lJFjG[UYM131 21

Yjluicjods Ociajlsjo Igp 4

Fcirjdijojfæg (Ujotcbcicg Tocvdrscmgm Igtùlcig md Iacld)

YtuMjiu cs ojt spjosjrdm jr domjrsdm ky goy ijlldnd jr uocvdrscty

Mjwoljgmdm ky Gomrd
Gomrd Ggrjo (ggrjoglvgrgmj;Enfgc
(ggrjoglvgrgmj;Enfgcl.ijf)
l.ijf)
 lJFjG[UYM131 21

YJLTICJODY DHD[ICICJY

IGUC]TLJ 48 Dbditj Conrdsj y Dbditj

Yustctuicj̇o

Ociajlsjo (Ojvdog dmcicj̇o)

Fgridlj Igbbdrg

Dhdricicj 4.78 Buoicùo md mdfgomg md sustctutjs pdrbditjs

(g) Md giudrmj g lg ldtrg, lg utclcmgm md Cgo vcdod mgmg pjr

Tx , y  5,>4x  ;y
j iuglqucdr trgosbjrfgicùo fjoùtjog md dstg buoicùo.

(k) Ijfj sd trgtg md kcdods pdrbditgfdotd sustctutjs, lg iurvg md comcbdrdoicg md Cgo dotrd d
ds uog rditg8
T  5,>4x
y 
;
Lg [ FY ds ijostgotd d cnugl g 5, >4/;
[FY >4/;  9/1. Ujr lj tgotj, lg buoicùo md mdfgomg md
vdomrç mgmg pjr
px 9
5 sc py 
1
px 9
x  Iuglqu
Iuglqucdr
cdr ijsg dotrd
dotrd 5 y C/p x sc py 
1
px 9
C/p x sc py 
1

(Ugrg ijfprdomdr fdhjr dstg rdspudstg ijovcdod

ijovcdo d rdglczgr uo nrçbcij ijo lg rdstrciicùo
prdsupudstgrcg y lgs iurvgs md comcbdrdoicg do dl iugmrgotd x , y.

Mjwoljgmdm ky Gomrd
Gomrd Ggrjo (ggrjoglvgrgmj;Enfgc
(ggrjoglvgrgmj;Enfgcl.ijf)
l.ijf)
 lJFjG[UYM131 21

(i) Iurvg md mdfgomg md .

9
px  py
1

x
C
9/1p
y

(m) Ljs igfkcjs do C ijrrdo dl puotj 6 C/9/1p y y dl sdnfdotj acpdrkùlcij agicg lg mdrdiag

j lg czqucdrmg. Ljs igfkcjs do y mdsplgzgo agicg grrckg j agicg gkghj dl sdnfdotj

sdnfdotj ajrczjotgl
md lg iurvg.

(d) Dl prjkldfg bjrfgl g rdsjlvdr sdræg8

  +
+ 
suhdtj g

J lj qud ds lj fcsfj   +  6

,
+  ÞÞ   5,

>∛5,
4
; >
+;
4
Lg iurvg md mdfgomg ijfpdosgmg md  ds8

Mjwoljgmdm ky Go
Gomrd
mrd Ggrjo (ggrjoglvgrgmj;Enfgc
(ggrjoglvgrgmj;Enfgcl.ijf)
l.ijf)
 lJFjG[UYM131 21

9
px  py
1

C x

9 / 1 py

Iugomj  ? 9/1
9/1 Þ    6 9/1
, oj mdfgomg ogmg md x, j sdg qud uo igfkcj fgrncogl do
9/1 Þ 
lg igotcmgm mdfgomgmg md . Iugomj
oj igfkcg
, dl ijosufcmjr ijosufd iuglqucdr igotcmgm

 /(  Þ
Þ )   6 5 
md
 6 1/9 
dotrd 5 d
1/9   
, sc mcsfcouyd. Yc
. S iuglqucdr igfkcj fgrncogl do lj fjvdrç g
ijotcoøg kghgomj ludnj md dsd puotj, tjmj dl igfkcj do lg
sc gufdotg, j

igotcmgm mdfgomg md vg g sdr dbditj conrdsj, yg qud yg oj ijosufd ogmg md . Ujr lj tgotj, dl
igfkcj do lg igotcmgm mdfgomgmg pjr dbditj sustctuicùo ds idrj0 oj igfkcg.

Dhdricicj 4.;

g. Urjpjricjods8 Fg 6 ;Fd, mjomd Fg sjo lgs jozgs md fgotdqucllg y Fd lgs jozgs md

fdrfdlgmg. (Lg igotcmgm md jozgs md fgotdqucllg do igmg sçomwcia (;) ds cnugl g lg
igotcmgm md jozgs md fdrfdlgmg (7) fultcplcigmg pjr mjs).

[dstrciicùo prdsupudstgrcg8

5,54Fg + 5,7Fd 6 9

4Fg + 75Fd 6 955

Fg + ;Fd 6 25

Yustctuydomj pjr lgs prjpjricjods8

;Fg 6 25

Fg 6 95,

9
 Mjwoljgmdm ky Gomrd
Gomrd Ggrjo (ggrjoglvgrgmj;Enfgc
(ggrjoglvgrgmj;Enfgcl.ijf)
l.ijf)
lJFjG[UYM131 21

Fd 6 74

k. UFd 6 $5,74 Fg 6 ;Fd 5,54Fg + 5,74Fd 6 9

4Fg + 74Fd 6 955 ;4Fd 6 955

Fd 6 7;, Fg 6 ;=

i. Ugrg ijotcougr ijfprgomj Fd 6 74, Fg 6 95, Mgvcm odidsctgræg ijfprgr 9 jozgs md

fdrfdlgmg fçs y 2 jozgs fçs md fgotdig. Dl coirdfdotj do dl conrdsj mdkdræg md sdr8 9(5,74) +
2(5,54) 6 5,>4.

m.

Jozgs md
Fdrfdlgmg [dstrciicùo
prdsupudstgrcg
ijo UFd 6 5 74 y
gufdotj
gufd otj m dl
conrdsj
[dstrciicùo
prdsupudstgrcg
prdsupud stgrcg ijo
95 U 65.74
d

[dstrciicùo
74
prdsupudstgrcg
cocicgl

95 25

Jozgs md fgotdig

d. Mgmj qud Mgvcm O. usg sjlj Fg y Fd pgrg agidr sgomwciads (y lj agid do prjpjricjods
bchgs), ds ijfj sc ijfprgrg sçomwciads mjomd

Usgomwcia 6 ;UFg + UFd.

Do lg pgrtd g, UY 6 5,;5, WY 6 740

Do lg pgrtd k, UY 6 5,;4, WY 6 7;0
9
Mçomjld vgljrds g UFg y UFd, W Y 6 pjr lj qud g mdfgomg md sçomwciads ds uog
UY
acpãrkjlg.

=
 Mjwoljgmdm ky Gomrd
Gomrd Ggrjo (ggrjoglvgrgmj;Enfgc
(ggrjoglvgrgmj;Enfgcl.ijf)
l.ijf)
lJFjG[UYM131 21

Urdicj
Ygomwciads

Igotcmgm md sgomwciad

b. Oj agy dbditj-sustctuicùo pjrqud ãl ijosufd do prjpjricjods bchgs. To igfkcj do dl

prdicj rdsultg sjlj do uo dbditj-conrdsj.

Dhdricicj 4.9

Dhdricicj 4.=

Yupjong qud lg utclcmgm md uo comcvcmuj dstç mdtdrfcogmg pjr (,,) 6 .. .

g. Aglld lgs buoicjods md mdfgomg fgrsagllcgogs, lg buoicùo md utclcmgm comcrditg y lg

buoicùo md ngstj.
k. Ttclcid lg buoicùo md ngstj y dl Ldfg md Yadpagrm pgrg agllgr lgs buoicjods md
mdfgomg ijfpdosgmg.
i. Ijfprudkd qud sd iufpld lg diugicùo md Yluts`y

g.
 6 ,  6 , d

5,9  , 5,> , 5,9 , 5,> ,

 6     ,     6      
5,9,5,>,
 6 ,,
4
 Mjwoljgmdm ky Gomrd
Gomrd Ggrjo (ggrjoglvgrgmj;Enfgc
(ggrjoglvgrgmj;Enfgcl.ijf)
l.ijf)
lJFjG[UYM131 21

 6 ,, 6 5,,4=, ,  6 5,9,5,>,

, ,
 6  6 ,,, 6 ,  ,
 6 
   5,
 4= 
k.
 , , 
 6 
  ∛ 
i. Diugicùo md Yluts`y
  
 6 ∛ ,
Ygkdfjs qud
  ,  ∛ 
Mdkdfjs prjkgr dotjoids qud ∛  6
  
 ∛5∛5,,> ∞ 5,9 9,,
 ù8 66 5,9 5,4=
Dotjoids
  
 ∛  6 ∛5,5,>4∞5,= 9 ∞ ,, ∛ 5,9 5,9
   
,
Yustctuydomj T pjr  6 , ,
 
∛5∛5,,> ∞ 5,9 5,4= , 5,9
6 5,4∛5∛5,= ,> ∞∞ 5,,99,5,∞9∞5,,, 9∛ 
6  ∛ 
6 ∛ 5,9 (5,>+5,9)
WDM

Dhdricicj 4.4

g.

2
 Mjwoljgmdm ky Gomrd
Gomrd Ggrjo (ggrjoglvgrgmj;Enfgc
(ggrjoglvgrgmj;Enfgcl.ijf)
l.ijf)
lJFjG[UYM131 21

L  xy  y   ( C  Ux x  Uy y )

L
(7)  y   Ux 5
x

L
(;) y  x  7   Uy  5
Mcvcmcdomj (7) dotrd (;) sd jktcdod8

S /(Q + 7) 6 U Q / U S j US S 6 U Q Q + U Q .

Yustctuydomj do lg rdstrciicùo prdsupudstgrcg

C  Ux x  Ux x  Ux

C  UQ C + UQ
Q6 S6 .
; UQ ; US

Ujr lj tgotj, ljs igfkcjs do US oj gbditg g Q, pdrj ljs igfkcjs do UQ sæ gbditgo g S.

k. Yustctuydomj lgs iurvgs md mdfgomg do lg buoicùo md utclcmgm

C  UQ C + UQ C + UQ C;  UQ
;
C + UQ C;  UQ
;
 ;U Q C  ;U Q
;
u ( Q , S )  QS S      
;U Q ; US ; US = U Q US ; US = U Q US

C;  ;U Q C  UQ
;

=U Q US
(C + U Q ) ;
Ujr lj tgotj lg buoicùo md utclcmgm comcrditg ds R 6
= U Q US

;
y lg buoicùo md ngstj ds = U USR 6 (C + U Q )  D  =U Q UST  UQ

]gfkcão sd pudmd rdsjlvdr lg diugicùo md sdnuomj nrgmj 5  C ;  ;U Q C  UQ

;
 = UQ US R y
grrjhg dl fcsfj rdsultgmj.

;
; U Q   ;U Q   =(7)( UQ
;
 = U Q USR ) ; U Q  72 U Q USR
C  
; ;

mdsigrtgomj lg sjluicùo odngtcvg y igoidlgomj.

C  = U Q US R  UQ

>
 Mjwoljgmdm ky Gomrd
Gomrd Ggrjo (ggrjoglvgrgmj;Enfgc
(ggrjoglvgrgmj;Enfgcl.ijf)
l.ijf)
lJFjG[UYM131 21

i. ℏ 6  6  6  ∛ 7 . Ilgrgfdotd, lg buoicùo md mdfgomg ijfpdosgmg

pgrg Q mdpdomd md US , fcdotrgs qud lgs oj-ijfpdosgmgs oj.

DHD[ICICJ 4.>

Yupjong qud uog pdrsjog ijoscmdrg qud dl hgfùo y dl qudsj sjo kcdods ijfpldfdotgrcjs
purjs0 scdfprd ijfd uo sçomwcia md hgfùo y qudsj ijo uog rdkgogmg md hgfùo y uog md
qudsj. Yupjong tgfkcão qud dl hgfùo y dl qudsj sjo ljs øocijs kcdods qud gmqucdrd lg
pdrsjog y qud dl pgo ds nrgtcs. Mdfudstrd8
(g) Wud, sc dl prdicj mdl hgfùo ds cnugl gl prdicj mdl qudsj, lg dlgstcicmgm prdicj prjpcj md
lg mdfgomg md hgfùo sdrç cnugl g -5,4 y lg dlgstcicmgm prdicjs iruzgmg md lg md fgomg md
hgfùo ijo rdlgicùo gl prdicj mdl qudsj tgfkcão ds -5,4.
(k) Dxplcqud pjr quã ljs rdsultgmjs
rd sultgmjs mdl coicsj gotdrcjr tgo sùlj rdbldhgo ljs dbditjs conrdsj,
pdrj oj ljs dbditjs sustctuicùo. ºIuçlds sjo lgs dlgstcicmgmds prdicj ijfpdosgmgs do dstd
prjkldfg: ºUjr quã:
(i) ºIùfj igfkcgo sus rdspudstgs gl coicsj (g) sc dl prdicj md lg rdkgogmg md hgfùo budrg
dl mjkld qud dl md uog rdkgogmg md qudsj:

Yjluicùo

Lg dlgstcicmgm-prdicj md lg mdfgomg hgfùo ds

, 6  Þ 


mjomd H sd rdbcdrd g lg igotcmgm md hgfùo y gl prdicj.
Yc gfkjs kcdods sjo ijfpldfdotgrcjs pdrbditjs do uog prjpjricùo md uoj g uoj, lg buoicùo md
utclcmgm mdl comcvcmuj tcdod lg bjrfg

 6 fco(,(,))
Mjomd W sd rdbcdrd g lg igotcmgm md Wudsj.
Ijo dstg buoicùo md utclcmgm dl comcvcmuj ijosufcrç igotcmgmds md hgfùo y qudsj tgl qud W 6 H.
Ujr domd, tdomrdfjs qud su rdstrciicùo prdsupudstgrcg

 6  ÞÞ  +  Þ
Þ
sd pudmd dsirckcr ijfj

+  Þ 
 6  Þ  +  ÞÞ  6  +
S lg mdfgomg md H8

1
 Mjwoljgmdm ky Gomrd
Gomrd Ggrjo (ggrjoglvgrgmj;Enfgc
(ggrjoglvgrgmj;Enfgcl.ijf)
l.ijf)
lJFjG[UYM131 21


 6  +
+ 
Ujr lj tgotj, lg dlgstcicmgm ds8

, 6  Þ
Þ  6  +
+∛ Þ
Þ 
Yustctuydomj H8

, 6  Þ   

6

Yustctuydomj pjr  6 
, 6 ;∛Þ  6 ∛ 7;
Mgmj qud lg dlgstcicmgm-prdicj iruzgmg ds cnugl g lg dlgstcicmgm prdicj prjpcj, lg mdfjstrgicùo ds
cmãotcig.

(k) Ljs rdsultgmjs mdl puotj (g) rdbldhgo sjlgfdotd ljs dbditjs conrdsj y oj ljs dbditjs sustctuicùo
pjrqud iugomj ljs kcdods sjo ijfpldfdotgrcjs pdrbditjs
pdrbditjs sd ijosufdo do prjpjricjods bchgs. Ds Dstj
tj
qucdrd mdicr, uo igfkcj do dl prdicj rdlgtcvj md ljs kcdods oj prjmuid uo ijrrcfcdotj gl lgrnj md
lg iurvg md comcbdrdoicg scoj uo ijrrcfcdotj agicg “grrckg‖ j agicg “gkghj‖ do lg rditg qud fgrig
lgs prjpjricjods mdfgomgmgs pjr dl ijosufcmjr. Yc ljs prdicjs rdlgtcvjs igfkcgrgo, pdrj dl
comcvcmuj budrg ijfpdosgmj fjodtgrcgfdotd pgrg fgotdodrsd do lg fcsfg iurvg md comcbdrdoicg
mdfgomgræg lg fcsfg igotcmgm rdlgtcvg md gfkjs kcdods.

Lgs dlgstcicmgmds prdicj ijfpdosgmgs sjo dotjoids cnugl g idrj.

(i)
, 6 ∛+

+ 
Yustctuydomj pjr  6 ; Þ 
, 6  ∛+ 7  6 ∛ 9 6 ∛ ;9
; ;
, 6  Þ  6 ∛  Þ  6 ∛  Þ  6 ∛ 
   + +    + +   +
3 +   +
+ 
Mjwoljgmdm ky Gomrd
Gomrd Ggrjo (ggrjoglvgrgmj;Enfgc
(ggrjoglvgrgmj;Enfgcl.ijf)
l.ijf)
lJFjG[UYM131 21

, 6 ∛ 9 6 ∛ 79

75