गणित कक्षा ३

d/] f] l;h{gf

x/] f}‘, d}n] slt l;s]F <

!= ?n/sf] ;xfotfn] lbOPsf bO' { laGb'x¿nfO{ hf8] ]/ /v] fv08

lvRg'xf];\ / gfk lng'xf;] \ M

-s_ D -v_

P

C ;]lG6ld6/ /]vfv08 PQ ≠ Q

/]vfv08 CD ≠ ;]lG6ld6/

@= lbOPsf] gfkcg';f/sf] /v] fv08 lvRgx' f];\ / gfd lbgx' f];\ M

-!_ % ;]=ld= -@_ !) ;]=ld=

#= tkfO‘{sf] sIffsf]7fdf sxfF sxfF sf]0f ag]sf 5g\ < vfh] L ugx{' f];\
/ sf0] f ags] f] 7fpFsf] gfd nV] g'xf];\ M

$= ?n/sf] ko| fu] u/L lbOPsf laGbx' ¿ hf8] /] sf0] f agfpgx' f];\,
zLif{laGb,' eh' fx¿ / sf0] fsf] gfd n]Vgx' f];\ M

AL

B C N M

e'hfx¿ M / eh' fx¿ M /
zLif{laGb' M zLif{laGb' M
sf]0f M sf]0f M d]/f] ul0ft, sIff # 95



%= sg' } b'O{cf6] f sf0] fx¿ lvRg'xf];\ . tL sf]0fx¿sf] gfd ABC / DEF
/fvL t'ngf ug'x{ f;] \ M

;fgf] sf]0f M 7n' f] sf]0f M

^= tn lbOPsf hf8] f sf]0fx¿ tn' gf ug{'xf];\ / ;dsf]0feGbf 7n' f]

sf0] fsf] gfd nV] g'xf;] \ M A

R

P QB C

;dsf]0feGbf 7'nf] sf]0f M

&= lbOPsf laGb'x¿nfO{ jm| d;} Fu hf]8]/ aGb cfsl[ t agfpgx' f;] \ .

o;/L ag]sf] cfs[ltsf] gfd nV] gx' f;] \ M E
FH
AD E

B CF GG

*= tn lbOPsf j:tx' ¿df sxfF sxfF cfotfsf/ jf juf{sf/ ;tx
5g\, 5'6\ofpg'xf];\ / nV] g'xf;] \ M

cleefjssf] b:tvt

lzIfssf] b:tvt 

96 d/] f] ul0ft, sIff #

ul0ftsf cfwf/et" lj|mof %

kf7 !@ hf]8

xl/;uF !@ cf]6f cG8f 5g\ . ;Ltf;Fu !^ cf]6f cG8f 5g\ .
bj' };uF u/L hDdf sltcf6] f cG8f xG' 5g\ <

xl/ ;Ltf

ul0ftLo jfSodf n]Vbf !@±!^ ≠ @* xG' 5 .
ca o;nfO{ :tDedf x]/f¹ x} t .

xl/;Fu cf]6f cG8f 5g\ . ;Ltf;Fu cf]6f cG8f 5g\ .

xl/;Fu cf]6f cG8f 5g\ . ;Ltf;Fu cf]6f cG8f 5g\ .

hDdf cG8f 5g\ .

cf]xf] ¤ lx;fa ug]{ of] tl/sf t lgs} ;lhnf] /x]5 .

d/] f] ul0ft, sIff # 97

/fdufk] fn / wlgofnF ] dn] fdf ?= !@) kg{] Pp6f d/' nL / ?= #^* kg{]
Pp6f emfn] f lsg] . pgLx¿n] hDdf slt ?lkofF vr{ u/] xfn] fg\ <

;o bz Ps ;o bz Ps

!@)

±# ^ *

$**

hDdfM $** ?lkofF
vr{ u/]5g\ .

98 d/] f] ul0ft, sIff #

lx;fa ugx{' f;] \ M ;o bz Ps ;o bz Ps
#* $@ ^
;o bz Ps ±% % #
$% ! ±@ ^ !
±#)^

;o bz Ps ;o bz Ps ;o bz Ps
*@^ %# @$%
±% ± #)
±@ # ^

!= /fl] xtsf] aur}‘ fdf #!& lnrLsf ?v / @$@ cfFksf ?v 5g\ .
pgsf au}r‘ fdf hDdf sltcf6] f ?vx¿ 5g\ <

@= gj;'{ Fu ##@ e8] f lyP . pgn] cfªl5l/ª;Fu #!$ e]8f lsg] . ca
pgL;uF hDdf slt e8] f eP < Anssf] k|ofu] u/]/ hf]8\gx' f];\ M

d/] f] ul0ft, sIff # 99

xftnfuL cfpg] hf8]

g]kfn cfbz{ dfWolds ljBfnodf ^%* ljBfyL{ 5g\ . /fli6«o cfwf/e"t
ljBfnodf @*# ljBfyL{ 5g\ . hDdf slt hgf ljBfyL{ eP <
;o bz Ps

!!
^%*
;o bz Ps ± @ * #
($!
* Ps / # Ps hf]8\bf !!
Ps eof] . !! Ps eg]sf]
! bz / ! Ps xf] .

ca ! bznfO{ bzsf]
:yfgdf /fv]/ hf]8f¹ .

bzsf] :yfgsf ;ª\Vofx¿
hf]8\bf !$ bz eof] . !$
bz eg]sf] ! ;o / $ bz
xf] . To;}n] hDdf ( ;o,
$ bz / ! Ps eof] . of]
($! xf] .

100 d]/f] ul0ft, sIff #

lx;fa ug'x{ f;] \ M

@(! @%^ #^&
± %* ±$ & ^ ± ^(

! (* &* (
±% ) % ±% @ * ±^ ( %

7f8f] ¿kdf /fvL lx;fa ug{x' f;] \ M

!%^ + $$* *% + &(# *%& + (#

±±±

d]/f] ul0ft, sIff # 101

tLg cªs\ sf ;ª\Vofsf] hf8]

cWoog u/L 5nkmn ug'x{ f];\ M
tLg cª\ssf ;ª\Vofsf] hf]8 s;/L ug]{ xf]nf <

Pssf] :yfgaf6 hf]8\g ;?' ug{'k5{ .

;osf] :yfgaf6 bz ;o cyjf ! xhf/nfO{
xhf/sf] :yfgdf n]Vg'k5{ .

^#% ^#% ^#%
±* ^ @ ±* ^ @ ±* ^ @
!$( &
& (&

lx;fa ug{'xf];\ M

&@$ ^#% *() %&^
±% !$ ±( @ ^ ±$( # ±% ^ &

@($ *^@ ()* &
±& % ! ±( # * ± (% ±((&

102 d]/f] ul0ft, sIff #

lx;fa ugx'{ f];\ M #%&
±* ^ #
%$%
±! $ ^

^%& %*(
±^ % $ ± $^

% &&
±! $ ^ ±# # #

d]/f] ul0ft, sIff # 103

lx;fa ug'{xf;] \ M
!= cfOtaf/ #^% cf]6f dfOjm| f]a;

sf7df8f¹af6 gfu9ª' \uf x'Fb} aflxl/P .
;f]daf/ $^* dfOj|mf]a; aflxl/P .
cfOtaf/ / ;f]daf/ u/L hDdf sltcf]6f
dfOj|mf]a; sf7df8fa¹ f6 aflxl/P <

@= lx/fn] ?= %^) df Pp6f l:j6/,
?= ^#) df Pshf]/ hQ' f / ?= (&)
df Pp6f Hofs]6 lslgg\ . pgLn] hDdf
slt ?lkofF vr{ ul/g\ <

?= %^) ?= ^#) ?= (&)

#= c?0fsf aa' fn] xf6ahf/df Pp6f
v;L ?= ^,$() df / Ps af]/f rfdn
?= @,%$) df a]r] . pgn] hDdf slt
?lkofF sdfP <

104 d/] f] ul0ft, sIff #

rf/ cª\ssf ;ª\Vofsf] hf8]

s'g} Pp6f ufpFkflnsfsf b'Oc{ f6] f j8fsf] hg;ªV\ of jm| dzM @$!%
/ !#^& 5 . b'j} j8fdf u/L hDdf slt hg;ª\Vof /x5] <

@$!%
±! # ^ &

! ! ! !
@$!% @$!% @$!% @$!%
±! # ^ & ±! # ^ & ±! # ^ & ±! # ^ &

@ *@ &*@ # &*@

lx;fa ugx{' f;] \ M

# #*$ ! ^ @( @ & ^* %%(@
±%)(@ ±$^ # % ±@(%& ±! # !*

$&%$ ^)& ^$*( *&

±# @*$ ±&*(^ ± % ! ( ±@(*(

d/] f] ul0ft, sIff # 105

hf]8\g'xf];\ M %*&
#^%
# (* ± $)#
@%$
± !&$ @(%$
!^^&
^% ±%%*&
*#%
± %*% $^(&
$# ^%
%(&^ ± $*^
!**(
± (%* ^!@#
@))&
$% ^ & ±! $% ^
#@%

±!$)@

106 d/] f] ul0ft, sIff #

lx;fa ugx'{ f;] \ M
!= Pp6f kf]v/Ldf !%)) cf]6f df5f 5g\ .

csf]{ kf]v/Ldf @%)) cf]6f df5f 5g\ .
b'j} kf]v/Ldf u/L hDdf slt df5fx¿
eP <

@= ;GWofsf] 3/ agfpg &*^) OF6fx¿
nfUof] . To;} u/L kvf{n agfpg @!$)
OF6f nfUof] . b'j}df u/L hDdf slt OF6f
nfUof] <

#= Pp6f afs;df !!$ cf]6f :ofp / csf]{
afs;df *^ cf]6f :ofp 5g\ . b'j} afs;df
u/L hDdf sltcf]6f :ofp eP <

d/] f] ul0ft, sIff # 107

$= eStk/' gu/kflnsfdf $^( cf]6f ;8saQL
5g\ . rfFu'gf/fo0f gu/kflnsfdf !)%
cf]6f ;8saQL 5g\ . bj' } gu/kflnsfdf
u/L hDdf sltcf]6f ;8saQL /x]5g\ <

%= Pp6f vf]/df (&% cf]6f sv' /' f 5g\ .
csf]{ vf]/df (* cf]6f sv' /' f 5g\ .
b'j} vf]/df u/L hDdf slt cf]6f
s'v/' f eP <

^= klxnf] cfFksf] ?vaf6 @*^ cf]6f
cfFk, bf];f| af6 #)( / t];|faf6
@(^ cf]6f cfFk l6lkof] . tLgcf]6f ?vaf6
u/L hDdf sltcf]6f cfFk l6lkof] <

108 d/] f] ul0ft, sIff #

kf7 !# 36fp
36fpgx' f;] \ M

!)) – !) ≠ () () – !) ≠ *)

!%& – !) ≠ – !) ≠

() – % ≠ – % ≠
– % ≠
^@ – % ≠

!)) – @) ≠ – @) ≠

!&# – @) ≠ – @) ≠

d]/f] ul0ft, sIff # 109

hf8] / 36fpsf] ;DaGw

5nkmn ug'{xf];\ / vfnL 7fpF eg{'xf];\ M

km/s kQf nufpg
of]ukmn kQf nufpg b'O{cf]6f 7'nf] ;ª\Vofaf6
;ª\Vofx¿ hf]8\gk' 5{ .
;fgf] ;ª\Vof 36fpg'k5{ .

!$ / ( sf] of]ukmn xG' 5 . !$ ± (

olb of]ukmn @# xg' ] bO' {cf]6f ;ª\VofdWo] Pp6f ;ª\Vof !$

eP csf]{ ;ª\Vof x'G5 . @#

olb Pp6f ;ª\Vof ( eP csf]{ ;ª\Vof x'G5 . @#
@# / ( sf] km/s x'G5 . –!$ –(

@# / !$ sf] km/s x'G5 . ( !$

%@ / $* sf] of]ukmn xG' 5 .
olb of]ukmn !)) x'g] b'O{cf]6f ;ªVofdWo] Pp6f ;ª\Vof $* eP
csf]{ ;ª\Vof xG' 5 .

olb Pp6f ;ª\Vof %@ eP csf]{ ;ª\Vof xG' 5 .
!)) / %@ sf] km/s x'G5 .
!)) / $* sf] km/s xG' 5 .

^) ± $% ≠ !)% — $% ≠
± $% ≠ !)% !)% — ^) ≠
!)% – ≠ ^)
^) ± ≠ !)%

110 d/] f] ul0ft, sIff #

tnsf ul0ftLo jfSox¿ k/" f ugx'{ f];\ M

#) ± $) ≠ &) %) ± $) ≠
&) – #) ≠
&) – $) ≠ – %) ≠ $)

() – ≠ %)

&) ± *) ≠ !)) ± () ≠
– !)) ≠ ()
– &) ≠ *) – () ≠ !))

!%) – ≠ &)

@) ± @) ≠
@) ± !) ≠

#) – ≠ @) – @) ≠ @)

– @) ≠ $) – ≠ @)

%) ± %) ≠ !))
%) ± $) ≠

() – ≠ $) !)) – ≠ %)

– $) ≠ %) – %) ≠ %)

&) ± &) ≠ () ± () ≠ !*)

!$) – ≠ &) !*) – ≠ ()

– &) ≠ &) – () ≠ ()

d]/f] ul0ft, sIff # 111

hf]8 / 36fp

Pp6f kfv] /Ldf @& cf]6f Eofu'tf 5g\ . kf]v/LeGbf aflx/ /x]sf
!$ cf6] f Eofu'tf kf]v/Ldf ylkP eg] kfv] /Ldf hDdf sltcf6] f
Eofut' f eP <

sf]7fdf Eofut' f ;ª\Vof
n]Vgx' f];\ .

hDdf Eofu'tf

kf]v/Ldf klxnf ePsf Eofut' f ylkPsf Eofu'tf

ul0ftLo jfSodf ± ≠

hDdf eof] .
Pp6f kmnkmn" Jofkf/L;uF #% lsnfu] f| d :ofp lyof] . pgn] *

lsnfu] f| d :ofp ar] ] . ca pgL;uF slt lsnfu] f| d :ofp afsF L /x] <

;'?df ePsf] :ofp afFsL :ofp lgsfNg 36fpsf]
laj|mL u/]sf] ko| f]u ul/G5 .
ul0ftLo jfSodf
112 d]/f] ul0ft, sIff # —≠

:ofp afFsL /x]sf] :ofp

cf]6f :ofp afFsL /x]5 .

hf8] / 36fp

Pp6f sIffdf @% hgf ljBfyL{ 5g\ . To;dWo] !# hgf 5fq
5g\ eg] slt hgf 5fqf /x5] g\ <

5fq 5fqf

ul0ftLo jfSoM − ≠ hDdf 5fqf

Ps hgf dflg; 3/af6 sfofn{ o hfbF f 6]Dkf]n] ?= @) / a;n]
?= !% ef8f lnG5 . b'O{cf]6f oftfoftsf ;fwgn] lng] ef8fdf
slt km/s /x]5 <
?lkofF

6]Dkf]

a; km/s ?lkofF
?lkofF

oxfF ef8fdf ePsf] km/s
kQf nufpg 36fpgk' 5{ .

ul0ftLo jfSoM ?lkofF
−≠

d]/f] ul0ft, sIff # 113

kflsª{ A df !! cf]6f df]6/;fOsn kfls{ª ul/Psf] 5 . kflsª{ B df
kflsª{ A df eGbf & cf6] f w/] } df6] /;fOsn kflsª{ ul/Psf] 5
eg] kfls{ª B df sltcf]6f df6] /;fOsn kfls{ª u/s] f] /x]5 .

kfls{ª A kfls{ª B

kfls{ª A ePsf df]6/;fOsn
kfls{ª B ePsf df]6/;fOsn

sg' rfFlx kfls{ªdf a9L df]6/;fOsn 5g\ <

kfls{ª B df df]6/;fOsn 5g\ .
cAb'nsf] pd]/ $* jif{ 5 . pxfsF L >LdtL pxfFeGbf ( jif{ sfG5L

l5g\ eg] pxfFsf] >LdtL slt pd/] sL eOg\ <

cAb'nsf] pd]/ jif{ jif{ sfG5L 5g\ .
>LdtLsf] pd]/ jif{

pgsf] >LdtLsf] pd]/ jif{

:tDelrqsf] k|of]u ubf{ sg' lj|mof ug]{ ;xh} 56' \ofpg ;lsG5 .

klxnf] kZ| gdf kfls{ª B df bf];|f] k|Zgdf cAbn' sf]
ePsf df]6/;fOsnsf] >LdtLsf] pd]/ kQf
;ª\Vof kQf nufpg !!±& nufpg $*–( ul/of] .
ul/of] .

114 d/] f] ul0ft, sIff #

k/ljgn] lxhf] d/] f] ul0ft k:' tssf] !$#
ki[ 7;Dd k9/] ;s] . k/ljgn] cfh !&( ki[ 7;Dd
k9/] ;s] eg] pgn] cfh slt ki[ 7 k9] <

lxhf];Dd k9]sf] k[i7 ;ª\Vof !$#
lxhf];Dd k9]sf] k[i7 ;ª\Vof
!$# ±
cfh k9]sf] k[i7 ;ª\Vof

!$# ± ≠ !&(

lxhf];Dd k9]sf] k[i7 ;ª\Vof cfh k9]sf] k[i7 ;ª\Vof

xfdLn] kQf nufpgk' g]{ k[i7 ;ª\Vof

t;y{, !$# ± ≠ !&(

dflysf lrqaf6 36fpsf] ko| f]u ug{ ;lsG5 .
æhDdf k[i7 ;ª\VofÆ – ælxhf];Dd k9]sf] k[i7 ;ª\VofÆ
≠ æcfh k9]sf] k[i7 ;ª\VofÆ

!&( – !$# ≠

k/ljgn] cfh k[i7 k9]5g\ .

d/] f] ul0ft, sIff # 115

xl/;uF %% dg} aQL 5g\ . pgn] sx] L d}gaQL
alxgLnfO{ lbP kl5 xl/;uF $@ cf6] f
dg} aQL afFsL /x]5g\ eg] xl/n] sltcf]6f
d}gaQL alxgLnfO{ lbPsf /x]5g\ <

;?' df ePsf] d}gaQL

%%

;?' df ePsf] d}gaQL

%% –

alxgLnfO{ lbPsf] d}gaQL

;?' df ePsf] hDdf d}gaQL

%% – ≠ $@
alxgLnfO{ lbPsf] afFsL /x]sf] d}gaQL

dflysf lrqaf6 ul0ftLo jfSodf n]Vbf %% – $@ ≠

alxgLn] cf]6f d}gaQL kfO5g\ .

ul0ftLo jfSodf ± $@ ≠ %% a'em¸ .

w]/} /fdf| ] . xfdLn] hf]8sf ;d:ofx¿nfO{ 36fpsf] ul0ftLo
jfSodf / 36fpsf ;d:ofx¿ hf]8sf] ul0ftLo jfSodf
n]vL lx;fa ug{ ;lsG5 .

116 d]/f] ul0ft, sIff #

sf7df8fa‘} f6 kfv] /f hfg] Pp6f a;df
sx] L dflg;x¿n] ofqf ul//xs] f
5g\ . dl' Ungdf ku' k] l5 & hgf
dflg;x¿ a;af6 cfl] nP{ . ca a;df
!( hgf dflg;x¿ afsF L 5g\ eg] hDdf
slt hgf dflg;x¿ sf7df8fa‘} f6 a;df r9s] f /x5] g\ <

a;df sf7df8f¹af6 r9]sf dflg;x¿

a;df sf7df8fa¹ f6 r9]sf dflg;x¿

–&

d'lUngdf a;af6 cf]ln{Psf dflg;x¿

a;df sf7df8f¹af6 r9]sf dflg;x¿

dl' Ungdf a;af6 a;df afFsL /x]sf – & ≠ !(
dflg;x¿ r9]sf /x]5g\ .
cf]ln{Psf dflg;x¿ dflg;x¿

a;df sf7df8f¹af6 hDdf

o; ;d:ofnfO{ ul0ftLo jfSodf s;/L n]Vg] <

& ± !( ≠ – !( ≠ &

– & ≠ !(

;d:of ;dfwfgsf nflu ;Gbe{ ae' m\g' dxãjk"0f{ x'G5 .

d]/f] ul0ft, sIff # 117

xl/n] lxhf;] Dd s]xL ;G' tnfsf la?jf /f]ks] f lyP . cfh pgn]
yk @) cf]6f ;G' tnfsf la?jf /f]k] . olb xl/sf] au‘r} fdf hDdf
&% cf]6f ;G' tnfsf la?jf /fl] k;lsP5 eg] lxhf;] Dd /f]lkPsf
la?jf slt lyP <

lxhf];Dd /f]lkPsf la?jfx¿

lxhf];Dd /f]lkPsf la?jfx¿ ± @)
hDdf ;G' tnfsf la?jfx¿ cfh /f]lkPsf la?jfx¿

lxhf];Dd /f]lkPsf la?jfx¿ ± @) ≠ &%
cfh /f]lkPsf la?jfx¿

pgn] lxhf];Dd hDdf la?jf /f]k]5g\ .

o; ;d:ofnfO{ ul0ftLo jfSodf n]Vbf,

&% – ≠ @) &% – @) ≠

118 d]/f] ul0ft, sIff #

/fdk/' ufpFdf ePsf !#^$ e}‘;LdWo] !@$@ e‘};Lsf] kzw' g ladf
ul/Psf] 5 . ca slt e;}‘ Lsf] ladf ug{ afsF L 5 <

kz' ljsf; a}ªs\

af/f, g]kfn

kzw' g ladfn]v

kzw' gLsf] gfd M /fdnvg ofbj

kz'sf] lsl;d M ufO{÷uf]?÷e¹;L÷/fFuf

k~hLs/0f ;ª\Vof M !#)!

ladf cjlw M @)&*÷!÷! b]lv @)&*÷!@÷#)

ladf /sd M ?= (),)))

pSt cjlwdf kz' d/]df ?= (),))) lbOg] 5 .

ldltM clwsf/Lsf] b:tvtM

e¹;Lsf] ;ª\VofnfO{ bz cfwf/ Ans k|of]u u/L 36fpFbf,

xhf/ ;o bz Ps

! #^$

afFsLM — ! @ $ @

!@@ e¹;Lsf] ladf ug{ afFsL 5 . !@@

36fpg'xf;] \ M

xhf/ ;o bz Ps xhf/ ;o bz Ps
&*)( (*&%

—$ # ) $ —$ # # %

d/] f] ul0ft, sIff # 119

k/z/' fdsf af/Lsf !@#% ;G' tnfsf af6] x¿dWo] &*# df
kmn nfus] f] 5 eg] sltcf6] f ;G' tnfsf af6] x¿df kmn nfus] f] 5g} <

;'Gtnfsf af]6x¿nfO{ bz cfwf/ Ans k|of]u u/L kg' M;d"xLs/0f ubf{,
! ;onfO{ !) bz /
! xhf/nfO{ !) ;o agfpFbf

afFsL M xhf/ ;o bz Ps
!! !#
$%@ cf]6f
af]6df kmn !@#%
nfu]sf] 5}g .
– &*#
120 d]/f] ul0ft, sIff # $%@

36fpg'xf;] \ M

@$# ^ !))) !))) !)) !)) !) !) !!!
– ! ! ^* !)) !)) !) !!!

!!!
!!!
!!!

!

!))) !))) !)) !)) !) !) !!!
!)) !)) !!!
!) !) !) !!
!) !) !)
!) !) !)
!)

klxnf] r/0f bf];f| ] r/0f t];|f] r/0f rf}yf] r/0f

@ !^ # !@ !^ # !@ !^ # !@ !^

@$# ^ @$# ^ @$# ^ @$# ^

–!!^* –!!^* –!!^* –!!^*

* ^ * @ ^ * !, @ ^ *
Pssf] :yfgdf bzsf] :yfgdf ; osf] :yfgdf xhf/sf] :yfgdf
36fpgx' f];\ . 36fpg'xf];\ . 36fpg'xf];\ . 36fpg'xf];\ .

d/] f] ul0ft, sIff # 121

36fpgx' f;] \ M % # @$
– #^$
^ # (%
–# & $ )

^#%^ (^#@
–$ # $ * – *(

*&^% &#^)
–^ # ( * –@ # ) *

&#^) %)))
–@ & ( ^ –!

122 d]/f] ul0ft, sIff #

sg' } Ps k|hfltsf r/fx¿ pQ/
Pl;ofaf6 hf8f] 5Ng gk] fn
cfp5F g\ hg' ;d'b| ;txaf6 (,#(%
ld6/sf] prfOdf p85\ g\ . df5fkR' 5«]
lxdfnsf] prfO ^((# ld6/ eP
tL r/fx¿ df5fkR' 5«] lxdfneGbf
slt dfly p85\ g\ <

Po/a; A380 df *%# hgf ofqx' ¿ ofqf ug{ ;S5g\ . gk] fn
Po/nfOG;sf] ;ae} Gbf 7n' f] xjfOh{ xfh A330 df @&& hgf ofqx' ¿
ofqf ug{ ;S5g\ . A380 df slt a9L ofq' ofqf ug{ ;S5g\ <

d]/f] ul0ft, sIff # 123

lx;fa ug'{xf];\ M
!= sg' } Ps jifd{ f @,(&@ cf6] f gofF a; gk] fn lelqP . kl5Nnf] jif{

@,#%$ cf]6f gofF a; g]kfn lelqP . klxnf] jifd{ f bf];|f] jifd{ f
eGbf slt w/] } lelqPsf /x]5g\ <

@= Pp6f df]afOnsf] ?= !,(%) k5{ . kd] f;uF ?= !%*) dfq 5 .
pSt dfa] fOn lsGg pgnfO{ yk slt ?lkofF cfjZos 5 <

#= h'Dnfdf ls;fgn] Ps afs; :ofp ?= !,@%) df Jofkf/LnfO{
aR] 5g\ . pSt :ofp gk] fnuGhsf pkef]Stfn] ?= !,(@) df lsG5g\ .
Jofkf/Ln] pSt :ofp slt a9Ldf laj|mL u/5] g\ <

124 d]/f] ul0ft, sIff #

$= hg cfbz{ dfWolds ljBfnosf @,!^# hgf ljBfyLd{ Wo] @(^ hgf
5fqjf;df a:b5g\ eg] slt hgf ljBfyL{ 5fqjf;df a:bg} g\ <

%= /dz] ;Fu ePsf] ?= $,))) dWo] pgn] ?= !,%%) df Pp6f
Hofs]6 lsg] eg] ca pgL;uF slt afsF L xf]nf <

^= /fdl/emgn] cfkmg\ f] au‘r} fdf !,@)) dj] fsf af]6 nufpg] ljrf/
u/] . olb pgsf] aur‘} fdf xfn &*$ af]6 5g\ eg] slt gofF af6]
/fK] gk' nf{ <

d]/f] ul0ft, sIff # 125

tflnsfdf g]kfnsf cf7 xhf/ ld6/eGbf sd prfO ePsf
sx] L lxdfnsf ljj/0f lbOPsf] 5 . To;sf cfwf/df tnsf
kZ| gx¿sf] pQ/ kQf nufpgx' f;] \ M

jm| =;= lxdfnsf] gfd prfO -ld6/df_
!= u0f]z lxdfn &,!^#

@= uf}/Lzª\s/ lxdfn &,!#$

#= ;}kfn lxdfn &,)#!

$= hu' n lxdfn ^,%#%

%= e[s6' L lxdfn ^,#^$

-s_ u0f]z lxdfnsf] prfO uf}/Lzª\s/
lxdfnsf] prfOeGbf slt a9L
/x]5 <

-v_ hu' n lxdfnsf] prfO ;}kfn
lxdfnsf] prfOeGbf sltn] sd
/x]5 <

-u_ u0f]z lxdfn / e[s6' L lxdfnsf]
prfOlarsf] km/s slt /x]5 <

126 d]/f] ul0ft, sIff #

lx;fa ugx'{ f];\ M
!= Pp6f r'ªu\ L agfpg :g]xfn] !%* / ;u' tn] *( /a/aofG8

ld;fP . pSt r'ªu\ Ldf slt /a/AofG8x¿ eP <

@= bO' { ;ªV\ ofx¿sf] ofu] kmn !,$#% 5 . olb Pp6f ;ªV\ of *@%
xf] eg] csf{] ;ªV\ of slt xf]nf <

!$#%
*@%

#= Pp6f sfo{jm| ddf !,#%) hgfsf nflu s';L{ /flvPsf] lyof] . olb
!,@&# dflg;x¿ dfq cfP eg] sltcf]6f s;' L{ vfnL /x] <

d/] f] ul0ft, sIff # 127

lx;fa ug{'xf];\ M
!= tn bO' { vfnsf 38Lx¿sf] d"No lbOPsf] 5 . 38L …sÚ sf] dN" o

38L …vÚ sf] eGbf slt ?lkofnF ] a9L 5 <

38L ‘s’ ?= #,^%( 38L ‘v’ ?= @,!^$

@= xfdf| ] ;xsf/Lsf #,^&* ;b:ox¿dWo] !,(*( dlxnf 5g\ eg]
k?' if ;b:ox¿sf] ;ªV\ of slt xfn] f <

128 d]/f] ul0ft, sIff #

#= jz} fv dlxgfdf cfndn] bw' a]r]/ ?= $,^*&, t/sf/L a]r]/
?= #,@%^ / kmnkmn" ar] /] ?= @,)%& sdfP . pgsf] j}zfv
dlxgfsf] hDdf cfDbfgL slt eof] <

$= hnhnf ufpdF f /x]sf] !) cf]6f kfx'gf3/x¿df @)&^ ;fndf
!,(%* g]kfnL ko6{ s / *^& ljb]zL ko{6sx¿ cfP eg] hDdf slt
ko6{ s cfP <

%= Pp6f l;gd] f 3/n] zlgaf/ @,!@# l6s6 a]Rof] . To;} u/L cfOtaf/
!,(*@ l6s6 aR] of] . ;fd] af/ !,*&^ l6s6 aR] of] . # lbgdf hDdf
slt l6s6 laj|mL eof] . olb # lbgdf %,))) l6s6 aR] g] nIo
/fvs] f] eP slt l6s6 a9L a]r]5 <

d/] f] ul0ft, sIff # 129

kf7 !$ u0' fg

u'0fg

Ps hgf lzIfsn] $ cf]6f l;;fsndx¿ /xs] f @÷@ cf6] f Kofs6]
# hgf ljBfyLn{ fO{ pknAw u/fP . ;f]sf nflu hDdf sltcf6] f
l;;fsnd cfjZos knf{ <

Ps hgf ljBfyL{sf nflu hDdf Kofs]6 ;ª\Vof

$ × @ ≠ * @ × # ≠ ^
# hgf ljBfyL{sf nflu
* × # ≠ ! ;]6df $ cf]6f
l;;fsnd 5g\ . t;y{,
$ × ^ ≠

hDdf l;;fsnd cfjZos kb5{ .

$ × @ × # $×@×#
-$ × @_ × # ≠ $ × -@× #_

pko{'St b'j} tl/sfaf6 u0' fg ubf{ u'0fgkmn Pp6} xG' 5 .

130 d]/f] ul0ft, sIff #

# × @ × @ nfO{ b'j} tl/sfn] lx;fa ug{x' f;] \ M

tl/sf !M -# × @ _ × @ ≠ ×@≠

tl/sf @M # × -@ × @_ ≠ # × ≠

@ × @ × $ nfO{ bj' } tl/sfn] lx;fa ugx{' f;] \ M

tl/sf !M -@ × @ _ × $ ≠ ×$≠

tl/sf @M @ × -@ × $_ ≠ @ × ≠

lx;fa ug'x{ f;] \ M @ ×@ × # ≠
# ×# × @ ≠ @ ×$ × @ ≠
# ×@ × $ ≠

lrq x/] L u'0fgsf ¿kdf nV] gx' f;] \ M

@×$≠*

d]/f] ul0ft, sIff # 131

z"Go;uF sf] u0' fg

;Ltf / xl/ u0' fg tflnsf xb] {} 5g\ . # × ) sf] u0' fgkmn slt xfn] f <

) ! @ # $ % ^ & * ( !) !! !@ !# !$ !%

# × ) sf] u0' fgkmn slt x'G5 ;f]rf¹ .

M u'0fg tflnsf # n]
M
# × % ≠ !% # n] 36]sf] 5 . 36\b} uPsf] 5 .
# × $ ≠ !@
# × # ≠ ( # n] 36]sf] 5 .
# × @ ≠ ^
# × ! ≠ # # n] 36]sf] 5 .
# × ) ≠ < # n] 36]sf] 5 . # × ) sf] u0' fgkmn # df #

36fpFbf cfpg] z"Go x'G5 .
# n] 36]sf] 5 .

sg' } ;ª\Vofn] z"GonfO{ u0' fg ubf{ u'0fgkmn z"Go x'G5 .
×)≠)
df h'g;s' } ;ª\Vof /fVg ;lsG5 .

;fy}, ) × ≠ ) x'G5 .

lx;fa ug{x' f;] \ M ( × ) ≠
) × & ≠
* × ) ≠ ) × % ≠
) × @ ≠ ) × ) ≠
! × ) ≠
) × $ ≠

132 d]/f] ul0ft, sIff #

u0' fg tflnsf

vfnL 7fpF eg{'xf];\ M

;f]rf¹ Û
u0' fgkmn s;/L kQf nufpg ;lsG5 .

× ! @ # $ % ^ & * ( !)
!!@
@
#(
$
% $%
^
&
*
( $%
!)

d]/f] ul0ft, sIff # 133

bO' { cªs\ sf] ;ª\VofnfO{ Ps cª\ssf] ;ª\Vofn] u'0fg

Pp6f kl/jf/df # hgf aRrfx¿ 5g\ . xh/' cfdfn] kT| os] nfO{
@! ?lkofsF f b/n] lbFbf hDdf slt ?lkofF rflxG5 <

hDdf ^# ?lkofF rflxG5 .
dflysf] ;d:ofnfO{ o;/L klg xn ug{ ;lsG5 .

k|To]s aRrfnfO{ @! ?lkofF # hgf aRrfnfO{
! ?lkofF × # ≠ # ?lkofF

@) ?lkofF × # hDdf ^# ?lkofF
≠ ^) ?lkofF

kZ| gnfO{ ul0ftLo jfSodf n]Vbf, @! × #

@! × # ≠ ^# x'G5 .

o;nfO{ 7f8f] ¿kdf /fv]/ u0' fg ug]{ tl/sf l;sf¹ .

134 d/] f] ul0ft, sIff #

@! × # nfO{ 7f8f] ¿kdf /fv]/ u0' fg ubf{, @!
@! × # ×#
!×#
#
@) × # ^)
^#

@! × # ! × # ≠ #

@) × # ≠ ^)
hDdf ^#

@! × # nfO{ o;/L klg u'0fg ug{ ;lsG5,

klxnf] r/0f M Pssf] :yfg bf];|f] r/0f M bzsf] :yfg

@! @! @!
×# ×# ×#

:yfgdfgcg';f/ # ^#
!×# @×#
Pssf] :yfgdf u'0fg bzsf] :yfgdf u0' fg

n]Vg] u/L n]Vg], ! × # ≠ # u/L k|fKt u'0fgkmn
bzsf] :yfgdf n]Vg]

lx;fa ug'x{ f;] \ M

!@ !! #@
×$ ×& ×#

d]/f] ul0ft, sIff # 135

dfNaf]s s/] fsf] Pp6f sfOoF f]df @% cf6] f s]/f /x5] g\ . olb
ljjs] n] # sfOoF f] s/] f lsg] eg] hDdf sltcf6] f s]/f lsg] <

Pp6f sfOFof]df ePsf s]/fsf] ;ª\Vof ≠ @%
hDdf lsg]sf] s]/fsf] sfOFof] ;ª\Vof ≠ #
hDdf lsg]sf] s]/fsf] ;ª\Vof ≠ @% × #

@% @% @% @%
×# ×# ×# ×#

!% !% !%
+^
% × # ≠ !% ^
@×#≠^ &%
Pssf] :yfgdf /x]sf] %
nfO{ # n] u0' fg ubf{ !% bzsf] :yfgdf /x]sf] @ ca Pssf] :yfgdf
x'G5 . !% df ! bz / % nfO{ # n] u'0fg ubf{ ^ /x]sf] u'0fgkmn /
Ps xG' 5 . ! nfO{ bz xG' 5 . ^ nfO{ bzsf] bzsf] :yfgdf /x]sf]
/ % nfO{ Pssf] :yfgdf :yfgdf n]Vg'kb{5 . u0' fgkmnnfO{ j|mdzM
n]Vgk' b{5 . hf]8\gk' b{5 .

o;nfO{ o;/L klg u0' fg ug{ ;lsG5 M ! -3_ @ (
@% ×^
×#
&%

u'0fg ugx{' f;] \ M

-s_ # % -v_ $ ^ -u_ @ *
×@ ×@ ×#

-ª_ $ @ -r_ # % -5_ @ ^
×& ×# ×$

136 d/] f] ul0ft, sIff #

Pp6f sf6'{g afs;df #@ cf]6f ;G' tnf 5g\ . To:t} vfnsf $
cf6] f sf6{g' df sltcf6] f ;'Gtnf xf]nfg\ <

#@ × $ #@ !@* cf6] f
≠ !@* ×$ ;'Gtnf 5g\ .
!@*


lx;fa ug{'xf];\ M

!= Pp6f cfFksf] ?vaf6 &#
cf]6fsf b/n] # cf]6f ?vaf6
cfFk l6Kbf sltcf]6f cfFk
l6lkof] <

@= Pp6f j|m6] df #) cf6] f cG8f 5g\
eg] & cf]6f jm| 6] df sltcf6] f
cG8f xf]nfg\ <

#= tLg sIffdf @ cf6] f ;]S;g
5g\ . Pp6f ;S] ;gdf ## hgf
ljBfyL{ 5g\ eg] hDdf slt
hgf ljBfyL{ xfn] fg\ <

d]/f] ul0ft, sIff # 137

u'0fg ugx'{ f;] \ M @ * bz ^ Ps

# ! bz ( Ps *^ × $
!( × $
×$ ×$
& ^ $ bz #^ Ps
# $ $ #@ bz @$ Ps
$) ± #^ ≠ &^
#@) ± @$ ≠ #$$

& % *@
×% ×$

*$ (#
×% ×$

$% &^
×^ ×^

138 d]/f] ul0ft, sIff #

u0' fg ug{x' f];\ M %
@% × & slt x'G5 <
@) s] & × @% eg]sf]
@% × & xf] <
&

@% % @) # dflysf] tflnsfnfO{
×& ×& @% csf]{ tl/sfn] /fVbf,
×& # % !$) & × @% xG' 5 .
#% @% × & ≠ !&% ×&
!&%
+ !$)
!&% @% × & / & × @% bj' }n]
u0' fgkmn !&% lbG5g\ .

u0' fg ug{x' f;] \ M

#* * × ^ ≠ $* #* #*
#) × ^ ≠ !*) ×^ ×^
× ^ @@*

^& ^& ^&
×% ×% ×%

d/] f] ul0ft, sIff # 139

u0' fg ug'{xf;] \ M @* @*
×& ×&
@*
×& #(
×$
#( #(
×$ ×$ &*
×^
&* &*
×^ ×^ %&
×*
%& %&
×* ×* ^$
×&
^$ ^$
×& ×&

140 d/] f] ul0ft, sIff #

bO' { cª\ssf] ;ªV\ ofnfO{ Ps cª\ssf] ;ªV\ ofn] u0' fg

u'0fg ugx{' f];\ M

!% × # !% $%
!) %
# #) !%

!* × $ !*
!) *
$

@ ! × % @!
@) !
%

# @ × @ #@
#) @
@

$* × ^ $*
$) *
^

*% × # *%
*) %
#

d]/f] ul0ft, sIff # 141

b'O{ cªs\ sf] ;ªV\ ofnfO{ Ps cªs\ sf] ;ªV\ ofn] u0' fg

Pp6f afs;df p:t} vfnsf $) k':tsx¿ c6fp5F g\ eg] *
cf6] f afs;x¿df sltcf]6f k':tsx¿ c6fpnfg\ <

$)±$)±$)± ===== ±$) -* k6s_ ≠ #@)

o;f] ubf{ a9L ;do nfU5 .
$) × * ubf{ sd ;do nfU5 .

klxn] z"Goafx]s cGo $) df ePsf] z"GonfO{ k|fKt
cª\sx¿ u0' fg u/f¹ . ;ª\Vofsf] k5fl8 /fVgx' f];\ .

$ × * ≠ #@ $) × * ≠ #@)

$) × * ≠ #@) k:' tsx¿ 5g\ .

u'0fg ugx{' f];\ M

@) × @ ≠ $) × ^ ≠
@) × $ ≠ %) × ^ ≠
#) × ! ≠ *) × & ≠
#) × & ≠ () × * ≠

142 d]/f] ul0ft, sIff #

vfnL 7fpdF f l7s cªs\ /fVg'xf;] \ M

! ! $
× # × $ × $
#( $*
%^

@ @ @
× $ × # × #
*) &@ &*

( ( *
× # × ^ × #
@&( %$^ @^$

^ ^ ^
× ^ × $ × ^
#() @&@ #(^

d]/f] ul0ft, sIff # 143

b'O{ cªs\ sf] ;ª\VofnfO{ b'O{ cªs\ sf] ;ªV\ ofn] u'0fg

;Ltf;uF #) Kofs]6 rsn]6 5g\ . Ps Kofs6] df @) cf]6f rsn]6
eP pgL;uF hDdf sltcf6] f rsn]6 xG' 5g\ <

@) × #) s;/L ug]{ xf]nf, ;f]rf¹ .

≠ @ × !) × # × !)

≠ @ × # × !) × !) @) × #) ubf{ @ × #
≠ ^ × !) × !) u/]/ k5fl8 ePsf bO' {cf]6f

≠ ^) × !) z"Gox¿ yKbf klg xg' ] /x]5 .

≠ ^))

lx;fa ug{x' f];\ M #) × @) ≠
#) × $) ≠
!) × @) ≠ $) × %) ≠
#) × #) ≠ ^) × $) ≠
$) × @) ≠ ^) × &) ≠
^) × %) ≠ %) × () ≠
#) × *) ≠ &) × () ≠
&) × *) ≠ () × () ≠
() × ^) ≠
() × *) ≠

144 d]/f] ul0ft, sIff #