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8/13/2019 Jee 2014 Booklet7 Hwt Magnetic Effects of Current
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Uchymdmbhcr Jgmssos
UDJ/Dmnbotcj Oeeojt ̀e Jurrobt :? K^X->/Ykyscjs
HMXO < XCDO < 89 Dcbutos DMVAZ < Q PPP /=?T XOZX J@HO < DMN Q=T
ZXMVX XCDO < OBH XCDO < XCDO XMAOB< YMVOBX–Z ZCNBMX_VO <
Xkcs tost j`btmcbs m t`tmg ̀ e =? @flojtcvo Xypo ]uostc̀ bs. Omjk quostc`b jmrrcos = dmra. Xkoro cs B@ BONMXCUO dmracbn.
Jk``so tko j`rrojt mgtorbmtcvo. @bgy `bo jk`cjo cs j`rrojt.
=. M g`bn cbsugmtoh j`ppor wcro cs jg`sogy w`ubh ms m spcrmg `e B turbs. Xko spcrmg kms cbborrmhcus m mbh `utor rmhcus f. Xko spcrmg gcos cb tko [ -S pgmbo mbh m stomhy jurrobt C eg`wstkr`unk tko wcro. Xko \ -j`dp`bobt `e tko dmnbotcj ecogh mt tko jobtro `e tko spcrmg cs <
(M)
9
6
BC fgb
f m m
¿
(F)
96
BC f mgb
f m f m
¿
(J) 96
BC fgb
f m
¿
(H) 96 BC f m
gbf f m
¿
6. Jurrobt tkr`unk MFJ mbh M F J cs C . ^kmt cs tko dmnbotcj ecogh mt Y 0 FY FY r (Koro J F YFJ mro j`ggcbomr)
(M)= 6
8 C
Fr ω
(F) 9 6
8
C F
r
¿
ω
(J) 98
C F
r
¿
ω
(H) zor`
:. Xko dmnbotcj ecogh mt tko p`cbt `e cbtorsojtc`b `e hcmn`bmgs `e m squmro wcro g``p `e scho G jmrrycbn m jurrobt C cs <
(M) 9 C
G
¿
ω(F) 96 C
G
¿
ω(J) 96 C
G
¿
ω(H) 96 6 C
G
¿
ω
8. M jurrobt c cs eg`wcbn tkr`unk tko g``p. Xko hcrojtc`b `e tko jurrobt mbh tko skmpo `e tko g``p mro mssk`wb cb tko ecnuro. Xko dmnbotcj ecogh mt tko jobtro `e tko g``p cs 9c
V
¿ tcdo
6 39 DM V DF V HDM
(M)=5
fut `ut `e tko pgmbo `e tko pmpor (F)?
=5 , fut cbt` tko pgmbo `e tko pmpor
(J)>
=5, fut `ut `e tko pgmbo `e tko pmpor (H)
>
=5, fut cbt` tko pgmbo `e tko pmpor
?. Y] mbh VZ mro g`bn pmrmggog j`bhujt`rs sopmrmtoh fy jortmcbhcstmbjo. D . cs tko dch-p`cbt fotwoob tkod (soo tko ecnuro). Xko botdmnbotcj ecogh mt D cs F. B`w, tko jurrobt 6 M cs swctjkoh `ee. Xko
ecogh mt D b`w foj`dos.(M) 6 F (F) F
(J)6
F(H) : F
5. Dmnbotcj ecogh mt tko jobtro `e m jcrjugmr j`cg `e rmhcus V huo t` c eg`wcbn tkr`unk ct cs F. Xko dmnbotcj ecogh mt m p`cbt mg`bn tkomxcs mt hcstmbjo V er`d tko jobtro cs <
(M)6
F(F)
8
F(J)
;
F(H) ; F
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Uchymdmbhcr Jgmssos
UDJ/Dmnbotcj Oeeojt ̀e Jurrobt :> K^X->/Ykyscjs
HMXO < XCDO < 89 Dcbutos DMVAZ < Q PPP /=?T XOZX J@HO < DMN Q6T
ZXMVX XCDO < OBH XCDO < XCDO XMAOB< YMVOBX–Z ZCNBMX_VO <
Xkcs tost j`btmcbs m t`tmg ̀ e =? @flojtcvo Xypo ]uostc`bs. Omjk quostc`b jmrrcos = dmra. Xkoro cs B@ BONMXCUO dmracbn.
Jk``so tko j`rrojt mgtorbmtcvo. @bgy `bo jk`cjo cs j`rrojt.
=. M jcrjugmr j`cg `e 69 turbs mbh rmhcus =9jd cs pgmjoh cb ubce`rd dmnbotcj ecogh `e 9.=9 X b`rdmg t` tko pgmbo `e tko j`cg. Ce tkojurrobt cb j`cg cs ? M, tkob tko t`rquo mjtcbn `b tko j`cg wcgg fo <(M) :=.8 Bd (F) :.=8 Bd (J) 9.:=8 Bd (H) zor`
6. M jurrobt C obtors m ubce`rd jcrjugmr j`cg `e rmhcus V, frmbjkos cbt` tw` pmrts mbh tkobroj`dfcbos ms sk`wb cb tko jcrjuct hcmnrmd. Xko rosugtmbt dmnbotcj ecogh mt tko jobtro `e tko j̀ cg cs <
(M) zor` (F) 96
C
V
¿(J) 9:
8 6 C
V
¿
(H) 9=8 6
C
V
¿
:. Xw` g`bn pmrmggog wcros jmrry jurrobts c= mbh c6 sujk tkmt = 6c c . ^kob tko jurrobts mro cb tko smdo hcrojtc`b, tko dmnbotcj ecogh mt
m p`cbt dchwmy fotwoob tko wcros cs 55 =9 X . Ce tko hcrojtc`b `e c6 cs rovorsoh, tko ecogh foj`dos ?: =9 X . Xko rmtc` =
6
c
c cs <
(M) =/6 (F) 6 (J) 6/: (H) :/6
8. M k`rcz`btmg `vorkomh p`worgcbo cs mt m kocnkt `e 8 d er`d tko nr`ubh mbh jmrrcos m jurrobt `e =99 M er`d omst t` wost.
Xko dmnbotcj ecogh hcrojtgy fog`w ct `bo tko nr`ubh cs < > =9 8 =9 XdM¿ ω
(M)>6 ? =9.
s`utkwmrh (F)5? 9 =9. X
b`rtkwmrh
(J)5? 9 =9. X
, s`utkwmrh (H)>6 ? =9. X
, b`rtkwmrh
?. Xw` j`bjobtrcj j`cgs `e =9 turbs omjk mro pgmjoh cb tko smdo pgmbo. Xkocr rmhcc mro 69 jd mbh 89 jd mbh jmrry 9.6 M mbh 9.: Mjurrobt rospojtcvogy cb `pp`scto hcrojtc`bs. Xko dmnbotcj cbhujtc`b (cb tosmgm) mt tko jobtro cs <
(M) 9:
8¿ (F) 9
?
8¿ (J) 9
>
8¿ (H) 9
3
8¿
5. Jurrobt C cs eg`wcbn cb j`bhujt`r skmpoh ms sk`wb cb tko ecnuro. Xko rmhcus `e tko jurvoh pmrt cs r mbh tko gobntk `e strmcnkt p`rtc`bcs vory gmrno. Xko vmguo `e tko dmnbotcj ecogh mt tko jobtro @ wcgg fo <
(M) 9 :=
8 6
C
r
¿ ω
ω
(F) 9 :
=8 6
C
r
¿ ω
ω
(J) 9 =8 6
C
r
¿ ω
ω
(H) 9 =
8 6 C
r
¿ ω
ω
>. M jcrjugmr jurrobt jmrrycbn j`cg kms m rmhcus V. Xko hcstmbjo er`d tko jobtro `e tko j`cg `b tko mxcs `e tko j`cg, wkoro tko dmnbotcjcbhujtc`b cs =/;tk `e cts vmguo mt tko jobtro ̀ e tko j`cg cs <
(M) : V (F) : V / (J)6
: V
(H)6 :
V
;. Xko hcrojtc`b `e cbhujoh dmnbotcj ecogh hF huo t` jurrobt ogodobt c hG , mt m p`cbt `e hcstmbjo r er`d ct, wkob m jurrobt c pmssos
tkr`unk m g`bn j`bhujt`r cs cb tko hcrojtc`b.
(M) `e p`sctc`b vojt`r r `e tko p`cbt (F) `e jurrobt ogodobt hG
(J) porpobhcjugmr t` f`tk mbhhG r (H) porpobhcjugmr t` hG `bgy
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Uchymdmbhcr Jgmssos
UDJ/Dmnbotcj Oeeojt ̀e Jurrobt :; K^X->/Ykyscjs
3. M jcrjugmr j`cg `e ? turbs mbh `e =9 jd domb hcmdotor cs j`bbojtoh t` m v`gtmno s`urjo. Ce tko roscstmbjo ̀ e j`cg cs =9Σ , tko v`gtmno
`e tko s`urjo s` cs t` buggcey tko k`rcz`btmg j`dp`bobt `e omrtk's dmnbotcj ecogh `e :9 M turb =d
mt tko jobtro `e tko j`cg sk`ughfo<(M) 5U , pgmbo `e tko j`cg b`rdmg t` dmnbotcj dorchcmb(F) 6 U , pgmbo `e tko j`cg b`rdmg t` dmnbotcj dorchcmb
(J) 6 U , pgmbo `e tko j`cg mg`bn tko dmnbotcj dorchcmb(H) 8U , pgmbo `e tko j`cg b`rdmg t` dmnbotcj dorchcmb
=9. Cb tko ecnuro sk`wb, tko dmnbotcj ecogh cbhujtc`b mt tko p`cbt @ wcgg fo <
(M) 96
c
r
¿
ω(F) 9 6
8
c
r
¿ω
ω
(J) 9 =8
c
r
¿ω
ω
(H) 9 6
8c
r
¿ω
ω
==. ^kcjk `e tko e`gg`wcbn nrmpks roprosobt vmrcmtc`b `e dmnbotcj ecogh F wctk hcstmbjo r e`r m strmcnkt g`bn wcro jmrrycbn jurrobt0
(M) (F) (J) (H)
=6. Cb m kyhr`nob mt`d, tko ogojtr`b cb dmacbn =?5 5 =9. rps cb m jcrjugmr pmtk `e rmhcus9 ?:Ï. . ^kmt cs tko dmnbotcj cbhujtc`b
pr`hujoh mt tko jobtro ̀ e `rfct0(M) 5.: X (F) =6.5 X (J) =;.3 X (H) 6?.6 X
=:. M jogg cs j`bbojtoh fotwoob tko p`cbts. M mbh J `e m jcrjugmr j`bhujt`r MFJH wctk @ ms
jobtro mbh mbngo M@J 4 59 . Ce = 6mbh F F mro tko dmnbctuhos `e tko dmnbotcj ecoghs mt
@ huo t` tko jurrobts cb MFJ mbh MHJ rospojtcvogy, tkob rmtc` =
6 F
Ζ .
(M) =(F) 6(J) ?(H) 5
=8. Cb tko mhlmjobt ecnuro cs sk`wb m jg`soh pmtk Y. M g`bn strmcnktj`bhujt`r jmrrycbn m jurrobt C pmssos tkr`unk @ mbh porpobhcjugmr t`tko pgmbo `e tko pmpor. Xkob wkcjk `e tko e`gg`wcbn k`ghs n``hs0
(M) 9
p
F . hg (F) 9
Y
F . hg C ¿
(J) 9
Y
F .hg C ¿ (H) B`bo `e tkoso
=?. M wcro `rcobtoh cb tko omst-wost hcrojtc`b jmrrcos m jurrobt omstwmrh. Hcrojtc`b `e tko dmnbotcj ecogh mt m p`cbt t` tko s`utk `e tkowcro cs<(M) vortcjmggy h`wb (F) vortcjmggy up (J) b`rtk-omst (H) s`utk-omst
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Uchymdmbhcr Jgmssos
UDJ/Dmnbotcj Oeeojt ̀e Jurrobt :3 K^X->/Ykyscjs
HMXO < XCDO < 89 Dcbutos DMVAZ < Q PPP /=?T XOZX J@HO < DMN Q:T
ZXMVX XCDO < OBH XCDO < XCDO XMAOB< YMVOBX–Z ZCNBMX_VO <
Xkcs tost j`btmcbs m t`tmg ̀ e =? @flojtcvo Xypo ]uostc̀ bs. Omjk quostc`b jmrrcos = dmra. Xkoro cs B@ BONMXCUO dmracbn.
Jk``so tko j`rrojt mgtorbmtcvo. @bgy `bo jk`cjo cs j`rrojt.
=. Xko ecnuro sk`ws tko jr`ss-sojtc`b `e m g`bn jygcbhrcjmg j`bhujt`r `e rmhcus m jmrrycbn m ubce`rdgy hcstrcfutoh jurrobt c. Xko dmnbotcjecogh huo t` jurrobt mt Y cs <
(M) 69 6cr / m¿ ω (F) 6
9 6cr / m¿ ω
(J) 69 6cm / r ¿ ω (H) 6 6
9cm / r ¿ ω
6. E`r tko dmnbotcj ecogh t` fo dmxcdud huo t` m sdmgg ogodobt `e jurrobt jmrrycbn j`bhujt`r mt m p`cbt, tko mbngo fotwoob tkoogodobt mbh tko gcbo l`cbcbn tko ogodobt t` tko ncvob p`cbt dust fo <
(M) 9 (F) 39 (J) =;9 (H) 8?
:. Dmnbotcj cbhujtc`b mt tko jobtro `e m jcrjugmr g``p `e mrom 6dω cs 9.= X . Xko dmnbotcj d`dobt `e tko g``p cs < ( 9¿ 4 pordomfcgcty
`e mcr)
(M)9 =
9. ω
¿(F)
9
9 6. ω
¿(J)
9
9 :. ω
¿(H)
9
9 8. ω
¿
8. M g`bn strmcnkt wcro jmrrycbn m jurrobt `e :9 M cs pgmjoh cb mb oxtorbmg ubce`rd dmnbotcj ecogh `e cbhujtc`b 88 =9 X .
Xko dmnbotcj ecogh cs mjtcbn pmrmggog t` tko hcrojtc`b `e jurrobt. Xko dmnbctuho `e tko rosugtmbt dmnbotcj cbhujtc`b cb tosgm mt m
p`cbt 6.9 jd mwmy er`d tko wcro >9 8 =9 K / d¿ ω
(M)8=9
(F)8: =9 (J)
8? =9 (H)85 =9
?. Mb ogojtr`b mbh m pr`t`b mro d`vcbn `b strmcnkt pmrmggog pmtk wctk smdo vog`jcty. Xkoy obtor m sodc-cbecbcto ronc`b `e ubce`rddmnbotcj ecogh porpobhcjugmr t` tko vog`jcty. ^kcjk `e tko e`gg`wcbn stmtodobt(s) cs/mro truo0(M) Xkoy wcgg bovor j`do `ut tko dmnbotcj ecogh ronc`b(F) Xkoy wcgg j`do `ut trmvoggcbn mg`bn pmrmggog pmtks(J) Xkoy wcgg j`do `ut mt tko smdo tcdo(H) Xkocr pmtk wcgg b`t fo pmrmggog
5. ^kcjk `e tko ecogh pmttorb ncvob cb tko ecnuro cs vmgch e`r ogojtrcj ecogh ms wogg ms e`r dmnbotcj ecogh0
(M) (F) (J) (H)
>. M tkcb jcrjugmr hcsa `e rmhcus V cs ubce`rdgy jkmrnoh wctk hobscty 9ϊ por ubct mrom. Xko hcsa r`tmtos mf`ut cts mxcs wctk m
ubce`rd mbnugmr spooh σ . Xko dmnbotcj d`dobt `e tko hcsa cs <
(M)66 Vω ϊσ (F)
86 Vω ϊσ (J)
8
6
Vωϊσ (H)
8
8
Vωϊσ
;. M pr`t`b, m houtr`b mbh mb pmrtcjgoν kmvcbn tko smdo acbotcj oborny mro d`vcbn cb jcrjugmr trmlojt`rcos cb m j`bstmbt dmnbotcj
ecogh. mbh p h r r r ν hob`to rospojtcvogy tko rmhcc `e tko trmlojt`rcos `e tkoso pmrtcjgo, tkob
(M) h pr r r ν (F) m h pr r r (J) h pr r r ν (H) h pr r r ν
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Uchymdmbhcr Jgmssos
UDJ/Dmnbotcj Oeeojt ̀e Jurrobt 89 K^X->/Ykyscjs
3. M boutr`b, m pr`t`b, mb ogojtr`b mbh mb pmrtcjgoν obtor m ronc`b `e ubce`rd dmnbotcj ecogh wctk
tko smdo vog`jctcos. Xko dmnbotcj ecogh cs porpobhcjugmr mbh hcrojtoh cbt` tko pgmbo `e tko pmpor.tko trmjas `e tko pmrtcjgos mro gmfoggoh cb tko ecnuro. Xko ogojtr`b e`gg`ws tko trmja(M) H (F) J
(J) F (H) M
=9. Xko ecnuro sk`ws tkroo sctumtc`bs wkob mb ogojtr`b wctk vog`jcty v trmvogs tkr`unk m ubce`rd dmnbotcj ecogh F . Cb omjk jmso,
wkcjk cs tko hcrojtc`b `e dmnbotcj e`rjo `b tko ogojtr`b 0
(M) mxcs mxcs mxcsvo z vo x vo y (F) mxcs mxcs mbh zor`vo z vo x (J) mxcs mxcs mbh zor`vo z vo y (H) mxcs mxcs mbh zor`vo z vo y
==. Xko rmhcus `e tko pmtk `e mb ogojtr`b d`vcbn mt m spooh `e >: =9 d / s porpobhcjugmr t` m dmnbotcj ecogh cs bomrgy
(M) =? jd (F) 8? jd (J) 6> jd (H) :8 jd
=6. Xko dmxcdud vog`jcty t` wkcjk m pr`t`b jmb fo mjjogormtoh cb m jyjg`tr`b `e =9 DK z eroquobjy mbh rmhcus ?9 jd cs
(M);5 6; =9. d / s (F)
;: =8 =9. d / s (J)>5 6; =9. d / s (H)
>: =8 =9. d / s
=:. Xw` pmrtcjgos `e dmssos mbhm fd d mbh smdo jkmrno mro pr`lojtoh cb m porpobhcjugmr dmnbotcj ecogh. Xkoy trmvog mg`bn jcrjugmr
pmtks `e rmhcus mbhm fr r sujk tkmt m fr r . Xkob wkcjk cs truo0
(M) m m f fd v d v (F) mbhm f m fd d v v
(J) mbhm f m fd d v v (H) f f m md v d v
=8. Mb ogojtr`b d`vos cb m jcrjugmr `rfct wctk m ubce`rd spooh v. Ct pr`hujos m dmnbotcj ecogh F mt tko jobtro `e tko jcrjgo. Xko rmhcus`e tko jcrjgo cs pr`p`rtc`bmg t` <
(M) F
v(F)
v
F(J)
v
F(H)
F
v
=?. Xw` jkmrnoh pmrtcjgos mro pr`lojtoh cbt` m ronc`b cb wkcjk m dmnbotcj ecogh cs porpobhcjugmr t` tkocr vog`jctcos. Metor tkoy obtor tkodmnbotcj ecogh, y`u jmb j`bjguho tkmt(M) tko jkmrnos mro hoegojtoh cb `pp`scto hcrojtc`bs (F) tko jkmrnos j`btcbuo d`vo cb m strmcnkt gcbo(J) tko jkmrnos d`vo cb jcrjugmr pmtks (H) tko jkmrnos d`vo cb jcrjugmr pmtk fut cb `pp`scto hcrojtc`bs
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Uchymdmbhcr Jgmssos
UDJ/Dmnbotcj Oeeojt ̀e Jurrobt 8= K^X->/Ykyscjs
HMXO < XCDO < 89 Dcbutos DMVAZ < Q PPP /=?T XOZX J@HO < DMN Q8T
ZXMVX XCDO < OBH XCDO < XCDO XMAOB< YMVOBX–Z ZCNBMX_VO <
Xkcs tost j`btmcbs m t`tmg ̀ e =? @flojtcvo Xypo ]uostc̀ bs. Omjk quostc`b jmrrcos = dmra. Xkoro cs B@ BONMXCUO dmracbn.
Jk``so tko j`rrojt mgtorbmtcvo. @bgy `bo jk`cjo cs j`rrojt.
=. ^kcjk `e tko e`gg`wcbn pmrtcjgos wcgg hosjrcfo tko sdmggost jcrjgo wkob pr`lojtoh wctk tko smdo vog`jcty porpobhcjugmr t` tkodmnbotcj ecogh0
(M) Ogojtr`b (F) Yr`t`b (J) pmrtcjgoν (H) Houtor`b
6. E`ur jkmrnoh pmrtcjgos mro pr`lojtoh porpobhcjugmrgy cbt` tko dmnbotcj ecogh wctk oqumg. ^kcjk wcgg kmvo dcbcdud eroquobjy `e r`rmtc`b0(M) Yr`t`b (F) Ogojtr`b (J) Gc+
(H) Ko+
:. Mb ogojtr`b mbh m pr`t`b `e oqumg gcbomr d`dobtud obtor cb tko hcrojtc`b porpobhcjugmr t` ubce`rd dmnbotcj ecogh. Ce tko rmhcc `e
tkocr jcrjugmr pmtk fo or mbh pr rospojtcvogy, tkob o
p
r
r , cs oqumg t` od dmss `e ogojtr`b, pd dmss `e pr`t`b.
(M)
= 6 / p
o
d
d
(F) p
o
d
d(J)
= 6 /
o
p
d
d
(H) =
8. ^kcjk `e tko e`gg`wcbn wkcgo cb d`tc`b jmbb`t fo hoegojtoh fy dmnbotcj ecogh0(M) Yr`t`bs (F) Jmtk`ho rmys (J) Mgpkm pmrtcjgos (H) Boutr`bs
?. Xw` pmrtcjgos `e oqumg jkmrnos metor focbn mjjogormtoh tkr`unk tko smdo p`tobtcmg hceeorobjo obtor m ubce`rd trmbsvorso dmnbotcjecogh mbh hosjrcfo jcrjugmr pmtk `e rmhcc V= mbh V6 rospojtcvogy. Xkob tko rmtc` `e tkocr dmss ( D = / D 6) cs
(M) =
6
V
V(F)
6=
6
V
V
(J) 6
=
V
V(H)
66
=
V
V
5. Xko dmnbotcj e`rjo `b m jkmrnoh pmrtcjgo d`vcbn cb tko ecogh h`os b`t w`ra, fojmuso.
(M) acbotcj oborny `e tko jkmrnoh pmrtcjgo h`os b`t jkmbno(F) tko jkmrno ̀ e tko pmrtcjgo rodmcbs smdo(J) tko dmnbotcj e`rjo cs pmrmggog t` vog`jcty `e tko pmrtcjgo(H) tko dmnbotcj e`rjo cs pmrmggog t` dmnbotcj ecogh
>. Chobtcey tko j`rrojt stmtodobt er`d tko e`gg`wcbn(M) Jyjg`tr`b eroquobjy cs hopobhobt `b spooh `e tko jkmrnoh pmrtcjgo(F) Acbotcj oborny `e jkmrnoh pmrtcjgo cb jyjg`tr`b h`os b`t hopobhobt `b cts dmss(J) Jyjg`tr`b eroquobjy h`os b`t hopobh `b spooh `e jkmrnoh pmrtcjgo(H) Acbotcj oborny `e jkmrnoh pmrtcjgo cb jyjg`tr`b cs cbhopobhobt `e cts jkmrno
;. M pr`t`b wctk oborny `e 6 DoU obtors m ubce`rd dmnbotcj ecogh `e 6.? X b`rdmggy. Xkob dmnbotcj e`rjo `b tko pr`t`b cs
(Xmao dmss `e pr`t`b fo6>
= 5 =9. an
)(M)
=6: =9 B (F)=9; =9 B (J)
=6; =9 B (H)=96 =9 B
3. Xko jyjg`tr`b eroquobjy `e mb ogojtr`b nrmtcbn cb m dmnbotcj ecogh `e = X cs mppr`xcdmtogy(M) 6; DKz (F) 6;9 DKz (J) 6.; NKz (H) 6; NKz
=9. M j`bhujtc`b r`h `e = d gobntk mbh = an dmss cs suspobhoh fy t`w vortcjmg wcros tkr`unk cts obhs. Mb oxtorbmg dmnbotcj ecogh `e 6X cs mppgcoh b`rdmg t` tko r`h. B`w tko jurrobt t` fo pmssoh tkr`unk tko r`h s` ms t` dmao tko tobsc`b cb tko wcros zor` cs <
QXmao n 4 =9 6ds
T
(M) 9.? M (F) =? M (J) ? M (H) =.? M
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Uchymdmbhcr Jgmssos
UDJ/Dmnbotcj Oeeojt ̀e Jurrobt 86 K^X->/Ykyscjs
==. M wcro Y]V cs fobt ms sk`wb cb ecnuro mbh cs pgmjoh cb m ronc`b `e ubce`rd dmnbotcj ecogh F. Xko gobntk `e Y] 4 ]V 4 g. M jurrobt C mdporo eg`ws tkr`unk tko wcro ms sk`wb. Xko dmnbctuho `e tko e`rjo`b Y] mbh ]V wcgg fo <(M) FCg, 9 (F) 6 FCg, 9
(J) 9, FCg (H) 9, 9
=6. ^kob m p`sctcvogy jkmrnoh pmrtcjgo obtors m ubce`rd dmnbotcj ecogh wctk ubce`rd vog`jcty, cts trmlojt`ry jmb fo <=. m strmcnkt gcbo 6. m jcrjgo :. m kogcx(M) = `bgy (F) = `r 6 (J) = `r : (H) Mby `bo `e =, 6 mbh :
=:. M jkmrno pmrtcjgo `e dmss d mbh jkmrno q obtors m ronc`b `e ubce`rd dmnbotcj ecogh F porpobhcjugmr `e cts vog`jcty v. Xko pmrtcjgocbctcmggy mt rost wms mjjogormtoh fy m p`tobtcmg hceeorobjo U (v`gts) foe`ro ct obtoroh tko ronc`b `e dmnbotcj ecogh. ^kmt cs tkohcmdotor `e tko jcrjugmr pmtk e`gg`woh fy tko jkmrnoh pmrtcjgo cb tko ronc`b `e dmnbotcj ecogh0
(M)6:
dU
q(F)
6 6dU
F q(J)
6dU F
q(H)
6 F dU
q F
=8. M tkcb egoxcfgo wcro `e gobntk G cs j`bbojtoh t` tw` mhlmjobt ecxohp`cbts mbh jmrrcos m jurrobt C cb tko jg`jawcso hcrojtc`b, ms sk`wb cbtko ecnuro. ^kob tko systod cs put cb m ubce`rd dmnbotcj ecogh `e strobntk F n`cbn cbt` tko pgmbo `e tko pmpor, tko wcro tmaos tko skmpo`e m jcrjgo. Xko tobsc`b cb tko wcro cs <
(M) CFG (F) CFG
ω(J)
6 CFG
ω(H)
8 CFG
ω
=?. Xw` g`bn pmrmggog wcros mro mt m hcstmbjo 6h mpmrt. Xkoy jmrry stomhy oqumg jurrobt eg`wcbn `ut `e tko pgmbo `e tko pmpor ms sk`wb.
Xko vmrcmtc`b `e tko dmnbotcj ecogh mg`bn tko gcbo [[ cs ncvob fy
(M) (F)
(J) (H)
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Uchymdmbhcr Jgmssos
UDJ/Dmnbotcj Oeeojt ̀e Jurrobt 8: K^X->/Ykyscjs
HMXO < XCDO < 89 Dcbutos DMVAZ < Q PPP /=?T XOZX J@HO < DMN Q?T
ZXMVX XCDO < OBH XCDO < XCDO XMAOB< YMVOBX–Z ZCNBMX_VO <
Xkcs tost j`btmcbs m t`tmg ̀ e =? @flojtcvo Xypo ]uostc̀ bs. Omjk quostc`b jmrrcos = dmra. Xkoro cs B@ BONMXCUO dmracbn.
Jk``so tko j`rrojt mgtorbmtcvo. @bgy `bo jk`cjo cs j`rrojt.
=. Xw` jcrjugmr j`bjobtrcj g``ps `e rmhcc = 669 mbh :9r jd r jd mro
pgmjo cb tko [ -S pgmbo ms sk`wb cb tko ecnuro. M jurrobt C 4 > M cseg`wcbn tkr`unk tkod. Xko dmnbotcj d`dobt `e tkcs g``p systod cs <
(M) 69 8. a Md (F) 6= ?. a Md
(J) 6= =. a Md (H) 6= :. l M d
6. :M `e jurrobt cs eg`wcbn cb m gcbomr j`bhujt`r kmvcbn m gobntk `e 89 jd. Xko j`bhujt`r cs pgmjoh cb m dmnbotcj ecogh `e strobntk
?99 nmuss mbh dmaos mb mbngo `e :9 wctk hcrojtc`b `e tko ecogh, Ct oxporcobjo m e`rjo `e dmnbctuho
(M)8: =9 B (F)
6:?=9 B (J)6: =9 B (H)
8: =9 B
:. M jcrjugmr g``p mbh m squmro g``p mro e`rdoh er`d tko smdo wcro mbh tko smdo jurrobt cs pmssoh tkr`unk tkod. Ecbh tko rmtc` `e tkocr hcp`go d`dobts.
(M) 8ω (F)8ω
(J)6ω
(H) 6ω
8. M tmbnobt nmgvmb`dotor cs j`bbojtoh hcrojtgy t` mb chomg fmttory. Ce tko budfor `e turbs cb tko j`cg cs h`ufgoh, tko hoegojtc`b wcgg.(M) cbjromso (F) hojromso (J) rodmcb ubjkmbnoh (H) octkor cbjromso ̀ r hojromso
?. Ztmtodobt = < ^kob rmhcus `e jcrjugmr g``p jmrrycbn jurrobt cs h`ufgoh, cts dmnbotcj d`dobt foj`dos e`ur tcdos.
Ztmtodobt 6 < Dmnbotcj d`dobt hopobhs `b mrom `e tko g``p.(M) Ztmtodobt-= cs Xruo, Ztmtodobt-6 cs Xruo mbh Ztmtodobt-6 cs m j`rrojt oxpgmbmtc`b e`r Ztmtodobt-=.(F) Ztmtodobt-= cs Xruo, Ztmtodobt-6 cs Xruo mbh Ztmtodobt-6 cs B@X m j`rrojt oxpgmbmtc`b e`r Ztmtodobt-=.(J) Ztmtodobt-= cs Xruo, Ztmtodobt-6 cs Emgso.(H) Ztmtodobt-= cs Emgso, Ztmtodobt-6 cs Xruo.
5. Ce m jurrobt cs pmssoh cb m sprcbn, ct <(M) nots j`dprossoh (F) not oxpmbhoh (J) `sjcggmtos (H) rodmcbs ubjkmbnoh
>. M jurrobt jmrrycbn strmcnkt wcro cs aopt mg`bn tko mxcs `e m jcrjugmr g``p jmrrycbn m jurrobt. Xko strmcnkt wcro.(M) wcgg oxort mb cbwmrh e`rjo `b tko jcrjugmr g``p(F) wcgg oxort mb `utwmrh e`rjo `b tko jcrjugmr g``p
(J) wcgg oxort m e`rjo `b tko jcrjugmr g``p pmrmggog t` ctsoge (H) wcgg b`t oxort mby e`rjo `b tko jcrjugmr g``p
;. Xko dmnbotcj hcp`go d`dobt `e m jurrobt g``p cs cbhopobhobt `e (M) dmnbotcj ecogh cb wkcjk ct cs gmycbn (F) budfor `e turbs(J) mrom `e tko g``p (H) jurrobt cb tko g``p
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Uchymdmbhcr Jgmssos
UDJ/Dmnbotcj Oeeojt ̀e Jurrobt 88 K^X->/Ykyscjs
3. Mb ogojtr`b cs d`vcbn cb mb ̀ rfct `e rmhcus V wctk m tcdo porc`h X ms sk`wb cb tko ecnuro. Xko dmnbotcj d`dobt pr`hujoh dmy foncvob fy <
(M)6 | o | M
D X
ω (F)
6 | o | M D
X
ω
(J)
| o | M
D X (H)
| o | M
D X | o | roprosobts tko dmnbctuho `e tko ogojtr`b jkmrno
=9. Xko rosugtmbt e`rjo `b tko jurrobt g``p Y]VZ huo t` m g`bn jurrobt jmrrycbn j`bhujt`r wcgg fo
(M)8=9 B
(F)8: 5 =9. B (J)
8= ; =9. B (H)8? =9 B
==. X`r`ch cs <(M) rcbn skmpoh jg`soh s`gob`ch (F) rojtmbnugmr skmpoh s`gob`ch(J) rcbn skmpoh `pob s`gob`ch (H) squmro skmpoh s`gob`ch
=6. Fc`t - Zmvmrt's gmw dmy fo roprosobtoh cb vojt`r er`d ms
(M) 9:8
hg r hF cr
¿
ω (F) 9
8hF c hg r ¿
ω (J) 9
68hg r hF c
r
¿
ω (H) 9
8hg r hF c
r
¿
ω
=:. Xko dmnbotcj ecogh `b tko mxcs `e m g`bn s`gob`ch kmvcbn b turbs por ubct gobntk mbh jmrrycbn m jurrobt c cs <
(M) 9bc¿ (F)6
9b c¿ (J)6
9bc¿ (H) B`bo `e tkoso
=8. M jurrobt c cs pmsscbn tkr`unk m strmcnkt j`bhujt`r `e cbecbcto gobntk. Xko dmnbotcj ecogh mt m p`cbt sctumtoh mt m hcstmbjo V er`d tkoj`bhujt`r cs <
(M) 96
c¿
ω(F) 9
66
c
V
¿
ω(J) 9
:6
c
V
¿
ω(H) 9
6
c
V
¿
ω
=?. ^kob jurrobt cs pmssoh tkr`unk m jcrjugmr wcro propmroh er`d m g`bn j`bhujtcbn wcro, tko dmnbotcj ecogh pr`hujoh mt cts jobtro cs F. B`w m g``p kmvcbn tw` turbs cs propmroh er`d tko smdo wcro mbh tko smdo jurrobt cs pmssoh tkr`unk ct. Xko dmnbotcj ecogh mt ctsjobtro wcgg fo <
(M) 8 F (F)8
F(J)
6
F(H) =5 F