mwÜ
3. mwÜ
3.1 mwÜ
3.2 wemM© mwÜ
Kg©-Abykxjb
3. mwÜ
3.1 mwÜ
gvbyl K_v ejvi mgq K_vi
MwZ e„w× cvq| `ªyZ K_v ejvi mgq KL‡bv KL‡bv KvQvKvwQ _vKv `y‡Uv
aŸwbi D”PviY
GKwÎZ n‡q hvq| e¨vKi‡Y G‡K mwÜ ejv nq| †hgb : ÔAvwg we`¨v Avj‡q hve|Õ
evK¨wU ejvi mgq ÔAvwg
we`¨vj‡q hveÕ n‡q hvq| GLv‡b Ôwe`¨vÕ-Gi ÔAvÕ-aŸwb Ges ÔAvjqÕ-Gi
ÔAvÕ-aŸwb wg‡j †M‡Q| W.
gyn¤§` knx`yjøvn& wj‡L‡Qb : `ywU eY© AZ¨šÍ wbKUeZ©x n‡j D”Pvi‡Yi
myweavi
Rb¨ Df‡qi wgjb n‡q GK eY©
ev G‡Ki i~cvšÍi ev G‡Ki †jvc ev Df‡qi i~cvšÍi n‡jÑ Gi~c wgjb‡K
mwÜ e‡j|
Z‡e Zv‡K mwÜ e‡j|
mwÜ k‡ãi A_© wgjb| `y‡Uv
aŸwbi mw܇Z cÖ_g k‡ãi †kl aŸwb Ges c‡ii k‡ãi cÖ_g aŸwbi wgjb
N‡U| mwÜi d‡j bZzb bZzb
k‡ãi m„wó nq| K_v ejvi mgq k‡ãi D”PviY mnR nq| fvlv mswÿß
nq Ges ïb‡Z fv‡jv jv‡M|
mw܇Z aŸwbi Pvi ai‡bi wgjb
nq :
1. Dfq aŸwb wg‡j GKwU aŸwb
nq|
2. GKwU aŸwb e`‡j hvq|
3. GKwU aŸwb †jvc cvq|
4. Dfq aŸwbi e`‡j bZzb
aŸwbi m„wó nq|
evsjv mwÜ `y cÖKvi : 1.¯^
imwÜ, 2. e¨ÄbmwÜ|
1. ¯^
imwÜ :¯^
iaŸwbi m‡½¯^
iaŸwbi wgj‡b †h mwÜ nq,
Zv‡K¯^
imwÜ e‡j| †hgb :
†mvbv + Avwj = †mvbvwj
iycv + Avwj = iycvwj
wg_¨v + DK = wg_y¨K
Kzwo+GK= KzwoK
b`x + Gi = b`xi gv + Gi =
gv‡qi
2. e¨ÄbmwÜ :¯^
iaŸwbi m‡½ e¨ÄbaŸwb wKsev
e¨ÄbaŸwbi m‡½ e¨ÄbaŸwb wgwjZ n‡q †h
mwÜ nq, Zv‡K e¨ÄbmwÜ e‡j|
†hgb :
KuvPv + Kjv = KuvPKjv bvwZ
+ †eŠ = bvZ‡eŠ
†QvU + `v = †Qvo`v Dr +
PviY = D”PviY
Avi + bv = Avbœv Pvi + wU
= PvwÆ
evsjv fvlvq A‡bK ms¯‹…Z kã
†Kv‡bv iKg cwieZ©b Qvov e¨eüZ nq| G¸‡jv‡K Zrmg kã e‡j|
Gme Zrmg k‡ãi mwÜ ms¯‹…‡Zi
wbq‡gB nq| ZvB ms¯‹…‡Zi wbqg †g‡b evsjv fvlvq e¨eüZ Zrmg
k‡ãi mwÜ 3 cÖKvi|h_v :
K.¯^
imwÜ L. e¨ÄbmwÜ 3.
wemM©mwÜ
K.¯^
imwÜ : mnRm~Î I D`vniY
1. A, Av aŸwbi mwÜ
m~Î D`vniY D`vniY
A+A=Av(v) be+Abœ= bevbœ
cig + AYy = cigvYy
A+Av=Av(v) Rj+ Avkq =
Rjvkq cvV + AvMvi = cvVvMvi
Av+A=Av(v) K_v+ Ag„Z =
K_vg„Z Avkv + AZxZ = AvkvZxZ
Av+Av=Av(v) we`¨v + Avjq =
we`¨vjq gnv + Avkq = gnvkq
2. B, C aŸwbi mwÜ
B+B=C(x) AwZ+BZ= AZxZ iwe
+ B›`ª = iex›`ª
B+C=C(x) cwi+C ÿv = cixÿv
Awa + Ck¦i = Aaxk¦i
C+B=C(x) kPx+ B›`ª =
kPx›`ª gnx + B›`ª = gnx›`ª
C+C=C(x) kªx+Ck= kªxk
c„w_ex + Ck¦i = c„w_exk¦i
3. D, E aŸwbi mwÜ
D+D=E(~) KUz+Dw³= KU~w³
giy + D`¨vb = gi~`¨vb
D+E=E(~) jNy+ Ewg© =
jN~wg© eû + EaŸ© = en~aŸ©
mwÜ 17
E+D=E(~) ea~+ Drme =
ea~rme ea~ + Dw³ = ea~w³
E+E=E(~) f~+ Ea٩ = f~a٩
2. A/Av, B/C aŸwbi mwÜ
A/Av + B/C = G (†) ¯^ + B”Qv
= †¯^”Qv h_v + Bó = h‡_ó
Ac+C ÿv = A‡cÿv gnv + Ck¦i
= g‡nk¦i
3. A/Av, D/E aŸwbi mwÜ
A/Av+D/E=I(†v) m~h© + D`q
= m~‡h©v`q K_v + DcK_b = K‡_vcK_b
Pj + Ewg© = P‡jvwg© gnv +
Ewg© = g‡nvwg©
4. A/Av, F aŸwbi mwÜ
A/Av+F=Ai(©) †`e+Fwl=
†`ewl© ivRv + Fwl = ivRwl©
A/Av + FZ = Avi kxZ + FZ =
kxZvZ© ÿzav+FZ= ÿzavZ©
5. A/Av, G/H aŸwbi mwÜ
A/Av + G/H = H (ˆ) Rb + GK
= R‰bK gZ + HK¨ = g‰ZK¨
Z_v + Ge = Z‰_e gnv +
Hk¦h© = g‰nk¦h©
6. A/Av, I/J aŸwbi mwÜ
A/Av+I/J=J(†Š) Rj+IKv=
R‡jŠKv gnv + Ilwa = g‡nŠlwa
wPË + J`vh© = wP‡ËŠ`vh©
gnv + Jla = g‡nŠla
7. B/C-Gi ci wfbœ aŸwbi
mwÜ
B/C + A/Av = h (¨/¨v) AwZ
+ AšÍ= AZ¨šÍ BwZ + Avw` = BZ¨vw`
b`x + A¤^y = b`¨¤^y gmx +
Avavi = gm¨vavi
B/C + D/E = h (¨y/¨~) AwZ
+ Dw³ = AZy¨w³ cÖwZ + El = cÖZz¨l
B/C+G=h(†¨) cÖwZ + GK =
cÖ‡Z¨K
8. D/E-Gi ci wfbœ aŸwbi
mwÜ
E/E + A/Av = e/ev my + Aí
=¯^
í my + AvMZ =¯^
vMZ
D/E + B/C = we/ex Aby + BZ
= Awš^
Z Zby+C=Z š^
x
D/E+G=†e Aby+GlY =A‡š¦lY
18 evsjv e¨vKiY wbwg©wZ
9. G/H-Gi ci wfbœ aŸwbi
mwÜ
G/H + A/Av = Aq/Avq †b +
Ab = bqb ˆb + AK = bvqK
10.I/J-Gi ci wfbœ aŸwbi
mwÜ
I/J + A/Av = Ae/Ave †cv +
Ab = ceb †Mv + Avw` = Mevw`
†cŠ + AK = cveK
I/J+B= Awe/Avwe †cv + BÎ =
cweÎ †bŠ + BK = bvweK
I/J+D= Avey †fŠ + DK =
fveyK
I/J+G=A‡e †Mv+ GlYv =
M‡elYv
L. e¨ÄbmwÜ : cwiwPZ m~Î I
D`vniY
1. ¯^
iaŸwb + e¨ÄbaŸwbi mwÜ
A+Q=”Q
cÖ+Q`= cÖ”Q`
Av + Q =”Q
K_v + Q‡j = K_v”Q‡j
B+Q=”Q
cwi+Q`= cwi”Q`
D+Q=”Q
Ziy+ Qvqv = Ziy”Qvqv
2. e¨ÄbaŸwb +¯^
iaŸwbi mwÜ
K+A=M w`K& + AšÍ=
w`MšÍ
U + Av = W> o lU& +
Avbb = lovbb
Z+A=` Zr+A šÍ=Z` šÍ
c+A=e myc& + AšÍ=
myešÍ
3. e¨ÄbaŸwb + e¨ÄbaŸwbi
mwÜ
Z + P/Q = ”P/”Q
Dr + PviY = D”PviY Dr + †Q` = D‡”Q`
` + P/Q = ”P/”Q
wec` + Pq = wec”Pq wec` + Qvqv = wec”Qvqv
Z + R/S = ¾/À hver + Rxeb
= hve¾xeb Kzr + SwUKv = KzÀwUKv
`+R=¾ wec` + RbK = wec¾bK
Z+W=Ç Dr+Wxb= DÇxb
Z+j=j ø Dr + wjwLZ =
DwjøwLZ
Z+k=”Q
Pjr+kw³= Pj”Qw³
Z+n= × Zr+wnZ=Zw ×Z
`+n= × c`+nwZ=c ×wZ
mwÜ 19
K&+`=M&+` evK&
+ `vb = evM&`vb
K&+e=M&+e w`K&
+ weRq = w`wM¦Rq
U&+h=W&>o+h
lU&+h š¿= lohš¿
Z&+M=`&+M Dr+ wMiY
= D`&wMiY
Z&+N=`&+N Dr+ NvUb
= D`&NvUb
Z&+e=`&+e Dr+eÜb=
DØÜb
Z&+f=`&+f Dr+fe=
D™¢e
Z&+i=`&+i Zr+i~c=
Z`ª~c
K&+b=O+b w`K& +
wbY©q = w`KwbY©q
Z&+g=b+g Zr+ g‡a¨ =
Zb¥‡a¨
Z&+b=bœ Dr+bwZ= DbœwZ
RMr + bv_ = RMbœv_
Z&+g=b¥ Zr+gq= Zb¥q
g„r + gq = g„b¥q
g&+K=O/s+K
kg&+Kv=k¼v mg&+ KxY© = msKxY©
g&+L=s+L mg&+L¨v=
msL¨v
g&+M=s+M mg&+MxZ=
msMxZ
g&+N=s+N mg&+NvZ=
msNvZ
g&+P=Â mg&+Pq=mÂq
g&+Z= šÍ mg&+Zvc=m
šÍvc
g&+`=›` mg&+ `k©b
= m›`k©b
g&+a=Ü mg&+avb=
mÜvb
g&+b=bœ wKg& + bi
= wKbœi
g&+h=s+h mg&+hg=
mshg
g&+i=s+i mg&+i ÿY
= msiÿY
g&+k=s+k mg&+kq=
mskq
g&+n=s+n mg&+nvi=
msnvi
20 evsjv e¨vKiY wbwg©wZ
3.2 wemM©mwÜ
wemM© (t )-Gi m‡½¯^
iaŸwb wKsev e¨ÄbaŸwbi †h
mwÜ nq, Zv‡K wemM©mwÜ e‡j| D”Pvi‡Yi
w`K †_‡K wemM© `y iKg :
1. i&-RvZ wemM© :k‡ãi
†k‡l i& _vK‡j D”Pvi‡Yi mgq i& †jvc cvq Ges i&-Gi RvqMvq
wemM© (t)
nq| D”Pvi‡Y
i& eRvq _v‡K| †hgb : AšÍi>A šÍt+MZ=A šÍM©Z
( Ab&‡Zvi&M‡Zv )|
6. m&-RvZ wemM© :k‡ãi
†k‡l m& _vK‡j mwÜi mgq m& †jvc cvq Ges m&-Gi RvqMvq wemM©
( t ) nq| D”Pvi‡Y
m& eRvq _v‡K| †hgb : bgm& > bgt + Kvi = bg¯‹vi ( b‡gvk&Kvi&
)|
wemM©mwÜ `y fv‡e mvwaZ nq
: 1. wemM©(t)I ¯^
iaŸwb wg‡j
2. wemM©(t)I e¨ÄbaŸwb
wg‡j|
1. wemM© I¯^
iaŸwbi mwÜ
K. A-aŸwbi m‡½ wemM© Ges
c‡i A-aŸwb _vK‡j wemM© I A-aŸwb ¯’‡j I-Kvi nq|†hgb :
ZZt + AwaK = Z‡ZvwaK hkt +
Awfjvl = h‡kvwfjvl
eqt + AwaK = e‡qvwaK
L. A-aŸwbi m‡½ wemM© Ges
c‡i A, Av, D-aŸwb _vK‡j wemM© I A-aŸwb wg‡j i nq|†hgb :
cybt + AwaKvi = cybiwaKvi
cÖvZt + Avk = cÖvZivk
cybt + Ave„wË = cybive„wË
cybt + D³ = cybiy³
2. wemM© I e¨ÄbaŸwbi mwÜ
K. A-aŸwbi m‡½ wemM© Ges
c‡i e‡M©i 3q/ 4_©/ 5g aŸwb A_ev h, i, j, n _vK‡j wemM© I
A-aŸwb ¯’‡j i-RvZ wem‡M©
i/ †id(©)Ges m-RvZ wem‡M© I-Kvi nq|†hgb :
i-RvZ wemM© : i
AšÍt+MZ=A šÍM©Z cybt + Rb¥
= cybR©b¥
AšÍt+avb=A šÍa©vb cybt +
evi = cybe©vi
AšÍt+fz³=A šÍfz©³ cybt +
wgjb = cybwg©jb
mwÜ 21
m-RvZ wemM© : I
gbt + MZ = g‡bvMZ m`¨t +
RvZ = m‡`¨vRvZ
wZit + avb = wZ‡ivavb Zct
+ eb = Z‡cveb
Aat + gyL = A‡avgyL gbt +
†hvM = g‡bv‡hvM
gbt + ig = g‡bvig gbt +
†jvfv = g‡bv‡jvfv
gbt + ni = g‡bvni
L. wem‡M©i c‡i P/Q _vK‡j
wem‡M©i ¯’‡j k; U/V _vK‡j l Ges Z/_ _vK‡j m nq|†hgb :
wbt + Pq = wbðq wkit + †Q`
= wki‡ñ`
`yt + PwiÎ = `yðwiÎ wbt +
wQ`ª = wbwñ`ª
abyt + U¼vi = abyó¼vi wbt
+ Vzi = wbôzi
PZzt + Uq = PZzóq
`yt + Zi = `y¯Íi wbt + †ZR
= wb‡¯ÍR
BZt + ZZ = BZ¯ÍZ `yt+_=
`y¯’
M. A/Av wfbœ Ab¨¯^
‡ii m‡½ wemM© Ges c‡i¯^
iaŸwb, e‡M©i 3q / 4_© / 5g
aŸwb A_ev h, i,
j, n _vK‡j wemM© ¯’‡j i
nq|†hgb :
wbt + Aewa = wbiewa wbt +
Avc` = wbivc`
wbt + MZ = wbM©Z wbt + NÈ
= wbN©È
wbt + evK = wbe©vK wbt +
fq = wbf©q
Avwet + fve = Avwef©ve
Avkxt + ev` = Avkxe©v`
`yt + Ae¯’v = `yie¯’v `yt
+ AvPvi = `yivPvi
`yt + MwZ = `yM©wZ `yt +
†eva = `y‡e©va
cÖv`yt + fve = cÖv`yf©ve
`yt + gi = `yg©i
`yt + †hvM = `y‡h©vM `yt +
jf = `yj©f
22 evsjv e¨vKiY wbwg©wZ
N. i-RvZ wem‡M©i c‡i i
_vK‡j wemM© †jvc cvq Ges cÖ_g ¯^i `xN© nq|†hgb :
wbt + ie = bxie wbt + im =
bxim
wbt + †ivM = bx‡ivM
O. A/Av aŸwbi m‡½ wemM©
Ges c‡i K, L, c, d _vK‡j wemM© ¯’‡j m nq|†hgb :
bgt + Kvi = bg¯‹vi wZit +
Kvi = wZi¯‹vi
cyit + Kvi = cyi¯‹vi fvt +
Ki = fv¯‹i
P. B/D aŸwbi m‡½ wemM© Ges
c‡i K, L, c, d _vK‡j wemM© ¯’‡j l nq|†hgb :
wbt + Kvg = wb®‹vg wbt +
cvc = wb®úvc
wbt + dj = wbõj ewnt + Kvi
= ewn®‹vi
PZzt + c` = PZz®ú` PZzt +
†KvY = PZz‡®‹vY
Q. †Kv‡bv †Kv‡bv †ÿ‡Î mwÜi
wemM© †jvc cvq bv|†hgb :
cÖvZt + Kvj = cÖvZtKvj gbt
+ Kó = gbtKó
wkit + cxov = wkitcxov
AšÍt+KiY=A šÍtKiY
R. †Kv‡bv †Kv‡bv †ÿ‡Î
wemM© †jvc †c‡jI mwÜ nq bv|†hgb :
AZt + Ge = AZGe
mwÜ 23
Kg©-Abykxjb
1. mZx Ck evey gva¨wgK
we`¨v Avj‡qi wkÿK| Zvi Ang& Kvi †bB, wZwb mg& MxZ fv‡jvev‡mb|
ÑDׄZvs‡k †gvUv `vM †`qv
k㸇jvi mwÜ Ki| †Zvgvi cQ›` g‡Zv GiKg Av‡iv K‡qKwU
k‡ãi evK¨ †jL|
2. Zzwg AvR mviv w`b hv‡`i
K_vevZ©v ï‡bQ, †mme K_vi g‡a¨ †_‡K g‡b K‡i 10wU mwÜe×
kã †ei Ki| Zvici †m¸‡jv‡K
Avjv`v K‡i †Kvb cÖKvi mwÜ Zv †jL| †hgb: †Zvgvi evev
ej‡jb, Ò†`vKvb †_‡K
cuvk‡mi Avjy, †MvUv cuv‡PK KuvPKjv Avi GK †KŠUv Me¨-N„Z wb‡q Gm
ZvovZvwo| †`L, †hb †ekKg
bv nq|Ó Ñ GLv‡b mwÜe×k㸇jv n‡”QÑ
cuvk‡mi, cuv‡PK, KuvPKjv,
Me¨, †ekKg|
cuvk‡mi = cuvP + †mi =
e¨ÄbmwÜ
cuv‡PK = cuvP + GK =
e¨ÄbmwÜ
KuvPKjv = KuvPv + Kjv =
e¨ÄbmwÜ
Me¨ = †Mv + h = e¨ÄbmwÜ
†ekKg = †ewk + Kg =
e¨ÄbmwÜ
3. ewn®‹vi, Avwe®‹vi,
bg¯‹vi, cyi¯‹vi k㸇jv mwÜi †Kvb wbqg †g‡b MwVZ n‡q‡Q?
wbqg¸‡jv †jL Ges †m wbqg
Abyhvqx †Zvgvi cvV¨ eB †_‡K Av‡iv K‡qKwU kã Luy‡R wb‡q
GKwU ZvwjKv ˆZwi Ki|
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