errors in measurement theory_e

7/21/2019 Errors in Measurement Theory_E

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"keogslcukerplysgfs.go" 5

MPPAPV GO KMEV]PMKMOX5 . MPPAPV GO KMEV]PMKMOX

Xa imt sakm avmrvgmw an mrrar, dmest fauot eoh sgiogngfeot ngiurms, dmts faosghmr tlm mxekpdm igvmo `mdaw.

Vuppasm wm levm ta kmesurm tlm dmoitl an e rah. Law feo wm!

(e) Dmts usm e fk. sfedm: (e sfedm ao wlgfl aody fk. kercs erm tlmrm)

Ym wgdd kmesurm dmoitl ; 8 fk.

Edtlauil tlm dmoitl wgdd `m e `gt karm tleo 8 , `ut wm feooat sey gts dmoitl ta `m 8.5 fk ar 8.0 fk.,

es tlm sfedm feo kmesurm upta fks aody , oat fdasmr tleo tlet.

* Gt (tlgs sfedm) feo kmesurm upta fks effurefy aody.

* sa wmdd sey tlet gts d m es t f a u o t gs 5 fk.

(`) Dmts usm eo kk sfedm : (e sfedm ao wlgfl kk. kercs erm tlmrm)

Ym wgdd kmesurm dmoitl ; 8.0 fk. , wlgfl gs e karm fdasmr kmesurmkmot. Lmrm edsa gn wm a`smrvm fdasmdy,wmdd ngoh tlet tlm dmoitl gs e `gt karm tleo 8.0, `ut wm feooat sey gts dmoitl ta `m 8.05, ar 8.00, ar 8.02 es

tlgs sfedm feo kmesurm upta 2.5 fks (5 kk) aody, oat fdasmr tleo tlet.

* Gt (tlgs sfedm) feo kmesurm upta 2.5 fk effurefy

Gts d m es t f a u o t gs 2.5 fk.

Kex u o f m r t eg o t y go feo `m ; 2.5fkK ex p a s s g ` d m m r r a r go feo `m ; 2.5fkKmesurmkmot an dmoitl ; 8.0 fk. les twa s g i o g n g f e o t n g i u r m s 3 8 eoh 0 , go wlgfl 8 gs e`sadutmdy farrmft ,

eoh 0 gs rmesaoe`dy farrmft (Hau`tnud) `mfeusm uofmrtegoty an 2.5 fk gs tlmrm.

(f) Ym feo usm ^mrogmr feddgpmrs : ( wlgfl feo kmesurm karm fdasmdy , upta 2.25 fk )

Xlmo wmdd kmesurm dmoitl d ; 8.06 fk wlgfl gs karm fdasmr kmesurmkmot.

* Gt feo kmesurm upta 2.25 fk effurefy

Dmest fauot ; 2.25 fk Kex u o f m r t eg o t y go dfeo `m ; 2.25fk

Kex passg`dm mrrar go d feo `m ; 2.25fk

Kmesurmkmot an dmoitl ; 8.06 fk. les tlrmms g i o g n g f e o t n g i u r m s 3 8 , 0 eoh 6, go wlgfl 8 eoh 0 erm

e`sadutmdy farrmft , eoh 6 gs rmesaoe`dm farrmft (Hau tnud) `mfeusm uofmrtegoty an 2.25 fk gs tlmrm.


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Xa imt nurtlmr karm fdasmr kmesurmkmot :-

(h) Ym feo usm Vfrmw Ieuim : ( wlgfl feo kmesurm karm fdasmdy , upta 2.225 fk )

wmdd kmesurm dmoitl d ; 8.068 fk.

* Kex passg`dm uofmrtegoty (mrrar) go dfeo `m ; 2.225 fk

* dmoitl ; 8.068 fk. les naurs g i o g n g f e o t n g i u r m s 3 8 , 0 , 6 eoh 8.

e`sadutmdy e`sadutmdy e`sadutmdy Pmesaoe`dyfarrmft farrmft farrmft farrmft

Xa imt nurtlmr karm fdasmr kmesurmkmot

(m) Ym feo ]sm kgfrasfapm :

wmdd kmesurm dmoitl d ; 8.0680 fk.

* Kex passg`dm uofmrtegoty (mrrar) go d feo `m ; 2.2225fk

* dmoitl ; 8.0680fk. les ngvms g i o g n g f e o t n g i u r m s 3 8 , 0 , 6 ,8 eoh 0

0 . VGIOGNGFEOX NGI]PMVNrak tlm e`avm mxekpdm , wm feo faofduhm tlet ,go e kmesurmh queotgty,

Vgiogngfeot ngiurms erm ; Ngiurms wlgfl erm e`sadutmdy farrmft + Xlm ngrst uofmrtego ngiurm

0.5 Fakkao rudms an fauotg oi s gio gngfeot ngiurms :Pudm 5 :

Edd oao-zmra hgigts erm sgiogngfeot

m.g. 506.=> les ngvm V.N.

Pudm 0 :

Edd zmras affurrgoi `mtwmmo twa oao-zmras hgigts erm sgiogngfeot (a`vgausdy)

m.g. 5062.2= les sgx V.N.

Pudm 6 :

; 6.= fk ; 6.=2 fk ; 6.=22 fk

2.5 fk effurefy 2.25 fk effurefy 2.225 fk effurefy

fdasmr! karm fdasmr!

dgms `mtwmmo (6.877 6.=25) dgms `mtwmmo(6.8 6.>) dgms `mtwmmo (6.87 6.=5)

Va tregdgoi zmrams entmr hmfgked pdefm erm sgiogngfeot (Vlaws tlm nurtlmr effurefy)

Aofm e kmesurmkmot gs haom, sgiogngfeot ngiurms wgdd `m hmfghmh effarhgoi ta fdasmomss an kmesurmkmot.

Oaw gn wm weot ta hgspdey tlm kmesurmkmot go sakm hgnnmrmot uogts, tlm V.N. slaudhot fleoim (V.N. hmpmohs

aody ao effurefy an kmesurmkmot)

Ouk `mr an V .N. gs edweys f aos mrvmh, fleoim an uo gts feoo at fl eoim V .N.


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Vuppasm kmesurmkmot wes haom usgoi kk sfedm, eoh wm imt;


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Vadutgao : V.N. ; Xwa, tlm tregdgoi zmras erm oat sgiogngfeot.

Mxekpdm 8. Fauot tated ouk`mr an V.N. go 622.22

Vadutgao : V.N. ; Ngvm, tregdgoi zmras entmr hmfgked pagot erm sgiogngfeot.

Mxekpdm =. Fauot tated ouk`mr an V.N. go =.226202

Vadutgao : V.N. ; Vmvmo, tlm tregdgoi zmras entmr hmfgked pdefm erm sgiogngfeot.

Mxekpdm >. Fauot tated ouk`mr an V.N. go >.202 52

06

Vadutgao : V.N. ; Naur 3 >, 2, 0, 2 3 rmkegogoi 06 zmras erm oat sgiogngfeot.

Mxekpdm 9. Fauot tated ouk`mr an V.N. go 5.>2 5257

Vadutgao : V.N. ; Xlrmm 3 5, >, 2 3 rmkegogoi 57 zmras erm oat sgiogngfeot.

0 . 0 Apmretgaos effarhgoi ta sgiogngfeot ngiurms:Oaw dmts smm law ta ha ergtlkmtgf apmretgaos gm. ehhgtgao, su`treftgao, kudtgpdgfetgao eoh

hgvgsgao effarhgoi ta sgiogngfeot ngiurms

(e) Ehhgtgaosu`treftgaoNar tlgs, dmts faosghmr tlm mxekpdm igvmo `mdaw.

Go e sgkpdm pmohuduk, dmoitl an tlm tlrmeh gs kmesurmh ( nrak kk sfedm) es

9=.8 fk. eoh tlm rehgus an tlm `a` gs kmesurmh (nrak vmrogmr) es 0.=6 fk.

Ngohmq

;+ r

; 9=.8 fk; 9=.81 fk

r ; 0.=6 fk

gs coawo upta 2.5 fk( ngrst hmfgked pdefm) aody. Ym haot coaw wlet gs et tlm omxt hmfgked pdefm.

Va wm feo wrgtm;9=.8 fk ; 9=.81 fk eoh tlm rehgus r ; 0.=6 fk .

Gn wm ehh eoh r, wmhaot coawwlgfl ouk`mr wgdd`m ehhmh wgtl 6. Va wmlevm ta dmevm tlet pasgtgao.

mq

; 9=.81 + 0.=6 ; 99.71 fk ; 99.7 fk

Pudms nar Ehhgtgaosu`treftgao : (`esmh ao tlm prmvgaus mxekpdm)* Ngrst rauoh ann edd queotgtgms ta tlm hmfgked pdefm an dmest effuretm queotgty.

* Xlmoha tlm ehhgtgao/su`treftgao go oarked keoomr

m.g.

Pudms nar KudtgpdyHgvgsgaoVuppasm wm levm ta kudtgpdy 0.55 x 5.0 ; 0.55 1 x 5.0 1

0 . 55 1

x 5.0 1

1 1 1 1

8 0 0 1 x

0 5 5 1 x x

0.= 1 1 1 1 ; 0.=

Va eoswmr wgdd fakm go dmest sgiogngfeot ngiurms aut an tlm twa ouk`mrs.

Kudtgpdy hgvghm go oarked keoomr.

Pauoh ann tlm eoswmr ta tlm wmecmst dgoc (ouk`mr levgoi dmest V.N.)

Mxekpdm


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"keogslcukerplysgfs.go" =

Xwa V.N. Xwa V.N. Xwa V.N.

; 5.90< 52> k6

Pauoh ann ta 0 V.N..

; 5.9 52> k6 Eos.

Pudms an Pauohgoi ann Gn rmkave`dm hgigt gs dmss tleo = (=2%) 3 hrap gt.

89.0.=

6 VN

0 VN rauoh annta 0 V.N.

>0

Mxekpdm 52. ]sgoi sfrmw ieuim rehgus an wgrm wes nauoh ta `m 0.=2 kk. Xlm dmoitl an wgrm nauoh `y kk. sfedm

gs =2.2 fk. Gn kess an wgrm wes kmesurmh es 0= ik, tlm hmosgty an tlm wgrm go farrmft V.N. wgdd `m

(usm ; 6.58 mxeftdy )

Vadutgao : ;

0r

k

;

; 0.=8>=

6. DMEVX FA]OX :Ym levm stuhgmh (nrak peim 5 ) tlet oa kmesurmkmot gs pmrnmft. Mvmry gostrukmot feo kmesurm upta e

fmrtego effurefy3 feddmh dmest fauot.

Dmest fauot : XlmVkeddmst queotgty eo gostukmot feo kmesurm

kk. sfedm ^mrogmr Vfrmw ieuim stapwetfl

Xmkp tlmrkakmtmrD.F. ; 5FD.F. ; 5 kk D.F. ;2.5 kk D.F. ; 2.25 kk

D.F. ; 2.5 smf


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RLWVGFV

"keogslcukerplysgfs.go" >

8 . RMPKGVVG@DM MPPAPMrrar go kmesurmkmot hum ta tlm dgkgtetgao (dmest fauot) an tlm gostrukmot, gs feddmh pmrkgssg`dm mrrar.

Nrak kk sfedmwm feo kmesurm upta 5 kk effurefy (dmest fauot). Nrak tlgs wm wgdd imt kmesurmkmotdgcm ; 68 kk

Kex uofmrtegoty feo `m 5 kk.

Kex pmrkgssg`dm mrrar () ; 5 kk.@ut gn nrak eoy atlmr gostrukmot, wm imt ; 68.= kk tlmo kex pmrkgssg`dm mrrar () ; 2.5 kk

eoh gn nrak e karm effuretm gostrukmot, wm imt ; 68.=09 kk tlmo kex pmrkgssg dm mrrar () ; 2.225 kk; pdefm vedum an dest ouk`mr

Kex pmrkgssg` dm mrrar go e kmesurmh q ueotgty gs ; pdefm vedum an tlm dest ouk `mr

=. KE\. RMPKGVVG@DM MPPAP GO PMV]DX H]M XA MPPAP GO MEFLKMEV]PE@DM U]EOXGXW :

Dmt Pmsudt n(x, y) faotegos twa kmesure`dm queotgty x eoh y

Dmt mrrar go x gs ; x g.m. x (x x, x +x)

mrrar go y gs ; y g.m. y (y y, y +y)

Fesm : () Gn n(x, y) ; x + yhn ; hx + hy

mrrar go n ;n ; x ykex passg`dm mrrar go n ; (n)

kex; kex an (x y)

(n)kex

;x + y

Fesm : ( ) Gn n ; x yhn ; hx hy

(n) ; xykex passg`dm mrrar go n ; (n)

kex; kex an (xy)

(n)kex;x + yNar imtt goi kexgku k pmrkgs sg`dm mrrar , sgio slau dh `m ehjustmh, sa tlet mrrars imt ehhmh up ta i gvmkexgkuk mnnmft

g.m. n ; 0x 6y - z

(n)kex

; 0x + 6y +z

Mxekpdm 55. Go rmsaoeofm tu`m mxp. wm ngoh 5

; 0=.2 fk eoh 0

; 9=.2 fk. Gn tlmrm gs oa mrrar go nrmqumofy.Ylet

wgdd `m kex pmrkgssg`dm mrrar go spmmh an sauoh (tecm n2; 60= Lz.)

^ ; 0n2(

0

5)

Vadutgao : (h^) ; 0n2(h

0 h

5)

(^)kex

; kex an S0n2

(0

0

[ ; 0n2

(0

+5

)

5; 0=.2 fk

5; 2.5 fk (pdefm vedum an dest ouk`mr )

5; 9=.2 fk

0; 2.5 fk (pdefm vedum an dest ouk`mr )

Va kex pmrkgssg dm mrrar go spmmh an sauoh (^)kex

; 0(60=Lz) (2.5 fk + 2.5 fk) ; 5.6 k/s

êdum an ^ ; 0n2(

0

5) ; 0(60=Lz) (9=.2 fk - 0=.2 fk) ; 60= k/s

sa ^ ; ( 60= 5.6 ) k/s

Fesm ( ) Gn n(x, y, z) ; (faosteot) xey`zfta sfettmr edd tlm tmrks, Dmts tecm dai ao àtl sghms

o n ;o(faosteot) + eo x + ò y + fo z

Hgnnmrmotgetgoi àtl sghms

n

hn; 2 + e

x

hx+ `

y

hy+ f

z

hz


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n

n; e

x

x `

y

y f

z

z

kexn

n

; kex an ( e

x

x `

y

y f

z

z)

g.m. n ; 5= x0 y6/0 z=

n

hn

; 2 + 0 x

hx

0

6

y

hy

= z

hz

n

n; 0

x

x

0

6

y

y =

z

z

kexn

n

; kex an ( 0

x

x

0

6

y

y =

z

z)

kexn

n

; 0

x

x+

0

6

y

y+=

z

z

s gio sla udh `m ehjustmh, sa tlet mrrars imt ehhmh up

Mxekpdm 50. Gn kmesurmh vedum an rmsgsteofm P ; 5.2= , wgrm hgekmtmr h ; 2.>2 kk, eoh dmoitl ; 9=.6 fk, tlmo

ngoh kex. pmrkgssg`dm mrrar go rmsgstgvgty ;

8

hP

0

Vadutgao :kex

;P

P+ 0

h

h+

P ; 5.2= P ; 2.25(dmest fauot)h ; 2.>2 kk h ; 2.25 kk (dmest fauot); 9=.6 ; 2.5 fk (dmest fauot)

kex

;

2=.5

25.2+ 0

kk>2.2

kk25.2+

fk6.9=

fk5.2 522 ; 8 %.

Mxekpdm 56. Go alks dew mxpmrgkmot, patmotged hrap efrass e rmsgsteofm wes kmesurmh es v ; =.2 vadt eoh

furrmot wes kmesurmh es g ; 0.22 ekp. Ngoh tlm kexgkuk pmrkgssg`dm mrrar go rmsgsteofm.

Vadutgao : P ;g

v; v g5

kexP

P

;

v

v+

g

g

v ; =.2 vadt v ; 2.5 vadtg ; 0.22 ekp g ; 2.25 ekp

%kexP

P

;

22.0

25.2

2.=

5.2 522 % ; 0.= %

vedum an P nrak tlm a`smrvetgao P ;g

v; =.0

22.0

2.=

Va wm feo wrgtm P ; ( 0.= 0.=% )

Mxekpdm 58. Go Vmerdms mxp ta ngoh Wauois kahudus, tlm hgekmtmr an wgrm gs kmesurmh es H ; 2.2= fk, dmoitl an

wgrm gs D ; 50= fk, eoh wlmo e wmgilt, k ; 02.2 ci gs put, mxtmosgao go wgrm wes nauoh ta `m 2.522

fk. Ngoh kexgkuk pmrkgssg`dm mrrar go yauois kahudus (W).

Vadutgao :8/h

ki0

; W (

x ) W ;xh)8/(

ki0


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"keogslcukerplysgfs.go" . Go sakm a`smrvetgaos, vedum an i erm fakgoi es

7.


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sa i ; ( 7.


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RLWVGFV


a`jmft

2

(kk)

=

82

=2

Ngxmh JewKego sfedm

Fgrfuder sfedm

kave`dm Jew(spgohdm)

Xlgk`dm

Ngxmh nrekmNgxmh sdmmvm

Xlm a jmft ta `m kmesurmh gs put `mtwmmo tlm jews. Xlm sdmmvm gs laddaw pert, ngxmh wgtl tlm nrekm eoh kego

sfedm gs prgotmh ao gt.

Xlm spgohdm eoh tlgk`dm erm wmdhmh, eoh kavm taimtlmr `y kmeos an e sfrmw. Xlm fgrfuder sfedm gs prgotmh ao

tlm tlgk`dm es slawo. Gt imomreddy faosgsts an 522 hgvgsgaos (sakmtgkm =2 hgvgsgaos edsa)

Xlm kego sfedm les kk kercs (Vakmtgkms gt edsa les 5/0 kk kercs `mdaw kk kercs.)

(]sueddy gn pgtfl an tlm sfrmw ieuim gs 5kk tlmo tlmrm erm 5kk kercs ao kego sfedm eoh gn pgtfl gs5/0 kk tlmo tlmrm erm 5/0 kk kercs edsa)

Xlgs gostrukmot feo rmeh upta 2.25 kk (52k) effurefy tlet gs wly gt gs feddmh kgfrakmtmr

Mxekpdm 59. Pmeh tlm oarked sfrmwieuim

*Kego sfedm les aody kk kercs.

*Fgrfuder sfedm les 522 hgvgsgaos

*Go fakpdmtm ratetgao, tlm sfrmw ehveofms `y 5 kk.

Vadutgao :

2

(kk)

=

>2

9252

A`jmft tlgfcomss ; 55 kk + >= (522

5 kk

522

5 kk)

; 55.>= kk

Vado:

Mxekpdm 5.= kk + 86 (=2

5/0 kk )

; >.76 kk

A`jmft tlgfcomss ; >.= - - - -

>.=

Vado:

Mxekpdm 57. Pmeh tlm sfrmwieuim slawo `mddaw:

*Kego sfedm les0

5kk kercs.


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*Fgrfuder sfedm les =2 hgvgsgao.

*Go fakpdmtm ratetgao, tlm sfrmw ehveofms `y0

5kk.

Vadutgao :

Mxekpdm 02. E wgrm an rmsgsteofm P ; 522.2 eoh dmoitl d ; =2.2 fk gs put `mtwmmo tlm jews an sfrmw ieuim. Gtsrmehgoi gs slawo go ngiurm. Ngoh gts rmsgstgvgty go farrmft sgiogngfeot ngiurms eoh kexgkuk pmrkgssg dm

mrrar go(rmsgstgvgty).Vadutgao :

2 52

A`jmft tlgfcomss ; < kk + 80 5/0 kk

=2

82

2

;


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Zmra Mrrar :Gn tlmrm gs oa a`jmft `mtwmmo tlm jews (g.m. jews erm go faoteft), tlm sfrmwieuim slaudh igvm zmra rmehgoi. @ut

hum ta mxtre ketmrged ao jews, mvmo gn tlmrm gs oa a`jmft, gt igvms sakm mxfmss rmehgoi. Xlgs mxfmss.

Pmehgoi gs feddmhzmra mrrar :

Mxekpdm 06. Ngoh tlm tlgfcomss an tlm wgrm.

Vadutgao : Mxfmss rmehgoi (Zmra mrrar) ; 2.26 kk Gt gs igvgoi 9.>9 kk go wlgfl tlmrm gs 2.26 kk mxfmssrmehgoi , wlgfl les ta `m rmkavmh ( su`treftmh)

sa eftued rmehgoi ; 9.>9 - 2.26 ; 9.>8 kk

Mxekpdm 08. Ngoh tlm tlgfcomss an tlm wgrm.

Vadutgao : Mxfmss rmehgoi (Zmra mrrar) ; - 2.29kk Gt gs igvgoi 9.7= kk go wlgfl tlmrm gs -2.29 kk mxfmssrmehgoi , wlgfl les ta `m rmkavmh ( su`treftmh)

sa eftued rmehgoi ; 9.7= -(- 2.29) ;


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RLWVGFV


M\RMPGKMOX # 0

^mrogmr feddgpmrs

Gt gs usmh ta kmesurm effuretmdy upta 2.5 kk.

2 kk 52 02 62 82 =2

2.5 kk

*Ao tlm uppmr pdetm, kego sfedm gs prgotmh wlgfl gs sgkpdy eo kk sfedm.

*Ao tlm dawmr pdetm , vmrogmr sfedm gs prgotmh, wlgfl gs e `gt fakprmssmh sfedm. Gts aom pert gs an 2.7 kk

Xlm a`jmft wlgfl gs ta `m kmesurmh , gs ngttmh `mtwmmo tlm jews es slawo.

Law ta rmeh ^mrogmr Feddgpmrs:

Oaw dmts smm Law tlm sdgilt hgnnmrmofm `mtwmmo 5 KVH eoh 5 ^VH rmndmfts es dmest fauot


15/44

RLWVGFV

"keogslcukerplysgfs.go" 5=

Pmqugrmh dmoitl ; 56 kk + x ; 1

et pagot E , kego sfedm eoh vmrogmr sfedm erm ketflgoi

sa dmoitl AE edaoi kego Vfedm ; dmoitl AE edaoi mrogmr Vfedm

56 kk +6 (Kego sfedm hgvgsgao) ; ( 56 kk + x ) + 6 (vmrogmr Vfedm hgvgsgao )

Imt 56 kk + x ; 56 kk + 6 (Kego sfedm hgvgsgao - vmrogmr Vfedm hgvgsgao)

; 56 kk + 6 ( 5 kk - 2.7 kk )

; 56 kk + 6 (2.5 kk ) ; 56.6 kk

rmehgoi

sfedm

kego

+

rmehgoi

sfedm

vmrogmr

fauot

Dmest

(Kego sfedm hgvgsgao - vmrogmr Vfedm hgvgsgao)

Lmofm tlm sdgilt hgnnmrmofm `mtwmmo 5 KVH (5 kk ) eoh 5 ^VH (2.7 kk ) rmndmfts es dmest fauot (2.5 kk)

feddgpmrsvmrogmranehgoiPm

a`jmftanXlgfcoms

;

rmehgoi

sfedm

kego

+

rmehgoi

sfedm

vmrogmr

fauotDmest

Mxekpdm 0=. Pmeh tlm vmrogmr


16/44

RLWVGFV

"keogslcukerplysgfs.go" 5>

Vadutgao: Xlgfcomss an tlm a`jmft ; (kego sfedm rmehgoi) + (vmrogmr sfedm Pmehgoi) (dmest fauot )

wlmrm dmest fauot ; (Kego sfedm hgvgsgao - vmrogmr Vfedm hgvgsgao)

; 5 kk - 2.7 kk (nrak ngiurm)

; 2.5 kk

Va tlgfcomss an tlm a`jmft ; 5= kk + (>) (2.5kk ) ; 5=.> kk Eos.

Mxekpdm 0>. Pmeh tlm spmfged typm an vmrogmr

Vadutgao : Xlgfcomss an tlm a jmft ; (kego sfedm rmehgoi) + (vmrogmr sfedm Pmehgoi) (dmest fauot )

wlmrm dmest fauot ; (Kego sfedm hgvgsgao - vmrogmr Vfedm hgvgsgao)

; 5 kk - 57/02 kk ( nrak ngi.)

; 2.2= kk

Va tlgfcomss an tlm a`jmft ; 56 kk + (50) (2.2=kk )

; 56.>2 kk Eos.

Zmra Mrrar:

Gn tlmrm gs oa a`jmft mtwmmo tlm jews (gm. jews erm go faoteft ),tlm vmrogmr slaudh igvm zmra rmehgoi.@ut hum ta sakm mxtre ketmrged ao jews, mvmo gn tlmrm gs oa a`jmft `mtwmmo tlm jews, gt igvms sakm

mxfmss Pmehgoi . Xlgs mxfmss rmehgoi gs feddmh zmra mrrar


17/44

RLWVGFV


Mxekpdm 09. Ngoh tlm tlgfcomss an tlm a`jmft usgoi tlm hmnmftmh vmrogmr feddgpmrs


18/44


19/44

RLWVGFV


Mxekpdm 07. Xlm kego sfedm an e vmrogmr feddgpmrs rmehs 52 kk go 52 hgvgsgaos. 52 hgvgsgaos an ^mrogmr sfedm

fagofghm wgtl 7 hgvgsgaos an tlm kego sfedm. Ylmo tlm twa jews an tlm feddgpmrs taufl mefl atlmr, tlm

ngntl hgvgsgao an tlm vmrogmr fagofghms wgtl 7 kego sfedm hgvgsgaos eoh tlm zmra an tlm vmrogmr gs ta tlm

rgilt an zmra an kego sfedm. Ylmo e fydgohmr gs tgiltdy pdefmh `mtwmmo tlm twa jews, tlm zmra an

vmrogmr sfedm dgms sdgiltdy ta tlm dmnt an 6.0 fk eoh tlm naurtl vmrogmr hgvgsgao fagofghms wgtl e kego

sfedm hgvgsgao. Xlm hgekmtmr an tlm fydgohmr gs.

Vadutgao : Zmra mrrar ; 2.= kk ; 2.2= fk.

A`smrvmh rmehgoi an fydgohmr hgekmtmr ; 6.5 fk + (8) (2.25 fk).

; 6.58 fk

Eftued tlgfcomss an fydgohmr ; (6.58) (2.2=).

; 6.27 fk Eos.

Mxekpdm 62. Go tlm prmvgaus qumstgao gn tlm dmoitl an tlm fydgohmr gs kmesurmh es 0= kk, eoh kess an tlm fydgohmr

gs kmesurmh es =2.2 ik, ngoh tlm hmosgty an tlm fydgohmr go prapmr sgiogngfeot ngiurms.

Vadutgao :l)8/h(

k0

; 650 fk)520=()0/27.6(58.6

ik)2.=2(

; 0.9 ik/fk6 (go twa V.N.) Eos.

M\RMPGKMOX # 6

Hmtmrkgogoi tlm vedum an i usgoi e sgkpdm pmohuduk

////////

D

r

Go tlgs mxp. e skedd splmrgfed à` gs leoimh wgtl e fattao tlrmeh. Xlgs erreoimkmot gs feddmh sekpdm pmohu-

duk. Xlm à` gs hgspdefmh sdgiltdy eoh eddawmh ta asfgddetm. Xa ngoh tgkm pmrgah, tgkm tecmo nar =2 asfgddetgaosgs oatmh usgoi e stap wetfl.

Xlmarmtgfeddy X ;i

D0 i ; 80 0X

D....(5)

wlmrm D ; Mqugvedmot dmoitl an pmohuduk ; dmoitl an tlrmeh () + rehgus (r) an à`,

X ; tgkm pmrgah an tlm sgkpdm pmohuduk ;=2

osasfgddetga=2nartecmoXgkm

sa i feo `m mesgdy hmtmrkgomh `y mquetgao ...(5).

Ireplgfed kmtlah ta ngoh i :

X0 ; Di

8 0

............(0) sa, X0 D

* Ngoh X nar hgnnmrmot vedums an D.

* Rdat X0 v/s D furvm. Nrak mquetgao (0), gt slaudh `m e stregilt dgom, wgtl sdapm ;

i

8 0

.

Ngoh sdapm an X0 v/s D irepl eoh mquetm gt ta

i

8 0

eoh imt i .


20/44

RLWVGFV


Mxekpdm 65. Go fmrtego a`smrvetgao wm iat; 06.0 fk, r ; 5.60 fk, eoh tgkm tecmo nar 52 asfgddetgao wes 52.2 smf.

Mstgketm tlm vedum an i go prapmr sgiogngfeot ngiurm. (tecm0 ; 52)Vadutgao : Mqugvedmot dmoitl an pmohuduk D ; 06.0 fk + 5.60 fk

; 08.= fk (effarhgoi ta ehhgtgao rudm an V.N.)

Eoh tgkm pmrgah X ; 22.552

2.52 (Xlrmm sgiogngfeot ngiurms)

imt i ; 80

0X

D

; 8 52 022.5

fk=.08

(go 6 V.N.)

; 8 52 00

0

smf)22.5(

k52=.08 ; 7.


21/44

RLWVGFV


M\RMPGKMOX # 8

Hmtmrkgogoi Wauois Kahudus an e igvmo wgrm `y Vmerdms Kmtlah :Eo mdmkmotery kmtlah:

Xa hmtmrkgom Wauois Kahudus, wm feo pmrnark eo arhgoery mxpmrgkmot. Dmts leoi e wmgilt k nrak e wgrm

nrak Laacs dew:

2

xW

E

ki

x ; ki

Wr02

Gn wm fleoim tlm wmgilt, tlm mdaoietgao an wgrm wgdd gofrmesm prapartgaoeddy.

Gn wm pdat mdaoietgao v/s ki, wm wgdd imt e stregilt dgom.

@y kmesurgoi gts sdapm eoh mquetgoi gt ta

Wr02

, wm feo mstgketm W..

Dgkgtetgaos go tlgs arhgoery kmtlah

(5) Nar skedd daeh, tlmrm key `m sakm `mohs ar cgocs go wgrm.

Va wm leh `ettmr stert wgtl sakm gogtged wt (sey 0 ci). Va tlet wgrm `mfakm stregilt.

(0) Xlmrm gs sdgilt hgnnmrmofm go `mlevgar an wgrm uohmr daehgoi eoh uodaehgoi

Va wm leh `mttmr tecm evmreim hurgoi daehgoi eoh uodaehgoi.

Xlm evmreim daeh wgdd `m karm eoh karm dgomer ar effuretm.

Kahgngfetgao go Vmerdms Kmtlah.

Xa cmmp tlm mxpmrgkmoted wgrm stregilt eoh cgoc nrmm, wm stert wgtl sakm hmeh wmgilt (0 ci)


22/44

RLWVGFV


Oaw wm irehueddy ehh karm eoh karm wmgilt. Xlm mxtre mdaoietgao (x) wgdd `m prapartgaoed ta mxtre wmgilt(w).

x ;Wr0

2

x ;Wr0

2

(w)

sa dmts pdatx v/sw, tlm sdapm an wlgfl wgdd `m ;

Wr02

Kmesurmkmot an Wauois kahudus.

Xa kmesurm mxtre mdaoietgao, fakpermh ta gogtged daehmh pasgtgao, wm usm e rmnmrmofm wgrm, edsa ferrygoi 0 ci

daeh (hmeh wmgilt). Xlgs kmtlah an kmesurgoi mdaoietgao `y fakpergsao, edsa feofmds tlm sghm mnnmft an

tekp eoh ygmdhgoi an suppart.A`smrvetgaos:

(g) Gogtged Pmehgoi ; x2; 2.=82 kk.

(Kgfrakmtmr Pmehgoi wgtlaut mxtre daeh)

(gg) Pehgus an wgrm ; 2.022 kk.

Kmesurmkmot an mxtre mxtmosgao hum ta mxtre daeh.

Kgfrakmtmr rmehgoi

V.Oa.

Mxtre daehao leoimr

k (ci)

Daehgofrmesgoi(p) (kk)

Daehhmfrmesgoi

(q) (kk)

Kmeormehgoi (x)

(p + q)/0(kk)

x mxtremdaoietgao

(xx2)(kk)

5 2.= 2.=== 2.=>5 2.==< 2.25= 2.=95 2.=>< 2.20 2.=22 2.2>2

> 6.2 2.>2< 2.>50 2.>52 2.292

9 6.= 2.>02 2.>00 2.>05 2.262 2.>60 2.>65 2.275

7 8.= 2.>85 2.>86 2.>80 2.520

52 =.2 2.>=0 2.>=0 2.>=0 2.550

Kmtlah # 5

Rdatx v/sw (k i)


23/44

RLWVGFV


M.R.M.R.

Mxtre mdaoietgao v/s mxtre daeh

2.=

2.=0

2.=8

2.=>

2.=

2.>0

2.>8

2.>>

= 52 5= 02 0= 62 6= 82 8= =2

mxtre daeh 'Y' (O)

Mxtremdaoietgao

'

'(kk)

E R

@

x

M.R.M.R.M.R.M.R.M.R.M.R.M.R.M.R.M.R.M.R.M.R.M.R.M.R.M.R.M.R.M.R.M.R.M.R.M.R.

* sdapm ; ........ER

@R

;)r(W 0

W ; .................

Kmtlah : 0@mtwmmo a`smrvetgao (5) (>)

eoh (0) (9)

eoh (6) () , (0) (9) , (6) ( x

5, x

= x

0, x

< x

6, ar x

52 x

=

eoh lmosm wm feo ngoh evmreim mdaoietgao hum ta 0.= ci wt.

Kgfrakmtmr rmehgoi

V.Oa.

Mxtre daeh ao

leoimrk(ci)

Daehgofrmesgoi (p)

(kk)

Daehhmfrmesgoi

(q) (kk)

Kmeormehgoi (x) (p

+ q)/0(kk)

x mxtremdaoietgao

hum ta 0.= cimxtre daeh

(kk)

5 2.= 2.=== 2.=>5 2.==< 2.2=0

0 5.2 2.=>= 2.=95 2.=>< 2.2=6

6 5.= 2.=9> 2.=22 2.2=0

> 6.2 2.>2< 2.>50 2.>52

9 6.= 2.>02 2.>00 2.>05

< 8.2 2.>62 2.>60 2.>657 8.= 2.>85 2.>86 2.>80

52 =.2 2.>=0 2.>=0 2.>=0

nar w ; 0.= i , evmreim mdaoietgao ;x ; 2.2=0 kk

x ;

Wr02

(w) wlmrmw ;k i ; 0= O eoh x) evmreim ; 2.= fk

Ruttgoi tlm vedums ngoh W ; .........

Mxekpdm 6=. Xlm ehjefmot irepl slaws tlm mxtre mxtmosgao (x) an e wgrm an dmoitl 5k suspmohmh nrak tlm tap ane raan et aom moh wgtl eo mxtre daeh w faoomftmh ta tlm atlmr moh. Gn tlm frass smftgaoed erme an tlmwgrm gs 52> k0, fedfudetm tlm Wauois kahudus an tlm ketmrged an tlm wgrm.


24/44

RLWVGFV


(E) 0 5255 O/k0 (@) 0 5255 O/k0 (F) 6 5256 O/k6 (H) 0 525> O/k0

Vadutgao : (E) ;

EW

2

w,, sdapm ;EW

2;

02

525 8

W)52(

5> ;

02

525 8

W ; 02 5252 ; 0 5255 O/k0

Mxekpdm 6>. Go tlm mxpmrgkmot, tlm furvm `mtwmmox eohw gs slawo eshattmh dgom (5). Gn wm usm eo eoatlmr wgrm an sekm ketmrged, ùt wgtl

hau dm dmoitl eoh hau dm rehgus. Ylgfl an tlm furvm gs mxpmftmh.

05

6

Y

x

(E) 5 (@) 0 (F*) 6 (H) 8

Vadutgao : Gogtgeddy sdapm ;

w

x

;

) (W)r( 0

2

go smfaoh fesm (sdapm)5 ;W)r0(

)0(0

2

;W)r(0

50

2

sa sdapm wgdd `m ledvmh, Eos. wgdd `m (6)

Mxekpdm 69. Essmrtgao : Go Vmerdms mxpmrgkmot ta ngoh yauois kahudus, e rmnmrmofm wgrm gs edsa usmh edaoi wgtl

tlm mxpmrgkmot wgrm.

Pmesao : Pmnmrmofm wgrm omutredgzms tlm mnnmft an tmkpmreturm, ygmdhgoi an suppart eoh atlmr mxtmroed

neftars

(E) Gn àtl Essmrtgao eoh Pmesao erm trum eoh tlm Pmesao gs e farrmft mxpdeoetgao an Essmrtgao

(@) Gn àtl Essmrtgao eoh Pmesao eoh trum ùt Pmesao gs oat e farrmft mxpdeoetgao an Essmrtgao.(F) Gn Essmrtgao gs trum ùt Pmesao gs nedsm.

(H) Gn àtlEssmrtgao eoh Pmesao erm nedsm.

Eos. (E)

Mxekpdm 6


25/44

RLWVGFV


Vad.8/h

ki0

; W (

x) W ;

xh)8/(

ki0

..................(5)

kexW

hW

;

k

k+

+ 0

h

h+

x

x

k ; 02.2 ci k ; 2.5 ci; 50= k ; 5 fk

h ; 2.2=2 fk h ; 2.225 fkx ; 2.522 fk x ; 2.225 fk

kexW

hW

; %522

fk522.2

fk225.2

fk2=.2

fk225.2

fk50=

fk5

ci2.02

ci5.2

; 8.6 %

Hmtegdmh Epperetus eoh kmtlah an smerds mxpmrgkmot

Vmerdms Epperetus Vtetgf Kmtlah)Xlm ngiurm slaws e Vmerdms epperetus. Gt faosgsts an twa kmted nrekmsN

5

eoh N0

lgoimh taimtlmr sa tlet tlmy feo levm aody vmrtgfed rmdetgvm katgao.

E spgrgt dmvmdD gs suppartmh et aom moh ao rgigh frass èr nrekm wlasmatlmr moh rmsts ao tlm tgp an e kgfrakmtmr sfrmw V, wlgfl kavms vmrtgfeddy

tlrauil rgigh frass èr.

Gn tlmrm gs eoy rmdetgvm katgao `mtwmmo tlm twa nrekms, tlm spgrgt dmvmd oa

daoimr rmkegos largzaoted eoh tlm ù``dm gs hgspdefmh. Xa `rgoi tlm ù``dm

èfc ta gts argigoed pasgtgao, tlm sfrmw les ta `m kavmh up eoh hawo. Xlm

hgsteofm tlrauil wlgfl tlm sfrmw les ta `m kavmh igvms tlm rmdetgvm katgao

`mtwmmo tlm twa nrekms.

Xlm nrekms erm suspmohmh `ytwa ghmotgfed daoi wgrms an tlm stmmd nrak tlm

sekm rgigh largzaoted suppart. Xlm wgrm @ gs eo mxpmrgkmoted wgrm eoh tlm

wgrm Eefts es e rmnmrmofm wgrm. Xlm twa nrekms erm pravghmh wgtl twa

laacs L5

eoh L0

et tlmgr dawmr mohs. Xlm laac L5

ferrgms e faosteot wmgilt

Yta cmmp tlm wgrm teut. Xa tlm laac L0, e leoimr gs etteflmh avmr wlgfl

sdattmh wmgilts feo `m put ta eppdy tlm strmtflgoi narfm.

Rrafmhurm :(g) Kmesurm tlm dmoitl an tlm mxpmrgkmoted wgrm nrak tlm pagot wlmrm gt dmevms tlm ngxmh suppart ta tlm pagot

wlmrm gt gs ngxmh go tlm nrekm.

(gg) Xlm hgekmtmr an tlm mxpmrgkmoted wgrm gs kmesurmh wgtl tlm lmdp an e sfrmw ieuim et eàut ngvm hgnnmrmot

pdefms eoh et mefl pdefm go twa kutueddy pmrpmohgfuder hgrmftgaos.

(ggg) Ngoh tlm pgtfl eoh tlm dmest fauot an tlm kgfrakmtmr eoh ehjust gt sufl tlet tlm ù``dm go tlm spgrgt dmvmd

gs mxeftdy go tlm fmotmr. Xlm gogtged rmehgoi an tlm kgfrakmtmr gs oatmh.

(gv) Xlm daeh ao tlm leoimrL0

gs irehueddy gofrmesmh go stmps an 2.= ci. A`smrvm tlm rmehgoi ao tlm kgfrakmtmr

et mefl steim entmr dmvmdgoi tlm gostrukmot wgtl tlm lmdp an tlm spgrgt dmvmd. Xa evagh tlm èfcdesl mrrar, edd tlm

ngoed ehjustkmots slaudh `m kehm `y kavgoi tlm sfrmw go tlm upwerh hgrmftgao aody. Gn et eoy tgkm tlm sfrmw

gs regsmh taa kufl, dawmr gt `mdaw tlm fmotred pasgtgao eoh tlmo regsm gt sdawdy ta tlm prapmr pasgtgao.

(v) ]odaeh tlm wgrm `yrmkavgoi tlm wmgilts go tlm sekm arhmr eoh tecm tlm rmehgoi ao tlm kgfrakmtmr sfrmw

mefl tgkm. Xlm rmehgoi tecmo nar e pertgfuder daeh wlgdm daehgoi tlm wgrm ar uodaehgoi tlm wgrm, slaudh eirmm

fdasmdy.

M\RMPGKMOX =Hmtmrkgogoi spmfgngf lmet fepefgty an eo uocoawo dgqugh usgoi fedargkmtmr :

Ngiurm slaws tlm Pmioeudt's epperetus ta hmtmrkgom tlm spmfgngf lmet fepefgty an e uocoawo dgqugh.

E sadgh splmrm an coawo spmfgng f lmet fepefgty s5

levgoi kess k5

eoh gogtged tmkpmreturm5

gs kgxmhwgtl tlm uocoawo dgqugh ngddmh go e fedargkmtmr. Dmt kessms an dgqugh eoh fedargkmtmr erm k

0eoh k

6

rmspmftgvmdy, spmfgngf lmet fepefgtgms erm s0eoh s

6eoh gogtgeddy tlmy wmrm et raak tmkpmreturm

0.

Ylmo tlm lat splmrm gs hrappmh go gt, tlm splmrm daasms lmet eoh tlm dgqugh fedargkmtmr systmk tecms

lmet. Xlgs prafmss faotgoums tgdd tlm tmkpmreturm an edd tlm mdmkmots `mfakms sekm (sey ).


26/44

RLWVGFV


Lmet dast `y lat splmrm ; k5s

5(

5 )

Lmet tecmo `y dgqugh & fedargkmtmr ; k0s

0(

0) + k

6s

6(

0)

Gn tlmrm wmrm oa mxtmroed lmet dass

Lmet igvmo `y splmrm ; Lmet tecmo `y dgqugh-Fedargkmtmr systmk

k5s

5(

5) ; k

0s

0(

0) + k

6s

6(

0)

Imt s0

;k s

k

5 5 5

0 0

( )

( )

k s

k

6 6

0

@ y k m es u rg oi t lm n g oe d ( s tm eh y s te tm )

tmkpmreturm an tlm kgxturm, wm feo mstgketm

s0: spmfgngf lmet fepefgty an tlm uocoawo dgqugh.

Xa igvm gogtged tmkpmreturm (5) ta tlm splmrm,

wm cmmp gt go stmek flek`mr ("A"), leoimh `y

tlrmeh. Yg tlgo s akm t gkm (sey 5= kg o) g t

eflgmvms e faosteot tmkpmreturm 5.

Oaw tlm fedargkmtmr, ngddmh wgtl wetmr (pert F)

gs tecmo `mdaw tlm stmek flek`mr, tlm waahmo

rmkave`dm hgsf H gs rmkavmh, eoh tlm tlrmeh

g s f ut . X l m s pl mr m h r ap s g o t lm w et mr -

fedargkmtmr systmk eoh tlm kgxgoi sterts.

Gn sp. lmet fepefgty an dgqugh (s0) wmrm coawo eoh tlet an tlm sadgh `edd (s

5) gs uocoawo tlmo wm feo ngoh

s5

; ( ) ( )

( )

k s k s

k

0 0 6 6 0

5 5

Mxekpdm 82. Xlm kess, spmfgngf lmet fepefgty eoh gogtged tmkpmreturm an tlm splmrm wes 5222 ik, 5/0

fed/ikF eoh


27/44

RLWVGFV


Vadutgao : (F) Lmet imomretmh ; k5

s5

(5

) +J

ilk5

Kexgkuk Rmrkgssg dm mrrar go V5

hum ta mrrar go kmesurgoi5,

0 eoh :

Xa hmtmrkgom tlm spmfgngf lmet fepefgty an uocoawo sadgh,

wm usm ssadgh

;5

0055

k

sksk

ss5

0ss

s ;5

0055

k

sksk

ss5

0ss

s

hs;

)(

)(h

0ss

0ss

ss5

ss5 )(h

s

s;

0ss

+ss5

kexs

s

; 0

ss50ss

55; 0

))(( 5ss0ss

05

Gn kess eoh sp. lmet fepefgtgms an wetmr eoh fedargkmtmr gs prmfgsmdy coawo, eoh dmest fauot an tmkp.

kmesurgoi tlmrkakmtmr gs 2.5F, tlmo ;5

;0; 2.5

kexs

s

wgdd `m dmest wlmo (

ss

0) (

ss

5)gs kex g.m.

0

05ss

Gn k5, s

5, k

0, s

0 e r m p r m f g s m d y c o a w o , t l m k e x g k u k p m r k g s s g ` d m % m r r a r g o s

s a d g h w g d d ` m d m e s t w l m o

stmehy stetm tmkpmreturm

ss ; 0 05

Mxekpdm 8>. Go tlm mxp. an ngohgoi sp. lmet fepefgty an eo uocoawo splmrm (V0), kess an tlm splmrm eoh

fedargkmtmr erm 5222 ik eoh 022 ik rmspmftgvmdy eoh sp. lmet fepefgty an fedargkmtmr gs mqued

ta05

fed/ik/F.

Xlm kess an dgqugh (wetmr) usmh gs 722 ik. Gogtgeddy `atl tlm wetmr eoh tlm fedargkmtmr wmrm et

raak tmkp 02.2F wlgdm tlm splmrm wes et tmkp


28/44

RLWVGFV


Hmtmrkgogoi spmmh an sauoh usgoi rmsaoeofm tu`m kmtlah :

Pmsaoeofmtu`m

Ngiurm slaws tlm mxpmrgkmot ta ngoh vmdafgty an sauoh go egr usgoi Pmsaoeofm tu`m kmtlah.

Rrgofgpdm : Pmsaoeofm tu`m gs e cgoh an fdasmh arieo pgpm.

Va gts oetured nrmq. wgdd `m

^

mq8 ,

6

8

^

mq ,

=

8

^

mq ........ ar imomreddy n o ; (0o 5)

^

mq8

Gn gt gs narfmh wgtl e tuogoi narc an nrmqumofy n2 3

nar rmsaoeofm, Oetured nrmq ; narfgoi nrmq.

(0o 5)^

mq8 ; n

2

mq ; (0o 5)

^

n8 2

Nar tlm ngrst Pmsaoeofmmq

;^

n8 2; (farrmspaohgoi ta 5st kahm)

Nar tlm smfaoh Pmsaoeofmmq

;6

8 2

^

n ; (farrmspaohgoi ta 0oh kahm)

Yarcgoi : Pmsaoeofm tu`m gs e 522 fk tu`m. Gogtgeddy gt gs ngddmh wgtl wetmr. Xa gofrmesm tlm dmoitl an egr

faduko go tlm tu`m, wetmr dmvmd gs dawmrmh. Xlm egr faduko gs narfmh wgtl e tuogoi narc an nrmqumofy

n2. Dmt et dmoitl

5, wm imt e ngrst rmsaoeofm (dauh vagfm) tlmo

mq5

;

^

n8 2 5 + ;^

n8 2.........(g) wlmrm gs moh farrmftgao

Gn wm nurtlmr dawmr tlm wetmr dmvmd, tlm oagsm `mfakms kahmretm. @ut et0. Ym, eiego imt e dauh oagsm

(smfaoh rmsaoeofm) tlmo


30/44

RLWVGFV


mq

0;

6

8 2

^

n

0 + ;

6

8 2

^

n .........(gg)

Nar (g) eoh (gg)

^ ; 0n2

(0

5)

A`smrvetgao te`dm :

Paak tmkp. go `mioogoi ; 0>F, Paak tmkp. et moh ; 0


31/44

RLWVGFV


Mxekpdm =0. Xecgoi tlm apmo moh an tu`m es y ; 2, pasgtgao an prmssurm oahms wgdd `m

(E) y ; 5 fk, y ; 87 fk

(@) y ; 2 fk, y ; =2 fk y

(F) y ; 5 fk, y ; =5 fk

(H) Oaom an tlmsm

Vadutgao : (E)

Mxekpdm =6. Xlm mquetgao an steohgoi wevm nar tlm smfaoh rmsaoeofm feo `m

(E) Rmx

; 0E sgo 0 (y + 5fk) fas 0 (682) t (@) Rmx

; 0E sgo 0 (y 5fk) fas 0 (682) t(F) R

mx ; 0E fas 0 (y + 5fk) fas 0 (682) t (H) Rmx ; 0E fas 0 (y 5fk) fas 0 (682) t

Vadutgao : (E) c ;0

;5

0; 0 ; 0n ; ( 0) (682)

ngrst oahm wgdd `m narkmh et y ; 5 gostmeh an y ; 2 sa mqo. an steohgoi wevm gs

Rmx

; 0E sgo 0(y + 5fk) fas 0(682) t

Kex Rmrkgssg`dm Mrrar go spmmh an sauoh hum ta mrrar go n2,

5,

0 :

nar Pmsaoeofm tu`m mxpmrgkmot

^ ; 0n2

(0

5)

do ^ ; do 0 + do n2 + do (

0

5)

kex. pmrkgssg`dm mrrar go spmmh an sauoh ;kex^

^

;

n

n

2

2

+

0 5

0 5

( )

Mxekpdm =8. Gn e tuogoi narc an (682 Lz 5%) gs usmh go tlm rmsaoeofm tu`m kmtlah, eoh tlm ngrst eoh smfaoh

rmsaoeofm dmoitls erm 08.2 fk eoh 98.2 fk rmspmftgvmdy. Ngoh kex. pmrkgssg`dm mrrar go spmmh

an sauoh.

Vadutgao : 5

; 02.2 fk 5

; 2.5 fk

0

; 98.2 fk 0

; 2.5 fk

n2

; (682 Lz 5%)nn

2

2; 5% ; 5

522

kex^

^

;

nn

2

2

+

0 5

0 5

; 2.022.98

5.25.2

522

5

;2.=8

0.2

522

5

M\RMPGKMOX # 9

^mrgngfetgao an Alk's dew usgoi vadtkmtmr eoh ekkmtmr

Alk's dew stetms tlet tlm mdmftrgf furrmot G ndawgoi tlrauil e faohuftar gs hgrmftdy prapartgaoed ta tlm R.H.

(^) efrass gts mohs pravghmh tlet tlm plysgfed faohgtgaos an tlm faohuftar (sufl es tmkpmreturm, hgkmosgaos,mtf.) erm cmpt faosteot. Ketlmketgfeddy.

^ G ar ^ ; GPLmrm P gs e faosteot coawo es rmsgsteofm an tlm faohuftar eoh hmpmohs ao tlm oeturm eoh hgkmosgaos

an tlm faohuftar.

Fgrfugt Hgeirek : Xlm fgrfugt hgeirek gs es slawo `mdaw :


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Rrafmhurm :@y slgntgoi tlm rlmastet faoteft, rmehgoi an ekkmtmr eoh vadtkmtmr erm oatmh hawo. Et dmest

sgx smt an a`smrvetgaos erm tecmo. Xlmo e irepl gs pdattmh `mtwmmo patmotged hgnnmrmofm (^) efrass P eoh

furrmot () tlrauil P. Xlm irepl fakms ta `m e stregilt dgom es slawo go ngiurm.Pmsudt : Gt gs nauoh nrak tlm irepl tlet tlm retga ^/G gs faosteot. Lmofm, furrmot vadteim rmdetgao slgp gs

mste`dgslmh, g.m., ^ G. Gt kmeos Alk's dew gs mste`dgslmh.

es ; ^P

5

, ngoh tlm sdapm an ^ furvm eoh mquetm gt ta

P

5.

sdapm ;ER

@R;

P

5Imt P ; ......

Mxekpdm ==. Gn mkn an `ettmry gs 522 v, tlmo wlet wes tlm rmsgsteofm an Plmastet ehjustmh et 0oh rmehgoi

( ; 0E, ^ ; 02^).

(E) 52 (@) 02 (F) 62 (H) 82

Vadutgao : Nrak tlm furvm sdapm ;v ;

P

5;

52

5P ; 52

nar smfaoh rmehgoi ;rlPP

Mkn

0 ; rlP52522

Prl; 82

Mxekpdm =>. Gn tlrmm wgrms an sekm ketmrged `ut hgnnmrmot hgkmosgao wmrm usmh go pdefm an uocoawo rmsgsteofm,

wm imt tlmsm G-^ furvm


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Ketfl tlm faduko effarhgoi ta farrmft furvm :

Ygrm hgkmosgao Farrmspaohgoi furvm

(p) ; 5k, rehgus ; 5 kk (g) Furvm (5)

(q) ; 5k, rehgus ; 0 kk (gg) Furvm (0)

(r) ;5

0 k, rehgus ;

5

0 kk (ggg) Furvm (6)

(E) (p)-(gg)3 (q)-(ggg)3 (r)-(g) (@) (p)-(ggg)3 (q)-(gg)3 (r)-(g)(F) (p)-(g)3 (q)-(gg)3 (r)-(ggg) (H) Oaom an tlmsm

Vadutgao : P ;E

; 0r

nar fesm(p) P 0)5(

)5(nar fesm (q) P 0)0(

)5(

nar fesm (r) P 0)0/5(

)0/5(sa P

r ? P

p ? P

q

eoh sdapm an G v furm ;v

;

P

5

sa sdapmr

4 sdapmp

4 sdapmq

qdgom (6), pdgom (0), r dgom (5)

Mxekpdm =9. v/s ^ furvm nar e oao-alkgf rmsgsteofm gs slawo.Xlm hyoekgf rmsgsteofm gs kexgkuk et pagot

(E) e (@)

(F) f (H) sekm nar edd

Vadutgao : (E)Hyoekgf rmsgsteofm P ;h

hv;

hv/h

5

;

sdapm

5

Et Rt. e, sdapm gs kgo , Va P gs kex

Mxekpdm =


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Ym feo ngoh tlm spmfgngf rmsgsteofm an e ketmrged usgoi alk's dew mxpmrgkmot.

;PE

D ;

HD

0

8

^

G

do ; do8

+ 0 do H do D + do do G

h ; 0

hH

H

hD

D +

h^

^

hG

G

; 0

HH

DD

^^

GG

kex

; kex an

^

^

D

D

H

H0

kex

; +0HH

+DD

+^^

+GG

; kex. pmrkgssg`dm mrrar go .

Mxekpdm >2. Go tlm Alk's mxpmrgkmot, wlmo patmotged hgnnmrmofm 52.2 ^ gs eppdgmh, furrmot kmesurmh gs 5.22

E. Gn dmoitl an wgrm gs nauoh ta `m 52.2 fk, eoh hgekmtmr an wgrm gs 0.=2 kk, tlmo tlm kexgkuk

pmrkgssg`dm mrrar go rmsgstgvgty wgdd `m -

(E) 5.6. Go tlm prmvgaus qumstgao, kexgkuk pmrkgssg`dm mrrar go rmsgstgvgty eoh rmsgsteofm kmesurmkmot

wgdd `m (rmspmftgvmdy)

(E) 0.58%, 5.=% (@) 5.=%, 0.8=% (F) 0.85%, 5.5% (H) Oaom an tlmsm

Vadutgao :kexP

P

;

g

g+

v

v;

2.52

5.2+

2.522

5.2; 5.5%

kexP

P

; 0.85%


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M\RMPGKMOX # 8. Gn rmsgsteofm V go P@ ; 622, tlmo tlm `edeofmh dmoitl gs nauoh ta `m 0=.2 fk nrak moh E .

Xlm hgekmtmr an uocoawo wgrm gs 5kk eoh dmoitl an tlm uocoawo wgrm gs 65.8 fk. Xlm spmfgngfrmsgstgvgty an tlm wgrm slaudh `m

(E) 0.= 528 -k (@) 6.= 528 -k (F) 8.= 528 -k (H) Oaom an tlmsm

Vadutgao : (E)622

P;

9=

0= P ; 522 ;

D8

hP 0; 0.= 528 -k

Mxekpdm >=. Go tlm prmvgaus qumstgao. Gn P eoh V erm gotmrfleoimh, tlm `edeofmh pagot gs slgntmh `y

(E) 62 fk (@) 82 fk (F) =2 fk (H) Oaom an tlmsm

Vadutgao : Gn P eoh V wevm gotmrfleoimh ,

; 9= , 522 -; 0=

@edeofm pagot wgdd `m slgntmh `y 9= 0= ; =2 fk

Mxekpdm >>. Go e kmtmr `rghim, oudd pagot gs et ; 66.9 fk, wlmo tlm rmsgsteofm V gs sluotmh `y 50rmsgsteofm tlm oudd pagot gs nauoh ta `m slgntmh `y e hgsteofm an 5


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Vuppasm wm imt oudd pagot et5

hgsteofm tlmo

0

5

P

P;

5

5522

......(g)

Oaw wm gotmrfleoim tlm pasgtgao an P5

eoh P0, eoh imt oudd pagot et

0hgsteofm tlmo

5

0

P

P;

0

0522

......(gg)

Vadvgoi mquetgao (g) eoh (gg) imt

05

0550

PP

PP

eoh 05

0055

PP

PP

522

Xlmsm moh farrmftgaos ( eoh) erm usmh ta kahgny tlm a`smrvetgaos

Mxekpdm >9. Gn wm usmh 522eoh 022 rmsgsteofm go pdefm an P eoh V, wm imt oudd hmndmftgao et5; 66.2fk.

Gn wm gotmrfleoim tlm Pmsgsteofm, tlm oudd hmnndmftgao wes nauoh ta `m et0

; >9.2 fk. Xlm moh

farrmftgao eoh slaudh `m :

(E) ; 5fk, ; 5fk (@) ; 0fk, ; 5fk(F) ; 5fk, ; 0fk (H) Oaom an tlmsmEos. (E)

Vadutgao : ;05

0550

PP

PP

;022522

)>9)(522()66)(022(

; 5 fk

; 522PP

PP

05

0055

; 522022522

)>9)(022()522)(66(

; 5 fk

Mxekpdm >fk, mstgketm tlm

uocoawo rmsgsteofm.(E) 52522(

50>

;9=

09

P ;

9=

09622 ; 527. Gn tlm uocoawo Pmsgsteofm fedfudetmh wgtlaut usgoi tlm moh farrmftgao, gs P5

eoh wgtl usgoi tlm

moh farrmftgaos gs P0

tlmo

(E) P5

? P0

wlmo èdeofmh pagot gs go ngrst ledn

(@) P5

4 P0 wlmo èdeofmh pagot gs go ngrst ledn

(F) P5

? P0

wlmo èdeofmh pagot gs go smfaoh ledn

(H) P5

? P0

edweys

Eos. (E)

Vadutgao : P5;

522V , P

0;

522V

Gn èdeofm pagot gs go ngrst ledn sey d; 82

P5;

>2

82V P

0;

>5

85V sa P

0? P

5


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Gn èdeofm pagot gs go smfaoh ledn sey d ; 92

P5;

62

92V P

0;

65

95V sa P

04 P

5.

Kexgkuk Rmrkgssg`dm Mrrar go :Xlm spmfgngf rmsgstgvgty an wgrm, nrak kmtmr `rghim gs ;

H VD

0

8

522 Essukm tlet coawo rmsgsteofm go P@(V), eoh tated dmoitl an wgrm gs prmfgsmdy coawo, tlmo dmts ngoh

kexgkuk pmrkgssg`dm mrrar go hum ta mrrar go kmesurmkmot an (èdeofm dmoitl) eoh H (hgekmtmr anwgrm).

do ; do

D8

V+ 0 do H + do do (522 )

h ; 0

hH

H +

h

h ( )

( )

522

522

; 0

hH

H +

h

+

h

522

kex

; 0 H

H +

+

522

kex

hum ta mrrar go aody gs ;

522 ;

)522(

)522(

kex

wgdd `m dmest wlmo(522 ) gs kexgkuk , g.m. ; =2 fk

M\RMPGKMOX # 7

RAVX ANNGFM @A\

R

\]oco

awo

U

PEF

I

@

H +

M

F @

C5

C0I

H

0252 = 0 0 5

x02 =2 522 022 0220222 0222 5222

522252 22 5 22 52252 52

=22=222

R @

UFE

uocoawormsgsteofm

P

Ngiurm-5 Ngiurm-0

Go e wlmet staom's @rghim fgrfugt, GnR

U ;

P

\ tlmo tlm `rghim gs èdeofmh. Va uocoawo rmsgsteofm

\ ;U

R P. Xa uohmrsteoh tlm wlmet staom's @rghim fgrfugt, e pax anngfm @ax gs hmsfrg`mh.

Pmsgsteofm R eoh U erm smt go erks E@ eoh @F wlmrm wm feo levm 52 , 522 ar 5222 rmsgsteofm,

ta smt eoy retgaU

R .


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Mxekpdm 90. Ylet gs tlm kexgkuk eoh kgogkuk passg`dm rmsgsteofm, wlgfl feo `m hmtmrkgomh usgoi tlm RA

@ax slawo go eàvm ngiurm-0

(E) 5555 c , 2.5 (@*) 5555 c , 2.25 (F) 5555 c , 2.225 (H) Oaom an tlmsmEos. (@)

Vadutgao : \ ; PR

U (\)

kex;

kgo

kex

)R(

)U((P)

kex ;

52

5222(55552) ; 5555 c

(\)kgo;kex

kgo

)R(

)U((P)kgo ; 5222

52

5222

52(5) ; 2.25.

Mxekpdm 96. Go e fmrtego mxpmrgkmot gnU

R;

5

52eoh go P, gn 570 gn usmh wm erm imttgoi hmndmftgao tawerh

rgilt, et 576 , eiego tawerh rgilt ùt et 578 , hmndmftgao gs tawerh dmnt. tlm uocoawormsgsteofm slaudh dgm `mtwmmo

(E) 57.0 ta 57.6 (@) 57.6 ta 57.8 (F) 57 ta 02 (H) 57.8 ta 57.=Eos. (@)

Vadutgao : \ ;R

U(P) ;

52

5(576 578) ; 57.6 57.8

Mxekpdm 98. Gn @y kgstecm, @ettmry gs faoomftmh `mtwmmo @ eoh F Iedveoakmtmr gs faoomftmh efrass E eohF tlmo

(E) Ym feooat imt èdeofmh pagot.

(@) Mxpmrgkmot wgdd `m dmss effuretm

(F) Mxpmrgkmot feo `m haom go sgkgder keoomr.

(H) Mxpmrgkmot feo `m haom go sgkgder keoomr ùt oaw, C0

slaudh `m prmssmh ngrst, tlmo C5

.

Eos. (H)

M\RMPGKMOX # 52

XA NGOH NAFED DMOIXL AN E FAOFE M KGPPAP ]VGOI ]-^ KMXLAH.

Rrgofgpdm : Nar hgnnmrmot u, wm kmesurm hgnnmrmot v, eoh ngoh n usgoi kgrrars narkude5

n ;

5

v +

5

u.

Go tlgs mxpmrgkmot, e faofevm kgrrar gs ngxmh et pasgtgao KK eoh e cogttgoi ommhdm gs usmh es eo

a jmft, kauotmh go nraot an tlm faofevm kgrrar. Xlgs ommhdm gs feddmh a jmft ommhdm (A go ngi)

Gkeim

Gkeim

ommhdm A`jmftommhdm

Ngrst an edd wm kecm e rauil mstgketgao an n. Nar mstgketgoi n rauildy, kecm e slerp gkeim an e

ner ewey a`jmft (dgcm suo) ao e ngdtmr pepmr. Xlm gkeim hgsteofm an tlm ner a`jmft wgdd `m eo epprax

mstgketgao an nafed dmoitl).

Oaw, tlm a`jmft ommhdm gs cmpt `myaoh n, sa tlet gts rmed eoh govmrtmh gkeim (G go ngi) feo `m narkmh.

Wau feo smm tlgs govmrtmh gkeim go tlm kgrrar `y fdasgoi yaur aom mym eoh cmmpgoi tlm atlmr mym

edaoi tlm padm an tlm kgrrar.

Xa dafetm tlm pasgtgao an tlm gkeim, usm e smfaoh ommhdm, eoh slgnt tlgs ommhdm sufl tlet gts pmec

Fagofghm wgtl tlm gkeim. Xlm smfaoh ommhdm igvms tlm hgsteofm an gkeim (v), sa gt gs feddmh gkeim

ommhdm (G' go ngiurm). Oatm tlm a jmft hgsteofm u eoh gkeim hgsteofm v nrak tlm kk sfedm ao

aptgfed `mofl.

Vgkgderdy tecm 8-= karm a`smrvetgaos.


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(gv) Nrak gotmrsmftgao an dgoms jagogoi uo

eoh vo

:

Gohgfetm u5, u

0, u

6 ....... u

o ao x-exgs, eoh v

5, v

0, v

6 ........ v

o ao y-exgs. Gn wm jago u

5 wgtl v

5,

u0

wgtl v0, u

6 wgtl v

6 eoh ............... sa ao. Edd d gom gotmrsmfts et e fakkao pagot (n, n).

Irepl an v vs. u nar e Faofevm Kgrrar

M\RDEOEXGAO

Dgom jagogoi u5 eoh v5 gs

5u

x+

5v

y; 5 ..............(5)

wlmrm,5

5u +

5

5v ;

n

5ar

n

u5+

n

v5; 5 .. .. ... .. ..... (5 )

Dgom jagogoi u0

eoh v0

gs

x

u0+

y

v0; 5 ..............(0)

wlmrm nu0

+ nv0

; 5 ..............(0)

Vgkgderdy Dgom jagogoi uo

eoh vo

gs

x

vo+

y

uo; 5 ..............(6)

wlmrmn

uo+

n

vo; 5 ..............(6)

Nrak mquetgao (5), (0), (6), wm feo sey tlet x ; n eoh y ; n wgdd setgsny edd mquetgaos (5),

(0), (6). Va pagot (n, n) wgdd `m tlm fakkao gotmrsmftgao pagot an edd tlm dgoms.

Nrak u v hetes hrew u5, u0 ...... uo ao x-exgs eoh v5, v0, ........ vo hetes ao y-exgs. Jago u5wgtl v

5, u

0 wgtl v

0 ........ u

o wgtl v

o. Ngoh fakkao gotmrsmftgao pagot eoh mquetm gt ta (n, n).


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u (go fk)

^ (go fk)

(-52,-52)

(n,n);

Va n;-52 fk

GOHM\ MPPAP

Go u v kmtlah, wm rmqugrm tlm hgsteofm `mtwmmo a`jmft ar gkeim nrak tlm padm (vmrtmx) an tlm

kgrrar (eftued hgsteofm).

@ut preftgfeddy wm kmesurm tlm hgsteofm `mtwmmo tlm gohgfms E eoh @. (A`smrvmh hgsteofm),

wlgfl ommh oat mxeftdy fagofghm wgtl a`jmft eoh padm, tlmrm feo `m e sdgilt kgsketfl feddmh

gohmx mrrar, wlgfl wgdd `m faosteot nar mvmry a`smrvetgao.

Gohmx mrrar ; A`smrvmh hgsteofm Eftued hgsteofm

(Just dgcm zmra mrrar go sfrmw ieuim, gt gs tlm mxfmss rmehgoi).

Xa hmtmrkgom gohmx mrrar, kgrrar eoh a`jmft ommhdm erm pdefmh et er`gtrery pasgtgao. Nar kmesur-

goi eftued hgsteofm, e cogttgoi ommhdm gs just ngttmh `mtwmmo tlm padm an kgrrar eoh a`jmft ommhdm

A. Xlm dmoitl an cogttgoi ommhdm wgdd igvm tlm eftued a`jmft hgsteofm wlgdm tlm smperetgao

`mtwmmo gohgfms E eoh @ et tlet gosteot gs tlm a`smrvmh hgsteofm.

Va gohmx mrrar gs -

m ; A`smrvmh hgsteofm Eftued hgsteofm

; Vmperetgao `mtwmmo gohgfms E eoh @ Dmoitl an cogttgoi ommhdm

aofm wm imt m, go mvmry a`smrvetgao, wm imt

Eftued hgsteofm ; A`s mrvmh hgsteof m (smperetgao `mtwmmo tlm gohgf ms) Mxfmss rmehgoi (m)

*Xlmrm gs eo eoatlmr tmrk , Gohmx farrmftgao wlgfl gs govmrsm an gohmx mrrar.

Gohmx farrmftgao ; - gohmx mrrar

Mxekpdm 9=. Xa ngoh gohmx mrrar nar u , wlmo e cogttgoi ommhdm an dmoitl 02.2 fk gs ehjustmh `mtwmmopadm eoh a`jmft ommhdm, tlm smperetgao `mtwmmo tlm gohgfms an a`jmft ommhdm eoh kgrrar

wes a`smrvmh ta `m 02.0 fk.

Gohmx farrmftgao nar u gs -

(E) 2.0 fk (@) 2.0 fk (F) 2.5 fk (H) 2.5 fk

Vadutgao : (@) Gohmx mrrar (Mxfmss rmehgoi) ; A`smrvmh rmehgoi Eftued rmehgoi

; 02.0 02.2 ; 2.0 fk

Mxekpdm 9>. Xa ngoh gohmx mrrar nar v, wlmo tlm sekm cogttgoi ommhdm gs ehjustmh `mtwmmo tlm padm

eoh tlm gkeim ommhdm, tlm smperetgao `mtwmmo tlm gohgfms an gkeim ommhdm eoh kgrrar

wes nauoh ta `m 57.7 fk. Gohmx mrrar nar v gs

(E) 2.5 fk (@) 2.5 fk (F) 2.0 fk (H) 2.0 fk

Vadutgao : (@) m ; 57.7 fk 02.2 fk ; 2.5 fk

Mxekpdm 99. Go sakm a`smrvetgao, tlm a`smrvmh a`jmft hgsteofm (Vmperetgao `mtwmmo gohgfms an a`jmftommhdm eoh kgrrar) gs 62.0 fk, eoh tlm a`smrvmh gkeim hgsteofm gs 57.7 fk. ]sgoi

gohmx farrmftgao nrak prmvgaus twa qumstgaos, mstgketm tlm nafed dmoitl an tlm faofevm

kgrrar!


43/44

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Vadutgao : u ; 62.0 2.0 (mxfmss rmehgoi)

; 62.2 f k.

v ; 57.7 (2.5) (mxfmss rmehgoi)

; 02.2 f k.

5

n ;

5

v +

5

u n ; 50.2 fk .

Kexgkuk pmrkgssg`dm mrrar go n hum ta gkpmrnmft kmesurmkmot anu & v :Go tlgs mxpmrgkmot, nrak e smt (u, v), nafus hgsteofm n feo `m fedfudetmh nrak mquetgao.

n

5;

u

5+

v

5 0n

hn; 0u

hu+ 0v

hv

0n

n; 0u

u 0v

v

kex0n

n

; + 0u

u+ 0v

v

0

00kex n

v

v

u

un

Mxekpdm 9


44/44

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(gg) Nrak5

v v/s

5

u furvm :

5

v -

5

u ;

5

n

5

5

/

/

u

n +5

5

/

/

v

n ; 5

x

e +

y

` ; 5

Va furvm `mtwmmo5

v

v/s5

u

slaudh `m e stregilt dgom levgoi x

eoh y gotmrfmpts ; -5

n eoh

5

n

5/u

5/v

5/n

-5/n

Irepl anv

5vs.

u

5nar e faovmx dmos

nrak tlm a`smrvetgaos an u eoh v, pdat5

v v/s

5

u furvm

es e stregilt dgom, ngoh tlm x eoh y gotmrfmpts, eoh mquetm

tlmk ta - 5n

eoh 5n

.

5/u (go 5/k)

5/v (go 5/k)

52 ;5/n

;-5/n

Va n ; 5/52

-52

Va n ; 5/52

(ggg) Nrak u v furvm :

Pmdetgao `mtwmmo u eoh v gs

5

v -

5

u ;

5

n ..............(5)

Va furvm `mtwmmo v v/s u gs e rmfteoiuder lypmr`ade es slawo `mdaw.

Gn wm hrew e dgom `gsmftgoi `atl tlm exgs, g.m. dgom

u ; - v ..............(0)

tlmo tlmgr gotmrsmftgao pagots slaudh `m ^ ; 0n, u ; - 0n (@y sadvgoi mquetgao (5) eoh

mquetgao (0))

nrak u v hete, pdat v v/s u furvm, eoh hrew e dgom y ; -x . Ngoh tlm gotmrsmftgao pagot eoh

mquetm tlmk ta (-0n, 0n).

Irepl an v vs. u nar e Faovmx dmos

]Dgom

v;-u

5/v -5/u ; 5/n(-0n,0n)

(gv) Nrak gotmrsmftgao an dgoms jagogoi uo

eoh vo

:

Gohgfetm u5, u

0, u

6 ....... u

o ao x-exgs, eoh v

5, v

0, v

6 ........ v

o ao

y-exgs. Gn wm jago u5

wgtl v5, u

0 wgtl v

0, u

6 wgtl v

6 eoh ...............

sa ao. Edd d gom gotmrsmfts et e fakkao pagot (-n, n).

u

v

(-n,n)

u5 u0 u6 u8

v5v0

v6

v8

nrak u v hetes hrew u5, u

0...... u

oao x-exgs eoh v

5, v

0, ........ v

ohetes

ao y-exgs Jago u eoh v u wgtl v u eoh v Ngoh fakkao

errors in measurement theory_e

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