एनसीईआरटी पाठ्यपुस्तक: उपभोक्ता के व्यवहार का सिद्धांत | Must Read Old & New NCERTs for UPSC CSE (Hindi) PDF Download

 Page 1


miHkksDrk osQ
O;ogkj dk fl¼kar
bl vè;k; esa ge ,d O;fDrxr miHkksDrk osQ O;ogkj dk vè;;u djsaxsA
miHkksDrk dks ;g fu.kZ; djuk gksrk gS fd og viuh vk; dks fofHkUu oLrqvksa
1
ij fdl çdkj O;; djsaA vFkZ'kkL=kh bls pquko dh leL;k dgrs gSaA vr%
LokHkkfod :i ls] dksbZ Hkh miHkksDrk oLrqvksa osQ ,sls la;kstu dks çkIr djuk
pkgsxk tks mls vf/dre larks"k çnku djrk gSA ;g ^loksZÙke la;ksx* D;k gksxk\
;g miHkksDrk dh #fp;ksa vkSj og fdruk O;; dj ldrk gS] oLrqvksa dh
dherksa vkSj mldh vk; ij fuHkZj djrk gSA ;g vè;k;] miHkksDrk O;ogkj dks
le>kus okys nks fHkUu fl¼karksa dks çLrqr djrk gSA
izkjafHkd laosQru rFkk vfHkxzg
miHkksDrk lkekU; :i ls cgqr&lh oLrqvksa dk miHkksx djrk gS] ijarq
ljyhdj.k osQ fy, ge miHkksDrk dh p;u leL;k ij ,slh fLFkfr esa fopkj
djsaxs] tgk¡ osQoy nks gh oLrq,¡
2
 gksaA ge bu nksuksa oLrqvksa dks osQyk rFkk vke
dgsaxsA nksuksa oLrqvksa dh ek=kkvksa dh dksbZ Hkh lfEefyr jkf'k dks miHkksDrk
caMy vFkok la{ksi esa caMy dg ldrs gSaA lkekU;r% ge osQys dh ek=kk dks
O;Dr djus osQ fy, x
1
 ifjorZ dk vkSj vke dh ek=kk dks O;Dr djus osQ
fy, x
2
 ifjorZ dk mi;ksx djsaxsA x
1
 vkSj x
2
 /ukRed ;k 'kwU; gks ldrs gSaA
(x
1
,x
2
), dk rkRi;Z gksxk fd osQyk dh x
1
 ek=kk rFkk vke dh x
2
 ek=kkA x
1
rFkk x
2
 osQ fdlh fo'ks"k ewY; osQ fy, (x
1
, x
2
)] gesa ,d fo'ks"k caMy iznku
djrh gSA mnkgj.kkFkZµ caMy (5] 10) esa OksQys dh 5 bdkb;k¡ vkSj vke dh
10 bdkb;k¡ gSa_ caMy (10]5) esa osQys dh 10 bdkb;k¡ vkSj vke dh 5
bdkb;k¡ gSaA
2-1 mi;ksfxrk
lkekU;r% ,d miHkksDrk fdlh oLrq osQ fy, viuh ek¡x dk vuqeku ml
mi;ksfxrk (vFkok larks"k) OksQ vk/kj ij yxkrk gS tks og mlls çkIr djrk gSA
mi;ksfxrk D;k gS\ ,d oLrq dh mi;ksfxrk] mldh fdlh vko';drk dks
larq"V djus dh {kerk gSA oLrq dh ftruh T;knk vko';drk gksrh gS vFkok
1
^oLrqvksa* 'kCn dk iz;ksx loZ=k oLrqvksa rFkk lsokvksa nksuksa osQ fy, fd;k x;k gSA
2
;g /kj.kk gS fd oLrq,¡ osQoy nks gh gSa fo'ys"k.k dks ljy dj nsrh gSa vkSj ljy vkjs[kksa osQ ”kfj, egÙoiw.kZ
ladYiukvksa dks le>us esa lgk;d gSaA
2018-19
Page 2


miHkksDrk osQ
O;ogkj dk fl¼kar
bl vè;k; esa ge ,d O;fDrxr miHkksDrk osQ O;ogkj dk vè;;u djsaxsA
miHkksDrk dks ;g fu.kZ; djuk gksrk gS fd og viuh vk; dks fofHkUu oLrqvksa
1
ij fdl çdkj O;; djsaA vFkZ'kkL=kh bls pquko dh leL;k dgrs gSaA vr%
LokHkkfod :i ls] dksbZ Hkh miHkksDrk oLrqvksa osQ ,sls la;kstu dks çkIr djuk
pkgsxk tks mls vf/dre larks"k çnku djrk gSA ;g ^loksZÙke la;ksx* D;k gksxk\
;g miHkksDrk dh #fp;ksa vkSj og fdruk O;; dj ldrk gS] oLrqvksa dh
dherksa vkSj mldh vk; ij fuHkZj djrk gSA ;g vè;k;] miHkksDrk O;ogkj dks
le>kus okys nks fHkUu fl¼karksa dks çLrqr djrk gSA
izkjafHkd laosQru rFkk vfHkxzg
miHkksDrk lkekU; :i ls cgqr&lh oLrqvksa dk miHkksx djrk gS] ijarq
ljyhdj.k osQ fy, ge miHkksDrk dh p;u leL;k ij ,slh fLFkfr esa fopkj
djsaxs] tgk¡ osQoy nks gh oLrq,¡
2
 gksaA ge bu nksuksa oLrqvksa dks osQyk rFkk vke
dgsaxsA nksuksa oLrqvksa dh ek=kkvksa dh dksbZ Hkh lfEefyr jkf'k dks miHkksDrk
caMy vFkok la{ksi esa caMy dg ldrs gSaA lkekU;r% ge osQys dh ek=kk dks
O;Dr djus osQ fy, x
1
 ifjorZ dk vkSj vke dh ek=kk dks O;Dr djus osQ
fy, x
2
 ifjorZ dk mi;ksx djsaxsA x
1
 vkSj x
2
 /ukRed ;k 'kwU; gks ldrs gSaA
(x
1
,x
2
), dk rkRi;Z gksxk fd osQyk dh x
1
 ek=kk rFkk vke dh x
2
 ek=kkA x
1
rFkk x
2
 osQ fdlh fo'ks"k ewY; osQ fy, (x
1
, x
2
)] gesa ,d fo'ks"k caMy iznku
djrh gSA mnkgj.kkFkZµ caMy (5] 10) esa OksQys dh 5 bdkb;k¡ vkSj vke dh
10 bdkb;k¡ gSa_ caMy (10]5) esa osQys dh 10 bdkb;k¡ vkSj vke dh 5
bdkb;k¡ gSaA
2-1 mi;ksfxrk
lkekU;r% ,d miHkksDrk fdlh oLrq osQ fy, viuh ek¡x dk vuqeku ml
mi;ksfxrk (vFkok larks"k) OksQ vk/kj ij yxkrk gS tks og mlls çkIr djrk gSA
mi;ksfxrk D;k gS\ ,d oLrq dh mi;ksfxrk] mldh fdlh vko';drk dks
larq"V djus dh {kerk gSA oLrq dh ftruh T;knk vko';drk gksrh gS vFkok
1
^oLrqvksa* 'kCn dk iz;ksx loZ=k oLrqvksa rFkk lsokvksa nksuksa osQ fy, fd;k x;k gSA
2
;g /kj.kk gS fd oLrq,¡ osQoy nks gh gSa fo'ys"k.k dks ljy dj nsrh gSa vkSj ljy vkjs[kksa osQ ”kfj, egÙoiw.kZ
ladYiukvksa dks le>us esa lgk;d gSaA
2018-19
10
O;f"V vFkZ'kkL=k
,d ifjp;
mldks çkIr djus dh ftruh -T;knk bPNk gksrh gS] ml oLrq ls mruh gh vf/d mi;ksfxrk çkIr
gksrh gSA
mnkgj.k OksQ fy,] ,d O;fDr tks pkWdysV ilan djrk gS mls ,d pkWdysV ls vf/d mi;ksfxrk çkIr
gksxh ml O;fDr dks fdlh oLrq ls feyus okyh mi;ksfxrk] LFkku ,oa le; OksQ lkFk Hkh cny ldrh
gSA tSls ,d ^:e ghVj* ls feyus okyh mi;ksfxrk bl ckr ij fuHkZj djsxh fd og O;fDr yík[k
esa gS vFkok psUubZ esa (LFkku) vFkok ogk¡ xehZ dk ekSle gS ;k B.M dk ekSleA
2-1-1 x.kukokpd mi;ksfxrk fo'ys"k.k
x.kukokpd mi;ksfxrk fo'ys"k.k dh ekU;rk gS fd mi;ksfxrk OksQ Lrj dks la[;kvksa esa O;Dr
fd;k tk ldrk gSA mnkgj.k OksQ fy,] ,d deh”k ls çkIr mi;ksfxrk dks eki ldrs gS] vkSj
dgsa fd ;g deh”k eq>s 50 bdkbZ mi;ksfxrk çnku djrh gSA vkxs ppkZ djus ls iwoZ] ;g
mi;ksxh gksxk fd ge mi;ksfxrk OksQ nks egRoiw.kZ ekiksa dks le>saA
mi;ksfxrk OksQ mik;
OkqQy mi;ksfxrk% OkqQy mi;ksfxrk (TU) ,d oLrq dh ,d fuf'pr ek=kk ls çkIr mi;ksfxrk dk
;ksx gS tks fdlh oLrq (X) dh nh xbZ ek=kk dks mi;ksx djus ls çkIr gksrh gSA oLrq X dh
vf/d ek=kk] miHkksDrk dks vf/d larks"k çnku djrh gSA vr% TU oLrq dh mi;ksx dh xbZ
ek=kk ij fuHkZj djrh gSA vr% TU
n 
oLrq X dh n bdkb;ksa OksQ miHkksx ls çkIr OkqQy mi;ksfxrk
gksrh gSA
lhekar mi;ksfxrk
lhekar mi;ksfxrk (MU) dqy mi;ksfxrk esa og ifjorZu gS tks oLrq dh ,d vfrfjDr bdkbZ
OksQ miHkksx ls gksrk gSA mnkgj.k OksQ fy,] eku yhft;s 4 OksQyksa ls gesa 28 bdkbZ dqy mi;ksfxrk
çkIr gksrh gS vkSj 5 OksQyksa ls OkqQy mi;ksfxrk 30 bdkbZ feyrh gSA Li"V gS fd ik¡pos OksQys OksQ
miHkksx ls OkqQy mi;ksfxrk 2 bdkbZ c<+ xbZ (30 bdkb;k¡&28 bdkb;k¡)] blfy, ik¡pos OksQys dh
lhekar mi;ksfxrk 2 bdkbZ gSA
MU
5 
= TU
5 
– TU
4 
= 30-28 =2
lk/kj.k :i esa] MU
n 
= TU
n 
– TU
n-1
] tgk¡ ij n dk vFkZ gS] oLrq dh n
th
 
bdkbZA
OkqQy mi;ksfxrk vkSj lhekar mi;ksfxrk dks fuEu rjhosQ ls Hkh lEcfU/r fd;k tk ldrk gS%&
TU
n 
= MU
1 
+ MU
2 
+ ....... + MU
n-1 
+ MU
n
ljy :i esa bldk vFkZ gS fd OksQyksa dh n bdkb;ksa OksQ miHkksx ls çkIr mi;ksfxrk] lhekar
mi;ksfxrk çFke OksQyk (MU
n
), lhekar mi;ksfxrk f}rh; OksQys (MU
2
) vkSj blh çdkj n
th
 bdkbZ
rd dk OkqQy ;ksx gSA
rkfydk la[;k 2-1 rFkk js[kkfp=k 2-1] ,d oLrq dh fofHkUu ek=kkvksa osQ miHkksx ls feyus
okyh lhekar rFkk OkqQy mi;ksfxrk OksQ ewY;ksa dk ,d dkYifud mnkgj.k dks fn[kkrh gSA
lkekU;r% ;g ns[kk tkrk gS fd lhekar mi;ksfxrk] oLrq OksQ mi;ksx esa o`f¼ OksQ lkFk fxj tkrh
gSA ,slk blfy, gksrk gS D;ksafd ,d oLrq dh dqN ek=kk miyC/ gks tkus ij] miHkksDrk dh ml
oLrq dks vkSj vf/d çkIr djus dh bPNk de gks tkrh gSA rkfydk rFkk js[kkfp=k esa blh dks
fn[kk;k x;k gSA
nsf[k;s] fd MU
3, 
MU
2 
ls de gSA vki ;g Hkh ns[k ldrs gSa fd dqy mi;ksfxrk esa o`f¼
gksrh gS ysfdu ?kVrh gqbZ nj ijA fdlh oLrq osQ mi;ksx dh ek=kk esa ifjorZu osQ iQyLo:i]
OkqQy mi;ksfxrk esa o`f¼ dh nj] mldh lhekar mi;ksfxrk dh vk; gSA ;g lhekar mi;ksfxrk]
2018-19
Page 3


miHkksDrk osQ
O;ogkj dk fl¼kar
bl vè;k; esa ge ,d O;fDrxr miHkksDrk osQ O;ogkj dk vè;;u djsaxsA
miHkksDrk dks ;g fu.kZ; djuk gksrk gS fd og viuh vk; dks fofHkUu oLrqvksa
1
ij fdl çdkj O;; djsaA vFkZ'kkL=kh bls pquko dh leL;k dgrs gSaA vr%
LokHkkfod :i ls] dksbZ Hkh miHkksDrk oLrqvksa osQ ,sls la;kstu dks çkIr djuk
pkgsxk tks mls vf/dre larks"k çnku djrk gSA ;g ^loksZÙke la;ksx* D;k gksxk\
;g miHkksDrk dh #fp;ksa vkSj og fdruk O;; dj ldrk gS] oLrqvksa dh
dherksa vkSj mldh vk; ij fuHkZj djrk gSA ;g vè;k;] miHkksDrk O;ogkj dks
le>kus okys nks fHkUu fl¼karksa dks çLrqr djrk gSA
izkjafHkd laosQru rFkk vfHkxzg
miHkksDrk lkekU; :i ls cgqr&lh oLrqvksa dk miHkksx djrk gS] ijarq
ljyhdj.k osQ fy, ge miHkksDrk dh p;u leL;k ij ,slh fLFkfr esa fopkj
djsaxs] tgk¡ osQoy nks gh oLrq,¡
2
 gksaA ge bu nksuksa oLrqvksa dks osQyk rFkk vke
dgsaxsA nksuksa oLrqvksa dh ek=kkvksa dh dksbZ Hkh lfEefyr jkf'k dks miHkksDrk
caMy vFkok la{ksi esa caMy dg ldrs gSaA lkekU;r% ge osQys dh ek=kk dks
O;Dr djus osQ fy, x
1
 ifjorZ dk vkSj vke dh ek=kk dks O;Dr djus osQ
fy, x
2
 ifjorZ dk mi;ksx djsaxsA x
1
 vkSj x
2
 /ukRed ;k 'kwU; gks ldrs gSaA
(x
1
,x
2
), dk rkRi;Z gksxk fd osQyk dh x
1
 ek=kk rFkk vke dh x
2
 ek=kkA x
1
rFkk x
2
 osQ fdlh fo'ks"k ewY; osQ fy, (x
1
, x
2
)] gesa ,d fo'ks"k caMy iznku
djrh gSA mnkgj.kkFkZµ caMy (5] 10) esa OksQys dh 5 bdkb;k¡ vkSj vke dh
10 bdkb;k¡ gSa_ caMy (10]5) esa osQys dh 10 bdkb;k¡ vkSj vke dh 5
bdkb;k¡ gSaA
2-1 mi;ksfxrk
lkekU;r% ,d miHkksDrk fdlh oLrq osQ fy, viuh ek¡x dk vuqeku ml
mi;ksfxrk (vFkok larks"k) OksQ vk/kj ij yxkrk gS tks og mlls çkIr djrk gSA
mi;ksfxrk D;k gS\ ,d oLrq dh mi;ksfxrk] mldh fdlh vko';drk dks
larq"V djus dh {kerk gSA oLrq dh ftruh T;knk vko';drk gksrh gS vFkok
1
^oLrqvksa* 'kCn dk iz;ksx loZ=k oLrqvksa rFkk lsokvksa nksuksa osQ fy, fd;k x;k gSA
2
;g /kj.kk gS fd oLrq,¡ osQoy nks gh gSa fo'ys"k.k dks ljy dj nsrh gSa vkSj ljy vkjs[kksa osQ ”kfj, egÙoiw.kZ
ladYiukvksa dks le>us esa lgk;d gSaA
2018-19
10
O;f"V vFkZ'kkL=k
,d ifjp;
mldks çkIr djus dh ftruh -T;knk bPNk gksrh gS] ml oLrq ls mruh gh vf/d mi;ksfxrk çkIr
gksrh gSA
mnkgj.k OksQ fy,] ,d O;fDr tks pkWdysV ilan djrk gS mls ,d pkWdysV ls vf/d mi;ksfxrk çkIr
gksxh ml O;fDr dks fdlh oLrq ls feyus okyh mi;ksfxrk] LFkku ,oa le; OksQ lkFk Hkh cny ldrh
gSA tSls ,d ^:e ghVj* ls feyus okyh mi;ksfxrk bl ckr ij fuHkZj djsxh fd og O;fDr yík[k
esa gS vFkok psUubZ esa (LFkku) vFkok ogk¡ xehZ dk ekSle gS ;k B.M dk ekSleA
2-1-1 x.kukokpd mi;ksfxrk fo'ys"k.k
x.kukokpd mi;ksfxrk fo'ys"k.k dh ekU;rk gS fd mi;ksfxrk OksQ Lrj dks la[;kvksa esa O;Dr
fd;k tk ldrk gSA mnkgj.k OksQ fy,] ,d deh”k ls çkIr mi;ksfxrk dks eki ldrs gS] vkSj
dgsa fd ;g deh”k eq>s 50 bdkbZ mi;ksfxrk çnku djrh gSA vkxs ppkZ djus ls iwoZ] ;g
mi;ksxh gksxk fd ge mi;ksfxrk OksQ nks egRoiw.kZ ekiksa dks le>saA
mi;ksfxrk OksQ mik;
OkqQy mi;ksfxrk% OkqQy mi;ksfxrk (TU) ,d oLrq dh ,d fuf'pr ek=kk ls çkIr mi;ksfxrk dk
;ksx gS tks fdlh oLrq (X) dh nh xbZ ek=kk dks mi;ksx djus ls çkIr gksrh gSA oLrq X dh
vf/d ek=kk] miHkksDrk dks vf/d larks"k çnku djrh gSA vr% TU oLrq dh mi;ksx dh xbZ
ek=kk ij fuHkZj djrh gSA vr% TU
n 
oLrq X dh n bdkb;ksa OksQ miHkksx ls çkIr OkqQy mi;ksfxrk
gksrh gSA
lhekar mi;ksfxrk
lhekar mi;ksfxrk (MU) dqy mi;ksfxrk esa og ifjorZu gS tks oLrq dh ,d vfrfjDr bdkbZ
OksQ miHkksx ls gksrk gSA mnkgj.k OksQ fy,] eku yhft;s 4 OksQyksa ls gesa 28 bdkbZ dqy mi;ksfxrk
çkIr gksrh gS vkSj 5 OksQyksa ls OkqQy mi;ksfxrk 30 bdkbZ feyrh gSA Li"V gS fd ik¡pos OksQys OksQ
miHkksx ls OkqQy mi;ksfxrk 2 bdkbZ c<+ xbZ (30 bdkb;k¡&28 bdkb;k¡)] blfy, ik¡pos OksQys dh
lhekar mi;ksfxrk 2 bdkbZ gSA
MU
5 
= TU
5 
– TU
4 
= 30-28 =2
lk/kj.k :i esa] MU
n 
= TU
n 
– TU
n-1
] tgk¡ ij n dk vFkZ gS] oLrq dh n
th
 
bdkbZA
OkqQy mi;ksfxrk vkSj lhekar mi;ksfxrk dks fuEu rjhosQ ls Hkh lEcfU/r fd;k tk ldrk gS%&
TU
n 
= MU
1 
+ MU
2 
+ ....... + MU
n-1 
+ MU
n
ljy :i esa bldk vFkZ gS fd OksQyksa dh n bdkb;ksa OksQ miHkksx ls çkIr mi;ksfxrk] lhekar
mi;ksfxrk çFke OksQyk (MU
n
), lhekar mi;ksfxrk f}rh; OksQys (MU
2
) vkSj blh çdkj n
th
 bdkbZ
rd dk OkqQy ;ksx gSA
rkfydk la[;k 2-1 rFkk js[kkfp=k 2-1] ,d oLrq dh fofHkUu ek=kkvksa osQ miHkksx ls feyus
okyh lhekar rFkk OkqQy mi;ksfxrk OksQ ewY;ksa dk ,d dkYifud mnkgj.k dks fn[kkrh gSA
lkekU;r% ;g ns[kk tkrk gS fd lhekar mi;ksfxrk] oLrq OksQ mi;ksx esa o`f¼ OksQ lkFk fxj tkrh
gSA ,slk blfy, gksrk gS D;ksafd ,d oLrq dh dqN ek=kk miyC/ gks tkus ij] miHkksDrk dh ml
oLrq dks vkSj vf/d çkIr djus dh bPNk de gks tkrh gSA rkfydk rFkk js[kkfp=k esa blh dks
fn[kk;k x;k gSA
nsf[k;s] fd MU
3, 
MU
2 
ls de gSA vki ;g Hkh ns[k ldrs gSa fd dqy mi;ksfxrk esa o`f¼
gksrh gS ysfdu ?kVrh gqbZ nj ijA fdlh oLrq osQ mi;ksx dh ek=kk esa ifjorZu osQ iQyLo:i]
OkqQy mi;ksfxrk esa o`f¼ dh nj] mldh lhekar mi;ksfxrk dh vk; gSA ;g lhekar mi;ksfxrk]
2018-19
11
miHkksDrk osQ
O;ogkj dk fl¼kar
oLrq dh miHkksx ek=kk esa o`f¼ osQ lkFk fxjrh
tkrh gS& 12 ls 6] 6 ls 4 vkSj blh izdkj
vkxsA ;g fu"d"kZ] âkleku mi;ksfxrk fu;e
ls fudyrk gSA âkleku mi;ksfxrk fu;e
crkrk gS fd tSls&tSls vU; oLrqvksa OksQ
mi;ksx dks fLFkj j[krs gq, fdlh oLrq OksQ
miHkksx dks c<+k;k tkrk gS] oLrq dh gj
vxyh bdkbZ osQ miHkksx ls izkIr lhekar
mi;ksfxrk fxjrh tkrh gSA
MU ml fcUnw ij 'kwU; gks tkrh gS tc
TU fLFkj jgrk gSA fn;s x;s mnkgj.k esa ik¡poha bdkbZ osQ miHkksx ij TU vifjofrZr jgrh gS]
vr% MU
5 
= 0 blosQ i'pkr TU fxjus yxkrh gS vkSj MU ½.kkRed gks tkrh gSA
,dy oLrq dh n'kk esa ekax oØ dh mRifÙk (âkleku mi;ksfxrk fu;e)
fdlh oLrq osQ ekax oØ dk] x.kukokpd fo'ys"k.k osQ }kjk cuk;k tk ldrk gSA ekax D;k gS
rFkk ekax oØ D;k gS\ oLrq dh ek=kk ftls ,d miHkksDrk fn;s x;s oLrq ewY;ksa vkSj vk; ij]
[kjhnus osQ fy;s bPNqd ,oa leFkZ gS dks ml oLrq dh ekax dgrs gSaA oLrq X dh ekax] X oLrq
dh Lo;a dh dher osQ vfrfjDr] vU; oLrqvksa dh dherksa] (çfrLFkkukiUu ,oa iwjd oLrq,a
2-4 ns[ksa)] miHkksDrk dh #fp;ksa vkSj ojh;rk tSls dkjdksa ij fuHkZj djrh gSA ekax oØ] ,d
oLrq dh] fofHkUu dherksa ij] ekaxs tkus okyh ek=kkvksa dh js[kh; çLrqfr gSA tks ,d miHkksDrk
fdlh oLrq dh fofHkUu dherksa ij [kjhnus
osQ fy;s bPNqd gS] vU; lacaf/r oLrqvksa
dh dherksa dks vkSj miHkksDrk dh vk; dks
fLFkj j[krs gq,A
fp=k 2-2] ,d O;fDr dk oLrq X osQ
fy,] fofHkUu dherksa ij] mlosQ  dkYifud
ekax oØ dks çnf'kZr djrk gSA ek=kkvksa dks
{kSfrt v{k ij rFkk dherksa dks mèokZ/j v{k
ij fn[kk;k x;k gSA
uhps dh vksj <yok ekax oØ çnf'kZr
djrk gS fd uhps ewY;ksa ij ,d O;fDr X
fdlh oLrq dh fofHkUu ek=kkvksa osQ miHkksx ls çkIr lhekar ,oa
OkqQy mi;ksfxrk osQ ewY; oLrq osQ miHkksx esa o`f¼ OksQ lkFk
lhekUr mi;ksfxrk ?kVrh tkrh gSA
oLrq X OksQ fy, ,d O;fDrxr dk ek¡x oØ
rkfydk 2-1% ,d oLrq dh fofHkUu ek=kkvksa osQ miHkksx ls çkIr lhekar rFkk dqy mi;ksfxrk osQ ewY;
        bdkbZ oqQy mi;ksfxrk lhekar mi;ksfxrk
1 12 12
2 18 6
3 22 4
4 24 2
5 24 0
6 22 &2
2018-19
Page 4


miHkksDrk osQ
O;ogkj dk fl¼kar
bl vè;k; esa ge ,d O;fDrxr miHkksDrk osQ O;ogkj dk vè;;u djsaxsA
miHkksDrk dks ;g fu.kZ; djuk gksrk gS fd og viuh vk; dks fofHkUu oLrqvksa
1
ij fdl çdkj O;; djsaA vFkZ'kkL=kh bls pquko dh leL;k dgrs gSaA vr%
LokHkkfod :i ls] dksbZ Hkh miHkksDrk oLrqvksa osQ ,sls la;kstu dks çkIr djuk
pkgsxk tks mls vf/dre larks"k çnku djrk gSA ;g ^loksZÙke la;ksx* D;k gksxk\
;g miHkksDrk dh #fp;ksa vkSj og fdruk O;; dj ldrk gS] oLrqvksa dh
dherksa vkSj mldh vk; ij fuHkZj djrk gSA ;g vè;k;] miHkksDrk O;ogkj dks
le>kus okys nks fHkUu fl¼karksa dks çLrqr djrk gSA
izkjafHkd laosQru rFkk vfHkxzg
miHkksDrk lkekU; :i ls cgqr&lh oLrqvksa dk miHkksx djrk gS] ijarq
ljyhdj.k osQ fy, ge miHkksDrk dh p;u leL;k ij ,slh fLFkfr esa fopkj
djsaxs] tgk¡ osQoy nks gh oLrq,¡
2
 gksaA ge bu nksuksa oLrqvksa dks osQyk rFkk vke
dgsaxsA nksuksa oLrqvksa dh ek=kkvksa dh dksbZ Hkh lfEefyr jkf'k dks miHkksDrk
caMy vFkok la{ksi esa caMy dg ldrs gSaA lkekU;r% ge osQys dh ek=kk dks
O;Dr djus osQ fy, x
1
 ifjorZ dk vkSj vke dh ek=kk dks O;Dr djus osQ
fy, x
2
 ifjorZ dk mi;ksx djsaxsA x
1
 vkSj x
2
 /ukRed ;k 'kwU; gks ldrs gSaA
(x
1
,x
2
), dk rkRi;Z gksxk fd osQyk dh x
1
 ek=kk rFkk vke dh x
2
 ek=kkA x
1
rFkk x
2
 osQ fdlh fo'ks"k ewY; osQ fy, (x
1
, x
2
)] gesa ,d fo'ks"k caMy iznku
djrh gSA mnkgj.kkFkZµ caMy (5] 10) esa OksQys dh 5 bdkb;k¡ vkSj vke dh
10 bdkb;k¡ gSa_ caMy (10]5) esa osQys dh 10 bdkb;k¡ vkSj vke dh 5
bdkb;k¡ gSaA
2-1 mi;ksfxrk
lkekU;r% ,d miHkksDrk fdlh oLrq osQ fy, viuh ek¡x dk vuqeku ml
mi;ksfxrk (vFkok larks"k) OksQ vk/kj ij yxkrk gS tks og mlls çkIr djrk gSA
mi;ksfxrk D;k gS\ ,d oLrq dh mi;ksfxrk] mldh fdlh vko';drk dks
larq"V djus dh {kerk gSA oLrq dh ftruh T;knk vko';drk gksrh gS vFkok
1
^oLrqvksa* 'kCn dk iz;ksx loZ=k oLrqvksa rFkk lsokvksa nksuksa osQ fy, fd;k x;k gSA
2
;g /kj.kk gS fd oLrq,¡ osQoy nks gh gSa fo'ys"k.k dks ljy dj nsrh gSa vkSj ljy vkjs[kksa osQ ”kfj, egÙoiw.kZ
ladYiukvksa dks le>us esa lgk;d gSaA
2018-19
10
O;f"V vFkZ'kkL=k
,d ifjp;
mldks çkIr djus dh ftruh -T;knk bPNk gksrh gS] ml oLrq ls mruh gh vf/d mi;ksfxrk çkIr
gksrh gSA
mnkgj.k OksQ fy,] ,d O;fDr tks pkWdysV ilan djrk gS mls ,d pkWdysV ls vf/d mi;ksfxrk çkIr
gksxh ml O;fDr dks fdlh oLrq ls feyus okyh mi;ksfxrk] LFkku ,oa le; OksQ lkFk Hkh cny ldrh
gSA tSls ,d ^:e ghVj* ls feyus okyh mi;ksfxrk bl ckr ij fuHkZj djsxh fd og O;fDr yík[k
esa gS vFkok psUubZ esa (LFkku) vFkok ogk¡ xehZ dk ekSle gS ;k B.M dk ekSleA
2-1-1 x.kukokpd mi;ksfxrk fo'ys"k.k
x.kukokpd mi;ksfxrk fo'ys"k.k dh ekU;rk gS fd mi;ksfxrk OksQ Lrj dks la[;kvksa esa O;Dr
fd;k tk ldrk gSA mnkgj.k OksQ fy,] ,d deh”k ls çkIr mi;ksfxrk dks eki ldrs gS] vkSj
dgsa fd ;g deh”k eq>s 50 bdkbZ mi;ksfxrk çnku djrh gSA vkxs ppkZ djus ls iwoZ] ;g
mi;ksxh gksxk fd ge mi;ksfxrk OksQ nks egRoiw.kZ ekiksa dks le>saA
mi;ksfxrk OksQ mik;
OkqQy mi;ksfxrk% OkqQy mi;ksfxrk (TU) ,d oLrq dh ,d fuf'pr ek=kk ls çkIr mi;ksfxrk dk
;ksx gS tks fdlh oLrq (X) dh nh xbZ ek=kk dks mi;ksx djus ls çkIr gksrh gSA oLrq X dh
vf/d ek=kk] miHkksDrk dks vf/d larks"k çnku djrh gSA vr% TU oLrq dh mi;ksx dh xbZ
ek=kk ij fuHkZj djrh gSA vr% TU
n 
oLrq X dh n bdkb;ksa OksQ miHkksx ls çkIr OkqQy mi;ksfxrk
gksrh gSA
lhekar mi;ksfxrk
lhekar mi;ksfxrk (MU) dqy mi;ksfxrk esa og ifjorZu gS tks oLrq dh ,d vfrfjDr bdkbZ
OksQ miHkksx ls gksrk gSA mnkgj.k OksQ fy,] eku yhft;s 4 OksQyksa ls gesa 28 bdkbZ dqy mi;ksfxrk
çkIr gksrh gS vkSj 5 OksQyksa ls OkqQy mi;ksfxrk 30 bdkbZ feyrh gSA Li"V gS fd ik¡pos OksQys OksQ
miHkksx ls OkqQy mi;ksfxrk 2 bdkbZ c<+ xbZ (30 bdkb;k¡&28 bdkb;k¡)] blfy, ik¡pos OksQys dh
lhekar mi;ksfxrk 2 bdkbZ gSA
MU
5 
= TU
5 
– TU
4 
= 30-28 =2
lk/kj.k :i esa] MU
n 
= TU
n 
– TU
n-1
] tgk¡ ij n dk vFkZ gS] oLrq dh n
th
 
bdkbZA
OkqQy mi;ksfxrk vkSj lhekar mi;ksfxrk dks fuEu rjhosQ ls Hkh lEcfU/r fd;k tk ldrk gS%&
TU
n 
= MU
1 
+ MU
2 
+ ....... + MU
n-1 
+ MU
n
ljy :i esa bldk vFkZ gS fd OksQyksa dh n bdkb;ksa OksQ miHkksx ls çkIr mi;ksfxrk] lhekar
mi;ksfxrk çFke OksQyk (MU
n
), lhekar mi;ksfxrk f}rh; OksQys (MU
2
) vkSj blh çdkj n
th
 bdkbZ
rd dk OkqQy ;ksx gSA
rkfydk la[;k 2-1 rFkk js[kkfp=k 2-1] ,d oLrq dh fofHkUu ek=kkvksa osQ miHkksx ls feyus
okyh lhekar rFkk OkqQy mi;ksfxrk OksQ ewY;ksa dk ,d dkYifud mnkgj.k dks fn[kkrh gSA
lkekU;r% ;g ns[kk tkrk gS fd lhekar mi;ksfxrk] oLrq OksQ mi;ksx esa o`f¼ OksQ lkFk fxj tkrh
gSA ,slk blfy, gksrk gS D;ksafd ,d oLrq dh dqN ek=kk miyC/ gks tkus ij] miHkksDrk dh ml
oLrq dks vkSj vf/d çkIr djus dh bPNk de gks tkrh gSA rkfydk rFkk js[kkfp=k esa blh dks
fn[kk;k x;k gSA
nsf[k;s] fd MU
3, 
MU
2 
ls de gSA vki ;g Hkh ns[k ldrs gSa fd dqy mi;ksfxrk esa o`f¼
gksrh gS ysfdu ?kVrh gqbZ nj ijA fdlh oLrq osQ mi;ksx dh ek=kk esa ifjorZu osQ iQyLo:i]
OkqQy mi;ksfxrk esa o`f¼ dh nj] mldh lhekar mi;ksfxrk dh vk; gSA ;g lhekar mi;ksfxrk]
2018-19
11
miHkksDrk osQ
O;ogkj dk fl¼kar
oLrq dh miHkksx ek=kk esa o`f¼ osQ lkFk fxjrh
tkrh gS& 12 ls 6] 6 ls 4 vkSj blh izdkj
vkxsA ;g fu"d"kZ] âkleku mi;ksfxrk fu;e
ls fudyrk gSA âkleku mi;ksfxrk fu;e
crkrk gS fd tSls&tSls vU; oLrqvksa OksQ
mi;ksx dks fLFkj j[krs gq, fdlh oLrq OksQ
miHkksx dks c<+k;k tkrk gS] oLrq dh gj
vxyh bdkbZ osQ miHkksx ls izkIr lhekar
mi;ksfxrk fxjrh tkrh gSA
MU ml fcUnw ij 'kwU; gks tkrh gS tc
TU fLFkj jgrk gSA fn;s x;s mnkgj.k esa ik¡poha bdkbZ osQ miHkksx ij TU vifjofrZr jgrh gS]
vr% MU
5 
= 0 blosQ i'pkr TU fxjus yxkrh gS vkSj MU ½.kkRed gks tkrh gSA
,dy oLrq dh n'kk esa ekax oØ dh mRifÙk (âkleku mi;ksfxrk fu;e)
fdlh oLrq osQ ekax oØ dk] x.kukokpd fo'ys"k.k osQ }kjk cuk;k tk ldrk gSA ekax D;k gS
rFkk ekax oØ D;k gS\ oLrq dh ek=kk ftls ,d miHkksDrk fn;s x;s oLrq ewY;ksa vkSj vk; ij]
[kjhnus osQ fy;s bPNqd ,oa leFkZ gS dks ml oLrq dh ekax dgrs gSaA oLrq X dh ekax] X oLrq
dh Lo;a dh dher osQ vfrfjDr] vU; oLrqvksa dh dherksa] (çfrLFkkukiUu ,oa iwjd oLrq,a
2-4 ns[ksa)] miHkksDrk dh #fp;ksa vkSj ojh;rk tSls dkjdksa ij fuHkZj djrh gSA ekax oØ] ,d
oLrq dh] fofHkUu dherksa ij] ekaxs tkus okyh ek=kkvksa dh js[kh; çLrqfr gSA tks ,d miHkksDrk
fdlh oLrq dh fofHkUu dherksa ij [kjhnus
osQ fy;s bPNqd gS] vU; lacaf/r oLrqvksa
dh dherksa dks vkSj miHkksDrk dh vk; dks
fLFkj j[krs gq,A
fp=k 2-2] ,d O;fDr dk oLrq X osQ
fy,] fofHkUu dherksa ij] mlosQ  dkYifud
ekax oØ dks çnf'kZr djrk gSA ek=kkvksa dks
{kSfrt v{k ij rFkk dherksa dks mèokZ/j v{k
ij fn[kk;k x;k gSA
uhps dh vksj <yok ekax oØ çnf'kZr
djrk gS fd uhps ewY;ksa ij ,d O;fDr X
fdlh oLrq dh fofHkUu ek=kkvksa osQ miHkksx ls çkIr lhekar ,oa
OkqQy mi;ksfxrk osQ ewY; oLrq osQ miHkksx esa o`f¼ OksQ lkFk
lhekUr mi;ksfxrk ?kVrh tkrh gSA
oLrq X OksQ fy, ,d O;fDrxr dk ek¡x oØ
rkfydk 2-1% ,d oLrq dh fofHkUu ek=kkvksa osQ miHkksx ls çkIr lhekar rFkk dqy mi;ksfxrk osQ ewY;
        bdkbZ oqQy mi;ksfxrk lhekar mi;ksfxrk
1 12 12
2 18 6
3 22 4
4 24 2
5 24 0
6 22 &2
2018-19
12
O;f"V vFkZ'kkL=k
,d ifjp;
oLrq dh vf/d ek=kk [kjhnus dks bPNqd gS rFkk m¡Qps ewY;ksa ij] og oLrq dh dher rFkk ek¡xas
tkus okyh ek=kk osQ chp ½.kkRed laca/ gSA bls ekax dk fu;e dgrs gSaA
uhps dh vksj <yok ekax oØ dh Li"Vhdj.k] âkleku lhekar mi;ksfxrk osQ fopkj ij
vk/kfjr gSA âkleku lhekar mi;ksfxrk dk fu;e crkrk gS fd oLrq dh çR;sd vxyh bdkbZ]
de lhekar mi;ksfxrk çnku djrh gSA blfy;s] O;fDr çR;sd vxyh bdkbZ osQ fy;s mruk nsus
osQ fy;s bPNqd ugha gksxk vkSj blh dkj.k ekax oØ uhps dh vksj <yok oØ gksrk gSA X oLrq
dh :- 40 çfr bdkbZ dher ij] O;fDr dh X oLrq dh ikap bdkb;ksa dh ekax FkhA oLrq dh
NBoha bdkbZ] ikapoh bdkbZ ls de egRo dh gksxhA O;fDr NBh bdkbZ dks RkHkh [kjhnsxk tc
dher :- 40 çfr bdkbZ ls uhps fxj tk,A vr% âkleku lhekar mi;ksfxrk fu;e] ;g Li"V
djrk gS fd ekax oØksa dk ½.kkRed nke D;ksa gksrk gSaA
2-1-2 Øeokpd mi;ksfxrk fo'ys"k.k
x.kukokpd mi;ksfxrk fo'ys"k.k le>us esa ljy gS] ysfdu mi;ksfxrk dk la[;kvksa osQ :i esa
ifjek.ku] bldk ,d cM+k nks"k gSA okLrfod thou esa] ge mi;ksfxrk dks la[;kvksa osQ :i esa
dHkh O;Dr ugha djrsA vf/d ls vf/d ge de ;k vf/d mi;ksfxrk osQ vk/kj ij Øe
ns ldrs gSaA nwljs 'kCnksa esa] miHkksDrk mi;ksfxrk dks la[;kvksa esa ugha ekirk gS] ;|fi og
fofHkUu miHkksDrk caMyksa dks Øe nsrk gSA ;gh] Øeokpd mi;ksfxrk fo'ys"k.k çlax dk çkjafHkd
fcUnq gSA
caMyks osQ lsV esa ,d miHkksDrk osQ vfHkekuksa dks fp=k }kjk Hkh çnf'kZr fd;k tk ldrk gSA
ge igys gh ns[k pqosQ gSa] fd ,d miHkksDrk dk miyC/ caMyksa dks ,d f}rh; fp=k }kjk
fn[kk;k tk ldrk gSA fcUnqvksa tks miHkksDrk dk
leku mi;ksfxrk gS] çnku djus okys caMyksa dk
çfrfuf/Ro djrs gSa] dks js[kkfp=k 2-3 esa fn[kk;s
x;s oØ dks çkIr djus osQs fy;s lkekU;r% tksM+k
tk ldrk gSA
,d vuf/eku oØ ij lHkh fcUnq tSls A, B,
C rFkk D miHkksDrk dks leku Lrj dk larks"k çnku
djrs gSaA ;g Li"V gS fd tc miHkksDrk dks ,d
vkSj osQyk çkIr gksrk gS rks mldks dqN vkeksa dk
R;kx djuk iM+rk gS rkfd mldh mi;ksfxrk dk
Lrj ogh jgs vkSj og mlh vuf/eku oØ ij jgsA
,d vfrfjDr osQyk çkIr djus osQ fy;s] dqy
mi;ksfxrk dk Lrj leku jgrs gq,] vkeksa dh og ek=kk] ftldk miHkksDrk dks R;kx djuk iM+rk
gS] lhekar çfrLFkkiu nj dgrs gSaA nwljs 'kCnksa esa] MRS] ljy:i esa og nj gS ftl ij
miHkksDrk vkeksa osQ LFkku ij osQyksa dks çfrLFkkfir djsxkA rkfd mldh mi;ksfxrk leku jgsA
vr% / MRS Y X =| ? ? |
3
vuf/eku oØ& ,d vuf/eku oØ mu lHkh
fcUnqvksa dk çfrfuf/Ro djus okys caMyks dks ftudks
miHkksDrk rVLFk le>rk gS vkSj mUgh dks tksM+rk gSA
3 
/ / ( / ) 0 Y X Y Xif Y X | ? ? |= ? ? ? ? =
           / ( / ) 0 Y Xif Y X = -? ? ? ? <
/ MRS Y X =| ? ? | dk vFkZ gS fd MRS cjkcj gksrk gSA OksQoy / Y X ? ? ds ifjek.k OksQ ;fn / 3/1 Y X ? ? = - ]
rks bldk vFkZ gS fd MRS=3
2018-19
Page 5


miHkksDrk osQ
O;ogkj dk fl¼kar
bl vè;k; esa ge ,d O;fDrxr miHkksDrk osQ O;ogkj dk vè;;u djsaxsA
miHkksDrk dks ;g fu.kZ; djuk gksrk gS fd og viuh vk; dks fofHkUu oLrqvksa
1
ij fdl çdkj O;; djsaA vFkZ'kkL=kh bls pquko dh leL;k dgrs gSaA vr%
LokHkkfod :i ls] dksbZ Hkh miHkksDrk oLrqvksa osQ ,sls la;kstu dks çkIr djuk
pkgsxk tks mls vf/dre larks"k çnku djrk gSA ;g ^loksZÙke la;ksx* D;k gksxk\
;g miHkksDrk dh #fp;ksa vkSj og fdruk O;; dj ldrk gS] oLrqvksa dh
dherksa vkSj mldh vk; ij fuHkZj djrk gSA ;g vè;k;] miHkksDrk O;ogkj dks
le>kus okys nks fHkUu fl¼karksa dks çLrqr djrk gSA
izkjafHkd laosQru rFkk vfHkxzg
miHkksDrk lkekU; :i ls cgqr&lh oLrqvksa dk miHkksx djrk gS] ijarq
ljyhdj.k osQ fy, ge miHkksDrk dh p;u leL;k ij ,slh fLFkfr esa fopkj
djsaxs] tgk¡ osQoy nks gh oLrq,¡
2
 gksaA ge bu nksuksa oLrqvksa dks osQyk rFkk vke
dgsaxsA nksuksa oLrqvksa dh ek=kkvksa dh dksbZ Hkh lfEefyr jkf'k dks miHkksDrk
caMy vFkok la{ksi esa caMy dg ldrs gSaA lkekU;r% ge osQys dh ek=kk dks
O;Dr djus osQ fy, x
1
 ifjorZ dk vkSj vke dh ek=kk dks O;Dr djus osQ
fy, x
2
 ifjorZ dk mi;ksx djsaxsA x
1
 vkSj x
2
 /ukRed ;k 'kwU; gks ldrs gSaA
(x
1
,x
2
), dk rkRi;Z gksxk fd osQyk dh x
1
 ek=kk rFkk vke dh x
2
 ek=kkA x
1
rFkk x
2
 osQ fdlh fo'ks"k ewY; osQ fy, (x
1
, x
2
)] gesa ,d fo'ks"k caMy iznku
djrh gSA mnkgj.kkFkZµ caMy (5] 10) esa OksQys dh 5 bdkb;k¡ vkSj vke dh
10 bdkb;k¡ gSa_ caMy (10]5) esa osQys dh 10 bdkb;k¡ vkSj vke dh 5
bdkb;k¡ gSaA
2-1 mi;ksfxrk
lkekU;r% ,d miHkksDrk fdlh oLrq osQ fy, viuh ek¡x dk vuqeku ml
mi;ksfxrk (vFkok larks"k) OksQ vk/kj ij yxkrk gS tks og mlls çkIr djrk gSA
mi;ksfxrk D;k gS\ ,d oLrq dh mi;ksfxrk] mldh fdlh vko';drk dks
larq"V djus dh {kerk gSA oLrq dh ftruh T;knk vko';drk gksrh gS vFkok
1
^oLrqvksa* 'kCn dk iz;ksx loZ=k oLrqvksa rFkk lsokvksa nksuksa osQ fy, fd;k x;k gSA
2
;g /kj.kk gS fd oLrq,¡ osQoy nks gh gSa fo'ys"k.k dks ljy dj nsrh gSa vkSj ljy vkjs[kksa osQ ”kfj, egÙoiw.kZ
ladYiukvksa dks le>us esa lgk;d gSaA
2018-19
10
O;f"V vFkZ'kkL=k
,d ifjp;
mldks çkIr djus dh ftruh -T;knk bPNk gksrh gS] ml oLrq ls mruh gh vf/d mi;ksfxrk çkIr
gksrh gSA
mnkgj.k OksQ fy,] ,d O;fDr tks pkWdysV ilan djrk gS mls ,d pkWdysV ls vf/d mi;ksfxrk çkIr
gksxh ml O;fDr dks fdlh oLrq ls feyus okyh mi;ksfxrk] LFkku ,oa le; OksQ lkFk Hkh cny ldrh
gSA tSls ,d ^:e ghVj* ls feyus okyh mi;ksfxrk bl ckr ij fuHkZj djsxh fd og O;fDr yík[k
esa gS vFkok psUubZ esa (LFkku) vFkok ogk¡ xehZ dk ekSle gS ;k B.M dk ekSleA
2-1-1 x.kukokpd mi;ksfxrk fo'ys"k.k
x.kukokpd mi;ksfxrk fo'ys"k.k dh ekU;rk gS fd mi;ksfxrk OksQ Lrj dks la[;kvksa esa O;Dr
fd;k tk ldrk gSA mnkgj.k OksQ fy,] ,d deh”k ls çkIr mi;ksfxrk dks eki ldrs gS] vkSj
dgsa fd ;g deh”k eq>s 50 bdkbZ mi;ksfxrk çnku djrh gSA vkxs ppkZ djus ls iwoZ] ;g
mi;ksxh gksxk fd ge mi;ksfxrk OksQ nks egRoiw.kZ ekiksa dks le>saA
mi;ksfxrk OksQ mik;
OkqQy mi;ksfxrk% OkqQy mi;ksfxrk (TU) ,d oLrq dh ,d fuf'pr ek=kk ls çkIr mi;ksfxrk dk
;ksx gS tks fdlh oLrq (X) dh nh xbZ ek=kk dks mi;ksx djus ls çkIr gksrh gSA oLrq X dh
vf/d ek=kk] miHkksDrk dks vf/d larks"k çnku djrh gSA vr% TU oLrq dh mi;ksx dh xbZ
ek=kk ij fuHkZj djrh gSA vr% TU
n 
oLrq X dh n bdkb;ksa OksQ miHkksx ls çkIr OkqQy mi;ksfxrk
gksrh gSA
lhekar mi;ksfxrk
lhekar mi;ksfxrk (MU) dqy mi;ksfxrk esa og ifjorZu gS tks oLrq dh ,d vfrfjDr bdkbZ
OksQ miHkksx ls gksrk gSA mnkgj.k OksQ fy,] eku yhft;s 4 OksQyksa ls gesa 28 bdkbZ dqy mi;ksfxrk
çkIr gksrh gS vkSj 5 OksQyksa ls OkqQy mi;ksfxrk 30 bdkbZ feyrh gSA Li"V gS fd ik¡pos OksQys OksQ
miHkksx ls OkqQy mi;ksfxrk 2 bdkbZ c<+ xbZ (30 bdkb;k¡&28 bdkb;k¡)] blfy, ik¡pos OksQys dh
lhekar mi;ksfxrk 2 bdkbZ gSA
MU
5 
= TU
5 
– TU
4 
= 30-28 =2
lk/kj.k :i esa] MU
n 
= TU
n 
– TU
n-1
] tgk¡ ij n dk vFkZ gS] oLrq dh n
th
 
bdkbZA
OkqQy mi;ksfxrk vkSj lhekar mi;ksfxrk dks fuEu rjhosQ ls Hkh lEcfU/r fd;k tk ldrk gS%&
TU
n 
= MU
1 
+ MU
2 
+ ....... + MU
n-1 
+ MU
n
ljy :i esa bldk vFkZ gS fd OksQyksa dh n bdkb;ksa OksQ miHkksx ls çkIr mi;ksfxrk] lhekar
mi;ksfxrk çFke OksQyk (MU
n
), lhekar mi;ksfxrk f}rh; OksQys (MU
2
) vkSj blh çdkj n
th
 bdkbZ
rd dk OkqQy ;ksx gSA
rkfydk la[;k 2-1 rFkk js[kkfp=k 2-1] ,d oLrq dh fofHkUu ek=kkvksa osQ miHkksx ls feyus
okyh lhekar rFkk OkqQy mi;ksfxrk OksQ ewY;ksa dk ,d dkYifud mnkgj.k dks fn[kkrh gSA
lkekU;r% ;g ns[kk tkrk gS fd lhekar mi;ksfxrk] oLrq OksQ mi;ksx esa o`f¼ OksQ lkFk fxj tkrh
gSA ,slk blfy, gksrk gS D;ksafd ,d oLrq dh dqN ek=kk miyC/ gks tkus ij] miHkksDrk dh ml
oLrq dks vkSj vf/d çkIr djus dh bPNk de gks tkrh gSA rkfydk rFkk js[kkfp=k esa blh dks
fn[kk;k x;k gSA
nsf[k;s] fd MU
3, 
MU
2 
ls de gSA vki ;g Hkh ns[k ldrs gSa fd dqy mi;ksfxrk esa o`f¼
gksrh gS ysfdu ?kVrh gqbZ nj ijA fdlh oLrq osQ mi;ksx dh ek=kk esa ifjorZu osQ iQyLo:i]
OkqQy mi;ksfxrk esa o`f¼ dh nj] mldh lhekar mi;ksfxrk dh vk; gSA ;g lhekar mi;ksfxrk]
2018-19
11
miHkksDrk osQ
O;ogkj dk fl¼kar
oLrq dh miHkksx ek=kk esa o`f¼ osQ lkFk fxjrh
tkrh gS& 12 ls 6] 6 ls 4 vkSj blh izdkj
vkxsA ;g fu"d"kZ] âkleku mi;ksfxrk fu;e
ls fudyrk gSA âkleku mi;ksfxrk fu;e
crkrk gS fd tSls&tSls vU; oLrqvksa OksQ
mi;ksx dks fLFkj j[krs gq, fdlh oLrq OksQ
miHkksx dks c<+k;k tkrk gS] oLrq dh gj
vxyh bdkbZ osQ miHkksx ls izkIr lhekar
mi;ksfxrk fxjrh tkrh gSA
MU ml fcUnw ij 'kwU; gks tkrh gS tc
TU fLFkj jgrk gSA fn;s x;s mnkgj.k esa ik¡poha bdkbZ osQ miHkksx ij TU vifjofrZr jgrh gS]
vr% MU
5 
= 0 blosQ i'pkr TU fxjus yxkrh gS vkSj MU ½.kkRed gks tkrh gSA
,dy oLrq dh n'kk esa ekax oØ dh mRifÙk (âkleku mi;ksfxrk fu;e)
fdlh oLrq osQ ekax oØ dk] x.kukokpd fo'ys"k.k osQ }kjk cuk;k tk ldrk gSA ekax D;k gS
rFkk ekax oØ D;k gS\ oLrq dh ek=kk ftls ,d miHkksDrk fn;s x;s oLrq ewY;ksa vkSj vk; ij]
[kjhnus osQ fy;s bPNqd ,oa leFkZ gS dks ml oLrq dh ekax dgrs gSaA oLrq X dh ekax] X oLrq
dh Lo;a dh dher osQ vfrfjDr] vU; oLrqvksa dh dherksa] (çfrLFkkukiUu ,oa iwjd oLrq,a
2-4 ns[ksa)] miHkksDrk dh #fp;ksa vkSj ojh;rk tSls dkjdksa ij fuHkZj djrh gSA ekax oØ] ,d
oLrq dh] fofHkUu dherksa ij] ekaxs tkus okyh ek=kkvksa dh js[kh; çLrqfr gSA tks ,d miHkksDrk
fdlh oLrq dh fofHkUu dherksa ij [kjhnus
osQ fy;s bPNqd gS] vU; lacaf/r oLrqvksa
dh dherksa dks vkSj miHkksDrk dh vk; dks
fLFkj j[krs gq,A
fp=k 2-2] ,d O;fDr dk oLrq X osQ
fy,] fofHkUu dherksa ij] mlosQ  dkYifud
ekax oØ dks çnf'kZr djrk gSA ek=kkvksa dks
{kSfrt v{k ij rFkk dherksa dks mèokZ/j v{k
ij fn[kk;k x;k gSA
uhps dh vksj <yok ekax oØ çnf'kZr
djrk gS fd uhps ewY;ksa ij ,d O;fDr X
fdlh oLrq dh fofHkUu ek=kkvksa osQ miHkksx ls çkIr lhekar ,oa
OkqQy mi;ksfxrk osQ ewY; oLrq osQ miHkksx esa o`f¼ OksQ lkFk
lhekUr mi;ksfxrk ?kVrh tkrh gSA
oLrq X OksQ fy, ,d O;fDrxr dk ek¡x oØ
rkfydk 2-1% ,d oLrq dh fofHkUu ek=kkvksa osQ miHkksx ls çkIr lhekar rFkk dqy mi;ksfxrk osQ ewY;
        bdkbZ oqQy mi;ksfxrk lhekar mi;ksfxrk
1 12 12
2 18 6
3 22 4
4 24 2
5 24 0
6 22 &2
2018-19
12
O;f"V vFkZ'kkL=k
,d ifjp;
oLrq dh vf/d ek=kk [kjhnus dks bPNqd gS rFkk m¡Qps ewY;ksa ij] og oLrq dh dher rFkk ek¡xas
tkus okyh ek=kk osQ chp ½.kkRed laca/ gSA bls ekax dk fu;e dgrs gSaA
uhps dh vksj <yok ekax oØ dh Li"Vhdj.k] âkleku lhekar mi;ksfxrk osQ fopkj ij
vk/kfjr gSA âkleku lhekar mi;ksfxrk dk fu;e crkrk gS fd oLrq dh çR;sd vxyh bdkbZ]
de lhekar mi;ksfxrk çnku djrh gSA blfy;s] O;fDr çR;sd vxyh bdkbZ osQ fy;s mruk nsus
osQ fy;s bPNqd ugha gksxk vkSj blh dkj.k ekax oØ uhps dh vksj <yok oØ gksrk gSA X oLrq
dh :- 40 çfr bdkbZ dher ij] O;fDr dh X oLrq dh ikap bdkb;ksa dh ekax FkhA oLrq dh
NBoha bdkbZ] ikapoh bdkbZ ls de egRo dh gksxhA O;fDr NBh bdkbZ dks RkHkh [kjhnsxk tc
dher :- 40 çfr bdkbZ ls uhps fxj tk,A vr% âkleku lhekar mi;ksfxrk fu;e] ;g Li"V
djrk gS fd ekax oØksa dk ½.kkRed nke D;ksa gksrk gSaA
2-1-2 Øeokpd mi;ksfxrk fo'ys"k.k
x.kukokpd mi;ksfxrk fo'ys"k.k le>us esa ljy gS] ysfdu mi;ksfxrk dk la[;kvksa osQ :i esa
ifjek.ku] bldk ,d cM+k nks"k gSA okLrfod thou esa] ge mi;ksfxrk dks la[;kvksa osQ :i esa
dHkh O;Dr ugha djrsA vf/d ls vf/d ge de ;k vf/d mi;ksfxrk osQ vk/kj ij Øe
ns ldrs gSaA nwljs 'kCnksa esa] miHkksDrk mi;ksfxrk dks la[;kvksa esa ugha ekirk gS] ;|fi og
fofHkUu miHkksDrk caMyksa dks Øe nsrk gSA ;gh] Øeokpd mi;ksfxrk fo'ys"k.k çlax dk çkjafHkd
fcUnq gSA
caMyks osQ lsV esa ,d miHkksDrk osQ vfHkekuksa dks fp=k }kjk Hkh çnf'kZr fd;k tk ldrk gSA
ge igys gh ns[k pqosQ gSa] fd ,d miHkksDrk dk miyC/ caMyksa dks ,d f}rh; fp=k }kjk
fn[kk;k tk ldrk gSA fcUnqvksa tks miHkksDrk dk
leku mi;ksfxrk gS] çnku djus okys caMyksa dk
çfrfuf/Ro djrs gSa] dks js[kkfp=k 2-3 esa fn[kk;s
x;s oØ dks çkIr djus osQs fy;s lkekU;r% tksM+k
tk ldrk gSA
,d vuf/eku oØ ij lHkh fcUnq tSls A, B,
C rFkk D miHkksDrk dks leku Lrj dk larks"k çnku
djrs gSaA ;g Li"V gS fd tc miHkksDrk dks ,d
vkSj osQyk çkIr gksrk gS rks mldks dqN vkeksa dk
R;kx djuk iM+rk gS rkfd mldh mi;ksfxrk dk
Lrj ogh jgs vkSj og mlh vuf/eku oØ ij jgsA
,d vfrfjDr osQyk çkIr djus osQ fy;s] dqy
mi;ksfxrk dk Lrj leku jgrs gq,] vkeksa dh og ek=kk] ftldk miHkksDrk dks R;kx djuk iM+rk
gS] lhekar çfrLFkkiu nj dgrs gSaA nwljs 'kCnksa esa] MRS] ljy:i esa og nj gS ftl ij
miHkksDrk vkeksa osQ LFkku ij osQyksa dks çfrLFkkfir djsxkA rkfd mldh mi;ksfxrk leku jgsA
vr% / MRS Y X =| ? ? |
3
vuf/eku oØ& ,d vuf/eku oØ mu lHkh
fcUnqvksa dk çfrfuf/Ro djus okys caMyks dks ftudks
miHkksDrk rVLFk le>rk gS vkSj mUgh dks tksM+rk gSA
3 
/ / ( / ) 0 Y X Y Xif Y X | ? ? |= ? ? ? ? =
           / ( / ) 0 Y Xif Y X = -? ? ? ? <
/ MRS Y X =| ? ? | dk vFkZ gS fd MRS cjkcj gksrk gSA OksQoy / Y X ? ? ds ifjek.k OksQ ;fn / 3/1 Y X ? ? = - ]
rks bldk vFkZ gS fd MRS=3
2018-19
13
miHkksDrk osQ
O;ogkj dk fl¼kar
rkfydk 2-2% ßkleku lhekar izfrLFkkiuk nj osQ fu;e dk izn'kZu
vki ns[k ldrs gSa fd mijksDr rkfydk esa] tSls ge osQyksa dh la[;k dks c<+krs gSa] osQys
dh izR;sd vfrfjDr bdkbZ osQ fy;s R;kx fd;s tkus okys vkeksa dh ek=kk fxjrh tkrh gSA vU;
'kCnksa esa] osQyksa dh la[;k esa o`f¼ osQ lkFk MRS de gks tkrk gSA miHkksDrk osQ ikl tSls&tSls
osQyksa dh la[;k c<+rh gS osQys ls izkIr MU fxjrh tkrh gSA blh Hkkafr] vkeksa dh la[;k esa deh
ls] vkeksa ls izkIr lhekar mi;ksfxrk c<+rh tkrh gSA bl izdkj] osQyksa dh la[;k esa o`f¼ gksus ls]
miHkksDrk dh] de vkSj de] vkeksa dh ek=kk R;kxus dh izo`fr c<+sxhA osQyksa dh ek=kk esa o`f¼
osQ iQyLo:i] MRS osQ fxjus dh bl izo`fr dks] ßkleku lhekar izfrLFkkiuk dk fu;e dgrs
gSaA bls fp=k 2-3 esa ns[kk tk ldrk gSA A A fcUnq ls B fcUnq ij tkus esa] miHkksDrk 3 vkeksa
dk R;kx djrk gS] B fcUnq ls C fcUnq ij tkus esa] og 2 vkeksa dk R;kx djrk gS rFkk C fcUnq
ls D fcUnq ij tkus esa] og ,d vke dk R;kx djrk gSA vr% ;g Li"V gS fd izR;sd vxys
Øsrk osQ fy;s miHkksDrk FkksM+h vkSj FkksM+h] vkeksa dh ek=kk dk R;kx djrk gSA
vuf/eku oØ dh vkd`fr
mYys[kuh; gS] fd ßkleku lhekar izfrLFkku dk fu;e] vuf/eku oØ dks mn~xe osQ izfr mÙky
cuk nsrk gS] (D;ksa\ ,d vuf/eku oØ cukdj bldh tk¡p dhft;s)A ,d vuf/eku oØ dh
;g vfr lkekU; vkd`fr gSA ysfdu iw.kZ LFkkiUu oLrqvksa
4
 esa lhekar izfrLFkkiUu nj ugha fxjrh
gS] ;g leku jgrh gSA ,d mnkgj.k ysrs gSa&
rkfydk 2-3% ßkleku lhekar izfrLFkkiu nj dk izn'kZu
;gka] bu lHkh la;ksxksa osQ chp] tc rd dh ik¡p #i;s osQ flDosQ rFkk ik¡p #i;s osQ uksVksa dk
;ksx ogh jgrk gS] miHkksDrk rVLFk gSA miHkksDrk osQ fy;s] blls dksbZ iQdZ ugha iM+rk fd mls
ik¡p #i;s dk ,d flDdk feys vFkok ikap #i;s dk ,d uksVA blfy;s] bl ckr dk fopkj
fd;s fcuk fd mlosQ ikl ik¡p #i;s osQ fdrus uksV gSa] miHkksDrk] ik¡p #i;s osQ ,d uksV osQ
la;ksx OksQyksa dh ek=kk (Qx) vkeksa dh ek=kk (Qy) lh-iz- nj (MRS)
A 1 15 &
B 2 12 3-1
C 3 10 2-1
D 4 9 1-1
la;ksx ik¡p #i;s ds uksVksa ik¡p #i;s OksQ flDdksa lh-iz- nj
dh la[;k (Qy) dh la[;k (Qy) (MRS)
A 1 8 &
B 2 7 1%1
C 3 6 1%1
D 4 5 1%1
4
lgh fodYi mu oLrqvksa osQ fy;s gS] tks ,d nwljs osQ LFkku ij iz;ksx fd;k tk ldrk gs vkSj miHkksDrk dks mlh
rjg dh mi;ksfxrk iznku djrk gSA
2018-19