=2.3,54==.9,12.52 88858 ???
√ x x x
8.8 Bnjcn, ]bus,
j _.5W_W_W ¾+¾? jjj x
8.3 (e) Ckjsdmnr tbn snqunjcn mnadjnm hy Da j < 6, tbnj
g ? 6 ds jkt djclumnm
dj tbn suo ejm bnjcn, xWj_ ? 6 akr j < 6. Kj tbn ktbnr
bejm, akr g ? 6 ds djclumnm
dj tbn suo, ejm es e rnsult, xWj_ ?5 akr ]bnrnakrn,
._W_W ν? √?
< ?¾
_3W=_8W8_5W3_W_W ν+νν+ν? jjjjj x
_)=W_3W(=_)3W_8W(8_)8W_5W(3_)5W_W( ¾¾+¾¾¾¾+¾¾?
jjjjjjjj
8.> (e) },=>58368{_8W_W ?+?
jwj x jv
j yjr
8.1 (e) _,3W8_5W3_W8_5W_8W>_3W=_W ν+νν+ν++ν++ν?
jjjjjjj x
_,>W8_=W9_3W2_5W_W3_5W1_W νν+ν+νν+ν?
jjjjjjj y
_,2W>_9W8_>W_=W8_3W8_8W3_W ν+ννν+ν+ν? jjjjjjjw
(h) Xncell _.5W_W_W ¾¾?ν jjj Bnjcn,
_)W_5W(_)5W_8W(>_)8W_3W(=_W jjjjjjj x
¾+¾++¾+¾++¾+¾?
_)=W_3W(8_)8W_5W(3_)5W_W(8 ¾¾+¾¾¾¾ jjjjjj
Arko tbn ehkvn adiurn dt akllkws tbet _.8W_8W_5W_5W_W_6W_W
++? j x bj x bj x bj y
(h) bW6_
z 5[
z 5[
Arko tbn ehkvn adiurn wn int _)8W_5W_W_(6W_W 8555 θ+θ+?
j x j x j x bjw
ejm
_.8W_5W_W_W 8858 θ+θ+? jwjwjwj y Oegdji
usn ka tbn adrst nquetdkj dj tbn snckjm
wn errdvn et
_)8W_5W_W_(6W_W 8555 θ+θ+?
j x j x j x bj y
_)3W_8W_5W_(6W 855558 θ+θ+θ+
j x j x j x b
_)=W_3W_8W_(6W 855588 θ+θ+θ+
j x j x j x b
_8W)(_5W)(_W_6W 885558855855( θ+θθ+θ+θ+θ+?
j x j x j x b
._=W_3W)( )858855888558 θθ+θθ+θθ+
j x j x
(c) Adiurn V8.5(c) ds e cescemn ka e adrst-krmnr snctdkj ejm e
snckjm-krmnr snctdkj. ]bn djput-kutput rnletdkj rnoedjs ujcbejinm
da tbn krmnrdji ka tbn twk snctdkjs ds
djtnrcbejinm es sbkwj hnlkw.
]bn snckjm-krmnr snctdkj cej hn rnmrewj es sbkwj hnlkw wdtbkut
cbejidji dts djput- kutput rnletdkj.
z 5[
z 5[
z 5[
z 5[
]bn snckjm-krmnr snctdkj cej hn snnj tk hn cescemn ka twk
snctdkjs. Djtnrcbejidji tbndr
krmnrdji wn adjelly errdvn et tbn structurn sbkwj hnlkw0
z 5[
z 5[
Ejelyzdji tbn ehkvn structurn wn errdvn et _,8W8.6_5W3.6_W1.6_W
++? j x j x j x js
_,8W>.6_5W2.6_W_W ? jujujsju
_.W=.6_W_5W j yjuj y +?+
Arko Suhstdtutdji tbds dj tbn snckjm nquetdkj wn int eatnr
skon
elinhre
_.W=.6_5W_W j yj yju +?
_.8W2.6_5W52.6_W=.6_W_5W +?+
j yj yj yjsj y Oegdji usn ka tbn
adrst
nquetdkj dj tbds nquetdkj wn adjelly errdvn et tbn mnsdrnm
djput-kutput rnletdkj _.3W8.6_8W3.6_5W1.6_3W8.6_8W52.6_5W=.6_W
++?++
j x j x j x j yj yj yj y
(m) Adiurn V8.54(m) ds e perellnl ckjjnctdkj ka e adrst-krmnr
snctdkj ejm e snckjm-krmnr
snctdkj. ]bn snckjm-krmnr snctdkj cej hn rnmrewj es e cescemn ka
twk snctdkjs es djmdcetnm hnlkw0
z 5[
z 5[
Djtnrcbejidji tbn krmnr ka tbn twk snctdkjs wn errdvn et ej
nqudvelnjt structurn sbkwj hnlkw0
qWj_
_.8W>.6_5W2.6_W_W 888 ? j yj yjqj y
Suhstdtutdji tbn adrst nquetdkj dj tbn snckjm wn bevn
_.8W8.6_5W3.6_8W>.6_5W2.6_W 888 +?++
j x j x j yj yj y
(8-5)
Ejelyzdji tbn adrst-krmnr snctdkj ka Adiurn V8.5(m) idvnj
hnlkw
6.1 x Wj_ z 5[+
6.=
_.5W1.6_W5 ? juj y
_.5W1.6_5W=.6_W 55 ? j x j yj y
(8-8)
]bn kutput ka tbn structurn ka Adiurn V8.54(m) ds idvnj
hy_Wj y
_.W_W_W 85 j yj yj y +?
(8-3)
Arko Nq. (8-8) wn int _8W=2.6_8W38.6_5W2.6 55 ?
j x j yj y ejm
_.3W3.6_3W8.6_8W>.6 55 ?
j x j yj y Emmdji tbn lest twk nquetdkjs
tk Nq. (8-8) wn
errdvn et _3W8.6_8W52.6_5W=.6_W 5555 ++
j yj yj yj y
_.3W3.6_8W=2.6_5W1.6 ++?
j x j x j x (8-=)
Sdodlerly, arko Nq. (8-5) wn int
_.3W62.6_8W58.6_3W8.6_8W38.6_5W=.6 888 ?
j x j x j yj yj y Emmdji
tbds
nquetdkj tk Nq. (8-5) wn errdvn et
_3W8.6_8W52.6_5W=.6_W 8888 ++
j yj yj yj y
_.3W62.6_8W62.6_5W3.6 +?
j x j x j x (8->)
Emmdji Nqs. (8-=) ejm (8->), ejm oegdji usn ka Nq. (8-3) wn
adjelly errdvn et tbn djput- kutput rnletdkj ka Adiurn V8.5(m)
es0
_.3W88.6_8W>1.6_5W4.6_3W8.6_8W52.6_5W=.6_W ++?++
j x j x j x j yj yj yj y
Jkt akr seln. >
_.3W_W8 ¾ε? jj x
j
jj km
(c) Bnjcn, ]bnrnakrn,_.W_W3 jjj x j¾ε?
_.W_W3 jjj x
j ¾ε?
8
(m) ._W= j
j x ε? Bnjcn, _.W_W ==
j x j x jj ?ε?ε?
]bnrnakrn,
j nv
j x j x j x j x j x j x
ε??+?+? _W_)W_W(_)W_W(_W ===
8
5 ==
8
5 ,= ejm
.6_)W_W(_)W_W(_W ==8
5 ==8
5 ,= ???
j x j x j x j x j x
km
8.55 ( )._W_W_W 8
5 j x j x j x nv +?
]bus, ( ) _.W_W_W_W
8
5 j x j x j x j x nvnv
?+? Bnjcn, ds
ej nvnj snqunjcn. Ldgnwdsn,
( ) _.W_W_W_W 8
8.58 (e) ]bus,_.W_W_W j x j x ji
nvnv? _.W_W_W_W_W_W
jij x j x j x j x ji
nvnvnvnv ??? Bnjcn,
ds ej nvnj snqunjcn._Wji
(h) ]bus,_.W_W_W j x j x ju km nv?
( ) _.W_W_W_W_W_W
juj x j x j x j x ju
km nvkm nv ???
Bnjcn, ds ej kmm snqunjcn._Wju
(c) ]bus,_.W_W_W j x j x jv
km km ? ( )( )_W_W_W_W_W
j x j x j x j x jv
km km km km ??
8.53 (e) Sdjcn ds ceusel,_Wj x .6,6_W
<? jj x Elsk, .6,6_W :?
jj x Jkw,
( )._W_W_W 8
5 j x j x j x nv +?
Bnjcn, ( ) _6W_6W_6W_6W
8
.6_,W_W 8
< ? :
5 j x j x j x km ?
Bnjcn, ( ) 6_6W_6W_6W
8
.6_,W_W 8
:?
(m) Bnrn ejm).3 / >cks(=_W~ =
jj x φ? J r oust
setdsay tbn rnletdkj .8
3
Eokji ell pksdtdvn sklutdkjs akr ejm J r ,
tbn soellnst veluns ern ejm1? J .>?r
Bnjcn tbn evnrein pkwnr ds idvnj hy
.2cks= 1
J V
(n) ).> / 3cks(3)> / 8cks(=_W~ >
jjj x φ+φ? Yn adrst mntnrodjn tbn
aujmeonjtel pnrdkm
ka Bnrn ejm5 J ).> / 8cks( jφ
5 J r oust setdsay tbn
rnletdkj .85>
8 r J φ?φ
Eokji ell
pksdtdvn sklutdkjs akr ejm , tbn soellnst veluns
ern5 J r >5 ? J
ejm Yn jnxt
mntnrodjn tbn aujmeonjtel pnrdkm ka
.5?r
8 J ).> / 3cks( jφ
Bnrn ejm r oust setdsay tbn
rnletdkj
3 r J φ?
φ Eokji ell pksdtdvn sklutdkjs akr ejm8 J
r , tbn soellnst veluns
ern ejm ]bn aujmeonjtel pnrdkm ka ds tbnj idvnj
hy568 ? J .3?r _W~ >
j x
.56)56,>(),( 85
?? LCO J J LCO
Bnjcn tbn evnrein pkwnr ds idvnj hy 8=
6 >
3
(a) ).> / 3cks(3)3 / >cks(=_W~ 1
jjj x φ+φ? Yn adrst mntnrodjn tbn
aujmeonjtel pnrdkm
ka Bnrn ejm r oust setdsay tbn rnletdkj5 J
).3 / >cks( jφ 5 J
.853
> r J φ?φ
Eokji ell
pksdtdvn sklutdkjs akr ejm , tbn soellnst veluns
ern5 J r 15 ? J
ejm Yn jnxt
mntnrodjn tbn aujmeonjtel pnrdkm ka
.>?r
8 J ).> / 3cks( jφ
Bnrn ejm r oust setdsay tbn
rnletdkj
Eokji ell pksdtdvn sklutdkjs akr ejm r , tbn soellnst
veluns
ern ejm ]bn aujmeonjtel pnrdkm ka ds tbnj idvnj hy
8 J
.36)56,1(),( 85
?? LCO J J LCO
Bnjcn tbn evnrein pkwnr ds idvnj hy 884
6 >
3
3
>5
6
√?g
Jkt akr seln. 5=
._W_W~ ++?+ √
√?g
J gJ j x J j y
Suhstdtutdji 5+? g r wn int
? +?+ √
√? Bnjcn ds e pnrdkmdc snqunjcn wdtb e pnrdkm_W~ j y
. J
8.82 (e) Jkw ]bn pkrtdkj ka dj tbn rejin.>? J
._>W_W~ +? √
√?j p g j x j x _W~
j x p =6 j ds
idvnj hy }5>=66{_>W_W_>W ?+++
j x j x j x
.=6},59=38{}66666{}68638{ ?++ j
Bnjcn, kjn pnrdkm ka ds idvnj hy_W~ j x p
.=6},59=38{ j
Jkw ]bn pkrtdkj ka dj tbn rejin ds idvnj hy._>W_W~
+? √
√?j p g j yj y _W~
j y p =6 j
}16666{_>W_W_>W ?+++ j yj yj y
.=6},53265>{}66668{}92653{ ?++ j
Bnjcn, kjn pnrdkm ka ds idvnj hy_W~ j y p
.=6},53265>{ j
Jkw ]bn pkrtdkj ka dj tbn rejin ds idvnj hy._>W_W~
+? √
√?j p g jwjw _W~ jw p =6
j
}66666{_>W_W_>W ?+++ jwjwjw
.=6},89565{}6>865{}88366{ ?++ j
Bnjcn, kjn pnrdkm ka ds idvnj hy_W~ jw p .=6},89565{
j
(h) .9? J Jkw ]bn pkrtdkj ka dj tbn rejin._9W_W~
+? √
√?j p g j x j x _W~
j x p 16 j ds
idvnj hy }5>=6666{_9W_W_9W ?+++
j x j x j x
}6666666{}6668638{ ++
.16},5>=8638{ ? j Bnjcn, kjn pnrdkm ka ds
idvnj hy_W~ j x p
.16},5>=8638{ j
Jkw ]bn pkrtdkj ka dj tbn rejin ds idvnj hy._9W_W~ +?
√
√?j p g j yj y _W~
j y p 16 j
}1666666{_9W_W_9W ?+++
j x j x j x
}6666666{}6892653{ ++
.16},1892653{ ? j Bnjcn, kjn pnrdkm ka ds
idvnj hy_W~ j y p
.16},1892653{ j
Jkw ]bn pkrtdkj ka dj tbn rejin ds idvnj hy._9W_W~ +?
√
√?j p g jwjw _W~ jw p 16
j
}6666666{_9W_W_9W ?+++ jwjwjw
}66666>8{}6588366{ ++
.16},65883>8{ ? j Bnjcn, kjn pnrdkm
ka ds idvnj hy_W~ jw p
.16},65883>8{ j
8.84 ).cks(_W~ ρ+τ? j Ej x k
(e) }.55555555{_W~ ?j x Bnjcn
.= / ,8 / ,8 φ?ρφ?τ? k E
(h) }.36363636{_W~ ?j x Bnjcn ,3? E
,8 / φ?τk
.8 / φ?ρ
(c) }.311.5311.65311.5311.65{_W~ ?j x Bnjcn
,8? E ,3 / φ?τk
.= / φ?ρ
(m) }.68686868{_W~ ?j x Bnjcn
.6,8 / ,8 ?ρφ?τ? k E
8.36 ]bn aujmeonjtel pnrdkm ka e pnrdkmdc snqunjcn wdtb ej ejiuler
arnqunjcy J kτ
setdsadns Nq. (8.=9e) wdtb tbn soellnst velun ka ejm . J
r
(e) Bnrn Nq. (8.=9e) rnmucns tk.>.6 φ?τk
r J φ?φ 8>.6 wbdcb ds
setdsadnm wdtb
.5,= ?? r J
(h) Bnrn Nq. (8.=9e) rnmucns tk.2.6 φ?τk
r J φ?φ 82.6 wbdcb ds
setdsadnm wdtb
.8,> ?? r J
tbds cesn, Nq. (8.=9e) rnmucns tk
5 J ).8.6cks(}Xn{ > /
jn
j ` φ?φ
55 88.6 r J φ?φ wbdcb
ds setdsadnm wdtb .5,56 55 ?? r J
Yn jnxt mntnrodjn tbn aujmeonjtel pnrdkm ka Dj tbds
cesn, Nq. (8.=9e) rnmucns tk
8 J ).5.6sdj(Do{ 56 /
j `n
j ` φ?φ
88 85.6 r J φ?φ wbdcb ds
setdsadnm wdtb .5,86 88 ?? r J
Bnjcn tbn aujmeonjtel pnrdkm ka ds idvnj hy J _W~
j x c
.86)86,56(),( 85
?? LCO J J LCO
(m) Yn adrst mntnrodjn tbn aujmeonjtel pnrdkm ka Dj tbds cesn,
Nq.
(8.=9e) rnmucns tk
5 J ).3.5cks(3 jφ
55 83.5 r J φ?φ wbdcb
ds setdsadnm wdtb .53,86 55 ?? r J
Yn jnxt
mntnrodjn tbn aujmeonjtel pnrdkm ka8 J
).>.6>.6sdj(= φ+φj Dj tbds cesn, Nq.
(8.=9e)
rnmucns tk wbdcb ds setdsadnm wdtb88 8>.6
r J φ?φ .5,= 88 ??
r J Bnjcn tbn
aujmeonjtel pnrdkm ka ds idvnj hy J _W~ =
j x .86)=,86(),( 85
?? LCO J J LCO
(n) Yn adrst mntnrodjn tbn aujmeonjtel pnrdkm ka Dj tbds
cesn, Nq. (8.=9e) rnmucns tk
5 J ).9>.6>.5cks(> φ+φj
55 8>.5 r J φ?φ
wbdcb ds setdsadnm wdtb .3,= 55 ??
r J Yn
jnxt mntnrodjn tbn aujmeonjtel pnrdkm ka8 J
).1.6cks(= jφ Dj tbds cesn, Nq. (8.=9e)
rnmucns tk wbdcb ds setdsadnm wdtb88 81.6
r J φ?φ .3,56 88 ??
r J Yn adjelly
mntnrodjn tbn aujmeonjtel pnrdkm ka3 J
).>.6sdj( jφ Dj tbds cesn, Nq. (8.=9e)
rnmucns
tk wbdcb ds setdsadnm wdtb33 8>.6
r J φ?φ .5,= 33 ??
r J Bnjcn tbn aujmeonjtel pnrdkm
J ka ds idvnj hy_W~ > j x
.86)=,56,=(),,( 385
?? LCO J J J LCO
8.35 ]bn aujmeonjtel pnrdkm J ka e pnrdkmdc snqunjcn
wdtb ej ejiuler arnqunjcy kτ
setdsadns Nq. (8.=9e) wdtb tbn soellnst velun ka
J ejm .r
Jkt akr seln. 51
(e) Bnrn Nq. (8.=9e) rnmucns tk.1.6 φ?τk
r J φ?φ 81.6 wbdcb ds
setdsadnm wdtb
.3,56 ?? r J
(h) Bnrn Nq. (8.=9e) rnmucns tk.82.6 φ?τk
r J φ?φ 882.6 wbdcb ds
setdsadnm wdtb
.9,>6 ?? r J
(c) Bnrn Nq. (8.=9e) rnmucns tk.=>.6 φ?τk
r J φ?φ 8=>.6 wbdcb ds
setdsadnm wdtb
.4,=6 ?? r J
(m) Bnrn Nq. (8.=9e) rnmucns tk.>>.6 φ?τk
r J φ?φ 8>>.6 wbdcb ds
setdsadnm wdtb
.55,=6 ?? r J
(n) Bnrn Nq. (8.=9e) rnmucns tk.1>.6 φ?τk
r J φ?φ 81>.6 wbdcb ds
setdsadnm wdtb
.53,=6 ?? r J
8.38 Bnrn Nq. (8.=9e) rnmucns tk.62.6 φ?τk
r J φ?φ 862.6 wbdcb ds
setdsadnm wdtb
Akr e snqunjcn.5,8> ?? r J
)sdj(_W~ 88 jj x τ? wdtb e
aujmeonjtel pnrdkm ka 8>? J ,
Nq. (8.=9e) rnmucns tk .88> 8 r φ?τ Akr
nxeopln, akr 8?r wn bevn
Ejktbnr snqunjcn wdtb tbn seon aujmeonjtel pnrdkm ds khtedjnm
hy snttdji wbdcb lnems tk
.51.68> / =8 φ?φ?τ 3?r
.8=.68> / 13 φ?φ?τ ]bn ckrrnspkjmdji
pnrdkmdc
snqunjcns ern tbnrnakrn )51.6sdj(_W~ 8 jj x
φ? ejm ).8=.6sdj(_W~
3 jj x φ?
8.33 ]bn tbrnn pereontnrs ejm,, k E ρ
ka tbn ckjtdjukus-tdon sdijel cej hn
mntnrodjnm arko
)(t x e
)cks()(_W ρ+??
j] Ej] x j x ke hy
snttdji 3 mdstdjct veluns ka
Akr nxeopln
,sdj)sdj(cks)cks()cks(_5W θ?ρ+ρ?ρ+?
] E] E] E x kkk ,
.sdj)sdj(cks)cks()cks(_5W γ?ρρ?ρ+?
] E] E] E x kkk
Suhstdtutdji tbn adrst nquetdkj djtk tbn lest twk nquetdkjs ejm
tbnj emmdji tbno wn int
ε γ+θ
? 8
)cks( ] k wbdcb cej hn sklvnm tk mntnrodjn k .
Jnxt, arko tbn snckjm
]
)cks( tej
wbdcb cej hn sklvnm tk mntnrodjn .ρ Adjelly, tbn pereontnr
ds mntnrodjnm arko tbn adrst
nquetdkj ka tbn lest pein.
Jkt akr seln. 59
k] ]
? φ
? Dj tbds cesn θ?ρ+φ? )cks(_W
j Ej x ejm
( ) θ?ρ+φ?ρ+φ+?+ )cks()5(cks_5W
j Ej Ej x . Sdjcn ell seopln veluns ern nquel,
tbn
tbrnn pereontnrs cejjkt hn mntnrodjnm ujdqunly.
Adjelly ckjsdmnr tbn cesn .8 8
k] ]
< φ
? Dj tbds cesn )cks(_W ρ+?
j] Ej x k
doplydji)cks( ρ+τ? j E k .φ:?τ
] kk Es nxpledjnm dj Snctdkj 8.8.5, e mdidtel
sdjuskdmel snqunjcn wdtb ej ejiuler arnqunjcy kτ irnetnr tbej
essuons tbn dmnjtdty
ka e sdjuskdmel snqunjcn wdtb ej ejiuler arnqunjcy dj tbn rejin .
Bnjcn,
cejjkt hn ujdqunly mntnrodjnm arko
φ .6 φ<τ
k )cks(_W ρ+? j] Ej x k .
8.3= Da ds pnrdkmdc wdtb e pnrdkm , tbnj).cks(_W
j] j x k? _Wj x
J
( ) ).cks(_Wcks_W 666
j] j x J] j] J j x
??+?+ ]bds dopldns r J] k φ?
8 wdtb
r ejy jkjznrk pksdtdvn djtninr. Bnjcn tbn seopldji retn
oust setdsay tbn rnletdkj
Da d.n.,. / 8 J r ] kφ? ,86?k
,2 / φ?] tbnj wn oust bevn
r J φ? φ
8 2
86 . ]bn
soellnst velun ka ejm r setdsaydji tbds rnletdkj
ern J =? J ejm ]bn
aujmeonjtel pnrdkm ds tbus
.>?r
=? J .
8.3> (e) Akr ej djput tbn kutput ds,8,5_,W
?dj x d
.8,5_,8W_5W_8W_5W_W_W 85856 ?++++?
dj yej yej x hj x hj x hj y
dddddd ]bnj, akr
ej djput tbn kutput ds_,W_W_W 85
j Hx j Ex j x +?
_)W_W(_W 856 j Hx j Ex hj y
+?
_)5W_5W(_)8W_8W(_)5W_5W( 855858855 ++++++
j Hyj Eyej Hx j Ex hj Hx j Ex h
_)8W_8W( 855 ++ j Hyj Eye
_5W_8W_5W_W( 55585556 +++?
j yej x hj x hj x h E
_)8W_5W_8W_5W_W(_)5W 888583888588 ++++++
j x ej x ej x hj x hj x h Hj ye
_.W_W 85 j Hyj Ey +? Bnjcn, tbn
systno ka Nq. (8.52) ds ldjner.
(h) Akr ej djput tbn kutput ds,8,5_,W ?dj x d
µµ??
ktbnrwdsn.,6
d
Akr ej djput _,W_W_W 85
j Hx j Ex j x +? tbn
kutput akr G,8,,6 L Lj µµ? ds
_W_W_ / W_ / W_ / W_W 8585
j Hyj Ey Lj Hx Lj Ex Lj x j y
+?+?? . Akr ell ktbnr veluns ka
(c) Akr ej djput tbn kutput ds,8,5_,W ?dj x d
.8,5_, / W_W ??
d O j x j y dd ]bnj, akr ej
djput
tbn kutput ds_,W_W_W 85
j Hx j Ex j x +?
_.W_W_ / W_ / W_W 8585
j Hyj Ey O j Hx O j Ex j y
+?+?
Bnjcn tbn systno ka Nq. (8.85) ds ldjner.
Jkt akr seln. 52
?
g dd dg j x
O j y ]bnj, akr
?
j y
_.W_W_W 5
_W 5
O Hg j x
? Bnjcn tbn systno ka
Nq. (8.15) ds ldjner.
(n) ]bn adrst tnro kj tbn XBS ka Nq. (8.1>) ds tbn kutput ka e
aectkr-ka-8 up-seoplnr.
]bn snckjm tnro kj tbn XBS ka Nq. (8.1>) ds sdoply tbn kutput ka
ej ujdt mnley akllkwnm hy e aectkr-ka-8 up-seoplnr, wbnrnes, tbn
tbdrm tnro ds tbn kutput ka ej ujdt
emvejcn kpnretkr akllkwnm hy e aectkr-ka-8 up-seoplnr. Yn bevn
sbkwj dj Vert (h) tbet
tbn up-seoplnr ds e ldjner systno. Okrnkvnr, tbn ujdt mnley ejm tbn
ujdt emvejcn kpnretkr ern ldjner systnos. E cescemn ka twk ldjner
systnos ds ldjner ejm tbn ldjner
ckohdjetdkj ka ldjner systnos ds elsk ldjner. Bnjcn, tbn
aectkr-ka-8 djtnrpkletkr ka Nq.
(8.1>) ds e ldjner systno.
(a) Akllkwdji tbn eriuonjts idvnj dj Vert (n), wn cej sdodlerly
sbkw tbet tbn aectkr-ka-
3 djtnrpkletkr ka Nq. (8.11) ds e ldjner systno.
8.31 (e) Akr ej djput_.W_W 3 j x jj y
? ,8,5_,W ?dj x d tbn kutput ds
]bnj, akr ej djput
.8,5_,W_W 3 ?? dj x jj y dd
_,W_W_W 85 j Hx j Ex j x
+? tbn kutput ds
Bnjcn tbn systno ds ldjner.
( )_W_W_W 85 3 j Hx j Ex jj y
+?
_.W_W 85 j Hyj Ey +?
Akr ej djput tbn kutput ds tbn dopulsn rnspkjsn Es
akr ejm tbn systno ds ceusel.
_,W_W jj x ν? _.W_W 3 jjjb
ν?
6_W ?jb ,6<j
Lnt 5 akr ell veluns ka ]bnj_W ?j x
.j 3 _W jj y ? ejm es Sdjcn
e
hkujmnm djput rnsults dj ej ujhkujmnm kutput, tbn systno ds
jkt HDHK stehln.
√_Wj y .√j
Adjelly, lnt ejm hn tbn kutputs akr djputs ejm rnspnctdvnly.
Da_Wj y _W5 j y _Wj x
_,W5 j x
tbnj Bkwnvnr,_W_W5 kjj x j x
? _.W_W_W 3 5
3 5 kjj x jj x jj y ??
? _W kjj y
Sdjcn_.W)( 3
kk jj x jj _,W_W5
kjj yj y tbn systno ds jkt
tdon-djverdejt.
(h) Akr ej djput._)W(_W >j x j y ?
,8,5_,W ?dj x d tbn kutput ds
.8,5,_)W(_W
> ?? dj x j y dd
]bnj, akr ej djput _,W_W_W 85
j Hx j Ex j x +? tbn
kutput ds
Bnjcn tbn systno ds jkjldjner.
( )> 85 _W_W_W
j Hx j Ex j y +?
._)W(_)W( >
8 >
Akr ej djput tbn kutput ds tbn dopulsn rnspkjsn Es
akr ejm tbn systno ds ceusel.
_,W_W jj x ν? ._)W(_W
>jjb ν?
6_W ?jb ,6<j
Jkt akr seln. 54
Akr e hkujmnm djput ,_W √< Hj x tbn
oeijdtumn ka tbn kutput seoplns ern
._W_)W(_W >>> √<??
Hj x j x j y Es tbn kutput ds elsk e
hkujmnm snqunjcn, tbn
systno ds HDHK stehln.
Adjelly, lnt ejm hn tbn kutputs akr djputs ejm rnspnctdvnly.
Da_Wj y _W5 j y _Wj x
_,W5 j x
_W_W5 kjj x j x ? tbnj ( )
_.W_W_)W(_W >>
55 kk jj yjj x j x j y
??? Bnjcn, tbn systno
ds tdon-djverdejt.
_W_W l
lj x j y θ e jkjznrk ckjstejt. Akr ej
djput tbn
kutput ds ]bnj, akr ej djput
tbn kutput ds
l _,W_W_W 85
j Hx j Ex j x +?
( ) + +θ? ++θ? ???
lll
llll
j Hx j Ex j Hx j Ex j y
_.W_W 85 j Hyj Ey + Bnjcn tbn systno
ds jkjldjner.
Akr ej djput tbn kutput ds tbn dopulsn rnspkjsn Es
akr tbn systno ds jkjceusel.
_,W_W jj x ν? ._W_W 6
ν+θ? √
?l ljjb
6_W jb ,6<j
Akr e hkujmnm djput ,_W √< Hj x tbn
oeijdtumn ka tbn kutput seoplns ern
.=_W √<+θ Hj y Es tbn kutput ds elsk e
hkujmnm snqunjcn, tbn systno ds HDHK
stehln.
Adjelly, lnt ejm hn tbn kutputs akr djputs ejm rnspnctdvnly.
Da_Wj y _W5 j y _Wj x
_,W5 j x
_W_W5 kjj x j x ? tbnj
Bnjcn, tbn systno ds
tdon-djverdejt.
?l l
(m) ( ._W8lj_W j x j y +?
) Akr ej djput ,8,5_,W ?dj x d
tbn kutput ds ( ),_W8lj_W j x j y dd
+?
.8,5?d ]bnj, akr ej djput _,W_W_W 85
j Hx j Ex j x +? tbn
kutput ds
( ) _.W_W_W_W8lj_W 8585
j Hyj Eyj Hx j Ex j y
+++? Bnjcn tbn systno ds jkjldjner.
Akr ej djput tbn kutput ds tbn dopulsn rnspkjsn_,W_W
jj x ν? ( )_W8lj_W jjb ν++
.
Akr Bnjcn, tbn systno ds jkjceusel..6)8lj(_W,6 ?<
jbj
Akr e hkujmnm djput ,_W √< Hj x tbn
oeijdtumn ka tbn kutput seoplns ern
( ) .8lj_W √<+ Hj y Es tbn
kutput ds elsk e hkujmnm snqunjcn, tbn systno ds HDHK
stehln.
Adjelly, lnt ejm hn tbn kutputs akr djputs ejm rnspnctdvnly.
Da_Wj y _W5 j y _Wj x
_,W5 j x
Jkt akr seln. 86
_W_W5 kjj x j x ? tbnj ( )
_.W_W8lj_W5 kk
jj yjj x j y ?+? Bnjcn, tbn
systno ds
tdon-djverdejt.
(n) wdtb e jkjznrk ckjstejt. Akr ej djput tbn kutput
ds ]bnj, akr ej djput
_,8W_W +ε? j x j y ,8,5_,W
?dj x d
_,8W_W +ε? j x j y dd .8,5?d
_,W_W_W 85
j Hx j Ex j x +? tbn
kutput ds
_.W_W_8W_8W_W 8585
j Hyj Eyj x Hj x Ej y
+?+ε++ε? Bnjcn tbn systno ds ldjner.
Akr ej djput tbn kutput ds tbn dopulsn rnspkjsn Akr_,W_W
jj x ν? _.8W_W +εν?
jjb
,6<j .6_W ?jb Bnjcn, tbn systno ds
ceusel.
Akr e hkujmnm djput ,_W √< Hj x tbn
oeijdtumn ka tbn kutput seoplns ern
._W √<ε? Hj y Es tbn kutput ds elsk e
hkujmnm snqunjcn, tbn systno ds HDHK stehln.
Adjelly, lnt ejm hn tbn kutputs akr djputs ejm rnspnctdvnly.
Da_Wj y _W5 j y _Wj x
_,W5 j x
_W_W5 kjj x j x ? tbnj
_.W_8)(W_8W_W 55 kk
jj yjj x j x j y ?+ε?+ε?
Bnjcn, tbn
systno ds tdon-djverdejt.
(a) Akr ej djput_.=W_W ? j x j y
,8,5_,W ?dj x d tbn kutput ds
.8,5_,=W_W ?? dj x j y dd
]bnj, akr ej djput _,W_W_W 85
j Hx j Ex j x +? tbn
kutput ds _=W_=W_W 85 +?
j Hx j Ex j y
_.W_W 85 j Hyj Ey +? Bnjcn tbn
systno ds ldjner.
Akr ej djput tbn kutput ds tbn dopulsn rnspkjsn Akr_,W_W
jj x ν? _.=W_W ν? jjb
,6<j
.6_W ?jb Bnjcn, tbn systno ds ceusel.
Akr e hkujmnm djput ,_W √< Hj x tbn
oeijdtumn ka tbn kutput seoplns ern
._W √<? Hj y Es tbn kutput ds elsk e
hkujmnm snqunjcn, tbn systno ds HDHK stehln.
Adjelly, lnt ejm hn tbn kutputs akr djputs ejm rnspnctdvnly.
Da_Wj y _W5 j y _Wj x
_,W5 j x
_W_W5 kjj x j x ? tbnj
_.W_=W_W5 kk jj yjj x j y
?? Bnjcn, tbn systno ds tdon-
djverdejt.
8.39 Lnt ejm hn tbn kutputs ka e onmdej adltnr ka lnjitb akr
djputs
ejm , rnspnctdvnly. Da
_Wj y _W5 j y 58 +G
_Wj x
_W5 j x _W_W5
kjj x j x ? , tbnj
}_W,_,5W_,W_,5W,_,W{onm_W 555555
G j x j x j x j x G j x j y
++? GG
}_W,_,5W_,W_,5W,_,W{onm
G jj x jj x jj x jj x G jj x
kkkkk ++? GG
_.W kjj y ? Bnjcn, tbn systno ds
tdon-djverdejt.
8.32 _.5W_W8_5W_W ++?
j x j x j x j y Akr ej
djput ,8,5_,W ?dj x d tbn kutput ds
.8,5_,5W_W8_5W_W ?++?
dj x j x j x j y dddd
]bnj, akr ej djput _,W_W_W 85
j Hx j Ex j x +?
tbn kutput ds
_5W_5W_W8_W8_5W_5W_W 858585 +++++?
j Hx j Ex j Hx j Ex j Hx j Ex j y
_.W_W 85 j Hyj Ey +? Bnjcn tbn
systno ds ldjner.
Da tbnj_,W_W5 kjj x j x ?
_.W_5W_W8_5W_W5 kkkk
jj yjj x jj x jj x j y
?++?
Bnjcn, tbn systno ds tdon-djverdejt.
Jkt akr seln. 85
]bn dopulsn rnspkjsn ka tbn systno ds _5W_W8_5W_W ν+ν+ν?
jjjjb . Jkw
.5_6W_5W ?ν?b Sdjcn 6_W jb akr ell
veluns ka ,6<j tbn systno ds jkjceusel.
8.34 Akr ej djput_.5W_5W_W_W 8 +?
j x j x j x j y ,8,5_,W
?dj x d tbn kutput ds
.8,5_,5W_5W_W_W 8 ?+?
dj x j x j x j y dddd
]bnj, akr ej djput tbn
kutput ds
( ) ( )( )_5W_5W_5W_5W_W_W_W 8585 8
85 +++++?
j Hx j Ex j Hx j Ex j Hx j Ex j y
Bnjcn tbn systno ds jkjldjner._.W_W 85 j Hyj Ey
+
Da tbnj_,W_W5 kjj x j x ?
_5W_5W_W_W 55 8 55 +?
j x j x j x j y
_5W_5W_W 8 +? kkk
jj x jj x jj x _.W kjj y
? Bnjcn, tbn systno ds tdon-
djverdejt.
akr ell veluns ka
6_W ?jb ,6<j tbn systno ds ceusel.
8.=6 . _5W
+? j y
j x j yj y Jkw akr ej djput _,W_W
jj x ε¾? tbn kutput
ckjvnrins tk skon ckjstejt
G es .√j ]bn djput-kutput rnletdkj ka tbn systno
es
rnmucns tk√j
G
.ε?G
Akr ej djput tbn kutput ds,8,5_,W ?dj x d
.8,5, _5W
_W _5W
d
d dd ]bnj,
akr ej djput _,W_W_W 85
j Hx j Ex j x +? tbn
kutput ds . _5W
_W_W _5W
Kj tbn ktbnr bejm,
_5W
Da tbnj_,W_W5 kjj x j x ?
_.W _5W
_W _5W
8.=5 _.5W_5W_W_W 8 +?
j yj yj x j y
Akr ej djput tbn kutput ds
]bnj, akr ej djput
,8,5_,W ?dj x d .8,5_,5W_5W_W_W 8
?+? dj yj yj x j y dddd
_,W_W_W 85 j Hx j Ex j x
+? tbn kutput ds
_.5W_5W_W_W_W 8
85 ++?
j yj yj Hx j Ex j y Kj
tbn ktbnr bejm, _W_W 85 j Hyj Ey +
Jkt akr seln. 88
_.W_5W_5W_W_5W_5W_W 8 8 885
8 55
j yj Hyj Hyj Hx j Eyj Eyj Ex
+++? Bnjcn tbn
systno ds jkjldjner.
Lnt hn tbn kutput akr ej djput , d.n., ]bnj, akr ej
djput tbn kutput ds idvnj hy , kr
dj ktbnr wkrms, tbn systno ds tdon-djverdejt.
_Wj y _Wj x _.W_W_W_W 5 8 +?
j yj yj xj y
Jkw, akr ej djput _W_W jj x ε¾? , tbn
kutput ckjvnrins tk skon ckjstejt G es_Wj y
√j .
]bn mdaanrnjcn nquetdkj mnscrdhdji tbn systno es √j rnmucns
tk kr
d.n.,
,ε?8 G .ε? G
8.=8 ]bn dopulsn rnspkjsn ka tbn aectkr-ka-3 djtnrpkletkr ka Nq.
(8.11) ds tbn kutput akr ej
djput ejm ds idvnj hy_W_W jj x u ν?
_)8W_8W(_)5W_5W(_W_W 3
8 +ν+ν++ν+ν+ν? jjjjjjb kr nqudvelnjtly hy
.88,,,5,_}W{ 3
5 ? jjb
8.=3 ]bn djput-kutput rnletdkj ka e aectkr-ka- L
djtnrpkletkr ds idvnj hy
( ._W_W_W_W 5
L
? ) Dts dopulsn rnspkjsn ds tbn kutput akr
ej djput ejm ds tbus idvnj hy_W_W jj x u ν?
( )_W_W_W_W 5
5
8.=> Es ejm ern rdibt-sdmnm snqunjcns, essuon_Wj x
_Wjb 6_W ?j x akr ell
ejm
ejm Bnjcn,
5 J j < 6_W ?jb
.8 J j < 6_WK_W_W * ??
j x jbj y akr ell ejm tbus85
J J j +<
Jkt akr seln. 83
8585
? √
+?
√
+?
? ? ? √
+?
√
?
√
?
√
+?
√
+?
√
? 8588 8585 8
_W_W_W_W_W_W
J J j J g J g
J J j J J j
J g
g j x g bg j x g bg j x g b
es?? √
? 58858
_W_W_W_W
J o J g g J J o J g
o x g bo x g b 6_W
?o x akr ell .
Bnjcn,5 J o <
._W_W_W
√
?
√
√? ,,
j g
√
?
√
j g
8.=9 Jkw arko Nq. (8.98) ej erhdtrery djput cej hn nxprnssnm
es
wbdcb cej hn rnwrdttnj usdji Nq. (8.=5h) es
_Wj x
( ) ¾¾? √
√?g
√?g
√?g
g jg x
Sdjcn ds tbn rnspkjsn ka ej L]D systno akr ej
djput_Wjs _,Wj¾ ds tbn rnspkjsn
akr ej djput ejm
_W g js _W g j ¾ _5W
g js ds tbn rnspkjsn akr ej djput Bnjcn,
tbn kutput akr ej djput ds idvnj hy
_.5W ¾ g j
√?g
√?
√
√?
jsj x jsj x g jsg x g jsg x j y
g g
._W~_W_W ? √
oj x obj y
? ?+?+ √
√?
√
. J
8.=4 Dj tbds prkhlno wn oegn usn ka tbn dmnjtdty _.W_WK_W *
r ojr joj ν?νν
Jkt akr seln. 8=
(m) ( ) ( )_5W8_W=_8WK_5W8_3W>_WK_W_W *5*8=
νν++ν+ν+ν?? jjjjjjbj x j y
_8WK_5W8_5WK_3W56_WK_3W86_8WK_3W> ****
+ν+ννννν++νν? jjjjjjjj
_5WK_5W=_WK_5W2 ** ν+νν+ν+ jjjj
_3W8_=W56_3W86_5W> +ννν+ν? jjjj
_W=_5W2 jj ν+ν+ _.=W56_3W86_5W>_W=_5W2_3W8
νν+νν+ν++ν? jjjjjj
8.>6 (e) _WK_W_W * j yj x ju
?
.2=},=,5=,88,59,=8,8>,11,83,=>,86,>,=8,8={ ?
j
(h) _WK_W_W * jwj x jv ?
.555},56,=,5>,1,53,36,82,3,51,56,>,9,58{ ?
j
(c) _WK_W_W * j yjwji ?
Jkt akr seln. 8>
akr ds mnadjnm akr
85 O oj O
_W,5 ojb J o ? 855
O J j O , kr nqudvelnjtly,
akr
. Ldgnwdsn, akr5855
J O j J O ++
_W,8 ojb J o ? ds mnadjnm akr
, kr nqudvelnjtly, akr885 O J j O
.8885
J O j J O ++
Akr tbn spncdadnm
snqunjcns .1,8,=,3 8585 ????
O O J J (e) ]bn lnjitb
ka ds_Wj y
585)3(8=155588 ?++?++
J O J O . (h) ]bn rejin ka
akrj 6_W j y ds
),oex(),odj( 88558855
J O J O j J O J O
++++ , d.n.,
Akr tbn spncdadnm snqunjcns tbn rejin ka ds.8855
J O j J O ++ j
.565 j
8.>8 Jkw,
√?g
?? √
√?g
J g x g J j x J j x J j x j
_.W_W_WK_W_W 885588*55v .8
o J g ?
]bnj ?? √
√?o
8.>3 _WK_W_WK_WK_W_W 3*3*8*5
j x j yj x j x j x ji
?? wbnrn _.WK_W_W 8*5
j x j x j y ? Jkw
_.WK_W_W 88*55
J j x J j x jv ?
Mnadjn _.WK_W_W 33* J j x jvjb ?
]bnj arko tbn
rnsults ka Vrkhlno 8.>8, _.W_W 85
J J j yjv ? Bnjcn,
_.WK_W_W 33*85
J j x J J j yjb ?
]bnrnakrn, oegdji usn ka tbn rnsults ka Vrkhlno
8.>8 eiedj wn int _.W_W 385
J J J j yjb ?
8.>= Suhstdtutdji hy ?? √
√?g
g bg j x jbj x j y
_.W_W_WK_W_W * g oj dj tbds
nxprnssdkj, wn
int Bnjcn tbn ckjvklutdkj kpnretdkj ds
ckooutetdvn.
Lnt ( ) ( +?+? √
√?g
g bg bg j x jbjbj x j y
_W_W_W_W_WK_W_W 8585* )
Bnjcn tbn
+ ?+? √
√?
√
√?g g
snqunjcn, tbn rnsult ka tbds ckjvklutdkj cejjkt hn mntnrodjnm.
Hut
_).WK_W(K_W_WK_WK_W 5*8*35*8*3
j x j x j x j x j x j x
? _WK_W 5*8 j x j x
Jkt akr seln. 81
_).WK_W(K_W_WK_WK_W 5*3*85*3*8
j x j x j x j x j x j x
? Jkw akr ell veluns
ka , ejm bnjcn tbn kvnrell rnsult ds znrk. Es e rnsult, akr tbn
idvnj snqunjcns
6_WK_W 5*3 ?j x j x
j
g jbg x jbj x j y
_W_W_WK_W_W * ejm
._W_W_WK_W_W * ?? g
g jbg ijijbj a Ckjsdmnr _WK_)WK_W(_W
**5 jijbj x jw ?
_.W_W_W_WK_W *
g ojbg x oijij y g o
?? Jkw ckjsdmnr _)WK_W(K_W_W **8
jijbj x jw ?
_.W_W_W_WK_W *
og jboig x j a j x
og
?? ]bn mdaanrnjcn hntwnnj tbn
dopldns ]bus,
< ?
g ojbg x oi 66
_.W_W_W Ell suos bevn kjly e adjdtn juohnr ka tnros.
Bnjcn,
tbn djtnrcbejin ka tbn krmnr ka tbn suooetdkjs ds `ustdadnm ejm
wdll idvn ckrrnct rnsults.
H) Essuoptdkjs0 ejm ern stehln snqunjcns, ejm ds e hkujmnm
snqunjcn
wdtb
5 _W_W_W_W_W g
g g g
g o x g bg o x g bo y√< Hj x
_W
_W 85 , og g ξ+ wdtb ._W
85 , Ho jg g ξξ Dj tbds cesn, ell
suos bevn naanctdvnly kjly e adjdtn
juohnr ka tnros ejm tbn nrrkr _W 85 , og g ξ
cej hn rnmucnm hy cbkksdji ejm
suaadcdnjtly lerin. Es e rnsult, dj tbds cesn tbn prkhlno ds eiedj
naanctdvnly rnmucnm
tk tbet ka tbn kjn-sdmnm snqunjcns. ]bus, tbn djtnrcbejin ka tbn
krmnr ka tbn
suooetdkjs ds eiedj `ustdadnm ejm wdll idvn ckrrnct rnsults.
5g
8g
Bnjcn, akr tbn ckjvklutdkj tk hn esskcdetdvn, dt ds
suaadcdnjt tbet tbn snqunjcns hn stehln
ejm sdjiln-sdmnm.
8.>9 Sdjcn ds ka lnjitb._W_W_W ? √
√?g g bg j x j y
_Wg b O ejm mnadjnm akr ,56
O g
tbn ckjvklutdkj suo rnmucns tk wdll hn jkjznrk akr
ell tbksn veluns ka ejm akr g wbdcb
._W_W_W )5( 6 ?
g g bg j x j y
_Wj y
j g j setdsadns .56 J g j
Odjdouo
velun ka ejm kccurs akr lkwnst et6? g j j 6?j
ejm .6?g Oexdouo velun ka
ejm kccurs akr oexdouo velun ka et5? J g j g
.5 O ]bus 5? O g j
8.>2 ]bn oexdouo velun ka kccurs et wbnj ell
prkmuct tnros ern prnsnjt. ]bn oexdouo velun ds idvnj
hy
._W_W_W 5 6 ?
? J g
g x g j x j y
_Wj y 5? J j
._5W 5 6 5?
8.>4 ]bn oexdouo velun ka kccurs et wbnj ell
prkmuct tnros ern prnsnjt. ]bn oexdouo velun ds idvnj
hy
._W_W_W 5 6 ?
? J g
g bg j x j y _Wj y
5? J j
._5W 5 6 5?
?? √ √?g nvnvnvnv
g ig jbjbjij y _.W_W_WK_W_W *
? √ √?g nvnv
g ig jbj y _.W_W_W g
.g ]bnj tbn suooetdkj kj tbn lnat
hnckons ?+? √ √?
√ √?g g nvnvnvnv
g ig jbg ig jbj y
_W)_(W_W_W_W
_.Wj y? Bnjcn ds ej nvnj snqunjcn._WK_W *
jbji nvnv
+? ? √ √?
√ √?
Bnjcn ds ej kmm snqunjcn. _WK_W * jbji
km nv
+? ? √ √?
√ √?
Bnjcn ds ej nvnj snqunjcn. _WK_W * jbji
km km
8.15 ]bn dopulsn rnspkjsn ka tbn cescemn ds idvnj hy _WK_W_W 8*5
jbjbjb ? wbnrn
ejm Bnjcn,_W_W5 jjb j¾ε? _.W_W8
jjb
j¾θ? ( _.W_W 6
jjb j g
g jg ¾ θε? ?
8.18 Jkw ]bnrnakrn_.W_W jjb j¾ε?
_W_W_W_W 6
g j x g j x g bj y
g
g g ε? ? √
_.5W_W_5W_W_W_W 65 ε+? εε+? ε+?
√ ?
√ ?
j yj x g j x j x g j x j x
g
g g
g
Bnjcn, _.5W_W_W ε? j yj yj x
]bus tbn djvnrsn systno ds idvnj hy
]bn dopulsn rnspkjsn ka tbn djvnrsn systno ds idvnj hy_.5W_W_W
ε? j x j x j y
.56},,5{_W ε? jjb
8.13 Arko tbn rnsults ka Vrkhlno 8.18 wn bevn ( _.W_W
6
jjb j g
Jkt akr seln. 82
_W_W_W_W_W_W 6 66
g j x g j x og bg j x j y
g
g
o
g
o
_5W_W_5W_W_W 6
√
?
+
?
+√
?
+
?
+
r j x j x r j x j x j y
r
r r
r
r
r
o
or o
._3W_8W_5W_5W_W 38
G+ε+ε+ε+θ+?
j x j x j x j yj x
]bn djvnrsn systno ds
tbnrnakrn idvnj hy _.8W_5W)(_W_W εθ+θ+ε?
j yj yj yj x
8.1= (e) _.WK_W_WK_W_WK_WK_WK_W_W =*38*53*3*8*5
jbjbjbjbjbjbjbjbjb ++?
(h) . _WK_WK_W5
_)8W8_5W(K_)5W3_8W8(_WK_W *8*5 +ν+ν+νν?
jjjjjbjb
_WK_W_WK_W_WK_W_WK_W **** 88538888553588
+ν+ν+νν+ν+ννν? jjjjjjjj
? _W_W_W_W 31=338 +νν+νν jjjj ejm
_.W_W_W_W_W_W 535839>>3 +ν+νν+ν+ν?
jjjjjjb ]bnrnakrn,
_W_W_W_W_W_W_W_W_W_W 535839>>31=338
+ν+νν+ν+ν++νν+νν? jjjjjjjjjjb
_.W_W_W_W_W 31535834>> +ν+ν+ν+ν+ν?
jjjjj
? ?
j x wn
(h) ]bn lnjitb ka ds_Wj x .=5=9 ?+
^sdji
? ?
j x wn
(c) ]bn lnjitb ka ds_Wj x .=5>2 ?+
^sdji
? ?
=
j x wn
Jkt akr seln. 84
8.19 _.W_5W_W jhx jeyj y +? Bnjcn,
_.6W_5W_6W hx ey y +? Jnxt,
Ckjtdjudji aurtbnr dj e sdodler
wey wn khtedjnm
._W_5W_W 6
5 +? ? + j
(e) Lnt hn tbn kutput mun tk ej djput ]bnj
Da
._W_5W_W 6 5
5 5 +? ?
+ j g
g jj g hx e yej y _,W_W5
kjj x j x ? tbnj
Bkwnvnr,
g jj
r hx e yejg hx e yej y
._W_5W_W_5W_W 6 6
k kk
r hx e yejg hx e yejj y
Bnjcn da_W_W5 kjj yj y ,6_5W
y d.n., tbn systno ds tdon-verdejt. ]bn systno
ds
tdon-djverdejt da ejm kjly da ,6_5W ? y es tbnj
_W_W5 kjj yj y ? .
(h) Lnt ejm hn tbn kutputs mun tk djputs ejm , rnspnctdvnly.
Lnt hn tbn kutput mun tk ej djput
_W5 j y _W5 j y _W5
j x _W5 j x
_Wj y _.W_W 85 j x j x
θ+ε Bkwnvnr, ?θ+ε _W_W 85
j yj y
wbnrnes,,_W_W_5W_5W 6 86 5
? ++ j
g g jjj
g hx eg hx e ye ye
Bnjcn, tbn systno ds
._W_W_5W_W 6 86 5
g g jj
g hx eg hx e yej y
6_5W y .6_5W ? y
(c) Injnreldzdji tbn ehkvn rnsult dt cej hn sbkwj tbet ej –tb krmnr
ceusel mdscrntn-
tdon systno ds ldjner ejm tdon-djverdejt da ejm kjly da
J
.5,6_W J r r y ?
8.12 _5W_5W_W_W 556 +?
j ym j x pj x pj y
lnems tk _,5W_5W_W_W 6
5
6
5
6
p
p
m
p
wbdcb ds tbn mdaanrnjcn nquetdkj cberectnrdzdji tbn djvnrsn systno.
Dj ktbnr wkrms, sdoply sklvn tbn nquetdkj akr xWj_ dj tnros ka
prnsnjt ejm pest veluns ka yWj_ ejm xWj_.
8.14 ejm,6_,W_W_W_W 66
j g
.6,6_W <? jjs Sdjcn ds
jkjjnietdvn, ds e okjktkjdcelly djcrnesdji aujctdkj ka akr , ejm ds
jkt
kscdlletkry. Bnjcn, tbnrn ds jk kvnrsbkkt.
_Wg b
8.96 (e) _.8W_5W_W +?
j a j a j a Lnt tbnj tbn
mdaanrnjcn nquetdkj rnmucns tk,_W jr j a
ε?
6 85 ?εεε jjj r r r wbdcb rnmucns
aurtbnr tk rnsultdji dj65
8 ? r r
Es bnjcn,6_6W ? a .685 ?ε+ε
Elsk ,5_5W ? a , ejm bnjcn
.5> 88
5 85 ?ε?ε Bnjcn,
Jkt akr seln. 36
? +
(h) _.5W_8W_5W_W ++?
j x j yj yj y Es tbn systno ds L]D,
tbn djdtdel ckjmdtdkjs ern
nquel tk znrk. Lnt _.W_W jj x ν?
]bnj _.5W_8W_5W_W ν++?
jj yj yj y Bnjcn,
6_8W_5W_6W ?+? y y y ejm
.5_6W_8W_6W_5W ?ν++? y y y Akr
tbn
ckrrnspkjmdji mdaanrnjcn nquetdkj ds
_8W_5W_W +? j yj yj y wdtb
djdtdel ckjmdtdkjs
ejm wbdcb ern tbn seon es tbksn akr tbn sklutdkj ka tbn
Adhkjeccd—s
snqunjcn. Bnjcn
]bus mnjktns tbn dopulsn
rnspkjsn ka e ceusel L]D systno mnscrdhnm hy tbn mdaanrnjcn
nquetdkj _.5W_8W_5W_W ++?
j x j yj yj y
8.95 Mnjktdji_.W_5W_W j x j yj y
+ε? _,W_W_W j y `j yj y dorn
+? ejm ,h `e +?ε wn int
_.W_)5W_5W)((_W_W
j x j y `j yh `ej y `j y
dorndorn +++?+ Nquetdji tbn rnel ejm tbn
doeidjery perts, ejm jktdji tbet ds rnel, wn
int_Wj x
_,W_5W_5W_W j x jhyjeyj y dornrn +?
_.5W_5W_W +? jeyjhyj y dorndo Arko
tbn
snckjm nquetdkj wn bevn _.5W_W_5W 5 ?
j yj yj y rn
e
_,W_5W_W_5W_W 8
h doe
_.5W_8W)(_5W_5W 88 +++? jex j yhejeyjhy
rnrndo Suhstdtutdji tbds nquetdkj
dj tbn nquetdkj _W_5W_5W_W j x jhyjeyj y
dornrn +? wn errdvn et
_5W_W_8W)(_5W8_W 88 ++?
jex j x j yhejeyj y rnrnrn wbdcb ds
e snckjm-krmnr
mdaanrnjcn nquetdkj rnprnsnjtdji dj tnros ka_Wj yrn
_.Wj x
8.98 ]bn adrst-krmnr ceusel L]D systno ds cberectnrdznm hy tbn
mdaanrnjcn nquetdkj
_.5W_5W_W_W 556 +?
j ym j x pj x pj y
Lnttdji _W_W jj x ν? wn
khtedj tbn mdaanrnjcn
nquetdkj rnprnsnjtetdkj ka dts dopulsn rnspkjsn _.5W_5W_W_W 556
ν+ν? jbm j pj pjb
Sklvdji dt akr wn int,8,5,6?j ,_6W_5W,_6W 655556
pm pbm pb pb ??? ejm
).(_5W_8W 655655 pm p pm bm b
?? Sklvdji tbnsn nquetdkjs wn int _,6W6
b p ?
, _5W
O
g g
g j ym g j x p _.W_W
jj x ν? ]bnj ._W_W
66
? ν ??
J
g g g jbm g j p
]bus, Sdjcn tbn systno ds essuonm tk hn ceusel, ? ?
J g g r
g r bm p
6 _.W 6_W ? g r b
Jkt akr seln. 35
akr ell Bnjcn,.r g : ?? ? ?
J g
J g g r g r
m g bg r bm p
6 6 ._W_W
8.9= Akr e adltnr wdtb e ckoplnx-velunm dopulsn rnspkjsn, tbn adrst
pert ka tbn prkka ds tbn
seon es tbet akr e adltnr wdtb e rnel-velunm dopulsn rnspkjsn.
Arko
√? √
√? g g
g j x g bg j x g bj y
_W_W_W_W_W .
Sdjcn tbn djput ds hkujmnm ._W x Hj x
]bnrnakrn ._W_W √ √?g x
g b Hj y Sk da
,_W √<?√ √? S g bg tbnj
S Hj y x _W djmdcetdji tbet ds
elsk hkujmnm._Wj y
]k prkvn tbn ckjvnrsn wn jnnm tk sbkw tbet da e hkujmnm
djput ds prkmucnm hy e
hkujmnm djput tbnj Ckjsdmnr tbn akllkwdji hkujmnm djput
mnadjnm hy.√<S
. _W
g b
g g
√? Jkw sdjcn tbn kutput
ds hkujmnm, ]bus akr e adltnr wdtb e ckoplnx dopulsn rnspkjsn ds
HDHK stehln da.√<S
ejm kjly da ._W √<?√ √?
S g bg
8.9> ]bn dopulsn rnspkjsn ka tbn cescemn ds ]bus._W_W_W 85
? √ √?g
r br g bg i
._W_W_W_W_W 8585
r bg br br g bg i Sdjcn
_W5 jb ejm ern stehln,_W8 jb √<
g
g b _W5 ejm ._W8 √< g
g b Bnjcn √< g
g i _W ejm es e
rnsult, e cescemn ka twk stehln L]D systnos ds elsk stehln.
8.91 ]bn dopulsn rnspkjsn ka tbn perellnl structurn ds _.W_W_W 85
jbjbji +? Jkw,
._W_W_W_W_W 8585 + +? √
ern stehln,
_W5 jb
g b Bnjcn √< g
g i _W ejm es e rnsult, e
perellnl ckjjnctdkj ka twk stehln L]D systnos ds elsk
stehln.
8.99 Ckjsdmnr e cescemn ckjjnctdkj ka twk pessdvn L]D systnos wdtb
ej djput ejm ej
kutput Lnt ejm hn tbn kutputs ka tbn twk systnos akr tbn
djput
Jkw
_.Wj x √ √?
88 8 √
setdsaydji tbn ehkvn djnqueldtdns. ]bnj
ejm es e rnsult,
_W_W_W 85 j x j yj y ??
_W_W_W 85 j yj yj y +?
_W8 j x ? 888
√ √? jjj j x j x j y .
Bnjcn, tbn
perellnl ckjjnctdkj ka twk pessdvn L]D systnos oey jkt hn
pessdvn.
8.92 Ckjsdmnr e perellnl ckjjnctdkj ka twk pessdvn L]D systnos wdtb
ej djput ejm ej
kutput Lnt ejm hn tbn kutputs ka tbn twk systnos akr tbn
djput
_Wj x
Jkt akr seln. 38
88 8 √
setdsaydji tbn ehkvn djnqueldtdns. ]bnj
ejm es e rnsult,
_W_W_W 85 j x j yj y ??
_W_W_W 85 j yj yj y +?
_W8 j x ? 888
√ √? jjj j x j x j y .
Bnjcn, tbn
perellnl ckjjnctdkj ka twk pessdvn L]D systnos oey jkt hn
pessdvn.
8.94 Lnt tbn mdaanrnjcn nquetdkj rnprnsnjts tbn
ceusel DDX mdidtel adltnr. Akr ej djput
+? ?? J
g g O g g
g j ym j yg j x p
56 _W_W_W
_,W_W jj x ν? tbn ckrrnspkjmdji kutput ds
tbnj
tbn dopulsn rnspkjsn ka tbn adltnr. Es tbn juohnr ka cknaadcdnjts
ds
ejm tbn juohnr ka cknaadcdnjts ds
_,W_W jbj y ? }{ g p
5+ O }{ g m J , tbnrn ern
e tktel ka 5++ O J
ujgjkwjs. ]k mntnrodjn tbnsn cknaadcdnjts arko tbn dopulsn rnspkjsn
seoplns, wn
ckoputn kjly tbn adrst 5++ O J dopulsn
rnspkjsn seoplns. ]k dllustretn tbn ontbkm,
wdtbkut ejy lkss ka injnreldty, wn essuon .3??
O J ]bnj, arko tbn mdaanrnjcn
nquetdkj wn errdvn et tbn akllkwdji 95
?++ O J nquetdkjs0
,_6W 6 pb ?
,_6W_5W_8W_3W 8885 pm bm bm bb ?+++
,6_5W_8W_3W_=W 885 ?+++ m bm bm bb
,6_8W_3W_=W_>W 885 ?+++ m bm bm bb
.6_3W_=W_>W_1W 885 ?+++ m bm bm bb
Yrdtdji tbn lest tbrnn nquetdkjs dj oetrdx akro wn errdvn et
, 6 6 6
_1W _>W _=W
_3W_=W_>W _8W_3W_=W _5W_8W_3W
5
3
8
5
m
m
m
.
(e) Akr ._W,6_5W 8
8.25 ( ) ( ) ( ).)5()5()()5()( )5(
] j x ] ] j ym x ] j yj] y
j] ] j +∧ ?+? ]bnrnakrn,
tbn
mdaanrnjcn nquetdkj rnprnsnjtetdkj ds idvnj hy _5W_5W_W +?
j x ] j yj y wbnrn
ejm)(_W j] yj y ? ).(_W
j] x j x ?
8.28 + ?? ??
j jj x x x j y ll ll
Jkw ?
? 5 55
l l d.n,
_.W_5W_W 55
j +
?
8.23 wdtb_W=.8_5W3>.6_W jj yj y ¾?
.3_5W ? y ]bn tktel sklutdkj ds idvnj
hy
_,W_W_W j yj yj y pc +?
wbnrn ds tbn ckoplnonjtery sklutdkj ejm ds tbn
pertdculer sklutdkj.
_Wj yc _Wj y p
_Wj yc ds khtedjnm hy sklvdji .6_5W3>.6_W ?
j yj y cc ]k tbds njm wn snt
wbdcb ydnlms rnsultdji dj tbn sklutdkj
,_W j
.3>.6?η Bnjcn
.)3>.6(_W j
c j y ε?
Akr tbn pertdculer sklutdkj wn cbkksn ._W θ?j y p
Suhstdtutdji tbds sklutdkj dj tbn
mdaanrnjcn nquetdkj rnprnsnjtdji tbn systno wn int _.W=.83>.6
j¾?θθ Akr 6?j wn
int , d.n.,=.83>.6 ?θθ =.8)3>.65( ?θ
ejm bnjcn .53 / =21>.6 / =.8
??θ
]bnrnakrn .6,)3>.6(_W_W_W 53
=2 +ε?+? jj yj yj y j pc
Akr ,5?j wn tbus bevn
53
=25 )3>.6(3_5W +ε?? y doplydji
.8=83.6?ε ]bn tktel sklutdkj ds tbus idvnj hy
.6,)3>.6(8=83.6_W 53
=2 +? jj y j
8.2= wdtb_W3_8W6=.6_5W3.6_W jj yj yj y
j ¾? 8_5W ? y ejm ]bn tktel
sklutdkj ds idvnj hy
.5_8W ? y
_,W_W_W j yj yj y pc +?
wbnrn ds tbn ckoplnonjtery sklutdkj
ejm ds tbn pertdculer sklutdkj.
_Wj yc
_Wj y p
_Wj yc ds khtedjnm hy sklvdji .6_8W6=.6_5W3.6_W ?
j yj yj y ccc ]k tbds njm wn
snt wbdcb ydnlms rnsultdji dj tbn sklutdkjs
kr Bnjcn
,_W j
c j y η? 66=.63.6 85 ?ηηη
jjj
=.6?η .5.6?η .)5.6()=.6(_W 85 jj
c j y ε+ε?
Akr tbn pertdculer sklutdkj wn cbkksn Suhstdtutdji tbds sklutdkj dj
tbn
mdaanrnjcn nquetdkj rnprnsnjtdji tbn systno wn int
Akr
6?j wn bevn
.5511.5?θ ]bnrnakrn
pc Akr ejm
int ejm>=24.65 ?ε .68>>.68 ?ε
Bnjcn,
.6,)3(5511.5)5.6(68>>.6)=.6(>=24.6_W +?
jj y jjj
8.2> _5W8_W_8W6=.6_5W3.6_W +?
j x j x j yj yj y wdtb
_,W3_W jj x j ¾? 8_5W
? y ejm
]bn tktel sklutdkj ds idvnj hy.5_8W ? y _,W_W_W
j yj yj y pc +? wbnrn ds
tbn
ckoplnonjtery sklutdkj ejm ds tbn pertdculer sklutdkj. Arko tbn
sklutdkj ka
Vrkhlno 8.2=, tbn ckoplnonjtery sklutdkj ds ka tbn akro
_Wj yc
]k mntnrodjn wn khsnrvn tbet tbn dt ds idvnj hy tbn suo ka tbn
pertdculer sklutdkj
ka tbn mdaanrnjcn nquetdkj
Arko tbn sklutdkj ka
]bnrnakrn,
_Wj y p
_W5 j y p _W3_W_8W6=.6_5W3.6_W 555
jj x j yj yj y j ¾??
_W8 j y p
.)3(_W5 j
58 θ?? j
_.5W38_W3)3(8)3(_W8_W_W 55
85 ¾+¾?θ+θ?+? jjj yj yj y jjjj
p p p
Akr tbn ehkvn nquetdkj rnmucns tk5?j .8383 +?θ+θ
]bus, ]bnrnakrn, tbn
tktel sklutdkj ds idvnj hy
Akr
.5?θ
.6,)3(8)3()5.6()=.6(_W_W_W 5
85 ++ε+ε?+? jj yj yj y jjjj
pc 5?j ejm
wn tbus bevn ejm8?j 8)3(8)3()5.6()=.6(_5W 855
8 5
int ejm6.=2235 ?ε .6.688=8 ?ε Bnjcn,
.6,)3(8)3()5.6(6.688=)=.6(6.=223_W 5 ++?
8.21 ]bn sklutdkj ds idvnj hy_.W_5W3>.6_W jjbjb ν?
_,W_W_W jbjbjb pc +? wbnrn
ds tbn ckoplnonjtery sklutdkj ejm ds tbn pertdculer sklutdkj. Da ds
tbn
dopulsn rnspkjsn, tbnj
_Wjb p _Wjb
.6_W ?jb p Arko Vrkhlno 8.23 wn jktn tbet
.)3>.6(_W j
c jb ε?
8.29 ]bn kvnrell systno cej hn rniermnm es tbn cescemn ka twk
ceusel L]D systnos0
S50 _W_8W6=.6_5W3.6_W 5 j xj yj yj y
? ejm S80 _5W8_W_W5 +?
j xj xj x .
Jkt akr seln. 3>
]bn dopulsn rnspkjsn ka tbn systno S5 cej hn akujm hy sklvdji
tbn
ckoplnonjtery sklutdkj ka
ckoplnonjtery sklutdkj hn , wn bevn
bnjcn . ]bnrnakrn, tbn dopulsn rnspkjsn ds idvnj hy
. Sklvdji ckjstejts wn int
85 ?ηηη jjj
{ 5.6,=.6 ?η } _W5 jb
c ,, H E 2.6? E
ejm Bnjcn ..8.6? H .6,)5.6(8.6)=.6(2.6_W5 +?
jjb jj
8.22 Stnp rnspkjsn ds tbnj idvnj hy.56_,W)(_W <ε<¾ε?
jjb j
_WK_W_W * jjbjs ¾?
√
55 d
j Sdjcn tbnrn
nxdsts e pksdtdvn djtninr sucb tbet akr ellk J .5
8
5 6, 5 <
,>5},8,9,2,6,5,3,1{_}W{ ? jj y
.28},>,8,6,5,8,8,3{_}W{ ? jjw
(e) .11,_W_W_W 3 3 ? ? lll
j xx j x j x r
,11},2,56,5=,55,83,55,>4,55,83,55,5=,56,2{_}W{ ?
ll xx r
.11,_W_W_W > >
,11},58,=2,84,35,36,89,513,89,36,35,84,=2,58{_}W{ ?
ll yyr
.11,_W_W_W 1 1
? ? lll jww jwjwr
Jkt akr seln. 31
.555,_W_W_W 3 2 ? ? lll
j xw j yj x r
,555},1,=,>,5=,9,58,58,9,8=,2,>,33,86{_}W{ ?
ll xwr
8.45 (e) _.W_W5 jj x j¾ε?
_W_W_W_W_W 5555
lll l ¾ε¾ ε?? √
√? √
j j
ε ?
l
855 _W,6
dj tbn snckjm nxprnssdkj wn int . 55
_W_W 5588
ε
ε
ε ??
Bnjcn,
ds ej nvnj aujctdkj ka Oexdouo velun ka kccurs et sdjcn ds e
mnceydji aujctdkj akr djcrnesdji wbnj
_W 55 l x x r
.l _W 55 l x x r
6?l lε
.5<ε
? ?
+?
l
ll
l
ll
l
l
Dt akllkws arko tbn ehkvn tbet ds e trdejiuler aujctdkj ka , ejm
bnjcn ds ej
nvnj aujctdkj wdtb e oexdouo velun ka et
_W 88 l x x r l
J .6?l
8.48 (e)
? φ
O
j j x cks_W5 wbnrn O ds e
pksdtdvn djtninr. Vnrdkm ka ds , ejm
bnjcn
O r
.6 5
8 cks
ll l
(h) { } { } .>6,>,=,3,8,5,6_W 18 ??
jjj x Dt ds e pnrdkmdc snqunjcn wdtb e
pnrdkm ]bus,.1 .>6,_W_W_W 18 >
5
88 +? ? lll
j x j x r j x x
ds elsk e
pnrdkmdc snqunjcn wdtb e pnrdkm
_W 88 l x x r
.1
>> 888888888888
1
5
88 ?+++++?
x x x x x x x x x x x x r
x x
( ) ,_6W_>W_>W_=W_=W_3W_3W_8W_8W_5W_5W_6W_5W 1
=6 888888888888
1
5
88 ?+++++?
x x x x x x x x x x x x r
x x
( ) ,_5W_>W_6W_=W_>W_3W_=W_8W_3W_5W_8W_6W_8W 1
38 888888888888
1
5
88 ?+++++?
x x x x x x x x x x x x r
x x
( ) ,_8W_>W_5W_=W_6W_3W_>W_8W_=W_5W_3W_6W_3W 1
82 888888888888
1
5
88 ?+++++?
x x x x x x x x x x x x r
x x
( ) ,_=W_3W_8W_=W_5W_3W_6W_8W_>W_5W_=W_6W_=W 1
35 888888888888
1
5
88 ?+++++?
x x x x x x x x x x x x r
x x
( ) . 1
5 _=W_>W_3W_=W_8W_3W_5W_8W_6W_5W_>W_6W_>W
(c) ds e pnrdkmdc snqunjcn wdtb e pnrdkm ]bus,jj x
)5(_W3 ? .8
.56_,W_W_W 3 5
5
33 +? ? lll
j x j x r j x x
Bnjcn,
( ) ,5_5W_5W_6W_6W_6W 33338
33 ?+?
x x x x r
x x ( ) .5_6W_5W_5W_6W_5W
33338
5
33 ?+?
x x x x r
x x
O8.5 (e) ]bn djput mete njtnrnm murdji tbn nxncutdkj ka Vrkireo
8[8.o ern0 ]ypn dj rnel nxpkjnjt ? -5/58
]ypn dj doeidjery nxpkjnjt ? pd/1
]ypn dj iedj ckjstejt ? 5
]ypn dj lnjitb ka snqunjcn ? =5
(h) ]bn djput mete njtnrnm murdji tbn nxncutdkj ka Vrkireo
8[8.o ern0 ]ypn dj rnel nxpkjnjt ? -5/58
]ypn dj doeidjery nxpkjnjt ? pd/1
]ypn dj iedj ckjstejt ? 5
]ypn dj lnjitb ka snqunjcn ? =5
Jkt akr seln. 32
-5.>
-5
-6.>
6
6.>
-5.>
-5
-6.>
6