Aieee 2011 Paper

8/2/2019 Aieee 2011 Paper

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NHFDSF MHKNHKCFI HIDDD – :288

uisip ts hp fppr=!!www+cfdnispzyczdsp+can! Rhod 8




uisip ts hp fppr=!!www+cfdnispzyczdsp+can! Rhod :

Nhvintn Nhzls= 8:2

Xtdspiak rhrdz eaznhp hk` Nhzliko scfdnd=

8+ Pfis xtdspiak rhrdz fhs 32 xtdspiaks ae dxthg wdiofp hod+ Dhcf xtdspiak is hggappd` 0&eatz) nhzls eaz dhcf

cazzdcp zdsraksd+

:+ · &akd eatzpf) ae paphg nhzls hggappd` pa dhcf xtdspiak i+d+' 8 nhzl wigg jd `d`tcpd` eaz ikìchpiko ikcazzdcpzdsraksd

3+ Ka `d`tcpiak ezan pfd paphg scazd wigg jd nh`d ie ka zdsraksd is ikìchpd` eaz hk ipdn ik pfd hkswdz sfddp+

0+ Pfdzd is akgy akd cazzdcp zdsraksd eaz dhcf xtdspiak+ Eiggiko tr nazd pfhk akd zdsraksd ik dhcf xtdspiak

wigg jd pzdhpd` hs wzako zdsraksd hk` nhzls eaz wzako zdsraksd wigg jd `d`tcpd` hccazìkogy+





8+ Hnako pfd eaggawiko pfd nhvintn cauhgdkp cfhzhcpdz is sfawk jy pfd canratk` &8) SkCg: &:) HgCg3 &3) NoCg: &0) EdCg:

Sag+ &:)Hccazìko pa Ehmhk’s ztgds' Ozdhpdz pfd cfhzod hk` snhggdz pfd si}d ae chpiak' ozdhpdz is pfd raghzi}iko

rawdz hk` ozdhpdz is pfd cauhgdkp khptzd ae pfd canratk`+

:+ Pfd rzdsdkcd az hjsdkcd ae fy`zavyg ozatr ak wficf chzjak hpan ae stohz ìeedzdkpihpds ZKHhk` @KH+&8) :

k`&:) 3

z`&3) 0

pf&0) 8

sp

Sag+ &8)

Ik ZKH' pfd stohz is κ √@ √ Zijasd' wfdzd hs ik @KH pfd Stohz is κ .@.:.`davy Zijasd+Pfdsd pwa ìeedz ik pfd rasipiak ae fy`zavyg ozatr hp :k` chzjak+ κ .@.:.`davy Zijasd ghcls fy`zavyg

ozatr hp :k`

chzjak+

3+ Pzicfgazahcdphg`dfy`d whs stjmdcpd` pa Chkki}}hza’s zdhcpiak jy tsiko KhAF+ Pfd nivptzd ae pfdrza`tcps cakphiks saìtn pzicfgazahcdphpd hk` hkapfdz canratk`+ Pfd apfdz canratk` is&8) Pzicfgazandpfhkag &:) :' :' :.Pzicfgazarzarhkag

&3) Cfgazaeazn &0) :' :' :.Pzicfgazadpfhkag

Sag+ &0)

KhAF3 3 3 ::CCg CFA CCg CAAKh,CCg CF AF ↚

Chkki}hza zdhcpiak is h ìsrzarazpiakhpiak zdhcpiak & Sdge avi`hpiak hk` sdge zd`tcpiak)

Akd hg`dfy`d nagdctgd is aviì}d` pa shgp ae pfd chzjavygic hci` hk` pfd apfdz akd is zd`tcd` pa

Hgcafag+ Sa pfd canratk` eaznd` is 3 :CCg CF AF

ITRHC Khnd ae pfd canratk` eaznd` is :' :' :' . Pzicfgaza dpfhkag





0+ Saìtn dpfavi`d fhs zdhcpd` wipf dpfhkayg cfgazi`d+ Pfd canratk` pfhp is rza`tcd` ik pfd hjaudzdhcpiak is&8) :.Jtphkakd &:) Dpfyg cfgazi`d &3) Dpfyg dpfhkahpd &0) @idpfyg dpfdz

Sag+ &3)

4+ Pfd zd`tcpiak rapdkpihg ae fy`zaodk fhge cdgg wigg jd kdohpiud ie &8) r&F:) > 8 hpn hk` \F,^ >8+2 N &:) r&F:) >: hpn hk` \F,^ >8+2 N

&3) r&F:) > : hpn hk` \F,^ > :+2 N &0) r& F:~ >8 hpn hk` \F,^ > :+2 N

Sag+ &:)

& ), .: F , : d F o:

↚

RF :

2 ,:D> D . 2+249gao Fdzd D .ud wfdk R ; FF:

, :F

wiggjd

2+2498 :> . gao

82: 8

> kdohpiud uhgtd

5+ Pfd spzakodsp hci` hnakosp pfd eaggawiko canratk`s is&8) FCAAF &:) CF3CF:CF&Cg)CA:F

&3) CgCF:CF:CF:CAAF &0) CF3CAAF

Sag+ &:)+ ,I deedcpiko ozatrs `dczdhsd hciìc khptzd hk` ikczdhsd jhsic khptzd

.I deedcpiko ozatrs ikczdhsd hciìc khptzd hk` `dczdhsd jhsic khptzd +

F is phldk hs zdedzdkcd+

Hs pfd – I deedcpiko ozatr nauds ehzpfdz ezan pfd etkcpiakhg ozatr' hci`c khptzd `dczdhsds+

Cg fhs – I deedcp hk` hglyg ozatrs fhud ,I deedcp+

Pfdzdeazd pfd akd wipf – I kdhzdz pfd etkcpiakhg ozatr wigg fhud nazd hciìc khptzd+





?+ Pfd `dozdd ae `issacihpiak &θ) ae h wdhl dgdcpzagypd' HvJy is zdghpd` pa uhk’p Faee ehcpaz &i) jy pfddvrzdssiak

&8)8

8

i

v yθ √

>, ,

&:)8

8

v y

iθ , √

>√

&3)8

8

v y

iθ , ,

>√

&0) & )

8

8

i

v yθ √

>, √

Sag+ &0)

& )i > 8. θ , kθ > 8 , θ k .8

8θ

8

i

k

√>

√

y v H J vH yJ

v y, √

↚ ,

k > v,y

Sa 8θ >8

i v y

√

, √

6+ bh’ hk` bj’ hzd uhk `dz _hhgs’ caksphkps eaz ohsds+ Cfgazikd is nazd dhsigy gixtdeid` pfhk dpfhkdjdchtsd&8) h hk` j eaz Cg: 7 hk` j eaz C:F5

&:) h eaz Cg: 7 h eaz C:F5 jtp j eaz Cg: ; j eaz C:F5

&3) h eaz Cg: ; h eaz C:F5 jtp j eaz Cg: 7 j eaz C:F5

&0) h hk` j eaz Cg: ; h hk` j eaz C:F5

Sag+ &3)

Dhsd ae gixtdehcpiak h

jθ

Eaz dpfhkd h > 4+45: G: Jhz nagd.8' j > 2+2536 G nagd.8

Eaz Cg: h > 5+4?9 G: Jhz nagd.8' j > 2+245: G nagd.8

9+ H udssdg hp 8222 L cakphiks CA: wipf h rzdsstzd ae 2+4 hpn+ Sand ae pfd CA: is cakudzpd` ikpa CAak pfd h``ipiak ae ozhrfipd+ Ie pfd paphg rzdsstzd hp dxtigijzitn is 2+6 hpn' pfd uhgtd ae L is&8) 3 hpn &:) 2+3 hpn &3) 2+86 hpn &0) 8+6 hpn

Sag+ &0)CA:&o) , C &s) ↚ :CA&o)

Ikipihg nagds r ADxtigijzitn nagds r.v :v

Paphg rzdsstzd hp dxtigijzitn > 2+6 hpn< Paphg ka+ae nagds > r , v

Pfdzdeazd

<

2+42+3

2+6

r k

r v

r v

θ

> ⇚ >

,

:

CAr

:

R 2+5 ß 2+5L > > > 8+6 hpn

CA 2+:





82+ Jazak chkkap eazn wficf akd ae pfd eaggawiko hkiaks 1

&8) .JF0

&:) & ).

J AF0

&3) .JA:

&0) 3.JE5

Sag+ &0)Sikcd Jazak fhs akgy eatz azjiphgs ik pfd uhgdkcd sfdgg & i+d+ :s' :rv' :ry % :r})' ip chk sfaw h nhvintn

cauhgdkcy ae eatz+Pfdzdeazd \JE5^

3. is kap rassijgd

88+ _ficf ae pfd eaggawiko ehcps hjatp pfd canrgdv & )Cz KF Cg3 35

is wzako 1

&8) Pfd canrgdv is rhzhnhokdpic&:) Pfd canrgdv is hk atpdz azjiphg canrgdv

&3) Pfd canrgdv oiuds wfipd rzdciriphpd wipf sigudz kipzhpd sagtpiak

&0) Pfd canrgdv ikuaguds `:sr

3fyjzi`i}hpiak hk` is acphfd`zhg ik sfhrd+

Sag+ &:)

Ik \Cz&KF3^5^3, 'Cz3,

fhs 3`3

cakeiotzhpiak+

Ka+ae tkrhizd` dgdcpzaks > 3+Sa ip is rhzhnhokdpic+

& )Cz KF Cg3 35

ikuaguds `:sr3 fyjzi`i}hpiak hk` ip is hk ikkdz azjiphg canrgdv jdchtsd ip is nhliko

tsd ae &k .8) ` azjiphgs i+d+' ikkdz ` azjiphgs+

8:+ Dpfygdkd ogycag is tsd` hs hk hkpiezdd}d ik h cag` cginhpd+ Nhss ae dpfygdkd ogycag wficf sfatg` jdh``d` pa 0 lo ae whpdz pa rzdudkp ip ezan ezdd}iko hp √5 aC wigg jd=\Le eaz whpdz > 8+65 L lo nag√8 ' hk` naghz nhss ae dpfygdkd ogycag > 5:o nag√8 )

&8) :20+32 o &:) 022+22 o &3) 320+52 o &0) 620+3: o

Sag+ &0)

:e e e

8 :

ß 8222∅P > L ß n > L ß

ß n

w

w

w8 % w: > _p ae sagudkp % sagtpd zdsrdcpiudgy' n: > nag+wp ae sagtpd

& )2 2 :e

ß 8222∅P > a . . 5 > 5 > 8+65 ß

0222 ß 5:

w

w: > 622o




uisip ts hp fppr=!!www+cfdnispzyczdsp+can! Rhod ?

83+ _ficf akd ae pfd eaggawiko az`dz zdrzdsdkps pfd cazzdcp sdxtdkcd ae pfd ikczdhsiko jhsic khptzd ae pfd oiudk avi`ds1

&8): : 3 :

NoA L A Hg A Kh A7 7 7 &:): : : 3

Kh A L A NoA Hg A7 7 7

&3): : : 3

L A Kh A Hg A NoA7 7 7 &0): 3 : :

Hg A NoA Kh A L A7 7 7

Sag+ &0)Odkdzhggy ndphgs eazn jhsic avi`ds+

Hczass h rdzia` ndphggic spzdkopf `dczdhsds % àwk pfd ozatr ikczdhsds+

Pfdzdeazd Jhsic khptzd ae avi`ds `dczdhsds hczass hrdzia` hk` ikczdhsds àwk pfd ozatr+

80+ Pfd zhpd ae h cfdnichg zdhcpiak àtjgds eaz dudzy 82aC zisd ae pdnrdzhptzd+ Ie pfd pdnrdzhptzd is

zhisd` jy 42

a

C' pfd zhpd ae pfd zdhcpiak ikczdhsds jy hjatp =&8) :0 pinds &:) 3: pinds &3) 50 pinds &0) 82 pinds

Sag+ &:)Oiudk' Pdnrdzhptzd cadeeicidkp : ´ > <

∅P

:82

8

l ´ >

l

42

82

4

:

8

:

> :

l 3:l

> >

3: l 8 > l :

i+d+' Zhpd ae zdhcpiak ikczdhsds jy hjatp 3: pinds+

84+ Pfd nhokdpic nandkp &srik akgy) ae :

KiCg0

√

is

&8) 4+05 JN &:) :+63 JN &3) 8+80 JN &0) 8+6: JN

Sag+ &:)

:

0 KiCg

√

+ =

& ):,:Ki ,0Cg Ki Cg

03sr

√√ ↚





Kicldg ik , : Avi`hpiak sphpd fhs 3`6

cakeiotzhpiak +

Cg.

jdiko wdhl eidg` giohk` àds kap eazcijgy rhiz tr dgdcpzaks+

Sa' pfis canrgdv fhs pdpzhfd`zhg odandpzy wipf sr3

fyjziìshpiak

Ka ae tkrhizd` dgdcpzaks > :

& )´ > k k,: JN

& )> : :,: :+6:JN>

85+ Pfd fyjziì}hpiak ae azjiphgs ae K hpan ik3

KA√ hk`0

KF , hzd zdsrdcpiudgy+

&8) : 3sr 'sr'sr &:) 3 :sr'sr 'sr &3) : 3sr 'sr 'sr &0) : 3sr'sr 'sr

Sag+ &8)





:

3

4 2 8K 3

:A sr√

, ,> > >

: 4 2 8K ::

A sr√ , √> > >

3

0

4 0 8KF 0

:sr,

, √> > >

8?+ Ik cakpdvp ae pfd ghkpfhkai`s' wficf ae pfd eaggawiko sphpdndkps is kap cazzdcp 1&8) Hgg pfd ndnjdzs dvfijip ,3 avi`hpiak sphpd+&:) Jdchtsd ae sinighz rzardzpids pfd sdrhzhpiak ae ghkpfhkai`s is kap dhsy+

&3) Huhighjigipy ae 0e dgdcpzaks zdstgps ik pfd eaznhpiak ae canratk`s ik ,0 sphpd eaz hgg pfd ndnjdzs ae

pfd sdzids+

&0) Pfdzd is h ozh`thg `dczdhsd ik pfd zhìi ae pfd ndnjdzs wipf ikczdhsiko hpanic ktnjdz ik pfd sdzids+

Sag+ &3)Pfd odkdzhg avi`hpiak sphpd ae ghkpfhki`ds is ,3+ Sand Ghkpfhkai`s dvfijip ,0 avi`hpiak sphpd hgsa+

86+ H 4+: naghg hxtdats sagtpiak ae ndpfyg hgcafag' CF3AF is strrgid`+ _fhp is pfd nagd ezhcpiak ae ndpfyg hgcafag ik pfd sagtpiak 1&8) 2+892 &:) 2+265 &3) 2+242 &0) 2+822

Sag+ &:)

Nagdezhcpiak ae sagtpd &v:) ik hxtdats sagtpiak >n

8222

n, * N

_fdzd * N is pfd nag wp ae sagudkp

4+:2+2265

82224+: ,

86

> >

89+ _ficf ae pfd eaggawiko sphpdndkp is wzako1

&8) Kipzaodk chkkap eazn `ς . rς jak`+&:) Sikogd K. K jak` is wdhldz pfhk pfd sikogd R – R jak`'

&3) K:A0 fhs pwa zdsakhkcd spztcptzds+

&0) Pfd sphjigipy ae fy`zi`ds ikczdhsds ezan KF3 pa JiF3 ik ozatr 84 ae pfd rdziaìc phjgd+

Sag+ &0)Sphjigipy ae 84 pf ozatr fy`zi`ds `dczdhsds àwk pfd ozatr i+d+' ezan KF3 pa JiF3 hs N.F jak` dkdzoy

`dczdhsds wfdzd N is h UH az 84 pf ozatr dgdndkp+





:2+ Pfd atpdz dgdcpzak cakeiotzhpiak ae O` &Hpanic Ka = 50) is=

&8)

6 2 :

0e 4` 5s &:)

0 0 :

0e 4` 5s &3)

? 8 :

0e 4` 5s &0)

3 4 :

0e 4` 5s

Sag+ &3)

Atpdz dgdcpzak cakeiotzhpiak ae Ohàgikitn > \ Vd ^ ? 8 :0e 4` 5s

:8+ _ficf ae pfd eaggawiko sphpdndkps zdohzìko stgrftz is ikcazzdcp1&8) Pfd uhratz hp :22

2C caksisps naspgy ae S6 zikos

&:) Hp 5222C pfd ohs nhikgy caksisps ae S: nagdctgds

&3) Pfd avi`hpiak sphpd ae stgrftz is kdudz gdss pfhk ,0 ik ips canratk`s&0) S: nagdctgd is rhzhnhokdpic

Sag+ &3)Nikintn avi`hpiak sphpd dvfijipd` jy ‖S’ is .: &Dv+ F:S)Ik odkdzhg Nhv rasipiud avi`hpiak sphpd dvfijipd` jy hk dgdndkp is dxthg pa ips Ozatr ka

Nhv kdohpiud avi`hpiak sphpd dvfijipd` jy hk dgdndkp is dxthg pa 6 – ozatr ka

::+ Pfd spztcptzd ae IE? is=&8) pzioakhg jiryzhni` &:) acphfd`zhg&3) rdkphoakhg jiryzhni` &0) sxthzd ryzhni`

Sag+ &3)Ik IE?' I tk`dzoads sr

3`

3fyjziì}hpiak+

:3+ A}akagysis ae hk azohkic canratk` oiuds eaznhg`dfy`d hs akd ae pfd rza`tcps+ Pfis cakeizns pfdrzdsdkcd ae =&8) h uikyg ozatr &:) hk isarzaryg ozatr

&3) hk hcdpygdkic pzirgd jak` &0) pwa dpfygdkic àtjgd jak`s

Sag+ &8)Uikyg ozatr ak a}akagasys oiud eaznhg`dfy`d

Ik A}akagysis ae uikyg ozatr C > C is zdrghcd` jy : C > A ozatrs+

+





:0+ H ohs hjsazjs h rfapak ae 344 kn hk` dnips hp pwa whudgdkopfs+ Ie akd ae pfd dnissiaks is hp562 kn' pfd apfdz is hp = &8) 3:4 kn &:) ?03 kn &3) 486 kn &0) 8234 kn

Sag+ &:)D hjsazjd` > D dnippd`

> &D8 , D:) dnippd`

fc> ,

μ μ & ) μ & )hjsazjd` 8 :

fc fc

dnippd` dnippd`

8 8 8> ,μ μ & ) μ & )

hjsazjd` 8 :dnippd` dnippd`

8 8 8

344 562:

μ ⇚ > ,

?0:+6 ?03 kn:

μ ⇚ > ≄

:4+ Sigudz Nizzaz pdsp is oiudk jy wficf akd ae pfd eaggawiko canratk`s1&8) Hcdpakd &:) Eaznhg`dfy`d &3) Jdk}arfdkakd &0) Hcdphg`dfy`d

Sag+ &:' 0)Eaznhg`dfy`d hk` Hcdphg`dfy`d chk jd aviì}d` jy paggdk’s zdhodkp pa oiud sigudz nizzaz+

Paggdk’s pdsp is oiudk jy hgg hgirfhpic hk` hzanhpic hg`dfy`ds hk` dudk jy θ – Fy`zavy ldpakds+

Paggdks/ zdhodkp is hgsa h pdsp eaz hglykds wipf h pzirgd jak` ik pfd 8.rasipiak+ H ydggaw rzdciriphpd ae

pfd ndphg hcdpygi`d is eaznd` ik pfis chsd+

Japf Paggdks/ zdhodkp hk` Edfgiko’s zdhodkp hgsa oiud rasipiud pdsp wipf Eaznic hci` &ndpfhkaic hci` .FCAAF)' wficf is etggy aviìsd` pa whpdz hk` CA:

:5+ _ficf ae pfd eaggawiko zdhodkps nhy jd tsd` pa ìspikotisf jdpwddk rfdkag hk` jdk}aic hci`1&8) Paggdk’s zdhodkp &:) Nagiscf zdhodkp&3) Kdtpzhg EdCg3 &0) Hxtdats KhAF

Sag+ &3)Rfdkag oiuds uiagdp cagatzd` canrgdv wipf kdtpzhg EdCg3+ Rfdkag eazn cfhzhcpdzispic cagatzd` izak

canrgdvds wfdk pzdhpd` wipf kdtpzhg edzzic cfgazi`d sagtpiak+ Do+ Rfdkag % zdsazcikag . uiagdp cagatz'

chpdcfag.ozddk dpc+




uisip ts hp fppr=!!www+cfdnispzyczdsp+can! Rhod 8:

:?+ Rfdkag is fdhpd` wipf h sagtpiak ae nivptzd ae LJz hk` LJzA3+ Pfd nhmaz rza`tcp ajphikd` ik pfdhjaud zdhcpiak is&8) 3.Jzanarfdkag &:) 0.Jzanarfdkag&3) :' 0' 5. Pzijzanarfdkag &0) :.Jzanarfdkag

Sag+ &3)Ik hciìc ndìtn' LJz hk` LJzA3 rza`tcds Jz: +

LJzA3 , 4 LJz , 3 F:SA0 ↚ 3 L:SA0 , 3 Jz: , 3 F:A

Rfdkag zdhcps wipf Jz: &hx) pa oiud :' 0' 5.Pzikipzarfdkag+

:: F A

3Jz, ↚

AF

3FJz,

AF

Jz

Jz

Jz

:'0'5.pzijzanarfdkag

:6+ Ik h ehcd cdkpzd` ctjic ghppicd' hpan H acctrids pfd cazkdz rasipiaks hk` hpan J acctrids pfd ehcdcdkpzd rasipiaks+ Ie akd hpan ae J is nissiko ezan akd ae pfd ehcd cdkpzd` raikps' pfd eazntgh ae pfdcanratk` is&8) HJ: &:) H:J3 &3) H:J4 &0) H:J

Sag+ &3)H ctjd cakphiks 5 ehcds' 6 cazkdzs hk` 8: dòds

Deedcpiud ka+ ae H hpans >

8

6 86 ß > & 6 tkip cdggs sfhzd h cazkdz rhzpicgd)

Deedcpiud ka+ ae J hpans >8

4 4 ! ::

ß > & Atp ae 5' akd rhzpicgd is nissiko )&: tkip cdggs sfhzd h ehcd

cdkpzd` rhzpicgd)

Pfdzdeazd eazntgh ae pfd canratk` is H:J4

:9+ Pfd dkpzary cfhkod ikuagud` ik pfd isapfdznhg zdudzsijgd dvrhksiak ae : nagds ae hk i`dhg ohsezan h uagtnd ae 82 `n3 pa h uagtnd ae 822 `n

3hp :?

aC is=

&8) .8 .834+6M nag L &:) .8 .83:+3M nag L &3) .8 .80:+3M nag L &0) .8 .836+3M nag L

Sag+ &0)Eaz isapfdznhg zdudzsijgd dvrhksiak ae hk i`dhg ohs'

:

8

U∅S > :+323k Z gao

U

822∅S > :+323 v : v 6+380 v gao

82

> . 8 . 836+3 M nag L





32+ I`dkpiey pfd canratk` pfhp dvfijips phtpandzisn+

&8) Ghcpic hci` &:) :.Rdkphkakd &3) Rfdkag &0) :. Jtpdkd

Sag+ :Rdkphk.:.akd chk dvfijip phtpandzisn

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