xf.

kr
Hkkx & I

d{kk 12 osQ fy, ikB~;iqLrd

2018-19
 ISBN 81-7450-668-3

izFke laLdj.k lokZf/dkj lqjf{kr

q izdk'kd dh iwoZ vuqefr osQ fcuk bl izdk'ku osQ fdlh Hkkx dks
iQjojh 2007 iQkYxqu 1928 Nkiuk rFkk bySDVªkWfudh] e'khuh] iQksVksizfrfyfi] fjdkWfMZ±x vFkok
fdlh vU; fof/ ls iqu% iz;ksx i¼fr }kjk mldk laxzg.k vFkok
iqueqZæ.k izlkj.k oftZr gSA
vDrwcj 2007 dk£rd 1929 q bl iqLrd dh fcØh bl 'krZ osQ lkFk dh xbZ gS fd izdk'kd dh
iwoZ vuqefr osQ fcuk ;g iqLrd vius ewy vkoj.k vFkok ftYn osQ
tuojh 2009 ek?k 1930 vykok fdlh vU; izdkj ls O;kikj }kjk m/kjh ij] iqufoZØ; ;k
uoacj 2009 dkfrZd 1931 fdjk, ij u nh tk,xh] u csph tk,xhA
q bl izdk'ku dk lgh ewY; bl i`"B ij eqfæzr gSA jcM+ dh eqgj vFkok
fnlacj 2010 vxzgk;.k 1932 fpidkbZ xbZ iphZ (fLVdj) ;k fdlh vU; fof/ }kjk vafdr dksbZ Hkh
la'kksf/r ewY; xyr gS rFkk ekU; ugha gksxkA
tuojh 2012 ikS"k 1933
ekpZ 2013 iQkYxqu 1934 ,u lh bZ vkj Vh osQ izdk'ku izHkkx osQ dk;kZy;
iQjojh 2014 ek?k 1935 ,u-lh-bZ-vkj-Vh- oSaQil
fnlacj 2015 ikS"k 1937 Jh vjfoan ekxZ
u;h fnYyh 110 016 iQksu % 011&26562708
fnlacj 2016 ikS"k 1938 108, 100 iQhV jksM
tuojh 2018 ek?k 1939 gsyh ,DlVsas'ku] gksLMsosQjs
cuk'kadjh III bLVst
cSaxyq# 560 085 iQksu % 080&26725740
PD 6T RSP
uothou VªLV Hkou
Mkd?kj uothou
vgenkckn 380 014 iQksu % 079&27541446
© jk"Vªh; 'kSf{kd vuql/
a ku vkSj izf'k{k.k lh-MCY;w-lh- oSaQil
ifj"kn~ ] 2007 fudV% /udy cl LVkWi ifugVh
dksydkrk 700 114 iQksu % 033&25530454
lh-MCY;w-lh- dkWEIySDl
ekyhxkao
xqokgkVh 781021 iQksu % 0361&2674869

` 115.00 izdk'ku lg;ksx

vè;{k] izdk'ku izHkkx % ,e- fljkt vuoj
,u-lh-bZ-vkj-Vh- okVjekoZQ 80 th-,l-,e- eq[; laiknd % 'osrk mIiy
isij ij eqfnzrA eq[; O;kikj izca/d % xkSre xkaxqyh
eq[; mRiknu vfèkdkjh % v#.k fprdkjk
izdk'ku izHkkx esa lfpo] jk"Vªh; 'kSf{kd (izHkkjh)
vuqLakèkku vkSj izf'k{k.k ifj"kn~] Jh vj¯on
ekxZ] u;h fnYyh 110 016 }kjk izdkf'kr rFkk laiknd % js[kk vxzoky
jkt fizafVax oDlZ] 2&bZ] m|ksx fogkj] lwjtiqj lgk;d mRiknu vf/dkjh % tgku yky
baMfLVª;y ,fj;k] xzsVj uks,Mk (m-iz-) }kjk fp=kkadu lTtk ,oa vkoj.k
eqfnzrA vjfoanj pkoyk

2018-19
 vkeq[k
jk"Vªh; ikB~;p;kZ dh :ijs[kk (2005) lq>krh gS fd cPpksa osQ LowQyh thou dks ckgj osQ thou
ls tksM+k tkuk pkfg,A ;g fl¼kar fdrkch Kku dh ml fojklr osQ foijhr gS ftlosQ izHkkoo'k
gekjh O;oLFkk vkt rd LowQy vkSj ?kj osQ chp varjky cuk, gq, gSA u;h jk"Vªh; ikB~;p;kZ ij
vk/kfjr ikB~;Øe vkSj ikB~;iqLrosaQ bl cqfu;knh fopkj ij vey djus dk iz;kl gSA bl iz;kl
esa gj fo"k; dks ,d e”kcwr nhokj ls ?ksj nsus vkSj tkudkjh dks jVk nsus dh izo`fÙk dk fojks/ 'kkfey
gSA vk'kk gS fd ;s dne gesa jk"Vªh; f'k{kk uhfr (1986) esa of.kZr cky&osaQfnzr O;oLFkk dh fn'kk
esa dkiQh nwj rd ys tk,¡xsA
bl iz;Ru dh liQyrk vc bl ckr ij fuHkZj gS fd LowQyksa osQ izkpk;Z vkSj vè;kid cPpksa
dks dYiuk'khy xfrfof/;ksa vkSj lokyksa dh enn ls lh[kus vkSj lh[kus osQ nkSjku vius vuqHko
ij fopkj djus dk volj nsrs gSaA gesa ;g ekuuk gksxk fd ;fn txg] le; vkSj vkt+knh nh tk,
rks cPps cM+ksa }kjk lkSaih xbZ lwpuk&lkexzh ls tqM+dj vkSj tw>dj u, Kku dk l`tu dj ldrs gSaA
f'k{kk osQ fofo/ lk/uksa ,oa lzksrksa dh vuns[kh fd, tkus dk izeq[k dkj.k ikB~;iqLrd dks ijh{kk
dk ,dek=k vk/kj cukus dh izo`fÙk gSA ltZuk vkSj igy dks fodflr djus osQ fy, t+:jh gS fd
ge cPpksa dks lh[kus dh izfØ;k esa iwjk Hkkxhnkj ekusa vkSj cuk,¡] mUgsa Kku dh fu/kZfjr [kqjkd dk
xzkgd ekuuk NksM+ nsaA
;s mís'; LowQy dh nSfud f”kanxh vkSj dk;Z'kSyh esa dkiQh isQjcny dh ek¡x djrs gSaA nSfud
le;&lkj.kh esa yphykiu mruk gh ”k:jh gS] ftruk okf"kZd dSysaMj osQ vey esa pqLrh] ftlls
f'k{k.k osQ fy, fu;r fnuksa dh la[;k gdhdr cu losQA f'k{k.k vkSj ewY;kadu dh
fof/;k¡ Hkh bl ckr dks r; djsaxh fd ;g ikB~;iqLrd LowQy esa cPpksa osQ thou dks ekufld ncko
rFkk cksfj;r dh txg [kq'kh dk vuqHko cukus esa fdruh izHkkoh fl¼ gksrh gSA cks> dh leL;k
ls fuiVus osQ fy, miyC/ le; dk è;ku j[kus dh igys ls vf/d lpsr dksf'k'k dh gSA bl
dksf'k'k dks vkSj xgjkus osQ ;Ru esa ;g ikB~;iqLrd lksp&fopkj vkSj foLe;] NksVs lewgksa esa ckrphr
,oa cgl vkSj gkFk ls dh tkus okyh xfrfof/;ksa dks izkFkfedrk nsrh gSA
,u-lh-bZ-vkj-Vh- bl iqLrd dh jpuk osQ fy, cukbZ xbZ ikB~;iqLrd fuekZ.k lfefr osQ ifjJe
osQ fy, o`QrKrk O;Dr djrh gSA ifj"kn~ bl ikB~;iqLrd osQ lykgdkj lewg osQ vè;{k izksi+sQlj
t;ar fo".kq ukjyhdj vkSj bl iqLrd osQ lykgdkj izkis + sQlj iou oqQekj tSu dh fo'ks"k vkHkkjh gSA
bl ikB~;iqLrd osQ fodkl esa dbZ f'k{kdksa us ;ksxnku fn;k_ bl ;ksxnku dks laHko cukus osQ fy,

2018-19
 iv

ge muosQ izkpk;ks± osQ vkHkkjh gSaA ge mu lHkh laLFkkvksa vkSj laxBuksa osQ izfr o`QrK gSa ftUgksaus vius
lalk/uksa] lkexzh rFkk lg;ksfx;ksa dh enn ysus esa gesa mnkjrkiwoZd lg;ksx fn;kA ge] fo'ks"k :i
ls ekè;fed ,oa mPprj f'k{kk foHkkx] ekuo lalk/u fodkl ea=kky; }kjk izks- e`.kky fejh vkSj
izks- th-ih- ns'kikaMs dh vè;{krk esa xfBr] jk"Vªh; ekuhVfjax lfefr }kjk iznÙk cgqewY; le; ,oa
;ksxnku osQ fy, o`QrK gSaA O;oLFkkxr lq/kjksa vkSj vius izdk'kuksa esa fujarj fu[kkj ykus osQ izfr
lefiZ r ,u-lh-bZ - vkj-Vh- fVIif.k;ks a ,oa lq > koks a dk Lokxr djs x h ftuls Hkkoh
la'kks/uksa esa enn yh tk losQA

funs'kd
u;h fnYyh jk"Vªh; 'kSf{kd vuqla/ku
20 uoacj 2006 vkSj izf'k{k.k ifj"kn~

2018-19
 izLrkouk
jk"Vªh; 'kSf{kd vuqla/ku vkSj izf'k{k.k ifj"kn us fo|ky;h f'k{kk ls lacaf/r fofHkUu fo"k;ksa osQ
vè;;u osQ fy,] jk"Vªh; ikB~; p;kZ :ijs[kk dh leh{kk gsrq fo|ky;h f'k{kk&2000 (,u-lh-,iQ-
,l-bZ&2000) osQ varxZr vkfoHkkZo pqukSfr;ksa vkSj fo"k; oLrq osQ :ikarj.k] tks f'k{kk 'kkL=k osQ {ks=k
esa varfuZfgr gSa] mUgsa jk"Vªh; ,oa varjkZ"Vªh; Lrj ij fo|ky;h f'k{kk osQ fy, 21 iQksdl lewgksa dk
xBu fd;k gSA bl iQksdl lewg us fo|ky;h f'k{kk {ks=k osQ fofHkUu igyqvksa ij viuh O;kid vkSj
fo'ks"k fVIif.k;k¡ dh gSAa blh osQ iQyLo:i] bu lewgksa }kjk viuh fjiksVks± osQ vk/kj ij jk"Vªh; ikB~;
p;kZ :ijs[kk&2005 dks fodflr fd;k x;kA
u, fn'kk&funsZ'kksa osQ varxZr gh jk"Vªh; 'kSf{kd vuqla/ku vkSj izf'k{k.k ifj"kn us d{kk XI vkSj
XII dh xf.kr fo"k; dk ikB~;Øe rS;kj fd;k rFkk ikB~;iqLrosaQ rS;kj djus osQ fy, ,d Vhe dk
xBu fd;kA d{kk XI dh ikB~;&iqLrd igys ls gh iz;ksx esa gS tks 2005 esa izdkf'kr dh tk
pqdh gSA
iqLrd dk izFke izk:i (d{kk XII) ,u-lh-bZ-vkj-Vh- ladk;] fo'ks"kK vkSj dk;Zjr~ vè;kidksa
dh Vhe }kjk rS;kj dj fy;k x;kA rRi'pkr~ fodkl'khy Vhe us fofHkUu cSBosaQ vk;ksftr dj bl
izk:i dks ifj"o`Qr fd;k FkkA
iqLrd osQ bl izk:i dks ns'k osQ fofHkUu Hkkxksa esa mPprj ekè;fed Lrj ij xf.kr osQ vè;kiu
ls lac¼ vè;kiujr~ f'k{kdksa dh ,d Vhe osQ le{k izLrqr fd;k FkkA iqu% izk#i dh ,ulhbZvkjVh
}kjk vk;ksftr dk;Z'kkyk esa leh{kk dh xbZA lgHkkfx;ksa }kjk fn, x, lq>koksa ,oa fVIif.k;ksa dks izk:i
ikB~;iqLrd esa lek;ksftr dj fy;k x;kA fodkl'khy Vhe esa ls gh xfBr ,d laikndh; eaMy
us ikB~;&iqLrd osQ bl izk:i dks vafre :i ns fn;kA varr%] foKku ,oa xf.kr ds lykgdkj lewg
rFkk ekuo lalk/u ea=kky;] Hkkjr ljdkj }kjk xfBr fuxjkuh lfefr (Monitoring Committee)
us bl ikB~;iqLrd izk:i dks vuqeksfnr dj fn;kA
fo"k; dh izekf.kdrk dh n`f"V ls iqLrd dks izHkkfor djus okys oqQN vko';d rRoksa dk mYys[k
djrs gSaA ;s fof'k"Vrk,¡ yxHkx bl iqLrd osQ lHkh ikBksa esa ifjyf{kr gSaA izLrqr ikB~;iqLrd esa 13
eq[; vè;k; vkSj nks ifjf'k"V 'kkfey gSAa izR;sd vè;k; fuEufyf[kr fcanq lekfgr djrk gS%
• Hkwfedk % fo"k; osQ egRoiw.kZ fcanqvksa ij cy_ iwoZ esa i<+k, x, fo"k;&oLrqvksa dk ijLij
laca/_ vè;k; esa yxHkx u;h vo/kj.kkvksa dk lkj&:i esa foospukA
• vè;k; esa [kaMks dks 'kfey djrs gq, /kj.kkvksa vkSj vo/kj.kkvksa dk laxBuA
• /kj.kkvksa @ vo/kj.kkvksa dh tkudkjh dks izsj.kknk;d cukrs gq,] tgk¡ Hkh laHko gks ldk n`"Vakr
miyC/ djk, x, gSaA

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 vi

• miifÙk@leL;k osQ gy fl¼kar vkSj vuqiz;ksx nksuksa i{kksa ij cy nsrs gq, ;k rkfoZQd] cgqfo/
lk/u] tgk¡ Hkh bUgsa viukus dh vko';drk iM+h] viuk;k gSA
• T;kfefr; n`f"Vdks.k@ladYiukvksa dk izLrqrhdj.k vko';d gksus ij fn;k x;k gSA
• xf.krh; vo/kj.kkvksa vkSj blosQ lg&fo"k;ksa tSls% foKku ,oa lkekftd foKku ls Hkh tksM+k
x;k gSA
• fo"k; osQ izR;sd [kaM esa i;kZIr vkSj fofo/ mnkgj.k@vH;kl fn, x, gSaA
• leL;kvksa dks gy djus dh {kerk ;k dkS'ky ,oa vuqiz;ksx djus dh le> dks osaQnzhr ,oa
etcwr djus gsrq vè;k; osQ var esa nks ;k nks ls vf/d ladYiukvksa dks lekosf'kr djus okys
mnkgj.kksa rFkk vH;kl&iz'uksa dk lek;kstu fd;k x;k gS] tSlk fd jk"Vªh; ikB~;&p;kZ :i js[kk
2005 esa dgk x;k gS] blh osQ vuq:i es/koh Nk=kksa osQ fy, Hkh ikB~;&iqLrd esa pqukSrhiw.kZ
leL;kvksa dks 'kkfey fd;k x;k gSA
• fo"k; dks vkSj vf/d izsj.kknk;d cukus osQ mn~ns'; ls fo"k; dh laf{kIr ,sfrgkfld i`"BHkwfe
ikB osQ var esa nh xbZ gS vkSj izR;sd ikB osQ izkjaHk esa lacaf/r dFku ,oa lqizfl¼ xf.krKksa osQ
fp=k fn;s x, gSa ftUgksaus fo'ks"kr;k fo"k;&oLrq dks fodflr vkSj lqcks/ cukus osQ fy, viuk
;ksxnku fn;kA
• varr% fo"k; dh ladYukvksa osQ lw=k ,oa ifj.kke osQ izR;{k lkj&dFku osQ fy, ikB dk laf{kIr
lkjka'k Hkh izLrqr fd;k x;k gSA
eSa fo'ks"k :i ls jk"Vªh; 'kSf{kd vuqla/ku vkSj izf'k{k.k ifj"kn~ osQ funs'kd izks- o`Q".k oqQekj dk
vkHkkjh gw¡ ftUgksaus eq>s fueaf=kr dj xf.kr f'k{kk osQ jk"Vªh; iz;kl dh dM+h ls tksM+k gSA mUgksaus gesa
bl gsrq ckSf¼d ifjizs{; rFkk LoLF; okrkoj.k iznku fd;kA bl iqLrd dks rS;kj djus dk dk;Z
vR;Ur lq[kn ,oa iz'kaluh; jgkA eSa] foKku ,oa xf.kr dh lykgdkj lewg osQ vè;{k izks-ts-oh-
ukjyhdj dk o`QrK gw¡ ftUgksaus le;&le; ij bl iqLrd osQ fy, vius fo'ks"k lq>ko ,oa lg;ksx
nsdj iqLrd osQ lq/kj esa dk;Z fd;kA eSa ifj"kn~ osQ la;qDr funs'kd izks-th-johUnzk dks Hkh /U;okn
nsrk g¡w ftUgksus le;&le; ij ikB~;&iqLrd ls lacaf/r fØ;k&fof/ dks lapkfyr djus esa ;ksxnku
fd;kA
eSa izks- gqoqQe flag] eq[; la;kstd ,oa vè;{k] foKku ,oa xf.kr] MkW-oh-ih-flag] la;kstd rFkk
izks- ,l-osQ-flag xkSre osQ izfr lân; /U;okn O;Dr djrk gWw¡ ftUgkasus bl ifj;kstuk dks liQy cukus
gsrq 'kSf{kd vkSj iz'kklfud :i ls layXu jgsA eSa bl usd dk;Z ls lac¼ lHkh Vhe osQ lnL;ksa vkSj
f'k{kdksa dh iz'kalk djrk g¡w rFkk mUgsa /U;okn nsrk gw¡ tks bl dk;Z esa fdlh Hkh :i esa ;ksxnku
fd;k gksA

iou osQ- tSu

eq[; lykgdkj
ikB~;iqLrd lao/Zu lfefr

2018-19
 ikB~;iqLrd fodkl lfefr
foKku ,oa xf.kr lykgdkj lewg osQ vè;{k
t;ar fo".kq ukjyhdj behfjVl izkis + sQlj] vè;{k] vkbZ-;w-lh-,-] iwuk fo'ofo|ky;] iwukA
eq[; lykgdkj
ih-osQ- tSu] izksi+sQlj xf.kr foHkkx] fnYyh fo'ofo|ky;] fnYyhA
eq[; leUo;d
gqoqQe flag] izkis + sQlj ,oa foHkkxkè;{k] Mh-bZ-,l-,e-] ,u-lh-bZ-vkj-Vh-] u;h fnYyhA
lnL;
v#.k iky flag] lhfu;j izoDrk] xf.kr foHkkx] n;ky flag dkWyt s ] fnYyh fo'ofo|ky;] fnYyhA
,-osQ-jktiwr] ,lksf'k,V izksi+sQlj] {ks-f'k-l- ,u-lh-bZ-vkj-Vh-] HkksikyA
izksi+sQlj] ch-,l-ih- jktw] {ks-f'k-l- ,u-lh-bZ-vkj-Vh-] eSlwj] dukZVdA
lh-vkj-iznhi] lgk;d izksi+sQlj] xf.kr foHkkx] Hkkjrh; foKku laLFkku] caxykSj] dukZVdA
vkj-Mh- 'kekZ] ih-th-Vh-] tokgj uoksn; fo|ky;] eqaxs'kiqj] fnYyhA
jke vorkj] izksi+sQlj (vodk'k izkIr) ,oa lykgdkj] Mh-bZ-,l-,e-] ,u-lh-bZ-vkj-Vh-] u;h fnYyhA
vkj-ih-ekS;Z] ,lksf'k,V izksi+sQlj] Mh-bZ-,l-,e-] ,u-lh-bZ-vkj-Vh-] u;h fnYyhA
,l-,l-[ksj] izksi+sQlj] le mi oqQyifr] ,u-bZ-,l-;w-] rqjk oSaQil es?kky;A
,l-osQ-,l- xkSre] izksi+sQlj] Mh-bZ-,l-,e-] ,u-lh-bZ-vkj-Vh-] u;h fnYyhA
,l-osQ-dkSf'kd] ,lksf'k,V izksi+sQlj] xf.kr foHkkx] fdjksM+hey dkWyst] fnYyh fo'ofo|ky;] fnYyhA
laxhrk vjksM+k] ih-th-Vh-] ,ihts LowQy] lkosQr] u;h fnYyhA
'kSytk frokjh] ih-th-Vh-] osaQnzh; fo|ky;] cjdkdkuk] gtkjhckx] >kj[kaMA
fouk;d cqtkMs] ysDpjj] fonZHk cqfu;knh twfu;j dkWyst] lDdjnkjk pkSd] ukxiqj] egkjk"VªA
lqfuy ctkt] lhfu;j Lis'kfyLV] ,l-lh-bZ-vkj-Vh-] xqM+xk¡o] gfj;k.kkA
lnL; leUo;d
oh-ih-flag] ,lksf'k,V izksi+sQlj] Mh-bZ-,l-,e-] ,u-lh-bZ-vkj-Vh-] u;h fnYyhA

2018-19
 viii

fganh :ikarj.kdrkZ
Mh-vkj-'kekZ] ih-th-Vh-] tokgj uoksn; fo|ky;] eqaxs'kiqj] fnYyhA
ih-osQ- frokjh] lgk;d vk;qDr (v-izk-) oasQnzh; fo|ky; laxBuA
,l-ch-f=kikBh] ysDpjj] (xf.kr) jktdh; izfrHkk fodkl fo|ky;] lwjtey fogkj] fnYyhA
,-osQ- jktiwr] ,lksf'k,V izkis + sQlj] (xf.kr)] {ks-f'k-l- ,u-lh-bZ-vkj-Vh-] Hkksiky] eè; izns'kA
oh-ih-flag] ,lksf'k,V izksi+sQlj] (xf.kr)] Mh-bZ-,l-,e-] ,u-lh-bZ-vkj-Vh-] u;h fnYyhA
fganh leUo;d
,l-osQ-flag xkSre] izksi+sQlj] Mh-bZ-,l-,e-] ,u-lh-bZ-vkj-Vh-] u;h fnYyhA

2018-19
 vkHkkj
ifj"kn~ bl ikB~;iqLrd leh{kk dk;kZ'kkyk osQ fuEufyf[kr izfrHkkfx;ksa osQ cgqewY; lg;ksx osQ fy,
viuk gkfnZd vkHkkj O;Dr djrh gS% txnh'k lju] izksi+sQlj] lkaf[;dh; foHkkx] fnYyh
fo'ofo|ky;_ oqQíwl [kku] ysDpjj] f'kcyh us'kuy ih-th- dkWyst vktex<] (m-iz-)_ ih-osQ-
frokjh] lgk;d vk;qDr (v-izk-)] osaQnzh; fo|ky; laxBu_ ,l-ch- f=kikBh] ysDpjj] vkj-ih-ch-
fo- lwjtey fogkj] fnYyh_ vks-,u- flag] jhMj] vkj-vkbZ-bZ- Hkqous'oj] mM+hlk_ oqQekjh ljkst]
ysDpjj] xouZeVas xYlZ lhfu;j lsoQas Mjh LowQy] u- 1] :iuxj] fnYyh_ ih-HkkLdj oqQekj] ih-th-Vh-]
tokgj uoksn; fo|ky;] ysik{kh] vuariqj] (vka/z izns'k)_ Jherh dYikxe~] ih-th-Vh] osQ-oh- uky
oSQail] cSaxyksj_ jkgqy lksiQr] ysDpjj] ,vj iQkslZ xksYMu tqcyh bafLVV~;wV] lqczrks ikoZQ] u;h
fnYyh_ oafnrk dkyjk] loksZn; dU;k fo|ky;] fodkliqjh tuin osaQnz] u;h fnYyh_ tuknZu
f=kikBh] ysDpjj] xouZesaV vkj-,p-,l-,l- ,stkOy] fetksje vkSj lqJh lq"kek t;jFk] jhMj] Mh-
MCy;w-,l-] ,u-lh-bZ-vkj-Vh] u;h fnYyhA
ifj"kn~ ,u-lh-bZ-vkj-Vh- esa fganh :ikraj.k osQ iqujkoyksdu gsrq dk;Z'kkyk esa fuEufyf[kr
izfrHkkfx;ksa dh cgqewY; fVIif.k;ksa osQ fy, vkHkkjh gS_ th-Mh-<y] vodk'k izkIr jhMj] ,u-lh-bZ-
vkj-Vh-] u;h fnYyh_ th-,l-jkBkSj] vflLVSaV izksi+sQlj] xf.kr ,oa lakf[;dh foHkkx] ,e-,y-
lq[kkfM+;k fo'ofo|ky;] mn;iqj] jktLFkku_ eukst oqQekj BkoqQj] Mh-,-oh- ifCyd LowQy] jktsUnz
uxj] lkfgckckn] xkft;kckn (m-iz-)_ jkes'oj n;ky 'kekZ] jktdh; baVj dkWyst] eFkqjk (m-iz-)_
MkW-vkj-ih-fxgkjs] CykWd fjlksZl dksvkfMZusVj] tuin f'k{kk osaQnz] fppkSyh] csrqy (e-iz-)_ lquhy
ctkt] ,l-lh-bZ-vkj-Vh-] xqM+xk¡o] gfj;k.kk_ Jherh ohuk /haxjk] lj y{eh ckfydk lhfu;j
lsosaQMjh LowQy] [kkjh ckoyh] fnYyh_ ,-osQ-o>yokj] jhMj] ,u-lh-bZ-vkj-Vh] u;h fnYyhA
ifj"kn~ fp=kkadu vjfoanj pkoyk] daI;wVj LVs'ku izHkkjh nhid diwj_ jkosQ'k oqQekj ,oa
lTtkn gSnj valkjh] ujfxl bLyke] Mh-Vh-ih- vkWijsVj_ osQ-ih-,l-;kno] eukst eksgu]
dkWih ,fMVj vkSj :ch oqQekjh rFkk j.k/hj BkoqQj] izwi+ Q jhMj] }kjk fd, x, iz;klksa osQ izfr viuk
vkHkkj izdV djrh gSA ,-ih-lh- vkWfiQl] foKku ,oa xf.kr f'k{kk foHkkx ,oa izdk'ku foHkkx Hkh
vius lg;ksx osQ fy, vkHkkj osQ ik=k gSaA

2018-19
2018-19
 fo"k;&lwph
Hkkx & I
vkeq[k iii
izLrkouk v
1. laca/ ,oa iQyu 1
1.1 Hkwfedk 1
1.2 laca/ksa osQ izdkj 2
1.3 iQyuksa osQ izdkj 8
1.4 iQyuksa dk la;kstu rFkk O;qRØe.kh; iQyu 13
1.5 f}&vk/kjh lafØ;k,¡ 22
2. izfrykse f=kdks.kferh; iQyu 38
2.1 Hkwfedk 38
2.2 vk/kjHkwr ladYiuk,¡ 38
2.3 izfrykse f=kdks.kferh; iQyuksa osQ xq.k/eZ 48
3. vkO;wg 62
3.1 Hkwfedk 62
3.2 vkO;wg 62
3.3 vkO;wgksa osQ izdkj 67
3.4 vkO;wgksa ij lafØ;k,¡ 71
3.5 vkO;wg dk ifjorZ 91
3.6 lefer rFkk fo"ke lefer vkO;wg 93
3.7 vkO;wg ij izkjafHkd lafØ;k (vkO;wg :ikarj.k) 98
3.8 O;qRØe.kh; vkO;wg 99
4. lkjf.kd 112
4.1 Hkwfedk 112
4.2 lkjf.kd 113
4.3 lkjf.kdksa osQ xq.k/eZ 119

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4.4 f=kHkqt dk {ks=kiQy 131

4.5 milkjf.kd vkSj lg[kaM 133
4.6 vkO;wg osQ lg[kaMt vkSj O;qRØe 137
4.7 lkjf.kdksa vkSj vkO;wgksa osQ vuqiz;ksx 144
5. lkarR; rFkk vodyuh;rk 160
5.1 Hkwfedk 160
5.2 lkarR; 160
5.3 vodyuh;rk 176
5.4 pj?kkrakdh rFkk y?kqx.kdh; iQyu 185
5.5 y?kqx.kdh; vodyu 191
5.6 iQyuksa osQ izkpfyd :iksa osQ vodyt 195
5.7 f}rh; dksfV dk vodyt 197
5.8 ekè;eku izes; 200
6. vodyt osQ vuqiz;ksx 210
6.1 Hkwfedk 210
6.2 jkf'k;ksa osQ ifjorZu dh nj 210
6.3 o/Zeku vkSj ßkleku iQyu 215
6.4 Li'kZ js[kk,¡ vkSj vfHkyac 223
6.5 lfUudVu 229
6.6 mPpre vkSj fuEure 233
ifjf'k"V1% xf.kr esa miifÙk;k¡ 265
A.1.1 Hkwfedk 265
A.1.2 miifÙk D;k gS? 265
ifjf'k"V 2: xf.krh; fun'kZu 274
A.2.1 Hkwfedk 274
A.2.2 xf.krh; fun'kZu D;ksa\ 274
A.2.3 xf.krh; fun'kZu osQ fl¼kar 275
mÙkjekyk 286
iwjd ikB~; lkexzh 303

2018-19
 vè;k; 1
laca/ ,oa iQyu
(Relations and Functions)

vThere is no permanent place in the world for ugly mathematics ... . It may
be very hard to define mathematical beauty but that is just as true of
beauty of any kind, we may not know quite what we mean by a
beautiful poem, but that does not prevent us from recognising
one when we read it. — G. H. Hardy v

1.1 Hkwfedk (Introduction)

Lej.k dhft, fd d{kk XI esa] laca/ ,oa iQyu] izkar] lgizkar rFkk
ifjlj vkfn dh vo/kj.kkvksa dk] fofHkUu izdkj osQ okLrfod
ekuh; iQyuksa vkSj muosQ vkys[kksa lfgr ifjp; djk;k tk pqdk
gSA xf.kr esa 'kCn ^laca/ (Relation)* dh loaQYiuk dks vaxzs”kh
Hkk"kk esa bl 'kCn osQ vFkZ ls fy;k x;k gS] ftlosQ vuqlkj nks
oLrq,¡ ijLij lacaf/r gksrh gS] ;fn muosQ chp ,d vfHkKs;
(Recognisable) dM+h gksA eku yhft, fd A, fdlh LowQy dh
d{kk XII osQ fo|kfFkZ;ksa dk leqPp; gS rFkk B mlh LowQy dh
d{kk XI osQ fo|kfFkZ;ksa dk leqPp; gSaA vc leqPp; A ls
leqPp; B rd osQ laca/ksa osQ oqQN mnkgj.k bl izdkj gSa
(i) {(a, b) ∈ A × B: a, b dk HkkbZ gS}, Lejeune Dirichlet
(1805-1859)
(ii) {(a, b) ∈ A × B: a, b dh cgu gS},
(iii) {(a, b) ∈ A × B: a dh vk;q b dh vk;q ls vf/d gS},
(iv) {(a, b) ∈ A × B: fiNyh vafre ijh{kk esa a }kjk izkIr iw.kk±d b }kjk izkIr iw.kk±d ls
de gS },
(v) {(a, b) ∈ A × B: a mlh txg jgrk gS tgk¡ b jgrk gS}.
rFkkfi A ls B rd osQ fdlh laca/ R dks vewrZ:i (Abstracting) ls ge xf.kr esa
A × B osQ ,d LosPN (Arbitrary) mileqPp; dh rjg ifjHkkf"kr djrs gSaA

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2 xf.kr

;fn (a, b) ∈ R, rks ge dgrs gSa fd laca/ R osQ varxZr a, b ls lacaf/r gS vkSj ge bls
a R b fy[krs gSaA lkekU;r%] ;fn (a, b) ∈ R, rks ge bl ckr dh fpark ugha djrs gSa fd
a rFkk b osQ chp dksbZ vfHkKs; dM+h gS vFkok ugha gSA tSlk fd d{kk XI esa ns[k pqosQ gSa] iQyu
,d fo'ks"k izdkj osQ laca/ gksrk gSaA
bl vè;k; esa] ge fofHkUu izdkj osQ laca/ksa ,oa iQyuksa] iQyuksa osQ la;kstu (composition)]
O;qRØe.kh; (Invertible) iQyuksa vkSj f}vk/kjh lafØ;kvksa dk vè;;u djsaxsA
1.2 laca/ksa osQ izdkj (Types of Relations)
bl vuqPNsn esa ge fofHkUu izdkj osQ laca/ksa dk vè;;u djsaxsA gesa Kkr gS fd fdlh leqPp; A
esa laca/] A × A dk ,d mileqPp; gksrk gSA vr% fjDr leqPp; φ ⊂ A × A rFkk
A × A Lo;a] nks vUR; laca/ gSaA Li"Vhdj.k gsrq] R = {(a, b): a – b = 10} }kjk iznÙk leqPp;
A = {1, 2, 3, 4} ij ifjHkkf"kr ,d laca/ R ij fopkj dhft,A ;g ,d fjDr leqPp; gS] D;ksafd
,slk dksbZ Hkh ;qXe (pair) ugha gS tks izfrca/ a – b = 10 dks larq"V djrk gSA blh izdkj
R′ = {(a, b) : | a – b | ≥ 0}] laiw.kZ leqPp; A × A osQ rqY; gS] D;ksafd A × A osQ lHkh ;qXe
(a, b), | a – b | ≥ 0 dks larq"V djrs gSaA ;g nksuksa vUR; osQ mnkgj.k gesa fuEufyf[kr ifjHkk"kkvksa
osQ fy, izsfjr djrs gSaA
ifjHkk"kk 1 leqPp; A ij ifjHkkf"kr laca/ R ,d fjDr laca/ dgykrk gS] ;fn A dk dksbZ Hkh
vo;o A osQ fdlh Hkh vo;o ls lacaf/r ugha gS] vFkkZr~ R = φ ⊂ A × A.
ifjHkk"kk 2 leqPp; A ij ifjHkkf"kr laca/ R, ,d lkoZf=kd (universal) laca/ dgykrk gS] ;fn
A dk izR;sd vo;o A osQ lHkh vo;oksa ls lacaf/r gS] vFkkZr~ R = A × A.
fjDr laca/ rFkk lkoZf=kd laca/ dks dHkh&dHkh rqPN (trivial) laca/ Hkh dgrs gSaA
mnkgj.k 1 eku yhft, fd A fdlh ckydksa osQ LowQy osQ lHkh fo|kfFkZ;ksa dk leqPp; gSA n'kkZb,
fd R = {(a, b) : a, b dh cgu gS } }kjk iznÙk laca/ ,d fjDr laca/ gS rFkk R′ = {(a, b) :
a rFkk b dh Å¡pkbZ;ksa dk varj 3 ehVj ls de gS } }kjk iznÙk laca/ ,d lkoZf=kd laca/ gSA
gy iz'ukuqlkj] D;ksafd LowQy ckydksa dk gS] vr,o LowQy dk dksbZ Hkh fo|kFkhZ] LowQy osQ fdlh
Hkh fo|kFkhZ dh cgu ugha gks ldrk gSA vr% R = φ] ftlls iznf'kZr gksrk gS fd R fjDr laca/ gSA
;g Hkh Li"V gS fd fdUgha Hkh nks fo|kfFkZ;ksa dh Å¡pkb;ksa dk varj 3 ehVj ls de gksuk gh pkfg,A
blls izdV gksrk gS fd R′ = A × A lkoZf=kd laca/ gSA
fVIi.kh d{kk XI esa fo|kFkhZx.k lh[k pqosQ gSa fd fdlh laca/ dks nks izdkj ls fu:fir fd;k tk
ldrk gS] uker% jksLVj fof/ rFkk leqPp; fuekZ.k fof/A rFkkfi cgqr ls ys[kdksa }kjk leqPp;
{1, 2, 3, 4} ij ifjHkkf"kr laca/ R = {(a, b) : b = a + 1} dks a R b }kjk Hkh fu:fir fd;k tkrk
gS] ;fn vkSj osQoy ;fn b = a + 1 gksA tc dHkh lqfo/ktud gksxk] ge Hkh bl laosQru (notation)
dk iz;ksx djsaxsA

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 laca/ ,oa iQyu 3

;fn (a, b) ∈ R, rks ge dgrs gSa fd a,b ls lacaf/r gS* vkSj bl ckr dks ge a R b }kjk
izdV djrs gSaA
,d vR;Ur egRoiw.kZ laca/] ftldh xf.kr esa ,d lkFkZd (significant) Hkwfedk gS] rqY;rk
laca/ (Equivalence Relation) dgykrk gSA rqY;rk laca/ dk vè;;u djus osQ fy, ge igys
rhu izdkj osQ laca/ksa] uker% LorqY; (Reflexive)] lefer (Symmetric) rFkk laØked
(Transitive) laca/ksa ij fopkj djrs gSaA
ifjHkk"kk 3 leqPp; A ij ifjHkkf"kr laca/ R;
(i) LorqY; (reflexive) dgykrk gS] ;fn izR;sd a ∈ A osQ fy, (a, a) ∈ R,
(ii) lefer (symmetric) dgykrk gS] ;fn leLr a1, a2 ∈ A osQ fy, (a1, a2) ∈ R ls
(a2, a1) ∈ R izkIr gksA
(iii) laØked (transitive) dgykrk gS] ;fn leLr, a1, a2, a3 ∈ A osQ fy, (a1, a2) ∈ R rFkk
(a2, a3) ∈ R ls (a1, a3) ∈ R izkIr gksA
ifjHkk"kk 4 A ij ifjHkkf"kr laca/ R ,d rqY;rk laca/ dgykrk gS] ;fn R LorqY;] lefer rFkk
laØked gSA
mnkgj.k 2 eku yhft, fd T fdlh lery esa fLFkr leLr f=kHkqtksa dk ,d leqPp; gSA leqPp;
T esa R = {(T1, T2) : T1, T2osQ lokZxale gS} ,d laca/ gSA fl¼ dhft, fd R ,d rqY;rk
laca/ gSA
gy la c a / R Lorq Y ; gS ] D;ks a f d iz R ;s d f=kHkq t Lo;a os Q lokx± l e gks r k gS A iq u %
(T1, T2) ∈ R ⇒ T1 , T2 osQ lokZxale gS ⇒ T2 , T1 osQ lokZxale gS ⇒ (T2, T1) ∈ R. vr%
laca/ R lefer gSA blosQ vfrfjDr (T1, T2), (T2, T3) ∈ R ⇒ T1 , T2 osQ lokZxale gS rFkk
T2, T3 osQ lokZxale gS ⇒ T1, T3 osQ lokZxale gS ⇒ (T1, T3) ∈ R. vr% laca/ R laØked gSA
bl izdkj R ,d rqY;rk laca/ gSA
mnkgj.k 3 eku yhft, fd L fdlh lery esa fLFkr leLr js[kkvksa dk ,d leqPp; gS rFkk
R = {(L1, L2) : L1, L2 ij yac gS} leqPp; L esa ifjHkkf"kr ,d laca/ gSA fl¼ dhft, fd R
lefer gS ¯drq ;g u rks LorqY; gS vkSj u laØked gSA
gy R LorqY; ugha gS] D;ksafd dksbZ js[kk L1 vius vki ij yac ugha gks ldrh gS] vFkkZr~
(L1, L1) ∉ R- R lefer gS] D;ksafd (L1, L2) ∈ R
⇒ L1, L2 ij yac gS
⇒ L2 , L1 ij yac gS
⇒ (L2, L1) ∈ R

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R laØked ugha gSA fu'p; gh] ;fn L1, L2 ij yac gS rFkk L2 ,

L3 ij yac gS] rks L1 , L3 ij dHkh Hkh yac ugha gks ldrh gSA okLro
esa ,slh n'kk esa L1 , L3 osQ lekUrj gksxhA vFkkZr~] (L1, L2) ∈ R,
(L2, L3) ∈ R ijarq (L1, L3) ∉ R
mnkgj.k 4 fl¼ dhft, fd leqPp; {1, 2, 3} esa R = {(1, 1), (2, 2), vko`Qfr 1-1
(3, 3), (1, 2), (2, 3)} }kjk iznÙk laca/ LorqY; gS] ijarq u rks lefer gS vkSj u laØked gSA
gy R LorqY; gS D;ksafd (1, 1), (2, 2) vkSj (3, 3), R osQ vo;o gSaA R lefer ugha gS] D;ksafd
(1, 2) ∈ R ¯drq (2, 1) ∉ R. blh izdkj R laØked ugha gS] D;ksafd (1, 2) ∈ R rFkk (2, 3)∈R
ijarq (1, 3) ∉ R
mnkgj.k 5 fl¼ dhft, fd iw.kk±dksa osQ leqPp; Z esa R = {(a, b) : la[;k 2, (a – b) dks
foHkkftr djrh gS} }kjk iznÙk laca/ ,d rqY;rk laca/ gSA
gy R LorqY; gS] D;ksafd leLr a ∈ Z osQ fy, 2] (a – a) dks foHkkftr djrk gSA vr%
(a, a) ∈ R. iqu%] ;fn (a, b) ∈ R, rks 2] a – b dks foHkkftr djrk gS A vr,o b – a dks Hkh
2 foHkkftr djrk gSA vr% (b, a) ∈ R, ftlls fl¼ gksrk gS fd R lefer gSA blh izdkj] ;fn
(a, b) ∈ R rFkk (b, c) ∈ R, rks a – b rFkk b – c la [ ;k 2 ls HkkT; gS A vc]
a – c = (a – b) + (b – c) le (even) gS (D;ksa\)A vr% (a – c) Hkh 2 ls HkkT; gSA blls fl¼
gksrk gS fd R laØked gSA vr% leqPp; Z esa R ,d rqY;rk laca/ gSA
mnkgj.k 5 esa] uksV dhft, fd lHkh le iw.kk±d 'kwU; ls lacaf/r gSa] D;ksafd (0, ± 2),
(0, ± 4), ---vkfn R esa gSa vkSj dksbZ Hkh fo"ke iw.kk±d 0 ls lacaf/r ugha gS] D;ksafd (0, ± 1),
(0, ± 3), ---vkfn R esa ugha gSAa blh izdkj lHkh fo"ke iw.kk±d 1 ls lacfa /r gSa vkSj dksbZ Hkh le iw.kk±d
1 ls lacaf/r ugha gSA vr,o] leLr le iw.kk±dksa dk leqPp; E rFkk leLr fo"ke iw.kk±dksa dk
leqPp; O leqPp; Z osQ mi leqPp; gSa] tks fuEufyf[kr izfrca/ksa dks larq"V djrs gSaA
(i) E osQ leLr vo;o ,d nwljs ls lacaf/r gSa rFkk O osQ leLr vo;o ,d nwljs ls
lacaf/r gSaA
(ii) E dk dksbZ Hkh vo;o O osQ fdlh Hkh vo;o ls lacaf/r ugha gS vkSj foykser% O dk dksbZ
Hkh vo;o E osQ fdlh Hkh vo;o ls lacaf/r ugha gSA
(iii) E rFkk O vla;qDr gS vkSj Z = E ∪ O gSA
mileqPp; E, 'kwU; dks varfoZ"V (contain) djus okyk rqY;rk&oxZ (Equivalence Class)
dgykrk gS vkSj ftls izrhd [0] ls fu:fir djrs gSaA blh izdkj O, 1 dks varfoZ"V djus okyk
rqY;rk&oxZ gS] ftls [1] }kjk fu:fir djrs gSaA uksV dhft, fd [0] ≠ [1], [0] = [2r] vkSj

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 laca/ ,oa iQyu 5

[1]= [2r + 1], r ∈ Z. okLro esa] tks oqQN geus Åij ns[kk gS] og fdlh Hkh leqPp; X esa ,d
LosPN rqY;rk laca/ R osQ fy, lR; gksrk gSA fdlh iznÙk LosPN leqPp; X esa iznÙk ,d LosPN
(arbitrary) rqY;rk laca/ R, X dks ijLij vla;qDr mileqPp;ksa Ai esa foHkkftr dj nsrk gS] ftUgsa
X dk foHkktu (Partition) dgrs gSa vksj tks fuEufyf[kr izfrca/ksa dks larq"V djrs gSa%
(i) leLr i osQ fy, Ai osQ lHkh vo;o ,d nwljs ls lacaf/r gksrs gSaA
(ii) Ai dk dksbZ Hkh vo;o] Aj osQ fdlh Hkh vo;o ls lacaf/r ugha gksrk gS] tgk¡ i ≠ j
(iii) ∪ Aj = X rFkk Ai ∩ Aj = φ, i ≠ j
mileqPp; Ai rqY;rk&oxZ dgykrs gSaA bl fLFkfr dk jkspd i{k ;g gS fd ge foijhr fØ;k
Hkh dj ldrs gSaA mnkgj.k osQ fy, Z osQ mu mifoHkktuksa ij fopkj dhft,] tks Z osQ ,sls rhu
ijLij vla;qDr mileqPp;ksa A1, A2 rFkk A3 }kjk iznÙk gSa] ftudk lfEeyu (Union) Z gS]
A1 = {x ∈ Z : x la[;k 3 dk xq.kt gS } = {..., – 6, – 3, 0, 3, 6, ...}
A2 = {x ∈ Z : x – 1 la[;k 3 dk xq.kt gS } = {..., – 5, – 2, 1, 4, 7, ...}
A3 = {x ∈ Z : x – 2 la[;k 3 dk xq.kt gS } = {..., – 4, – 1, 2, 5, 8, ...}
Z esa ,d laca/ R = {(a, b) : 3, a – b dks foHkkftr djrk gS} ifjHkkf"kr dhft,A mnkgj.k
5 esa iz;qDr roZQ osQ vuqlkj ge fl¼ dj ldrs gSa fd R ,d rqY;rk laca/ gSaA blosQ vfrfjDr
A1, Z osQ mu lHkh iw.kk±dksa osQ leqPp; osQ cjkcj gS] tks 'kwU; ls lacaf/r gSa] A2, Z osQ mu lHkh
iw.kk±dksa osQ leqPp; osQ cjkcj gS] tks 1 ls lacaf/r gSa vkSj A3 , Z osQ mu lHkh iw.kk±dksa osQ leqPp;
cjkcj gS] tks 2 ls lacaf/r gSaA vr% A1 = [0], A2 = [1] vkSj A3 = [2]. okLro esa A1 = [3r],
A2 = [3r + 1] vkSj A3 = [3r + 2], tgk¡ r ∈ Z.
mnkgj.k 6 eku yhft, fd leqPp; A = {1, 2, 3, 4, 5, 6, 7} esa R = {(a, b) : a rFkk b nksuksa gh
;k rks fo"ke gSa ;k le gSa} }kjk ifjHkkf"kr ,d laca/ gSA fl¼ dhft, fd R ,d rqY;rk laca/ gSA
lkFk gh fl¼ dhft, fd mileqPp; {1, 3, 5, 7} osQ lHkh vo;o ,d nwljs ls lacaf/r gS] vkSj
mileqPp; {2, 4, 6} osQ lHkh vo;o ,d nwljs ls lacaf/r gS] ijarq mileqPp; {1, 3, 5,7} dk
dksbZ Hkh vo;o mileqPp; {2, 4, 6} osQ fdlh Hkh vo;o ls lacaf/r ugha gSA
gy A dk iznÙk dksbZ vo;o a ;k rks fo"ke gS ;k le gS] vr,o (a, a) ∈ R- blosQ vfrfjDr
(a, b) ∈ R ⇒ a rFkk b nksuksa gh] ;k rks fo"ke gSa ;k le gSa ⇒ (b, a) ∈ R- blh izdkj
(a, b) ∈ R rFkk (b, c) ∈ R ⇒ vo;o a, b, c, lHkh ;k rks fo"ke gSa ;k le gSa ⇒ (a, c) ∈ R.
vr% R ,d rqY;rk laca/ gSA iqu%] {1, 3, 5, 7} osQ lHkh vo;o ,d nwljs ls lacaf/r gSa] D;ksafd
bl mileqPp; osQ lHkh vo;o fo"ke gSaA blh izdkj {2, 4, 6,} osQ lHkh vo;o ,d nwljs ls
lacaf/r gSa] D;ksafd ;s lHkh le gSaA lkFk gh mileqPp; {1, 3, 5, 7} dk dksbZ Hkh vo;o
{2, 4, 6} osQ fdlh Hkh vo;o ls lacaf/r ugha gks ldrk gS] D;ksafd {1, 3, 5, 7} osQ vo;o fo"ke
gSa] tc fd {2, 4, 6}, osQ vo;o le gSaA

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iz'ukoyh 1-1
1. fu/kZfjr dhft, fd D;k fuEufyf[kr laca/ksa esa ls izR;sd LorqY;] lefer rFkk
laØked gSa%
(i) leqPp; A = {1, 2, 3, ..., 13, 14} esa laca/ R, bl izdkj ifjHkkf"kr gS fd
R = {(x, y) : 3x – y = 0}
(ii) izko`Qr la[;kvksa osQ leqPp; N esa R = {(x, y) : y = x + 5 rFkk x < 4}}kjk ifjHkkf"kr
laca/ R.
(iii) leqPp; A = {1, 2, 3, 4, 5, 6} esa R = {(x, y) : y HkkT; gS x ls} }kjk ifjHkkf"kr lac/
a RgSA
(iv) leLr iw.kk±dksa osQ leqPp; Z esa R = {(x, y) : x – y ,d iw.kk±d gS} }kjk ifjHkkf"kr
laca/ R-
(v) fdlh fo'ks"k le; ij fdlh uxj osQ fuokfl;ksa osQ leqPp; esa fuEufyf[kr
laca/ R
(a) R = {(x, y) : x rFkk y ,d gh LFkku ij dk;Z djrs gSa}
(b) R = {(x, y) : x rFkk y ,d gh eksgYys esa jgrs gSa}
(c) R = {(x, y) : x, y ls Bhd&Bhd 7 lseh yack gS}
(d) R = {(x, y) : x , y dh iRuh gS}
(e) R = {(x, y) : x, y osQ firk gSa}
2. fl¼ dhft, fd okLrfod la[;kvksa osQ leqPp; R esa R = {(a, b) : a ≤ b2}, }kjk
ifjHkkf"kr laca/ R] u rks LorqY; gS] u lefer gSa vkSj u gh laØked gSA
3. tk¡p dhft, fd D;k leqPp; {1, 2, 3, 4, 5, 6} esa R = {(a, b) : b = a + 1} }kjk ifjHkkf"kr
laca/ R LorqY;] lefer ;k laØked gSA
4. fl¼ dhft, fd R esa R = {(a, b) : a ≤ b}, }kjk ifjHkkf"kr laca/ R LorqY; rFkk laØked
gS ¯drq lefer ugha gSA
5. tk¡p dhft, fd D;k R esa R = {(a, b) : a ≤ b3} }kjk ifjHkkf"kr laca/ LorqY;] lefer
vFkok laØked gS\
6. fl¼ dhft, fd leqPp; {1, 2, 3} esa R = {(1, 2), (2, 1)} }kjk iznÙk laca/ R lefer
gS ¯drq u rks LorqY; gS vkSj u laØked gSA
7. fl¼ dhft, fd fdlh dkWyst osQ iqLrdky; dh leLr iqLrdksa osQ leqPp; A esa
R = {(x, y) : x rFkk y esa istksa dh la[;k leku gS} }kjk iznÙk laca/ R ,d rqY;rk
laca/ gSA

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8. fl¼ dhft, fd A = {1, 2, 3, 4, 5} esa] R = {(a, b) : |a – b| le gS} }kjk iznÙk laca/

R ,d rqY;rk laca/ gSA izekf.kr dhft, fd {1, 3, 5} osQ lHkh vo;o ,d nwljs ls
lacaf/r gSa vkSj leqPp; {2, 4} osQ lHkh vo;o ,d nwljs ls lacaf/r gSa ijarq {1, 3, 5} dk
dksbZ Hkh vo;o {2, 4} osQ fdlh vo;o ls lacaf/r ugha gSA
9. fl¼ fdft, fd leqPp; A = {x ∈ Z : 0 ≤ x ≤ 12}, esa fn, x, fuEufyf[kr laca/ksa R
esa ls izR;sd ,d rqY;rk laca/ gS%
(i) R = {(a, b) : |a – b|, 4 dk ,d xq.kt gS},
(ii) R = {(a, b) : a = b},
izR;sd n'kk esa 1 ls lacaf/r vo;oksa dks Kkr dhft,A
10. ,sls laca/ dk mnkgj.k nhft,] tks
(i) lefer gks ijarq u rks LorqY; gks vkSj u laØked gksA
(ii) laØked gks ijarq u rks LorqY; gks vkSj u lefer gksA .
(iii) LorqY; rFkk lefer gks ¯drq laØked u gksA
(iv) LorqY; rFkk laØked gks ¯drq lefer u gksA
(v) lefer rFkk laØked gks ¯drq LorqY; u gksA
11. fl¼ dhft, fd fdlh lery esa fLFkr fcanqvksa osQ leqPp; esa] R = {(P, Q) : fcanq P dh
ewy fcanq ls nwjh] fcanq Q dh ewy fcanq ls nwjh osQ leku gS} }kjk iznÙk laca/ R ,d rqY;rk
laca/ gSA iqu% fl¼ dhft, fd fcanq P ≠ (0, 0) ls lacaf/r lHkh fcanqvksa dk leqPp; P
ls gksdj tkus okys ,d ,sls o`Ùk dks fu:fir djrk gS] ftldk osaQnz ewyfcanq ij gSA
12. fl¼ dhft, fd leLr f=kHkqtksa osQ leqPp; A esa] R = {(T1, T2) : T1, T2 osQ le:i gS}
}kjk ifjHkkf"kr laca/ R ,d rqY;rk laca/ gSA Hkqtkvksa 3] 4] 5 okys ledks.k f=kHkqt T1 ]
Hkqtkvksa 5] 12] 13 okys ledks.k f=kHkqt T2 rFkk Hkqtkvksa 6] 8] 10 okys ledks.k f=kHkqt
T3 ij fopkj dhft,A T1, T2 vkSj T3 esa ls dkSu ls f=kHkqt ijLij lacaf/r gSa\
13. fl¼ dhft, fd leLr cgqHkqtksa osQ leqPp; A esa, R = {(P1, P2) : P1 rFkk P2 dh Hkqtkvksa
dh la[;k leku gS}izdkj ls ifjHkkf"kr laca/ R ,d rqY;rk laca/ gSA 3] 4] vkSj 5 yackbZ
dh Hkqtkvksa okys ledks.k f=kHkqt ls lacaf/r leqPp; A osQ lHkh vo;oksa dk leqPp; Kkr
dhft,A
14. eku yhft, fd XY-ry esa fLFkr leLr js[kkvksa dk leqPp; L gS vkSj L esa R = {(L1,L2)
: L1 lekUrj gS L2 osQ} }kjk ifjHkkf"kr laca/ R gSA fl¼ dhft, fd R ,d rqY;rk laca/
gSA js[kk y = 2x + 4 ls lacaf/r leLr js[kkvksa dk leqPp; Kkr dhft,A

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15. eku yhft, fd leqPp; {1, 2, 3, 4} esa] R = {(1, 2), (2, 2), (1, 1), (4,4),
(1, 3), (3, 3), (3, 2)} }kjk ifjHkkf"kr laca/ R gSA fuEufyf[kr esa ls lgh mÙkj pqfu,A
(A) R LorqY; rFkk lefer gS ¯drq laØked ugha gSA
(B) R LorqY; rFkk laØked gS ¯drq lefer ugha gSA
(C) R lefer rFkk laØked gS ¯drq LorqY; ugha gSA
(D) R ,d rqY;rk laca/ gSA
16. eku yhft, fd leqPp; N esa] R = {(a, b) : a = b – 2, b > 6} }kjk iznÙk laca/ R gSA
fuEufyf[kr esa ls lgh mÙkj pqfu,%
(A) (2, 4) ∈ R (B) (3, 8) ∈ R (C) (6, 8) ∈ R (D) (8, 7) ∈ R

1.3 iQyuksa osQ izdkj (Types of Functions)

iQyuksa dh vo/kj.kk] oqQN fo'ks"k iQyu tSls rRled iQyu] vpj iQyu] cgqin iQyu] ifjes;
iQyu] ekikad iQyu] fpg~u iQyu vkfn dk o.kZu muosQ vkys[kksa lfgr d{kk XI esa fd;k tk
pqdk gSA
nks iQyuksa osQ ;ksx] varj] xq.kk rFkk Hkkx dk Hkh vè;;u fd;k tk pqdk gSA D;ksafd iQyu dh
ladYiuk xf.kr rFkk vè;;u dh vU; 'kk[kkvksa (Disciplines) esa lokZf/d egRoiw.kZ gS] blfy,
ge iQyu osQ ckjs esa viuk vè;;u ogk¡ ls vkxs c<+kuk pkgrs gSa] tgk¡ bls igys lekIr fd;k FkkA
bl vuqPNsn esa] ge fofHkUu izdkj osQ iQyuksa dk vè;;u djsaxsA
fuEufyf[kr vko`Qfr;ksa }kjk n'kkZ, x, iQyu f1, f2, f3 rFkk f4 ij fopkj dhft,A
vko`Qfr 1.2 esa ge ns[krs gSa fd X1 osQ fHkUu (distinct) vo;oksa osQ] iQyu f1 osQ varxZr]
izfrfcac Hkh fHkUu gSa] ¯drq f2 osQ varxZr nks fHkUu vo;oksa 1 rFkk 2 osQ izfrfcac ,d gh gSa uker%
b gSA iqu% X2 esa oqQN ,sls vo;o gS tSls e rFkk f tks f1 osQ varxZr X1 osQ fdlh Hkh vo;o
osQ izfrfcac ugha gSa] tcfd f3 osQ varxZr X3 osQ lHkh vo;o X1 osQ fdlh u fdlh vo;o osQ
izfrfcac gSaA
mi;qZDr ifjppkZ ls gesa fuEufyf[kr ifjHkk"kk,¡ izkIr gksrh gSaA
ifjHkk"kk 5 ,d iQyu f : X → Y ,oSQdh (one-one) vFkok ,oSQd (injective) iQyu dgykrk
gS] ;fn f osQ varxZr X osQ fHkUu vo;oksa osQ izfrfcac Hkh fHkUu gksrs gSa] vFkkZr~ izR;sd
x1, x2 ∈ X, osQ fy, f (x1) = f (x2) dk rkRi;Z gS fd x1 = x2, vU;Fkk f ,d cgq,d
(many-one) iQyu dgykrk gSA
vko`Qfr 1-2 (i) esa iQyu f1 ,oSQdh iQyu gS rFkk vko`Qfr 1-2 (ii) esa f2 ,d cgq,d iQyu gSA

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vko`Qfr 1-2
ifjHkk"kk 6 iQyu f : X → Y vkPNknd (onto) vFkok vkPNknh (surjective) dgykrk gS] ;fn
f osQ varxZr Y dk izR;sd vo;o] X osQ fdlh u fdlh vo;o dk izfrfcac gksrk gS] vFkkZr~ izR;sd
y ∈ Y, osQ fy,] X esa ,d ,sls vo;o x dk vfLrRo gS fd f (x) = y.
vko`Qfr 1-2 (iii) esa] iQyu f3 vkPNknd gS rFkk vko`Qfr 1-2 (i) esa] iQyu f1 vkPNknd
ugha gS] D;ksafd X2 osQ vo;o e, rFkk f, f1 osQ varxZr X1 osQ fdlh Hkh vo;o osQ izfrfcac
ugha gSaA
fVIi.kh f : X → Y ,d vkPNknd iQyu gS] ;fn vkSj osQoy ;fn f dk ifjlj (range)= Y.
ifjHkk"kk 7 ,d iQyu f : X → Y ,d ,oSQdh rFkk vkPNknd (one-one and onto) vFkok ,oSQdh
vkPNknh (bijective) iQyu dgykrk gS] ;fn f ,oSQdh rFkk vkPNknd nksuksa gh gksrk gSA
vko`Qfr 1-2 (iv) esa] iQyu f4 ,d ,oSQdh rFkk vkPNknh iQyu gSA
mnkgj.k 7 eku yhft, fd d{kk X osQ lHkh 50 fo|kfFkZ;ksa dk leqPp; A gSA eku yhft,
f : A → N ] f (x) = fo|kFkhZ x dk jksy uacj] }kjk ifjHkkf"kr ,d iQyu gSA fl¼ dhft, fd
f ,oSQdh gS ¯drq vkPNknd ugha gSA
gy d{kk osQ nks fHkUu&fHkUu fo|kfFkZ;ksa osQ jksy uacj leku ugha gks ldrs gSaA vr,o f ,oSQdh gSA
O;kidrk dh fcuk {kfr fd, ge eku ldrs gSa fd fo|kfFkZ;ksa osQ jksy uacj 1 ls 50 rd gSaA bldk

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rkRi;Z ;g gqvk fd N dk vo;o 51] d{kk osQ fdlh Hkh fo|kFkhZ dk jksy uacj ugha gS] vr,o
f osQ varxZr 51] A osQ fdlh Hkh vo;o dk izfrfcac ugha gSA vr% f vkPNknd ugha gSA
mnkgj.k 8 fl¼ dhft, fd f (x) = 2x }kjk iznÙk iQyu f : N → N, ,oSQdh gS ¯drq vkPNknd
ugha gSA
gy iQyu f ,oSQdh gS] D;ksafd f (x1) = f (x2) ⇒ 2x1 = 2x2 ⇒ x1 = x2. iqu%, f vkPNnd ugha
gS] D;ksafd 1 ∈ N, osQ fy, N esa ,sls fdlh x dk vfLrRo ugha gS rkfd f (x) = 2x = 1 gksA
mnkgj.k 9 fl¼ dhft, fd f (x) = 2x }kjk iznÙk iQyu f : R → R, ,oSQdh rFkk vkPNknd gSA
gy f ,oSQdh gS] D;ksafd f (x1) = f (x2) ⇒ 2x1 = 2x2 ⇒ x1 = x2- lkFk gh] R esa iznÙk fdlh
y y y
Hkh okLrfod la[;k y osQ fy, R esa dk vfLrRo gS] tgk¡ f ( ) = 2 . ( ) = y gSA vr% f
2 2 2
vkPNknd Hkh gSA

vko`Qfr 1-3
mnkgj.k 10 fl¼ fdft, fd f (1) = f (2) = 1 rFkk x > 2 osQ fy, f (x) = x – 1 }kjk iznÙk
iQyu f : N → N, vkPNknd rks gS ¯drq ,oSQdh ugha gSA
gy f ,oSQdh ugha gS] D;kasfd f (1) = f (2) = 1, ijarq f vkPNknd gS] D;ksafd fdlh iznÙk
y ∈ N, y ≠ 1, osQ fy,] ge x dks y + 1 pqu ysrs gS]a rkfd f (y+ 1) = y + 1 – 1 = y lkFk gh
1 ∈ N osQ fy, f (1) = 1 gSA

mnkgj.k 11 fl¼ dhft, fd f (x) = x2 }kjk ifjHkkf"kr iQyu f : R → R, u rks ,oSQdh gS vkSj
u vkPNknd gSA

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gy D;ksafd f (– 1) = 1 = f (1), blfy, f ,oSQdh

ugha gSA iqu% lgizkar R dk vo;o &2] izkar R osQ
fdlh Hkh vo;o x dk izfrfcac ugha gS (D;ksa\)A
vr% f vkPNknd ugha gSA
mnkgj.k 12 fl¼ dhft, fd uhps ifjHkkf"kr iQyu
f : N → N, ,oSQdh rFkk vkPNknd nksuksa gh gS

 x + 1, ;fn x fo"ke gS
f ( x) = 
 x − 1, ;fn x le gS
vko`Qfr 1-4
gy eku yhft, f (x1) = f (x2) gSA uksV dhft, fd ;fn x1 fo"ke gS rFkk x2 le gS] rks x1 + 1
= x2 – 1, vFkkZr~ x2 – x1 = 2 tks vlEHko gSA bl izdkj x1 osQ le rFkk x2 osQ fo"ke gksus dh Hkh
laHkkouk ugha gSA blfy, x1 rFkk x2 nksuksa gh ;k rks fo"ke gksaxs ;k le gksaxsA eku yhft, fd x1
rFkk x2 nksuksa fo"ke gSa] rks f (x1) = f (x2) ⇒ x1 + 1 = x2 + 1 ⇒ x1 = x2. blh izdkj ;fn x1 rFkk
x2 nksuksa le gSa] rks Hkh f (x1) = f (x2) ⇒ x1 – 1 = x2 – 1 ⇒ x1 = x2- vr% f ,oSQdh gSA lkFk gh
lgizkar N dh dksbZ Hkh fo"ke la[;k 2r + 1, izkar N dh la[;k 2r + 2 dk izfrfcac gS vkSj lgizkar
N dh dksbZ Hkh le la[;k 2r, N dh la[;k 2r – 1 dk izfrfcac gSA vr% f vkPNknd gSA
mnkgj.k 13 fl¼ dhft, fd vkPNknd iQyu f : {1, 2, 3} → {1, 2, 3} lnSo ,oSQdh iQyu gksrk gSA
gy eku yhft, fd f ,oSQdh ugha gSA vr% blosQ izkar esa de ls de nks vo;o eku fy;k fd
1 rFkk 2 dk vfLrRo gS ftuosQ lgizkar esa izfrfcac leku gSA lkFk gh f osQ varxZr 3 dk izfrfcac
osQoy ,d gh vo;o gSA vr%] ifjlj esa] lgizkar {1, 2, 3} osQ] vf/dre nks gh vo;o gks ldrs
gSa] ftlls izdV gksrk gS fd f vkPNknd ugha gS] tks fd ,d fojks/ksfDr gSA vr% f dks ,oSQdh
gksuk gh pkfg,A
mnkgj.k 14 fl¼ dhft, fd ,d ,oSQdh iQyu f : {1, 2, 3} → {1, 2, 3}vfuok;Z :i ls
vkPNknd Hkh gSA
gy pw¡fd f ,oSQdh gS] blfy, {1, 2, 3} osQ rhu vo;o f osQ varxZr lgizkar {1, 2, 3} osQ rhu
vyx&vyx vo;oksa ls Øe'k% lacaf/r gksaxsA vr% f vkPNknd Hkh gSA
fVIi.kh mnkgj.k 13 rFkk 14 esa izkIr ifj.kke fdlh Hkh LosPN ifjfer (finite) leqPp; X, osQ
fy, lR; gS] vFkkZr~ ,d ,oSQdh iQyu f : X → X vfuok;Zr% vkPNknd gksrk gS rFkk izR;sd
ifjfer leqPp; X osQ fy, ,d vkPNknd iQyu f : X → X vfuok;Zr% ,oSQdh gksrk gSA blosQ

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foijhr mnkgj.k 8 rFkk 10 ls Li"V gksrk gS fd fdlh vifjfer (Infinite) leqPp; osQ fy, ;g
lgh ugha Hkh gks ldrk gSA okLro esa ;g ifjfer rFkk vifjfer leqPp;ksa osQ chp ,d vfHky{kf.kd
(characteristic) varj gSA

iz'ukoyh 1-2
1
1. fl¼ dhft, fd f (x) = }kjk ifjHkkf"kr iQyu f : R∗ → R∗ ,oSQdh rFkk vkPNknd
x
gS] tgk¡ R∗ lHkh ½.ksrj okLrfod la[;kvksa dk leqPp; gSA ;fn izkar R∗ dks N ls cny
fn;k tk,] tc fd lgizkar iwoZor R∗gh jgs] rks Hkh D;k ;g ifj.kke lR; gksxk\
2. fuEufyf[kr iQyuksa dh ,oSQd (Injective) rFkk vkPNknh (Surjective) xq.kksa dh tk¡p
dhft,%
(i) f (x) = x2 }kjk iznÙk f : N → N iQyu gSA
(ii) f (x) = x2 }kjk iznÙk f : Z → Z iQyu gSA
(iii) f (x) = x2 }kjk iznÙk f : R → R iQyu gSA
(iv) f (x) = x3 }kjk iznÙk f : N → N iQyu gSA
(v) f (x) = x3 }kjk iznÙk f : Z → Z iQyu gSA
3. fl¼ dhft, fd f (x) = [x] }kjk iznÙk egÙke iw.kk±d iQyu f : R → R, u rks ,oSQdh gS
vkSj u vkPNknd gS] tgk¡ [x], x ls de ;k mlosQ cjkcj egÙke iw.kk±d dks fu:fir
djrk gSA
4. fl¼ dhft, fd f (x) = | x | }kjk iznÙk ekikad iQyu f : R → R, u rks ,oSQdh gS vkSj
u vkPNknd gS] tgk¡ | x | cjkcj x, ;fn x /u ;k 'kwU; gS rFkk | x | cjkcj – x, ;fn
x ½.k gSA
5. fl¼ dhft, fd f : R → R,

1, ;fn x > 0


f ( x ) = 0, ;fn x = 0
µ1, ;fn x < 0,

}kjk iznÙk fpg~u iQyu u rks ,oSQdh gS vkSj u vkPNknd gSA
6. eku yhft, fd A = {1, 2, 3}, B = {4, 5, 6, 7} rFkk f = {(1, 4), (2, 5), (3, 6)} A ls B
rd ,d iQyu gSA fl¼ dhft, fd f ,oSQdh gSA

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7. fuEufyf[kr esa ls izR;sd fLFkfr esa crykb, fd D;k fn, gq, iQyu ,oSQdh] vkPNknd
vFkok ,oSQdh vkPNknh (bijective) gSaA vius mÙkj dk vkSfpR; Hkh crykb,A
(i) f (x) = 3 – 4x }kjk ifjHkkf"kr iQyu f : R → R gSA

(ii) f (x) = 1 + x2 }kjk ifjHkkf"kr iQyu f : R → R gSA

8. eku yhft, fd A rFkk B nks leqPp; gSaA fl¼ dhft, fd f : A × B → B × A, bl izdkj
fd f (a, b) = (b, a) ,d ,oSQdh vkPNknh (bijective) iQyu gSA

n +1
 2 , ;fn n fo"ke gS
9. eku yhft, fd leLr n ∈ N osQ fy,] f (n) =  n
, ;fn n le gS
 2
}kjk ifjHkkf"kr ,d iQyu f : N → N gSA crykb, fd D;k iQyu f ,oSQdh vkPNknh
(bijective) gSA vius mÙkj dk vkSfpR; Hkh crykb,A
 x−2
10. eku yhft, fd A = R – {3} rFkk B = R – {1} gSaA f (x) =   }kjk ifjHkkf"kr iQyu
 x−3
f : A → B ij fopkj dhft,A D;k f ,oSQdh rFkk vkPNknd gS\ vius mÙkj dk vkSfpR;
Hkh crykb,A
11. eku yhft, fd f : R → R , f(x) = x4 }kjk ifjHkkf"kr gSA lgh mÙkj dk p;u dhft,A
(A) f ,oSQdh vkPNknd gS (B) f cgq,d vkPNknd gS
(C) f ,oSQdh gS ¯drq vkPNknd ugha gS (D) f u rks ,oSQdh gS vkSj u vkPNknd gSA
12. eku yhft, fd f (x) = 3x }kjk ifjHkkf"kr iQyu f : R → R gSA lgh mÙkj pqfu,%
(A) f ,oSQdh vkPNknd gS (B) f cgq,d vkPNknd gS
(C) f ,oSQdh gS ijarq vkPNknd ugha gS (D) f u rks ,oSQdh gS vkSj u vkPNknd gS

1.4 iQyuksa dk la;kstu rFkk O;qRØe.kh; iQyu (Composition of Functions and

Invertible Function)
bl vuqPNsn esa ge nks iQyuksa osQ la;kstu rFkk fdlh ,oSQdh vkPNknh (bijective) iQyu osQ
izfrykse (Inverse) dk vè;;u djsaxsA lu~ 2006 dh fdlh cksMZ (ifj"kn~~) dh d{kk X dh ijh{kk
esa cSB pqosQ lHkh fo|kfFkZ;ksa osQ leqPp; A ij fopkj dhft,A cksMZ dh ijh{kk esa cSBus okys izR;sd
fo|kFkhZ dks cksMZ }kjk ,d jksy uacj fn;k tkrk gS] ftls fo|kFkhZ ijh{kk osQ le; viuh mÙkj iqfLrdk
ij fy[krk gSA xksiuh;rk j[kus osQ fy, cksMZ fo|kfFkZ;ksa osQ jksy uacjksa dks fo:i (deface) djosQ]

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izR;sd jksy uacj dks ,d udyh lakosQfrd uacj (Fake Code Number) esa cny nsrk gSaA eku
yhft, fd B ⊂ N leLr jksy uacjksa dk leqPp; gS] rFkk C ⊂ N leLr lkaosQfrd uacjksa dk
leqPp; gSA blls nks iQyu f : A → B rFkk g : B → C curs gSa tks Øe'k% f (a) = fo|kFkhZ a
dks fn;k x;k jksy uacj rFkk g (b) = jksy uacj b dks cny dj fn;k x;k lkaosQfrd uacj] }kjk
ifjHkkf"kr gSaA bl izfØ;k esa iQyu f }kjk izR;sd fo|kFkhZ osQ fy, ,d jksy uacj fu/kZfjr gksrk gS
rFkk iQyu g }kjk izR;sd jksy uacj osQ fy, ,d lkaosQfrd uacj fu/kZfjr gksrk gSA vr% bu nksuksa
iQyuksa osQ la;kstu ls izR;sd fo|kFkhZ dks varr% ,d lkaosQfrd uacj ls laca/ dj fn;k tkrk gsA
blls fuEufyf[kr ifjHkk"kk izkIr gksrh gSA
ifjHkk"kk 8 eku yhft, fd f : A → B rFkk g : B → C nks iQyu gSaA rc f vkSj g dk la;kstu]
gof }kjk fu:fir gksrk gS] rFkk iQyu gof : A → C, gof (x) = g (f (x)), ∀ x ∈ A }kjk
ifjHkkf"kr gksrk gSA

vko`Qfr 1-5
mnkgj.k 15 eku yhft, fd f : {2, 3, 4, 5} → {3, 4, 5, 9} vkSj g : {3, 4, 5, 9} → {7, 11, 15}
nks iQyu bl izdkj gSa fd f (2) = 3, f (3) = 4, f (4) = f (5) = 5 vkSj g (3) = g (4) = 7 rFkk
g (5) = g (9) = 11, rks gof Kkr dhft,A
gy ;gk¡ gof (2) = g (f (2)) = g (3) = 7, gof (3) = g (f (3)) = g (4) = 7, gof (4) = g (f (4))
= g (5) = 11 vkSj gof (5) = g (5) = 11.
mnkgj.k 16 ;fn f : R → R rFkk g : R → R iQyu Øe'k% f (x) = cos x rFkk g (x) = 3x2 }kjk
ifjHkkf"kr gS rks gof vkSj fog Kkr dhft,A fl¼ dhft, gof ≠ fog.
gy ;gk¡ gof (x) = g (f (x)) = g (cos x) = 3 (cos x)2 = 3 cos2 x. blh izdkj, fog (x) =
f (g (x)) = f (3x2) = cos (3x2) gSaA uksV dhft, fd x = 0 osQ fy, 3cos2 x ≠ cos 3x2 gSA vr%
gof ≠ fog.
3x + 4 7  3
mnkgj.k 17 ;fn f ( x) = }kjk ifjHkkf"kr iQyu f : R −   → R −   rFkk
5x − 7 5  5
7x + 4
}kjk ifjHkkf"kr iQyu g : R −   → R −   iznÙk gSa] rks fl¼ dhft, fd
3 7
g ( x) =
5x − 3 5  5

2018-19
 laca/ ,oa iQyu 15

fog = IA rFkk gof = IB, bl izdkj fd IA (x) = x, ∀ x ∈ A vkSj IB (x) = x, ∀ x ∈ B, tgk¡

3 7 
A = R –   , B = R –   gSaA IA rFkk IB dks Øe'k% leqPp; A rFkk B ij rRled (Identity)
 
5 5 
iQyu dgrs gSaA
gy ;gk¡ ij
 (3x + 4) 
7 +4
 3 x + 4   (5 x − 7)  21x + 28 + 20 x − 28 41x
gof ( x) = g  = = = =x
 5x − 7   (3x + 4)  15 x + 20 − 15 x + 21 41
5 −3
 (5 x − 7) 

 (7 x + 4) 
3 +4
 7x + 4   (5 x − 3)  21x + 12 + 20 x − 12 41x
blh izdkj] fog ( x) = f  = = = =x
 5x − 3   (7 x + 4)  35 x + 20 − 35 x + 21 41
5 −7
 (5 x − 3) 

vr% gof (x) = x, ∀ x ∈ B vkSj fog (x) = x, ∀ x ∈ A, ftldk rkRi;Z ;g gS fd

gof = IB vkSj fog = IA.

mnkgj.k 18 fl¼ dhft, fd ;fn f : A → B rFkk g : B → C ,oSQdh gSa] rks gof : A → C

Hkh ,oSQdh gSA
gy gof (x1) = gof (x 2)
⇒ g (f (x1)) = g(f (x 2))
⇒ f (x1) = f (x2), D;ksafd g ,oSQdh gS
⇒ x1 = x2, D;ksafd f ,oSQdh gS
vr% gof Hkh ,oSQdh gSA
mnkgj.k 19 fl¼ dhft, fd ;fn f : A → B rFkk g : B → C vkPNknd gSa] rks gof : A →C
Hkh vkPNknd gSA
gy eku yhft, fd ,d LosPN vo;o z ∈ C gSA g osQ varxZr z osQ ,d iwoZ izfrfcac
(Pre-image) y ∈ B dk vfLrRo bl izdkj gS fd] g (y) = z, D;ksafd g vkPNknd gSA blh izdkj
y ∈ B osQ fy, A esa ,d vo;o x dk vfLrRo bl izdkj gS fd] f (x) = y, D;ksafd f vkPNknd
gSA vr% gof (x) = g (f (x)) = g (y) = z, ftlls izekf.kr gksrk gS fd gof vkPNknd gSA

2018-19
16 xf.kr

mnkgj.k 20 f rFkk g ,sls nks iQyuksa ij fopkj dhft, fd gof ifjHkkf"kr gS rFkk ,oSQdh gSA
D;k f rFkk g nksuksa vfuok;Zr% ,oSQdh gSa\
gy iQyu f : {1, 2, 3, 4} → {1, 2, 3, 4, 5, 6} f (x) = x, ∀ x }kjk ifjHkkf"kr vkSj g (x) = x,
x = 1, 2, 3, 4 rFkk g (5) = g (6) = 5 }kjk ifjHkkf"kr g : {1, 2, 3, 4, 5, 6} → {1, 2, 3, 4, 5, 6} ij
fopkj dhft,A ;gk¡ gof : {1, 2, 3, 4} → {1, 2, 3, 4, 5, 6} ifjHkkf"kr gS rFkk gof (x) = x, ∀ x,
ftlls izekf.kr gksrk gS fd gof ,oSQdh gSA ¯drq g Li"Vr;k ,oSQdh ugha gSA
mnkgj.k 21 ;fn gof vkPNnd gS] rks D;k f rFkk g nksuksa vfuok;Zr% vkPNknd gSa?
gy f : {1, 2, 3, 4} → {1, 2, 3, 4} rFkk g : {1, 2, 3, 4} → {1, 2, 3} ij fopkj dhft,] tks]
Øe'k% f (1) = 1, f (2) = 2, f (3) = f (4) = 3, g (1) = 1, g (2) = 2 rFkk g (3) = g (4) = 3. }kjk
ifjHkkf"kr gSaA ;gk¡ ljyrk ls ns[kk tk ldrk gS fd gof vkPNknd gS] ¯drq f vkPNknd ugha gSA
fVIi.kh ;g lR;kfir fd;k tk ldrk gS fd O;kid :i ls gof osQ ,oSQdh gksus dk rkRi;Z gS
fd f ,oSQdh gksrk gSA blh izdkj gof vkPNknd gksus dk rkRi;Z gS fd g vkPNknd gksrk gSA
vc ge bl vuqPNsn osQ izkjaHk esa cksMZ dh ijh{kk osQ lanHkZ esa of.kZr iQyu f vkSj g ij ckjhdh
ls fopkj djuk pkgrs gSaA cksMZ dh d{kk X dh ijh{kk esa cSBus okys izR;sd fo|kFkhZ dks iQyu f
osQ varxZr ,d jksy uacj iznku fd;k tkrk gS vkSj izR;sd jksy uacj dks g osQ varxZr ,d lkaosQfrd
uacj iznku fd;k tkrk gSA mÙkj iqfLrdkvksa osQ ewY;kadu osQ ckn ijh{kd izR;sd ewY;kafdr iqfLrdk
ij lkaosQfrd uacj osQ le{k izkIrkad fy[k dj cksMZ osQ dk;kZy; esa izLrqr djrk gSA cksMZ osQ
vf/dkjh] g osQ foijhr izfØ;k }kjk] izR;sd lkaosQfrd uacj dks cny dj iqu% laxr jksy uacj iznku
dj nsrs gSa vkSj bl izdkj izkIrkad lkaosQfrd uacj osQ ctk, lh/s jksy uacj ls lacaf/r gks tkrk gSA
iqu%] f dh foijhr izfØ;k }kjk] izR;sd jksy uacj dks ml jksy uacj okys fo|kFkhZ ls cny fn;k
tkrk gSA blls izkIrkad lh/s lacaf/r fo|kFkhZ osQ uke fu/kZfjr gks tkrk gSA ge ns[krs gSa fd f rFkk
g, osQ la;kstu }kjk gof, izkIr djrs le;] igys f vkSj fiQj g dks iz;qDr djrs gSa] tc fd la;qDr
gof, dh foijhr izfØ;k esa] igys g dh foijhr izfØ;k vkSj fiQj f dh foijhr izfØ;k djrs gSaA

mnkgj.k 22 eku yhft, fd f : {1, 2, 3} → {a, b, c} ,d ,oSQdh rFkk vPNknd iQyu bl

izdkj gS fd f (1) = a, f (2) = b vkSj f (3) = c, rks fl¼ dhft, fd iQyu g : {a, b, c} → {1, 2, 3}
dk ,s l k vfLrRo gS ] rkfd gof = I X rFkk fog = IY, tgk¡ X = {1, 2, 3} rFkk
Y = {a, b, c}gksA

gy iQyu g : {a, b, c} → {1, 2, 3} gS tgk¡ g (a) = 1, g (b) = 2 vkSj g (c) = 3, ij fopkj

dhft,A ;g lR;kfir djuk ljy gS fd la;qDr iQyu gof = IX , X ij rRled iQyu gS vkSj
l;aqDr iQyu fog = IY , Y ij rRled iQyu gaSA

2018-19
 laca/ ,oa iQyu 17

fVIi.kh ;g ,d jkspd rF; gS fd mi;qZDr mnkgj.k esa of.kZr ifj.kke fdlh Hkh LosPN ,oSQdh
rFkk vkPNknd iQyu f : X → Y osQ fy, lR; gksrk gSA osQoy ;gh ugha vfirq bldk foykse
(converse) Hkh lR; gksrk gS] vFkkZr~] ;fn f : X → Y ,d ,slk iQyu gS fd fdlh iQyu
g : Y → X dk vfLrRo bl izdkj gS fd gof = IX rFkk fog = IY, rks f ,oSQdh rFkk vkPNknd
gksrk gSA
mi;qZDr ifjppkZ] mnkgj.k 22 rFkk fVIi.kh fuEufyf[kr ifjHkk"kk osQ fy, izsfjr djrs gSa%
ifjHkk"kk 9 iQyu f : X → Y O;qRØe.kh; (Invertible) dgykrk gS] ;fn ,d iQyu
g : Y → X dk vfLrRo bl izdkj gS fd gof = IX rFkk fog = IY gSA iQyu g dks iQyu f dk
izfrykse (Inverse) dgrs gSa vkSj bls izrhd f –1 }kjk izdV djrs gSaA
vr%] ;fn f O;qRØe.kh; gS] rks f vfuok;Zr% ,oSQdh rFkk vkPNknd gksrk gS vkSj foykser%]
;fn f ,oSQdh rFkk vkPNknd gS] rks f vfuok;Zr% O;qRØe.kh; gksrk gSA ;g rF;] f dks ,oSQdh
rFkk vkPNknd fl¼ djosQ] O;qRØe.kh; izekf.kr djus esa egRoiw.kZ :i ls lgk;d gksrk gS] fo'ks"k
:i ls tc f dk izfrykse okLro esa Kkr ugha djuk gksA
mnkgj.k 23 eku yhft, fd f : N → Y, f (x) = 4x + 3, }kjk ifjHkkf"kr ,d iQyu gS] tgk¡
Y = {y ∈ N : y = 4x + 3 fdlh x ∈ N osQ fy,}A fl¼ dhft, fd f O;qRØe.kh; gSA izfrykse
iQyu Hkh Kkr dhft,A
gy Y osQ fdlh LosPN vo;o y ij fopkj dhft,A Y, dh ifjHkk"kk }kjk] izkar N osQ fdlh vo;o
( y − 3) ( y − 3)
x osQ fy, y = 4x + 3 gSA blls fu"d"kZ fudyrk gS fd x = gSA vc g ( y) = }kjk
4 4
(4 x + 3 − 3)
g : Y → N dks ifjHkkf"kr dhft,A bl izdkj gof (x) = g (f (x)) = g (4x + 3) = =x
4
( y − 3)  4 ( y − 3)
rFkk fog (y) = f (g (y)) = f  = + 3 = y – 3 + 3 = y gSA blls Li"V gksrk
 4  4
gS fd gof = IN rFkk fog = IY, ftldk rkRi;Z ;g gqvk fd f O;qRØe.kh; gS vkSj iQyu g iQyu
f dk izfrykse gSA

mnkgj.k 24 eku yhft, fd Y = {n2 : n ∈ N } ⊂ N gSA iQyu f : N → Y tgk¡ f (n) = n2

ij fopkj dhft,A fl¼ dhft, fd f O;qRØe.kh; gSA f dk izfrykse Hkh Kkr dhft,A

gy Y dk ,d LosPN vo;o y, n2 osQ :i dk gS tgk¡ n ∈ N . bldk rkRi;Z ;g gS fd n = y

blls g (y) = y }kjk ifjHkkf"kr ,d iQyu g : Y → N iz k Ir gks r k gS A vc

2018-19
18 xf.kr

( y)=( y)
2
gof (n) = g (n2) = n 2 = n vkSj fog (y) = f = y , ftlls izekf.kr gksrk gS fd

gof = IN rFkk fog = IYgSA vr% f O;qRØe.kh; gS rFkk f –1 = g.

mnkgj.k 25 eku yhft, fd f : N → R , f (x) = 4x2 + 12x + 15 }kjk ifjHkkf"kr ,d iQyu

gSA fl¼ dhft, fd f : N → S, tgk¡ S, f dk ifjlj gS] O;qRØe.kh; gSA f dk izfrykse Hkh Kkr
dhft,A
gy eku yhft, fd f osQ ifjlj dk y ,d LosPN vo;o gSA blfy, y = 4x2 + 12x + 15, tgk¡

x ∈ N- bldk rkRi;Z ;g gS fd y = (2x + 3)2 + 6. vr,o x =

(( y −6)−3 )) .
2

(( y − 6)−3 ))
vc] ,d iQyu g : S → N , g (y) = }kjk ifjHkkf"kr dhft,A
2
bl izdkj gof (x) = g (f (x)) = g (4x2 + 12x + 15) = g ((2x + 3)2 + 6))

=
(( (2 x + 3) 2 + 6 − 6 − 3 ) ) ( 2 x + 3 − 3)
= =x
2 2

(( )
y − 6) − 3   2 (( y − 6) − 3 ) + 3 
2

vkSj fog (y) = f   =   +6
 2   2 

(( y −6)−3+3 )) + 6 = ( y − 6 ) + 6 = y – 6 + 6 = y.
2 2
=

vr% gof = IN rFkk fog =ISgSA bldk rkRi;Z ;g gS fd f O;qRØe.kh; gS rFkk f –1 = g gSA

mnkgj.k 26 rhu iQyu f : N → N, g : N → N rFkk h : N → R ij fopkj dhft, tgk¡

f (x) = 2x, g (y) = 3y + 4 rFkk h (z) = sin z, ∀ x, y rFkk z ∈ N- fl¼ dhft, fd
ho(gof) = (hog) of.
gy ;gk¡
ho(gof) (x) = h(gof (x)) = h(g (f (x))) = h (g (2x))
= h(3(2x) + 4) = h(6x + 4) = sin (6x + 4), ∀ x ∈ N
lkFk gh] ((hog) o f ) (x) = (hog) ( f (x)) = (hog) (2x) = h ( g (2x))
= h(3(2x) + 4) = h(6x + 4) = sin (6x + 4), ∀ x ∈ N

2018-19
 laca/ ,oa iQyu 19

blls izekf.kr gksrk gS fd ho(gof) = (hog) o f

;g ifj.kke O;kid fLFkfr esa Hkh lR; gksrk gSA
izes; 1 ;fn f : X → Y, g : Y → Z rFkkk h : Z → S rhu iQyu gSa] rks
ho(gof) = (hog) o f
miifÙk ;gk¡ ge ns[krs gSa fd
ho(gof) (x) = h(gof (x)) = h(g (f (x))), ∀ x in X
rFkk (hog) of (x) = hog (f (x)) = h(g(f (x))), ∀ x in X
vr% ho(gof) = (hog) o f
mnkgj.k 27 f : {1, 2, 3} → {a, b, c} rFkk g : {a, b, c} → {lsc] xsan] fcYyh} f (1) = a,
f (2) = b, f (3) = c, g(a) = lsc, g(b) = xsan rFkk g(c) = fcYyh }kjk ifjHkkf"kr iQyuksa ij fopkj
dhft,A fl¼ dhft, fd f, g vkSj gof O;qRØe.kh; gSaA f –1, g–1 rFkk (gof)–1 Kkr dhft, rFkk
izekf.kr dhft, fd (gof) –1 = f –1o g–1 gSA
gy uksV dhft, fd ifjHkk"kk }kjk f vkSj g ,oSQdh vkPNknh iQyu gSaA eku yhft, fd
f –1: {a, b, c} → (1, 2, 3} vkSj g–1 : {lsc] xsan] fcYyh} → {a, b, c} bl izdkj ifjHkkf"kr gSa fd
f –1{a} = 1, f –1{b} = 2, f –1{c} = 3, g –1{lsc} = a, g –1{xsan} = b vkSj g –1{fcYyh} = c.
;g lR;kfir djuk ljy gS fd f –1 o f = I{1, 2, 3}, f o f –1 = I{a, b, c}, g –1og = I{a, b, c} vkSj
g o g–1 = ID, tgk¡ D = {lsc, xsan, fcYyh}A vc, gof : {1, 2, 3} → {lsc, xsan, fcYyh}
gof (1) = lsc, gof (2) = xsan, gof (3) = fcYyh }kjk iznÙk gSA
ge (gof)–1 : {lsc] xsan] fcYyh} → {1, 2, 3} dks (gof)–1 (lsc) = 1, (gof)–1 (xsan) = 2
rFkk (g o f)–1 (fcYyh) = 3 }kjk ifjHkkf"kr dj ldrs gSaA ;g ljyrk ls izekf.kr fd;k tk ldrk
gS fd (g o f)–1 o (g o f) = I{1, 2, 3} rFkk (g o f) o (g o f)–1 = ID gksxkA
bl izdkj izekf.kr gksrk gS fd f, g rFkk gof O;qRØe.kh; gSaA
vc f og–1 (lsc) = f –1(g–1(lsc)) = f –1(a) = 1 = (gof)–1 (lsc)
–1

f –1og–1 (xsan) = f –1(g–1(xsan)) = f –1(b) = 2 = (gof)–1 (xsan) rFkk

f –1og–1 (fcYyh) = f –1(g–1(fcYyh)) = f –1(c) = 3 = (gof)–1 (fcYyh)
vr% (gof)–1 = f –1og –1
mi;qZDr ifj.kke O;kid fLFkfr esa Hkh lR; gksrk gSA
izes; 2 eku yhft, fd f : X → Y rFkk g : Y → Z nks O;qRØe.kh; iQyu gSa] rks gof Hkh
O;qRØe.kh; gksxk rFkk (gof)–1 = f –1og–1
miifÙk gof dks O;qRØe.kh; rFkk (gof)–1 = f –1og–1, dks fl¼ djus osQ fy, ;g izekf.kr djuk
i;kZIr gS fd ( f –1og–1)o(gof) = IX rFkk (gof)o( f –1og–1) = IZ gSA

2018-19
20 xf.kr

vc (f –1og–1 ) o (gof) = ((f –1og–1) og) of, izes; 1 }kjk

= (f –1o(g–1og)) of, izes; 1 }kjk
= (f –1 o IY) of, g–1 dh ifjHkk"kk }kjk
= IX
blh izdkj] ;g izekf.kr fd;k tk ldrk gS fd] (gof ) (f –1 o g –1) = IZ
mnkgj.k 28 eku yhft, fd S = {1, 2, 3}gSA fu/kZfjr dhft, fd D;k uhps ifjHkkf"kr iQyu
f : S → S osQ izfrykse iQyu gaSA f –1, Kkr dhft, ;fn bldk vfLrRo gSA
(a) f = {(1, 1), (2, 2), (3, 3)}
(b) f = {(1, 2), (2, 1), (3, 1)}
(c) f = {(1, 3), (3, 2), (2, 1)}

gy
(a) ;g ljyrk ls ns[kk tk ldrk gS fd f ,oSQdh vkPNknh gS] blfy, f O;qRØe.kh; gS rFkk
f dk izfrykse f –1 = {(1, 1), (2, 2), (3, 3)} = f }kjk izkIr gksrk gSA
(b) D;ksafd f (2) = f (3) = 1, vr,o f ,oSQdh ugha gS] vr% f O;qRØe.kh; ugha gSA
(c) ;g ljyrk iwoZd ns[kk tk ldrk gS fd f ,oSQdh rFkk vkPNknd gS] vr,o f O;qRØe.kh;
gS rFkk f –1 = {(3, 1), (2, 3), (1, 2)}gSA

iz'ukoyh 1-3
1. eku yhft, fd f : {1, 3, 4} → {1, 2, 5} rFkk g : {1, 2, 5} → {1, 3},
f = {(1, 2), (3, 5), (4, 1)} rFkk g = {(1, 3), (2, 3), (5, 1)} }kjk iznÙk gSaA gof Kkr
dhft,A
2. eku yhft, fd f, g rFkk h, R ls R rd fn, iQyu gSaA fl¼ dhft, fd
(f + g) o h = foh + goh
(f . g) o h = (foh) . (goh)
3. gof rFkk fog Kkr dhft,] ;fn
(i) f (x) = | x | rFkk g(x) = | 5x – 2 |
1

(ii) f (x) = 8x rFkk g(x) = x 3

3

2018-19
 laca/ ,oa iQyu 21

(4 x + 3) 2 2
4. ;fn f (x) = , x ≠ , rks fl¼ dhft, fd lHkh x ≠ osQ fy, fof (x) = x gSA
(6 x − 4) 3 3
f dk izfrykse iQyu D;k gS?
5. dkj.k lfgr crykb, fd D;k fuEufyf[kr iQyuksa osQ izfrykse gSa%
(i) f : {1, 2, 3, 4} → {10} tgk¡
f = {(1, 10), (2, 10), (3, 10), (4, 10)}
(ii) g : {5, 6, 7, 8} → {1, 2, 3, 4} tgk¡
g = {(5, 4), (6, 3), (7, 4), (8, 2)}
(iii) h : {2, 3, 4, 5} → {7, 9, 11, 13} tgk¡
h = {(2, 7), (3, 9), (4, 11), (5, 13)}
x
6. fl¼ dhft, fd f : [–1, 1] → R, f (x) = , }kjk iznÙk iQyu ,oSQdh gSA iQyu
( x + 2)
f : [–1, 1] → ( f dk ifjlj)] dk izfrykse iQyu Kkr dhft,A

x
(laosQr y ∈ ifjlj f, osQ fy,] [–1, 1] osQ fdlh x osQ varxZr y = f (x) = , vFkkZr~
x+2
2y
x= )
(1 − y )
7. f (x) = 4x + 3 }kjk iznÙk iQyu f : R → R ij fopkj dhft,A fl¼ dhft, fd f
O;qRØe.kh; gSA f dk izfrykse iQyu Kkr dhft,A
8. f (x) = x2 + 4 }kjk iznÙk iQyu f : R+ → [4, ∞) ij fopkj dhft,A fl¼ dhft, fd f
O;qRØe.kh; gS rFkk f dk izfrykse f –1 , f –1(y) = y − 4 , }kjk izkIr gksrk gS] tgk¡ R+
lHkh ½.ksrj okLrfod la[;kvksa dk leqPp; gSA
9. f (x) = 9x2 + 6x – 5 }kjk iznÙk iQyu f : R+ → [– 5, ∞) ij fopkj dhft,A fl¼ dhft,
 ( y + 6 ) −1 
fd f O;qRØe.kh; gS rFkk f –1(y) =   gSA
 3 
10. eku yhft, fd f : X → Y ,d O;qRØe.kh; iQyu gSA fl¼ dhft, fd f dk izfrykse
iQyu vf}rh; (unique) gSA (laosQr% dYiuk dhft, fd f osQ nks izfrykse iQyu g1
rFkk g2 gSaA rc lHkh y ∈ Y osQ fy, fog1(y) = 1Y(y) = fog2(y) gSA vc f osQ ,oSQdh xq.k
dk iz;ksx dhft,)

2018-19
22 xf.kr

11. f : {1, 2, 3} → {a, b, c}, f (1) = a, f (2) = b rFkk f (3) = c. }kjk iznÙk iQyu f ij fopkj
dhft,A f –1 Kkr dhft, vkSj fl¼ dhft, fd (f –1)–1 = f gSA
12. eku yhft, fd f : X → Y ,d O;qRØe.kh; iQyu gSa fl¼ dhft, fd f –1 dk izfrykse
f, gS vFkkZr~ (f –1)–1 = f gSA
1
13. ;fn f : R → R, f (x) = (3 − x3 ) 3 , }kjk iznÙk gS] rks fof (x) cjkcj gSA
1
(A) x 3 (B) x 3 (C) x (D) (3 – x3)

4x  4
14. eku yhft, fd f (x) = }kjk ifjHkkf"kr ,d iQyu f : R – −  → R gSA f dk
3x + 4  3

izfrykse] vFkkZr~ izfrfp=k (Map) g : ifjlj f → R – −  , fuEufyf[kr esa ls fdlosQ

4
 3
}kjk izkIr gksxk%
3y 4y
(A) g ( y) = (B) g ( y) =
3 − 4y 4 − 3y

4y 3y
(C) g ( y) = (D) g ( y) =
3 − 4y 4 − 3y

1.5 f}&vk/kjh lafØ;k,¡ (Binary Operations)

vius LowQy osQ fnuksa esa gh vki pkj ewy lafØ;kvksa] uker% ;ksx] varj] xq.kk rFkk Hkkx ls ifjfpr
gks pqosQ gSaA bu lafØ;kvksa dh eq[; fo'ks"krk ;g gS fd nks nh xbZ la[;kvksa a rFkk b, ls ge ,d
a
la[;k a + b ;k a – b ;k ab ;k , b ≠ 0 dks lac¼ (Associate) dj nsrs gSaA ;g ckr uksV
b
dhft, fd] ,d le; esa] osQoy nks la[;k,¡ gh tksM+h ;k xq.kk dh tk ldrh gSaA tc gesa rhu
la[;kvksa dks tksM+us dh vko';drk gksrh gS] rks ge igys nks la[;kvksa dks tksM+rs gSa vkSj izkIr
;ksxiQy dks fiQj rhljh la[;k esa tksM+ nsrs gSaA vr% ;ksx] xq.kk] varj rFkk Hkkx f}vk/kjh lafØ;k
osQ mnkgj.k gSa] D;ksafd ^f}vk/kjh* dk vFkZ gS ^nks vk/kj okyh*A ;fn ge ,d O;kid ifjHkk"kk
pkgrs gSa] ftless ;g pkjksa lafØ;k,¡ Hkh vk tkrh gSa] rks gesa la[;kvksa osQ leqPp; osQ LFkku ij ,d
LosPN leqPp; X ysuk pkfg, vkSj rc O;kid :i ls f}vk/kjh lafØ;k] oqQN vU; ugha vfirq]
X osQ nks vo;oksa a rFkk b dks X osQ gh fdlh vo;o ls lac¼ djuk gSA blls fuEufyf[kr O;kid
ifjHkk"kk izkIr gksrh gS%

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 laca/ ,oa iQyu 23

ifjHkk"kk 10 fdlh leqPp; A esa ,d f}vk/kjh lafØ;k ∗, ,d iQyu ∗ : A × A → A gSA ge

∗ (a, b) dks a ∗ b }kjk fu:fir djrs gSaA
mnkgj.k 29 fl¼ dhft, fd R esa ;ksx] varj vkSj xq.kk f}vk/kjh lafØ;k,¡ gSa] ¯drq Hkkx R esa
f}vk/kjh lafØ;k ugha gSA lkFk gh fl¼ dhft, fd Hkkx ½.ksrj okLrfod la[;kvksa osQ leqPp;
R esa f}vk/kjh lafØ;k gSA
gy + : R × R → R , (a, b) → a + b }kjk ifjHkkf"kr gS
– : R × R → R, (a, b) → a – b }kjk ifjHkkf"kr gS
× : R × R → R, (a, b) → ab }kjk ifjHkkf"kr gS
D;ksafd ‘+’, ‘–’ vkSj ‘×’ iQyu gSa] vr% ;s R esa f}vk/kjh lafØ;k,¡ gSaA
a a
ijarq ÷ : R × R → R, (a, b) → , ,d iQyu ugha gS] D;ksafd b = 0 osQ fy,
b b
ifjHkkf"kr ugha gSA
a
rFkkfi ÷ : R∗ × R∗ → R∗, (a, b) → }kjk ifjHkkf"kr ,d iQyu gS vkSj blfy,
b
;g R∗ esa ,d f}vk/kjh lafØ;k gSA
mnkgj.k 30 fl¼ dhft, fd varj (O;odyu) rFkk Hkkx N esa f}vk/kjh lafØ;k ugha gSA
gy – : N × N → N, (a, b) → a – b, }kjk iznÙk ,d f}vk/kjh lafØ;k ugha gS] D;ksafd ‘–’
a
osQ varxZr (3] 5) dk izfrfcac 3 – 5 = – 2 ∉ N. blh izdkj] ÷ N × N → N, (a, b) →
b
3
}kjk iznÙk ,d f}vk/kjh lafØ;k ugha gS] D;ksfa d ^÷* osQ varxZr (3 ¸ 5) dk izfrfcac 3 ÷ 5= ∉ N.
5
mnkgj.k 31 fl¼ dhft, fd ∗ : R × R → R, (a, b) → a + 4b2 }kjk iznÙk ,d f}vk/kjh
lafØ;k gSA
gy pw¡fd ∗ izR;sd ;qXe (a, b) dks R osQ ,d vf}rh; vo;o a + 4b2 rd ys tkrk gS] vr%
∗ R esa ,d f}vk/kjh lafØ;k gSA
mnkgj.k 32 eku yhft, fd P, fdlh iznÙk leqPp; X osQ leLr mi leqPp;ksa dk] leqPp; gSA
fl¼ dhft, fd ∪ : P × P → P, (A, B) → A ∪ B }kjk iznÙk rFkk ∩ : P × P → P,
(A, B) → A ∩ B }kjk ifjHkkf"kr iQyu] P esa f}vk/kjh lafØ;k,¡ gSaA
gy D;ksafd lfEeyu lafØ;k (Union Operation) ∪, P × P osQ izR;sd ;qXe (A, B) dks P osQ
,d vf}rh; vo;o A ∪ B rd ys tkrh gS] blfy, ∪, leqPp; P esa ,d f}vk/kjh lafØ;k

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24 xf.kr

gSA blh izdkj loZfu"B (Intersection) lafØ;k ∩ , P × P osQ izR;sd ;qXe (A, B) dks P osQ ,d
vf}rh; vo;o A ∩ B rd ys tkrh gS] vr,o ∩, leqPp; P esa ,d f}vk/kjh lafØ;k gSA
mnkgj.k 33 fl¼ dhft, fd (a, b) → vf/dre {a, b} }kjk ifjHkkf"kr ∨ : R × R → R rFkk
(a, b) → fuEure {a, b} }kjk ifjHkkf"kr ∧ : R × R → R f}vk/kjh lafØ;k,¡ gSaA
gy D;ksafd ∨ , R × R osQ izR;sd ;qXe (a, b) dks leqPp; R osQ ,d vf}rh; vo;o] uker%
a rFkk b esa ls vf/dre] ij ys tkrk gS] vr,o ∨ ,d f}vk/kjh lafØ;k gSa blh izdkj osQ roZQ
}kjk ;g dgk tk ldrk gS fd ∧ Hkh ,d f}vk/kjh lafØ;k gSA
fVIi.kh ∨ (4, 7) = 7, ∨ (4, – 7) = 4, ∧ (4, 7) = 4 rFkk ∧ (4, – 7) = – 7 gSA
tc fdlh leqPp; A esa vo;oksa dh la[;k de gksrh gS] rks ge leqPp; A esa ,d
f}vk/kjh lafØ;k ∗ dks ,d lkj.kh }kjk O;Dr dj ldrs gSa] ftls lafØ;k ∗ dh lafØ;k lkj.kh
dgrs gSaA mnkgj.kkFkZ A = {1, 2, 3} ij fopkj dhft,A rc mnkgj.k 33 esa ifjHkkf"kr A esa lafØ;k
∨ fuEufyf[kr lkj.kh (lkj.kh 1-1) }kjk O;Dr dh tk ldrh gSA ;gk¡ lafØ;k lkj.kh esa ∨ (1,
3) = 3, ∨ (2, 3) = 3, ∨ (1, 2) = 2.
lkj.kh 1.1
∨ 1 2 3
1 1 2 3
2 2 2 3
3 3 3 3
;gk¡ lafØ;k lkj.kh esa 3 iafDr;k¡ rFkk 3 LraHk gSa] ftlesa (i, j)oha izfof"V leqPp; A osQ
iosa rFkk josa vo;oksa esa ls vf/dre gksrk gSA bldk O;kidhdj.k fdlh Hkh lkekU; lafØ;k
* : A × A → A osQ fy, fd;k tk ldrk gSA ;fn A = {a1, a2, ..., an}gS rks lafØ;k lkj.kh esa
n iafDr;k¡ rFkk n LrEHk gksaxs rFkk (i, j)oha izfof"V ai ∗ aj gksxhA foykser% n iafDr;ksa RkFkk n LraHkksa
okys iznÙk fdlh lafØ;k lkj.kh] ftldh izR;sd izfof"V A = {a1, a2, ..., an}, dk ,d vo;o gS]
osQ fy, ge ,d f}vk/kjh lafØ;k ∗ : A × A → A ifjHkkf"kr dj ldrs gSa] bl izdkj fd
ai ∗ aj = lafØ;k lkj.kh dh ioha iafDr rFkk josa LrEHk dh izfof"V;k¡ gSaA
ge uksV djrs gSa fd 3 rFkk 4 dks fdlh Hkh Øe (order) esa tksM+sa] ifj.kke (;ksxiQy) leku
jgrk gS] vFkkZr~ 3 + 4 = 4 + 3, ijarq 3 rFkk 4 dks ?kVkus esa fofHkUu Øe fofHkUu ifj.kke nsrs gSa]
vFkkZr~ 3 – 4 ≠ 4 – 3. blh izdkj 3 rFkk 4 xq.kk djus esa Øe egRoiw.kZ ugha gS] ijarq 3 rFkk 4
osQ Hkkx esa fofHkUu Øe fofHkUu ifj.kke nsrs gSaA vr% 3 rFkk 4 dk ;ksx rFkk xq.kk vFkZiw.kZ gS ¯drq
3 Rkk 4 dk varj rFkk Hkkx vFkZghu gSA varj rFkk Hkkx osQ fy, gesa fy[kuk iM+rk gS fd ^3 esa

2018-19
 laca/ ,oa iQyu 25

ls 4 ?kVkb,* ;k ^4 esa ls 3 ?kVkb,* vFkok ^3 dks 4 ls Hkkx dhft,* ;k ^4 dks 3 ls Hkkx dhft,*A
blls fuEufyf[kr ifjHkk"kk izkIr gksrh gS%
ifjHkk"kk 11 leqPp; X esa ,d f}vk/kjh lafØ;k ∗ Øefofues; (Commutative) dgykrh gS]
;fn izR;sd a, b ∈ X osQ fy, a ∗ b = b ∗ a gksA
mnkgj.k 34 fl¼ dhft, fd + : R × R → R rFkk × : R × R → R Øefofues; f}vk/kjh
lafØ;k,¡ gS] ijarq – : R × R → R rFkk ÷ : R∗ × R∗ → R∗ Øefofues; ugha gSaA
gy D;ksafd a + b = b + a rFkk a × b = b × a, ∀ a, b ∈ R, vr,o ‘+’ rFkk ‘×’ Øefofues;
f}vk/kjh lafØ;k,¡ gSaA rFkkfi ‘–’ Øefofues; ugha gS] D;ksafd 3 – 4 ≠ 4 – 3.
blh izdkj 3 ÷ 4 ≠ 4 ÷ 3, ftlls Li"V gksrk gS fd ‘÷’ Øefofues; ugha gSA
mnkgj.k 35 fl¼ dhft, fd a ∗ b = a + 2b }kjk ifjHkkf"kr ∗ : R × R → R Øefofues;
ugha gSA
gy D;ksafd 3 ∗ 4 = 3 + 8 = 11 vkSj 4 ∗ 3 = 4 + 6 = 10, vr% lafØ;k ∗ Øefofues; ugha gSA
;fn ge leqPp; X osQ rhu vo;oksa dks X esa ifjHkkf"kr fdlh f}vk/kjh lafØ;k osQ }kjk
lac¼ djuk pkgrs gSa rks ,d LokHkkfod leL;k mBrh gSA O;atd a ∗ b ∗ c dk vFkZ
(a ∗ b) ∗ c vFkok a ∗ (b ∗ c) gks ldrk gS vkSj ;g nksuksa O;tad] vko';d ugha gS] fd leku
gksaA mnkgj.kkFkZ (8 – 5) – 2 ≠ 8 – (5 – 2). blfy,] rhu la[;kvksa 8] 5 vkSj 3 dk f}vk/kjh lafØ;k
^O;odyu* osQ }kjk laca/ vFkZghu gS tc rd fd dks"Bd (Bracket) dk iz;ksx ugha fd;k tk,A
ijarq ;ksx dh lafØ;k esa] 8 + 5 + 2 dk eku leku gksrk gS] pkgs ge bls ( 8 + 5) + 2 vFkok
8 + (5 + 2) izdkj ls fy[ksaA vr% rhu ;k rhu ls vf/d la[;kvksa dk ;ksx dh lafØ;k }kjk
laca/] fcuk dks"Bdksa osQ iz;ksx fd, Hkh] vFkZiw.kZ gSA blls fuEufyf[kr ifjHkk"kk izkIr gksrh gS%
ifjHkk"kk 12 ,d f}vk/kjh lafØ;k ∗ : A × A → A lkgp;Z (Associative) dgykrh gS] ;fn
(a ∗ b) ∗ c = a ∗ (b ∗ c), ∀ a, b, c, ∈ A.
mnkgj.k 36 fl¼ dhft, fd R esa ;ksx rFkk xq.kk lkgp;Z f}vk/kjh lafØ;k,¡ gSaA ijarq O;odyu
rFkk Hkkx R esa lkgp;Z ugha gSA
gy ;ksx rFkk xq.kk lkgp;Z gSa] D;ksafd (a + b) + c = a + (b + c) rFkk (a×b) × c = a × (b × c),
∀ a, b, c ∈ R gSA rFkkfi varj rFkk Hkkx lkgp;Z ugha gSa] D;ksafd (8 – 5) – 3 ≠ 8 – (5 – 3) rFkk
(8 ÷ 5) ÷ 3 ≠ 8 ÷ (5 ÷ 3).
mnkgj.k 37 fl¼ dhft, fd a ∗ b → a + 2b }kjk iznÙk ∗ : R × R → R lkgp;Z ugha gSA
gy lafØ;k ∗ lkgp;Z ugha gS] D;ksafd
(8 ∗ 5) ∗ 3 = (8 + 10) ∗ 3 = (8 + 10) + 6 = 24,
tcfd 8 ∗ (5 ∗ 3) = 8 ∗ (5 + 6) = 8 ∗ 11 = 8 + 22 = 30.

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fVIi.kh fdlh f}vk/kjh lafØ;k dk lkgp;Z xq.k/eZ bl vFkZ esa vR;ar egRoiw.kZ gS fd ge O;atd
a1 ∗ a2 ∗ ... ∗ an fy[k ldrs gSa] D;ksafd bl xq.k/eZ osQ dkj.k ;g lafnX/ ugha jg tkrk gSA ijarq
bl xq.k/eZ osQ vHkko esa] O;atd a1 ∗ a2 ∗ ... ∗ an lafnX/ (Ambiguous) jgrk gS] tc rd fd
dks"Bd dk iz;ksx u fd;k tk,A Lej.k dhft, fd iwoZorhZ d{kkvksa esa] tc dHkh varj ;k Hkkx dh
lafØ;k,¡ vFkok ,d ls vf/d lafØ;k,¡ laiUu dh xb± Fkha] rc dks"Bdksa dk iz;ksx fd;k x;k FkkA
R esa f}vk/kjh lafØ;k ^$* ls lacaf/r la[;k 'kwU; (zero) dh ,d jkspd fo'ks"krk ;g gS fd
a + 0 = a = 0 + a, ∀ a ∈ R, vFkkZr~, fdlh Hkh la[;k esa 'kwU; dks tksM+us ij og la[;k vifjofrZr
jgrh gSA ijarq xq.kk dh fLFkfr esa ;g Hkwfedk (Role) la[;k 1 }kjk vnk dh tkrh gS] D;ksafd
a × 1 = a = 1 × a, ∀ a ∈ R gSA blls fuEufyf[kr ifjHkk"kk izkIr gksrh gSA
ifjHkk"kk 13 fdlh iznÙk f}vk/kjh lafØ;k ∗ : A × A → A, osQ fy,] ,d vo;o e ∈ A, ;fn
bldk vfLrRo gS] rRled (Identity) dgykrk gS] ;fn a ∗ e = a = e ∗ a, ∀ a ∈ A gksA
mnkgj.k 38 fl¼ dhft, fd R esa 'kwU; (0) ;ksx dk rRled gS rFkk 1 xq.kk dk rRled gSA ijarq
lafØ;kvksa – : R × R → R vkSj ÷ : R∗ × R∗ → R∗ osQ fy, dksbZ rRled vo;o ugha gSA
gy a + 0 = 0 + a = a vkSj a × 1 = a = 1 × a, ∀ a ∈ R dk rkRi;Z gS fd 0 rFkk 1 Øe'k%
‘+’ rFkk ‘×’, osQ rRled vo;o gSaA lkFk gh R esa ,slk dksbZ vo;o e ugha gS fd a – e =
e – a, ∀ a ∈ R gksA blh izdkj gesa R∗ esa dksbZ ,slk vo;o e ugha fey ldrk gS fd
a ÷ e = e ÷ a, ∀ a ∈ R∗ gksA vr% ‘–’ rFkk ‘÷’ osQ rRled vo;o ugha gksrs gSaA

fVIi.kh R esa 'kwU; (0) /u lafØ;k dk rRled gS] ¯drq ;g N esa /u lafØ;k dk rRled ugha
gS] D;ksafd 0 ∉ N okLro esa N esa /u lafØ;k dk dksbZ rRled ugha gksrk gSA
ge iqu% ns[krs gSa fd /u lafØ;k + : R × R → R osQ fy,] fdlh iznÙk a ∈ R ls
lacaf/r R esa – a dk vfLrRo bl izdkj gS fd a + (– a) = 0 (‘+’ dk rRled) = (– a) + a.
1
blh izdkj R esa xq.kk lafØ;k osQ fy,] fdlh iznÙk a ∈ R, a ≠ 0 ls lacaf/r ge R esa dks
1 1 a
bl izdkj pqu ldrs gSa fd a × = 1(‘×’ dk rRled) = × a gksA blls fuEufyf[kr ifjHkk"kk
a a
izkIr gksrh gSA
ifjHkk"kk 14 A esa rRled vo;o e okys ,d iznÙk f}vk/kjh lafØ;k ∗ : A × A → A osQ fy,
fdlh vo;o a ∈ A dks lafØ;k ∗ osQ lanHkZ esa O;qRØe.kh; dgrs gSa] ;fn A esa ,d ,sls vo;o
b dk vfLrRo gS fd a ∗ b = e = b ∗ a gks rks b dks a dk izfrykse (Inverse) dgrs gSa] ftls
izrhd a–1 }kjk fu:fir djrs gSaA

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mnkgj.k 39 fl¼ dhft, fd R esa /u lafØ;k ‘+’ osQ fy, – a dk izfrykse a gS vkSj R esa
1
xq.kk lafØ;k ‘×’ osQ fy, a ≠ 0 dk izfrykse gSA
a
gy D;ksafd a + (– a) = a – a = 0 rFkk (– a) + a = 0, blfy, – a /u lafØ;k osQ fy, a
1 1 1
dk izfrykse gSA blh izdkj] a ≠ 0, osQ fy, a × = 1 = × a, ftldk rkRi;Z ;g gS fd
a a a
xq.kk lafØ;k osQ fy, a dk izfrykse gSA
mnkgj.k 40 fl¼ dhft, fd N esa /u lafØ;k '+' osQ fy, a ∈ N dk izfrykse – a ugha gS
1
vkSj N esa xq.kk lafØ;k ‘×’ osQ fy, a ∈ N, a ≠ 1 dk izfrykse ugha gSA
a
gy D;ksafd – a ∉ N, blfy, N esa /u lafØ;k osQ fy, a dk izfrykse – a ugha gks ldrk gS
;|fi – a, izfrca/ a + (– a) = 0 = (– a) + a dks larq"V djrk gSA blh izdkj] N esa a ≠ 1 osQ
1
fy, ∉ N, ftldk vFkZ ;g gS fd 1 osQ vfrfjDr N osQ fdlh Hkh vo;o dk izfrykse N
a
esa xq.kk lafØ;k osQ fy, ugha gksrk gSA
mnkgj.k 34] 36] 38 rFkk 39 ls Li"V gksrk gS fd R esa /u lafØ;k Øefofue; rFkk lkgp;Z
f}vk/kjh lafØ;k gS] ftlesa 0 rRled vo;o rFkk a ∈ R, ∀ a dk izfrykse vo;o – a gksrk
gSA
iz'ukoyh 1-4
1. fu/kZfjr dhft, fd D;k fuEufyf[kr izdkj ls ifjHkkf"kr izR;sd lafØ;k ∗ ls ,d
f}vk/kjh lafØ;k izkIr gksrh gS ;k ughaA ml n'kk esa tc ∗ ,d f}vk/kjh lafØ;k ugha gS]
vkSfpR; Hkh crykb,A
(i) Z+ esa, a ∗ b = a – b }kjk ifjHkkf"kr lafØ;k ∗
(ii) Z+ esa, a ∗ b = ab }kjk ifjHkkf"kr lafØ;k ∗
(iii) R esa] lafØ;k ∗] a ∗ b = ab2 }kjk ifjHkkf"kr
(iv) Z+ esa, lafØ;k ∗] a ∗ b = | a – b | }kjk ifjHkkf"kr
(v) Z+ esa, lafØ;k ∗, a ∗ b = a }kjk ifjHkkf"kr
2. fuEufyf[kr ifjHkkf"kr izR;sd f}vk/kjh lafØ;k ∗ osQ fy, fu/kZfjr dhft, fd D;k ∗
f}vk/kjh Øefofue; gS rFkk D;k ∗ lkgp;Z gSA

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28 xf.kr

(i) Z es,a a ∗ b = a – b }kjk ifjHkkf"kr

(ii) Q esa, a ∗ b = ab + 1 }kjk ifjHkkf"kr

ab
(iii) Q esa, a ∗ b = }kjk ifjHkkf"kr
2
(iv) Z+ esa, a ∗ b = 2ab }kjk ifjHkkf"kr
(v) Z+ esa, a ∗ b = ab }kjk ifjHkkf"kr

a
(vi) R – {– 1} esa, a ∗ b = }kjk ifjHkkf"kr
b +1
3. leqPp; {1, 2, 3, 4, 5} esa a ∧ b = fuEure {a, b} }kjk ifjHkkf"kr f}vk/kjh lafØ;k ij
fopkj dhft,A lafØ;k ∧ osQ fy, lafØ;k lkj.kh fyf[k,A
4. leqPp; {1, 2, 3, 4, 5} esa] fuEufyf[kr lafØ;k lkj.kh (lkj.kh 1-2) }kjk ifjHkkf"kr]
f}vk/kjh lafØ;k ∗ ij fopkj dhft, rFkk
(i) (2 ∗ 3) ∗ 4 rFkk 2 ∗ (3 ∗ 4) dk ifjdyu dhft,A
(ii) D;k ∗ Øefofues; gS\
(iii) (2 ∗ 3) ∗ (4 ∗ 5) dk ifjdyu dhft,A
(laosQr% fuEu lkj.kh dk iz;ksx dhft,A)
lkj.kh 1-2
* 1 2 3 4 5
1 1 1 1 1 1
2 1 2 1 2 1
3 1 1 3 1 1
4 1 2 1 4 1
5 1 1 1 1 5

5. eku yhft, fd leqPp; {1, 2, 3, 4, 5} esa ,d f}vk/kjh lafØ;k ∗′, a ∗′ b = a rFkk b

dk HCF }kjk ifjHkkf"kr gSA D;k lafØ;k ∗′ miZ;qDr iz'u 4 esa ifjHkkf"kr lafØ;k ∗ osQ
leku gS\ vius mÙkj dk vkSfpR; Hkh crykb,A
6. eku yhft, fd N esa ,d f}vk/kjh lafØ;k ∗] a ∗ b = a rFkk b dk LCM }kjk ifjHkkf"kr
gSA fuEufyf[kr Kkr dhft,%
(i) 5 ∗ 7, 20 ∗ 16 (ii) D;k lafØ; ∗ Øefofues; gS ?

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 laca/ ,oa iQyu 29

(iii) D;k ∗ lkgp;Z gS? (iv) N esa ∗ dk rRled vo;o Kkr dhft,
(v) N osQ dkSu ls vo;o ∗ lafØ;k osQ fy, O;qRØe.kh; gSa\
7. D;k leqPp; {1, 2, 3, 4, 5} esa a ∗ b = a rFkk b dk LCM }kjk ifjHkkf"kr ∗ ,d
f}vk/kjh lafØ;k gS\ vius mÙkj dk vkSfpR; Hkh crykb,A
8. eku yhft, fd N esa a ∗ b = a rFkk b dk HCF }kjk ifjHkkf"kr ,d f}vk/kjh lafØ;k
gSA D;k ∗ Øefofues; gS\ D;k ∗ lkgp;Z gS\ D;k N esa bl f}vk/kjh lafØ;k osQ rRled
dk vfLrRo gS\
9. eku yhft, fd ifjes; la[;kvksa osQ leqPp; Q esa fuEufyf[kr izdkj ls ifjHkkf"kr ∗ ,d
f}vk/kjh lafØ;k gS%
(i) a ∗ b = a – b (ii) a ∗ b = a2 + b2
(iii) a ∗ b = a + ab (iv) a ∗ b = (a – b)2
ab
(v) a ∗ b = (vi) a ∗ b = ab2
4
Kkr dhft, fd buesa ls dkSu lh lafØ;k,¡ Øefofues; gSa vkSj dkSulh lkgp;Z gSaA
10. iz'u 9 esa nh xbZ lafØ;kvksa esa fdlh dk rRled gS] og crykb,A
11. eku yhft, fd A = N × N gS rFkk A esa (a, b) ∗ (c, d) = (a + c, b + d) }kjk ifjHkkf"kr
,d f}vk/kjh lafØ;k gSA fl¼ dhft, fd ∗ Øefofue; rFkk lkgp;Z gSA A esa ∗ dk
rRled vo;o] ;fn dksbZ gS] rks Kkr dhft,A
12. crykb, fd D;k fuEufyf[kr dFku lR; gSa ;k vlR; gSaA vkSfpR; Hkh crykb,A
(i) leqPp; N esa fdlh Hkh LosPN f}vk/kjh lafØ;k ∗ osQ fy, a ∗ a = a, ∀ a ∈ N
(ii) ;fn N esa ∗ ,d Øefofues; f}vk/kjh lafØ;k gS] rks a ∗ (b ∗ c) = (c ∗ b) ∗ a
13. a ∗ b = a3 + b3 izdkj ls ifjHkkf"kr N esa ,d f}vk/kjh lafØ;k ∗ ij fopkj dhft,A vc
fuEufyf[kr esa ls lgh mÙkj dk p;u dhft,
(A) ∗ lkgp;Z rFkk Øefofues; nksuksa gS
(B) ∗ Øefofues; gS ¯drq lkgp;Z ugha gS
(C) ∗ lkgp;Z gS ¯drq Øefofues; ugha gS
(D) ∗ u rks Øefofues; gS vkSj u lkgp;Z gS

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30 xf.kr

fofo/ mnkgj.k
mnkgj.k 41 ;fn R1 rFkk R2 leqPp; A esa rqY;rk laca/ gSa] rks fl¼ dhft, fd R1 ∩ R2 Hkh ,d
rqY;rk laca/ gSA
gy D;ksafd R1 rFkk R2 rqY;rk laca/ gS blfy, (a, a) ∈ R1, rFkk (a, a) ∈ R2, ∀ a ∈ A bldk
rkRi;Z gS fd (a, a) ∈ R1 ∩ R2, ∀ a, ftlls fl¼ gksrk gS fd R1 ∩ R2 LorqY; gSA iqu%
(a, b) ∈ R1 ∩ R2 ⇒ (a, b) ∈ R1 rFkk (a, b) ∈ R2 ⇒ (b, a) ∈ R1 rFkk (b, a) ∈ R2 ⇒
(b, a) ∈ R1 ∩ R2, vr% R1 ∩ R2 lefer gSA blh izdkj (a, b) ∈ R1 ∩ R2 rFkk (b, c) ∈ R1 ∩ R2
⇒ (a, c) ∈ R1 rFkk (a, c) ∈ R2 ⇒ (a, c) ∈ R1 ∩ R2- blls fl¼ gksrk gS fd R1 ∩ R2 laØked
gSA vr% R1 ∩ R2 ,d rqY;rk laca/ gSA
mnkgj.k 42 eku yhft, fd leqPp; A esa /u iw.kk±dksa osQ Øfer ;qXeksa (ordered pairs)dk
,d laca/ R, (x, y) R (u, v), ;fn vkSj osQoy ;fn] xv = yu }kjk ifjHkkf"kr gSA fl¼ dhft,
fd R ,d rqY;rk laca/ gSA
gy Li"Vr;k (x, y) R (x, y), ∀ (x, y) ∈ A, D;ksafd xy = yx gSA blls Li"V gksrk gS fd R
LorqY; gSA iqu% (x, y) R (u, v) ⇒ xv = yu ⇒ uy = vx vkSj blfy, (u, v) R (x, y)gSA blls
Li"V gksrk gS fd R lefer gSA blh izdkj (x, y) R (u, v) rFkk (u, v) R (a, b) ⇒ xv = yu
a a b a
rFkk ub = va ⇒ xv = yu ⇒ xv = yu ⇒ xb = ya vkSj blfy, (x, y) R (a, b)gSA
u u v u
vr,o R laØked gSA vr% R ,d rqY;rk laca/ gSA
mnkgj.k 43 eku yhft, fd X = {1, 2, 3, 4, 5, 6, 7, 8, 9}gSA eku yhft, fd X esa
R1 = {(x, y) : x – y la[;k 3 ls HkkT; gS} }kjk iznÙk ,d laca/ R1 gS rFkk R2 = {(x, y): {x, y}
⊂ {1, 4, 7} ;k {x, y} ⊂ {2, 5, 8} ;k {(x, y} ⊂ {3, 6, 9} }kjk iznÙk X esa ,d vU; laca/ R2
gSA fl¼ dhft, fd R1 = R2gSA
gy uksV dhft, fd {1, 4, 7}, {2, 5, 8} rFkk {3, 6, 9} leqPp;ksa esa ls izR;sd dk vfHky{k.k
(characterstic) ;g gS fd buosQ fdlh Hkh nks vo;oksa dk varj 3 dk ,d xq.kt gSA blfy,
(x, y) ∈ R1 ⇒ x – y la[;k 3 dk xq.kt gS ⇒ {x, y} ⊂ {1, 4, 7} ;k {x, y} ⊂ {2, 5, 8}
;k {x, y} ⊂ {3, 6, 9} ⇒ (x, y) ∈ R2] vr% R1 ⊂ R2- blh izdkj {x, y} ∈ R2 ⇒ {x, y} ⊂
{1, 4, 7} ;k {x, y} ⊂ {2, 5, 8} ;k {x, y} ⊂ {3, 6, 9} ⇒ x – y la[;k 3 ls HkkT; gS ⇒ {x, y}
∈ R1- blls Li"V gksrk gS fd R2 ⊂ R1- vr% R1 = R2 gSA
mnkgj.k 44 eku yhft, fd f : X → Y ,d iQyu gSA X esa R = {(a, b): f (a) = f (b)} }kjk
iznÙk ,d laca/ R ifjHkkf"kr dhft,A tk¡fp, fd D;k R ,d rqY;rk laca/ gSA

2018-19
 laca/ ,oa iQyu 31

gy izR;sd a ∈ X osQ fy, (a, a) ∈ R, D;ksafd f (a) = f (a), ftlls Li"V gksrk gS fd R LorqY;
gSA blh izdkj] (a, b) ∈ R ⇒ f (a) = f (b) ⇒ f (b) = f (a) ⇒ (b, a) ∈ R- blfy, R lefer
gSA iqu% (a, b) ∈ R rFkk (b, c) ∈ R ⇒ f (a) = f (b) rFkk f (b) = f (c) ⇒ f (a) = f (c) ⇒
(a, c) ∈ R, ftldk rkRi;Z gS fd R laØked gSA vr% R ,d rqY;rk laca/ gSA
mnkgj.k 45 fu/kZfjr dhft, fd leqPp; R esa iznÙk fuEufyf[kr f}vk/kjh lafØ;kvksa esa ls dkSu
lh lkgp;Z gSa vkSj dkSu lh Øefofues; gSaA
(a + b )
(a) a ∗ b = 1, ∀ a, b ∈ R (b) a ∗ b = ∀ a, b ∈ R
2
gy
(a) Li"Vr;k ifjHkk"kk }kjk a ∗ b = b ∗ a = 1, ∀ a, b ∈ R- lkFk gh (a ∗ b) ∗ c =
(1 ∗ c) =1 rFkk a ∗ (b ∗ c) = a ∗ (1) = 1, ∀ a, b, c ∈ R vr% R lkgp;Z rFkk
Øefofues; nksuksa gSA
a +b b+a
(b) a ∗ b = = = b ∗ a, ∀ a, b ∈ R, ftlls Li"V gksrk gS fd ∗ Øefofues;
2 2
gSA iqu%
a+b
(a ∗ b) ∗ c =   ∗ c.
 2 

a+b
  + c a + b + 2c
 2  =
= .
2 4

b+c
¯drq a ∗ (b ∗ c) = a ∗  
 2 
b+c
a+
= 2 = 2a + b + c ≠ a + b + 2c (lkekU;r%)
2 4 4
vr% ∗ lkgp;Z ugha gSA
mnkgj.k 46 leqPp; A = {1, 2, 3} ls Lo;a rd lHkh ,oSQdh iQyu dh la[;k Kkr dhft,A
gy {1, 2, 3} ls Lo;a rd ,oSQdh iQyu osQoy rhu izrhdksa 1, 2, 3 dk Øep; gSA vr%
{1, 2, 3} ls Lo;a rd osQ izfrfp=kksa (Maps) dh oqQy la[;k rhu izrhdksa 1] 2 ] 3 osQ Øep;ksa
dh oqQy la[;k osQ cjkcj gksxh] tks fd 3! = 6 gSA

2018-19
32 xf.kr

mnkgj.k 47 eku yhft, fd A = {1, 2, 3} gSA rc fl¼ dhft, fd ,sls laca/ksa dh la[;k pkj
gS] ftuesa (1] 2) rFkk (2] 3) gSa vkSj tks LorqY; rFkk laØked rks gSa ¯drq lefer ugha gSaA
gy {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3), (1, 3)}, (1, 2) rFkk (2, 3) vo;oksa okyk og lcls NksVk
laca/ R1 gS] tks LorqY; rFkk laØked gS ¯drq lefer ugha gSA vc ;fn R1 esa ;qXe (2] 1) c<+k
nsa] rks izkIr laca/ R2 vc Hkh LorqY; rFkk laØked gS ijarq lefer ugha gSA blh izdkj, ge R1
esa (3, 2) c<+k dj R3 izkIr dj ldrs gSa] ftuesa vHkh"V xq.k/eZ gSaA rFkkfi ge R1 esa fdUgha nks ;qXeksa
(2, 1), (3, 2) ;k ,d ;qXe (3, 1) dks ugha c<+k ldrs gSa] D;ksafd ,slk djus ij ge] laØkedrk
cuk, j[kus osQ fy,] 'ks"k ;qXe dks ysus osQ fy, ckè; gks tk,¡xs vkSj bl izfØ;k }kjk izkIr laca/
lefer Hkh gks tk,xk] tks vHkh"V ugha gSA vr% vHkh"V laca/ksa dh oqQy la[;k rhu gSA
mnkgj.k 48 fl¼ dhft, fd leqPp; {1, 2, 3} esa (1, 2) rFkk (2, 1) dks vUrfoZ"V djus okys
rqY;rk laca/ksa dh la[;k 2 gSA
gy (1, 2) rFkk (2, 1) dks varfoZ"V djus okyk lcls NksVk rqY;rk laca/ R1, {(1, 1), (2, 2),
(3, 3), (1, 2), (2, 1)} gSA vc osQoy 4 ;qXe] uker% (2, 3), (3, 2), (1, 3) rFkk (3, 1) 'ks"k cprs gSaA
;fn ge buesa ls fdlh ,d dks] tSls (2] 3) dks R1 esa varfoZ"V djrs gSa] rks lefer osQ fy,
gesa (3] 2) dks Hkh ysuk iM+sxk] lkFk gh laØedrk gsrq ge (1] 3) rFkk (3] 1) dks ysus osQ
fy, ckè; gksaxsA vr% R1 ls cM+k rqY;rk laca/ osQoy lkoZf=kd laca/ gSA blls Li"V gksrk gS fd
(1] 2) rFkk (2] 1) dks varfoZ"V djus okys rqY;rk laca/ksa dh oqQy la[;k nks gSA
mnkgj.k 49 fl¼ dhft, fd {1, 2} esa ,slh f}vk/kjh lafØ;kvksa dh la[;k osQoy ,d gS] ftldk
rRled 1 gSa rFkk ftlosQ varxZr 2 dk izfrykse 2 gSA
gy {1, 2} esa dksbZ f}vk/kjh lafØ;k ∗, {1, 2} × {1, 2} ls {1, 2} esa ,d iQyu gS] vFkkZr~
{(1, 1), (1, 2), (2, 1), (2, 2)} ls {1, 2} rd ,d iQyuA D;ksafd vHkh"V f}vk/kjh lafØ;k ∗ osQ
fy, rRled vo;o 1 gS] blfy,, ∗ (1, 1) = 1, ∗ (1, 2) = 2, ∗ (2, 1) = 2 vkSj ;qXe (2, 2)
osQ fy, gh osQoy fodYi 'ks"k jg tkrk gSA D;ksafd 2 dk izfrykse 2 gS] blfy, ∗ (2, 2) vko';d
:i ls 1 osQ cjkcj gSA vr% vHkh"V f}vk/kjh lafØ;kvksa dh la[;k osQoy ,d gSA
mnkgj.k 50 rRled iQyu IN : N → N ij fopkj dhft,] tks IN (x) = x, ∀ x ∈ N }kjk
ifjHkkf"kr gSA fl¼ dhft, fd] ;|fi IN vkPNknd gS ¯drq fuEufyf[kr izdkj ls ifjHkkf"kr iQyu
IN + IN : N → N vkPNknd ugha gS
(IN + IN) (x) = IN (x) + IN (x) = x + x = 2x
gy Li"Vr;k IN vkPNknd gS ¯drq IN + IN vkPNknd ugha gSA D;ksafd ge lgizkar N esa
,d vo;o 3 ys ldrs gSa ftlosQ fy, izkar N esa fdlh ,sls x dk vfLrRo ugha gS fd
(IN + IN) (x) = 2x = 3 gksA

2018-19
 laca/ ,oa iQyu 33

mnkgj.k 51 f (x) = sin x }kjk iznÙk iQyu f : 0, π  → R rFkk g(x) = cos x }kjk iznÙk iQyu
 2
π
g :  0,  → R ij fopkj dhft,A fl¼ dhft, fd f rFkk g ,oSQdh gS] ijarq f + g ,oSQdh ugha
 2
gSA

gy D;ksafd 0, π  , osQ nks fHkUu&fHkUu vo;oksa x1 rFkk x2 osQ fy, sin x1 ≠ sin x2 rFkk
 2
cos x1 ≠ cos x2 blfy, f rFkk g nksuksa gh vko';d :i ls ,oSQdh gSaA ijarq (f + g) (0) =

 π π π
sin 0 + cos 0 = 1 rFkk (f + g)   = sin + cos = 1 gSA vr% f + g ,oSQdh ugha gSA
2 2 2

vè;k; 1 ij fofo/ iz'ukoyh

1. eku yhft, fd f : R → R , f (x) = 10x + 7 }kjk ifjHkkf"kr iQyu gSA ,d ,slk iQyu
g : R → R Kkr dhft, ftlosQ fy, g o f = f o g = 1R gksA
2. eku yhft, fd f : W → W, f (n) = n – 1, ;fn n fo"ke gS rFkk f (n) = n + 1, ;fn n
le gS] }kjk ifjHkkf"kr gSA fl¼ dhft, fd f O;qRØe.kh; gSA f dk izfrykse Kkr dhft,A
;gk¡ W leLr iw.kk±dksa dk leqPp; gSA
3. ;fn f : R → R tgk¡ f(x) = x2 – 3x + 2 }kjk ifjHkkf"kr gS rks f (f (x)) Kkr dhft,A

x
4. fl¼ dhft, fd f : R → {x ∈ R : – 1 < x < 1} tgk¡ f ( x ) = , x ∈ R }kjk
1+ | x |
ifjHkkf"kr iQyu ,oSQdh rFkk vkPNknd gSA
5. fl¼ dhft, fd f (x) = x3 }kjk iznÙk iQyu f : R → R ,oSQd (Injective) gSA
6. nks iQyuksa f : N → Z rFkk g : Z → Z osQ mnkgj.k nhft, tks bl izdkj gksa fd] g o f
,oSQd gS ijarq g ,oSQd ugha gSA
(laosQru: f (x) = x rFkk g (x) = | x | ij fopkj dhft,A)
7. nks iQyuksa f : N → N rFkk g : N → N osQ mnkgj.k nhft,] tks bl izdkj gksa fd]
g o f vkPNknd gS ¯drq f vkPNknu ugha gSA

x −1 , x > 1
(laosQr: f (x) = x + 1 rFkk g ( x) =  ij fopkj dhft,A
 1 , x =1

2018-19
34 xf.kr

8. ,d vfjDr leqPp; X fn;k gqvk gSA P(X) tks fd X osQ leLr mileqPp;ksa dk leqPp;
gS] ij fopkj dhft,A fuEufyf[kr rjg ls P(X) esa ,d laca/ R ifjHkkf"kr dhft,%
P(X) esa mileqPp;ksa A, B osQ fy,] ARB, ;fn vkSj osQoy ;fn A ⊂ B gSA D;k R, P(X)
esa ,d rqY;rk laca/ gS? vius mÙkj dk vkSfpR; Hkh fyf[k,A
9. fdlh iznÙk vfjDr leqPp; X osQ fy, ,d f}vk/kjh lafØ;k ∗ : P(X) × P(X) → P(X)
ij fopkj dhft,] tks A ∗ B = A ∩ B, ∀ A, B ∈ P(X) }kjk ifjHkkf"kr gS] tgk¡ P(X)
leqPp; X dk ?kkr leqPp; (Power set) gSA fl¼ dhft, fd bl lafØ;k dk rRled
vo;o X gS rFkk lafØ;k ∗ osQ fy, P(X) esa osQoy X O;qRØe.kh; vo;o gSA
10. leqPp; {1, 2, 3, ... , n} ls Lo;a rd osQ leLr vkPNknd iQyuksa dh la[;k Kkr dhft,A
11. eku yhft, fd S = {a, b, c} rFkk T = {1, 2, 3} gSA S ls T rd osQ fuEufyf[kr iQyuksa
F osQ fy, F–1 Kkr dhft,] ;fn mldk vfLrRo gS%
(i) F = {(a, 3), (b, 2), (c, 1)} (ii) F = {(a, 2), (b, 1), (c, 1)}
12. a ∗b = |a – b| rFkk a o b = a, ∀ a, b ∈ R }kjk ifjHkkf"kr f}vk/kjh la f Ø;kvks a
∗ : R × R → R rFkk o : R × R → R ij fopkj dhft,A fl¼ dhft, fd ∗ Øefofues;
gS ijarq lkgp;Z ugha gS] o lkgp;Z gS ijarq Øefofues; ugha gSA iqu% fl¼ dhft, fd lHkh
a, b, c ∈ R osQ fy, a ∗ (b o c) = (a ∗ b) o (a ∗ c) gSA [;fn ,slk gksrk gS] rks ge dgrs
gSa fd lafØ;k ∗ lafØ;k o ij forfjr (Distributes) gksrh gSA] D;k o lafØ;k ∗ ij forfjr
gksrh gS? vius mÙkj dk vkSfpR; Hkh crykb,A
13. fdlh iznÙk vfjDr leqPp; X osQ fy, eku yhft, fd ∗ : P(X) × P(X) → P(X), tgk¡
A * B = (A – B) ∪ (B – A), ∀ A, B ∈ P(X) }kjk ifjHkkf"kr gSA fl¼ dhft, fd fjDr
leqPp; φ] lafØ;k ∗ dk rRled gS rFkk P(X) osQ leLr vo;o A O;qRØe.kh; gS]a bl
izdkj fd A–1 = A. (laosQr : (A – φ) ∪ (φ – A) = A. rFkk (A – A) ∪ (A – A) =
A ∗ A = φ).
14. fuEufyf[kr izdkj ls leqPp; {0, 1, 2, 3, 4, 5} esa ,d f}vk/kjh lafØ;k ∗ ifjHkkf"kr dhft,

a + b, ;fn a + b < 6
a ∗b =
a + b − 6, ;fn a + b ≥ 6
fl¼ dhft, fd 'kwU; (0) bl lafØ;k dk rRled gS rFkk leqPp; dk izR;sd vo;o
a ≠ 0 O;qRØe.kh; gS] bl izdkj fd 6 – a, a dk izfrykse gSA

2018-19
 laca/ ,oa iQyu 35

15. eku yhft, fd A = {– 1, 0, 1, 2}, B = {– 4, – 2, 0, 2} vkSj f, g : A → B, Øe'k%

1
f (x) = x2 – x, x ∈ A rFkk g ( x) = 2 x −
− 1, x ∈ A }kjk ifjHkkf"kr iQyu gSaA D;k
2
f rFkk g leku gSa? vius mÙkj dk vkSfpR; Hkh crykb,A (laosQ r : uksV dhft, fd nks
iQyu f : A→ B rFkk g : A → B leku dgykrs gSa ;fn f (a) = g(a) ∀ a ∈ A gksA
16. ;fn A = {1, 2, 3} gks rks ,sls laca/ ftuesa vo;o (1] 2) rFkk (1] 3) gksa vkSj tks LorqY;
rFkk lefer gSa ¯drq laØked ugha gS] dh la[;k gS
(A) 1 (B) 2 (C) 3 (D) 4
17. ;fn A = {1, 2, 3} gks rks vo;o (1, 2) okys rqY;rk laca/ksa dh la[;k gSA
(A) 1 (B) 2 (C) 3 (D) 4
18. eku yhft, fd f : R → R gS rc fuEufyf[kr izdkj ls ifjHkkf"kr fpg~u iQyu (Signum
Function) gSA

 1, x > 0

f ( x ) =  0, x = 0
−1, x < 0

rFkk g : R → R] g (x) = [x], }kjk iznÙk egÙke iw.kkZad iQyu gS] tgk¡ [x]] x ls de ;k
x osQ cjkcj iw.kkZad gS] rks D;k fog rFkk gof , varjky [0, 1] esa laikrh (coincide) gSa?
19. leqPp; {a, b} esa f}vk/kjh lafØ;kvksa dh la[;k gS
(A) 10 (B) 16 (C) 20 (D) 8

lkjka'k
bl vè;k; esa] geus fofo/ izdkj osQ laca/ksa] iQyuksa rFkk f}vk/kjh lafØ;kvksa dk vè;;u fd;k
gSA bl vè;k; dh eq[; fo"k;&oLrq fuEufyf[kr gS%
® X esa] R = φ ⊂ X × X }kjk iznÙk laca/ R] fjDr laca/ gksrk gSA
® X esa] R = X × X }kjk iznÙk laca/ R] lkoZf=kd laca/ gSA
® X esa] ,slk laca/ fd ∀ a ∈ X] (a, a) ∈ R, LorqY; laca/ gSA
® X esa] bl izdkj dk laca/ R, tks izfrca/ (a, b) ∈ R dk rkRi;Z gS fd (b, a) ∈ R
dks larq"V djrk gS lefer laca/ gSA
® X esa] izfrca/ R, (a, b) ∈ R rFkk (b, c) ∈ R ⇒ (a, c) ∈ R ∀ a, b, c ∈ X dks larq"V
djus okyk laca/ R laØked laca/ gSA

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® X esa] laca/ R, tks LorqY;] lefer rFkk laØked gS] rqY;rk laca/ gSA
® X esa] fdlh rqY;rk laca/ R osQ fy, a ∈ X osQ laxr rqY;rk oxZ [a], X dk og
mileqPp; gS ftlosQ lHkh vo;o a ls lacaf/r gSaA
® ,d iQyu f : X → Y ,oSQdh (vFkok ,oSQd) iQyu gS] ;fn
f (x1) = f (x2) ⇒ x1 = x2, ∀ x1, x2 ∈ X
® ,d iQyu f : X → Y vkPNknd (vFkok vkPNknh) iQyu gS] ;fn fdlh iznÙk
y ∈ Y, ∃ x ∈ X, bl izdkj fd f (x) = y
® ,d iQyu f : X → Y ,oSQdh rFkk vkPNknd (vFkok ,oSQdh vkPNknh) iQyu
gS] ;fn f ,oSQdh rFkk vPNknd nksuksa gSA
® iQyu f : A → B rFkk g : B → C dk la;kstu] iQyu gof : A → C gS] tks gof (x)
= g(f (x)), ∀ x ∈ A }kjk iznÙk gSA

® ,d iQyu f : X → Y O;qRØe.kh; gS] ;fn ∃ g : Y → X, bl izdkj fd gof = 1X rFkk

fog = 1Y.

® ,d iQyu f : X → Y O;qRØe.kh; gS] ;fn vkSj osQoy ;fn f ,oSQdh rFkk vkPNknd gSA
® fdlh iznÙk ifjfer leqPp; X osQ fy, iQyu f : X → X ,oSQdh (rnkuqlkj
vkPNknd) gksrk gS] ;fn vkSj osQoy ;fn f vkPnNknd (rnkuqlkj ,oSQdh) gSA ;g
fdlh ifjfer leqPp; dk vfHkyk{kf.kd xq.k/eZ (Characterstic Property) gSA ;g
vifjfer leqPp; osQ fy, lR; ugha gSA
® A esa ,d f}vk/kjh lafØ;k ∗, A × A ls A rd ,d iQyu ∗ gSA
® ,d vo;o e ∈ X, f}vk/kjh lafØ;k ∗ : X × X → X, dk rRled vo;o gS] ;fn
a ∗ e = a = e ∗ a, ∀ a ∈ X

® dksbZ vo;o e ∈ X f}vk/kjh lafØ;k ∗ : X × X → X, osQ fy, O;qRØe.kh;

gksrk gS] ;fn ,d ,sls b ∈ X dk vfLrRo gS fd a ∗ b = e = b ∗ a gS tgk¡
e f}vk/kjh lafØ;k ∗ dk rRled gSA vo;o b, a dk izfrykse dgykrk gS] ftls
a–1 ls fu:fir djrs gSaA

® X dk ,d lafØ;k ∗, Øefofue; gS ;fn a ∗ b = b ∗ a, ∀ a, b ∈ X

® X esa] ,d lafØ;k ∗, lkgp;Z gS ;fn (a ∗ b) ∗ c = a ∗ (b ∗ c), ∀ a, b, c ∈ X

2018-19
 laca/ ,oa iQyu 37

,sfrgkfld i`"BHkwfe
iQyu dh ladYiuk] R. Descartes (lu~ 1596-1650 bZ-) ls izkjaHk gks dj ,d yacs
varjky esa fodflr gqbZ gSA Descartes us lu~ 1637 bZ- esa viuh ikaMqfyfi “Geometrie”
esa 'kCn ^iQyu* dk iz;ksx] T;kferh; oØksa] tSls vfrijoy; (Hyperbola)] ifjoy;
(Parabola) rFkk nh?kZoÙ` k (Ellipse), dk vè;;u djrs le;] ,d pj jkf'k x osQ /u iw.kk±d
?kkr xn osQ vFkZ esa fd;k FkkA James Gregory (lu~ 1636-1675 bZ-) us viuh o`Qfr “ Vera
Circuliet Hyperbolae Quadratura” (lu~ 1667 bZ-) esa] iQyu dks ,d ,slh jkf'k ekuk Fkk]
tks fdlh vU; jkf'k ij chth; vFkok vU; lafØ;kvksa dks mÙkjksÙkj iz;ksx djus ls izkIr gksrh
gSA ckn esa G. W. Leibnitz (1646-1716 bZ-) usa 1673 bZ- esa fyf[kr viuh ikaMqfyfi
“Methodus tangentium inversa, seu de functionibus” esa 'kCn ^iQyu* dks fdlh ,slh
jkf'k osQ vFkZ esa iz;ksx fd;k] tks fdlh oØ osQ ,d fcanq ls nwljs fcanq rd bl izdkj ifjofrZr
gksrh jgrh gS] tSls oØ ij fcanq osQ funsZ'kkad] oØ dh izo.krk] oØ dh Li'khZ rFkk vfHkyac
ifjofrZr gksrs gSaA rFkkfi viuh o`Qfr “Historia” (1714 bZ-) esa Leibnitz us iQyu dks ,d pj
ij vk/kfjr jkf'k osQ :i esa iz;ksx fd;k FkkA okD;ka'k ‘x dk iQyu’ iz;ksx esa ykus okys os
loZizFke O;fDr FksA John Bernoulli (1667-1748 bZ-) us loZizFke 1718 bZ- esa laosQru
(Notation) φx dks okD;ka'k ‘x dk iQyu’ dks izdV djus osQ fy, fd;k FkkA ijarq iQyu
dks fu:fir djus osQ fy, izrhdksa] tSls f, F, φ, ψ ... dk O;kid iz;ksx Leonhard Euler
(1707-1783 bZ-) }kjk 1734 bZ- esa viuh ikaMfq yfi “Analysis Infinitorium” osQ izFke [k.M
esa fd;k x;k FkkA ckn esa Joeph Louis Lagrange (1736-1813 bZ-) us 1793 bZ- esa viuh
ikaMfq yfi “Theorie des functions analytiques” izdkf'kr dh] ftlesa mUgksua s fo'ys"k.kkRed
(Analytic) iQyu osQ ckjs esa ifjppkZ dh Fkh rFkk laosQru f (x), F(x), φ(x) vkfn dk iz;ksx
x osQ fHkUu&fHkUu iQyuksa osQ fy, fd;k FkkA rnksijkar Lejeunne Dirichlet (1805-1859 bZ-) us
iQyu dh ifjHkk"kk nhA ftldk iz;ksx ml le; rd gksrk jgk tc rd orZeku dky esa iQyu
dh leqPp; lS¼kafrd ifjHkk"kk dk izpyu ugha gqvk] tks Georg Cantor (1845-1918 bZ)
}kjk fodflr leqPp; fl¼kar osQ ckn gqvkA orZeku dky esa izpfyr iQyu dh leqPp;
lS¼kafrd ifjHkk"kk Dirichlet }kjk iznÙk iQyu dh ifjHkk"kk dk ek=k vewrhZdj.k
(Abstraction) gSA

—v—

2018-19
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vè;k; 2
izfrykse f=kdks.kferh; iQyu
(Inverse Trigonometric Functions)

vMathematics, in general, is fundamentally the science of

self-evident things— FELIX KLEIN v

2.1 Hkwfedk (Introduction)

vè;k; 1 esa] ge i<+ pqosQ gSa fd fdlh iQyu f dk izrhd
f –1 }kjk fu:fir izfrykse (Inverse) iQyu dk vfLrRo osQoy
rHkh gS ;fn f ,oSQdh rFkk vkPNknd gksA cgqr ls iQyu ,sls gSa
tks ,oSQdh] vkPNknd ;k nksuksa gh ugha gSa] blfy, ge muosQ
izfrykseksa dh ckr ugha dj ldrs gSaA d{kk XI esa] ge i<+ pqosQ gSa
fd f=kdks.kferh; iQyu vius LokHkkfod (lkekU;) izkar vkSj
ifjlj esa ,oSQdh rFkk vkPNknd ugha gksrs gSa vkSj blfy, muosQ
izfrykseksa dk vfLrRo ugha gksrk gSA bl vè;k; esa ge f=kdks.kferh;
iQyuksa osQ izkra ksa rFkk ifjljksa ij yxus okys mu izfrca/ksa (Restrictions)
dk vè;;u djsaxs] ftuls muosQ izfrykseksa dk vfLrRo lqfuf'pr
gksrk gS vkSj vkys[kksa }kjk izfrykseksa dk voyksdu djsaxsA blosQ Arya Bhatta
(476-550 A. D.)
vfrfjDr bu izfrykseksa osQ oqQN izkjafHkd xq.k/eZ (Properties) ij
Hkh fopkj djsaxsA
izfrykse f=kdks.kferh; iQyu] dyu (Calculus) esa ,d egRoiw.kZ Hkwfedk fuHkkrs gSa] D;ksafd
mudh lgk;rk ls vusd lekdy (Integrals) ifjHkkf"kr gksrs gSaA izfrykse f=kdks.kferh; iQyuksa dh
ladYiuk dk iz;ksx foKku rFkk vfHk;kaf=kdh (Engineering) esa Hkh gksrk gSA
2.2 vk/kjHkwr ladYiuk,¡ (Basic Concepts)
d{kk XI, es]a ge f=kdks.kferh; iQyuksa dk vè;;u dj pqoQs gS]a tks fuEufyf[kr izdkj ls ifjHkkf"kr gSa
sine iQyu] vFkkZr~] sin : R → [– 1, 1]
cosine iQyu] vFkkZr~] cos : R → [– 1, 1]

2018-19
 izfrykse f=kdks.kferh; iQyu 39

π
tangent iQyu] vFkkZr~] tan : R – { x : x = (2n + 1) , n ∈ Z} → R
2
cotangent iQyu] vFkkZr~] cot : R – { x : x = nπ, n ∈ Z} → R
π
secant iQyu] vFkkZr~, sec : R – { x : x = (2n + 1) , n ∈ Z} → R – (– 1, 1)
2
cosecant iQyu] vFkkZr~] cosec : R – { x : x = nπ, n ∈ Z} → R – (– 1, 1)
ge vè;k; 1 esa ;g Hkh lh[k pqosQ gSa fd ;fn f : X→Y bl izdkj gS fd f (x) = y ,d
,oSQdh rFkk vkPNknd iQyu gks rks ge ,d vf}rh; iQyu g : Y→X bl izdkj ifjHkkf"kr dj
ldrs gSa fd g (y) = x, tgk¡ x ∈ X rFkk y = f (x), y ∈ Y gSA ;gk¡ g dk izkar = f dk ifjlj
vkSj g dk ifjlj = f dk izkarA iQyu g dks iQyu f dk izfrykse dgrs gSa vkSj bls f –1 }kjk
fu:fir djrs gSaA lkFk gh g Hkh ,oSQdh rFkk vkPNknd gksrk gS vkSj g dk izfrykse iQyu f gksrk
gSa vr% g –1 = (f –1)–1 = f blosQ lkFk gh
–1 –1
(f o f ) (x) = f (f (x)) = f –1(y) = x
vkSj (f o f –1) (y) = f (f –1(y)) = f (x) = y
D;ksafd sine iQyu dk izkar okLrfod la[;kvksa dk leqPp; gS rFkk bldk ifjlj lao`r varjky
−π π
[–1, 1] gSA ;fn ge blosQ izkar dks  ,  esa lhfer (izfrcaf/r) dj nsa] rks ;g ifjlj
 2 2
[– 1, 1] okyk] ,d ,oSQdh rFkk vkPNknd iQyu gks tkrk gSA okLro esa] sine iQyu] varjkyksa

 −3π − π   − π π   π 3π 
 2 , 2  ,  2 , 2  ,  2 , 2  bR;kfn esa] ls fdlh esa Hkh lhfer gksus ls] ifjlj [–1, 1]
okyk] ,d ,oSQdh rFkk vkPNknd iQyu gks tkrk gSA vr% ge buesa ls izR;sd varjky esa] sine
iQyu osQ izfrykse iQyu dks sin–1 (arc sine function) }kjk fu:fir djrs gSaA vr% sin–1 ,d
π 3π
iQyu gS] ftldk izkar [– 1, 1] gS] vkSj ftldk ifjlj  −3π , −π  ,  −π , π  ;k  , 
 2 2   2 2 2 2 
bR;kfn esa ls dksbZ Hkh varjky gks ldrk gSA bl izdkj osQ izR;sd varjky osQ laxr gesa iQyu
 −π π 
sin–1 dh ,d 'kk[kk (Branch) izkIr gksrh gSA og 'kk[kk] ftldk ifjlj  ,  gS] eq[; 'kk[kk
 2 2
(eq[; eku 'kk[kk) dgykrh gS] tc fd ifjlj osQ :i esa vU; varjkyksa ls sin–1 dh fHkUu&fHkUu
'kk[kk,¡ feyrh gSaA tc ge iQyu sin–1 dk mYys[k djrs gSa] rc ge bls izkar [–1, 1] rFkk ifjlj
 −π π   −π π 
 2 , 2  okyk iQyu le>rs gSaA bls ge sin : [–1, 1] →  2 , 2  fy[krs gSaA
–1

2018-19
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izfrykse iQyu dh ifjHkk"kk }kjk] ;g fu"d"kZ fudyrk gS fd sin (sin–1 x) = x , ;fn

π π
– 1 ≤ x ≤1 rFkk sin–1 (sin x) = x ;fn − ≤ x ≤ gSA nwljs 'kCnksa esa] ;fn y = sin–1 x gks rks
2 2
sin y = x gksrk gSA
fVIi.kh
(i) gesa vè;k; 1 ls Kkr gS fd] ;fn y = f (x) ,d O;qRØe.kh; iQyu gS] rks x = f –1 (y) gksrk
gSA vr% ewy iQyu sin osQ vkys[k esa x rFkk y v{kksa dk ijLij fofue; djosQ iQyu
sin–1 dk vkys[k izkIr fd;k tk ldrk gSA vFkkZr~] ;fn (a, b), sin iQyu osQ vkys[k dk
,d fcanq gS] rks (b, a), sin iQyu osQ izfrykse iQyu dk laxr fcanq gksrk gSA vr% iQyu

vko`Qfr 2-1 (i)

vko`Qfr 2-1 (ii) vko`Qfr 2-1 (iii)

2018-19
 izfrykse f=kdks.kferh; iQyu 41

y = sin–1 x dk vkys[k] iQyu y = sin x osQ vkys[k esa x rFkk y v{kksa osQ ijLij fofue;
djosQ izkIr fd;k tk ldrk gSA iQyu y = sin x rFkk iQyu y = sin–1 x osQ vkys[kksa dks
vko`Qfr 2.1 (i), (ii), esa n'kkZ;k x;k gSA iQyu y = sin–1 x osQ vkys[k esa xgjk fpfÉr Hkkx
eq[; 'kk[kk dks fu:fir djrk gSA
(ii) ;g fn[kyk;k tk ldrk gS fd izfrykse iQyu dk vkys[k] js[kk y = x osQ ifjr% (Along)]
laxr ewy iQyu osQ vkys[k dks niZ.k izfrfcac (Mirror Image)] vFkkZr~ ijkorZu
(Reflection) osQ :i esa izkIr fd;k tk ldrk gSA bl ckr dh dYiuk] y = sin x rFkk
y = sin–1 x osQ mUgha v{kksa (Same axes) ij] izLrqr vkys[kksa ls dh tk ldrh gS
(vko`Qfr 2.1 (iii))A
sine iQyu osQ leku cosine iQyu Hkh ,d ,slk iQyu gS ftldk izkar okLrfod la[;kvksa
dk leqPp; gS vkSj ftldk ifjlj leqPp; [–1, 1] gSA ;fn ge cosine iQyu osQ izkar dks varjky
[0, π] esa lhfer dj nsa rks ;g ifjlj [–1, 1] okyk ,d ,oSQdh rFkk vkPNknd iQyu gks tkrk
gSA oLrqr%] cosine iQyu] varjkyksa [– π, 0], [0,π], [π, 2π] bR;kfn esa ls fdlh esa Hkh lhfer
gksus ls] ifjlj [–1, 1] okyk ,d ,oSQdh vkPNknh (Bijective) iQyu gks tkrk gSA vr% ge bu
esa ls izR;sd varjky esa cosine iQyu osQ izfrykse dks ifjHkkf"kr dj ldrs gSaA ge cosine iQyu
osQ izfrykse iQyu dks cos–1 (arc cosine function) }kjk fu:fir djrs
gSaA vr% cos–1 ,d iQyu gS ftldk izkar [–1, 1] gS vkSj ifjlj [–π, 0],
[0, π], [π, 2π] bR;kfn esa ls dksbZ Hkh varjky gks ldrk gSA bl izdkj
osQ izR;sd varjky osQ laxr gesa iQyu cos–1 dh ,d 'kk[kk izkIr gksrh
gSA og 'kk[kk] ftldk ifjlj [0, π] gS] eq[; 'kk[kk (eq[; eku 'kk[kk)
dgykrh gS vkSj ge fy[krs gSa fd
cos–1 : [–1, 1] → [0, π]
y = cos–1 x }kjk iznÙk iQyu dk vkys[k mlh izdkj [khapk tk ldrk
gS tSlk fd y = sin–1 x osQ vkyss[k osQ ckjs esa o.kZu fd;k tk pqdk gSA
y = cos x rFkk y = cos–1 x osQ vkys[kksa dks vko`Qfr;ksa 2.2 (i) rFkk (ii)
esa fn[kyk;k x;k gSA

vko`Qfr 2-2 (i) vko`Qfr 2-2 (ii)

2018-19
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vkb, vc ge cosec–1x rFkk sec–1x ij fopkj djsaA

1
D;ksafd cosec x = , blfy, cosec iQyu dk izkar leqPp; {x : x ∈ R vkSj x ≠ nπ,
sin x
n ∈ Z} gS rFkk ifjlj leqPp; {y : y ∈ R, y ≥ 1 vFkok y ≤ –1}, vFkkZr~] leqPp;
R – (–1, 1) gSA bldk vFkZ gS fd y = cosec x, –1 < y < 1 dks NksM+ dj vU; lHkh okLrfod
ekuksa dks xzg.k djrk gS rFkk ;g π osQ iw.kk±d (Integral) xq.ktksa osQ fy, ifjHkkf"kr ugha gSA ;fn
 π π
ge cosec iQyu osQ izkar dks varjky  − 2 , 2  – {0}, esa lhfer dj nsa] rks ;g ,d ,oSQdh rFkk
 
vkPNknd iQyu gksrk gS] ftldk ifjlj leqPp; R – (– 1, 1). gksrk gSA oLrqr% cosec iQyu]
 −3π −π   −π π   π 3π 
varjkyksa  2 , 2  − {−π} ,  2 , 2  – {0},  2 , 2  − {π} bR;kfn esa ls fdlh esa Hkh
     
lhfer gksus ls ,oSQdh vkPNknh gksrk gS vkSj bldk ifjlj leqPp; R – (–1, 1) gksrk gSA bl izdkj
cosec–1 ,d ,sls iQyu osQ :i esa ifjHkkf"kr gks ldrk gS ftldk izkar R – (–1, 1) gS vkSj ifjlj
−π π
  −3π −π   π 3π 
varjkyksa  2 , 2  − {0} ,  2 , 2  − {−π} ,  ,  − {π} bR;kfn esa ls dksbZ Hkh ,d gks
   2 2 
−π π
ldrk gSA ifjlj  ,  − {0} osQ laxr iQyu dks cosec–1 dh eq[; 'kk[kk dgrs gSaA bl izdkj
 2 2
eq[; 'kk[kk fuEufyf[kr rjg ls O;Dr gksrh gS%

vko`Qfr 2-3 (i) vko`Qfr 2-3 (ii)

2018-19
 izfrykse f=kdks.kferh; iQyu 43

 −π π 
cosec–1 : R – (–1, 1) →  ,  − {0}
 2 2
y = cosec x rFkk y = cosec–1 x osQ vkys[kksa dks vko`Qfr 2-3 (i), (ii) esa fn[kyk;k x;k gSA

1 π
blh rjg] sec x = , y = sec x dk izkar leqPp; R – {x : x = (2n + 1) , n ∈ Z}
cos x 2
gS rFkk ifjlj leq P p; R – (–1, 1) gS A bldk vFkZ gS fd sec (secant) iQyu
–1 < y < 1 dks NksM+dj vU; lHkh okLrfod ekuksa dks xzg.k (Assumes) djrk gS vkSj ;g

π
osQ fo"ke xq.ktksa osQ fy, ifjHkkf"kr ugha gSA ;fn ge secant iQyu osQ izkar dks varjky
2
π
[0, π] – { }, esa lhfer dj nsa rks ;g ,d ,oSQdh rFkk vkPNknd iQyu gksrk gS ftldk ifjlj
2
−π π
leqPp; R – (–1, 1) gksrk gSA okLro esa secant iQyu varjkyksa [–π, 0] – { }, [0, π] –   ,
2 2 
3π
[π, 2π] – { } bR;kfn esa ls fdlh esa Hkh lhfer gksus ls ,oSQdh vkPNknh gksrk gS vkSj bldk
2
ifjlj R– (–1, 1) gksrk gSA vr% sec–1 ,d ,sls iQyu osQ :i esa ifjHkkf"kr gks ldrk gS
−π π
ftldk izkar (–1, 1) gks vkSj ftldk ifjlj varjkyksa [– π, 0] – { }, [0, π] – { },
2 2
3π
[π, 2π] – { } bR;kfn esa ls dksbZ Hkh gks ldrk gSA buesa ls izR;sd varjky osQ laxr gesa iQyu
2
π
sec–1 dh fHkUu&fHkUu 'kk[kk,¡ izkIr gksrh gSaA og 'kk[kk ftldk ifjlj [0, π] – { } gksrk gS]
2
iQyu sec dh eq[; 'kk[kk dgykrh gSA bldks ge fuEufyf[kr izdkj ls O;Dr djrs gSa%
–1

π
sec–1 : R – (–1,1) → [0, π] – { }
2
y = sec x rFkk y = sec–1 x osQ vkys[kksa dks vko`Qfr;ksa 2.4 (i), (ii) esa fn[kyk;k x;k gSA var
esa] vc ge tan–1 rFkk cot–1 ij fopkj djsaxsA
ges a Kkr gS fd] tan iQyu (tangent iQyu ) dk iz k a r leq P p; {x : x ∈ R rFkk
π π
x ≠ (2n +1) , n ∈ Z} gS rFkk ifjlj R gSA bldk vFkZ gS fd tan iQyu osQ fo"ke xq.ktksa
2 2

2018-19
44 xf.kr

vko`Qfr 2-4 (i) vko`Qfr 2-4 (ii)

−π π
osQ fy, ifjHkkf"kr ugha gSA ;fn ge tangent iQyu osQ izkar dks varjky  ,  esa lhfer dj
 2 2
nsa] rks ;g ,d ,oSQdh rFkk vkPNknd iQyu gks tkrk gS ftldk ifjlj leqPp; R gksrk gSA okLro
 −3π −π   −π π   π 3 π 
esa] tangent iQyu] varjkyksa  ,  ,  , ,  ,  bR;kfn esa ls fdlh esa Hkh
 2 2   2 2 2 2 
lhfer gksus ls ,oSQdh vkPNknh gksrk gS vkSj bldk ifjlj leqPp; R gksrk gSA vr,o tan–1 ,d
−3π −π 

,sls iQyu osQ :i esa ifjHkkf"kr gks ldrk gS] ftldk iazkr R gks vkSj ifjlj varjkyksa  2 , 2
,
 

 −π π   π 3π 
 ,  ,  ,  bR;kfn esa ls dksbZ Hkh gks ldrk gSA bu varjkyksa }kjk iQyu tan–1 dh
 2 2 2 2 
 −π π

fHkUu&fHkUu 'kk[kk,¡ feyrh gSaA og 'kk[kk] ftldk ifjlj  2 , 2  gksrk gS] iQyu tan–1 dh
 
eq[; 'kk[kk dgykrh gSA bl izdkj
 −π π 
tan–1 : R →  , 
 2 2

2018-19
 izfrykse f=kdks.kferh; iQyu 45

vko`Qfr 2-5 (i) vko`Qfr 2-5 (ii)

y = tan x rFkk y = tan–1x osQ vkys[kksa dks vko`Qfr;ksa 2.5 (i), (ii) esa fn[kyk;k x;k gSA
gesa Kkr gS fd cot iQyu (cotangent iQyu) dk izkar leqPp; {x : x ∈ R rFkk x ≠ nπ,
n ∈ Z} gS rFkk ifjlj leqPp; R gSA bldk vFkZ gS fd cotangent iQyu] π osQ iw.kk±dh; xq.ktksa

vko`Qfr 2-6 (i) vko`Qfr 2-6 (ii)

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46 xf.kr

osQ fy, ifjHkkf"kr ugha gSA ;fn ge cotangent iQyu osQ izkar dks varjky (0, π) esa lhfer dj
nsa rks ;g ifjlj R okyk ,d ,oSQdh vkPNknh iQyu gksrk gSA oLrqr% cotangent iQyu varjkyksa
(–π, 0), (0, π), (π, 2π) bR;kfn esa ls fdlh esa Hkh lhfer gksus ls ,oSQdh vkPNknh gksrk gS vkSj
bldk ifjlj leqPp; R gksrk gSA okLro esa cot –1 ,d ,sls iQyu osQ :i esa ifjHkkf"kr gks ldrk
gS] ftldk izkar R gks vkSj ifjlj] varjkyksa (–π, 0), (0, π), (π, 2π) bR;kfn esa ls dksbZ Hkh gksA bu
varjkyksa ls iQyu cot –1 dh fHkUu&fHkUu 'kk[kk,¡ izkIr gksrh gSaA og 'kk[kk] ftldk ifjlj (0, π)
gksrk gS] iQyu cot –1 dh eq[; 'kk[kk dgykrh gSA bl izdkj
cot–1 : R → (0, π)
y = cot x rFkk y = cot–1x osQ vkys[kksa dks vko`Qfr;ksa 2.6 (i), (ii) esa iznf'kZr fd;k x;k gSA
fuEufyf[kr lkj.kh esa izfrykse f=kdks.kferh; iQyuksa (eq[; ekuh; 'kk[kkvksa) dks muosQ izkarksa
rFkk ifjljksa osQ lkFk izLrqr fd;k x;k gSA

 π π
sin–1 →  − 2 , 2 
:
[–1, 1]

cos –1 : [–1, 1] → [0, π]

 π π
cosec–1 : R – (–1,1) →  − 2 , 2  – {0}

π
sec –1 : R – (–1, 1) → [0, π] – { }
2
 −π π 
tan–1 : R →  , 
 2 2
cot–1 : R → (0, π)

AfVIi.kh
1
1. sin–1x ls (sin x)–1 dh Hkzkafr ugha gksuh pkfg,A okLro esa (sin x)–1 = vkSj ;g rF;
sin x
vU; f=kdks.kferh; iQyuksa osQ fy, Hkh lR; gksrk gSA
2. tc dHkh izfrykse f=kdks.kferh; iQyuksa dh fdlh 'kk[kk fo'ks"k dk mYys[k u gks] rks gekjk
rkRi;Z ml iQyu dh eq[; 'kk[kk ls gksrk gSA
3. fdlh izfrykse f=kdks.kferh; iQyu dk og eku] tks mldh eq[; 'kk[kk esa fLFkr gksrk gS]
izfrykse f=kdks.kferh; iQyu dk eq[; eku (Principal value) dgykrk gSA

2018-19
 izfrykse f=kdks.kferh; iQyu 47

vc ge oqQN mnkgj.kksa ij fopkj djsaxs%

 1 
mnkgj.k 1 sin–1  2  dk eq[; eku Kkr dhft,A
 

 1  1
gy eku yhft, fd sin–1   = y. vr% sin y = .
 2 2

 –π π
 π 1
gesa Kkr gS fd sin–1 dh eq[; 'kk[kk dk ifjlj  2 , 2  gksrk gS vkSj sin   = gSA
4 2

 1  π
blfy, sin–1   dk eq[; eku gSA
 2 4

 −1 
mnkgj.k 2 cot–1   dk eq[; eku Kkr dhft,A
 3

 −1 
gy eku yhft, fd cot–1   = y . vr,o
 3

−1  π  π  2π 
cot y = = − cot   = cot  π −  = cot   gSA
3 3  3  3 

 2π  −1
gesa Kkr gS fd cot–1 dh eq[; 'kk[kk dk ifjlj (0, π) gksrk gS vkSj cot  = gSA vr%
 3  3

 −1  2π
cot–1   dk eq[; eku gSA
 3 3

iz'ukoyh 2-1
fuEufyf[kr osQ eq[; ekuksa dks Kkr dhft,%

 1  3
1. sin–1  −  2. cos–1  2  3. cosec–1 (2)
 2  

 1
4. tan–1 (− 3) 5. cos–1  −  6. tan–1 (–1)
 2

2018-19
48 xf.kr

 2   1 
7. sec–1   8. cot–1 ( 3) 9. cos–1  − 
 3  2
10. cosec–1 ( − 2 )
fuEufyf[kr osQ eku Kkr dhft,%
 1  1  1  1
11. tan–1(1) + cos–1  −  + sin–1  −  12. cos–1   + 2 sin–1  
 2  2  2  2
13. ;fn sin–1 x = y, rks
π π
(A) 0 ≤ y ≤ π (B) − ≤ y≤
2 2
π π
(C) 0 < y < π (D) − < y<
2 2
14. tan–1 3 − sec −1 ( − 2 ) dk eku cjkcj gS

π π 2π
(A) π (B) − (C) (D)
3 3 3
2.3 izfrykse f=kdks.kferh; iQyuksa osQ xq.k/eZ (Properties of Inverse Trigonometric
Functions)
bl vuqPNsn esa ge izfrykse f=kdks.kferh; iQyuksa osQ oqQN xq.k/eks± dks fl¼ djsaxsA ;gk¡ ;g mYys[k
dj nsuk pkfg, fd ;s ifj.kke] laxr izfrykse f=kdks.kferh; iQyuksa dh eq[; 'kk[kkvksa osQ varxZr
gh oS/ (Valid) gS] tgk¡ dgha os ifjHkkf"kr gSaA oqQN ifj.kke] izfrykse f=kdks.kferh; iQyuksa osQ izkarksa
osQ lHkh ekuksa osQ fy, oS/ ugha Hkh gks ldrs gSaA oLrqr% ;s mu oqQN ekuksa osQ fy, gh oS/ gksaxs]
ftuosQ fy, izfrykse f=kdks.kferh; iQyu ifjHkkf"kr gksrs gSaA ge izkar osQ bu ekuksa osQ foLr`r fooj.k
(Details) ij fopkj ugha djsaxs D;ksafd ,slh ifjppkZ (Discussion) bl ikB~; iqLrd osQ {ks=k ls
ijs gSA
Lej.k dhft, fd] ;fn y = sin–1x gks rks x = sin y rFkk ;fn x = sin y gks rks y = sin–1x
gksrk gSA ;g bl ckr osQ lerqY; (Equivalent) gS fd
 π π
sin (sin–1 x) = x, x ∈ [– 1, 1] rFkk sin–1 (sin x) = x, x ∈  − , 
 2 2
vU; ik¡p izfrykse f=kdks.kferh; iQyuksa osQ fy, Hkh ;gh lR; gksrk gSA vc ge izfrykse
f=kdks.kferh; iQyuksa osQ oqQN xq.k/eks± dks fl¼ djsaxsA

2018-19
 izfrykse f=kdks.kferh; iQyu 49

1
1. (i) sin–1 = cosec–1 x, x ≥ 1 ;k x≤–1
x
1
(ii) cos–1 = sec–1x, x ≥ 1 ;k x ≤ – 1
x

1
(iii) tan–1 = cot–1 x, x > 0
x
igys ifj.kke dks fl¼ djus osQ fy, ge cosec–1 x = y eku ysrs gSa] vFkkZr~
x = cosec y
1
vr,o = sin y
x
1
vr% sin–1 =y
x
1
;k sin–1 = cosec–1 x
x
blh izdkj ge 'ks"k nks Hkkxksa dks fl¼ dj ldrs gSaA
2. (i) sin–1 (–x) = – sin–1 x, x ∈ [– 1, 1]
(ii) tan–1 (–x) = – tan–1 x, x ∈ R
(iii) cosec–1 (–x) = – cosec–1 x, | x | ≥ 1
eku yhft, fd sin–1 (–x) = y, vFkkZr~ –x = sin y blfy, x = – sin y, vFkkZr~
x = sin (–y).
vr% sin–1 x = – y = – sin–1 (–x)
bl izdkj sin–1 (–x) = – sin–1x
blh izdkj ge 'ks"k nks Hkkxksa dks fl¼ dj ldrs gSaA
3. (i) cos–1 (–x) = π – cos–1 x, x ∈ [– 1, 1]
(ii) sec–1 (–x) = π – sec–1 x, | x | ≥ 1
(iii) cot–1 (–x) = π – cot–1 x, x ∈ R
eku yhft, fd cos–1 (–x) = y vFkkZr~ – x = cos y blfy, x = – cos y = cos (π – y)
vr,o cos–1 x = π – y = π – cos–1 (–x)
vr% cos–1 (–x) = π – cos–1 x
blh izdkj ge vU; Hkkxksa dks Hkh fl¼ dj ldrs gSaA

2018-19
50 xf.kr

π
4. (i) sin–1 x + cos–1 x = , x ∈ [– 1, 1]
2
π
(ii) tan–1 x + cot–1 x = ,x∈R
2
π
(iii) cosec–1 x + sec–1 x = , |x| ≥ 1
2
π 
eku yhft, fd sin–1 x = y, rks x = sin y = cos  2 − y 
 
π π
blfy, cos–1 x = −y = − sin –1 x
2 2
π
vr% sin–1 x + cos–1 x =
2
blh izdkj ge vU; Hkkxksa dks Hkh fl¼ dj ldrs gSaA
x+ y
5. (i) tan–1x + tan–1 y = tan–1 , xy < 1
1 – xy
x– y
(ii) tan–1x – tan–1 y = tan–1 , xy > – 1
1 + xy

 x+ y 
(iii) tan–1x + tan–1 y = π + tan–1   , x y > 1, x > 0, y > 0
 1 – xy 
eku yhft, fd tan–1 x = θ rFkk tan–1 y = φ rks x = tan θ rFkk y = tan φ
tan θ + tan φ x+ y
vc tan(θ + φ) = =
1 − tan θ tan φ 1 − xy

x+ y
vr% θ + φ = tan–1 1− xy

x+ y
vr% tan–1 x + tan–1 y = tan–1
1− xy
mi;qZDr ifj.kke esa ;fn y dks – y }kjk izfrLFkkfir (Replace) djsa rks gesa nwljk ifj.kke izkIr
gksrk gS vkSj y dks x }kjk izfrLFkkfir djus ls rhljk ifj.kke izkIr gksrk gSA

2018-19
 izfrykse f=kdks.kferh; iQyu 51

2x
6. (i) 2tan–1 x = sin–1 , |x| ≤ 1
1 + x2

1 – x2
(ii) 2tan–1 x = cos–1 ,x≥ 0
1 + x2

2x
(iii) 2tan–1 x = tan–1 ,–1< x<1
1 – x2
eku yhft, fd tan–1 x = y, rks x = tan y
2x 2 tan y
vc sin–1 2 = sin
–1
1 + tan 2 y
1+ x
= sin–1 (sin 2 y) = 2 y = 2 tan–1 x

1 − x2 1− tan 2 y
blh izdkj cos –1
= cos –1
1 + tan 2 y
= cos–1 (cos 2y) = 2y = 2tan–1 x
1 + x2
vc ge oqQN mnkgj.kksa ij fopkj djsaxsA
mnkgj.k 3 n'kkZb, fd
1 1
(i) sin–1 ( 2 x 1 − x 2 ) = 2 sin–1 x, − ≤ x≤
2 2

1
(ii) sin–1 ( 2 x 1 − x 2 ) = 2 cos–1 x, ≤ x ≤1
2
gy
(i) eku yhft, fd x = sin θ rks sin–1 x = θ bl izdkj

(
sin–1 ( 2 x 1 − x 2 ) = sin–1 2sin θ 1 − sin 2 θ )
= sin–1 (2sinθ cosθ) = sin–1 (sin2θ) = 2θ
= 2 sin–1 x
(ii) eku yhft, fd x = cos θ rks mi;qZDr fof/ osQ iz;ksx }kjk gesa
sin–1 ( 2 x 1 − x 2 ) = 2 cos–1 x izkIr gksrk gSA
1 2 3
mnkgj.k 4 fl¼ dhft, fd tan–1 + tan –1 = tan –1
2 11 4

2018-19
52 xf.kr

gy xq.k/eZ 5 (i), }kjk

1 2
+
1 –1 2 –1 2 11 15 −1 3
ck;k¡ i{k = tan + tan = tan −1 = nk;k¡ i{k
–1
= tan = tan
2 11 1 2 20 4
1− ×
2 11

cos x −3π π
mnkgj.k 5 tan −1 , − < x < dks ljyre :i esa O;Dr dhft,A
1 − sin x 2 2
gy ge fy[k ldrs gSa fd

 x x 
 cos2 − sin 2 
 cos x  2 2
tan −1   = tan 
–1

 1 − sin x  x x x
 cos2 + sin 2 − 2sin cos
x

 2 2 2 2 

 x x  x x 
  cos 2 + sin 2  cos 2 − sin 2  
–1   
= tan 
  x x2 
  cos − sin  
 2 2

 x x  x
 cos + sin   1 + tan 
–1 2 2 = tan –1 2
= tan 
x x  x
 cos − sin  1 − tan 
 2 2  2

–1   π x  π x
= tan  tan  +   = +
  4 2  4 2
fodYir%

 π     π − 2x  
 sin  − x    sin   
 cos x   2   = tan –1   2  
tan –1   = tan –1

 1 − sin x  1 − cos  π − x   1 − cos  π − 2 x  
     
2   2  

2018-19
 izfrykse f=kdks.kferh; iQyu 53

  π − 2x   π − 2x  
 2sin  4  cos  4  
   
= tan 
–1

  π − 2x  
2sin 2  
  4  

–1   π − 2 x   = tan –1 tan π − π − 2 x
= tan  cot  
  4  2 4

–1   π x  π x
= tan  tan  +   = +
  4 2  4 2

 
1
mnkgj.k 6 cot –1   , x > 1 dks ljyre :i esa fyf[k,A
 x −1 
2

gy eku yhft, fd x = sec θ, then x 2 − 1 = sec 2 θ − 1 = tan θ

1
blfy, cot –1 = cot–1 (cot θ) = θ = sec–1 x tks vHkh"V ljyre :i gSA
x −1
2

–1 2x  3 x − x3  1
mnkgj.k 7 fl¼ dhft, fd tan x + tan –1
= tan–1  2 , | x|<
1 − x2  1 − 3 x  3

gy eku yhft, fd x = tan θ. rks θ = tan–1 x gSA vc

3x − x3 3 tan θ − tan 3 θ
nk;k¡ i{k = tan –1 = tan –1
1 − 3x2 1 − 3 tan 2 θ

= tan–1 (tan3θ) = 3θ = 3tan–1 x = tan–1 x + 2 tan–1 x

2x
= tan–1 x + tan–1 = ck;k¡ i{k (D;ksa?)
1 − x2
mnkgj.k 8 cos (sec–1 x + cosec–1 x), | x | ≥ 1 dk eku Kkr dhft,A
 π
gy ;gk¡ ij cos (sec–1 x + cosec–1 x) = cos   = 0
2

2018-19
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iz'ukoyh 2-2

fuEufyf[kr dks fl¼ dhft,%

 1 1
1. 3sin–1 x = sin–1 (3x – 4x3), x ∈  – , 
 2 2

1 
2. 3cos–1 x = cos–1 (4x3 – 3x), x ∈  , 1
2 

2 7 1
3. tan –1 + tan −1 = tan −1
11 24 2
1 1 31
4. 2 tan −1 + tan −1 = tan −1
2 7 17
fuEufyf[kr iQyuksa dks ljyre :i esa fyf[k,%

1 + x2 − 1 1
5. tan −1 ,x≠0 6. tan −1 , |x| > 1
x x2 − 1

 1 − cos x   cos x − sin x  −π 3π

7. tan −1   , 0 < x < π 8. tan −1  , 4 <x < 4
 1 + cos x   cos x + sin x 

x
9. tan −1 , |x| < a
a2 − x2

 3a 2 x − x 3  −a a
10. tan −1  3  , a > 0; < x<
 a − 3ax 
2 3 3

fuEufyf[kr esa ls izR;sd dk eku Kkr dhft,%

µ1   µ1 1  
11. tan  2 cos  2 sin 
2  
12. cot (tan–1a + cot–1a)


1 2x –1 1 − y 
2

13. tan  sin –1 + cos  , | x | < 1, y > 0 rFkk xy < 1

2 1 + x2 1+ y2 

2018-19
 izfrykse f=kdks.kferh; iQyu 55

 µ1 1 
14. ;fn sin  sin + cosµ1 x = 1, rks x dk eku Kkr dhft,A
5 

x −1 x +1 π
15. ;fn tan
–1
+ tan –1 = , rks x dk eku Kkr dhft,A
x−2 x+2 4
iz'u la[;k 16 ls 18 esa fn, izR;sd O;atd dk eku Kkr dhft,%
–1  2π   3π 
16. sin  sin  17. tan –1  tan 
 3   4 

 3 3
18. tan  sin –1 + cot –1 
 5 2

−1  7π 
19. cos  cos  dk eku cjkcj gS
6
7π 5π π π
(A) (B) (C) (D)
6 6 3 6
π 1 
20. sin  − sin −1 ( − )  dk eku gS
3 2 
1 1 1
(A) gS (B) gS (C) gS (D) 1
2 3 4
21. tan −1 3 − cot −1 ( − 3) dk eku
π
(A) πgS (B) − gS (C) 0 gS (D) 2 3
2

fofo/ mnkgj.k
3π
mnkgj.k 9 sin −1 (sin ) dk eku Kkr dhft,A
5
3π 3π
gy gesa Kkr gS fd sin −1 (sin x) = x gksrk gSA blfy, sin −1 (sin )=
5 5
3π  π π 
fdarq ∉ − , , tks sin–1 x dh eq[; 'kk[kk gSA
5  2 2 

2018-19
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3π 3π 2π 2π  π π 
rFkkfi sin ( ) = sin( π − ) = sin rFkk ∈ − ,
5 5 5 5  2 2 
3π −1 2π 2π
vr% sin −1 (sin ) = sin (sin ) =
5 5 5
3 8 84
mnkgj.k 10 n'kkZb, fd sin −1 − sin −1 = cos −1
5 17 85
3
gy eku yhft, fd sin −1 = x vkSj sin −1 8 = y
5 17
3 8
blfy, sin x = rFkk sin y =
5 17
9 4
vc cos x = 1 − sin 2 x = 1 − = (D;ksa?)
25 5

64 15
vkSj cos y = 1 − sin 2 y = 1 − =
289 17
bl izdkj cos (x – y) = cos x cos y + sin x sin y
4 15 3 8 84
= × + × =
5 17 5 17 85

−1  84 
blfy, x – y = cos  
85

3 8 −1 84
vr% sin −1 − sin −1 = cos
5 17 85
12 4 63
mnkgj.k 11 n'kkZb, fd sin −1 + cos −1 + tan −1 =π
13 5 16
12 4 63
gy eku yhft, fd sin −1 = x, cos −1 = y, tan −1 =z
13 5 16
12 4 63
bl izdkj sin x = , cos y = , tan z =
13 5 16
5 3 12 3
blfy, cos x = , sin y = , tan x = vkSj tan y =
13 5 5 4

2018-19
 izfrykse f=kdks.kferh; iQyu 57

12 3
+
tan x + tan y = 5 4 = − 63
vc tan( x + y ) =
1 − tan x tan y 1 − 12 × 3 16
5 4

vr% tan( x + y ) = − tan z

vFkkZr~ tan (x + y) = tan (–z) ;k tan (x + y) = tan (π – z)
blfy, x + y = – z or x + y = π – z
D;ksafd x, y rFkk z /ukRed gSa] blfy, x + y ≠ – z (D;ksa?)

12 4 63
vr% x + y + z = π ;k sin
–1
+ cos –1 + tan –1 =π
13 5 16

 a cos x − b sin x  a
mnkgj.k 12 tan –1   dks ljy dhft,] ;fn b tan x > –1
 b cos x + a sin x 
gy ;gk¡
 a cos x − b sin x   a 
  − tan x 
–1  a cos x − b sin x  b
–1 cos x –1  b
tan   = tan  b cos x + a sin x  = tan  a 
 b cos x + a sin x    1 + tan x 
 b cos x   b 

a a
= tan
–1
− tan –1 (tan x) = tan –1 − x
b b
π
mnkgj.k 13 tan–1 2x + tan–1 3x = dks ljy dhft,A
4
π
gy ;gk¡ fn;k x;k gS fd tan–1 2x + tan–1 3x =
4

 2 x + 3x  π
;k tan –1   =
 1 − 2 x × 3x  4

 5x  π
;k tan –1  2  =
 1 − 6x  4

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5x π
blfy, 2 = tan =1
1 − 6x 4
;k 6x2 + 5x – 1 = 0 vFkkZr~ (6x – 1) (x + 1) = 0
1
ftlls izkIr gksrk gS fd] x= ;k x = – 1
6
D;ksafd x = – 1, iznÙk lehdj.k dks larq"V ugha djrk gS] D;ksafd x = – 1 ls lehdj.k dk
1
ck;k¡ i{k ½.k gks tkrk gSA vr% iznÙk lehdj.k dk gy osQoy x = gSA
6

vè;k; 2 ij fofo/ iz'ukoyh

fuEufyf[kr osQ eku Kkr dhft,%
–1  13π   7π 
1. cos  cos  2. tan –1  tan 
 6   6 
fl¼ dhft,
3 24 8 3 77
3. 2sin –1 = tan –1 4. sin
–1
+ sin –1 = tan –1
5 7 17 5 36

–1 4 12 33 12 3 56
5. cos + cos –1 = cos –1 6. cos
–1
+ sin –1 = sin –1
5 13 65 13 5 65
63 5 3
7. tan –1 = sin –1 + cos –1
16 13 5
1 1 1 1 π
8. tan –1 + tan −1 + tan −1 + tan −1 =
5 7 3 8 4
fl¼ dhft,%
1 1− x
tan µ1 x = cosµ1 
9.
2  1 + x  , x ∈ [0, 1]

µ1
 1 + sin x + 1 − sin x  x  π
10. cot   = 2 , x ∈  0, 
 1 + sin x − 1 − sin x   4

 1+ x − 1− x  π 1 1
11. tan µ1  µ1
 = 4 − 2 cos x , − ≤ x ≤ 1 [laoQs r: x = cos 2θ jf[k,]
 1+ x + 1− x  2

2018-19
 izfrykse f=kdks.kferh; iQyu 59

9π 9 1 9 2 2
12. − sin −1 = sin −1
8 4 3 4 3

fuEufyf[kr lehdj.kksa dks ljy dhft,%

1− x 1
13. 2tan–1 (cos x) = tan–1 (2 cosec x) 14. tan –1 = tan –1 x,( x > 0)
1+ x 2
15. sin (tan–1 x), | x | < 1 cjkcj gksrk gS%
x 1 1 x
(A) (B) (C) (D)
1 − x2 1 − x2 1 + x2 1 + x2
π
16. ;fn sin–1 (1 – x) – 2 sin–1 x = , rks x dk eku cjkcj gS%
2
1 1 1
(A) 0, (B) 1, (C) 0 (D)
2 2 2
x x− y
17. tan −1   − tan −1 dk eku gS%
y x+ y

π π π −3π
(A) gSA (B) gSA (C) gSA (D)
2 3 4 4

lkjka'k
® izfrykse f=kdks.kferh; iQyuksa (eq[; 'kk[kk) osQ izkar rFkk ifjlj fuEufyf[kr lkj.kh esa
of.kZr gSa%
iQyu izkra ifjlj
(eq[; 'kk[kk)
 −π π 
y = sin–1 x [–1, 1]  2 , 2 
y = cos–1 x [–1, 1] [0, π]
 −π π 
y = cosec–1 x R – (–1,1)  2 , 2  – {0}

π
y = sec–1 x R – (–1, 1) [0, π] – { }
2

2018-19
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 π π
y = tan–1 x R − , 
 2 2
y = cot–1 x R (0, π)
1
® sin–1x ls (sin x)–1 dh HkzkfUr ugha gksuh pkfg,A okLro esa (sin x)–1 = vkSj blh
sin x
izdkj ;g rF; vU; f=kdks.kferh; iQyuksa osQ fy, lR; gksrk gSA
® fdlh izfrykse f=kdks.kferh; iQyu dk og eku] tks mldh eq[; 'kk[kk esa fLFkr gksrk
gS] izfrykse f=kdks.kferh; iQyu dk eq[; eku (Principal Value) dgykrk gSA
mi;qDr izkarksa osQ fy,
® y = sin–1 x ⇒ x = sin y ® x = sin y ⇒ y = sin–1 x
® sin (sin–1 x) = x ® sin–1 (sin x) = x
1
® sin–1 = cosec–1 x ® cos–1 (–x) = π – cos–1 x
x
1
® cos–1 = sec–1x ® cot–1 (–x) = π – cot–1 x
x
1
® tan–1 = cot–1 x ® sec–1 (–x) = π – sec–1 x
x

® sin–1 (–x) = – sin–1 x ® tan–1 (–x) = – tan–1 x

π
® tan–1 x + cot–1 x = ® cosec–1 (–x) = – cosec–1 x
2
π π
® sin–1 x + cos–1 x = ® cosec–1 x + sec–1 x =
2 2
x+ y 2x
® tan–1x + tan–1y = tan–1 , xy < 1 ® 2tan–1x = tan–1 |x | < 1
1 − xy 1 − x2
x+ y
® tan–1x + tan–1y = π + tan–1 , xy > 1, x > 0, y > 0
1 − xy
x− y
® tan–1x – tan–1y = tan–1 , xy > –1
1 + xy
2x 1 − x2
® 2tan–1 x = sin–1 = cos–1
,0≤x≤1
1 + x2 1 + x2

2018-19
 izfrykse f=kdks.kferh; iQyu 61

,sfrgkfld i`"BHkwfe
,slk fo'okl fd;k tkrk gS fd f=kdks.kferh dk vè;;u loZizFke Hkkjr esa vkjaHk gqvk
FkkA vk;ZHkV ð (476 bZ-)] czãxqIr (598 bZ-) HkkLdj izFke (600 bZ-) rFkk HkkLdj f}rh;
(1114 bZ-)us izeq[k ifj.kkeksa dks izkIr fd;k FkkA ;g laiw.kZ Kku Hkkjr ls eè;iwoZ vkSj iqu%
ogk¡ ls ;wjksi x;kA ;wukfu;ksa us Hkh f=kdks.kfefr dk vè;;u vkjaHk fd;k ijarq mudh dk;Z
fof/ bruh vuqi;qDr Fkh] fd Hkkjrh; fof/ osQ Kkr gks tkus ij ;g laiw.kZ fo'o }kjk viukbZ
xbZA
Hkkjr esa vk/qfud f=kdks.kferh; iQyu tSls fdlh dks.k dh T;k (sine) vkSj iQyu
osQ ifjp; dk iwoZ fooj.k fl¼kar (laLo`Qr Hkk"kk esa fy[kk x;k T;ksfr"kh; dk;Z) esa fn;k
x;k gS ftldk ;ksxnku xf.kr osQ bfrgkl esa izeq[k gSA
HkkLdj izFke (600 bZ-) us 90° ls vf/d] dks.kksa osQ sine osQ eku osQ y, lw=k fn;k
FkkA lksygoha 'krkCnh dk ey;kye Hkk"kk esa sin (A + B) osQ izlkj dh ,d miifÙk gSA 18°,
36°, 54°, 72°, vkfn osQ sine rFkk cosine osQ fo'kq¼ eku HkkLdj f}rh; }kjk fn, x, gSaA
sin–1 x, cos–1 x, vkfn dks pki sin x, pki cos x, vkfn osQ LFkku ij iz;ksx djus dk
lq>ko T;ksfr"kfon Sir John F.W. Hersehel (1813 bZ-) }kjk fn, x, FksA Å¡pkbZ vkSj nwjh
lacaf/r iz'uksa osQ lkFk Thales (600 bZ- iwoZ) dk uke vifjgk;Z :i ls tqM+k gqvk gSA mUgsa
feJ osQ egku fijkfeM dh Å¡pkbZ osQ ekiu dk Js; izkIr gSA blosQ fy, mUgksaus ,d Kkr
Å¡pkbZ osQ lgk;d naM rFkk fijkfeM dh ijNkb;ksa dks ukidj muosQ vuqikrksa dh rqyuk dk
iz;ksx fd;k FkkA ;s vuqikr gSa
H h
= = tan (lw;Z dk mUurka'k)
S s
Thales dks leqnzh tgk”k dh nwjh dh x.kuk djus dk Hkh Js; fn;k tkrk gSA blosQ
fy, mUgksaus le:i f=kHkqtksa osQ vuqikr dk iz;ksx fd;k FkkA Å¡pkbZ vkSj nwjh lac/h iz'uksa dk
gy le:i f=kHkqtksa dh lgk;rk ls izkphu Hkkjrh; dk;ks± esa feyrs gSaA

—v—

2018-19
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vè;k; 3
vkO;wg (Matrices)

vThe essence of mathematics lies in its freedom — CANTOR v

3.1 Hkwfedk (Introduction)

xf.kr dh fofo/ 'kk[kkvksa esa vkO;wg osQ Kku dh vko';drk iM+rh gSA vkO;wg] xf.kr osQ
lokZf/d 'kfDr'kkyh lk/uksa esa ls ,d gSA vU; lh/h&lknh fof/;ksa dh rqyuk esa ;g xf.krh;
lk/u gekjs dk;Z dks dkiQh gn rd ljy dj nsrk gSA jSf[kd lehdj.kksa osQ fudk; dks gy djus
osQ fy, laf{kIr rFkk ljy fof/;k¡ izkIr djus osQ iz;kl osQ ifj.kkeLo:i vkO;wg dh ladYiuk
dk fodkl gqvkA vkO;wgksa dks osQoy jSf[kd lehdj.kksa osQ fudk; osQ xq.kkadksa dks izdV djus osQ
fy, gh ugha iz;ksx fd;k tkrk gS] vfirq vkO;wgksa dh mi;ksfxrk bl iz;ksx ls dgha vf/d gSA vkO;wg
laosQru rFkk lafØ;kvksa dk iz;ksx O;fDrxr oaQI;wVj osQ fy, bysDVªkfud LizsM'khV izksxzkeksa
(Electronic Spreadsheet Programmes) esa fd;k tkrk gS] ftldk iz;ksx] Øe'k% okf.kT; rFkk
foKku osQ fofHkUu {ks=kksa esa gksrk gS] tSls] ctV (Budgeting)] foØ; cfgosZ'ku (Sales
Projection)] ykxr vkdyu (Cost Estimation)] fdlh iz;ksx osQ ifj.kkeksa dk fo'ys"k.k bR;kfnA
blosQ vfrfjDr vusd HkkSfrd lafØ;k,¡ tSls vko/Zu (Magnification)] ?kw.kZu (Rotation) rFkk
fdlh lery }kjk ijkorZu (Reflection) dks vkO;wgksa }kjk xf.krh; <ax ls fu:fir fd;k tk ldrk
gSA vkO;wgksa dk iz;ksx xw<+ysf[kdh (Cryptography) esa Hkh gksrk gSA bl xf.krh;
lk/u dk iz;ksx u osQoy foKku dh gh oqQN 'kk[kkvksa rd lhfer gS] vfirq bldk iz;ksx
vuqoaf'kdh] vFkZ'kkL=k] vk/qfud euksfoKku rFkk vkS|kSfxd izca/u esa Hkh fd;k tkrk gSA
bl vè;k; esa vkO;wg rFkk vkO;wg chtxf.kr (Matrix algebra) osQ vk/kjHkwr fl¼karksa ls
voxr gksuk] gesa #fpdj yxsxkA
3.2 vkO;wg (Matrix)
eku yhft, fd ge ;g lwpuk O;Dr djuk pkgrs gSa fd jk/k osQ ikl 15 iqfLrdk,¡ gSaA bls ge
[15] :i esa] bl le> osQ lkFk O;Dr dj ldrs gSa] fd [ ] osQ vanj fyf[kr la[;k jk/k osQ ikl
iqfLrdkvksa dh la[;k gSA vc ;fn gesa ;g O;Dr djuk gS fd jk/k osQ ikl 15 iqfLrdk,¡ rFkk 6
dyesa gSa] rks bls ge [15 6] izdkj ls] bl le> osQ lkFk O;Dr dj ldrs gSa fd [ ] osQ vanj
dh izFke izfof"V jk/k osQ ikl dh iqfLrdkvksa dh la[;k] tcfd f}rh; izfof"V jk/k osQ ikl dyeksa

2018-19
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dh la[;k n'kkZrh gSA vc eku yhft, fd ge jk/k rFkk mlosQ nks fe=kksa iQksSft;k rFkk fleju osQ
ikl dh iqfLrdkvksa rFkk dyeksa dh fuEufyf[kr lwpuk dks O;Dr djuk pkgrs gSa%
jk/k osQ ikl 15 iqfLrdk,¡ rFkk 6 dye gSa]
iQkSft;k osQ ikl 10 iqfLrdk,¡ rFkk 2 dye gSa]
fleju osQ ikl 13 iqfLrdk,¡ rFkk 5 dye gSa]
vc bls ge lkjf.kd :i esa fuEufyf[kr izdkj ls O;ofLFkr dj ldrs gSa%
iqfLrdk dye
jk/k 15 6
iQkSft;k 10 2
fleju 13 5
bls fuEufyf[kr <ax ls O;Dr dj ldrs gSa%

vFkok
jk/k iQkSft;k fleju
iqfLrdk 15 10 13
dye 6 2 5
ftls fuEufyf[kr <ax ls O;Dr dj ldrs gSa%

igyh izdkj dh O;oLFkk esa izFke LraHk dh izfof"V;k¡ Øe'k% jk/k] iQkSft;k rFkk fleju osQ
ikl iqfLrdkvksa dh la[;k izdV djrh gSa vkSj f}rh; LraHk dh izfof"V;k¡ Øe'k% jk/k] iQkSft;k rFkk

2018-19
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fleju osQ ikl dyeksa dh la[;k izdV djrh gSaA blh izdkj] nwljh izdkj dh O;oLFkk esa izFke
iaafDr dh izfof"V;k¡ Øe'k% jk/k] iQkSft;k rFkk fleju osQ ikl iqfLrdkvksa dh la[;k izdV djrh
gSaA f}rh; iafDr dh izfof"V;k¡ Øe'k% jk/k] iQkSft;k rFkk fleju osQ ikl dyeksa dh la[;k izdV
djrh gSaA mi;qZDr izdkj dh O;oLFkk ;k izn'kZu dks vkO;wg dgrs gSaA vkSipkfjd :i ls ge vkO;wg
dks fuEufyf[kr izdkj ls ifjHkkf"kr djrs gSa%
ifjHkk"kk 1 vkO;wg la[;kvksa ;k iQyuksa dk ,d vk;rkdkj Øe&foU;kl gSA bu la[;kvksa ;k iQyuksa
dks vkO;wg osQ vo;o vFkok izfof"V;k¡ dgrs gSaA
vkO;wg dks ge vaxzsth o.kZekyk osQ cM+s (Capital) v{kjksa }kjk O;Dr djrs gSaA vkO;wgksa osQ oqQN
mnkgj.k fuEufyf[kr gSa%

 1
– 2 5  2 + i 3 − 2 
    1 + x x3 3 
A= 0 5  , B =  3.5 –1 2  , C =  
   cos x sin x + 2 tan x 
3  5
 6 
 3 5 
 7 

mi;qZDr mnkgj.kksa esa {kSfrt js[kk,¡ vkO;wg dh iafDr;k¡ (Rows) vksj ÅèoZ js[kk,¡ vkO;wg osQ
LraHk (Columns) dgykrs gSaA bl izdkj A esa 3 iafDr;k¡ rFkk 2 LraHk gSa vkSj B esa 3 iafDr;k¡ rFkk
3 LraHk tcfd C esa 2 iafDr;k¡ rFkk 3 LraHk gSaA
3.2.1 vkO;wg dh dksfV (Order of a matrix)
m iafDr;ksa rFkk n LraHkksa okys fdlh vkO;wg dks m × n dksfV (order) dk vkO;wg vFkok osQoy
m × n vkO;wg dgrs gSa A vr,o vkO;wgksa osQ mi;ZqDr mnkgj.kksa osQ lanHkZ esa A, ,d 3 × 2 vkO;wg]
B ,d 3 × 3 vkO;wg rFkk C, ,d 2 × 3 vkO;wg gSaA ge ns[krs gSa fd A esa 3 × 2 = 6 vo;o gS
vkSj B rFkk C esa Øe'k% 9 rFkk 6 vo;o gSaA
lkekU;r%] fdlh m × n vkO;wg dk fuEufyf[kr vk;krkdkj Øe&foU;kl gksrk gS%

vFkok A = [aij]m × n, 1≤ i ≤ m, 1≤ j ≤ n tgk¡ i, j ∈ N

2018-19
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bl izdkj ioha iafDr osQ vo;o ai1, ai2, ai3,..., ain gSa] tcfd josa LraHk osQ vo;o a1j, a2j,
a3j,..., amj gSaA
lkekU;r% aij, ioha iafDr vkSj josa LraHk esa vkus okyk vo;o gksrk gSA ge bls A dk (i, j)ok¡
vo;o Hkh dg ldrs gSaA fdlh m × n vkO;wg esa vo;oksa dh la[;k mn gksrh gSA

AfVIi.kh bl vè;k; esa]

1. ge fdlh m × n dksfV osQ vkO;wg dks izdV djus osQ fy,] laoQs r A = [aij]m × n dk iz;ksx
djsxa As
2. ge osQoy ,sls vkO;wgksa ij fopkj djsaxs] ftuosQ vo;o okLrfod la[;k,¡ gSa vFkok
okLrfod ekuksa dks xzg.k djus okys iQyu gSaA
ge ,d lery osQ fdlh fcanq (x, y) dks ,d vkO;wg (LraHk vFkok iafDr) }kjk izdV dj
 x 0
ldrs gSa] tSls   (vFkok [x, y])ls] mnkgj.kkFkZ] fcanq P(0, 1), vkO;wg fu:i.k esa P =   ;k
 y 1 
[0 1] }kjk izdV fd;k tk ldrk gSA

è;ku nhft, fd bl izdkj ge fdlh can jSf[kd vko`Qfr osQ 'kh"kks± dks ,d vkO;wg osQ :i
esa fy[k ldrs gSaA mnkgj.k osQ fy, ,d prqHkZt ABCD ij fopkj dhft,] ftlosQ 'kh"kZ Øe'k%
A (1, 0), B (3, 2), C (1, 3), rFkk D (–1, 2) gSaA
vc] prqHkZqt ABCD vkO;wg :i esa fuEufyf[kr izdkj ls fu:fir fd;k tk ldrk gS%

A B C D A 1 0
1 3 1 −1 B 3 2 
X= ;k Y= 
0 2 3 2 2 × 4 C 1 3
 
D −1 2  4× 2

vr% vkO;wgksa dk iz;ksx fdlh lery esa fLFkr T;kferh; vko`Qfr;ksa osQ 'kh"kks± dks fu:fir djus
osQ fy, fd;k tk ldrk gSA
vkb, vc ge oqQN mnkgj.kksa ij fopkj djsaA
mnkgj.k 1 rhu iSQfDVª;ksa I, II rFkk III esa iq#"k rFkk efgyk dfeZ;ksa ls lacaf/r fuEufyf[kr lwpuk
ij fopkj dhft,%

2018-19
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iq#"k dehZ efgyk dehZ

I 30 25
II 25 31
III 27 26
mi;qZDr lwpuk dks ,d 3 × 2 vkO;wg esa fu:fir dhft,A rhljh iafDr vkSj nwljs LraHk okyh
izfof"V D;k izdV djrh gS\
gy iznÙk lwpuk dks 3 × 2 vkO;wg osQ :i esa fuEufyf[kr izdkj ls fu:fir fd;k tk ldrk gS%
30 25
A =  25 31
27 26 
rhljh iafDr vkSj nwljs LraHk dh izfof"V iSQDVªh&III dkj[kkus esa efgyk dk;ZdrkZvksa dh la[;k
izdV djrh gSA
mnkgj.k 2 ;fn fdlh vkO;wg esa 8 vo;o gSa] rks bldh laHko dksfV;k¡ D;k gks ldrh gSa\
gy gesa Kkr gS fd] ;fn fdlh vkO;wg dh dksfV m × n gS rks blesa mn vo;o gksrs gSaA vr,o
8 vo;oksa okys fdlh vkO;wg osQ lHkh laHko dksfV;k¡ Kkr djus osQ fy, ge izko`Qr la[;kvksa osQ
mu lHkh Øfer ;qXeksa dks Kkr djsaxs ftudk xq.kuiQy 8 gSA
vr% lHkh laHko Øfer ;qXe (1, 8), (8, 1), (4, 2), (2, 4) gSaA
vr,o laHko dksfV;k¡ 1 × 8, 8 ×1, 4 × 2, 2 × 4 gSaA
1
mnkgj.k 3 ,d ,sls 3 × 2 vkO;wg dh jpuk dhft,] ftlosQ vo;o aij = | i − 3 j | }kjk
2
iznÙk gSaA
 a11 a12 
gy ,d 3 × 2 vkO;wg] lkekU;r% bl izdkj gksrk gS% A =  a21 a22 
 a31 a32 

1
vc] aij = | i − 3 j | , i = 1, 2, 3 rFkk j = 1, 2
2
blfy,
1 1 5
a11 = |1 − 3.1| = 1 a12 = |1 − 3.2 | =
2 2 2

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1 1 1
a21 = | 2 − 3.1| = a22 = | 2 − 3.2 | = 2
2 2 2
1 1 3
a31 = | 3 − 3.1| = 0 a32 = | 3 − 3.2 | =
2 2 2

1 5
 2
1 
vr% vHkh"V vkO;wg A =  2  gSA
2 3
0 
 2

3.3 vkO;wgksa osQ izdkj (Types of Matrices)

bl vuqPNsn esa ge fofHkUu izdkj osQ vkO;wgksa dh ifjppkZ djsaxsA
(i) LraHk vkO;wg (Column matrix)
,d vkO;wg] LraHk vkO;wg dgykrk gS] ;fn mlesa osQoy ,d LraHk gksrk gSA mnkgj.k osQ

 0 
 
 3
fy, A =  −1  , 4 × 1 dksfV dk ,d LraHk vkO;wg gSA O;kid :i ls] A= [aij] m × 1 ,d
 
1/ 2 

m × 1 dksfV dk LraHk vkO;wg gSA

(ii) iafDr vkO;wg (Row matrix)
,d vkO;wg] iafDr vkO;wg dgykrk gS] ;fn mlesa osQoy ,d iafDr gksrh gSA

mnkgj.k osQ fy, B = − 1 

5 2 3 , 1×4 dksfV dk ,d iafDr vkO;wg gSA O;kid
 2 1× 4
:i ls] B = [bij] 1 × n ,d 1 × n dksfV dk iafDr vkO;wg gSA

(iii) oxZ vkO;wg (Square matrix)

,d vkO;wg ftlesa iafDr;ksa dh la[;k LraHkksa dh la[;k osQ leku gksrh gS] ,d oxZ vkO;wg
dgykrk gSA vr% ,d m × n vkO;wg] oxZ vkO;wg dgykrk gS] ;fn m = n vkSj mls dksfV

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 3 −1 0
3 
‘n’ dk oxZ vkO;wg dgrs gSaA mnkgj.k osQ fy, A =  3 2 1  ,d 3 dksfV dk oxZ
2 
4 −1
 3

vkO;wg gSA O;kid :i ls A = [aij] m × m ,d m dksfV dk oxZ vkO;wg gSA

AfVIi.kh ;fn A = [aij] ,d n dksfV dk oxZ vkO;wg gS] rks vo;oksa (izfof"V;k¡)
a11, a22, ..., ann dks vkO;wg A osQ fod.kZ osQ vo;o dgrs gSaA

1 −3 1 
vr% ;fn A =  2 4 −1 gS rks A osQ fod.kZ osQ vo;o 1] 4] 6 gSaA
 3 5 6 

(iv) fod.kZ vkO;wg (Diagonal matrix)

,d oxZ vkO;wg B = [bij] m × m fod.kZ vkO;wg dgykrk gS] ;fn fod.kZ osQ vfrfjDr blosQ
vU; lHkh vo;o 'kwU; gksrs gSa vFkkZr~] ,d vkO;wg B = [bij] m × m fod.kZ vkO;wg dgykrk
gS] ;fn bij = 0, tc i ≠ j gksA
 −1.1 0 0 
 −1 0  
mnkgj.kkFkZ A = [4], B =  , C= 0 2 0  , Øe'k% dksfV 1] 2 rFkk 3 osQ
 0 2   0 0 3
fod.kZ vkO;wg gSaA
(v) vfn'k vkO;wg (Scalar matrix)
,d fod.kZ vkO;wg] vfn'k vkO;wg dgykrk gS] ;fn blosQ fod.kZ osQ vo;o leku gksrs
gSa] vFkkZr~] ,d oxZ vkO;wg B = [bij] n × n vfn'k vkO;wg dgykrk gS] ;fn
bij = 0, tc i ≠ j
bij = k, tc i = j, tgk¡ k dksbZ vpj gSA
mnkgj.kkFkZ]
 3 0 0
 −1 0   
A = [3], B= , C= 0 3 0  Øe'k%
 0 −1  
0 0 3

dksfV 1] 2 rFkk 3 osQ vfn'k vkO;wg gSaA

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(vi) rRled vkO;wg (Identity matrix)

,d oxZ vkO;wg] ftlosQ fod.kZ osQ lHkh vo;o 1 gksrs gSa rFkk 'ks"k vU; lHkh vo;o 'kwU;
gksrs gSa] rRled vkO;wg dgykrk gSA nwljs 'kCnksa esa] oxZ vkO;wg A = [aij] n × n ,d rRled
1 ;fn i = j
vkO;wg gS] ;fn aij = 
0 ;fn i ≠ j
ge] n dksfV osQ rRled vkO;wg dks In }kjk fu:fir djrs gSaA tc lanHkZ ls dksfV Li"V gksrh
gS] rc bls ge osQoy I ls izdV djrs gSaA
1 0 0 
1 0  0 1 0 
mnkgj.k osQ fy, [1], 0 1  ,   Øe'k% dksfV 1, 2 rFkk 3 osQ rRled vkO;wg gSaA
  0 0 1 
 

è;ku nhft, fd ;fn k = 1 gks rks, ,d vfn'k vkO;wg] rRled vkO;wg gksrk gS] ijarq izR;sd
rRled vkO;wg Li"Vr;k ,d vfn'k vkO;wg gksrk gSA
(vii) 'kwU; vkO;wg (Zero matrix)
,d vkO;wg] 'kwU; vkO;wg vFkok fjDr vkO;wg dgykrk gS] ;fn blosQ lHkh vo;o 'kwU;
gksrs gSaA
0 0
 
0 0 0
mnkgj.kkFkZ] [0],   ,  , [0, 0] lHkh 'kwU; vkO;wg gSaA ge 'kwU; vkO;wg dks
0 0 0 0 0
O }kjk fu:fir djrs gSaA budh dksfV;k¡] lanHkZ }kjk Li"V gksrh gSaA
3.3.1 vkO;wgksa dh lekurk (Equality of matrices)
ifjHkk"kk 2 nks vkO;wg A = [aij] rFkk B = [bij] leku dgykrs gSa] ;fn
(i) os leku dksfV;ksa osQ gksrs gksa] rFkk

(ii) A dk izR;sd vo;o] B osQ laxr vo;o osQ leku gks] vFkkZr~ i rFkk j osQ lHkh ekuksa osQ
fy, aij = bij gksa
 2 3  2 3  3 2  2 3
mnkgj.k osQ fy,]   rFkk  0 1 leku vkO;wg gSa ¯drq  0 1  rFkk  0 1
leku
 0 1     
vkO;wg ugha gSaA izrhdkRed :i esa] ;fn nks vkO;wg A rFkk B leku gSa] rks ge bls
A = B fy[krs gSaA

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 x y   −1.5 0 
;fn  z a  =  2 
6  , rks x = – 1.5, y = 0, z = 2, a = 6 , b = 3, c = 2
b c  3 2 

 x + 3 z + 4 2 y − 7  0 6 3y − 2 
 −6 a −1  
0  = − 6 −3 2c + 2
mnkgj.k 4 ;fn 
b − 3 − 21 0   2b + 4 − 21 0 
gks rks a, b, c, x, y rFkk z osQ eku Kkr dhft,A
gy pw¡fd iznÙk vkO;wg leku gSa] blfy, buosQ laxr vo;o Hkh leku gksaxsA laxr vo;oksa dh
rqyuk djus ij gesa fuEufyf[kr ifj.kke izkIr gksrk gS%
x + 3 = 0, z + 4 = 6, 2y – 7 = 3y – 2
a – 1 = – 3, 0 =2c + 2 b – 3 = 2b + 4,
bUgsa ljy djus ij gesa izkIr gksrk gS fd
a = – 2, b = – 7, c = – 1, x = – 3, y = –5, z = 2

 2a + b a − 2b  4 −3
mnkgj.k 5 ;fn 5c − d 4c + 3d  = 11 24  gks rks a, b, c, rFkk d osQ eku Kkr dhft,A
   

gy nks vkO;wgksa dh lekurk dh ifjHkk"kk }kjk] laxr vo;oksa dks leku j[kus ij gesa izkIr gksrk
gS fd
2a + b = 4 5c – d = 11
a – 2b = – 3 4c + 3d = 24
bu lehdj.kksa dks ljy djus ij a = 1, b = 2, c = 3 rFkk d = 4 izkIr gksrk gSA

iz'ukoyh 3-1

2 5 19 −7 
 5 
1. vkO;wg A =  35 −2 12  , osQ fy, Kkr dhft,%
 2 
 
 3 1 −5 17 
(i) vkO;wg dh dksfV (ii) vo;oksa dh la[;k
(iii) vo;o a13, a21, a33, a24, a23

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2. ;fn fdlh vkO;wg esa 24 vo;o gSa rks bldh laHko dksfV;k¡ D;k gSa\ ;fn blesa 13 vo;o
gksa rks dksfV;k¡ D;k gksaxh\
3. ;fn fdlh vkO;wg esa 18 vo;o gSa rks bldh laHko dksfV;k¡ D;k gSa\ ;fn blesa 5 vo;o
gksa rks D;k gksxk\
4. ,d 2 × 2 vkO;wg A = [aij] dh jpuk dhft, ftlosQ vo;o fuEufyf[kr izdkj ls iznÙk gSa
(i + j ) 2 i (i + 2 j ) 2
(i) aij = (ii) aij = (iii) aij =
2 j 2
5. ,d 3 × 4 vkO;wg dh jpuk dhft, ftlosQ vo;o fuEufyf[kr izdkj ls izkIr gksrs gSa%
1
(i) aij = | −3i + j | (ii) aij = 2i − j
2
6. fuEufyf[kr lehdj.kksa ls x, y rFkk z osQ eku Kkr dhft,%

 x + y + z  9 
 4 3  y z  x + y 2  6 2   x + z  = 5
(i)  =  (ii)  =
xy  5 8 
(iii)    
 x 5  1 5 5 + z
 y + z  7 

 a − b 2 a + c   −1 5 
7. lehdj.k  =  ls a, b, c rFkk d osQ eku Kkr dhft,A
 2a − b 3c + d   0 13
8. A = [aij]m × n\ ,d oxZ vkO;wg gS ;fn
(A) m < n (B) m > n (C) m = n (D) buesa ls dksbZ ugha
9. x rFkk y osQ iznÙk fdu ekuksa osQ fy, vkO;wgksa osQ fuEufyf[kr ;qXe leku gSa\
3x + 7 5  0 y − 2 
 y + 1 2 − 3 x  , 8 4 
  
−1
(A) x = , y=7 (B) Kkr djuk laHko ugha gS
3
−2 −1 −2
(C) y = 7 , x= (D) x = , y= .
3 3 3
10. 3 × 3 dksfV osQ ,sls vkO;wgksa dh oqQy fdruh la[;k gksxh ftudh izR;sd izfof"V 0 ;k 1 gS?
(A) 27 (B) 18 (C) 81 (D) 512
3.4 vkO;wgksa ij lafØ;k,¡ (Operations on Matrices)
bl vuqPNsn esa ge vkO;wgksa ij oqQN lafØ;kvksa dks izLrqr djsaxs tSls vkO;wgksa dk ;ksx] fdlh vkO;wg
dk ,d vfn'k ls xq.kk] vkO;wgksa dk O;odyu rFkk xq.kk%

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3.4.1 vkO;wgksa dk ;ksx (Addition of matrices)

eku yhft, fd iQkfrek dh LFkku A rFkk LFkku B ij nks iSQfDVª;k¡ gSaA izR;sd iSQDVªh esa yM+dksa
rFkk yM+fd;ksa osQ fy,] [ksy osQ twrs] rhu fHkUu&fHkUu ewY; oxks±] Øe'k% 1] 2 rFkk 3 osQ curs
gSaA izR;sd iSQDVªh esa cuus okys twrksa dh la[;k uhps fn, vkO;wgksa }kjk fu:fir gSa%

eku yhft, fd iQkfrek izR;sd ewY; oxZ esa cuus okys [ksy osQ twrksa dh oqQy la[;k tkuuk
pkgrh gSaA vc oqQy mRiknu bl izdkj gS%
ewY; oxZ 1 : yM+dksa osQ fy, (80 + 90), yM+fd;ksa osQ fy, (60 + 50)
ewY; oxZ 2 : yM+dksa osQ fy, (75 + 70), yM+fd;ksa osQ fy, (65 + 55)
ewY; oxZ 3 : yM+dksa osQ fy, (90 + 75), yM+fd;ksa osQ fy, (85 + 75)
80 + 90 60 + 50
vkO;wg osQ :i esa bls bl izdkj izdV dj ldrs gSa 75 + 70 65 + 55
90 + 75 85 + 75 

;g u;k vkO;wg] mi;qDZ r nks vkO;wgksa dk ;ksxiQy gSA ge ns[krs gSa fd nks vkO;wgksa dk ;ksxiQy,
iznÙk vkO;wgksa osQ laxr vo;oksa dks tksM+us ls izkIr gksus okyk vkO;wg gksrk gSA blosQ vfrfjDr] ;ksx
osQ fy, nksuksa vkO;wgksa dks leku dksfV dk gksuk pkfg,A

a a a  b b b 
bl izdkj] ;fn A =  11 12 13  ,d 2 × 3 vkO;wg gS rFkk B =  11 12 13  ,d
 a21 a22 a23  b21 b22 b23 

 a11 + b11 a12 + b12 a13 + b13 

vU; 2 × 3 vkO;wg gS] rks ge A + B =   }kjk ifjHkkf"kr djrs gSAa
 a21 + b21 a22 + b22 a23 + b23 
O;kid :i ls] eku yhft, fd A = [aij] rFkk B = [bij] nks leku dksfV] m × n okys vkO;wg
gSa rks A rFkk B nksuksa vkO;wgksa dk ;ksxiQy] vkO;wg C = [cij]m × n, }kjk ifjHkkf"kr gksrk gS] tgk¡
cij = aij + bij, i rFkk j osQ lHkh laHko ekuksa dks O;Dr djrk gSA

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2 5 1
 3 1 − 1  
mnkgj.k 6 A =   rFkk B =  1  gS rks A + B Kkr dhft,A
 2 3 0  −2 3
 2 

gy D;ksafd A rFkk B leku dksfV 2 × 3 okys vkO;wg gSa] blfy, A rFkk B dk ;ksx ifjHkkf"kr
gS] vkSj

2 + 3 1 + 5 1 − 1  2 + 3 1 + 5 0 
A+B =   
1 = 

1  }kjk izkIr gksrk gSA
2 − 2 3+3 0+ 0 6
 2  2 

AfVIi.kh
1. ge bl ckr ij cy nsrs gSa fd ;fn A rFkk B leku dksfV okys vkO;wg ugha gSa rks

2 3 1 2 3
A + B ifjHkkf"kr ugha gSA mnkgj.kkFkZ A =   , B=  , rks A + B ifjHkkf"kr
1 0 1 0 1 
ugha gSA
2. ge ns[krs gSa fd vkO;wgksa dk ;ksx] leku dksfV okys vkO;wgksa osQ leqPp; esa f}vk/kjh
lafØ;k dk ,d mnkgj.k gSA

3.4.2 ,d vkO;wg dk ,d vfn'k ls xq.ku (Multiplication of a matrix by a scalar)

vc eku yhft, fd i+Qkfrek us A ij fLFkr iSQDVªh esa lHkh ewY; oxZ osQ mRiknu dks nks xquk dj
fn;k gS (lanHkZ 3-4-1)

A ij fLFkr iSQDVªh esa mRiknu dh la[;k uhps fn, vkO;wg esa fn[kykbZ xbZ gSA

A ij fLFkr iSQDVªhs esa mRikfnr u;h (cnyh gqbZ) la[;k fuEufyf[kr izdkj gS%

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yM+ osQ yM+ fd;k¡

1  2 × 80 2 × 60
2  2 × 75 2 × 65
 
3  2 × 90 2 × 85 

160 120 
bls vkO;wg :i esa , 150 130  izdkj ls fu:fir dj ldrs gSaA ge ns[krs gSa fd ;g
180 170 
u;k vkO;wg igys vkO;wg osQ izR;sd vo;o dks 2 ls xq.kk djus ij izkIr gksrk gSA
O;kid :i esa ge] fdlh vkO;wg osQ ,d vfn'k ls xq.ku dks] fuEufyf[kr izdkj ls ifjHkkf"kr
djrs gSaA ;fn A = [aij] m × n ,d vkO;wg gS rFkk k ,d vfn'k gS rks kA ,d ,slk vkO;wg gS ftls
A osQ izR;sd vo;o dks vfn'k k ls xq.kk djosQ izkIr fd;k tkrk gSA
nwljs 'kCnksa esa] kA = k [aij] m × n = [k (aij)] m × n, vFkkZr~ kA dk (i, j)ok¡ vo;o] i rFkk j
osQ gj laHko eku osQ fy,] kaij gksrk gSA

 3 1 1.5
 
mnkgj.k osQ fy,] ;fn A =  5 7 −3  gS rks
2 0 5
 

 3 1 1.5  9 3 4.5
   
3A = 3  5 7 −3  = 3 5 21 −9 
2 0 5  6 0 15 
  
vkO;wg dk ½.k vkO;wg (Negative of a matrix) fdlh vkO;wg A dk ½.k vkO;wg –A
ls fu:fir gksrk gSA ge –A dks – A = (– 1) A }kjk ifjHkkf"kr djrs gSaA
 3 1
mnkgj.kkFkZ] eku yhft, fd A =   , rks – A fuEufyf[kr izdkj ls izkIr gksrk gS
 −5 x 

 3 1   −3 −1
– A = (– 1) A = (−1)  = 
 −5 x   5 − x 
vkO;wgksa dk varj (Difference of matrices) ;fn A = [aij], rFkk B = [bij] leku dksfV
m × n okys nks vkO;wg gSa rks budk varj A – B] ,d vkO;wg D = [dij] tgk¡ i rFkk j osQ leLr

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ekuksa osQ fy, dij = aij – bij gS, }kjk ifjHkkf"kr gksrk gSA nwljs 'kCnksa esa] D = A – B = A + (–1) B,
vFkkZr~ vkO;wg A rFkk vkO;wg – B dk ;ksxiQyA
1 2 3  3 −1 3 
mnkgj.k 7 ;fn A =  2 3 1 rFkk B =  −1 0 2 gSa rks 2A – B Kkr dhft,A
   
gy ge ikrs gSa
1 2 3  3 −1 3 
2A – B = 2  − 
 2 3 1  −1 0 2

 2 4 6  −3 1 −3
=  + 
 4 6 2   1 0 −2 

 2 − 3 4 + 1 6 − 3   −1 5 3
=  = 
 4 + 1 6 + 0 2 − 2  5 6 0 

3.4.3 vkO;wgksa osQ ;ksx osQ xq.k/eZ (Properties of matrix addition)

vkO;wgksa osQ ;ksx dh lafØ;k fuEufyf[kr xq.k/eks± (fu;eksa) dks larq"V djrh gS%
(i) Øe&fofues; fu;e (Commutative Law) ;fn A = [aij], B = [bij] leku dksfV
m × n, okys vkO;wg gSa] rks A + B = B + A gksxkA
vc A + B = [aij] + [bij] = [aij + bij]
= [bij + aij] (la[;kvksa dk ;ksx Øe&fofues; gSA)
= ([bij] + [aij]) = B + A
(ii) lkgp;Z fu;e (Associative Law) leku dksfV m × n okys fdUgha Hkh rhu vkO;wgksa
A = [aij], B = [bij], C = [cij] osQ fy, (A + B) + C = A + (B + C)
vc (A + B) + C = ([aij] + [bij]) + [cij]
= [aij + bij] + [cij] = [(aij + bij) + cij]
= [aij + (bij + cij)] (D;ksa ?)
= [aij] + [(bij + cij)] = [aij] + ([bij] + [cij]) = A + (B + C)
(iii) ;ksx osQ rRled dk vfLrRo (Existence of additive identity) eku yhft, fd
A = [aij] ,d m × n vkO;wg gS vkSj O ,d m × n 'kwU; vkO;wg gS] rks A+O = O+A= A
gksrk gSA nwljs 'kCnksa esa] vkO;wgksa osQ ;ksx lafØ;k dk rRled 'kwU; vkO;wg O gSA
(iv) ;ksx osQ izfrykse dk vfLrRo (The existence of additive inverse) eku yhft,
fd A = [aij]m × n ,d vkO;wg gS] rks ,d vU; vkO;wg – A = [– aij]m × n bl izdkj dk gS

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fd A + (– A) = (– A) + A= O] vr,o vkO;wg – A, vkO;wg A dk ;ksx osQ varxZr

izfrykse vkO;wg vFkok ½.k vkO;wg gSA
3.4.4 ,d vkO;wg osQ vfn'k xq.ku osQ xq.k/eZ (Properties of scalar multiplication of
a matrix)
;fn A = [aij] rFkk B = [bij] leku dksfV m × n, okys nks vkO;wg gSa vkSj k rFkk l vfn'k gSa] rks
(i) k(A +B) = k A + kB, (ii) (k + l)A = k A + l A
vc] A = [aij]m × n, B = [bij]m × n, vkSj k rFkk l vfn'k gSa] rks
(i) k (A + B) = k ([aij] + [bij])
= k [aij + bij] = [k (aij + bij)] = [(k aij) + (k bij)]
= [k aij] + [k bij] = k [aij] + k [bij] = kA + kB
(ii) ( k + l) A = (k + l) [aij]
= [(k + l) aij] = [k aij] + [l aij] = k [aij] + l [aij] = k A + l A.

8 0   2 −2
mnkgj.k 8 ;fn A =  4 −2 , B =  4 2  rFkk 2A + 3X = 5B fn;k gks rks vkO;wg X
 
 3 6   −5 1 
Kkr dhft,A
gy fn;k gS 2A + 3X = 5B
;k 2A + 3X – 2A = 5B – 2A
;k 2A – 2A + 3X = 5B – 2A (vkO;wg ;ksx Øe&fofues; gS)
;k O + 3X = 5B – 2A (– 2A, vkO;wg 2A dk ;ksx izfrykse gS)
;k 3X = 5B – 2A (O, ;ksx dk rRled gS)

1
;k X= (5B – 2A)
3
  2 −2  8 0     10 −10   −16 0  
1      1    −8 4  
;k X =  5  4 2  − 2  4 −2   =   20 10  +  
3 3
  −5 1 

 3 6     −25 5   −6 −12  

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 −10 
 −2 3 
 10 − 16 −10 + 0  − 6 −10   
1  1   4 14 
=  20 − 8 10 + 4  = 3 12 14  = 
3 3 
 −25 − 6 5 − 12   −31 −7   
 −31 −7 
 3 3 

5 2 3 6 
mnkgj.k 9 X rFkk Y, Kkr dhft,] ;fn X + Y = 0 9  rFkk X − Y =   gSA
   0 −1

5 2 3 6 
gy ;gk¡ ij (X + Y) + (X – Y) =  + 
 0 9   0 −1

8 8 8 8 
;k (X + X) + (Y – Y) =   ⇒ 2X =  
0 8 0 8 

1 8 8 4 4 
;k X= 0 8 = 0 4 
2    

5 2   3 6 
lkFk gh (X + Y) – (X – Y) =  − 
 0 9   0 −1

5 − 3 2 − 6  2 − 4
;k (X – X) + (Y + Y) =   ⇒ 2Y =  
 0 9 + 1  0 10 

1  2 − 4   1 −2 
;k Y= =
2  0 10   0 5 

mnkgj.k 10 fuEufyf[kr lehdj.k ls x rFkk y osQ ekuksa dks Kkr dhft,%

x 5   3 − 4  7 6 
2 +
y − 3 1 2  15 14 
=
7

gy fn;k gS
x 5   3 − 4  7 6  2x 10   3 − 4   7 6 
2 + ⇒ 
7
  =  
y − 3 1 2  15 14  + =
2  15 14
14 2 y − 6 1

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2x + 3 10 − 4  7 6 2x + 3 6  7 6
;k  14 + 1 2 y − 6 + 2  =  ⇒  =
  
15 14   15 2 y − 4  15 14 
;k 2x + 3 = 7 rFkk 2y – 4 = 14 (D;ksa?)
;k 2x = 7 – 3 rFkk 2y = 18
4 18
;k x= rFkk y=
2 2
vFkkZr~ x=2 rFkk y=9

mnkgj.k 11 nks fdlku jkefd'ku vkSj xqjpju flag osQoy rhu izdkj osQ pkoy tSls cklerh]
ijey rFkk umjk dh [ksrh djrs gSaA nksuksa fdlkuksa }kjk] flracj rFkk vDrwcj ekg esa] bl izdkj osQ
pkoy dh fcØh (#i;ksa esa) dks] fuEufyf[kr A r Fkk B vkO;wgksa esa O;Dr fd;k x;k gS%

(i) izR;sd fdlku dh izR;sd izdkj osQ pkoy dh flracj rFkk vDrwcj dh lfEefyr fcØh
Kkr dhft,A
(ii) flracj dh vis{kk vDrwcj esa gqbZ fcØh esa deh Kkr dhft,A

(iii) ;fn nksuksa fdlkuksa dks oqQy fcØh ij 2% ykHk feyrk gS] rks vDrwcj esa izR;sd izdkj osQ
pkoy dh fcØh ij izR;sd fdlku dks feyus okyk ykHk Kkr dhft,A
gy
(i) izR;sd fdlku dh izR;sd izdkj osQ pkoy dh flracj rFkk vDrwcj esa izR;sd izdkj osQ
pkoy dh fcØh uhps nh xbZ gS%

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(ii) flracj dh vis{kk vDrwcj esa gqbZ fcØh esa deh uhps nh xbZ gS]

2
(iii) B dk 2% = × B = 0. 02 × B
100

= 0.02

vr% vDrwcj ekg esa] jkefd'ku] izR;sd izdkj osQ pkoy dh fcØh ij Øe'k% `100]
`200] rFkk `120 ykHk izkIr djrk gS vkSj xqjpju flag] izR;sd izdkj osQ pkoy dh fcØh ij
Øe'k% `400] `200 rFkk `200 ykHk vftZr djrk gSA
3.4.5 vkO;wgksa dk xq.ku (Multiplication of matrices)
eku yhft, fd ehjk vkSj unhe nks fe=k gSaA ehjk 2 dye rFkk 5 dgkuh dh iqLrosaQ [kjhnuk pkgrh
gSa] tc fd unhe dks 8 dye rFkk 10 dgkuh dh iqLrdksa dh vko';drk gSA os nksuksa ,d nqdku
ij (dher) Kkr djus osQ fy, tkrs gSa] tks fuEufyf[kr izdkj gS%
dye & izR;sd `5] dgkuh dh iqLrd & izR;sd `50 gSA
mu nksuksa esa ls izR;sd dks fdruh /ujkf'k [kpZ djuh iM+sxh\ Li"Vr;k] ehjk dks
`(5 × 2 + 50 × 5) vFkkZr]~ `260 dh vko';drk gS] tcfd unhe dks `(8 × 5 + 50 × 10) vFkkZr~
`540 dh vko;drk gSA ge mi;qDZ r lwpuk dks vkO;wg fu:i.k esa fuEufyf[kr izdkj ls izdV dj
ldrs gS%

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vko';drk izfr ux nke (#i;ksa esa) vko';d /ujkf'k (#i;ksa esas)

2 5  5  5 × 2 + 5 × 50   260 
 8 10   50  8 × 5 + 10 × 50 =  540 
       
eku yhft, fd muosQ }kjk fdlh vU; nqdku ij Kkr djus ij Hkko fuEufyf[kr izdkj gSa%
dye & izR;sd ` 4] dgkuh dh iqLrd & izR;sd `40
vc] ehjk rFkk unhe }kjk [kjhnkjh djus osQ fy, vko';d /ujkf'k Øe'k% `(4 × 2 + 40 × 5)
= ` 208 rFkk `(8 × 4 + 10 × 40) = `432 gSA
iqu% mi;qZDr lwpuk dks fuEufyf[kr <ax ls fu:fir dj ldrs gSa%
vko';drk izfr ux nke (#i;ksa esa) vko';d /ujkf'k (#i;ksa esas)
2 5  4  4 × 2 + 40 × 5   208
 8 10   40   8 × 4 + 10 × 4 0 =  432 
       
vc] mi;qZDr nksuksa n'kkvksa esa izkIr lwpukvksa dks ,d lkFk vkO;wg fu:i.k }kjk fuEufyf[kr
izdkj ls izdV dj ldrs gSa%
vko';drk izfr ux nke (#i;ksa esa) vko';d /ujkf'k (#i;ksa esas)
2 5  5 4  5 × 2 + 5 × 50 4 × 2 + 40 × 5 
 8 10  50 40  8 × 5 + 10 × 5 0 8 × 4 + 10 × 4 0
     

 260 208
=  
 540 432 
mi;qZDr fooj.k vkO;wgksa osQ xq.ku dk ,d mnkgj.k gSA ge ns[krs gSa fd vkO;wgksa A rFkk B osQ
xq.ku osQ fy,] A esa LraHkksa dh la[;k B esa iafDr;ksa dh la[;k osQ cjkcj gksuh pkfg,A blosQ vfrfjDr
xq.kuiQy vkO;wg (Product matrix) osQ vo;oksa dks izkIr djus osQ fy,] ge A dh iafDr;ksa rFkk
B osQ LraHkksa dks ysdj] vo;oksa osQ Øekuqlkj (Element–wise) xq.ku djrs gSa vkSj rnksijkar bu
xq.kuiQyksa dk ;ksxiQy Kkr djrs gSaA vkSipkfjd :i ls] ge vkO;wgksa osQ xq.ku dks fuEufyf[kr rjg
ls ifjHkkf"kr djrs gSa%
nks vkO;wgksa A rFkk B dk xq.kuiQy ifjHkkf"kr gksrk gS] ;fn A esa LraHkksa dh la[;k] B esa iafDr;ksa
dh la[;k osQ leku gksrh gSA eku yhft, fd A = [aij] ,d m × n dksfV dk vkO;wg gS vkSj
B = [bjk] ,d n × p dksfV dk vkO;wg gSA rc vkO;wgksa A rFkk B dk xq.kuiQy ,d m × p dksfV
dk vkO;wg C gksrk gSA vkO;wg C dk (i, k)ok¡ vo;o cik izkIr djus osQ fy, ge A dh i oha iafDr
vkSj B osQ kosa LraHk dks ysrs gS vkSj fiQj muosQ vo;oksa dk Øekuqlkj xq.ku djrs gSaA rnksijkUr bu
lHkh xq.kuiQyksa dk ;ksxiQy Kkr dj ysrs gSaA nwljs 'kCnksa esa ;fn]

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 vkO;wg 81

A = [aij]m × n, B = [bjk]n × p gS rks A dh i oha iafDr [ai1 ai2 ... ain] rFkk B dk kok¡ LraHk

 b1k 
b  n
 .2 k 
 .  gS,a rc cik = ai1 b1k + ai2 b2k + ai3 b3k + ... + ain bnk = ∑ aij b jk
j =1
 . 
b 
 nk 

vkO;wg C = [cik]m × p, A rFkk B dk xq.kuiQy gSA

 2 7
1 −1 2
mnkgj.k osQ fy,] ;fn C =   rFkk D =
 −1 1  gS rks
 
0 3 4   5 − 4

2 7
 1 −1 2  
xq.kuiQy CD ifjHkkf"kr gS rFkk CD = 0 3 4  −1 1  ,d 2 × 2 vkO;wg gS ftldh
 5 −4
izR;sd izfof"V C dh fdlh iafDr dh izfof"V;ksa dh D osQ fdlh LraHk dh laxr izfof"V;ksa osQ
xq.kuiQyksa osQ ;ksxiQy osQ cjkcj gksrh gSA bl mnkgj.k esa ;g pkjksa ifjdyu fuEufyf[kr gSa]

13 −2 
vr% CD =  
17 −13

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6 9  2 6 0
mngkj.k 12 ;fn A =  2 3 rFkk B = 7 9 8  gS rks AB Kkr dhft,A
   

gy vkO;wg A esa 2 LraHk gSa tks vkO;wg B dh iafDr;ksa osQ leku gSaA vr,o AB ifjHkkf"kr gSA vc
 6( 2) + 9(7) 6(6) + 9(9) 6(0) + 9(8) 
AB =  
 2(2) + 3(7) 2(6) + 3(9) 2(0) + 3(8) 

12 + 63 36 + 81 0 + 72   75 117 72 
=   =  
 4 + 21 12 + 27 0 + 24  25 39 24 

A fVIi.kh ;fn AB ifjHkkf"kr gS rks ;g vko';d ugha gS fd BA Hkh ifjHkkf"kr gksA mi;qDZ r
mnkgj.k esa AB ifjHkkf"kr gS ijarq BA ifjHkkf"kr ugha gS D;ksafd B esa 3 LraHk gSa tcfd A esa osQoy
2 iafDr;k¡ (3 iafDr;k¡ ugha) gSaA ;fn A rFkk B Øe'k% m × n rFkk k × l dksfV;ksa osQ vkO;wg
gSa rks AB rFkk BA nksuksa gh ifjHkkf"kr gSa ;fn vkSj osQoy ;fn n = k rFkk l = m gksA fo'ks"k
:i ls] ;fn A vkSj B nksuksa gh leku dksfV osQ oxZ vkO;wg gSa] rks AB rFkk BA nksuksa ifjHkkf"kr
gksrs gSaA
vkO;wgksa osQ xq.ku dh vØe&fofues;rk (Non-Commutativity of multiplication of matrices)
vc ge ,d mnkgj.k osQ }kjk ns[ksaxs fd] ;fn AB rFkk BA ifjHkkf"kr Hkh gksa] rks ;g vko';d
ugha gS fd AB = BA gksA
2 3
 1 −2 3 
mnkgj.k 13 ;fn A =   vkSj B =  4 5 , rks AB rFkk BA Kkr dhft,A n'kkZb, fd
− 
 4 2 5   2 1
AB ≠ BA
gy D;ksafd fd A ,d 2 × 3 vkO;wg gS vkSj B ,d 3 × 2 vkO;wg gS] blfy, AB rFkk BA nksuksa
gh ifjHkkf"kr gSa rFkk Øe'k% 2 × 2 rFkk 3 × 3, dksfV;ksa osQ vkO;wg gSaA uksV dhft, fd
2 3
 1 −2 3 4  2−8+6 3 − 10 + 3   0 − 4 
5 =  =
−8 + 8 + 10 −12 + 10 + 5 10 3 
AB = 
− 4 2 5   

 2 1

 2 3  2 − 12 − 4 + 6 6 + 15  −10 2 21
   1 −2 3     
vkSj BA =  4 5   = 4 − 20 −8 + 10 12 + 25 = −16 2 37 
−
 2 1 
4 2 5  
 2 −4 −4+ 2 6 + 5   −2 −2 11 
Li"Vr;k AB ≠ BA.

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mi;qZDr mnkgj.k esa AB rFkk BA fHkUu&fHkUu dksfV;ksa osQ vkO;wg gSa vkSj blfy, AB ≠ BA
gSA ijarq dksbZ ,slk lksp ldrk gS fd ;fn AB rFkk BA nksuksa leku dksfV osQ gksrs rks laHkor% os
leku gkasxsA ¯drq ,slk Hkh ugha gSA ;gk¡ ge ,d mnkgj.k ;g fn[kykus osQ fy, ns jgs gSa fd ;fn
AB rFkk BA leku dksfV osQ gksa rks Hkh ;g vko';d ugha gS fd os leku gksaA
1 0 0 1 
gS rks AB =   0 1
mnkgj.k 14 ;fn A =   rFkk B =   
0 −1 1 0   −1 0 
0 −1
vkSj BA =   gSA Li"Vr;k AB ≠ BA gSA
1 0
vr% vkO;wg xq.ku Øe&fofues; ugha gksrk gSA

AfVIi.kh bldk rkRi;Z ;g ugha gS fd A rFkk B vkO;wgksa osQ mu lHkh ;qXeksa osQ fy,] ftuosQ
fy, AB rFkk BA ifjHkkf"kr gS] AB ≠ BA gksxkA mnkgj.k osQ fy,
1 0  3 0  3 0
;fn A =   , B = 0 , rks AB = BA =
0 2   4  0 8 
 
è;ku nhft, fd leku dksfV osQ fod.kZ vkO;wgksa dk xq.ku Øe&fofues; gksrk gSA
nks 'kwU;srj vkO;wgksa osQ xq.kuiQy osQ :i esa 'kwU; vkO;wg% (Zero matrix as the product
of two non-zero matrices)
gesa Kkr gS fd nks okLrfod la[;kvksa a rFkk b osQ fy,] ;fn ab = 0 gS rks ;k rks a = 0 vFkok
b = 0 gksrk gSA ¯drq vkO;wgksa osQ fy, ;g vfuok;Zr% lR; ugha gksrk gSA bl ckr dks ge ,d mnkgj.k
}kjk ns[ksaxsA
 0 −1 3 5 
mnkgj.k 15 ;fn A =   rFkk B =   gS rks AB dk eku Kkr dhft,
0 2 0 0 

 0 −1   3 5   0 0 
gy ;gk¡ ij AB =   = 
0 2   0 0   0 0 
vr% ;fn nks vkO;wgksa dk xq.kuiQy ,d 'kwU; vkO;wg gS rks vko';d ugha gS fd muesa ls ,d
vkO;wg vfuok;Zr% 'kwU; vkO;wg gksA
3.4.6 vkO;wgksa osQ xq.ku osQ xq.k/eZ (Properties of multiplication of matrices)
vkO;wgksa osQ xq.ku osQ xq.k/eks± dk ge uhps fcuk mudh miifÙk fn, mYys[k dj jgs gSa%
1. lkgp;Z fu;e% fdUgha Hkh rhu vkO;wgksa A, B rFkk C osQ fy,
(AB) C = A (BC), tc dHkh lehdj.k osQ nksuksa i{k ifjHkkf"kr gksrs gSaA

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2. forj.k fu;e % fdUgha Hkh rhu vkO;wgksa A, B rFkk C osQ fy,

(i) A (B+C) = AB + AC
(ii) (A+B) C = AC + BC, tc Hkh lehdj.k osQ nksuksa i{k ifjHkkf"kr gksrs gSaA
3. xq.ku osQ rRled dk vfLrRo % izR;sd oxZ vkO;wg A osQ fy, leku dksfV osQ ,d vkO;wg
I dk vfLrRo bl izdkj gksrk gS] fd IA = AI = A
vc ge mnkgj.kksa osQ }kjk mi;qZDr xq.k/ek± dk lR;kiu djsaxsA

 1 1 −1 1 3
1 2 3 − 4
mnkgj.k 16 ;fn A =  2 0 3  , B =  0
  2 rFkk C =  rks A(BC)
  2 0 − 2 1
 3 −1 2   −1 4
rFkk (AB)C Kkr dhft, vkSj fn[kykb, fd (AB)C = A(BC) gSA

 1 1 −1  1 3  1 + 0 + 1 3 + 2 − 4   2 1 
gy ;gk¡ AB =  2 0 3  0 2 =  2 + 0 − 3 6 + 0 + 12 = −1 18
 3 −1 2  −1 4  3 + 0 − 2 9 − 2 + 8   1 15

 2 1  2+2 4 + 0 6 − 2 − 8+1 
−1 18 1 2 3 − 4 =  −1 + 36 −2 + 0 −3 − 36 4 + 18
(AB) (C) =    2 0 −2 1  
 1 15   
 1 + 30 2 + 0 3 − 30 − 4 + 15

4 4 4 −7 
35 −2 −39 22 
=  
31 2 −27 11

 1 3  1 + 6 2 + 0 3 − 6 −4 + 3 
1 2 3 −4  
vc BC =  0


2   =  0 + 4 0 + 0 0 − 4 0 + 2 
 −1 2 0 −2 1 
4   −1 + 8 −2 + 0 −3 − 8 4 + 4 

7 2 −3 −1 
 
=  4 0 −4 2 
7 −2 −11 8 

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 1 1 −1   7 2 −3 −1 
vr,o A(BC) =  2 0 3   4 0 − 4 2 
   
 3 −1 2   7 −2 −11 8 

 7 + 4 − 7 2 + 0 + 2 −3 − 4 + 11 −1 + 2 − 8 
 
= 14 + 0 + 21 4 + 0 − 6 − 6 + 0 − 33 −2 + 0 + 24
 21 − 4 + 14 6 + 0 − 4 −9 + 4 − 22 −3 − 2 + 16 

4 4 4 −7 
35 −2 −39 22 
=  
31 2 −27 11

Li"Vr;k] (AB) C = A (BC)

 0 6 7 0 1 1  2 
mnkgj.k 17 ;fn A =  − 6 0 8  , B = 1 0 2  , C = −2
   

 7 − 8 0  1 2 0   3 

rks AC, BC rFkk (A + B)C dk ifjdyu dhft,A ;g Hkh lR;kfir dhft, fd

(A + B) C = AC + BC

 0 7 8
gy A + B = −5 0 10
 8 − 6 0 

 0 7 8 2   0 − 14 + 24  10 
 10  −2  = −10 + 0 + 30  =  20
vr,o] (A + B) C = −5 0      
 8 − 6 0   3   16 + 12 + 0   28

0 6 7   2   0 − 12 + 21  9
blosQ vfrfjDr AC = − 6 0 8  −2  = −12 + 0 + 24
  = 12 
       
 7 − 8 0  3   14 + 16 + 0  30

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0 1 1   2   0 − 2 + 3  1 
    =  2 + 0 + 6 =  8 
vkSj BC = 1 0 2  −2     
1 2 0   3   2 − 4 + 0 − 2 

9   1  10 
     
blfy, AC + BC = 12  +  8  = 20
30  −2   28

Li"Vr;k (A + B) C = AC + BC

1 2 3
mnkgj.k 18 ;fn A =  3 −2 1 gS rks n'kkZb, fd A3 – 23A – 40I = O
 4 2 1

1 2 3  1 2 3 19 4 8
gy ge tkurs gSa fd A = A.A = 3 −2
2
1  3 −2 1 = 1 12 8 
4 2 1  4 2 1 14 6 15 

1 2 3 19 4 8   63 46 69
 1 1 12 8  =  69 − 6 23
blfy, A3 = A A2 =  3 −2    
 4 2 1 14 6 15   92 46 63

 63 46 69 1 2 3  1 0 0
 69 − 6 23 – 23  3 −2 1 – 40 0 1 0

vc A – 23A – 40I = 
3
    
 92 46 63  4 2 1 0 0 1

 63 46 69  −23 − 46 − 69 − 40 0 0 

 69 − 6 23 +  − 69 46 −23  +  0 − 40 0 
=      
 92 46 63  − 92 − 46 − 23  0 0 − 40 

 63 − 23 − 40 46 − 46 + 0 69 − 69 + 0   0 0 0
 69 − 69 + 0 − 6 + 46 − 40 23 − 23 + 0   0 0 0 = O
=   =  
 92 − 92 + 0 46 − 46 + 0 63 − 23 − 40 0 0 0

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mnkgj.k 19 fdlh fo/ku lHkk pquko osQ nkSjku ,d jktuSfrd ny us vius mEehnokj osQ izpkj
gsrq ,d tu laioZQ iQeZ dks BsosQ ij vuqcaf¼r fd;kA izpkj gsrq rhu fof/;ksa }kjk laioZQ LFkkfir
djuk fuf'pr gqvkA ;s gSa% VsyhiQksu }kjk] ?kj&?kj tkdj rFkk ipkZ forj.k }kjkA izR;sd laioZQ dk
'kqYd (iSlksa esa) uhps vkO;wg A esa O;Dr gS]

izfr laioZQ ewY;

 40  VsyhiQksu }kjk
A=   ?kj tkdj
 100 
 50  ipkZ }kjkk

X rFkk Y nks 'kgjksa esa] izR;sd izdkj osQ lEidks± dh la[;k vkO;wg

VsyhiQksu ?kj tkdj ipkZ }kjk

1000 500 5000  → X
B=  esa O;Dr gSA X rFkk Y 'kgjksa esa jktuSfrd ny }kjk O;; dh
3000 1000 10, 000 → Y

xbZ oqQy /ujkf'k Kkr dhft,A

gy ;gk¡ ij
 40,000 + 50, 000 + 250,000  → X
BA =  
120,000 + 100,000 +500,000  → Y

 340, 000  → X
=  
 720,000  → Y
vr% ny }kjk nksuksa 'kgjksa esa O;; dh xbZ oqQy /ujkf'k Øe'k% 3]40]000 iSls o 7]20]000
iSls vFkkZr~ Rs 3400 rFkk Rs 7200 gSaA

iz'ukoyh 3-2

2 4  1 3 −2 5
1. eku yhft, fd A =   ,B=  ,C= , rks fuEufyf[kr Kkr dhft,%
3 2  −2 5 3 4
(i) A + B (ii) A – B (iii) 3A – C
(iv) AB (v) BA

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2. fuEufyf[kr dks ifjdfyr dhft,%

a b  a b a2 + b2 b2 + c 2   2ab 2bc 
(i)  + (ii)  2 + 
−b a   b a   a + c
2
a + b  −2ac −2ab 
2 2

−1 4 − 6 12 7 6
 cos 2 x sin 2 x   sin 2 x cos 2 x 
(iii)  8 5 16  +  8 0 5 (iv)  + 2 
   2 2 2
 2 8 5   3 2 4  sin x cos x  cos x sin x 

3. funf'kZr xq.kuiQy ifjdfyr dhft,%

1 
a b   a −b     1 −2  1 2 3
(i)     (ii)  2  [2 3 4] (iii)   
−b a  b a   2 3   2 3 1
 3 

 2 3 4  1 −3 5 2 1
3  1 0 1
(iv)  3 4 5  0 2 4 (v)  2 
   −1 2 1
 4 5 6  3 0 5  −1 1 

 2 −3
 3 −1 3  
(vi)    1 0  .
−1 0 2 
 3 1 
1 2 −3  3 −1 2 4 1 2
   5 rFkk C =  0 3
 2 , rks (A+B) rFkk
4. ;fn A = 5 0 2  , B =  4 2   
1 −1 1   2 0 3  1 −2 3

(B – C) ifjdfyr dhft,A lkFk gh lR;kfir dhft, fd A + (B – C) = (A + B) – C.

2 5 2 3 
3 1  5 1
3 5
   
1 2 4
rFkk B = 
1 2 4
5. ;fn A =   , rks 3A – 5B ifjdfyr dhft,A
3 3 3 5 5 5
   
 7 2 7 6 2
2
 3 3   5 5 5 

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 cos θ sin θ   sin θ − cos θ 

6. ljy dhft,] cosθ   + sinθ 
 − sin θ cos θ   cos θ sin θ

7. X rFkk Y Kkr dhft, ;fn

7 0  3 0
(i) X + Y =   rFkk X Y =  
 2 5  0 3

 2 3  2 −2
(ii) 2X + 3Y =   rFkk 3X + 2Y =  
 4 0  −1 5 

3 2   1 0
8. X rFkk Y Kkr dhft, ;fn Y =   rFkk 2X + Y =  −3 2 
1 4   

1 3   y 0 5 6 
9. x rFkk y Kkr dhft, ;fn 2 0 x  +  1 2  = 1 8 
     
10. iznÙk lehdj.k dks x, y, z rFkk t osQ fy, gy dhft, ;fn
x z 1 −1  3 5
2  +3  =3 
y t 0 2   4 6

2  −1  10 
11. ;fn x   + y   =   gS rks x rFkk y osQ eku Kkr dhft,A
3   1  5 

x y  x 6   4 x + y
12. ;fn 3   =  + gS rks x, y, z rFkk w osQ ekuksa dks Kkr
 z w  −1 2 w  z + w 3 
dhft,A
cos x − sin x 0 
13. ;fn F ( x ) =  sin x cos x 0  gS rks fl¼ dhft, fd F(x) F(y) = F(x + y)
 
 0 0 1 
14. n'kkZb, fd
 5 −1  2 1   2 1   5 −1
(i)   ≠  
6 7  3 4  3 4  6 7 

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1 2 3 −1 1 0   −1 1 0  1 2 3 
  1  ≠  0 −1 1  0 1 0 
(ii) 0 1 0   0 −1
1 1 0   2 3 4   2 3 4 1 1 0 

2 0 1
15. ;fn A =  2 1 3 gS rks A2 – 5A + 6I, dk eku Kkr dhft,A
1 −1 0 

1 0 2 
16. ;fn A = 0 2 1  gS rks fl¼ dhft, fd A3 – 6A2 + 7A + 2I = 0
 
2 0 3

 3 −2 1 0 
17. ;fn A =   rFkk I =   ,oa A = kA – 2I gks rks k Kkr dhft,A
2

 4 −2   0 1 

 α
 0 − tan
2
18. ;fn A =   rFkk I dksfV 2 dk ,d rRled vkO;wg gSA rks fl¼ dhft,
 tan α 0 
 2 

 cos α − sin α 
fd I + A = (I – A) 
 sin α cos α 
19. fdlh O;kikj la?k osQ ikl 30]000 #i;ksa dk dks"k gS ftls nks fHkUu&fHkUu izdkj osQ ckaMksa
esa fuosf'kr djuk gSA izFke ckaM ij 5% okf"kZd rFkk f}rh; ckaM ij 7% okf"kZd C;kt izkIr
gksrk gSA vkO;wg xq.ku osQ iz;ksx }kjk ;g fu/kZfjr dhft, fd 30]000 #i;ksa osQ dks"k dks
nks izdkj osQ ckaMksa esa fuos'k djus osQ fy, fdl izdkj ck¡Vas ftlls O;kikj la?k dks izkIr oqQy
okf"kZd C;kt
(a) Rs 1800 gksA (b) Rs 2000 gksA
20. fdlh LowQy dh iqLrdksa dh nqdku esa 10 ntZu jlk;u foKku] 8 ntZu HkkSfrd foKku rFkk
10 ntZu vFkZ'kkL=k dh iqLrosaQ gSaA bu iqLrdksa dk foØ; ewY; Øe'k% Rs 80] Rs 60 rFkk
Rs 40 izfr iqLrd gSA vkO;wg chtxf.kr osQ iz;ksx }kjk Kkr dhft, fd lHkh iqLrdksa dks
cspus ls nqdku dks oqQy fdruh /ujkf'k izkIr gksxhA
eku yhft, fd X, Y, Z, W rFkk P Øe'k% 2 × n, 3 × k, 2 × p, n × 3 rFkk p × k, dksfV;ksa
osQ vkO;wg gSaA uhps fn, iz'u la[;k 21 rFkk 22 esa lgh mÙkj pqfu,A

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21. PY + WY osQ ifjHkkf"kr gksus osQ fy, n, k rFkk p ij D;k izfrca/ gksxk\
(A) k = 3, p = n (B) k LosPN gS , p = 2
(C) p LosPN gS, k = 3 (D) k = 2, p = 3
22. ;fn n = p, rks vkO;wg 7X – 5Z dh dksfV gSA
(A) p × 2 (B) 2 × n (C) n × 3 (D) p × n
3.5. vkO;wg dk ifjorZ (Transpose of a Matrix)
bl vuqPNsn esa ge fdlh vkO;wg osQ ifjorZ rFkk oqQN fo'ks"k izdkj osQ vkO;wgksa] tSls lefer
vkO;wg (Symmetric Matrix) rFkk fo"ke lefer vkO;wg (Skew Symmetric Matrix) osQ
ckjs esa tkusaxsA
ifjHkk"kk 3 ;fn A = [aij] ,d m × n dksfV dk vkO;wg gS rks A dh iafDr;ksa rFkk LraHkksa dk ijLij
fofue; (Interchange) djus ls izkIr gksus okyk vkO;wg A dk ifjorZ (Transpose) dgykrk
gSA vkO;wg A osQ ifjorZ dks A′ (;k AT) ls fu:fir djrs gSaA nwljs 'kCnksa esa] ;fn
A = [aij]m × n, rks A′ = [aji]n × mgksxkA mnkgj.kkFkZ] ;fn

 3 5 3 3 0
   
A= 
3 1 gks rks A ′ =  −1  gksxkA
 0 −1 5 1
 5  2 × 3
 
 5 3 × 2

vkO;wgksa osQ ifjorZ osQ xq.k/eZ (Properties of transpose of matrices)

vc ge fdlh vkO;wg osQ ifjorZ vkO;wg osQ fuEufyf[kr xq.k/eks± dks fcuk miifÙk fn, O;Dr djrs
gSaA budk lR;kiu mi;qDr mnkgj.kksa }kjk fd;k tk ldrk gSaA mi;qDr dksfV osQ fdUgha vkO;wgksa
A rFkk B osQ fy,
(i) (A′)′ = A (ii) (kA)′ = kA′ (tgk¡ k dksbZ vpj gSA)
(iii) (A + B)′ = A′ + B′ (iv) (A B)′ = B′ A′

3 3 2  2 −1 2
mnkgj.k 20 ;fn A =   rFkk B =  rks fuEufyf[kr dks lR;kfir
4 2 0  1 2 4 

dhft,%
(i) (A′)′ = A (ii) (A + B)′ = A′ + B′
(iii) (kB)′ = kB′, tgk¡ k dksbZ vpj gSA

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gy
(i) ;gk¡

 3 4
3 3 2   ′ 3 3 2
A=   ⇒ A′ =  3 2 ⇒ ( A′ ) =  =A
4 2 0   2 0 4 2 0 
 
vr% (A′)′ = A
(ii) ;gk¡

3 3 2  2 −1 2  5 3 − 1 4
A=  , B =   ⇒A+B= 
4 2 0 1 2 4 5 4 4

 5 5
 
vr,o (A + B)′ =  3 − 1 4
 4 4


 3 4  2 1
   
vc A′ =  3 2 , B′ = −1 2
 2 0  2 4
 

 5 5
 
vr,o A′ + B′ =  3 −1 4
 4 4

vr% (A + B)′ = A′ + B′
(iii) ;gk¡
 2 −1 2  2k −k 2k 
kB = k  =
1 2 4   k 2k 4k 

 2k k   2 1
rc    
(kB)′ =  − k 2k  = k −1 2  = kB′
 2k 4k   2 4 

vr% (kB)′ = kB′

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 −2
mnkgj.k 21 ;fn A =  4  , B = [1 3 − 6] gS rks lR;kfir dhft, (AB)′ = B′A′ gSA
 
 5 
gy ;gk¡
 −2 
 4  , B = 1 3 −6
A=   [ ]
 5 
 −2  −2 −6 12 
blfy,  4 1 3 −6
AB =   [ ] =  4 12 −24 
 5   5 15 −30 

 −2 4 5 
 − 6 12 15 
vr% (AB)′ =  
 12 −24 −30

 1
vc A′ = [–2 4 5] , B′ =  3 
 
 − 6 

 1  −2 4 5
 3  −2 4 5 =  −6 12 15  = ( AB) ′
blfy, B′A′ =   [ ]  
 − 6  12 −24 −30
Li"Vr;k (AB)′ = B′A′

3.6 lefer rFkk fo"ke lefer vkO;wg (Symmetric and Skew Symmetric
Matrices)
ifjHkk"kk 4 ,d oxZ vkO;wg A = [aij] lefer dgykrk gS ;fn A′ = A vFkkZr~ i o j osQ gj laHko
ekuksa osQ fy, [aij] = [aji] gksA

 3 2 3
mnkgj.k osQ fy,] A =  2 −1.5 −1  ,d lefer vkO;wg gS] D;ksafd A′ = A

 3 −1 1 


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ifjHkk"kk 5 ,d oxZ vkO;wg A = [aij] fo"ke lefer vkO;wg dgykrk gS] ;fn A′ = – A, vFkkZr~
i rFkk j osQ gj laHko ekuksa osQ fy, aji = – aij gksA vc] ;fn ge i = j j[ksa] rks aii = – aii gksxkA
vr% 2aii = 0 ;k aii = 0 leLr i osQ fy,A
bldk vFkZ ;g gqvk fd fdlh fo"ke lefer vkO;wg osQ fod.kZ osQ lHkh vo;o 'kwU; gksrs
 0 e f
gSaA mnkgj.kkFkZ vkO;wg B =  −e 0 g  ,d fo"ke lefer vkO;wg gS] D;ksafd B′ = – B gSA
 − f −g 0 

vc] ge lefer rFkk fo"ke lefer vkO;wgksa osQ oqQN xq.k/eks± dks fl¼ djsaxsA

izes; 1 okLrfod vo;oksa okys fdlh oxZ vkO;wg A osQ fy, A + A′ ,d lefer vkO;wg rFkk
A – A′ ,d fo"ke lefer vkO;wg gksrs gSaA
miifÙk eku yhft, fd B = A + A′ rc
B′ = (A + A′)′
= A′ + (A′)′ (D;ksafd (A + B)′ = (A′ + B′)
= A′ + A (D;ksafd (A′)′ = A)
= A + A′ (D;ksafd A + B = B + A)
= B
blfy, B = A + A′ ,d lefer vkO;wg gSA
vc eku yhft, fd C = A – A′
C′ = (A – A′)′ = A′ – (A′)′ (D;ksa?)
= A′ – A (D;ksa?)
= – (A – A′) = – C
vr% C = A – A′ ,d fo"ke lefer vkO;wg gSA
izes; 2 fdlh oxZ vkO;wg dks ,d lefer rFkk ,d fo"ke lefer vkO;wgksa osQ ;ksxiQy osQ :i
esa O;Dr fd;k tk ldrk gSA
miifÙk eku yhft, fd A ,d oxZ vkO;wg gSA ge fy[k ldrs gSa fd
1 1
A = (A + A′) + (A − A′)
2 2

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ize;s 1 }kjk gesa Kkr gS fd (A + A′) ,d lefer vkO;wg rFkk (A – A′) ,d fo"ke lefer
vkO;wg gSA D;ksfa d fdlh Hkh vkO;wg A osQ fy, (kA)′ = kA′ gksrk gSA blls fu"d"kZ fudyrk gS fd
1 1
(A + A′) lefer vkO;wg rFkk (A − A′) fo"ke lefer vkO;wg gSA vr% fdlh oxZ vkO;wg dks
2 2
,d lefer rFkk ,d fo"ke lefer vkO;wgksa osQ ;ksxiQy osQ :i esa O;Dr fd;k tk ldrk gSA
 2 −2 − 4
mnkgj.k 22 vkO;wg B = −1 3 4  dks ,d lefer vkO;wg rFkk ,d fo"ke lefer
 1 −2 −3 
vkO;wg osQ ;ksxiQy osQ :i esa O;Dr dhft,A
 2 −1 1 
gy ;gk¡ B′ =  − 2 3 −2
 − 4 4 −3

 −3 −3 
 2 2 2
 4 −3 −3   
1  −3 1  gSA
2 = 
1
eku yhft, fd P = ( B + B′ ) = −3 6 3

2 2  2
 −3 2 − 6  −3 
 1 −3 
 2 
 −3 −3 
 2 2 2
 
−3
vc P′ =  3 1 = P
2 
 
 −3 1 −3 
 2 
1
vr% P = (B + B′) ,d lefer vkO;wg gSA
2
 −1 −5 
0 2 2
 0 −1 −5   
1 
Q = (B – B ) = 1 0 6  =  3  gSA
1 1
lkFk gh eku yhft, ′ 0
2 2 2 
5 −6 0   5 
 −3 0
 2 

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 1 5
0 2 3
 
−1
rc Q′ =  0 −3 = − Q
 2 
 
 −5 3 0
 2 

1
vr% Q= (B – B′) ,d fo"ke lefer vkO;wg gSA
2
 −3 −3   −1 −5 
 2 2 2 
0
2 2
    2 −2 − 4
−3
3  = −1 4  = B
1
vc P+Q= 3 1 + 0 3
 2  2 
     1 −2 −3 
 −3 1 −3  
5
−3 0
 2   2 
vr% vkO;wg B ,d lefer vkO;wg rFkk ,d fo"ke lefer vkO;wg osQ ;ksxiQy osQ :i esa
O;Dr fd;k x;kA

iz'ukoyh 3-3
1. fuEufyf[kr vkO;wgksa esa ls izR;sd dk ifjorZ Kkr dhft,%
5
1  −1 5 6 
 1 −1  
(i)   (ii)   (iii)  3 5 6 
2  2 3
−1   2 3 −1
   

 −1 2 3 − 4 1 −5
   2 0 gSa rks lR;kfir dhft, fd
2. ;fn A =  5 7 9 rFkk B =  1 
 −2 1 1  1 3 1
(i) (A + B)′ = A′ + B′ (ii) (A – B)′ = A′ – B′
 3 4
   −1 2 1
3. ;fn A ′ = −1 2 rFkk B =  gSa rks lR;kfir dhft, fd
 1 2 3
 0 1 
(i) (A + B)′ = A′ + B′ (ii) (A – B)′ = A′ – B′

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 −2 3  −1 0 
4. ;fn A ′ =   rFkk B =   gSa rks (A + 2B)′ Kkr dhft,A
 1 2  1 2
5. A rFkk B vkO;wgksa osQ fy, lR;kfir dhft, fd (AB)′ = B′A′, tgk¡

1  0 
  , B = −1 2 1
(i) A = − 4  [ ] (ii) A = 1  , B = [1 5 7 ]
 3  2 

 cos α sin α 
6. (i) ;fn A =   gks rks lR;kfir dhft, fd A′ A = I
 − sin α cos α 

 sin α cos α 
(ii) ;fn A =   gks rks lR;kfir dhft, fd A′ A = I
 − cos α sin α 

1 −1 5
7. (i) fl¼ dhft, fd vkO;wg A = −1 2 1 ,d lefer vkO;wg gSA
 5 1 3

 0 1 −1 
(ii) fl¼ dhft, fd vkO;wg A = −1 0 1  ,d fo"ke lefer vkO;wg gSA
 
 1 −1 0 

1 5 
8. vkO;wg A =   osQ fy, lR;kfir dhft, fd
6 7 
(i) (A + A′) ,d lefer vkO;wg gSA
(ii) (A – A′) ,d fo"ke lefer vkO;wg gSA

 0 a b
1 1
9. ;fn A = − a 0 c  rks ( A + A′ ) rFkk ( A − A′ ) Kkr dhft,A
2 2
 −b −c 0 
10. fuEufyf[kr vkO;wgksa dks ,d lefer vkO;wg rFkk ,d fo"ke lefer vkO;wg osQ ;ksxiQy
osQ :i esa O;Dr dhft,%
 6 −2 2
3 5  
(i)   (ii) −2 3 −1 
1 −1  2 −1 3 

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 3 3 −1
 1 5
(iii)  −2 −2 1
 (iv)  
 −1 2
 − 4 −5 2

iz'u la[;k 11 rFkk 12 esa lgh mÙkj pqfu,%

11. ;fn A rFkk B leku dksfV osQ lefer vkO;wg gSa rks AB – BA ,d
(A) fo"ke lefer vkO;wg gS (B) lefer vkO;wg gS
(C) 'kwU; vkO;wg gS (D) rRled vkO;wg gS
cos α − sin α 
12. ;fn A =  rFkk A + A′ = I, rks α dk eku gS
sin α cos α 

π π
(A) (B)
6 3
3π
(C) π (D)
2
3.7 vkO;wg ij izkjafHkd lafØ;k (vkO;wg :ikarj.k) [Elementary Operation
(Transformation) of a matrix]
fdlh vkO;wg ij N% izdkj dh lafØ;k,¡ (:ikarj.k) fd, tkrs gSa] ftuesa ls rhu iafDr;ksa rFkk rhu
LraHkksa ij gksrh gS] ftUgsa izkjafHkd lafØ;k,¡ ;k :ikarj.k dgrs gSaA
(i) fdlh nks iafDr;ksa ;k nks LraHkksa dk ijLij fofue;% izrhdkRed :i (symbolically) esa]
ioha rFkk joha iafDr;ksa osQ fofue; dks Ri ↔ Rj rFkk iosa rFkk josa LraHkksa osQ fofue; dks
Ci ↔ Cj }kjk fu:fir djrs gSaA mnkgj.k osQ fy,

1 2 1 −1 3 1
   
A = −1 3 1  , ij R1 ↔ R2 dk iz;ksx djus ij gesa vkO;wg  1 2 1  izkIr
5 6 7  5 6 7 
 
gksrk gSA
(ii) fdlh iafDr ;k LraHk osQ vo;oksa dks ,d 'kwU;srj la[;k ls xq.ku djuk% izrhdkRed :i
esa] ioha iafDr osQ izR;sd vo;o dks k, tgk¡ k ≠ 0 ls xq.ku djus dks Ri → kRi }kjk
fu:fir djrs gSaA
 1 2 1
laxr LraHk lafØ;k dks Ci → kCi }kjk fu:fir djrs gSaA mnkgj.kkFkZ B =  
 −1 3 1

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 1
 1 2
7
ij C3 → 1 C3 , dk iz;ksx djus ij gesa vkO;wg   izkIr gksrk gSA
7  −1 3
1
 7 
(iii) fdlh iafDr vFkok LraHk osQ vo;oksa esa fdlh vU; iafDr vFkok LraHk osQ laxr vo;oksa dks
fdlh 'kwU;srj la[;k ls xq.kk djosQ tksM+uk% izrhdkRed :i esa] ioha iafDr osQ vo;oksa esa
joha iafDr osQ laxr vo;oksa dks k ls xq.kk djosQ tksM+us dks Ri → Ri + kRj ls fu:fir djrs
gSAa
laxr LraHk lafØ;k dks Ci → Ci + k Cj ls fu:fir djrs gSaA
1 2
mnkgj.k osQ fy, C =   ij R2 → R2 – 2R1 dk iz;ksx djus ij] gesa vkO;wg
 2 −1
1 2
 0 −5 izkIr gksrk gSA
 
3.8 O;qRØe.kh; vkO;wg (Invertible Matrices)
ifjHkk"kk 6 ;fn A, dksfV m, dk] ,d oxZ vkO;wg gS vkSj ;fn ,d vU; oxZ vkO;wg dk vfLrRo
bl izdkj gS] fd AB = BA = I, rks B dks vkO;wg A dk O;qRØe vkO;wg dgrs gSa vkSj bls
A– 1 }kjk fu:fir djrs gSaA ,slh n'kk esa vkO;wg A O;qRØe.kh; dgykrk gSA
 2 3  2 −3 
mnkgj.kkFkZ] eku yhft, fd A=   rFkk B =  −1 2  nks vkO;wg gSaA
1 2  
 2 3   2 −3 
vc AB =   
 1 2   −1 2 
 4 − 3 −6 + 6   1 0 
=  = =I
 2 − 2 −3 + 4   0 1 
1 0 
lkFk gh BA =   = I gSA vr% B vkO;wg] A dk O;qRØe gSA
0 1 
nwljs 'kCnksa esa] B = A– 1 rFkk A vkO;wg B, dk O;qRØe gS] vFkkZr~ A = B–1

AfVIi.kh
1. fdlh vk;rkdkj (Rectangular) vkO;wg dk O;qRØe vkO;wg ugha gksrk gS] D;ksfa d xq.kuiQy
AB rFkk BA osQ ifjHkkf"kr gksus vkSj leku gksus osQ fy,] ;g vfuok;Z gS fd A rFkk B
leku dksfV osQ oxZ vkO;wg gksaA

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2. ;fn B, vkO;wg A dk O;qRØe gS] rks A, vkO;wg B dk O;qRØe gksrk gSA

izes; 3 [O;qRØe vkO;wg dh vf}rh;rk (Uniqueness of inverse)] fdlh oxZ vkO;wg dk O;qRØe
vkO;wg] ;fn mldk vfLrRo gS rks vf}rh; gksrk gSA
miifÙk eku yhft, fd A = [aij] dksfV m dk] ,d oxZ vkO;wg gSA ;fn laHko gks] rks eku yhft,
B rFkk C vkO;wg A osQ nks O;qRØe vkO;wg gSaA vc ge fn[kk,¡xsa fd B = C gSA
D;ksafd vkO;wg A dk O;qRØe B gS
vr% AB = BA = I ... (1)
D;ksafd vkO;wg A dk O;qRØe C Hkh gS vr%
AC = CA = I ... (2)
vc B = BI = B (AC) = (BA) C = IC = C
izes; 4 ;fn A rFkk B leku dksfV osQ O;qRØe.kh; vkO;wg gksa rks (AB)–1 = B–1 A–1
miifÙk ,d O;qRØe.kh; vkO;wg dh ifjHkk"kk ls
(AB) (AB)–1 =1
;k –1
A (AB) (AB)–1 = A–1I (A–1 dk nksuksa i{kksa ls iwoZxq.ku djus ij)
;k (A–1A) B (AB)–1 = A–1 (A–1 I = A–1] rFkk vkO;wg xq.ku lkgp;Z gksrk gS)
;k IB (AB)–1 = A –1
;k B (AB)–1 = A –1
;k –1
B B (AB)–1 = B–1 A –1
;k I (AB)–1 = B–1 A –1
vr% (AB)–1 = B–1 A –1

3.8.1 izkjfEHkd lafØ;kvksa }kjk ,d vkO;wg dk O;qRØe (Inverse of a matrix by

elementary operations)
eku yhft, fd X, A rFkk B leku dksfV osQ vkO;wg gSa rFkk X = AB gSA vkO;wg lehdj.k X
= AB ij izkjafHkd iafDr lafØ;kvksa dk iz;ksx djus osQ fy,] ge bu iafDr lafØ;kvksa dk ck,¡ i{k
esa X ij rFkk nk,¡ i{k esa izFke vkO;wg A ij] ,d lkFk iz;ksx djsaxsA
blh izdkj vkO;wg lehdj.k X = AB ij izkjafHkd LraHk lafØ;kvksa dk iz;ksx djus osQ fy,]
ge bu LraHk lafØ;kvksa dk ck,¡ i{k esa X ij rFkk nk,¡ i{k esa xq.kuiQy AB esa ckn okys
vkO;wg B ij] ,d lkFk iz;ksx djsaxsA
mi;qDZ r ifjppkZ dks è;ku esa j[krs gq, ge ;g fu"d"kZ fudkyrs gSa fd] ;fn A ,d ,slk vkO;wg
gS fd A–1 dk vfLrRo gS rks izkjafHkd iafDr lafØ;kvksa osQ iz;ksx }kjk A–1 Kkr djus osQ fy,]
A = IA fyf[k, vkSj iafDr lafØ;kvksa dk iz;ksx A = IA ij rc rd djrs jfg, tc rd fd
I = BA ugha fey tkrk gSA bl izdkj izkIr vkO;wg B, vkO;wg A dk O;qRØe gksxkA blh izdkj] ;fn

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ge LraHk lafØ;kvksa osQ iz;ksx }kjk A–1 Kkr djuk pkgrs gSa] rks A = AI fyf[k, vkSj
A = AI ij LraHk lafØ;kvksa dk iz;ksx rc rd djrs jfg, tc rd gesa I = AB izkIr ugha gks tkrk gSA
fVIi.kh ml n'kk esa tc A = IA (A = AI) ij ,d ;k vf/d izkjafHkd iafDr (LraHk) lafØ;kvksa
osQ djus ij ;fn ck,¡ i{k osQ vkO;wg A dh ,d ;k vf/d iafDr;ksa osQ lHkh vo;o 'kwU; gks tkrs
gSa rks A–1 dk vfLrRo ugha gksrk gSA
1 2
mnkgj.k 23 izkjafHkd lafØ;kvksa osQ iz;ksx }kjk vkO;wg A =  2 −1 dk O;qRØe Kkr dhft,A
 

gy izkjafHkd iafDr lafØ;kvksa osQ iz;ksx djus osQ fy, ge A = IA fy[krs gSa] vFkkZr~
 1 2   1 0 1 2   1 0 
 2 −1 =  0 1 A, rks 0 −5 =  −2 1 A (R2 → R2 – 2R1 osQ iz;ksx }kjk)
       

1 0
1 2   −1  A
1
;k 0 1  =  2 (R2 → – R osQ iz;ksx }kjk)
  5 2
5 5

1 2
 1 0  5 5
;k 0 1  =  2 A (R1 → R1 – 2R2 osQ iz;ksx }kjk)
   −1 
 5 5 

1 2
5 5
vr% A–1 =   gSA
2 −1 
 5 5 
fodYir% izkjafHkd LraHk lafØ;kvksa osQ iz;ksx gsrq] ge fy[krs gSa fd A = AI, vFkkZr~
1 2  1 0 
 2 −1 = A 0 1 
   
C2 → C2 – 2C1, osQ iz;ksx }kjk
1 0   1 −2
 2 −5 = A  0 1 
   

2018-19
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1
vc C2 → − C 2 , osQ iz;ksx }kjk
5

 2
1 0  1 5
2 1 = A  −1

  0
 5 
vUrr% C1 → C1 – 2C2, osQ iz;ksx }kjk
1 2
1 0  5 5
0 1  = A  2 
−1 
  
 5 5 

1 2
5 5
vr,o A–1 =  
2 −1 
 5 5 
mnkgj.k 24 izkjafHkd lafØ;kvksa osQ iz;ksx }kjk fuEufyf[kr vkO;wg dk O;qRØe izkIr dhft,%
0 1 2 
A = 1 2 3
3 1 1 

 0 1 2   1 0 0
 
gy ge tkurs gSa fd A = I A, vFkkZr~ 1 2 3  = 0 1 0 A
 3 1 1  0 0 1 

1 2 3   0 1 0
;k 0 1 2  = 1 0 0 A (R ↔ R }kjk)
    1 2
 3 1 1  0 0 1 

1 2 3  0 1 0
0 1  0  A (R3 → R3 – 3R1}kjk)
;k  2  = 1 0
0 −5 −8  0 −3 1 

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1 0 −1   −2 1 0
0 1 2  = 
;k   1 0 0  A (R1 → R1 – 2R2}kjk)
0 −5 −8   0 −3 1 

1 0 −1   −2 1 0
0 1 2  = 
;k   1 0 0 A (R3 → R3 + 5R2}kjk)
0 0 2   5 −3 1 

 −2 1 0
1 0 −1  
0 1 2   1 0 0  A 1
;k   =  5 −3 (R3 → R }kjk )
1 2 3
0 0 1   
2 2 2

 1 −1 1
1 0 0   2 2 2
;k  0 1 2 =  1 0 
0  A (R1 → R1 + R3 }kjk)
  
0 0 1   5 −3 1
 
2 2 2

 1 −1 1
1 0 0  2
0 1 0  2 2
 
;k   = − 4 3 −1 A (R2 → R2 – 2R3}kjk)
0 0 1   5 −3 1
 
2 2 2
 1 −1 1
2 2 2
 
vr% A =  −4 3
–1 −1 
 5 −3 1
 
2 2 2
fodYir%] A = AI fyf[k,, vFkkZr~
 0 1 2 1 0 0
1 2 3  A 0 1 0
  =  
3 1 1  0 0 1

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1 0 2   0 1 0
2 1 3  
;k   = A 1 0 0 (C1 ↔ C2)
1 3 1  0 0 1

1 0 0  0 1 0 
 2 1 −1 A 1 0 −2
;k   =   (C3 → C3 – 2C1)
1 3 −1  0 0 1 

1 0 0  0 1 1 
2 1 0  
;k   = A 1 0 −2 (C3 → C3 + C2)
1 3 2  0 0 1 

 1
1 0 0  0 1 2 
2 1 0   1
;k   = A 1 0 −1 (C3 → C)
2 3
1 3 1   1
0 0 
 2

 1
 −2 1
 1 0 0  2
 0 1 0  
;k   = A  1 0 −1 (C1 → C1 – 2C2)
 1
 −5 3 1  0 0 
 2

 1 1
 1
 1 0 0  2 2
0 1 0 A  − 4 0 −1 
;k   =   (C1 → C1 + 5C3)
 1
0 3 1 
5
0 
 2 2

 1 −1 1 
 1 0 0   2 2 2
 0 1 0 A  − 4 
3 −1 
;k   =  (C2 → C2 – 3C3)
0 0 1  5 −3 1 
 
 2 2 2

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 vkO;wg 105

1 −1 1
2 2 2
 
vr% A–1 =  − 4 3 −1 
 5 −3 1
 
 2 2 2

10 −2 
mnkgj.k 25 ;fn P =  gS rks P – 1 Kkr dhft,] ;fn bldk vfLrRo gSA
 −5 1 

10 −2   1 0 
gy P = I P fyf[k, vFkkZr~]  = P
 −5 1   0 1

 −1   1 
;k 1  
5 = 10
0
P (R1 →
1
R }kjk )
    10 1
 −5 1   0 1

 −1 1 
 0
;k 1 5  =  10  P (R2 → R2 + 5R1 }kjk)
   1 1
0 0
 2 
;gk¡ ck,¡ i{k osQ vkO;wg dh f}rh; iafDr osQ lHkh vo;o 'kwU; gks tkrs gSa] vr% P–1 dk
vfLrRo ugha gSA

iz'ukoyh 3-4
iz'u la[;k 1 ls 17 rd osQ vkO;wgksa osQ O;qRØe] ;fn mudk vfLrRo gS] rks izkjafHkd :ikarj.k osQ
iz;ksx ls Kkr dhft,%
 1 −1  2 1 1 3
1.  2 3 2.   3.  
   1 1 2 7

 2 3 2 1  2 5
4. 5 7 5.   6.  
  7 4  1 3

3 1  4 5  3 10 
7. 5 2 8.   9.  
   3 4 2 7 

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 3 −1  2 − 6  6 −3
10.  − 4 2 11.   12.  
   1 −2   −2 1 

 2 −3 3 
 2 −3 2 1  
13.  −1 2  14.   15.  2 2 3 
   4 2  3 −2 2 

1 3 −2   2 0 −1 
−3 0 −5  
16.  17.  5 1 0 
 2 5 0   0 1 3 
18. vkO;wg A rFkk B ,d nwljs osQ O;qRØe gksaxs osQoy ;fn
(A) AB = BA (B) AB = BA = 0
(C) AB = 0, BA = I (D) AB = BA = I

fofo/ mnkgj.k

 cos θ sin θ 
mnkgj.k 26 ;fn A =   gS rks fl¼ dhft, fd
 − sin θ cos θ

 cos nθ sin nθ 
An =  , n∈N
 − sin nθ cos nθ

gy ge bldks xf.krh; vkxeu osQ fl¼kar }kjk fl¼ djsaxsA

 cos θ sin θ   cos nθ sin nθ 
;gk¡ ij P(n) : ;fn A =   , rks A n =  ,n∈N
 − sin θ cos θ   − sin nθ cos nθ 

 cos θ sin θ   cos θ sin θ 

vc P(1) : A =   , blfy, A = 
1

 − sin θ cos θ   − sin θ cos θ 
vr%] ifj.kke n = 1 osQ fy, lR; gSA
eku yhft, fd ifj.kke n = k osQ fy, lR; gSA
 cos θ sin θ   cos k θ sin k θ 
, rks A = 
k
blfy, P(k) : A =   .
 − sin θ cos θ  − sin k θ cos k θ 

2018-19
 vkO;wg 107

vc ge fl¼ djsaxs fd ifj.kke n = k +1 osQ fy, Hkh lR; gSA

 cos θ sin θ   cos k θ sin k θ 
Ak + 1 = A ⋅ A = 
k
vc  
 − sin θ cos θ  − sin k θ cos k θ 

 cos θ cos k θ – sin θ sin k θ cos θ sin k θ + sin θ cos k θ 

=  
 − sin θ cos k θ + cos θ sin k θ − sin θ sin k θ + cos θ cos k θ 

 cos (θ + k θ) sin (θ + k θ)   cos ( k + 1)θ sin ( k + 1)θ 

=  = 
 − sin (θ + k θ) cos (θ + k θ)   − sin ( k + 1)θ cos (k + 1)θ 
blfy, ifj.kke n = k + 1 osQ fy, Hkh lR; gSA vr% xf.krh; vkxeu dk fl¼kar ls izekf.kr
 cos n θ sin n θ 
gksrk gS fd A n =  , leLr izko`Qr la[;kvksa n osQ fy, lR; gSA
− sin n θ cos n θ

mnkgj.k 27 ;fn A rFkk B leku dksfV osQ lefer vkO;wg gSa rks n'kkZb, fd AB lefer gS] ;fn
vkSj osQoy ;fn A rFkk B Øefofues; gS] vFkkZr~ AB = BA gSA
gy fn;k gS fd A rFkk B nksuksa lefer vkO;wg gSa] blfy, A′ = A rFkk B′ = B gSA
eku yhft, fd AB lefer gS rks (AB)′ = AB
¯drq (AB)′ = B′A′= BA (D;ksa?)
vr% BA = AB
foykser%] ;fn AB = BA gS rks ge fl¼ djsaxs fd AB lefer gSA
vc (AB)′ = B′A′
= B A (D;ksafd A rFkk B lefer gSa )
= AB
vr% AB lefer gSA

 2 −1 5 2  2 5
mnkgj.k 28 eku yhft, fd A =   ,B=  ,C=  gSA ,d ,slk vkO;wg
3 4 7 4  3 8
D Kkr dhft, fd CD – AB = O gksA
gy D;ksafd A, B, C lHkh dksfV 2, osQ oxZ vkO;wg gSa vkSj CD – AB Hkyh&Hkk¡fr ifjHkkf"kr gS]
blfy, D dksfV 2 dk ,d oxZ vkO;wg gksuk pkfg,A

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a b 
eku yhft, fd D=   gSA rc CD – AB = O ls izkIr gksrk gS fd
c d 

 2 5  a b   2 −1  5 2
 3 8  c d  − 3 4  7 4 = O
     

 2a + 5c 2b + 5d   3 0   0 0 
;k  3a + 8c 3b + 8d  −  43 22 =  
    0 0 

 2a + 5c − 3 2b + 5d   0 0 
;k  3a + 8c − 43 3b + 8d − 22 =  0 0 
   
vkO;wgksa dh lekurk ls gesa fuEufyf[kr lehdj.k izkIr gksrs gSa%
2a + 5c – 3 = 0 ... (1)
3a + 8c – 43 = 0 ... (2)
2b + 5d = 0 ... (3)
rFkk 3b + 8d – 22 = 0 ... (4)
(1) rFkk (2), dks ljy djus ij a = –191, c = 77 izkIr gksrk gSA
(3) rFkk (4), dks ljy djus ij b = – 110, d = 44 izkIr gksrk gSA
 a b   −191 −110 
vr% D=  =
 c d   77 44 

vè;k; 3 ij fofo/ iz'ukoyh

0 1 
1. eku yhft, fd A =   gks rks fn[kkb, fd lHkh n ∈ N osQ fy,
0 0
(aI + bA)n = an I + nan – 1 bA, tgk¡ I dksfV 2 dk rRled vkO;wg gSA

1 1 1 3n −1 3n−1 3n −1 
   
2. ;fn A = 1 1 1 , rks fl¼ dhft, fd A n = 3n −1 3n−1 3n −1  , n ∈ N
1 1 1  n −1 n−1 n −1 
3 3 3 

3 − 4  1 + 2n − 4n 
3. ;fn A =   rks fl¼ dhft, fd A n =  tgk¡ n ,d /u iw.kk±d gSA
1 −1   n 1 − 2n 

2018-19
 vkO;wg 109

4. ;fn A rFkk B lefer vkO;wg gSa rks fl¼ dhft, fd AB – BA ,d fo"ke lefer vkO;wg gSA
5. fl¼ dhft, fd vkO;wg B′AB lefer vFkok fo"ke lefer gS ;fn A lefer vFkok
fo"ke lefer gSA
0 2 y z
6. x, y, rFkk z osQ ekuksa dks Kkr dhft,] ;fn vkO;wg A =  x y − z  lehdj.k

 x − y z 
A′A = I dks larq"V djrk gSA
1 2 0   0 
7. x osQ fdl eku osQ fy, [1 2 1]  2 0 1   2  = O gS ?
   
1 0 2   x 

3 1
8. ;fn A =  gks rks fl¼ dhft, fd A2 – 5A + 7I = O gSA
−1 2

1 0 2   x 
9. ;fn [ x −5 −1] 0 2 1  4  = O gS rks x dk eku Kkr dhft,A
  
2 0 3  1 
10. ,d fuekZrk rhu izdkj dh oLrq,¡ x, y, rFkk z dk mRiknu djrk gS ftu dk og nks cktkjksa
esa foØ; djrk gSA oLrqvksa dh okf"kZd fcØh uhps lwfpr (funf'kZr) gS%
ck”kkj mRiknu
I 10,000 2,000 18,000
II 6,000 20,000 8,000
(a) ;fn x, y rFkk z dh izR;sd bdkbZ dk foØ; ewY; Øe'k% Rs 2-50] Rs 1-50 rFkk
Rs 1-00 gS rks izR;sd ck”kkj esa oqQy vk; (Revenue)] vkO;wg chtxf.kr dh lgk;rk
ls Kkr dhft,A
(b) ;fn mi;qZDr rhu oLrqvksa dh izR;sd bdkbZ dh ykxr (Cost) Øe'k% Rs 2-00]
Rs 1-00 rFkk iSls 50 gS rks oqQy ykHk (Gross profit) Kkr dhft,A
1 2 3  −7 −8 −9 
11. vkO;wg X Kkr dhft,] ;fn X  = gSA
4 5 6  2 4 6 
12. ;fn A rFkk B leku dksfV osQ oxZ vkO;wg bl izdkj gSa fd AB = BA gS rks xf.krh;
vkxeu }kjk fl¼ dhft, fd ABn = BnA gksxkA blosQ vfrfjDr fl¼ dhft, fd
leLr n ∈ N osQ fy, (AB)n = AnBn gksxkA

2018-19
110 xf.kr

fuEufyf[kr iz'uksa esa lgh mÙkj pqfu,%

α β 
13. ;fn A =  γ −α 
bl izdkj gS fd A² = I, rks

(A) 1 + α² + βγ = 0 (B) 1 – α² + βγ = 0
(C) 1 – α² – βγ = 0 (D) 1 + α² – βγ = 0
14. ;fn ,d vkO;wg lefer rFkk fo"ke lefer nksuksa gh gS rks%
(A) A ,d fod.kZ vkO;wg gSA (B) A ,d 'kwU; vkO;wg gSA
(C) A ,d oxZ vkO;wg gSA (D) buesa ls dksbZ ughaA
15. ;fn A ,d oxZ vkO;wg bl izdkj gS fd A2 = A, rks (I + A)³ – 7 A cjkcj gS%
(A) A (B) I – A (C) I (D) 3A

lkjka'k
® vkO;wg] iQyuksa ;k la[;kvksa dk ,d vk;rkdkj Øe&foU;kl gSA
® m iafDr;ksa rFkk n LraHkksa okys vkO;wg dks m × n dksfV dk vkO;wg dgrs gSaA
® [aij]m × 1 ,d LraHk vkO;wg gSA
® [aij]1 × n ,d iafDr vkO;wg gSA
® ,d m × n vkO;wg ,d oxZ vkO;wg gS] ;fn m = n gSA
® A = [aij]m × m ,d fod.kZ vkO;wg gS] ;fn aij = 0, tc i ≠ j
® A = [aij]n × n ,d vfn'k vkO;wg gS] ;fn aij = 0, tc i ≠ j, aij = k, (k ,d vpj gS),
tc i = j gSA
® A = [aij]n × n ,d rRled vkO;wg gS] ;fn aij = 1 tc i = j rFkk aij = 0 tc
i ≠ j gSA
® fdlh 'kwU; vkO;wg (;k fjDr vkO;wg) osQ lHkh vo;o 'kwU; gksrs gSaA
® A = [aij] = [bij] = B ;fn (i) A rFkk B leku dksfV osQ gSa rFkk (ii) i rFkk j osQ leLr
laHko ekuksa osQ fy, aij = bij gksA
® kA = k[aij]m × n = [k(aij)]m × n
® – A = (–1)A
® A – B = A + (–1) B
® A+ B = B +A

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 vkO;wg 111

® (A + B) + C = A + (B + C), tgk¡ A, B rFkk C leku dksfV osQ vkO;wg gSaA

® k(A + B) = kA + kB, tgk¡ A rFkk B leku dksfV osQ vkO;wg gS rFkk k ,d vpj gSA
® (k + l ) A = kA + lA, tgk¡ k rFkk l vpj gSaA n
® ;fn A = [aij]m × n rFkk B = [bjk]n × p rks AB = C = [cik]m × p, tgk¡ cik = ∑ aij b jk gSA
j =1
® (i) A(BC) = (AB)C, (ii) A(B + C) = AB + AC, (iii) (A + B)C = AC + BC
® ;fn A = [aij]m × n rks A′ ;k AT = [aji]n × m
® (i) (A′)′ = A (ii) (kA)′ = kA′ (iii) (A + B)′ = A′ + B′ (iv) (AB)′ = B′A′
® ;fn A′ = A gS rks A ,d lefer vkO;wg gSA
® ;fn A′ = –A gS rks A ,d fo"ke lefer vkO;wg gSA
® fdlh oxZ vkO;wg dks ,d lefer vkSj ,d fo"ke lefer vkO;wgksa osQ ;ksxiQy osQ :i
esa fu:fir fd;k tk ldrk gSA
® vkO;wgksa ij izkjafHkd lafØ;k,¡ fuEufyf[kr gSa%
(i) Ri ↔ Rj ;k Ci ↔ Cj
(ii) Ri → kRi ;k Ci → kCi
(iii) Ri → Ri + kRj ;k Ci → Ci + kCj
® ;fn A rFkk B nks oxZ vkO;wg gSa] bl izdkj fd AB = BA = I, rks vkO;wg A dk O;qRØe
vkO;wg B gS] ftls A–1 }kjk fu:fir djrs gSa vkSj vkO;wg B dk O;wRØe A gSA
® oxZ vkO;wg dk O;qRØe vkO;wg] ;fn mldk vfLrRo gS] vf}rh; gksrk gSA

—v—

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vè;k; 4

lkjf.kd (Determinants)
vAll Mathematical truths are relative and conditional — C.P. STEINMETZ v

4.1 Hkwfedk (Introduction)

fiNys vè;k; es]a geus vkO;wg vkSj vkO;wgksa osQ chtxf.kr osQ fo"k;
esa vè;;u fd;k gSA geus chtxf.krh; lehdj.kksa osQ fudk; dks
vkO;wgksa osQ :i esa O;Dr djuk Hkh lh[kk gSA blosQ vuqlkj jSf[kd
lehdj.kksa osQ fudk;
a1 x + b1 y = c 1
a2 x + b2 y = c 2

a b   x c 
dks  1 1    =  1  osQ :i esa O;Dr dj ldrs gSaA vc
 a2 b2   y   c2 
bu lehdj.kksa osQ fudk; dk vf}rh; gy gS vFkok ugha] bldks P.S. Laplace
a1 b2 – a2 b1 la[;k }kjk Kkr fd;k tkrk gSA (Lej.k dhft, fd (1749-1827)

a1 b1
;fn ≠ ;k a1 b2 – a2 b1 ≠ 0, gks rks lehdj.kksa osQ fudk; dk gy vf}rh; gksrk gS) ;g
a2 b2
la[;k a1 b2 – a2 b1 tks lehdj.kksa osQ fudk; osQ vf}rh; gy Kkr djrh gS] og vkO;wg
a b1 
A= 1 ls lacaf/r gS vkSj bls A dk lkjf.kd ;k det A dgrs gSaA lkjf.kdksa dk
 a2 b2 

bathfu;fjax] foKku] vFkZ'kkL=k] lkekftd foKku bR;kfn esa foLr`r vuqiz;ksx gSaA
bl vè;k; esa] ge osQoy okLrfod izfof"V;ksa osQ 3 dksfV rd osQ lkjf.kdksa ij fopkj djsaxsA
bl vè;k; esa lkjf.kdksa osQ xq.k /eZ] milkjf.kd] lg&[k.M vkSj f=kHkqt dk {ks=kiQy Kkr djus
esa lkjf.kdksa dk vuqiz;ksx] ,d oxZ vkO;wg osQ lg[kaMt vkSj O;qRØe] jSf[kd lehdj.k osQ fudk;ksa

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dh laxrrk vkSj vlaxrrk vkSj ,d vkO;wg osQ O;qRØe dk iz;ksx dj nks vFkok rhu pjkadksa osQ
jSf[kd lehdj.kksa osQ gy dk vè;;u djsaxsA
4.2 lkjf.kd (Determinant)
ge n dksfV osQ izR;sd oxZ vkO;wg A = [aij] dks ,d la[;k (okLrfod ;k lfEeJ) }kjk lacfa /r djk
ldrs gSa ftls oxZ vkO;wg dk lkjf.kd dgrs gSaA bls ,d iQyu dh rjg lkspk tk ldrk gS tks
izR;sd vkO;wg dks ,d vf}rh; la[;k (okLrfod ;k lfEeJ) ls lacaf/r djrk gSA
;fn M oxZ vkO;wgksa dk leqPp; gS] k lHkh la[;kvksa (okLrfod ;k lfEeJ) dk leqPp; gS
vkSj f : M → K, f (A) = k, osQ }kjk ifjHkkf"kr gS tgk¡ A ∈ M vkSj k ∈ K rc f (A) , A dk
lkjf.kd dgykrk gSA bls | A | ;k det (A) ;k ∆ osQ }kjk Hkh fu:fir fd;k tkrk gSA

a b a b
;fn A =  c d  , rks A osQ lkjf.kd dks | A| = c d = det (A) }kjk fy[kk tkrk gSA
 
fVIi.kh
(i) vkO;wg A osQ fy,] | A | dks A dk lkjf.kd i<+rs gSaA
(ii) osQoy oxZ vkO;wgksa osQ lkjf.kd gksrs gSaA
4.2.1 ,d dksfV osQ vkO;wg dk lkjf.kd (Determinant of a matrix of order one)
ekuk ,d dksfV dk vkO;wg A = [a ] gks rks A osQ lkjf.kd dks a osQ cjkcj ifjHkkf"kr fd;k tkrk gSA
4.2.2 f}rh; dksfV osQ vkO;wg dk lkjf.kd (Determinant of a matrix of order two)

 a11 a12 
ekuk 2 × 2 dksfV dk vkO;wg A =  a  gSA
 21 a22 
rks A osQ lkjf.kd dks bl izdkj ls ifjHkkf"kr fd;k tk ldrk gS%

det (A) = |A| = ∆ = = a11a22 – a21a12

2 4
mnkgj.k 1 –1 2 dk eku Kkr dhft,A

2 4
gy –1 2 = 2 (2) – 4(–1) = 4 + 4 = 8

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x x +1
mnkgj.k 2 x –1 x dk eku Kkr dhft,A
x x +1
gy x – 1 x = x (x) – (x + 1) (x – 1) = x2 – (x2 – 1) = x2 – x2 + 1 = 1

4.2.3 3 × 3 dksfV osQ vkO;wg dk lkjf.kd (Determinant of a matrix of order 3 × 3)

r`rh; dksfV osQ vkO;wg osQ lkjf.kd dks f}rh; dksfV osQ lkjf.kdksa esa O;Dr djosQ Kkr fd;k tkrk
gSA ;g ,d lkjf.kd dk ,d iafDr (;k ,d LraHk) osQ vuqfn'k izlj.k dgykrk gSA r`rh; dksfV
osQ lkjf.kd dks N% izdkj ls izlkfjr fd;k tkrk gS rhuksa iafDr;ksa (R1, R2 rFkk R3) esa ls izR;sd
osQ laxr vkSj rhuksa LraHk (C1, C2 rFkk C3) esa ls izR;sd osQ laxr n'kkZ, x, izlj.k leku ifj.kke
nsrs gSsa tSlk fd fuEufyf[kr fLFkfr;ksa esa Li"V fd;k x;k gSA
oxZ vkO;wg A = [aij]3 × 3 , osQ lkjf.kd ij fopkj djrs gSaA
a11 a12 a13
tgk¡ | A | = a21 a22 a23
a31 a32 a33

izFke iafDr (R1) osQ vuqfn'k izlj.k

a11 a12 a13
| A | = a21 a22 a23
a31 a32 a33
pj.k 1 R1 osQ igys vo;o a11 dks (–1)(1 + 1) [(–1) a11esa vuqyXukas dk ;kxs ] vkSj lkjf.kd |A| dh igyh
iafDr (R1) rFkk igyk LraHk (C1) osQ vo;oksa dks gVkus ls izkIr f}rh; dksfV osQ lkjf.kd ls xq.kk
dhft, D;ksafd a11, R1 vkSj C1 esa fLFkr gS
a22 a23
vFkkZr~ (–1)1 + 1 a11 a a33
32

pj.k 2 D;ksafd a12, R1 rFkk C2 esa fLFkr gS blfy, R1 osQ nwljs vo;o a12 dks (–1)1 + 2
[(–1) a esa vuqyXukas dk ;kxs ] vkSj lkjf.kd | A | dh igyh iafDr (R1) o nwljs LraHk (C2) dks gVkus ls izkIr
12

f}rh; Øe osQ lkjf.kd ls xq.kk dhft,

a21 a23
vFkkZr~ (–1)1 + 2a12 a a33
31

pj.k 3 D;ksafd a13, R1 rFkk C3 esa fLFkr gS blfy, R1 osQ rhljs vo;o dks (–1)1 + 3
[(–1) a esa vuqyXukas dk ;kxs ] vkSj lkjf.kd | A | dh igyh iafDr (R1) o rhljs LraHk (C3) dks gVkus ls izkIr
13

r`rh; dksfV osQ lkjf.kd ls xq.kk dhft,

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a21 a22
vFkkZr~ (–1)1 + 3 a13 a a
31 32

pj.k 4 vc A dk lkjf.kd vFkkZr~ | A | osQ O;atd dks mijksDr pj.k 1] 2 o 3 ls izkIr rhuksa
inksa dk ;ksx djosQ fyf[k, vFkkZr~
a22 a23 a21 a23
det A = |A| = (–1)1 + 1 a11 a + (–1)1 + 2 a12
32 a33 a31 a33

1+ 3 a21 a22
+ (–1) a13
a31 a32

;k |A| = a11 (a22 a33 – a32 a23) – a12 (a21 a33 – a31 a23)
+ a13 (a21 a32 – a31 a22)
= a11 a22 a33 – a11 a32 a23 – a12 a21 a33 + a12 a31 a23 + a13 a21 a32
– a13 a31 a22 ... (1)

AfVIi.kh ge pkjksa pj.kksa dk ,d lkFk iz;ksx djsaxsA

f}rh; iafDr (R2) osQ vuqfn'k izlj.k
a11 a12 a13
| A | = a21 a22 a23
a31 a32 a33
R2 osQ vuqfn'k izlj.k djus ij] gesa izkIr gksrk gS
2+1 a12 a13 a11 a13
| A | = (–1) a21 + (–1) 2 + 2 a22
a32 a33 a31 a33
a11 a12
+ (–1)2 + 3 a23
a31 a32
= – a21 (a12 a33 – a32 a13) + a22 (a11 a33 – a31 a13)
– a23 (a11 a32 – a31 a12)
| A | = – a21 a12 a33 + a21 a32 a13 + a22 a11 a33 – a22 a31 a13 – a23 a11 a32
+ a23 a31 a12
= a11 a22 a33 – a11 a23 a32 – a12 a21 a33 + a12 a23 a31 + a13 a21 a32
– a13 a31 a22 ... (2)

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igys LraHk (C1) osQ vuqfn'k izlj.k

a11 a12 a13
| A | = a21 a22 a23
a31 a32 a33

C1, osQ vuqfn'k izlj.k djus ij gesa izkIr gksrk gS

1+1 a22 a23 a12 a13

| A | = a11 (–1) + a21 ( −1) 2 + 1
a32 a33 a32 a33

3+1 a 12 a13
+ a31 (–1)
a22 a23

= a11 (a22 a33 – a23 a32) – a21 (a12 a33 – a13 a32) + a31 (a12 a23 – a13 a22)
| A | = a11 a22 a33 – a11 a23 a32 – a21 a12 a33 + a21 a13 a32 + a31 a12 a23
– a31 a13 a22
= a11 a22 a33 – a11 a23 a32 – a12 a21 a33 + a12 a23 a31 + a13 a21 a32
– a13 a31 a22 ... (3)
(1), (2) vkSj (3) ls Li"V gS fd | A | dk eku leku gSA ;g ikBdksa osQ vH;kl osQ fy, NksM+
fn;k x;k gS fd os ;g lR;kfir djsa fd |A| dk R3, C2 vkSj C3 osQ vuqfn'k izlj.k (1)] (2)
vkSj (3) ls izkIr ifj.kkeksa osQ leku gSA
vr% ,d lkjf.kd dks fdlh Hkh iafDr ;k LraHk osQ vuqfn'k izlj.k djus ij leku eku izkIr
gksrk gSA
fVIi.kh
(i) x.kuk dks ljy djus osQ fy, ge lkjf.kd dk ml iafDr ;k LraHk osQ vuqfn'k izlj.k djsaxs
ftlesa 'kwU;ksa dh la[;k vf/dre gksrh gSA
(ii) lkjf.kdksa dk izlj.k djrs le; (–1)i + j ls xq.kk djus osQ LFkku ij] ge (i + j) osQ le ;k
fo"ke gksus osQ vuqlkj +1 ;k –1 ls xq.kk dj ldrs gSaA
2 2 1 1
(iii) eku yhft, A =   vkSj B =   rks ;g fl¼ djuk ljy gS fd
4 0 2 0
A = 2B. ¯drq | A | = 0 – 8 = – 8 vkSj | B | = 0 – 2 = – 2 gSA

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voyksdu dhft, fd | A | = 4 (– 2) = 22 | B | ;k | A | = 2n | B |, tgk¡ n = 2, oxZ vkO;wgksa A

o B dh dksfV gSA
O;kid :i esa] ;fn A = kB, tgk¡ A o B oxZ vkO;wgksa dh dksfV n gS] rc | A| = kn | B |, tgk¡
n = 1, 2, 3 gSA

1 2 4
mnkgj.k 3 lkjf.kd ∆ = –1 3 0 dk eku Kkr dhft,A
4 1 0

gy è;ku nhft, fd rhljs LraHk esa nks izfof"V;k¡ 'kwU; gSaA blfy, rhljs LraHk (C3) osQ vuqfn'k
izlj.k djus ij geas izkIr gksrk gS fd
–1 3 1 2 1 2
∆= 4 –0 +0
4 1 4 1 –1 3
= 4 (–1 – 12) – 0 + 0 = – 52

0 sin α – cos α
mnkgj.k 4 ∆ = – sin α 0 sin β dk eku Kkr dhft,A
cos α – sin β 0

gy R1 osQ vuqfn'k izlj.k djus ij gesa izkIr gksrk gS fd

0 sin β – sin α sin β – sin α 0
∆= 0 – sin α – cos α
– sin β 0 cos α 0 cos α – sin β
= 0 – sin α (0 – sin β cos α) – cos α (sin α sin β – 0)
= sin α sin β cos α – cos α sin α sin β = 0
3 x 3 2
mnkgj.k 5 ;fn = rks x osQ eku Kkr dhft,A
x 1 4 1

3 x 3 2
gy fn;k gS fd =
x 1 4 1
vFkkZr~ 3 – x2 = 3 – 8
vFkkZr~ x2 = 8
vr% x= ±2 2

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iz'ukoyh 4-1
iz'u 1 ls 2 rd esa lkjf.kdksa dk eku Kkr dhft,
2 4
1.
–5 –1

cos θ – sin θ x2 – x + 1 x – 1
2. (i) (ii)
sin θ cos θ x +1 x +1

1 2
3. ;fn A=   , rks fn[kkb, | 2A | = 4 | A |
 4 2

1 0 1
 
4. ;fn A =  0 1 2  gks] rks fn[kkb, | 3 A | = 27 | A |
 0 0 4 

5. fuEufyf[kr lkjf.kdksa dk eku Kkr dhft,

3 –1 –2 3 –4 5 0 1 2
(i) 0 0 –1 (ii) 1 1 –2 (iii) –1 0 –3
3 –5 0 2 3 1 –2 3 0

2 –1 –2
(iv) 0 2 –1
3 –5 0

1 1 2
 3  , gks rks | A | Kkr dhft,A
6. ;fn A= 2 1 
 5 4 9 

7. x osQ eku Kkr dhft, ;fn

2 4 2x 4 2 3 x 3
(i) = (ii) =
5 1 6 x 4 5 2x 5

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x 2 6 2
8. ;fn = gks rks x cjkcj gS%
18 x 18 6
(A) 6 (B) ± 6 (C) – 6 (D) 0
4.3 lkjf.kdksa osQ xq.k/eZ (Properties of Determinants)
fiNys vuqPNsn esa geus lkjf.kdksa dk izlj.k djuk lh[kk gSA bl vuqPNsn esa ge lkjf.kdksa osQ oqQN
xq.k/eks± dks lwphc¼ djsaxs ftlls ,d iafDr ;k LraHk esa 'kwU; dh la[;kvksa dks vf/dre izkIr
djus ls budk eku Kkr djuk ljy gks tkrk gSA ;s xq.k/eZ fdlh Hkh dksfV osQ lkjf.kd osQ fy,
lR; gSa ¯drq ge Lo;a dks bUgsa osQoy rhljh dksfV rd osQ lkjf.kdksa rd lhfer j[ksaxsA
xq.k/eZ 1 fdlh lkjf.kd dk eku bldh iafDr;ksa vkSj LraHkksa osQ ijLij ifjofrZr djus ij
vifjofrZr jgrk gSA
a1 a2 a3
lR;kiu – eku yhft, ∆ = b1 b2 b3
c1 c2 c3
izFke iafDr osQ vuqfn'k izlj.k djus ij] ge izkIr djrs gSa fd
b2 b3 b1 b3 b1 b2
∆ = a1 − a2 + a3
c2 c3 c1 c3 c1 c2
= a1 (b2 c3 – b3 c2) – a2 (b1 c3 – b3 c1) + a3 (b1 c2 – b2 c1)
∆ dh iafDr;ksa dks LraHkksa esa ifjofrZr djus ij gesa lkjf.kd
a1 b1 c1
∆1 = a2 b2 c2 izkIr gksrk gSA
a3 b3 c3

∆1 dks izFke LraHk osQ vuqfn'k izlj.k djus ij ge ikrs gSa fd

∆1 = a1 (b2 c3 – c2 b3) – a2 (b1 c3 – b3 c1) + a3 (b1 c2 – b2 c1)
vr% ∆ = ∆1
fVIi.kh mi;qZDr O;k[;k ls Li"V gS fd ;fn A ,d oxZ vkO;wg gS rks det (A) = det (A′), tgk¡
A′, Α dk ifjorZ gSA

AfVIi.kh ;fn Ri = i oha iafDr vkSj Ci = i ok¡ LraHk gS] rks iafDr;ksa vkSj LraHkksa osQ ijLij
ifjorZu dks ge laosQru esa Ci ↔ Ri fy[ksaxsA
vkb, ge mijksDr xq.k/eZ dks mnkgj.k }kjk lR;kfir djsaA

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2 –3 5
mnkgj.k 6 ∆ = 6 0 4 osQ fy, xq.k/eZ 1 dk lR;kiu dhft,A
1 5 –7
gy lkjf.kd dk izFke iafDr osQ vuqfn'k izlj.k djus ij]
0 4 6 4 6 0
∆= 2 – (–3) +5
5 –7 1 –7 1 5
= 2 (0 – 20) + 3 (– 42 – 4) + 5 (30 – 0)
= – 40 – 138 + 150 = – 28
iafDr;ksa vkSj LraHkksa dks ijLij ifjorZu djus ij gesa izkIr gksrk gSA
2 6 1
∆1 = –3 0 5 (igys LraHk osQ vuqfn'k izlj.k djus ij)
5 4 –7

0 5 6 1 6 1
= 2 – (–3) +5
4 –7 4 –7 0 5
= 2 (0 – 20) + 3 (– 42 – 4) + 5 (30 – 0)
= – 40 – 138 + 150 = – 28
Li"Vr% ∆ = ∆1
vr% xq.k/eZ 1 lR;kfir gqvkA
xq.k/eZ 2 ;fn ,d lkjf.kd dh dksbZ nks iafDr;ksa (;k LraHkksa) dks ijLij ifjofrZr dj fn;k tkrk
gS] rc lkjf.kd dk fpÉ ifjofrZr gks tkrk gSA

a1 a2 a3
lR;kiu eku yhft, ∆ = b1 b2 b3
c1 c2 c3

izFke iafDr osQ vuqfn'k izlj.k djus ij ge ikrs gSa

∆ = a1 (b2 c3 – b3 c2) – a2 (b1 c3 – b3 c1) + a3 (b1 c2 – b2 c1)
igyh vkSj rhljh iafDr;ksa dks ijLij ifjofrZr djus vFkkZr~ R2↔R3 ls izkIr u;k lkjf.kd

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c1 c2 c3
∆1 = b1 b2 b3
a1 a2 a3

gSA bls rhljh iafDr osQ vuqfn'k izlj.k djus ij]

∆1 = a1 (c2 b3 – b2 c3) – a2 (c1 b3 – c3 b1) + a3 (b2 c1 – b1 c2)
= – [a1 (b2 c3 – b3 c2) – a2 (b1 c3 – b3 c1) + a3 (b1 c2 – b2 c1)] izkIr gksrk gSA
;g Li"V gS fd ∆1 = – ∆
blh izdkj] ge fdUgha nks LraHkksa dks ijLij ifjofrZr djds mDr ifj.kke dks lR;kfir dj ldrs gSAa

AfVIi.kh ge iafDr;ksa osQ ijLij ifjorZu dks Ri ↔ Rj vkSj LraHkksa osQ ijLij ifjorZu dks
Ci ↔ Cj osQ }kjk fufnZ"V djrs gSaA

2 –3 5
mnkgj.k 7 ;fn ∆ = 6 0 4 gS rks xq.k/eZ 2 dk lR;kiu dhft,A
1 5 –7

2 –3 5
gy ge Kkr dj pqosQ gSa fd ∆ = 6 0 4 = – 28 (nsf[k, mnkgj.k 6)
1 5 –7
R2 vkSj R3 dks ijLij ifjofrZr djus ij vFkkZr~ R2 ↔ R3 ls
2 –3 5
∆1 = 1 5 –7 izkIr gksrk gSA
6 0 4
lkjf.kd ∆1 dks igyh iafDr osQ vuqfn'k izlj.k djus ij ge izkIr djrs gSa fd
5 –7 1 –7 1 5
∆1 = 2 – (–3) +5
0 4 6 4 6 0
= 2 (20 – 0) + 3 (4 + 42) + 5 (0 – 30)
= 40 + 138 – 150 = 28
Li"Vr;k ∆1 = – ∆
vr% xq.k/eZ 2 lR;kfir gqvkA

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xq.k/eZ 3 ;fn ,d lkjf.kd dh dksbZ nks iafDr;k¡ (vFkok LraHk) leku gSa (lHkh laxr vo;o
leku gSa)] rks lkjf.kd dk eku 'kwU; gksrk gSA
miifÙk ;fn ge lkjf.kd ∆ dh leku iafDr;ksa (;k LraHkksa) dks ijLij ifjofrZr dj nsrs gaS rks ∆
dk eku ifjofrZr ugha gksrk gSA
rFkkfi] xq.k/eZ 2 osQ vuqlkj ∆ dk fpÉ cny x;k gSA
blfy, ∆=– ∆
;k ∆=0
vkb, ge mijksDr xq.k/eZ dk ,d mnkgj.k osQ }kjk lR;kiu djrs gSaA
3 2 3
mnkgj.k 8 ∆ = 2 2 3 dk eku Kkr dhft,A
3 2 3
gy igyh iafDr osQ vuqfn'k izlj.k djus ij ge izkIr djrs gSa fd
∆ = 3 (6 – 6) – 2 (6 – 9) + 3 (4 – 6)
= 0 – 2 (–3) + 3 (–2) = 6 – 6 = 0
;gk¡ R2 vkSj R3 leku gSaA
xq.k/eZ 4 ;fn ,d lkjf.kd osQ fdlh ,d iafDr (vFkok LraHk) osQ izR;sd vO;o dks ,d vpj
k, ls xq.kk djrs gSa rks mldk eku Hkh k ls xqf.kr gks tkrk gSA
a1 b1 c1
lR;kiu eku yhft, ∆ = a2 b2 c2
a3 b3 c3
bldh izFke iafDr osQ vo;oksa dks k ls xq.kk djus ij izkIr lkjf.kd ∆1 gS rks
k a1 k b1 k c1
∆1 = a2 b2 c2
a3 b3 c3
izFke iafDr osQ vuqfn'k izlj.k djus ij] ge izkIr djrs gSa fd
∆1 = k a1 (b2 c3 – b3 c2) – k b1 (a2 c3 – c2 a3) + k c1 (a2 b3 – b2 a3)
= k [a1 (b2 c3 – b3 c2) – b1 (a2 c3 – c2 a3) + c1 (a2 b3 – b2 a3)] = k ∆
k a1 k b1 k c1 a1 b1 c1
vr% a2 b2 c2 = k a2 b2 c2
a3 b3 c3 a3 b3 c3

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fVIi.kh
(i) bl xq.k/eZ osQ vuqlkj] ge ,d lkjf.kd dh fdlh ,d iafDr ;k LrHkksa ls lkoZ mHk;fu"B
xq.ku[kaM ckgj fudky ldrs gSaA
(ii) ;fn ,d lkjf.kd dh fdUgha nks iafDr;ksa (;k LraHkksa) osQ laxr vo;o lekuqikrh (mlh
vuqikr esa) gS] rc mldk eku 'kwU; gksrk gSA mnkgj.kr%
a1 a2 a3
∆= b1 b2 b3 = 0 (iafDr;k¡ R2 o R3 lekuqikrh gS)
k a1 k a2 k a3

102 18 36
mnkgj.k 9 lkjf.kd 1 3 4 dk eku Kkr dhft,
17 3 6

102 18 36 6(17) 6(3) 6(6) 17 3 6

gy è;ku nhft, fd 1 3 4 = 1 3 4 =6 1 3 4 =0
17 3 6 17 3 6 17 3 6
(xq.k/eZ 3 vkSj 4 )
xq.k/eZ 5 ;fn ,d lkjf.kd dh ,d iafDr ;k LraHk osQ oqQN ;k lHkh vo;o nks (;k vf/d)
inksa osQ ;ksxiQy osQ :i esa O;Dr gksa rks lkjf.kd dks nks (;k vf/d) lkjf.kdksa osQ ;ksxiQy osQ
:i esa O;Dr fd;k tk ldrk gSA
a1 + λ1 a2 + λ 2 a3 + λ 3 a1 a2 a3 λ1 λ2 λ3
mnkgj.kr;k b1 b2 b3 = b1 b2 b3 + b1 b2 b3
c1 c2 c3 c1 c2 c3 c1 c2 c3

a1 + λ1 a2 + λ 2 a3 + λ 3
lR;kiu ck¡;k i{k = b1 b2 b3
c1 c2 c3

izFke iafDr osQ vuqfn'k izlj.k djus ij ge ikrs gSa fd

∆ = (a1 + λ1) (b2 c3 – c2 b3) – (a2 + λ2) (b1 c3 – b3 c1)
+ (a3 + λ3) (b1 c2 – b2 c1)

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= a1 (b2 c3 – c2 b3) – a2 (b1 c3 – b3 c1) + a3 (b1 c2 – b2 c1)

+ λ1 (b2 c3 – c2 b3) – λ2 (b1 c3 – b3 c1) + λ3 (b1 c2 – b2 c1)
(inksa dks O;ofLFkr djus ij)

a1 a2 a3 λ1 λ2 λ3
= b1 b2 b3 + b1 b2 b3 = nk¡;k i{k
c1 c2 c3 c1 c2 c3

blh izdkj nwljh iafDr;ksa o LraHkksa osQ fy, ge xq.k/eZ 5 dk lR;kiu dj ldrs gSaA
a b c
mnkgj.k 10 n'kkZb, fd a + 2 x b + 2 y c + 2 z = 0
x y z

a b c a b c a b c
gy ge tkurs gSa fd a + 2 x b + 2 y c + 2 z = a b c + 2 x 2 y 2 z
x y z x y z x y z

(xq.k/eZ 5 osQ }kjk)

=0+0=0 (xq.k/eZ 3 vkSj 4 dk iz;ksx djus ij)
xq.k/eZ 6 ;fn ,d lkjf.kd osQ fdlh iafDr ;k LraHk osQ izR;sd vo;o esa] nwljh iafDr ;k LraHk
osQ laxr vo;oksa osQ leku xq.ktksa dks tksM+ fn;k tkrk gS rks lkjf.kd dk eku ogh jgrk gSA vFkkZr~]
;fn ge Ri → Ri + kRj ;k Ci → Ci + k Cj dk iz;ksx djsa rks lkjf.kd dk eku ogh jgrk gSA
lR;kiu
a1 a2 a3 a1 + k c1 a2 + k c2 a3 + k c3
eku yhft, ∆ = b1 b2 b3 vkSj ∆ =
1
b1 b2 b3 ,
c1 c2 c3 c1 c2 c3

tgk¡ ∆1 lafØ;k R1 → R1 + kR3 osQ iz;ksx }kjk izkIr gksrk gS

;gk¡ ge rhljh iafDr (R3) osQ vo;oksa dks vpj k ls xq.kk djosQ vkSj mUgsa igyh iafDr (R1) osQ
laxr vo;oksa esa tksM+rs gSaA
laosQru }kjk bl lafØ;k dks bl izdkj fy[krs gSa fd R1 → R1 + k R3

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vc iqu%
a1 a2 a3 k c1 k c2 k c3
∆1 = b1 b2 b3 + b1 b2 b3 (xq.k/eZ 5 osQ }kjk)
c1 c2 c3 c1 c2 c3

=∆+0 (tc fd R1 vkSj R3 lekuqikrh gSa)

vr% ∆ = ∆1
fVIi.kh
(i) ;fn lkjf.kd ∆ esa Ri → kRi ;k Ci → kCi osQ iz;ksx ls izkIr lkjf.kd ∆1 gS] rks
∆1 = k∆.
(ii) ;fn ,d lkFk Ri → Ri + kRj tSlh lafØ;kvksa dk ,d ls vf/d ckj iz;ksx fd;k x;k
gks rks è;ku nsuk pkfg, fd igyh lafØ;k ls izHkkfor iafDr dk vU; lafØ;k esa iz;ksx ugha
gksuk pkfg,A Bhd blh izdkj dh fVIi.kh LraHkksa dh lafØ;kvksa esa iz;ksx dh tkrh gSA
a a+b a+b+c
mnkgj.k 11 fl¼ dhft, fd ∆ = 2a 3a + 2b 4a + 3b + 2c = a
3

3a 6a + 3b 10a + 6b + 3c
gy lkjf.kd ∆ esa R2 → R2 – 2R1 vkSj R3 → R3 – 3R1 dk iz;ksx djus ij ge ikrs gSa fd
a a+b a+b+c
∆= 0 a 2a + b
0 3a 7a + 3b
iqu% R3 → R3 – 3R2 , dk iz;ksx djus ls ge ikrs gSa fd
a a+b a+b+c
∆= 0 a 2a + b
0 0 a
C1 osQ vuqfn'k izlj.k djus ij
a 2a + b
∆= a +0+0
0 a
= a (a2 – 0) = a (a2) = a3 izkIr gksrk gSA

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mnkgj.k 12 izlj.k fd, fcuk fl¼ dhft, fd

x+ y y+z z+x
∆= z x y =0
1 1 1

gy ∆ esa R1 → R1 + R2 dk iz;ksx djus ij ge ikrs gSa

x+ y+z x+ y+z x+ y+z

∆= z x y
1 1 1

vc R1 vkSj R3 osQ vo;o lekuqikrh gSaA

blfy, ∆=0

mnkgj.k 13 fuEufyf[kr dk eku Kkr dhft,

1 a bc
∆ = 1 b ca
1 c ab

gy R2 → R2 – R1 vkSj R3 → R3 – R1, dk iz;ksx djus ij ge ikrs gSa fd

1 a bc
∆= 0 b − a c ( a − b)
0 c − a b (a − c )

R2 vkSj R3 ls Øe'k% (b – a) vkSj (c – a) mHk;fu"B ysus ij ge ikrs gSa fd

1 a bc
∆ = (b − a ) (c − a ) 0 1 –c
0 1 –b

= (b – a) (c – a) [(– b + c)] (igys LraHk osQ vuqfn'k izlj.k djus ij)

= (a – b) (b – c) (c – a)

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b+c a a
mnkgj.k 14 fl¼ dhft, fd b c+a b = 4 abc
c c a+b
gy eku yhft,
b+c a a
∆= b c+a b
c c a+b
lkjf.kd ij R1 → R1 – R2 – R3 dk iz;ksx djus ij ge ikrs gSa fd
0 –2c –2b
∆= b c+ a b
c c a+b
R1 osQ vuqfn'k izlj.k djus ij ge ikrs gSa fd
c+a b b b b c+a
∆= 0 – (–2 c ) + (–2b)
c a+b c a+b c c
= 2 c (a b + b2 – bc) – 2 b (b c – c2 – ac)
= 2 a b c + 2 cb2 – 2 bc2 – 2 b2c + 2 bc2 + 2 abc
= 4 abc

x x2 1 + x3
mnkgj.k 15 ;fn x, y, z fofHkUu gksa vkSj ∆ = y y 2 1 + y 3 = 0 ,
z z2 1 + z3
rks n'kkZb, fd 1 + xyz = 0
x x2 1 + x3
gy gesa Kkr gS ∆ = y y 2 1 + y 3
z z2 1 + z3

x x2 1 x x2 x3
∆= y y2 1 + y y2 y 3 (xq.k/eZ 5 osQ iz;ksx }kjk)
z z2 1 z z2 z3

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1 x x2 1 x x2
= ( −1) 1 y
2
y 2 + xyz 1 y y 2 (C3 ↔ C2 vkSj rc C1 ↔ C2 osQ iz;ksx }kjk)
1 z z2 1 z z2

1 x x2
= 1 y y 2 (1 + xyz )
1 z z2

1 x x2
= (1 + xyz ) 0 y−x y 2 − x 2 (R2→R2– R1 vkSj R3 → R3–R1 dk iz;ksx djus ij)
0 z−x z 2 − x2
R2 ls (y – x) vkSj R3 ls (z – x) mHk;fu"B ysus ij ge izkIr djrs gSa fd

1 x x2
∆ = (1+xyz ) (y –x ) (z –x) 0 1 y+x
0 1 z+x

= (1 + xyz) (y – x) (z – x) (z – y) (C1 osQ vuqfn'k izlj.k djus ij)

pw¡fd ∆ = 0 vkSj x, y vkSj z lHkh fHkUu gSa]
vr% x – y ≠ 0, y – z ≠ 0, z – x ≠ 0, ls gesa 1 + xyz = 0 izkIr gksrk gSA
1+ a 1 1
 1 1 1
mnkgj.k 16 n'kkZb, fd 1 1 + b 1 = abc 1 + + +  = abc + bc + ca + ab
a b c
1 1 1+ c
gy R1, R2 vkSj R3 esa ls Øe'k% a, b vkSj c mHk;fu"B ysus ij ge izkIr djrs gSa fd
1 1 1
+1
a a a
1 1 1
ck¡;k i{k = abc b b + 1 b
1 1 1
+1
c c c
R1→ R1 + R2 + R3 dk iz;ksx djus ij ge ikrs gSa fd

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 lkjf.kd 129

1 1 1 1 1 1 1 1 1
1+ + + 1+ + + 1+ + +
a b c a b c a b c
1 1 1
∆ = abc +1
b b b
1 1 1
+1
c c c
1 1 1
 1 1 1 1 1 1
;k ∆ = abc 1+ + +  +1
 a b c b b b
1 1 1
+1
c c c
vc C2 → C2 – C1 vkSj C3 → C3 – C1 dk iz;ksx djus ij ge ikrs gSa fd
1 0 0
 1 1 1 1
∆ = abc  1+ + +  1 0
 a b c b
1
0 1
c
 1 1 1
= abc 1 + + +  1(1 – 0 )
 a b c

 1 1 1
= abc 1+ + +  = abc + bc + ca + ab = nk¡;k i{k
 a b c

AfVIi.kh vU; fof/ }kjk C1 → C1 – C2 o C3 → C3 – C2, dk vuqiz;ksx djosQ rFkk

C1→ C1 – a C3 dk iz;ksx djosQ mijksDr mnkgj.k dks gy djus dk iz;Ru djsaA

iz'ukoyh 4.2
fcuk izlj.k fd, vkSj lkjf.kdksa osQ xq.k/eks± dk iz;ksx djosQ fuEufyf[kr iz'u 1 ls 5 dks fl¼
dhft,A
x a x+a a −b b−c c−a 2 7 65
1. y b y +b =0 2. b − c c − a a − b = 0 3. 3 8 75 = 0
z c z+c c−a a−b b−c 5 9 86

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1 bc a ( b + c ) b+c q+r y+z a p x

4. 1 ca b ( c + a ) = 0 5. c+a r+ p z+x = 2 b q y
1 ab c ( a + b ) a+b p+q x+ y c r z

lkjf.kdksa osQ xq.k/eks± dk iz;ksx djosQ iz'u 6 ls 14 rd dks fl¼ dhft,%

0 a −b −a 2 ab ac
6. − a 0 −c = 0 7. ba −b 2
bc = 4 a 2 b 2 c 2
b c 0 ca cb −c 2

1 a a2
1 b b 2 = ( a − b )(b − c )(c − a )
8. (i)
1 c c2

1 1 1
a b c = ( a − b )(b − c )(c − a )( a + b + c )
(ii)
3 3
a b c3

x x2 yz
2
9. y y zx = (x – y) (y – z) (z – x) (xy + yz + zx)
2
z z xy

x+4 2x 2x
2x = (5 x + 4 )( 4 − x )
2
10. (i) 2x x+4
2x 2x x+4

y+k y y
(ii) y y+k y = k 2 (3 y + k )
y y y+k

a −b −c 2a 2a
b−c−a = (a + b + c )
3
11. (i) 2b 2b
2c 2c c − a −b

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 lkjf.kd 131

x + y + 2z x y
= 2( x + y + z )
3
(ii) z y + z + 2x y
z x z + x + 2y

1 x x2
x = (1 − x 3 )
2
12. x2 1
x x2 1

1 + a 2 − b2 2ab −2b
( )
3
13. 2ab 1− a + b
2 2
2a = 1 + a2 + b2
2b −2a 1 − a2 − b2

a2 + 1 ab ac
14. ab b +1
2
bc =1 + a 2 + b 2 + c 2
ca cb c2 + 1

iz'u la[;k 15 rFkk 16 esa lgh mÙkj pqfu,A

15. ;fn A ,d 3 × 3 dksfV dk oxZ vkO;wg gS rks | kA | dk eku gksxk%
(A) k | A | (B) k 2 | A | (C) k 3 | A | (D) 3k | A |
16. fuEufyf[kr esa ls dkSu lk dFku lgh gSA
(A) lkjf.kd ,d oxZ vkO;wg gSA
(B) lkjf.kd ,d vkO;wg ls lac¼ ,d la[;k gSA
(C) lkjf.kd ,d oxZ vkO;wg ls lac¼ ,d la[;k gSA
(D) buesa ls dksbZ ughaA
4.4 f=kHkqt dk {ks=kiQy (Area of a Triangle)
geus fiNyh d{kkvksa esa lh[kk gS fd ,d f=kHkqt ftlosQ 'kh"kZ¯cnq (x1, y1), (x2, y2) rFkk (x3, y3),
1
gksa rks mldk {ks=kiQy O;atd [x (y –y ) + x2 (y3–y1) + x3 (y1–y2)] }kjk O;Dr fd;k tkrk
2 1 2 3
gSA vc bl O;atd dks lkjf.kd osQ :i esa bl izdkj fy[kk tk ldrk gS%
x1 y1 1
1
∆ = x2 y2 1 ... (1)
2
x3 y3 1

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fVIi.kh
(i) D;ksafd {ks=kiQy ,d /ukRed jkf'k gksrh gS blfy, ge lnSo (1) esa lkjf.kd dk fujis{k
eku ysrs gSaA
(ii) ;fn {ks=kiQy fn;k gks rks x.kuk osQ fy, lkjf.kd dk /ukRed vkSj ½.kkRed nksuksa ekuksa dk
iz;ksx dhft,A
(iii) rhu lajs[k ¯cnqvksa ls cus f=kHkqt dk {ks=kiQy 'kwU; gksxkA

mnkgj.k 17 ,d f=kHkqt dk {ks=kiQy Kkr dhft, ftlosQ 'kh"kZ (3, 8), (– 4, 2) vkSj (5, 1) gSaA
gy f=kHkqt dk {ks=kiQy%
3 8 1
1 1
∆= –4 2 1 3 ( 2 – 1) – 8 ( – 4 – 5) + 1( – 4 – 10 )
2
=
2
5 1 1

1 61
= (3 + 72 – 14 ) =
2 2

mnkgj.k 18 lkjf.kdksa dk iz;ksx djosQ A(1, 3) vkSj B (0, 0) dks tksM+us okyh js[kk dk lehdj.k
Kkr dhft, vkSj k dk eku Kkr dhft, ;fn ,d ¯cnq D(k, 0) bl izdkj gS fd ∆ ABD dk
{ks=kiQy 3 oxZ bdkbZ gSA
gy eku yhft, AB ij dksbZ ¯cnq P (x, y) gS rc ∆ ABP dk {ks=kiQy = 0 (D;ksa?)
0 0 1
1
blfy, 1 3 1 = 0
2
x y 1

1
blls izkIr gS ( y – 3 x ) = 0 ;k y = 3x
2
tks vHkh"V js[kk AB dk lehdj.k gSA
fdarq ∆ ABD dk {ks=kiQy 3 oxZ bdkbZ fn;k gS vr%
1 3 1
1 − 3k
0 0 1 = ± 3 gesa izkIr gS = ± 3 , i.e., k = ∓ 2
2 2
k 0 1

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iz'ukoyh 4-3
1. fuEufyf[kr izR;sd esa fn, x, 'kh"kZ ¯cnqvksa okys f=kHkqtksa dk {ks=kiQy Kkr dhft,A
(i) (1, 0), (6, 0), (4, 3) (ii) (2, 7), (1, 1), (10, 8)
(iii) (–2, –3), (3, 2), (–1, –8)
2. n'kkZb, fd ¯cnq A (a, b + c), B (b, c + a) vkSj C (c, a + b) lajs[k gSaA
3. izR;sd esa k dk eku Kkr dhft, ;fn f=kHkqtksa dk {ks=kiQy 4 oxZ bdkbZ gS tgk¡ 'kh"kZ¯cnq
fuEufyf[kr gaS%
(i) (k, 0), (4, 0), (0, 2) (ii) (–2, 0), (0, 4), (0, k)
4. (i) lkjf.kdksa dk iz;ksx djosQ (1, 2) vkSj (3, 6) dks feykus okyh js[kk dk lehdj.k Kkr
dhft,A
(ii) lkjf.kdksa dk iz;ksx djosQ (3, 1) vkSj (9, 3) dks feykus okyh js[kk dk lehdj.k Kkr
dhft,A
5. ;fn 'kh"kZ (2, – 6), (5, 4) vkSj (k, 4) okys f=kHkqt dk {ks=kiQy 35 oxZ bdkbZ gks rks k dk
eku gS%
(A) 12 (B) –2 (C) –12, –2 (D) 12, –2
4.5 milkjf.kd vkSj lg[kaM (Minor and Co-factor)
bl vuqPNsn esa ge milkjf.kdksa vkSj lg[kaMksa dk iz;ksx djosQ lkjf.kdks osQ izlj.k dk foLr`r :i
fy[kuk lh[ksaxsA
ifjHkk"kk 1 lkjf.kd osQ vo;o aij dk milkjf.kd ,d lkjf.kd gS tks i oh iafDr vkSj j ok¡ LraHk
ftlesa vo;o aij fLFkr gS] dks gVkus ls izkIr gksrk gSA vo;o aij osQ milkjf.kd dks Mij osQ }kjk
O;Dr djrs gSaA
fVIi.kh n(n ≥ 2) Øe osQ lkjf.kd osQ vo;o dk milkjf.kd n – 1 Øe dk lkjf.kd gksrk gSA
1 2 3
mnkgj.k 19 lkjf.kd ∆ = 4 5 6 esa vo;o 6 dk milkjf.kd Kkr dhft,A
7 8 9
gy D;ksafd 6 nwljh iafDr ,oa r`rh; LraHk esa fLFkr gSA blfy, bldk milkfj.kd = M23
fuEufyf[kr izdkj ls izkIr gksrk gSA
1 2
M23 = = 8 – 14 = – 6 (∆ ls R2 vkSj C3 gVkus ij)
7 8

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ifjHkk"kk 2 ,d vo;o aij dk lg[kaM ftls Aij }kjk O;Dr djrs gSa] tgk¡
Aij = (–1)i + j Mij,
osQ }kjk ifjHkkf"kr djrs gSa tgk¡ aij dk milkjf.kd Mij gSA

1 –2
mnkgj.k 20 lkjf.kd osQ lHkh vo;oksa osQ milkjf.kd o lg[kaM Kkr dhft,A
4 3

gy vo;o aij dk milkjf.kd Mij gSA

;gk¡ a11 = 1, blfy, M11 = a11dk milkjf.kd = 3
M12 = vo;o a12 dk milkjf.kd = 4
M21 = vo;o a21 dk milkjf.kd = – 2
M22 = vo;o a22 dk milkjf.kd = 1
vc aij dk lg[kaM Aij gSA blfy,
A11 = (–1)1 + 1 M11 = (–1)2 (3) = 3
A12 = (–1)1 + 2 M12 = (–1)3 (4) = – 4
A21 = (–1)2 + 1 M21 = (–1)3 (–2) = 2
A22 = (–1)2 + 2 M22 = (–1)4 (1) = 1

a11 a12 a13

mnkgj.k 21 ∆ = a21 a22 a23 osQ vo;oksa a11 rFkk a21 osQ milkjf.kd vkSj lg[kaM
a31 a32 a33
Kkr dhft,A
gy milkjf.kd vkSj lg[kaM dh ifjHkk"kk }kjk ge ikrs gSa%
a22 a23
a11 dk milkjf.kd = M11 = = a22 a33– a23 a32
a32 a33
a11 dk lg[kaM = A11 = (–1)1+1 M11 = a22 a33 – a23 a32
a12 a13
a21 dk milkjf.kd = M21 = = a12 a33 – a13 a32
a32 a33
a21 dk lg[kaM = A21 = (–1)2+1 M21 = (–1) (a12 a33 – a13 a32) = – a12 a33 + a13 a32

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 lkjf.kd 135

fVIi.kh mnkgj.k 21 esa lkjf.kd ∆ dk R1 osQ lkis{k izlj.k djus ij ge ikrs gSa fd
a22 a23 a21 a23 a21 a22
∆ = (–1) 1+1
a11 a32 a33 + (–1)1+2
a 12 a a33 + (–1)
1+3
a13 a31 a32
31

= a11 A11 + a12 A12 + a13 A13, tgk¡ aij dk lg[kaM Aij gSaA
= R1 osQ vo;oksa vkSj muosQ laxr lg[kaMksa osQ xq.kuiQy dk ;ksxA
blh izdkj ∆ dk R2, R3, C1, C2 vkSj C3 osQ vuqfn'k 5 izlj.k vU; izdkj ls gSaA
vr% lkjf.kd ∆ , fdlh iafDr (;k LraHk) osQ vo;oksa vkSj muosQ laxr lg[kaMksa osQ xq.kuiQy
dk ;ksx gSA

AfVIi.kh ;fn ,d iafDr (;k LraHk) osQ vo;oksa dks vU; iafDr (;k LraHk) osQ lg[kaMksa
ls xq.kk fd;k tk, rks mudk ;ksx 'kwU; gksrk gSA mnkgj.kr;k] ekuk ∆ = a11 A21 + a12 A22
+ a13 A23 rc%

a12 a13 a11 a13 a a

∆ = a11 (–1)1+1 a 1+2
a33 + a12 (–1) a + a13 (–1)1+3 11 12
32
31 a33 a31 a32

a11 a12 a13

= a11 a12 a13 = 0 ( D;ksafd R vkSj R leku gSa)
1 2
a31 a32 a33

blh izdkj ge vU; iafDr;ksa vkSj LraHkksa osQ fy, iz;Ru dj ldrs gSaA
2 –3 5
mnkgj.k 22 lkjf.kd 6 0 4 osQ vo;oksa osQ milkjf.kd vkSj lg[kaM Kkr dhft, vkSj
1 5 –7
lR;kfir dhft, fd a11 A31 + a12 A32 + a13 A33= 0 gSA
0 4
gy ;gk¡ M11 = 5 –7 = 0 –20 = –20; blfy, A11 = (–1)1+1 (–20) = –20

6 4
M12 = 1 –7 = – 42 – 4 = – 46; blfy, A12 = (–1)1+2 (– 46) = 46

6 0
M13 = = 30 – 0 = 30; blfy, A13 = (–1)1+3 (30) = 30
1 5

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–3 5
M21 = = 21 – 25 = – 4; blfy, A21 = (–1)2+1 (– 4) = 4
5 –7

2 5
M22 = = –14 – 5 = –19; blfy, A22 = (–1)2+2 (–19) = –19
1 –7

2 –3
M23 = = 10 + 3 = 13; blfy, A23 = (–1)2+3 (13) = –13
1 5

–3 5
M31 = = –12 – 0 = –12; blfy, A31 = (–1)3+1 (–12) = –12
0 4

2 5
M32 = = 8 – 30 = –22; blfy, A32 = (–1)3+2 (–22) = 22
6 4

2 –3
vkSj M33 =
6 0
= 0 + 18 = 18; blfy, A33 = (–1)3+3 (18) = 18

vc a11 = 2, a12 = –3, a13 = 5; rFkk A31 = –12, A32 = 22, A33 = 18 gSA
blfy, a11 A31 + a12 A32 + a13 A33
= 2 (–12) + (–3) (22) + 5 (18) = –24 – 66 + 90 = 0

iz'ukoyh 4-4
fuEufyf[kr lkjf.kdksa osQ vo;oksa osQ milkjf.kd ,oa lg[kaM fyf[k,A
2 –4 a c
1. (i) (ii)
0 3 b d

1 0 0 1 0 4
2. (i) 0 1 0 (ii) 3 5 –1
0 0 1 0 1 2

5 3 8
3. nwljh iafDr osQ vo;oksa osQ lg[kaMksa dk iz;ksx djosQ ∆ = 2 0 1 dk eku Kkr dhft,A
1 2 3

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1 x yz
4. rhljs LraHk osQ vo;oksa osQ lg[kaMksa dk iz;ksx djosQ ∆ = 1 y zx dk eku Kkr dhft,A
1 z xy

a11 a12 a13

5. ;fn ∆ = a21 a22 a23 vkSj aij dk lg[kaM Aij gks rks ∆ dk eku fuEufyf[kr :i esa
a31 a32 a33

O;Dr fd;k tkrk gS%

(A) a11 A31+ a12 A32 + a13 A33 (B) a11 A11+ a12 A21 + a13 A31
(C) a21 A11+ a22 A12 + a23 A13 (D) a11 A11+ a21 A21 + a31 A31

4.6 vkO;wg osQ lg[kaMt vkSj O;qRØe (Adjoint and Inverse of a Matrix)
fiNys vè;k; esa geus ,d vkO;wg osQ O;qRØe dk vè;;u fd;k gSA bl vuqPNsn esa ge ,d
vkO;wg osQ O;qRØe osQ vfLrRo osQ fy, 'krks± dh Hkh O;k[;k djsaxsA
A–1 Kkr djus osQ fy, igys ge ,d vkO;wg dk lg[kaMt ifjHkkf"kr djsaxsA

4.6.1 vkO;wg dk lg[kaMt (Adjoint of a matrix)

ifjHkk"kk 3 ,d oxZ vkO;wg A = [aij] dk lg[kaMt] vkO;wg [Aij] osQ ifjorZ osQ :i esa ifjHkkf"kr gS]
tgk¡ Aij] vo;o aij dk lg[kaM gSA vkO;wg A osQ lg[kaMt dks adj A osQ }kjk O;Dr djrs gSAa

 a11 a12 a13 

 a23  gSA
eku yhft, A =  a21 a22
 a31 a32 a33 

 A11 A12 A13   A11 A 21 A 31 

rc adj A =  A 21   A 32  gksrk gSA
 A 22 A 23  dk ifjorZ =  A12 A 22
 A31 A32 A 33   A13 A 23 A33 

 2 3
mnkgj.k 23 vkO;wg A =   dk lg[kaMt Kkr dhft,A
 1 4

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gy ge tkurs gSa fd A11 = 4, A12 = –1, A21 = –3, A22 = 2

 A11 A 21   4 –3
vr% adj A =  =
A 22   –1 2 
 A12

 a11 a12 
fVIi.kh 2 × 2 dksfV osQ oxZ vkO;wg A =  dk lg[kaMt adj A, a11 vkSj a22 dks ijLij
 a21 a22 

cnyus ,oa a12 vkSj a21 osQ fpÉ ifjofrZr dj nsus ls Hkh izkIr fd;k tk ldrk gS tSlk uhps n'kkZ;k
x;k gSA

ge fcuk miifÙk osQ fuEufyf[kr izes; fufnZ"V djrs gSaA

izes; 1 ;fn A dksbZ n dksfV dk vkO;wg gS rks] A(adj A) = (adj A) A = A I , tgk¡ I, n dksfV
dk rRled vkO;wg gSA
lR;kiu% eku yhft,

 a11 a12 a13   A11 A 21 A31 

 
a23 , gS rc adj A =  A12
A =  a21 a22
 A 22 A32 
 a31 a32 a33   A13 A 23 A 33 
D;ksafd ,d iafDr ;k LraHk osQ vo;oksa dk laxr lg[kaMksa dh xq.kk dk ;ksx | A | osQ leku gksrk
gS vU;Fkk 'kwU; gksrk gSA
A 0 0 1 0 0
 
bl izdkj A (adj A) =  0 A 0  = A 0 1 0 = A I
0 0 A  0 0 1 

blh izdkj] ge n'kkZ ldrs gSa fd (adj A) A = | A | I
vr% A (adj A) = (adj A) A = | A | I lR;kfir gSA

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ifjHkk"kk 4 ,d oxZ vkO;wg A vO;qRØe.kh; (singular) dgykrk gS ;fn A = 0 gSA

1 2
mnkgj.k osQ fy, vkO;wg A =  4 8  dk lkjf.kd 'kwU; gSA vr% A vO;qRØe.kh; gSA
 

ifjHkk"kk 5 ,d oxZ vkO;wg A O;qRØe.kh; (non-singular) dgykrk gS ;fn A ≠ 0

1 2  1 2
eku yhft, A =   gks rks A = 3 4 = 4 – 6 = – 2 ≠ 0 gSA
3 4
vr% A O;qRØe.kh; gSA
ge fuEufyf[kr izes; fcuk miifÙk osQ fufnZ"V dj jgs gSaA
izes; 2 ;fn A rFkk B nksuksa ,d gh dksfV osQ O;qRØe.kh; vkO;wg gksa rks AB rFkk BA Hkh mlh
dksfV osQ O;qRØe.kh; vkO;wg gksrs gaSA
izes; 3 vkO;wgksa osQ xq.kuiQy dk lkjf.kd muosQ Øe'k% lkjf.kdksa osQ xq.kuiQy osQ leku gksrk
gS vFkkZr~ AB = A B , tgk¡ A rFkk B leku dksfV osQ oxZ vkO;wg gSaA
A 0 0
 
fVIi.kh ge tkurs gSa fd (adj A) A = A I =  0 A 0
 0 0 A 

nksuksa vksj vkO;wgksa dk lkjf.kd ysus ij]

A 0 0
(adj A) A = 0 A 0
0 0 A

1 0 0
3
vFkkZr~ |(adj A)| |A| = A 0 1 0 (D;ksa?)
0 0 1

vFkkZr~ |(adj A)| |A| = A (1)

3

vFkkZr~ |(adj A)| = A

2

O;kid #i ls] ;fn n dksfV dk ,d oxZ vkO;wg A gks rks | adj A| = | A |n – 1 gksxkA

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izes; 4 ,d oxZ vkO;wg A osQ O;qRØe dk vfLrRo gS] ;fn vkSj osQoy ;fn A O;qRØe.kh; vkO;wg gSA
miifÙk eku yhft, n dksfV dk O;qRØe.kh; vkO;wg A gS vkSj n dksfV dk rRled vkO;wg I gSA
rc n dksfV osQ ,d oxZ vkO;wg B dk vfLrRo bl izdkj gks rkfd AB = BA = I
vc AB = I gS rks | AB | = | I | ;k | A | | B | = 1 (D;ksafd | I | = 1, | AB | = | A | | B |)
blls izkIr gksrk gS | A | ≠ 0. vr% A O;qRØe.kh; gSA
foykser% eku yhft, A O;qRØe.kh; gSA rc | A | ≠ 0
vc A (adj A) = (adj A) A = A I (izes; 1)

 1   1 
;k A adj A  =  adj A  A = I
|A|   |A| 

1
;k AB = BA = I, tgk¡ B =
|A|
adj A

1
vr% A osQ O;qRØe dk vfLrRo gS vkSj A–1 = adj A
|A|

1 3 3
 
mnkgj.k 24 ;fn A = 1 4 3 gks rks lR;kfir dhft, fd A. adj A = A . I vkSj A–1
1 3 4

Kkr dhft,A
gy ge ikrs gSa fd A = 1 (16 – 9) –3 (4 – 3) + 3 (3 – 4) = 1 ≠ 0
vc A11 = 7, A12 = –1, A13 = –1, A21 = –3, A22 = 1, A23 = 0, A31 = –3, A32 = 0, A33 = 1
 7 −3 −3
 
blfy, adj A =  −1 1 0 
 −1 0 1 

1 3 3  7 −3 −3
  
vc A.(adj A) = 1 4 3  −1 1 0 
1 3 4  −1 0 1 

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 7 − 3 − 3 −3 + 3 + 0 −3 + 0 + 3
 
=  7 − 4 − 3 −3 + 4 + 0 −3 + 0 + 3
 7 − 3 − 4 −3 + 3 + 0 −3 + 0 + 4

1 0 0  1 0 0
 0 1 0
=  0 1 0 = (1)   = A .I
 
 0 0 1 0 0 1

 7 −3 −3  7 −3 −3
⋅ adj A = −1 1 0  =  −1 1 0 
−11 1
vkSj A =
A 1   
 −1 0 1   −1 0 1 

2 3 1 −2 
mnkgj.k 25 ;fn A = 1 − 4 , B =  −1 3  , rks lR;kfir dhft, fd (AB)–1 = B–1A–1 gSA
   

2 3  1 −2   −1 5 
gy ge tkurs gSa fd AB =   = −14 
 1 − 4   −1 3   5

D;ksafd AB = –11 ≠ 0, (AB)–1 dk vfLrRo gS vkSj bls fuEufyf[kr izdkj ls O;Dr fd;k
tkrk gSA
1 1  −14 −5 = 1 14 5
. adj (AB) = − 
(AB)–1 =
AB 11  −5 −1 11  5 1

vkSj A = –11 ≠ 0 o B = 1 ≠ 0. blfy, A–1 vkSj B–1 nksuksa dk vfLrRo gS vkSj ftls
fuEufyf[kr :i esa O;Dr fd;k tk ldrk gSA
1  − 4 −3 −1 3 2 
A −1 = − ,B = 
11  −1 2  
1 1 

1 3 2  −4 −3 1  −14 −5 1 14 5

blfy, B−1 A −1 = −     =−   = 
11 1 1   −1 2  11  −5 −1 11  5 1

vr% (AB)–1 = B–1 A–1 gSA

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 2 3
mnkgj.k 26 iznf'kZr dhft, fd vkO;wg A = 1 2 lehdj.k A2 – 4A + I = O, tgk¡ I
 
2 × 2 dksfV dk ,d rRled vkO;wg gS vkSj O, 2 × 2 dksfV dk ,d 'kwU; vkO;wg gSA bldh
lgk;rk ls A–1 Kkr dhft,A
 2 3   2 3   7 12
gy ge tkurs gSa fd A 2 = A.A =    = 
1 2  1 2   4 7 
 7 12  8 12 1 0  0 0
vr% A2 – 4A + I =  −  +  = =O
 4 7   4 8  0 1  0 0
vc A2 – 4A + I = O
blfy, A A – 4A = – I
;k A A (A ) – 4 A A–1 = – I A–1 (nksuksa vksj A–1 ls mÙkj xq.ku }kjk D;ksafd |A| ≠ 0)
–1

;k A (A A–1) – 4I = – A–1
;k AI – 4I = – A–1

 4 0   2 3  2 −3
;k A–1 = 4I – A =   −  =  
 0 4  1 2  −1 2 

 2 −3
vr% A–1 =  
 −1 2 

iz'ukoyh 4-5
iz'u 1 vkSj 2 esa izR;sd vkO;wg dk lg[kaMt (adjoint) Kkkr dhft,

 1 −1 2
 1 2  
1.  3 4 2.  2 3 5 
   −2 0 1 

iz'u 3 vkSj 4 esa lR;kfir dhft, fd A (adj A) = (adj A) .A = A . I gSA

1 −1 2 
2 3 3 0 −2
3.  −4 −6 4.  
  1 0 3 

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iz'u 5 ls 11 esa fn, x, izR;sd vkO;wgksa osQ O;qRØe (ftudk vfLrRo gks) Kkr dhft,A
1 2 3
 2 −2  −1 5 0 2 4
5. 4 3  6.   7.  
   −3 2  0 0 5 

1 0 0   2 1 3  1 −1 2 
3 3 0   4 −1 0  
8.   9.   10.  0 2 −3
5 2 −1  −7 2 1   3 −2 4 

1 0 0 
0 cos α sin α 
11.  
0 sin α − cos α 

3 7  6 8
12. ;fn A =   vkSj B = 7 9 gS rks lR;kfir dhft, fd (AB)–1 = B–1 A–1 gSA
 2 5  
 3 1
13. ;fn A =   gS rks n'kkZb, fd A2 – 5A + 7I = O gS bldh lgk;rk ls A–1 Kkr dhft,A
 −1 2

 3 2
14. vkO;wg A =   osQ fy, a vkSj b ,slh la[;k,¡ Kkr dhft, rkfd
1 1 
A2 + aA + bI = O gksA

1 1 1 
 
15. vkO;wg A = 1 2 −3 osQ fy, n'kkZb, fd A3– 6A2 + 5A + 11 I = O gSA
 2 −1 3 

bldh lgk;rk ls A–1 Kkr dhft,A

 2 −1 1 
 
16. ;fn A =  −1 2 −1 , rks lR;kfir dhft, fd A3 – 6A2 + 9A – 4I = O gS rFkk
 1 −1 2 

bldh lgk;rk ls A–1 Kkr dhft,A

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17. ;fn A, 3 × 3 dksfV dk oxZ vkO;wg gS rks | adj A | dk eku gS%

(A) | A | (B) | A | 2 (C) | A | 3 (D) 3 | A |
18. ;fn A dksfV nks dk O;qRØeh; vkO;wg gS rks det (A–1) cjkcj%

1
(A) det (A) (B) det (A) (C) 1 (D) 0

4.7 lkjf.kdksa vkSj vkO;wgksa osQ vuqiz;ksx (Applications of Determinants and

Matrices)
bl vuqPNsn esa ge nks ;k rhu vKkr jkf'k;ksa osQ jSf[kd lehdj.k fudk; osQ gy vkSj jSf[kd
lehdj.kksa osQ fudk; dh laxrrk dh tk¡p esa lkjf.kdksa vkSj vkO;wgksa osQ vuqiz;ksxksa dk o.kZu djsaxsA
laxr fudk;% fudk; laxr dgykrk gS ;fn blosQ gyksa (,d ;k vf/d) dk vfLrRo gksrk gSA
vlaxr fudk;% fudk; vlaxr dgykrk gS ;fn blosQ fdlh Hkh gy dk vfLrRo ugha gksrk gSA

AfVIi.kh bl vè;k; esa ge vf}rh; gy osQ lehdj.k fudk; rd lhfer jgsaxsA

4.7.1 vkO;wg osQ O;qRØe }kjk jSf[kd lehdj.kksa osQ fudk; dk gy (Solution of a system
of linear equations using inverse of a matrix)
vkb, ge jSf[kd lehdj.kksa osQ fudk; dks vkO;wg lehdj.k osQ :i esa O;Dr djrs gSa vkSj vkO;wg
osQ O;qRØe dk iz;ksx djosQ mls gy djrs gSaA
fuEufyf[kr lehdj.k fudk; ij fopkj dhft,
a1 x + b1 y + c1 z = d 1
a2 x + b2 y + c2 z = d 2
a3 x + b3 y + c3 z = d 3

 a1 b1 c1   x  d1 
eku yhft, A =  a2 b2 c2  , X =  y  vkSj B =  d 
   2
 a3 b3 c3   z   d3 

rc lehdj.k fudk; AX = B osQ :i esa fuEufyf[kr izdkj ls O;Dr dh tk ldrh gSA

 a1 b1 c1   x   d1 
a b2 c2   y d 
 2   =  2
 a3 b3 c3   z   d3 

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fLFkfr 1 ;fn A ,d O;qRØe.kh; vkO;wg gS rc blosQ O;qRØe dk vfLrRo gSA vr% AX = B ls

ge ikrs gSa fd
A–1 (AX) = A–1 B (A–1 ls iwoZ xq.ku osQ }kjk)
;k (A–1A) X = A–1 B (lkgp;Z xq.ku }kjk)
;k IX=A B –1

;k X = A–1 B
;g vkO;wg lehdj.k fn, x, lehdj.k fudk; dk vf}rh; gy iznku djrk gS D;ksafd ,d
vkO;wg dk O;qRØe vf}rh; gksrk gSA lehdj.kksa osQ fudk; osQ gy djus dh ;g fof/ vkO;wg fof/
dgykrh gSA
fLFkfr 2 ;fn A ,d vO;qRØe.kh; vkO;wg gS rc | A | = 0 gksrk gSA
bl fLFkfr esa ge (adj A) B Kkr djrs gSaA
;fn (adj A) B ≠ O, (O 'kwU; vkO;wg gS), rc dksbZ gy ugha gksrk gS vkSj lehdj.k fudk;
vlaxr dgykrh gSA
;fn (adj A) B = O, rc fudk; laxr ;k vlaxr gksxh D;ksafd fudk; osQ vuar gy gksaxs ;k
dksbZ Hkh gy ugha gksxkA
mnkgj.k 27 fuEufyf[kr lehdj.k fudk; dks gy dhft,%
2x + 5y = 1
3x + 2y = 7
gy lehdj.k fudk; AX = B osQ :i esa fy[kk tk ldrk gS] tgk¡
2 5  x 1 
A=   , X =   vkSj B =  
3 2  y 7 
vc, A = –11 ≠ 0, vr% A O;qRØe.kh; vkO;wg gS blfy, blosQ O;qRØe dk vfLrRo gSA vkSj
bldk ,d vf}rh; gy gSA
1  2 −5
è;ku nhft, fd A–1 = −
11  −3 2 
1  2 −5  1
blfy, X = A–1B = −
11  −3 2   7
 x 1  −33  3 
vFkkZr~  y  = − 11  11  =  −1
     
vr% x = 3, y = – 1

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mnkgj.k 28 fuEufyf[kr lehdj.k fudk;

3x – 2y + 3z = 8
2x + y – z = 1
4x – 3y + 2z = 4
dks vkO;wg fof/ ls gy dhft,A
gy lehdj.k fudk; dks AX = B osQ :i esa O;Dr fd;k tk ldrk gS tgk¡
 3 −2 3   x 8
A = 2 1 −1 , X =  y  vkSj B =
  1
     
 4 −3 2   z   4

ge ns[krs gSa fd
A = 3 (2 – 3) + 2(4 + 4) + 3 (– 6 – 4) = – 17 ≠ 0 gSA
vr% A O;qRØe.kh; gS] vkSj blosQ O;qRØe dk vfLrRo gSA
A11 = –1, A12 = – 8, A13 = –10
A21 = –5, A22 = – 6, A23 = 1
A31 = –1, A32 = 9, A33 = 7
 −1 − 5 −1
1  
blfy, A = −  −8 − 6 9 
–1
17
 −10 1 7 

 −1 − 5 −1  8 
1    
vkSj X = A B = −  −8 − 6 9  1 
–1

17
 −10 1 7   4

 x  −17   1 
 y 1    
vr%   = − 17  −34  =  2 
 z   −51  3 
vr% x = 1, y = 2 o z = 3
mnkgj.k 29 rhu la[;kvksa dk ;ksx 6 gSA ;fn ge rhljh la[;k dks 3 ls xq.kk djosQ nwljh la[;k
esa tksM+ nsa rks gesa 11 izkIr gksrk gSA igyh vksj rhljh dks tksM+us ls gesa nwljh la[;k dk nqxquk izkIr
gksrk gSA bldk chtxf.krh; fu:i.k dhft, vkSj vkO;wg fof/ ls la[;k,¡ Kkr dhft,A

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 lkjf.kd 147

gy eku yhft, igyh] nwljh o rhljh la[;k Øe'k% x, y vkSj z, }kjk fu:fir gSA rc nh xbZ
'krks± osQ vuqlkj gesa izkIr gksrk gS%
x+y+z=6
y + 3z = 11
x + z = 2y
;k x – 2y + z = 0
bl fudk; dks A X = B osQ :i esa fy[kk tk ldrk gS tgk¡
 1 1 1  x 6
   y 11
A =  0 1 3 , X =   vkSj B =   gSA
1 –2 1  z   0 

;gk¡ A = 1 (1 + 6) + 0 + 1 (3 – 1) = 9 ≠ 0 gSA vc ge adj A Kkr djrs gSaA

A11 = 1 (1 + 6) = 7, A12 = – (0 – 3) = 3, A13 = – 1
A21 = – (1 + 2) = – 3, A22 = 0, A23 = – (– 2 – 1) = 3
A31 = (3 – 1) = 2, A32 = – (3 – 0) = – 3, A33 = (1 – 0) = 1
 7 –3 2 
 
vr% adj A =  3 0 –3
 –1 3 1 

 7 –3 2 
1 1 
bl izdkj A –1
= adj. (A) =  3 0 –3
A 9
 –1 3 1 

D;ksafd X = A–1 B

 7 –3 2   6 
1  
X =  3 0 –3 11
9
 –1 3 1   0 

 x  42 − 33 + 0  9 1 
 y 1  18 + 0 + 0  1 18   2
;k   =   =   =  
 z  9  −6 + 33 + 0 9  27  3
   
vr% x = 1, y = 2, z = 3

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iz'ukoyh 4-6
fuEufyf[kr iz'uksa 1 ls 6 rd nh xbZ lehdj.k fudk;ksa dk laxr vFkok vlaxr osQ :i esa oxhZdj.k
dhft,
1. x + 2y = 2 2. 2x – y = 5 3. x + 3y = 5
2x + 3y = 3 x+y=4 2x + 6y = 8
4. x + y + z = 1 5. 3x–y – 2z = 2 6. 5x – y + 4z = 5
2x + 3y + 2z = 2 2y – z = –1 2x + 3y + 5z = 2
ax + ay + 2az = 4 3x – 5y = 3 5x – 2y + 6z = –1
fuEufyf[kr iz'u 7 ls 14 rd izR;sd lehdj.k fudk; dks vkO;wg fof/ ls gy dhft,A
7. 5x + 2y = 4 8. 2x – y = –2 9. 4x – 3y = 3
7x + 3y = 5 3x + 4y = 3 3x – 5y = 7
10. 5x + 2y = 3 11. 2x + y + z = 1 12. x – y + z = 4

3
3x + 2y = 5 x – 2y – z = 2x + y – 3z = 0
2
3y – 5z = 9 x+y+z=2
13. 2x + 3y +3 z = 5 14. x – y + 2z = 7
x – 2y + z = – 4 3x + 4y – 5z = – 5
3x – y – 2z = 3 2x – y + 3z = 12

 2 –3 5 
 
15. ;fn A =  3 2 – 4  gS rks A –1 Kkr dhft,A A–1 dk iz;ksx djosQ fuEufyf[kr
 1 1 –2 

lehdj.k fudk; dks gy dhft,A

2x – 3y + 5z = 11
3x + 2y – 4z = – 5
x + y – 2z = – 3
16. 4 kg I;kt] 3 kg xsgw¡ vkSj 2 kg pkoy dk ewY; Rs 60 gSA 2 kg I;kt] 4 kg xsgw¡ vkSj
6 kg pkoy dk ewY; Rs 90 gSA 6 kg I;kt] 2 kg vkSj 3 kg pkoy dk ewY;
Rs 70 gSA vkO;wg fof/ }kjk izR;sd dk ewY; izfr kg Kkr dhft,A

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 lkjf.kd 149

fofo/ mnkgj.k
mngkj.k 30 ;fn a, b, c /ukRed vkSj fHkUu gSa rks fn[kkb, fd lkjf.kd
a b c
∆= b c a dk eku ½.kkRed gSA
c a b
gy C1 → C1 + C2 + C3 dk iz;ksx djus ij
a+b+c b c 1 b c
∆ = a + b + c c a = (a + b + c) 1 c a
a+b+c a b 1 a b

1 b c
= (a + b + c) 0 c – b a – c (R2→ R2–R1, vkSj R3 → R3 – R1 dk iz;ksx djus ij)
0 a–b b–c

= (a + b + c) [(c – b) (b – c) – (a – c) (a – b)] (C1 osQ vuqfn'k izlj.k djus ij)

= ( a + b + c)(– a2 – b2 – c2 + ab + bc + ca)
–1
= (a + b + c) (2a2 + 2b2 + 2c2 – 2ab – 2bc – 2ca)
2
–1
= (a + b + c) [(a – b)2 + (b – c)2 + (c – a)2]
2
tks ½.kkRed gS ( D;ksafd a + b + c > 0 vkSj (a – b)2 + (b – c)2 + (c – a)2 > 0)
mnkgj.k 31 ;fn a, b, c lekarj Js<+h esa gkas rks fuEufyf[kr lkjf.kd dk eku Kkr dhft,
2y + 4 5y + 7 8y + a
∆ = 3y + 5 6 y + 8 9 y + b
4 y + 6 7 y + 9 10 y + c
gy R1 → R1 + R3 – 2R2 dk iz;ksx djus ij
0 0 0
∆ = 3y + 5 6 y + 8 9 y + b = 0 (D;ksafd 2b = a + c)
4 y + 6 7 y + 9 10 y + c

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mnkgj.k 32 n'kkZb, fd lkjf.kd

( y+ z )
2
xy zx
( x+ z )
2
∆= xy yz = 2xyz (x + y + z)3
( x+ y )
2
xz yz

gy lkjf.kd esa R1 → xR1, R2 → yR2 , R3 → z R3 dk iz;ksx djus vkSj xyz, ls Hkkx djus ij
ge izkIr djrs gSa fd lkjf.kd
x ( y+ z)
2
x2 y x2 z
1
y ( x+ z )
2
∆= xy 2 y2 z
xyz
z ( x+ y )
2
xz 2 yz 2

C1 , C2 vkSj C3 ls Øe'k% x, y, z mHk;fu"B ysus ij]

(y+ z)
2
x2 x2
xyz
(x+ z)
2

∆= y2 y2
xyz
(x+ y)
2
z2 z2

C2 → C2– C1, C3 → C3– C1, dk iz;ksx djus ij ge izkIr djrs gSa fd

( y + z )2 x2 – ( y + z )
2
x2 − ( y + z )
2

∆= y2 ( x + z )2 − y 2 0
z2 0 ( x + y )2 – z 2
vc C2 vkSj C3 ls (x + y + z) mHk;fu"B ysus ij] izkIr lkjf.kd
(y + z) x – ( y + z) x – ( y + z)
2

∆ = (x + y + z)2 y 2
(x + z) – y 0
z2 0 (x + y) – z
R1 → R1 – (R2 + R3) dk iz;ksx djus ij ge fuEufyf[kr lkjf.kd izkIr djrs gaS

2 yz –2z –2y
∆ = (x + y + z) 2 y2 x− y+z 0
z2 0 x+ y –z

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 lkjf.kd 151

1  1 
C2 → (C2 + C1) vkSj C3 →  C 3 + C1  dk iz;ksx djus ij izkIr lkjf.kd
y  z 

2 yz 0 0
y2
y2 x+ z
∆ = (x + y + z)2 z
z2
z2 x+ y
y

R1 osQ vuqfn'k izlj.k djus ij

∆ = (x + y + z)2 (2yz) [(x + z) (x + y) – yz] = (x + y + z)2 (2yz) (x2 + xy + xz)
= (x + y + z)3 (2xyz) izkIr gksrk gSA

 1 –1 2   – 2 0 1 
mnkgj.k 33 vkO;wgksa osQ xq.kuiQy 0 2 –3  9 2 –3 dk iz;ksx djrs gq, fuEufyf[kr
 3 –2 4   6 1 –2
lehdj.k fudk; dks gy dhft,%
x – y + 2z = 1
2y – 3z = 1
3x – 2y + 4z = 2

1 –1 2  –2 0 1 
gy fn;k x;k xq.kuiQy  0 2 – 3  9
 2 – 3 
 3 –2 4   6 1 – 2 

 − 2 − 9 + 12 0 − 2 + 2 1 + 3 − 4 1 0 0
   
=  0 + 18 − 18 0 + 4 − 3 0 − 6 + 6 = 0 1 0
 − 6 − 18 + 24 0 − 4 + 4 3 + 6 − 8 0 0 1
–1
 1 –1 2   –2 0 1 
vr%  0 2 –3  =  9 2 – 3
   
 3 –2 4   6 1 –2

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vc fn, x, lehdj.k fudk; dks vkO;wg osQ :i fuEufyf[kr :i esa fy[kk tk ldrk gS
 1 –1 2   x  1 
 0 2 –3  y  =  1 
     
 3 –2 4   z   2

−1
 x 1 −1 2  1   –2 0 1  1 
;k  y
  = 0
 2 −3 1  =  9 2 –3 1 
 
 z   3 −2 4  2   6 1 –2  2

 −2 + 0 + 2  0 
   
=  9 + 2 − 6  = 5 
 6 + 1 − 4   3
vr% x = 0, y = 5 vkSj z = 3
mnkgj.k 34 fl¼ dhft, fd lkjf.kd
a + bx c + dx p + qx a c p
∆ = ax + b cx + d px + q = (1 − x ) b d
2
q
u v w u v w
gy lkjf.kd ∆ ij R1 → R1 – x R2 dk iz;ksx djus ij gesa

a (1 − x 2 ) c (1 − x 2 ) p (1 − x 2 )
D= ax + b cx + d px + q izkIr gksrk gS
u v w

a c p
= (1 − x ) ax + b cx + d px + q
2

u v w

R2 → R2 – x R1, dk iz;ksx djus ij gesa lkjf.kd

a c p
∆ = (1 − x ) b d q izkIr gksrk gSA
2

u v w

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 lkjf.kd 153

vè;k; 4 ij fofo/ iz'ukoyh

x sin θ cos θ
1. fl¼ dhft, fd lkjf.kd – sin θ – x 1 , θ ls Lora=k gSA
cos θ 1 x

a a2 bc 1 a2 a3
2. lkjf.kd dk izlj.k fd, fcuk fl¼ dhft, fd b b2 ca = 1 b 2 b3
c c2 ab 1 c2 c3

cos α cos β cos α sin β – sin α

3. – sin β cos β 0 dk eku Kkr dhft,A
sin α cos β sin α sin β cos α

4. ;fn a, b vkSj c okLrfod la[;k,¡ gks vkSj lkjf.kd

b+ c c+ a a+ b
∆ = c + a a+ b b+ c =0
a+b b+ c c+ a
gks rks n'kkZb, fd ;k rks a + b + c = 0 ;k a = b = c gSA
x+a x x
5. ;fn a ≠ 0 gks rks lehdj.k x x+a x = 0 dks gy dhft,A
x x x+a

a2 bc ac + c 2
6. fl¼ dhft, fd a + ab
2
b2 ac = 4a2b2c2
ab b2 + bc c 2

 3 –1 1   1 2 –2 
   
7. ;fn A–1 =  –15 6 –5 vkSj B =  –1 3 0  , gks rks ( AB) dk eku Kkr dhft,A
−1

 5 –2 2   0 –2 1 

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154 xf.kr

 1 –2 1 
 
8. eku yhft, A =  –2 3 1 gks rks lR;kfir dhft, fd
 1 1 5

(i) [adj A]–1 = adj (A–1) (ii) (A–1)–1 = A

x y x+ y
9. y x+ y x dk eku Kkr dhft,A
x+ y x y

1 x y
10. 1 x+ y y dk eku Kkr dhft,A
1 x x+ y

lkjf.kdksa osQ xq.k/eks± dk iz;ksx djosQ fuEufyf[kr 11 ls 15 rd iz'uksa dks fl¼ dhft,%

α α2 β+γ
11. β β 2
γ + α = (β – γ) (γ – α) (α – β) (α + β + γ)
γ γ 2
α +β

x x 2 1 + px 3
12. y y 2 1 + py 3 = (1 + pxyz) (x – y) (y – z) (z – x),
z z2 1 + pz 3

3a – a+ b – a+ c
13. –b+ a 3b – b + c = 3(a + b + c) (ab + bc + ca)
–c+ a – c+ b 3c

1 1+ p 1+ p+ q
14. 2 3+ 2 p 4 + 3 p + 2q = 1
3 6 + 3 p 10 + 6 p + 3 q

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 lkjf.kd 155

sin α cos α cos (α + δ )

15. sin β cos β cos (β + δ ) = 0
sin γ cos γ cos ( γ + δ )

16. fuEufyf[kr lehdj.k fudk; dks gy dhft,

2 3 10
+ + =4
x y z

4 6 5
– + =1
x y z

6 9 20
+ – =2
x y z
fuEufyf[kr iz'uksa 17 ls 19 esa lgh mÙkj dk pquko dhft,A
17. ;fn a, b, c lekarj Js<+h esa gksa rks lkjf.kd

x + 2 x + 3 x + 2a
x + 3 x + 4 x + 2b dk eku gksxk%
x + 4 x + 5 x + 2c

(A) 0 (B) 1 (C) x (D) 2x

 x 0 0
 
18. ;fn x, y, z 'kwU;srj okLrfod la[;k,¡ gksa rks vkO;wg A =  0 y 0 dk O;qÙØe gS%
 0 0 z 

 x −1 0 0   x −1 0 0 
   
(A)  0 y −1 0  (B) xyz  0 y −1 0 
   
 0 0 z −1   0 0 z −1 

 x 0 0 1 0 0 
1 
0 y 0 
1 
(C)  (D)  0 1 0 
xyz xyz
0 0 z  0 0 1 

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156 xf.kr

 1 sin θ 1 
 − sin θ sin θ, tgk¡ 0 ≤ θ ≤ 2π gks rks%
19. ;fn A =  1

 −1 − sin θ 1 
(A) det (A) = 0 (B) det (A) ∈ (2, ∞)
(C) det (A) ∈ (2, 4) (D) det (A) ∈ [2, 4].

lkjka'k

® vkO;wg A = [a11 ] 1×1 dk lkjf.kd a11 1×1 = a11 osQ }kjk fn;k tkrk gSA

 a11 a12 
® vkO;wg A =
a22 
dk lkjf.kd
 a21

a11 a12
A = = a11 a22 – a12 a21 osQ }kjk fn;k tkrk gSA
a21 a22

 a1 b1 c1 
® vkO;wg A = a2 b2 c2  osQ lkjf.kd dk eku (R1 osQ vuqfn'k izlj.k ls) fuEufyf[kr
 a3 b3 c3 

:i }kjk fn;k tkrk gSA

a1 b1 c1
b c2 a2 c2 a2 b2
A = a2 b2 c2 = a1 2 − b1 + c1
b3 c3 a3 c3 a3 b3
a3 b3 c3
fdlh oxZ vkO;wg A osQ fy,] |A| fuEufyf[kr xq.k/eks± dks larq"V djrk gSA
® A′ = A , tgk¡ A′ = A dk ifjorZ gSA

® ;fn ge nks iafDr;ksa ;k LraHkksa dks ijLij cny nsa rks lkjf.kd dk fpÉ cny tkrk gSA
® ;fn lkjf.kd dh dksbZ nks iafDr ;k LraHk leku ;k lekuqikrh gksa rks lkjf.kd dk eku
'kwU; gksrk gSA
® ;fn ge ,d lkjf.kd dh ,d iafDr ;k LraHk dks vpj k, ls xq.kk dj nsa rks lkjf.kd
dk eku k xquk gks tkrk gSA

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 lkjf.kd 157

® ,d lkjf.kd dks k ls xq.kk djus dk vFkZ gS fd mlosQ vanj osQoy fdlh ,d iafDr
;k LraHk osQ vo;oksa dks k ls xq.kk djukA
® ;fn A = [aij ]3×3 , rks k .A = k 3 A
® ;fn ,d lkjf.kd osQ ,d iafDr ;k LraHk osQ vo;o nks ;k vf/d vo;oksa osQ ;ksx osQ
:i esa O;Dr fd, tk ldrs gksa rks ml fn, x, lkjf.kd dks nks ;k vf/d lkjf.kdksa
osQ ;ksx osQ :i esa O;Dr fd;k tk ldrk gSA
® ;fn ,d lkjf.kd osQ fdlh ,d iafDr ;k LraHk osQ izR;sd vo;o osQ lexq.kt vU; iafDr
;k LraHk osQ laxr vo;oksa esa tksM+ fn, tkrs gSa rks lkjf.kd dk eku vifjofrZr jgrk gSA
® (x1, y1), (x2, y2) vkSj (x3, y3) 'kh"kks± okyh f=kHkqt dk {ks=kiQy fuEufyf[kr :i }kjk
fn;k tkrk gS%
x1 y1 1
1
∆= x2 y2 1
2
x3 y3 1

® fn, x, vkO;wg A osQ lkjf.kd osQ ,d vo;o aij dk milkjf.kd] i oha iafDr vkSj
j oka LraHk gVkus ls izkIr lkjf.kd gksrk gS vkSj bls Mij }kjk O;Dr fd;k tkrk gSA
® aij dk lg[kaM Aij = (– 1)i+j Mij }kjk fn;k tkrk gSA

® A osQ lkjf.kd dk eku A = a11 A11 + a12 A12 + a13 A13 gS vkSj bls ,d iafDr ;k
LraHk osQ vo;oksa vkSj muosQ laxr lg[kaMksa osQ xq.kuiQy dk ;ksx djosQ izkIr fd;k tkrk
gSA
® ;fn ,d iafDr (;k LraHk) osQ vo;oksa vkSj vU; nwljh iafDr (;k LraHk) osQ lg[kaMksa
dh xq.kk dj nh tk, rks mudk ;ksx 'kwU; gksrk gS mnkgj.kr;k
a11 A21 + a12 A22 + a13 A23 = 0

 a11 a12 a13   A11 A 21 A31 

® ;fn vkO;wg A = a21 a22 a23  , rks lg[kaMt adj A = A
 12 A 22 A32  gksrk
 a31 a32 a33   A13 A 23 A 33 
gS] tgk¡ aij dk lg[kaM Aij gSA
® A (adj A) = (adj A) A = A I, tgk¡ A, n dksfV dk oxZ vkO;wg gSA
® ;fn dksbZ oxZ vkO;wg Øe'k% vO;qRØe.kh; ;k O;qRØe.kh; dgykrk gS ;fn A =0
;k A ≠ 0

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158 xf.kr

® ;fn–1 AB = BA = I, tgk¡ B ,d oxZ vkO;wg gS rc A dk O;qRØe B gksrk gS vkSj

A = B ;k B–1 = A vkSj blfy, (A–1)–1 = A
® fdlh oxZ vkO;wg A dk O;qRØe gS ;fn vkSj osQoy ;fn A O;qRØe.kh; gSA
1
® A –1 = ( adj A)
A
® ;fn a1 x + b1 y + c1 z = d1
a2 x + b2 y + c2 z = d2
a3 x + b3 y + c3 z = d3
rc bu lehdj.kksa dks A X = B osQ :i esa fy[kk tk ldrk gSA
 a1 b1 c1   x  d1 
tgk¡ A = a2 b2 c2  , X = y vkSj B=  d 2 
 
 

 a3 b3 c3   z   d3 

® lehdj.k AX = B dk vf}rh; gy X = A–1 B }kjk fn;k tkrk gS tgk¡ A ≠ 0

® lehdj.kksa dk ,d fudk; laxr ;k vlaxr gksrk gS ;fn blosQ gy dk vfLrRo gS vFkok
ugha gSA
® vkO;wg lehdj.k AX = B esa ,d oxZ vkO;wg A osQ fy,
(i) ;fn A ≠ 0 , rks vf}rh; gy dk vfLrRo gSA

(ii) ;fn A = 0 vkSj (adj A) B ≠ O, rks fdlh gy dk vfLrRo ugha gSA

(iii) ;fn A = 0 vkSj (adj A) B = O, rks fudk; laxr ;k vlaxr gksrh gSA

,sfrgkfld i`"BHkwfe
x.kuk cksMZ ij NM+ksa dk iz;ksx djosQ oqQN jSf[kd lehdj.kksa dh vKkr jkf'k;ksa osQ
xq.kkadksa dks fu:fir djus dh phuh fof/ us okLro esa foyksiu dh lk/kj.k fof/ dh [kkst
djus esa lgk;rk dh gSA NM+ksa dh O;oLFkk Øe ,d lkjf.kd esa la[;kvksa dh mfpr O;oLFkk
Øe tSlh FkhA blfy, ,d lkjf.kd dh ljyhdj.k esa LraHkksa ;k iafDr;ksa osQ ?kVkus dk fopkj
mRiUu djus esa phuh izFke fopkjdksa esa Fks (‘Mikami, China, pp 30, 93).
l=kgoha 'krkCnh osQ egku tkikuh xf.krK Seki Kowa }kjk 1683 esa fyf[kr iqLrd
'Kai Fukudai no Ho' ls Kkr gksrk gS fd mUgsa lkjf.kdksa vkSj muosQ izlkj dk Kku FkkA ijarq

2018-19
 lkjf.kd 159

mUgksaus bl fof/ dk iz;ksx osQoy nks lehdj.kksa ls ,d jkf'k osQ foyksiu esa fd;k ijarq ;qxir
jSf[kd lehdj.kksa osQ gy Kkr djus esa bldk lh/k iz;ksx ugha fd;k FkkA ‘T. Hayashi,
“The Fakudoi and Determinants in Japanese Mathematics,” in the proc. of the
Tokyo Math. Soc., V.
Vendermonde igys O;fDr Fks ftUgksuas lkjf.kdksa dks Lora=k iQyu dh rjg ls igpkuk
bUgsa fof/or bldk vUos"kd (laLFkkid) dgk tk ldrk gSA Laplace (1772) us lkjf.kdksa
dks blosQ iwjd milkjf.kdksa osQ :i esa O;Dr djosQ izlj.k dh O;kid fof/ nhA 1773 esa
Lagrange us nwljs o rhljs Øe osQ lkjf.kdksa dks O;oâr fd;k vkSj lkjf.kdksa osQ gy osQ
vfrfjDr mudk vU;=k Hkh iz;ksx fd;kA 1801 esa Gauss us la[;k osQ fl¼karksa esa lkjf.kdksa
dk iz;ksx fd;kA
vxys egku ;ksxnku nsus okys Jacques - Philippe - Marie Binet, (1812) Fks ftUgksaus
m-LraHkksa vkSj n-iafDr;ksa osQ nks vkO;wgksa osQ xq.kuiQy ls lacaf/r izes; dk mYys[k fd;k tks
fo'ks"k fLFkfr m = n esa xq.kuiQy izes; esa cny tkrh gSA
mlh fnu Cauchy (1812) us Hkh mlh fo"k;&oLrq ij 'kks/ izLrqr fd,A mUgksaus vkt
ossQ O;kogkfjd lkjf.kd 'kCn dk iz;ksx fd;kA mUgksaus Binet ls vf/d larq"V djus okyh
xq.kuiQy izes; dh miifÙk nhA
bu fl¼karksa ij egkure ;ksxnku okys Carl Gustav Jacob Jacobi FksA blosQ i'pkr
lkjf.kd 'kCn dks vafre Loho`Qfr izkIr gqbZA

—v—

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160 xf.kr

vè;k; 5
lkarR; rFkk vodyuh;rk
(Continuity and Differentiability)

vThe whole of science is nothing more than a refinement

of everyday thinking.” — ALBERT EINSTEIN v

5.1 Hkwfedk (Introduction)

;g vè;k; vfuok;Zr% d{kk 11 esa i<+s x, iQyuksa osQ vodyu
(differentiation) dk Øekxr gSA ge oqQN fuf'pr cgqinh; iQyuksa
,oa f=kdks.kferh; iQyuksa dk vodyu djuk lh[k pqosQ gSaA
bl vè;k; es a ge lka r R; (continuity), vodyuh;rk
(differentiability) rFkk buosQ ikjLifjd laca/ksa dh egRoiw.kZ
ladYiukvksa dks izLrqr djsaxsA ;gk¡ ge izfrykse f=kdks.kferh;
(inverse trigonometric) iQyuksa dk vodyu djuk Hkh lh[ksaxsA
vc ge oqQN u, izdkj osQ iQyuksa dks izLrqr dj jgs gSa] ftudks
pj?kkrkadh (exponential) vkSj y?kqx.kdh; (logarithmic) iQyu
dgrs gSAa bu iQyuksa }kjk gesa vodyu dh l'kDr izfof/;ksa dk Kku
gksrk gSA vody xf.kr (differential calculus) osQ ekè;e ls ge
T;kferh; :i ls lqLi"V (obvious) oqQN fLFkfr;ksa dks le>krs gSaA
bl izfØ;k] esa ge bl fo"k; dh oqQN vk/kjHkwr (ewy) izes;ksa Sir Issac Newton
(theorems) dks lh[ksaxsA (1642-1727)
5.2 lkarR; (Continuity)
lkarR; dh ladYiuk dk oqQN vuqeku (cks/) djkus osQ
fy,] ge vuqPNsn dks nks vukSipkfjd mnkgj.kksa ls
izkjaHk djrs gSaA fuEufyf[kr iQyu ij fopkj dhft,%
1, ;fn x ≤ 0
f ( x) = 
2, ;fn x > 0
;g iQyu okLro esa okLrfod js[kk (real line) osQ
izR;sd ¯cnq ij ifjHkkf"kr gSA bl iQyu dk vkys[k
vko`Qfr 5-1 esa n'kkZ;k x;k gSA dksbZ Hkh bl vkys[k ls
fu"d"kZ fudky ldrk gS fd x = 0 osQ vfrfjDr] x&v{k vko`Qfr 5-1

2018-19
 lkarR; rFkk vodyuh;rk 161

osQ vU; lfUudV ¯cnqvksa osQ fy, iQyu osQ laxr eku Hkh x = 0 dks NksM+dj ,d nwljs osQ lehi
(yxHkx leku) gSaA 0 osQ lfUudV ck;ha vksj osQ ¯cnqvksa] vFkkZr~ – 0.1, – 0.01, – 0.001, izdkj
osQ ¯cnqvksa] ij iQyu dk eku 1 gS rFkk 0 osQ lfUudV nk;ha vksj osQ ¯cnqvksa] vFkkZr~ 0.1, 0.01,
0.001, izdkj osQ ¯cnqvksa ij iQyu dk eku 2 gSA ck,¡ vkSj nk,¡ i{k dh lhekvksa (limits) dh Hkk"kk
dk iz;ksx djosQ] ge dg ldrs gSa fd x = 0 ij iQyu f osQ ck,¡ rFkk nk,¡ i{k dh lhek,¡ Øe'k%
1 rFkk 2 gSaA fo'ks"k :i ls ck,¡ rFkk nk,¡ i{k dh lhek,¡ leku @ laikrh (coincident) ugha gSaA
ge ;g Hkh ns[krs gSa fd x = 0 ij iQyu dk eku ck,¡ i{k dh lhek osQ laikrh gS (cjkcj gS)A
uksV dhft, fd bl vkys[k dks ge yxkrkj ,d lkFk (in one stroke)] vFkkZr~ dye dks bl
dkx”k dh lrg ls fcuk mBk,] ugha [khap ldrsA okLro esa] gesa dye dks mBkus dh vko';drk
rc gksrh gS tc ge 'kwU; ls ck;ha vksj vkrs gSaA ;g ,d mnkgj.k gS tgk¡ iQyu
x = 0 ij larr (continuous) ugha gSA
vc uhps n'kkZ, x, iQyu ij fopkj dhft,%
1, ;fn x ≠ 0
f ( x) = 
 2, ;fn x = 0
;g iQyu Hkh izR;sd ¯cnq ij ifjHkkf"kr gSA
x = 0 ij nksuksa gh] ck,¡ rFkk nk,¡ i{k dh lhek,¡ 1 osQ
cjkcj gSaA fdarq x = 0 ij iQyu dk eku 2 gS] tks ck,¡
vkSj nk,¡ i{k dh lhekvksa osQ mHk;fu"B eku osQ cjkcj
ugha gSA
iqu% ge uksV djrs gSa fd iQyu osQ vkys[k dks
fcuk dye mBk, ge ugha [khap ldrs gSaA ;g ,d
nwljk mnkgj.k gS ftlesa x = 0 ij iQyu larr ugha gSA
vko`Qfr 5-2
lgt :i ls (naively) ge dg ldrs gSa fd
,d vpj ¯cnq ij dksbZ iQyu larr gS] ;fn ml ¯cnq osQ vkl&ikl (around) iQyu osQ vkys[k
dks ge dkx”k dh lrg ls dye mBk, fcuk [khap ldrs gSaA bl ckr dks ge xf.krh; Hkk"kk esa]
;FkkrF; (precisely)] fuEufyf[kr izdkj ls O;Dr dj ldrs gSa%
ifjHkk"kk 1 eku yhft, fd f okLrfod la[;kvksa osQ fdlh mileqPp; esa ifjHkkf"kr ,d okLrfod
iQyu gS vkSj eku yhft, fd f osQ izkra esa c ,d ¯cnq gSA rc f ¯cnq c ij larr gS] ;fn
lim f ( x) = f (c ) gSA
x→ c

foLr`r :i ls ;fn x = c ij ck,¡ i{k dh lhek] nk,¡ i{k dh lhek rFkk iQyu osQ eku dk
;fn vfLrRo (existence) gS vkSj ;s lHkh ,d nwljs osQ cjkcj gksa] rks x = c ij f larr dgykrk
gSA Lej.k dhft, fd ;fn x = c ij ck,¡ i{k rFkk nk,¡ i{k dh lhek,¡ laikrh gSa] rks buosQ mHk;fu"B

2018-19
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eku dks ge x = c ij iQyu dh lhek dgrs gSaA bl izdkj ge lkarR; dh ifjHkk"kk dks ,d vU;
izdkj ls Hkh O;Dr dj ldrs gSa] tSlk fd uhps fn;k x;k gSA
,d iQyu x = c ij larr gS] ;fn iQyu x = c ij ifjHkkf"kr gS vkSj ;fn x = c ij iQyu
dk eku x = c ij iQyu dh lhek osQ cjkcj gSA ;fn x = c ij iQyu larr ugha gS rks ge dgrs
gSa fd c ij f vlarr (discontinuous) gS rFkk c dks f dk ,d vlkarR; dk ¯cnq (point of
discontinuity) dgrs gSAa
mnkgj.k 1 x = 1 ij iQyu f (x) = 2x + 3 osQ lkarR; dh tk¡p dhft,A
gy igys ;g è;ku nhft, fd iQyu] x = 1 ij ifjHkkf"kr gS vkSj bldk eku 5 gSA vc iQyu
dh x = 1 ij lhek Kkr djrs gSaA Li"V gS fd
lim f ( x ) = lim (2 x + 3) = 2(1) + 3 = 5 gSA
x →1 x →1

vr% lim f ( x ) = 5 = f (1)

x →1

vr,o x = 1 ij f larr gSA

mnkgj.k 2 tk¡fp, fd D;k iQyu f (x) = x2, x = 0 ij larr gS\
gy è;ku nhft, fd iznÙk ¯cnq x = 0 ij iQyu ifjHkkf"kr gS vkSj bldk eku 0 gSA vc
x = 0 ij iQyu dh lhek fudkyrs gSaA Li"Vr;k

lim f ( x) = lim x 2 = 02 = 0
x→ 0 x→0

bl izdkj lim f ( x ) = 0 = f (0)

x→ 0

vr% x = 0 ij f larr gSA

mnkgj.k 3 x = 0 ij iQyu f (x) = | x | osQ lkarR; ij fopkj dhft,A
gy ifjHkk"kk }kjk
− x, ;fn x < 0
f (x) = 
 x, ;fn x ≥ 0
Li"Vr;k x = 0 ij iQyu ifjHkkf"kr gS vkSj f (0) = 0 gSA ¯cnq x = 0 ij f dh ck,¡ i{k dh lhek
lim f ( x ) = lim− (– x) = 0 gSA
x → 0− x→0

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 lkarR; rFkk vodyuh;rk 163

blh izdkj 0 ij f dh nk,¡ i{k dh lhek osQ fy,

lim f ( x ) = lim+ x = 0 gSA
x → 0+ x→0

bl izdkj x = 0 ij ck,¡ i{k dh lhek] nk,¡ i{k dh lhek rFkk iQyu dk eku laikrh gaSA vr%
x = 0 ij f larr gSA
mnkgj.k 4 n'kkZb, fd iQyu
 x3 + 3, ;fn x ≠ 0
f (x) = 
1, ;fn x = 0

x = 0 ij larr ugha gSA

gy ;gk¡ x = 0 ij iQyu ifjHkkf"kr gS vkSj x = 0 ij bldk eku 1 gSA tc x ≠ 0, rc iQyu
cgqinh; gSA blfy,
lim f ( x) = lim ( x 3 + 3) = 03 + 3 = 3
x→ 0 x→0

D;ksafd x = 0 ij f dh lhek] f (0) osQ cjkcj ugha gS] blfy, x = 0 ij iQyu larr ugha
gSA ge ;g Hkh lqfuf'pr dj ldrs gSa fd bl iQyu osQ fy, vlkarR; dk ¯cnq osQoy x = 0 gSA
mnkgj.k 5 mu ¯cnqvksa dh tk¡p dhft, ftu ij vpj iQyu (Constant function)
f (x) = k larr gSA

gy ;g iQyu lHkh okLrfod la[;kvksa osQ fy, ifjHkkf"kr gS vkSj fdlh Hkh okLrfod la[;k osQ
fy, bldk eku k gSA eku yhft, fd c ,d okLrfod la[;k gS] rks
lim f ( x) = lim k = k
x→ c x→ c

pw¡fd fdlh okLrfod la[;k c osQ fy, f (c) = k = lim

x → c f (x) gS blfy, iQyu f izR;sd

okLrfod la[;k osQ fy, larr gSA

mnkgj.k 6 fl¼ dhft, fd okLrfod la[;kvksa osQ fy, rRled iQyu (Identity function)
f (x) = x, izR;sd okLrfod la[;k osQ fy, larr gSA

gy Li"Vr;k ;g iQyu izR;sd ¯cnq ij ifjHkkf"kr gS vkSj izR;sd okLrfod la[;k c osQ fy,
f (c) = c gSA

lkFk gh lim f ( x) = lim x = c

x→ c x→c

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bl izdkj] lim
x→c
f(x) = c = f(c) vkSj blfy, ;g iQyu f osQ izkar osQ lHkh ¯cnqvksa ij larr gS A

,d iznÙk ¯cnq ij fdlh iQyu osQ lkarR; dks ifjHkkf"kr djus osQ ckn vc ge bl ifjHkk"kk
dk LokHkkfod izlkj (extension) djosQ fdlh iQyu osQ] mlosQ izkar esa] lkarR; ij fopkj djsaxsA
ifjHkk"kk 2 ,d okLrfod iQyu f larr dgykrk gS ;fn og f osQ izkra osQ izR;sd ¯cnq ij larr gSA
bl ifjHkk"kk dks oqQN foLrkj ls le>us dh vko';drk gSA eku yhft, fd f ,d ,slk iQyu gS]
tks lao`r varjky (closed interval) [a, b] esa ifjHkkf"kr gS] rks f osQ larr gksus osQ fy, vko';d
gS fd og [a, b] osQ vaR; ¯cnqvksa (end points) a rFkk b lfgr mlosQ izR;sd ¯cnq ij larr gksA
f dk vaR; ¯cnq a ij lakrR; dk vFkZ gS fd

lim f ( x ) = f (a)
x→ a+

vkSj f dk b ij lkarR; dk vFkZ gS fd

lim f ( x) = f(b)
x→b –

iz{s k.k dhft, fd lim f ( x) rFkk lim f ( x) dk dksbZ vFkZ ugha gSA bl ifjHkk"kk osQ ifj.kkeLo:i]
x→ a− x → b+

;fn f osQoy ,d ¯cnq ij ifjHkkf"kr gS] rks og ml ¯cnq ij larr gksrk gS] vFkkZr~ ;fn f dk
izkar ,dy (leqPp;) gS] rks f ,d larr iQyu gksrk gSA
mnkgj.k 7 D;k f (x) = | x | }kjk ifjHkkf"kr iQyu ,d larr iQyu gS\

 − x, ;fn x < 0
gy f dks ge ,sls fy[k ldrs gSa fd f (x) = 
 x, ;fn x ≥ 0

mnkgj.k 3 ls ge tkurs gSa fd x = 0 ij f larr gSA

eku yhft, fd c ,d okLrfod la[;k bl izdkj gS fd c < 0 gSA vr,o f (c) = – c
lkFk gh lim f ( x) = lim (− x) = – c (D;ksa?)
x→ c x→c

pw¡fd lim f ( x) = f (c ) , blfy, f lHkh ½.kkRed okLrfod la[;kvksa osQ fy, larr gSA
x→ c

vc eku yhft, fd c ,d okLrfod la[;k bl izdkj gS fd c > 0 gSA vr,o f (c) = c

lkFk gh lim f ( x) = lim x = c (D;ksa?)
x→ c x→ c

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 lkarR; rFkk vodyuh;rk 165

D;ksafd lim f ( x) = f (c ) , blfy, f lHkh /ukRed okLrfod la[;kvksa osQ fy, larr gSA
x→ c

pw¡fd f lHkh ¯cnqvksa ij larr gS] vr% ;g ,d larr iQyu gSA

mnkgj.k 8 iQyu f (x) = x3 + x2 – 1 osQ lkarR; ij fopkj dhft,A
gy Li"Vr;k f izR;sd okLrfod la[;k c osQ fy, ifjHkkf"kr gS vkSj c ij bldk eku
c3 + c2 – 1 gSA ge ;g Hkh tkurs gSa fd

lim f ( x) = lim ( x 3 + x 2 − 1) = c 3 + c 2 − 1
x→ c x→c

vr% lim f ( x) = f (c ) gS blfy, izR;sd okLrfod la[;k osQ fy, f larr gSA bldk vFkZ
x→ c

gS fd f ,d larr iQyu gSA

1
mnkgj.k 9 f (x) = , x ≠ 0 }kjk ifjHkkf"kr iQyu f osQ lkarR; ij fopkj dhft,A
x
gy fdlh ,d 'kwU;srj ( Non-zero) okLrfod la[;k c dks lqfuf'pr dhft,
1 1
vc lim f ( x) = lim =
x→c x→ c x c
1
lkFk gh] pw¡fd c ≠ 0, blfy, f (c) = gSA bl izdkj lim
x→ c
f ( x) = f (c ) vkSj blfy, f vius
c
izkar osQ izR;sd ¯cnq ij larr gSA bl izdkj f ,d larr iQyu gSA
ge bl volj dk ykHk] vuar (infinity) dh ladYiuk (concept) dks le>kus osQ fy,]
1
mBkrs gSaA ge blosQ fy, iQyu f (x) = dk fo'ys"k.k x = 0 osQ fudVLFk ekukas ij djrs gSaA
x
blosQ fy, ge 0 osQ lfUudV dh okLrfod la[;kvksa osQ fy, iQyu osQ ekuksa dk vè;;u djus
dh izpfyr ;qfDr dk iz;ksx djrs gSaA vfuok;Zr% (essentially) ge x = 0 ij f osQ nk,¡ i{k dh
lhek Kkr djus dk iz;kl djrs gSaA bldks ge uhps lkj.khc¼ djrs gSaA (lkj.kh 5.1)
lkj.kh 5-1
x 1 0.3 0.2 0.1 = 10–1 0.01 = 10–2 0.001 = 10–3 10–n
f (x) 1 3.333... 5 10 100 = 102 1000 = 103 10n

ge ns[krs gSa fd tSls&tSls x nk;ha vksj ls 0 osQ fudV vxzlj gksrk gS f (x) dk eku mÙkjksÙkj
vfr 'kh?kzrk ls c<+rk tkrk gSA bl ckr dks ,d vU; izdkj ls Hkh O;Dr fd;k tk ldrk gS] tSls%

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,d /u okLrfod la[;k dks 0 osQ vR;ar fudV pqudj] f (x) osQ eku dks fdlh Hkh iznÙk la[;k
ls vf/d fd;k tk ldrk gSA izrhdksa esa bl ckr dks ge fuEufyf[kr izdkj ls fy[krs gSa fd
lim f ( x ) = + ∞
x → 0+

(bldks bl izdkj i<+k tkrk gS% 0 ij] f (x) osQ nk,¡ i{k dh /ukRed lhek vuar gS)A ;gk¡ ij
ge cy nsuk pkgrs gSa fd + ∞ ,d okLrfod la[;k ugha gS vkSj blfy, 0 ij f osQ nk,¡ i{k dh
lhek dk vfLrRo ugha gS (okLrfod la[;kvksa osQ :i esa)A
blh izdkj ls 0 ij f osQ ck,¡ i{k dh lhek Kkr dh tk ldrh gSA fuEufyf[kr lkj.kh ls
Lor% Li"V gSA
lkj.kh 5-2
x –1 – 0.3 – 0.2 – 10–1 – 10–2 – 10–3 – 10–n
f (x) – 1 – 3.333... –5 – 10 – 102 – 103 – 10n

lkj.kh 5-2 ls ge fu"d"kZ fudkyrs gSa fd ,d

½.kkRed okLrfod la[;k dks 0 osQ vR;ar fudV
pqudj] f (x) osQ eku dks fdlh Hkh iznÙk la[;k ls
de fd;k tk ldrk gSA izrhdkRed :i ls ge
lim f ( x) = − ∞ fy[krs gSa
−
x→0
(ftls bl izdkj i<+k tkrk gS% 0 ij f (x) osQ ck,¡
i{k dh lhek ½.kkRed vuar gSA) ;gk¡ ge bl ckr
ij cy nsuk pkgrs gSa fd – ∞ ,d okLrfod la[;k
ugha gS vr,o 0 ij f osQ ck,¡ i{k dh lhek dk
vfLrRo ugha gS (okLrfod la[;kvksa osQ :i esa)A
vko`Qfr 5.3 dk vkys[k mi;qDZ r rF;ksa dk T;kferh;
fu:i.k gSA vko`Qfr 5-3
mnkgj.k 10 fuEufyf[kr iQyu osQ lkarR; ij fopkj dhft,%
 x + 2, ;fn x ≤ 1
f (x) = 
 x − 2, ;fn x > 1

gy iQyu f okLrfod js[kk osQ izR;sd ¯cnq ij ifjHkkf"kr gSA

n'kk 1 ;fn c < 1, rks f (c) = c + 2 gSA bl izdkj lim f ( x) = lim x + 2 = c + 2 gSA
x →c x →c

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 lkarR; rFkk vodyuh;rk 167

vr% 1 ls de lHkh okLrfod la[;kvksa ij f larr gSA

n'kk 2 ;fn c > 1, rks f (c) = c – 2 gSA
blfy, lim f ( x) = lim (x – 2) = c – 2 = f (c) gSA
x →c x→c

vr,o mu lHkh ¯cnqvksa ij tgk¡ x > 1 gS] f larr gSA

n'kk 3 ;fn c = 1, rks x = 1 ij f osQ ck,¡ i{k dh lhek]
vFkkZr~
lim f ( x ) = lim– ( x + 2) = 1 + 2 = 3
x →1– x →1

x = 1 ij f osQ nk,¡ i{k dh lhek] vFkkZr~

vko`Qfr 5-4
lim f ( x) = lim– ( x − 2) = 1 − 2 = −1
x →1+ x→1

vc pw¡fd x = 1 ij f osQ ck,¡ rFkk nk,¡ i{k dh lhek,¡ laikrh (coincident) ugha gSa] vr%
x = 1 ij f larr ugha gSA bl izdkj f osQ vlkarR; dk ¯cnq osQoy ek=k x = 1 gSA bl iQyu
dk vkys[k vko`Qfr 5-4 esa n'kkZ;k x;k gSA
mnkgj.k 11 fuEufyf[kr izdkj ls ifjHkkf"kr iQyu f osQ leLr (lHkh) vlkarR; ¯cnqvksa dks Kkr dhft,

 x + 2, ;fn x < 1

f (x) =  0 , ;fn x = 1
 x − 2, ;fn x > 1


gy iwoZorhZ mnkgj.k dh rjg ;gk¡ Hkh ge ns[krs gSa izR;sd okLrfod la[;k x ≠ 1 osQ fy, f larr
gSA x = 1 osQ fy, f osQ ck,¡ i{k dh lhek] lim− f ( x) = lim– ( x + 2) = 1 + 2 = 3 gSA
x→1 x→1
x = 1 osQ fy, f osQ nk,¡ i{k dh lhek] lim− f ( x) = lim– ( x − 2) = 1 − 2 = −1 gSA
x→1 x→1

pw¡fd x = 1 ij f osQ ck,¡ rFkk nk,¡ i{k dh lhek,¡ laikrh ugha gSa] vr% x = 1 ij f larr
ugha gSA bl izdkj f osQ vlkarR; dk ¯cnq osQoy ek=k x = 1 gSA bl iQyu dk vkys[k vko`Qfr
5-5 esa n'kkZ;k x;k gSA
mnkgj.k 12 fuEufyf[kr iQyu osQ lkarR; ij fopkj dhft,%

 x + 2,;fn x < 0
f (x) = 
− x + 2, ;fn x > 0

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gy è;ku nhft, fd fopkjk/hu iQyu 0 ('kwU;) osQ

vfrfjDr vU; leLr okLrfod la[;kvksa osQ fy, ifjHkkf"kr
gSA ifjHkk"kkuqlkj bl iQyu dk izkar
D1 ∪ D2 gS tgk¡ D1 = {x ∈ R : x < 0} vkSj
D2 = {x ∈ R : x > 0}gSA
n'kk 1 ;fn c ∈ D1, rks lim f ( x) = lim (x + 2) =
x→c x→c
c + 2 = f (c) gS vr,o D1 esa f larr gSA

n'kk 2 ;fn c ∈ D2, rks lim f ( x) = lim (– x + 2) =

x→c x→c
vko`Qfr 5-5
– c + 2 = f (c) gS vr,o D2 esa Hkh f larr gSA
D;ksafd f vius izkar osQ leLr ¯cnqvksa ij larr gS
ftlls ge fu"d"kZ fudkyrs gSa fd f ,d larr iQyu gSA
bl iQyu dk vkys[k vko`Qfr 5.6 esa [khapk x;k gSA è;ku
nhft, fd bl iQyu osQ vkys[k dks [khapus osQ fy, gesa
dye dks dkx”k dh lrg ls mBkuk iM+rk gS] fdarq gesa
,slk osQoy mu ¯cnqvksa ij djuk iM+rk gS tgk¡ ij iQyu
ifjHkkf"kr ugha gSA
mnkgj.k 13 fuEufyf[kr iQyu osQ lkarR; ij fopkj
dhft,%
vko`Qfr 5-6
 x, ;fn x ≥ 0
f (x) =  2
 x , ;fn x < 0
gy Li"Vr;k] iznÙk iQyu izR;sd okLrfod la[;k osQ
fy, ifjHkkf"kr gSA bl iQyu dk vkys[k vko`Qfr 5.7
esa fn;k gSA bl vkys[k osQ fujh{k.k ls ;g roZQlaxr
yxrk gS fd iQyu osQ izkra dks okLrfod js[kk osQ rhu
vla;Dq r (disjoint) mi leqPp;ksa esa foHkkftr dj
fy;k tk,A eku fy;k fd
D1 = {x ∈ R : x < 0}, D2 = {0} rFkk
D3 = {x ∈ R : x > 0}gSA
vko`Qfr 5-7

2018-19
 lkarR; rFkk vodyuh;rk 169

n'kk 1 D1 osQ fdlh Hkh ¯cnq ij f (x) = x2 gS vkSj ;g ljyrk ls ns[kk tk ldrk gS fd D1 esa
f larr gSA (mnkgj.k 2 nsf[k,)
n'kk 2 D3 osQ fdlh Hkh ¯cnq ij f (x) = x gS vkSj ;g ljyrk ls ns[kk tk ldrk gS fd D3 esa
f larr gSA (mnkgj.k 6 nsf[k,)
n'kk 3 vc ge x = 0 ij iQyu dk fo'ys"k.k djrs gSaA 0 osQ fy, iQyu dk eku f (0) = 0 gSA
0 ij f osQ ck,¡ i{k dh lhek
lim f ( x) = lim− x2 = 02 = 0 gS rFkk
x→0– x→0
0 ij f osQ nk,¡ i{k dh lhek
lim f ( x) = lim+ x = 0 gSA
x→0+ x→0

vr% lim f ( x) = 0 = f (0) vr,o 0 ij f larr gSA bldk vFkZ ;g gqvk fd f vius izkar osQ
x→0
izR;sd ¯cnq ij larr gSA vr% f ,d larr iQyu gSA
mnkgj.k 14 n'kkZb, fd izR;sd cgqin iQyu larr gksrk gSA
gy Lej.k dhft, fd dksbZ iQyu p, ,d cgqin iQyu gksrk gS ;fn og fdlh izko`Qr la[;k n
osQ fy, p(x) = a0 + a1 x + ... + an xn }kjk ifjHkkf"kr gks] tgk¡ ai ∈ R rFkk an ≠ 0 gSA Li"Vr;k
;g iQyu izR;sd okLrfod la[;k osQ fy, ifjHkkf"kr gSA fdlh fuf'pr okLrfod la[;k c osQ fy,
ge ns[krs gSa fd
lim p ( x) = p (c)
x→c
blfy, ifjHkk"kk }kjk c ij p larr gSA pw¡fd c dksbZ Hkh okLrfod la[;k gS blfy, p fdlh
Hkh okLrfod la[;k osQ fy, larr gS]
vFkkZr~ p ,d larr iQyu gSA
mnkgj.k 15 f (x) = [x] }kjk ifjHkkf"kr
egÙke iw.kk±d iQyu osQ vlkarR; osQ leLr
¯cnqvksa dks Kkr dhft,] tgk¡ [x] ml
egÙke iw.kk±d dks izdV djrk gS] tks x ls
de ;k mlosQ cjkcj gSA
gy igys rks ge ;g ns[krs gSa fd f lHkh
okLrfod la[;kvksa osQ fy, ifjHkkf"kr gSA
bl iQyu dk vkys[k vko`Qfr 5.8 esa
fn[kk;k x;k gSA vko`Qfr 5-8

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vkys[k ls ,slk izrhr gksrk gS fd iznÙk iQyu x osQ lHkh iw.kk±d ekuksa osQ fy, vlarr gSA uhps ge
Nkuchu djsaxs fd D;k ;g lR; gSA
n'kk 1 eku yhft, fd c ,d ,slh okLrfod la[;k gS] tks fdlh Hkh iw.kk±d osQ cjkcj ugha gSA
vkys[k ls ;g Li"V gS fd c osQ fudV dh lHkh okLrfod la[;kvksa osQ fy, fn, gq, iQyu dk
eku [c]; gSa] vFkkZr~ lim f ( x) = lim [ x] = [c] lkFk gh f (c) = [c] vr% iznÙk iQyu] mu lHkh
x →c x→c

okLrfod la[;kvksa osQ fy, larr gS] tks iw.kk±d ugha gSA
n'kk 2 eku yhft, fd c ,d iw.kk±d gSA vr,o ge ,d ,slh i;kZIrr% NksVh okLrfod la[;k
r > 0 izkIr dj ldrs gSa tks fd [c – r] = c – 1 tcfd [c + r] = c gSA
lhekvksa osQ :i esa] bldk vFkZ ;g gqvk fd
lim f (x) = c – 1 rFkk lim+ f (x) = c
x →c− x →c

pw¡fd fdlh Hkh iw.kk±d c osQ fy, ;s lhek,¡ leku ugha gks ldrh gSa] vr% iznÙk iQyu x lHkh
iw.kk±d ekuksa osQ fy, vlarr gSA
5.2.1 larr iQyuksa dk chtxf.kr (Algebra of continuous functions)
fiNyh d{kk esa] lhek dh ladYiuk le>us osQ mijkar] geusa lhekvksa osQ chtxf.kr dk oqQN
vè;;u fd;k FkkA vuq:ir% vc ge larr iQyuksa osQ chtxf.kr dk Hkh oqQN vè;;u djsaxsA pw¡fd
fdlh ¯cnq ij ,d iQyu dk lkarR; iw.kZ:i ls ml ¯cnq ij iQyu dh lhek }kjk
fu/kZfjr gksrk gS] vr,o ;g rdZlaxr gS fd ge lhekvksa osQ ln`'; gh ;gk¡ Hkh chth; ifj.kkeksa
dh vis{kk djsaA
izes; 1 eku yhft, fd f rFkk g nks ,sls okLrfod iQyu gSa] tks ,d okLrfod la[;k c osQ fy,
larr gSaA rc]
(1) f + g , x = c ij larr gS
(2) f – g , x = c ij larr gS
(3) f . g , x = c ij larr gS
f 
(4)   , x = c ij larr gS (tcfd g (c) ≠ 0 gSA)
g
miifÙk ge ¯cnq x = c ij (f + g) osQ lkarR; dh tk¡p djrs gSaA ge n[krs gSa fd
lim( f + g ) ( x ) = lim [ f ( x ) + g ( x )] (f + g dh ifjHkk"kk }kjk)
x →c x →c

= lim f ( x) + lim g ( x) (lhekvksa osQ izes; }kjk)

x →c x →c

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 lkarR; rFkk vodyuh;rk 171

= f (c) + g(c) (D;ksa f rFkk g larr iQyu gaS)

= (f + g) (c) (f + g dh ifjHkk"kk }kjk)
vr%] f + g Hkh x = c osQ fy, larr gSA
izes; 1 osQ 'ks"k Hkkxksa dh miifÙk blh osQ leku gS ftUgas ikBdksa osQ fy, vH;kl gsrq NksM+
fn;k x;k gSA
fVIi.kh
(i) mi;qZDr izes; osQ Hkkx (3) dh ,d fo'ks"k n'kk osQ fy,] ;fn f ,d vpj iQyu
f (x) = λ gks] tgk¡ λ, dksbZ vpj okLrfod la[;k gS] rks (λ . g) (x) = λ . g (x) }kjk
ifjHkkf"kr iQyu (λ . g) Hkh ,d larr iQyu gSA fo'ks"k :i ls] ;fn λ = – 1, rks f osQ
lkarR; esa – f dk lkarR; varfuZfgr gksrk gSA
(ii) mi;qZDr izes; osQ Hkkx (4) dh ,d fo'ks"k n'kk osQ fy,] ;fn f ,d vpj iQyu

λ λ λ
f (x) = λ, rks ( x) = }kjk ifjHkkf"kr iQyu g Hkh ,d larr iQyu gksrk gS] tgk¡
g g ( x)
1
g(x) ≠ 0 gSA fo'ks"k :i ls] g osQ lkarR; esa dk lkarR; varfuZfgr gSA
g
mi;qZDr nksuksa izes;ksa osQ mi;ksx }kjk vusd larr iQyuksa dks cuk;k tk ldrk gSA buls ;g
fuf'pr djus esa Hkh lgk;rk feyrh gS fd dksbZ iQyu larr gS ;k ughaA fuEufyf[kr mnkgj.kksa esa
;g ckr Li"V dh xbZ gSA
mnkgj.k 16 fl¼ dhft, fd izR;sd ifjes; iQyu larr gksrk gSA
gy Lej.k dhft, fd izR;sd ifjes; iQyu f fuEufyf[kr :i dk gksrk gS%
p( x )
f ( x) = , q ( x) ≠ 0
q ( x)

tgk¡ p vkSj q cgqin iQyu gSaA f dk izkar] mu ¯cnqvksa dks NksM+dj ftu ij q 'kwU; gS] leLr
okLrfod la[;k,¡ gSaA pw¡fd cgqin iQyu larr gksrs gSa (mnkgj.k 14)] vr,o izes; 1 osQ Hkkx (4)
}kjk f ,d larr iQyu gSA
mnkgj.k 17 sine iQyu osQ lkarR; ij fopkj dhft,A
gy bl ij fopkj djus osQ fy, ge fuEufyf[kr rF;ksa dk iz;ksx djrs gSa%
lim sin x = 0
x →0

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geus bu rF;ksa dks ;gk¡ izekf.kr rks ugha fd;k gS] fdUrq sine iQyu osQ vkys[k dks 'kwU; osQ
fudV ns[k dj ;s rF; lgtkuqHkwfr (intuitively) ls Li"V gks tkrk gSA
vc nsf[k, fd f (x) = sin x lHkh okLrfod la[;kvksa osQ fy, ifjHkkf"kr gSA eku yhft, fd
c ,d okLrfod la[;k gSA x = c + h j[kus ij] ;fn x → c rks ge ns[krs gSa fd h → 0 blfy,
lim f ( x ) = lim sin x
x →c x →c

= lim sin(c + h)
h →0

= lim [sin c cos h + cos c sin h]

h →0

= lim [sin c cos h] + lim [cos c sin h]

h →0 h →0
= sin c + 0 = sin c = f (c)
bl izdkj lim f (x) = f (c) vr% f ,d larr iQyu gSA
x →c

fVIi.kh blh izdkj cosine iQyu osQ lkarR; dks Hkh izekf.kr fd;k tk ldrk gSA
mnkgj.k 18 fl¼ dhft, fd f (x) = tan x ,d larr iQyu gSA
sin x
gy fn;k gqvk iQyu f (x) = tan x = gSA ;g iQyu mu lHkh okLrfod la[;kvksa osQ fy,
cos x
π
ifjHkkf"kr gS] tgk¡ cos x ≠ 0, vFkkZr~ x ≠ (2n +1)
gSA geus vHkh izekf.kr fd;k gS fd sine vkSj
2
cosine iQyu] larr iQyu gSaA blfy, tan iQyu] bu nksuksa iQyuksa dk HkkxiQy gksus osQ dkj.k] x
osQ mu lHkh ekuksa osQ fy, larr gS ftu osQ fy, ;g ifjHkkf"kr gSA
iQyuksa osQ la;kstu (composition) ls lacaf/r] larr iQyuksa dk O;ogkj ,d jkspd rF; gSA
Lej.k dhft, fd ;fn f vkSj g nks okLrfod iQyu gSa] rks
(f o g) (x) = f (g (x))
ifjHkkf"kr gS] tc dHkh g dk ifjlj f osQ izkar dk ,d mileqPp; gksrk gSA fuEufyf[kr izes;
(izek.k fcuk osQoy O;Dr)] la;qDr (composite) iQyuksa osQ lkarR; dks ifjHkkf"kr djrh gSA
izes; 2 eku yhft, fd f vkSj g bl izdkj osQ nks okLrfod ekuh; (real valued) iQyu gSa
fd c ij (f o g) ifjHkkf"kr gSA ;fn c ij g rFkk g (c) ij f larr gS] rks c ij (f o g) larr
gksrk gSA
fuEufyf[kr mnkgj.kksa esa bl izes; dks Li"V fd;k x;k gSA

2018-19
 lkarR; rFkk vodyuh;rk 173

mnkgj.k 19 n'kkZb, fd f (x) = sin (x2) }kjk ifjHkkf"kr iQyu] ,d larr iQyu gSA
gy isz{k.k dhft, fd fopkjk/hu iQyu izR;sd okLrfod la[;k osQ fy, ifjHkkf"kr gSsA iQyu
f dks] g rFkk h nks iQyuksa osQ la;kstu (g o h)osQ :i esa lkspk tk ldrk gS] tgk¡ g (x) = sin x
rFkk h (x) = x2 gSA pw¡fd g vkSj h nksuksa gh larr iQyu gSa] blfy, izes; 2 }kjk ;g fu"d"kZ fudkyk
tk ldrk gS] fd f ,d larr iQyu gSA
mnkgj.k 20 n'kkZb, fd f (x) = |1 – x + | x | | }kjk ifjHkkf"kr iQyu f] tgk¡ x ,d okLrfod la[;k
gS] ,d larr iQyu gSA
gy lHkh okLrfod la[;kvksa x osQ fy, g dks g (x) = 1 – x + | x | rFkk h dks h (x) = | x | }kjk
ifjHkkf"kr dhft,A rc]
(h o g) (x) = h (g (x))
= h (1– x + | x |)
= | 1– x + | x | | = f (x)
mnkgj.k 7 esa ge ns[k pqosQ gSa fd h ,d larr iQyu gSA blh izdkj ,d cgqin iQyu vkSj ,d
ekikad iQyu dk ;ksx gksus osQ dkj.k g ,d larr iQyu gSA vr% nks larr iQyuksa dk la;qDr iQyu
gksus osQ dkj.k f Hkh ,d larr iQyu gSA

iz'ukoyh 5-1
1. fl¼ dhft, fd iQyu f (x) = 5x – 3, x = 0, x = – 3 rFkk x = 5 ij larr gSA

2. x = 3 ij iQyu f (x) = 2x2 – 1 osQ lkarR; dh tk¡p dhft,A

3. fuEufyf[kr iQyuksa osQ lkarR; dh tk¡p dhft,%

1
(a) f (x) = x – 5 (b) f (x) = ,x≠5
x−5

x 2 − 25
(c) f (x) = , x ≠ –5 (d) f (x) = | x – 5 |
x+5
4. fl¼ dhft, fd iQyu f (x) = xn , x = n, ij larr gS] tgk¡ n ,d /u iw.kk±d gSA

 x, ;fn x ≤ 1
5. D;k f ( x ) =  }kjk ifjHkkf"kr iQyu f
5, ;fn x > 1
x = 0, x = 1, rFkk x = 2 ij larr gS\

2018-19
174 xf.kr

f osQ lHkh vlkarR; osQ ¯cnqvksa dks Kkr dhft,] tc fd f fuEufyf[kr izdkj ls ifjHkkf"kr gS%

| x | +3, ;fn x ≤ − 3
 2 x + 3, ;fn x ≤ 2 
6. f ( x) =  7. f ( x ) =  −2 x, ;fn − 3 < x < 3
 2 x − 3, ;fn x > 2 6 x + 2, ;fn x ≥ 3


| x |  x
 , ;fn x ≠ 0  , ;fn x < 0
8. f ( x) =  x 9. f ( x) =  | x |
 0, ;fn x = 0  −1, ;fn x ≥ 0
 

 x + 1, ;fn x ≥ 1  x 3 − 3, ;fn x ≤ 2
10. f ( x) =  2 11. f ( x) = 
 x + 1, ;fn x < 1  x + 1, ;fn x > 2
2

 x − 1, ;fn x ≤ 1
10
12. f ( x) = 
2
 x , ;fn x > 1

 x + 5, ;fn x ≤ 1
13. D;k f ( x ) =  }kjk ifjHkkf"kr iQyu] ,d larr iQyu gS\
 x − 5, ;fn x > 1
iQyu f, osQ lkarR; ij fopkj dhft,] tgk¡ f fuEufyf[kr }kjk ifjHkkf"kr gS%

3, ;fn 0 ≤ x ≤ 1  2 x, ;fn x < 0

 
14. f ( x ) =  4, ;fn 1< x < 3 15. f ( x ) = 0, ;fn 0 ≤ x ≤ 1
5, ;fn 3 ≤ x ≤ 10  4 x, ;fn x > 1
 

 −2, ;fn x ≤ − 1

16. f ( x ) =  2 x, ;fn − 1 < x ≤ 1
 2, ;fn x > 1


17. a vkSj b osQ mu ekuksa dks Kkr dhft, ftuosQ fy,

ax + 1, ;fn x ≤ 3
f ( x) = 
bx + 3, ;fn x > 3
}kjk ifjHkkf"kr iQyu x = 3 ij larr gSA

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 lkarR; rFkk vodyuh;rk 175

18. λ osQ fdl eku osQ fy,

 λ ( x − 2 x), ;fn x ≤ 0
2
f ( x) = 
 4 x + 1, ;fn x > 0
}kjk ifjHkkf"kr iQyu x = 0 ij larr gSA x = 1 ij blosQ lkarR; ij fopkj dhft,A
19. n'kkZb, fd g (x) = x – [x] }kjk ifjHkkf"kr iQyu leLr iw.kk±d ¯cnqvksa ij vlarr gSA ;gk¡
[x] ml egÙke iw.kk±d fu:fir djrk gS] tks x osQ cjkcj ;k x ls de gSA
20. D;k f (x) = x2 – sin x + 5 }kjk ifjHkkf"kr iQyu x = π ij larr gS?
21. fuEufyf[kr iQyuksa osQ lkarR; ij fopkj dhft,%
(a) f (x) = sin x + cos x (b) f (x) = sin x – cos x
(c) f (x) = sin x . cos x
22. cosine, cosecant, secant vkSj cotangent iQyuksa osQ lkarR; ij fopkj dhft,A
23. f osQ lHkh vlkarR;rk osQ ¯cnqvksa dks Kkr dhft,] tgk¡

 sin x
 , ;fn x < 0
f ( x) =  x
 x + 1, ;fn x ≥ 0

24. fu/kZfjr dhft, fd iQyu f
 2 1
 x sin , ;fn x ≠ 0
f ( x) =  x
0, ;fn x = 0

}kjk ifjHkkf"kr ,d larr iQyu gSA
25. f osQ lkarR; dh tk¡p dhft,] tgk¡ f fuEufyf[kr izdkj ls ifjHkkf"kr gS
sin x − cos x, ;fn x ≠ 0
f ( x) = 
 −1, ;fn x = 0
iz'u 26 ls 29 esa k osQ ekuksa dks Kkr dhft, rkfd iznÙk iQyu fufnZ"V ¯cnq ij larr gks%
 k cos x π
 π − 2 x , ;fn x ≠ 2 π
26. f ( x) =  }kjk ifjHkkf"kr iQyu x = ij
3, π 2
;fn x =
 2

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kx 2 , ;fn x ≤ 2
27. f ( x) =  }kjk ifjHkkf"kr iQyu x = 2 ij
3, ;fn x > 2

kx + 1, ;fn x ≤ π
28. f ( x) =  }kjk ifjHkkf"kr iQyu x = π ij
cos x, ;fn x > π

kx + 1, ;fn x ≤ 5
29. f ( x) =  }kjk ifjHkkf"kr iQyu x = 5 ij
3x − 5, ;fn x > 5
30. a rFkk b osQ ekuksa dks Kkr dhft, rkfd

5, ;fn x ≤ 2

f ( x ) =  ax + b, ;fn 2 < x < 10
 21, ;fn x ≥ 10

}kjk ifjHkkf"kr iQyu ,d larr iQyu gksA
31. n'kkZb, fd f (x) = cos (x2) }kjk ifjHkkf"kr iQyu ,d larr iQyu gSA
32. n'kkZb, fd f (x) = | cos x | }kjk ifjHkkf"kr iQyu ,d larr iQyu gSA
33. t¡kfp, fd D;k sin | x | ,d larr iQyu gSA
34. f (x) = | x | – | x + 1 | }kjk ifjHkkf"kr iQyu f osQ lHkh vlkaR;rk osQ ¯cnqvksa dks Kkr
dhft,A
5.3. vodyuh;rk (Differentiability)
fiNyh d{kk esa lh[ks x, rF;ksa dks Lej.k dhft,A geusa ,d okLrfod iQyu osQ vodyt
(Derivative) dks fuEufyf[kr izdkj ls ifjHkkf"kr fd;k FkkA
eku yhft, fd f ,d okLrfod iQyu gS rFkk c blosQ izkar esa fLFkr ,d ¯cnq gSA c ij f
dk vodyt fuEufyf[kr izdkj ls ifjHkkf"kr gS%
f (c + h ) − f (c)
lim
h →0 h
d
;fn bl lhek dk vfLrRo gks rks c ij f osQ vodyt dks f′(c) ;k ( f ( x)) | c }kjk izdV
dx
djrs gSaA
f ( x + h) − f ( x )
f ′( x) = lim
h →0 h

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 lkarR; rFkk vodyuh;rk 177

}kjk ifjHkkf"kr iQyu] tc Hkh bl lhek dk vfLrRo gks] f osQ vodyt dks ifjHkkf"kr djrk gSA
d dy
f osQ vodyt dks f ′ (x) ;k ( f ( x )) }kjk izdV djrs gSa vkSj ;fn y = f (x) rks bls ;k y′
dx dx
}kjk izdV djrs gSaA fdlh iQyu dk vodyt Kkr djus dh izfØ;k dks vodyu
(differentiation)dgrs gSAa ge okD;ka'k ¶x osQ lkis{k f (x) dk vodyu dhft, (differentiate)¸
dk Hkh iz;ksx djrs gSa] ftldk vFkZ gksrk gS fd f ′(x) Kkr dhft,A
vodyt osQ chtxf.kr osQ :i esa fuEufyf[kr fu;eksa dks izekf.kr fd;k tk pqdk gS%
(1) (u ± v)′ = u′ ± v′.
(2) (uv)′ = u′v + uv′ (yscuh”k ;k xq.kuiQy fu;e)
′
(3)  u  = u′v − uv′ , tgk¡ v ≠ 0 (HkkxiQy fu;e)
v v2
uhps nh xbZ lkj.kh esa oqQN izkekf.kd (standard) iQyuksa osQ vodytksa dh lwph nh xbZ gS%
lkj.kh 5-3
f (x) xn sin x cos x tan x
f ′(x) nx n – 1 cos x – sin x sec2 x

tc dHkh Hkh geus vodyt dks ifjHkkf"kr fd;k gS rks ,d lq>ko Hkh fn;k gS fd ¶;fn lhek
dk vfLrRo gks A¸ vc LokHkkfod :i ls iz'u mBrk gS fd ;fn ,slk ugha gS rks D;k gksxk\ ;g
iz'u furkar izklafxd gS vkSj bldk mÙkj HkhA ;fn lim f (c + h) − f (c) dk vfLrRo ugha gS] rks
h →0 h
ge dgrs gSa fd c ij f vodyuh; ugha gSA nwljs 'kCnksa esa] ge dgrs gSa fd vius izkar osQ fdlh
f (c + h) − f (c )
¯cnq c ij iQyu f vodyuh; gS ] ;fn nks u ks a lhek,¡ lim rFkk
h →0 – h
f (c + h ) − f (c )
lim+ ifjfer (finite) rFkk leku gSaA iQyu varjky [a, b] esa vodyuh;
h →0 h
dgykrk gS] ;fn og varjky [a, b] osQ izR;sd ¯cnq ij vodyuh; gSA tSlk fd lkarR; osQ lanHkZ
esa dgk x;k Fkk fd vaR; ¯cnqvksa a rFkk b ij ge Øe'k% nk,¡ rFkk ck,¡ i{k dh lhek,¡ ysrs gSa]
tks fd vkSj oqQN ugha] cfYd a rFkk b ij iQyu osQ nk,¡ i{k rFkk ck,¡ i{k osQ vodyt gh gaSA
blh izdkj iQyu varjky (a, b) esa vodyuh; dgykrk gS] ;fn og varjky (a, b) osQ izR;sd
¯cnq ij vodyuh; gSA

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izes; 3 ;fn iQyu fdlh ¯cnq c ij vodyuh; gS] rks ml ¯cnq ij og larr Hkh gSA
miifÙk pw¡fd ¯cnq c ij f vodyuh; gS] vr%
f ( x) − f (c)
lim = f ′(c)
x →c x−c
fdarq x ≠ c osQ fy,
f ( x ) − f ( c)
f (x) – f (c) = . ( x − c)
x−c

 f ( x) − f ( c) 
blfy, lim [ f ( x ) − f (c)] = lim  . ( x − c) 
x →c x →c  x−c 

;k lim [ f ( x)] − lim [ f (c )] = lim  f ( x ) − f (c )  . lim [( x − c )]

x →c x →c x →c  x−c  x→c
= f ′(c) . 0 = 0
;k lim f ( x ) = f (c)
x →c

bl izdkj x = c ij iQyu f larr gSA

miizes; 1 izR;sd vodyuh; iQyu larr gksrk gSA
;gk¡ ge è;ku fnykrs gSa fd mi;qZDr dFku dk foykse (converse) lR; ugha gSA fu'p; gh ge
ns[k pqosQ gSa fd f (x) = | x | }kjk ifjHkkf"kr iQyu ,d larr iQyu gSA bl iQyu osQ ck,¡ i{k dh
lhek ij fopkj djus ls
f (0 + h) − f (0) − h
lim– = = −1
h →0 h h
rFkk nk¡, i{k dh lhek
f (0 + h) − f (0) h
lim = = 1 gSA
h → 0+ h h
f (0 + h) − f (0)
pwf¡ d 0 ij mi;qDZ r ck,¡ rFkk nk,¡ i{k dh lhek,¡ leku ugha ga]S blfy, lim
h →0 h
dk vfLrRo ugha gS vkSj bl izdkj 0 ij f vodyuh; ugha gSA vr% f ,d vodyuh; iQyu
ugha gSA

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 lkarR; rFkk vodyuh;rk 179

5.3.1 la;qDr iQyuksa osQ vodyt (Differentials of composite functions)

la;qDr iQyuksa osQ vodyt osQ vè;;u dks ge ,d mnkgj.k }kjk Li"V djsaxsA eku yhft, fd
ge f dk vodyt Kkr djuk pkgrs gSa] tgk¡
f (x) = (2x + 1)3
,d fof/ ;g gS fd f}in izes; osQ iz;ksx }kjk (2x + 1)3 dks izlkfjr djosQ izkIr cgqin iQyu
dk vodyt Kkr djsa] tSlk uhps Li"V fd;k x;k gS_
d d
f ( x) = (2 x + 1)3 
dx dx
d
= (8 x 3 + 12 x 2 + 6 x + 1)
dx
= 24x2 + 24x + 6
= 6 (2x + 1)2
vc] è;ku nhft, fd
f (x) = (h o g) (x)
tgk¡ g(x) = 2x + 1 rFkk h(x) = x3 gSA eku yhft, t = g(x) = 2x + 1. rks f (x) = h(t) = t3.
df dh dt
vr% = 6 (2x + 1)2 = 3(2x + 1)2 . 2 = 3t2 . 2 = ⋅
dx dt dx
bl nwljh fof/ dk ykHk ;g gS fd oqQN izdkj osQ iQyu] tSls (2x + 1)100 osQ vodyt dk
ifjdyu djuk bl fofèk }kjk ljy gks tkrk gSA mi;qZDr ifjppkZ ls gesa vkSipkfjd :i ls
fuEufyf[kr izes; izkIr gksrk gS] ftls Ük`a[kyk fu;e (chain rule) dgrs gSaA
izes; 4 (Ük`a[kyk fu;e ) eku yhft, fd f ,d okLrfod ekuh; iQyu gS] tks u rFkk v nks iQyuksa
dt dv
dk la;kstu gS_ vFkkZr~ f = v o u. eku yhft, fd t = u (x) vkSj] ;fn rFkk nksuksa dk
dx dt
df dv dt
vfLrRo gS] rks = ⋅
dx dt dx
ge bl izes; dh miifÙk NksM+ nsrs gSaA Ük`a[kyk fu;e dk foLrkj fuEufyf[kr izdkj ls fd;k
tk ldrk gSA eku yhft, fd f ,d okLrfod ekuh; iQyu gS] tks rhu iQyuksa u, v vkSj w dk
la;kstu gS] vFkkZr~
f = (w o u) o v gS ;fn t = u (x) rFkk s = v (t) gS rks
df d dt dw ds dt
= (w o u ) ⋅ = ⋅ ⋅
dx dt dx ds dt dx

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;fn mi;qZDr dFku osQ lHkh vodytksa dk vfLrRo gks rks ikBd vkSj vf/d iQyuksa osQ la;kstu
osQ fy, Ük`a[kyk fu;e dks iz;qDr dj ldrs gSaA
mnkgj.k 21 f (x) = sin (x2) dk vodyt Kkr dhft,A
gy è;ku nhft, fd iznÙk iQyu nks iQyuksa dk la;kstu gSA okLro esa] ;fn u(x) = x2 vkSj
v(t) = sin t gS rks
f (x) = (v o u) (x) = v(u(x)) = v(x2) = sin x2
dv dt
t = u(x) = x2 j[kus ij è;ku nhft, fd = cos t rFkk = 2 x vkSj nksuksa dk vfLrRo Hkh
dt dx
gSaA vr% Ük`a[kyk fu;e }kjk
df dv dt
= ⋅ = cos t . 2 x
dx dt dx
lkekU;r% vafre ifj.kke dks x osQ inksa esa O;Dr djus dk izpyu gS vr,o
df
= cos t ⋅ 2 x = 2 x cos x 2
dx
fodYir% ge lh/s Hkh bldk eku fudky ldrs gSa tSls uhps of.kZr gS]
dy d
y = sin (x2) ⇒ = (sin x2)
dx dx
d 2
= cos x2 (x ) = 2x cos x2
dx
mnkgj.k 22 tan (2x + 3) dk vodyt Kkr dhft,A
gy eku yhft, fd f (x) = tan (2x + 3), u (x) = 2x + 3 rFkk v(t) = tan t gSA
(v o u) (x) = v(u(x)) = v(2x + 3) = tan (2x + 3) = f (x)

dv
bl izdkj f nks iQyuksa dk la;kstu gSA ;fn t = u(x) = 2x + 3. rks = sec2 t rFkk
dt
dt
= 2 rFkk nksuksa dk gh vfLrRo gSA vr% Ük`a[kyk fu;e }kjk
dx

df dv dt
= ⋅ = 2sec2 (2 x + 3)
dx dt dx

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 lkarR; rFkk vodyuh;rk 181

mnkgj.k 23 x osQ lkis{k sin (cos (x2)) dk vodyu dhft,A

gy iQyu f (x) = sin (cos (x2)) , u, v rFkk w, rhu iQyuksa dk la;kstu gSA bl izdkj
f (x) = (w o v o u) (x), tgk¡ u(x) = x2, v(t) = cos t rFkk w(s) = sin s gSA t = u(x) = x2 vkSj
dw ds dt
s = v(t) = cos t j[kus ij ge ns[krs gSa fd = cos s, = − sin t rFkk = 2 x vkSj bu lHkh
ds dt dx
dk] x osQ lHkh okLrfod ekuksa osQ fy, vfLrRo gSA
vr% Ük`a[kyk fu;e osQ O;kidhdj.k }kjk
df dw ds dt
= ⋅ ⋅ = (cos s) (– sin t) (2x) = – 2x sin x2 cos (cos x2)
dx ds dt dx
fodYir%
y = sin (cos x2)
dy d d
blfy, = sin (cos x2) = cos (cos x2) (cos x2)
dx dx dx

d
= cos (cos x2) (– sin x2) (x2)
dx
= – sin x2 cos (cos x2) (2x)
= – 2x sin x2 cos (cos x2)

iz'ukoyh 5-2
iz'u 1 ls 8 esa x osQ lkis{k fuEufyf[kr iQyuksa dk vodyu dhft,%
1. sin (x2 + 5) 2. cos (sin x) 3. sin (ax + b)
sin (ax + b)
4. sec (tan ( x )) 5. 6. cos x3 . sin2 (x5)
cos (cx + d )

7. 2 cot ( x 2 ) 8. cos ( x )

9. fl¼ dhft, fd iQyu f (x) = | x – 1 |, x ∈ R, x = 1 ij vodfyr ugha gSA

10. fl¼ dhft, fd egÙke iw.kk±d iQyu f (x) =[x], 0 < x < 3, x = 1 rFkk x = 2 ij
vodfyr ugha gSA

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5.3.2 vLi"V iQyuksa osQ vodyt (Derivatives of Implicit Functions)

vc rd ge y = f (x) osQ :i osQ fofo/ iQyuksa dk vodyu djrs jgs gSa ijarq ;g vko';d
ugha gS fd iQyuksa dks lnSo blh :i esa O;Dr fd;k tk,A mnkgj.kkFkZ] x vkSj y osQ chp fuEufyf[kr
laca/ksa esa ls ,d ij fo'ks"k :i ls fopkj dhft,%
x–y–π=0
x + sin xy – y = 0
igyh n'kk esa] ge y osQ fy, ljy dj ldrs gSa vkSj laca/ dks y = x – π osQ :i esa fy[k
ldrs gSaA nwljh n'kk esa] ,slk ugha yxrk gS fd laca/ y dks ljy djus dk dksbZ vklku rjhdk gSA
fiQj Hkh nksuksa esa ls fdlh Hkh n'kk esa] y dh x ij fuHkZjrk osQ ckjs esa dksbZ lansg ugha gSA tc x
vkSj y osQ chp dk laca/ bl izdkj O;Dr fd;k x;k gks fd mls y osQ fy, ljy djuk vklku
gks vkSj y = f (x) osQ :i esa fy[kk tk losQ] rks ge dgrs gSa fd y dks x osQ Li"V (explicit)iQyu
osQ :i esa O;Dr fd;k x;k gSA mi;qZDr nwljs laca/ esa] ge dgrs gSa fd y dks x osQ vLi"V
(implicity) iQyu osQ :i esa O;Dr fd;k x;k gSA
dy
mnkgj.k 24 ;fn x – y = π rks Kkr dhft,A
dx

gy ,d fof/ ;g gS fd ge y osQ fy, ljy djosQ mi;qZDr laca/ dks fuEu izdkj fy[ksa ;Fkk
y=x – π
dy
rc =1
dx
fodYir% bl laca/ dk x, osQ lkis{k lh/s vodyu djus ij
d dπ
( x − y) =
dx dx

dπ
;kn dhft, fd dk vFkZ gS fd x osQ lkis{k ,d vpj π dk vodyu djukA bl izdkj
dx
d d
( x) − ( y ) = 0
dx dx
ftldk rkRi;Z gS fd
dy dx
= =1
dx dx

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 lkarR; rFkk vodyuh;rk 183

mnkgj.k 25 ;fn y + sin y = cos x rks dy Kkr dhft,A

dx
gy ge bl laca/ dk lh/s vodyt djrs gSaA
dy d d
+ (sin y ) = (cos x)
dx dx dx
Ük`a[kyk fu;e dk iz;ksx djus ij
dy dy
+ cos y ⋅ = – sin x
dx dx
blls fuEufyf[kr ifj.kke feyrk gS]
dy sin x
= −
dx 1 + cos y
tgk¡ y ≠ (2n + 1) π
5.3.3 izfrykse f=kdks.kferh; iQyuksa osQ vodyt (Derivatives of Inverse Trigonometric
Functions)
ge iqu% è;ku fnykrs gSa fd izfrykse f=kdks.kferh; iQyu larr gksrs gSa] ijarq ge bls izekf.kr ugha
djsaxsA vc ge bu iQyuksa osQ vodytksa dks Kkr djus osQ fy, Ük`a[kyk fu;e dk iz;ksx djsaxsA
mnkgj.k 26 f (x) = sin –1 x dk vodyt Kkr dhft,A ;g eku yhft, fd bldk
vfLrRo gSA

gy eku yhft, fd y = f (x) = sin–1 x gS rks x = sin y

nksuksa i{kksa dk x osQ lkis{k vodyu djus ij
dy
1 = cos y
dx

dy 1 1
⇒ = =
dx cos y cos (sin −1 x )

π π
è;ku nhft, fd ;g osQoy cos y ≠ 0 osQ fy, ifjHkkf"kr gS] vFkkZr~ , sin–1 x ≠ − , , vFkkZr~
2 2
x ≠ – 1, 1, vFkkZr~ x ∈ (– 1, 1)

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bl ifj.kke dks oqQN vkd"kZd cukus gsrq ge fuEufyf[kr O;ogkj dkS'ky (manipulation)
djrs gSaA Lej.k dhft, fd x ∈ (– 1, 1) osQ fy, sin (sin–1 x) = x vkSj bl izdkj

cos2 y = 1 – (sin y)2 = 1 – (sin (sin–1 x))2 = 1 – x2

π π
lkFk gh pw¡fd y ∈  − ,  , cos y ,d /ukRed jkf'k gS vkSj blfy, cos y = 1 − x 2
 2 2
bl izdkj x ∈ (– 1, 1) osQ fy,
dy 1 1
= =
dx cos y 1 − x2
mnkgj.k 27 f (x) = tan–1 x dk vodyt Kkr dhft,] ;g ekurs gq, fd bldk vfLrRo gSA

gy eku yhft, fd y = tan–1 x gS rks x = tan y gSA x osQ lkis{k nksuksa i{kksa dk vodyu
djus ij

dy
1 = sec2 y
dx

dy 1 1 1 1
⇒ = = = −1
=
dx sec y 1 + tan y 1 + (tan (tan x)) 1 + x2
2 2 2

vU; izfrykse f=kdks.kferh; iQyuksa osQ vodytksa dk Kkr djuk vkiosQ vH;kl osQ fy, NksM+
fn;k x;k gSA 'ks"k izfrykse f=kdks.kferh; iQyuksa osQ vodytksa dks fuEufyf[kr lkj.kh 5-4 esa fn;k
x;k gSA
lkj.kh 5.4
f (x) cos –1x cot –1x sec –1x cosec–1x

−1 −1 1 −1
f ′(x) x x2 −1
1 − x2 1 + x2 x x2 − 1

Domain of f ′ (–1, 1) R (– ∞, –1) ∪ (1, ∞) (– ∞, –1) ∪ (1, ∞)

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 lkarR; rFkk vodyuh;rk 185

iz'ukoyh 5-3
dy
fuEufyf[kr iz'uksa esa Kkr dhft,
dx
1. 2x + 3y = sin x 2. 2x + 3y = sin y 3. ax + by2 = cos y
4. xy + y2 = tan x + y 5. x2 + xy + y2 = 100 6. x3 + x2y + xy2 + y3 = 81

 2x 
7. sin2 y + cos xy = k 8. sin2 x + cos2 y = 1 9. y = sin–1  
 1 + x2 

 3x − x3  1 1
10. y = tan–1  2 
, − <x<
 1 − 3x  3 3

 1 − x2 
11. y = cos −1  , 0 < x < 1
 1 + x2 

 1 − x2 
12. y = sin −1  , 0 < x < 1
 1 + x2 

 2x ,
13. y = cos −1   −1 < x <1
 1 + x2 

14. (
y = sin −1 2 x 1 − x 2 , −)1
< x<
1
2 2

 1 , 1
15. y = sec −1  2  0< x<
 2x −1  2

5.4 pj?kkrakdh rFkk y?kqx.kdh; iQyu (Exponential and Logarithmic Functions)

vHkh rd geus iQyuksa] tSls cgqin iQyu] ifjes; iQyu rFkk f=kdks.kferh; iQyu] osQ fofHkUu oxks±
osQ oqQN igyqvksa osQ ckjs esa lh[kk gSA bl vuqPNsn esa ge ijLij lacaf/r iQyuksa osQ ,d u, oxZ
osQ ckjs esa lh[ksxa ]s ftUgsa pj?kkrkadh (exponential) rFkk y?kqx.kdh; (logarithmic) iQyu dgrs
gSaA ;gk¡ ij fo'ks"k :i ls ;g crykuk vko';d gS fd bl vuqPNsn osQ cgqr ls dFku izsjd rFkk
;FkkrF; gSa vkSj mudh miifÙk;k¡ bl iqLrd dh fo"k;&oLrq osQ {ks=k ls ckgj gSaA

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186 xf.kr

vko`Qfr 5-9 esa y = f1(x) = x, y = f2(x) = x2, y = f3(x) = x3 rFkk y = f4(x) = x4 osQ vkys[k
fn, x, gSaA è;ku nhft, fd T;ksa&T;ksa x dh ?kkr c<+rh tkrh gS oØ dh izo.krk Hkh c<+rh tkrh
gSA oØ dh izo.krk c<+us ls o`f¼ dh nj rst
gksrh tkrh gSA bldk vFkZ ;g gS fd x (>1) osQ
eku esa fuf'pr o`f¼ osQ laxr y = fn(x) dk
eku c<+rk tkrk gS tSls&tSls n dk eku 1] 2]
3] 4 gksrk tkrk gSA ;g dYiuh; gS fd ,slk
dFku lHkh /ukRed eku osQ fy, lR; gS tgk¡
fn(x) = xn gSA vko';d:i ls] bldk vFkZ ;g
gqvk fd tSls&tSls n esa o`f¼ gksrh tkrh gS
y = fn (x) dk vkys[k y-v{k dh vksj vf/d
>qdrk tkrk gSA mnkgj.k osQ fy, f10(x) = x10
rFkk f15(x) = x15 ij fopkj dhft,A ;fn x dk
eku 1 ls c<+dj 2 gks tkrk gS] rks f10 dk eku vko`Qfr 5-9
1 ls c<+dj 210 gks tkrk gS] tcfd f15 dk eku
1 ls c<+dj 215 gks tkrk gSA bl izdkj x esa leku o`f¼ osQ fy,] f15 dh o`f¼ f10 dh o`f¼ osQ
vis{kk vf/d rhozrk ls gksrh gSA
mi;qZDr ifjppkZ dk fu"d"kZ ;g gS fd cgqin iQyuksa dh o`f¼ muosQ ?kkr ij fuHkZj djrh gS]
vFkkZr~ ?kkr c<+krs tkb, o`f¼ c<+rh tk,xhA blosQ mijkar ,d LokHkkfod iz'u ;g mBrk gS fd]
D;k dksbZ ,slk iQyu gS tks cgqin iQyuksa dh vis{kk vf/d rsth ls c<+rk gS\ bldk mÙkj
ldkjkRed gS vkSj bl izdkj osQ iQyu dk ,d mnkgj.k y = f (x) = 10x gS
gekjk nkok ;g gS fd fdlh /u iw.kk±d n osQ fy, ;g iQyu f ] iQyu fn (x) = xn dh
vis{kk vf/d rsth ls c<+rk gSA mnkgj.k osQ fy, ge fl¼ dj ldrs gSa fd f100 (x) = x100 dh
vis{kk 10x vf/d rsth ls c<+rk gSA ;g uksV dhft, fd x osQ cM+s ekuksa osQ fy,] tSls x = 103,
f100 (x) = (103)100 = 10300 tcfd f (103) = 1010 = 101000 gSA Li"Vr% f100 (x) dh vis{kk f (x)
3

dk eku cgqr vf/d gSA ;g fl¼ djuk dfBu ugha gS fd x osQ mu lHkh ekuksa osQ fy, tgk¡
x > 103 , f (x) > f100 (x) gSA fdarq ge ;gk¡ ij bldh miifÙk nsus dk iz;kl ugha djsaxsA blh izdkj
x osQ cM+s ekuksa dks pqudj ;g lR;kfir fd;k tk ldrk gS fd] fdlh Hkh /u iw.kk±d n osQ fy,
fn (x) dh vis{kk f (x) dk eku vf/d rsth ls c<+rk gSA

ifjHkk"kk 3 iQyu y = f (x) = bx,/ukRed vk/kj b > 1 osQ fy, pj?kkrkadh iQyu dgykrk gSA
vko`Qfr 5-9 esa y = 10x dk js[kkfp=k n'kkZ;k x;k gSA

2018-19
 lkarR; rFkk vodyuh;rk 187

;g lykg nh tkrh gS fd ikBd bl js[kkfp=k dks b osQ fof'k"V ekuksa] tSls 2, 3 vkSj 4 osQ fy,
[khap dj ns[ksAa pj?kkrkadh iQyu dh oqQN ize[q k fo'ks"krk,¡ fuEufyf[kr gS%a
(1) pj?kkrkadh iQyu dk izkar] okLrfod la[;kvksa dk leqPp; R gksrk gSA
(2) pj?kkrkadh iQyu dk ifjlj] leLr /ukRed okLrfod la[;kvksa dk leqPp; gksrk gSA
(3) ¯cnq (0, 1) pj?kkrkadh iQyu osQ vkys[k ij lnSo gksrk gS (;g bl rF; dk iqu% dFku
gS fd fdlh Hkh okLrfod la[;k b > 1 osQ fy, b0 = 1)
(4) pj?kkrkadh iQyu lnSo ,d o/Zeku iQyu (increasing function) gksrk gS] vFkkZr~
tSls&tSls ge ck,¡ ls nk,¡ vksj c<+rs tkrs gSa] vkys[k Åij mBrk tkrk gSA
(5) x osQ vR;f/d cM+s ½.kkRed ekuksa osQ fy, pj?kkrkadh iQyu dk eku 0 osQ vR;ar fudV
gksrk gSA nwljs 'kCnksa esa] f}rh; prqFkk±'k esa] vkys[k mÙkjksÙkj x-v{k dh vksj vxzlj gksrk
gS (fdarq mlls dHkh feyrk ugha gSA)
vk/kj 10 okys pj?kkrkadh iQyu dks lk/kj.k pj?kkrkadh iQyu (common exponential
Function) dgrs gSaA d{kk XI dh ikB~;iqLrd osQ ifjf'k"V A.1.4 esa geus ns[kk Fkk fd Js.kh

1 1
1+ + + ... gSA
1! 2!
dk ;ksx ,d ,slh la[;k gS ftldk eku 2 rFkk 3 osQ eè; gksrk gS vkSj ftls e }kjk izdV djrs gSAa
bl e dks vk/kj osQ :i esa iz;ksx djus ij] gesa ,d vR;ar egRoiw.kZ pj?kkrkadh iQyu
y = ex izkIr gksrk gSA bls izko`Qfrd pj?kkrkadh iQyu (natural exponential function)
dgrs gSAa
;g tkuuk #fpdj gksxk fd D;k pj?kkrkadh iQyu osQ izfrykse dk vfLrRo gS vkSj ;fn ^gk¡*
rks D;k mldh ,d leqfpr O;k[;k dh tk ldrh gSA ;g [kkst fuEufyf[kr ifjHkk"kk osQ fy, izsfjr
djrh gSA
ifjHkk"kk 4 eku yhft, fd b > 1 ,d okLrfod la[;k gSA rc ge dgrs gSa fd]
b vk/kj ij a dk y?kqx.kd x gS] ;fn bx = a gSA
b vk/kj ij a osQ y?kqx.kd dks izrhd logba ls izdV djrs gSaA bl izdkj ;fn bx = a, rks
logb a = x bldk vuqHko djus osQ fy, vkb, ge oqQN Li"V mnkgj.kksa dk iz;ksx djsaA gesa Kkr
gS fd 23 = 8 gSA y?kqx.kdh; 'kCnksa esa ge blh ckr dks iqu% log2 8 = 3 fy[k ldrs gSaA blh izdkj
104 = 10000 rFkk log10 10000 = 4 lerqY; dFku gSaA blh rjg ls 625 = 54 = 252 rFkk log5
625 = 4 vFkok log25 625 = 2 lerqY; dFku gSaA
FkksM+k lk vkSj vf/d ifjiDo n`f"Vdks.k ls fopkj djus ij ge dg ldrs gSa fd b > 1 dks
vk/kj fu/kZfjr djus osQ dkj.k ^y?kqx.kd* dks /u okLrfod la[;kvksa osQ leqPp; ls lHkh

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188 xf.kr

okLrfod la[;kvksa osQ leqPp; esa ,d iQyu

osQ :i esa ns[kk tk ldrk gSA ;g iQyu] ftls
y?kqx.kdh; iQyu (logarithmic function)
dgrs gSa] fuEufyf[kr izdkj ls ifjHkkf"kr gS%
logb : R+ → R
x → logb x = y ;fn by = x
iwoZ dfFkr rjg ls] ;fn vk/kj b = 10 gS
rks bls ^lk/kj.k y?kqx.kd* vkSj ;fn b = e gS
rks bls ^izko`Qfrd y?kqx.kd* dgrs gSaA cgq/k
izko`Qfrd y?kqx.kd dks ln }kjk izdV djrs gSaA vko`Qfr 5-10
bl vè;k; esa log x vk/kj e okys y?kqx.kdh; iQyu dks fu:fir djrk gSA vko`Qfr 5-10 esa 2,
rFkk 10 vk/kjh; y?kqx.kdh; iQyuksa osQ vkys[k n'kkZ, x, gSaA
vk/kj b > 1 okys y?kqx.kdh; iQyuksa dh oqQN egRoiw.kZ fo'ks"krk,¡ uhps lwphc¼ gSa%
(1) /usrj (non-positive) la[;kvksa osQ fy, ge y?kqx.kd dh dksbZ vFkZiw.kZ ifjHkk"kk ugha cuk
ldrs gSa vkSj blfy, y?kqx.kdh; iQyu dk izkar R+ gSA
(2) y?kqx.kdh; iQyu dk ifjlj leLr okLrfod la[;kvksa dk leqPp; gSA
(3) ¯cnq (1, 0) y?kqx.kdh; iQyuksa osQ vkys[k ij lnSo jgrk gSA
(4) y?kqx.kdh; iQyu ,d o/Zeku iQyu gksrs gSa] vFkkZr~ T;ksa&T;ksa ge ck,¡ ls nk,¡ vksj pyrs
gSa] vkys[k mÙkjksÙkj Åij mBrk tkrk gSA
(5) 0 osQ vR;kf/d fudV okys x osQ fy,]
log x osQ eku dks fdlh Hkh nh xbZ
okLrfod la[;k ls de fd;k tk ldrk
gSA nwljs 'kCnksa es]a pkSFks (prqFkZ) prqFkk±'k esa
vkys[k y-v{k osQ fudVre vxzlj gksrk gS
(fdarq blls dHkh feyrk ugha gS)A
(6) vko`Qfr 5.11 esa y = ex rFkk y = loge x
osQ vkys[k n'kkZ, x, gaSA ;g è;ku nsuk
jkspd gS fd nksuksa oØ js[kk y = x esa ,d
nwljs osQ niZ.k izfrfcac gSaA vko`Qfr 5-11
y?kqx.kdh; iQyuksa osQ nks egRoiw.kZ xq.k uhps izekf.kr fd, x, gSa%
(1) vk/kj ifjorZu dk ,d ekud fu;e gS] ftlls loga p dks logb p osQ inksa esa Kkr fd;k
tk ldrk gSA eku yhft, fd loga p = α, logb p = β rFkk logb a = γ gSA bldk vFkZ ;g

2018-19
 lkarR; rFkk vodyuh;rk 189

gS fd aα = p, bβ = p rFkk bγ = a gSA vc rhljs ifj.kke dks igys esa j[kus ls

(bγ)α = bγα = p
bldks nwljs lehdj.k esa iz;ksx djus ij
bβ = p = bγα
β
vr% β = αγ vFkok α = gSA bl izdkj
γ
log b p
loga p =
log b a
(2) xq.kuiQyuksa ij log iQyu dk izHkko bldk ,d vU; jkspd xq.k gSA eku yhft, fd
logb pq = α gSA blls bα = pq izkIr gksrk gSA blh izdkj ;fn logb p = β rFkk logb q = γ
gS rks bβ = p rFkk bγ = q izkIr gksrk gSA ijarq bα = pq = bβbγ = bβ + γ gSA
bldk rkRi;Z gS fd α = β + γ, vFkkZr~
logb pq = logb p + logb q
blls ,d fo'ks"k jkspd rFkk egRoiw.kZ ifj.kke rc fudyrk gS tc p = q gSA ,slh n'kk esa]
mi;qZDr dks iqu% fuEufyf[kr izdkj ls fy[kk tk ldrk gS
logb p2 = logb p + logb p = 2 logb p
bldk ,d ljy O;kidhdj.k vH;kl osQ fy, NksM+ fn;k x;k gS vFkkZr~ fdlh Hkh /u iw.kk±d
n osQ fy,
logb pn = n logb p
okLro esa ;g ifj.kke n osQ fdlh Hkh okLrfod eku osQ fy, lR; gS] fdarq bls ge izekf.kr
djus dk iz;kl ugha djsaxsA blh fof/ ls ikBd fuEufyf[kr dks lR;kfir dj ldrs gSa%
x
log b = logb x – logb y
y
mnkgj.k 28 D;k ;g lR; gS fd x osQ lHkh okLrfod ekuksa osQ fy, x = elog x gS?
gy igys rks è;ku nhft, fd log iQyu dk izkar lHkh /u okLrfod la[;kvksa dk leqPp; gksrk
gSA blfy, mi;qZDr lehdj.k /usrj okLrfod la[;kvksa osQ fy, lR; ugha gSA vc eku yhft,
fd y = elog x gSA ;fn y > 0 rc nksuks i{kksa dk y?kqx.kd ysus ls log y = log (elog x) = log x . log
e = log x gSA ftlls y = x izkIr gksrk gSA vr,o x = elog x osQoy x osQ /u ekuksa osQ fy, lR; gSA
vody xf.kr (differential calculus) es]a izko`Qfrd pj?kkrkadh iQyu dk ,d vlk/kj.k xq.k
;g gS fd] vodyu dh izfØ;k esa ;g ifjofrZr ugha gksrk gSA bl xq.k dks uhps izes;ksa esa O;Dr
fd;k x;k gS] ftldh miifÙk dks ge NksM+ nsrs gSaA

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190 xf.kr

izes; 5*
d x
(1) x osQ lkis{k ex dk vodyt ex gh gksrk gS] vFkkZr~ (e ) = ex
dx
1 d 1
(2) x osQ lkis{k log x dk vodyt gksrk gS] vFkkZr~ (log x) =
x dx x
mnkgj.k 29 x osQ lkis{k fuEufyf[kr dk vodyu dhft,%
(i) e –x (ii) sin (log x), x > 0 (iii) cos–1 (ex) (iv) ecos x
gy
(i) eku yhft, y = e – x gSA vc Ük`a[kyk fu;e osQ iz;ksx }kjk
dy d
= e− x ⋅ (– x) = – e– x
dx dx
(ii) eku yhft, fd y = sin (log x) gSA vc Ük`a[kyk fu;e }kjk
dy d cos (log x )
= cos(log x ) ⋅ (log x) =
dx dx x
(iii) eku yhft, fd y = cos (e ) gSA vc Ük`a[kyk fu;e }kjk
–1 x

dy −1 d −e x
= ⋅ (e x ) = .
dx 1 − (e x ) 2 dx 1 − e2 x
(iv) eku yhft, fd y = ecos x gSA vc Ük`a[kyk fu;e }kjk
dy
= ecos x ⋅ ( − sin x) = − (sin x) ecos x
dx

iz'ukoyh 5-4
fuEufyf[kr dk x osQ lkis{k vodyu dhft,%
ex −1 3
1. 2. esin x 3. e x
sin x
2 5
4. sin (tan–1 e–x) 5. log (cos ex) 6. e x + e x + ... + e x
cos x
7. e x
, x>0 8. log (log x), x > 1 9. log x , x > 0

10. cos (log x + ex)

*o`Qi;k iwjd ikB~; lkexzh i`"B 303&304 ij ns[ksa

2018-19
 lkarR; rFkk vodyuh;rk 191

5.5. y?kqx.kdh; vodyu (Logarithmic Differentiation)

bl vuqPNsn esa ge fuEufyf[kr izdkj osQ ,d fof'k"V oxZ osQ iQyuksa dk vodyu djuk lh[ksaxs%
y = f (x) = [u(x)]v (x)
y?kqx.kd (e vk/kj ij ) ysus ij mi;qZDr dks fuEufyf[kr izdkj ls iqu% fy[k ldrs gSa
log y = v (x) log [u(x)]
Ük`a[kyk fu;e osQ iz;ksx }kjk
1 dy 1
⋅ = v( x) ⋅ . u′(x) + v′(x) . log [u(x)]
y dx u ( x)
bldk rkRi;Z gS fd
dy  v( x) 
= y ⋅ u ′ ( x) + v′ ( x) ⋅ log [u ( x) ]
dx  u( x) 
bl fof/ esa è;ku nsus dh eq[; ckr ;g gS fd f (x) rFkk u(x) dks lnSo /ukRed gksuk pkfg,
vU;Fkk muosQ y?kqx.kd ifjHkkf"kr ugha gksxa As bl izfØ;k dks y?kqx.kdh; vodyu (logarithmic
differentiation) dgrs gSa vkSj ftls fuEufyf[kr mnkgj.kksa }kjk Li"V fd;k x;k gSA

( x − 3) ( x 2 + 4)
mnkgj.k 30 x osQ lkis{k dk vodyu dhft,A
3x 2 + 4 x + 5

( x − 3) ( x 2 + 4)
gy eku yhft, fd y =
(3 x 2 + 4 x + 5)
nksuksa i{kksa osQ y?kqx.kd ysus ij
1
log y = [log (x – 3) + log (x2 + 4) – log (3x2 + 4x + 5)]
2
nksuksa i{kksa dk x, osQ lkis{k voydu djus ij
1 dy 1 1 2x 6x + 4 
⋅ =  + 2 − 2
y dx 2  ( x − 3) x + 4 3x + 4 x + 5 

dy y 1 2x 6x + 4 
vFkok =  + 2 − 2
dx 2  ( x − 3) x + 4 3 x + 4 x + 5 

1 ( x − 3) ( x 2 + 4)  1 2x 6x + 4 
+ 2 − 2
3x + 4 x + 5  ( x − 3) x + 4 3 x + 4 x + 5 

= 2
2

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192 xf.kr

mnkgj.k 31 x osQ lkis{k ax dk vodyu dhft,] tgk¡ a ,d /u vpj gSA

gy eku yhft, fd y = ax, rks
log y = x log a
nksuksa i{kksa dk x, osQ lkis{k vodyu djus ij
1 dy
y dx = log a
dy
vFkok = y log a
dx
d x
bl izdkj (a ) = ax log a
dx
d x d x log a ) d
fodYir% (a ) = (e = e x log a ( x log a)
dx dx dx
= ex log a . log a = ax log a
mnkgj.k 32 x osQ lkis{k xsin x, dk vodyu dhft,] tc fd x > 0 gSA
gy eku yhft, fd y = xsin x gSA vc nksuksa i{kksa dk y?kqx.kd ysus ij
log y = sin x log x
1 dy d d
vr,o . sin x (log x ) + log x (sin x )
y dx = dx dx
1 dy 1
;k (sin x ) + log x cos x
y dx = x
dy  sin x 
;k = y + cos x log x 
dx  x 
sin x  sin x 
= x  + cos x log x 
 x 
= xsin x −1 ⋅ sin x + x sin x ⋅ cos x log x
dy
mnkgj.k 33 ;fn yx + xy + xx = ab gSA rks Kkr dhft,A
dx
gy fn;k gS fd yx + xy + xx = ab
u = yx, v = xy rFkk w = xx j[kus ij gesa u + v + w = ab izkIr gksrk gSA

2018-19
 lkarR; rFkk vodyuh;rk 193

du dv dw
blfy, + + =0 ... (1)
dx dx dx
vc u = yx gSA nksuksa i{kksa dk y?kqx.kd ysus ij
log u = x log y
nksuksa i{kksa dk x osQ lkis{k vodyu djus ij
1 du d d
⋅ = x (log y ) + log y ( x)
u dx dx dx
1 dy
= x ⋅ + log y ⋅ 1 izkIr gksrk gSA
y dx

du  x dy   x dy 
blfy, = u + log y  = y x  + log y  ... (2)
dx  y dx   y dx 
blh izdkj v=x y

nksuksa i{kksa dk y?kqx.kd ysus ij

log v = y log x
nksuksa i{kksa dk x osQ lkis{k vodyu djus ij
1 dv d dy
. = y (log x) + log x
v dx dx dx
1 dy
= y ⋅ + log x ⋅ izkIr gksrk gSA
x dx
dv y dy 
vr,o = v  + log x 
dx  x dx 

y y dy 
= x  + log x  ... (3)
x dx 
iqu% w=x x

nksuksa i{kksa dk y?kqx.ku djus ij

log w = x log x
nksuksa i{kksa dk x osQ lkis{k vodyu djus ij
1 dw d d
⋅ = x (log x ) + log x . ( x )
w dx dx dx
1
= x ⋅ + log x ⋅ 1 izkIr gksrk gSA
x

2018-19
194 xf.kr

dw
vFkkZr~ = w (1 + log x)
dx
= xx (1 + log x) ... (4)
(1), (2), (3) rFkk (4), }kjk

 x dy  y dy 
yx  + log y  + x y  + log x  + xx (1 + log x) = 0
 y dx   x dx 

dy
;k (x . yx – 1 + xy . log x) = – xx (1 + log x) – y . xy–1 – yx log y
dx

dy − [ y x log y + y . x y −1 + x x (1 + log x)]

vr% =
x . y x −1 + x y log x
dx

iz'ukoyh 5-5
1 ls 11 rd osQ iz'uksa esa iznÙk iQyuksa dk x osQ lkis{k vodyu dhft,%
( x − 1) ( x − 2)
1. cos x . cos 2x . cos 3x 2.
( x − 3) ( x − 4) ( x − 5)
3. (log x)cos x 4. xx – 2sin x
x  1
 1 1+ 
2 3
5. (x + 3) . (x + 4) . (x + 5) 4
6.  x +  + x  x 
 x
7. (log x)x + xlog x 8. (sin x)x + sin–1 x
x2 + 1
9. xsin x + (sin x)cos x 10. x x cos x +
x2 − 1
1
11. (x cos x) + x
( x sin x) x
dy
12 ls 15 rd osQ iz'uksa esa iznÙk iQyuksa osQ fy, Kkr dhft,%
dx
12. xy + yx = 1 13. yx = xy
14. (cos x)y = (cos y)x 15. xy = e(x – y)
16. f (x) = (1 + x) (1 + x2) (1 + x4) (1 + x8) }kjk iznÙk iQyu dk vodyt Kkr dhft, vkSj
bl izdkj f ′(1) Kkr dhft,A

2018-19
 lkarR; rFkk vodyuh;rk 195

17. (x2 – 5x + 8) (x3 + 7x + 9) dk vodyu fuEufyf[kr rhu izdkj ls dhft,%

(i) xq.kuiQy fu;e dk iz;ksx djosQ
(ii) xq.kuiQy osQ foLrkj.k }kjk ,d ,dy cgqin izkIr djosQ
(iii) y?kqx.kdh; vodyu }kjk
;g Hkh lR;kfir dhft, fd bl izdkj izkIr rhuksa mÙkj leku gSaA
18. ;fn u, v rFkk w , x osQ iQyu gaS, rks nks fof/;ksa vFkkZr~ izFke&xq.kuiQy fu;e dh iqujko`fÙk
}kjk] f}rh; & y?kqx.kdh; vodyu }kjk n'kkZb, fd
d du dv dw
(u. v. w) = v. w + u . .w+u.v
dx dx dx dx

5.6 iQyuksa osQ izkpfyd :iksa osQ vodyt (Derivatives of Functions in

Parametric Forms)
dHkh&dHkh nks pj jkf'k;ksa osQ chp dk laca/ u rks Li"V gksrk gS vkSj u vLi"V] fdarq ,d vU;
(rhljh) pj jkf'k ls i`Fko~Q&i`Fko~Q laca/ksa }kjk izFke nks jkf'k;ksa osQ eè; ,d laca/ LFkkfir gks tkrk
gS ,slh fLFkfr esa ge dgrs gSa fd mu nksuksa osQ chp dk laca/ ,d rhljh pj jkf'k osQ ekè;e ls
of.kZr gSA ;g rhljh pj jkf'k izkpy (Parameter) dgykrh gSA vf/d lqLi"V rjhosQ ls nks pj
jkf'k;ksa x rFkk y osQ chp] x = f (t), y = g (t) osQ :i esa O;Dr laca/] dks izkpfyd :i esa O;Dr
laca/ dgrs gSa] tgk¡ t ,d izkpy gSA
bl :i osQ iQyuksa osQ vodyt Kkr djus gsrq] Ük`a[kyk fu;e }kjk
dy dy dx
= ⋅
dt dx dt
dy
dy dt  dx 
;k = dx  tc dHkh ≠ 0 izkIr gksrk gSA

dx dt
dt
dy g ′ (t )  dy dx 
bl izdkj =  D;ksa fd = g ′ (t ) rFkk = f ′ (t ) [c'krsZ f ′(t) ≠ 0]
dx f ′ (t )  dt dt 
dy
mnkgj.k 34 ;fn x = a cos θ, y = a sin θ, rks Kkr dhft,A
dx
gy fn;k gS fd
x = a cos θ, y = a sin θ
dx dy
blfy, = – a sin θ, = a cos θ
dθ dθ

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dy
dy d θ = a cos θ = − cot θ
vr% =
dx − a sin θ
dx
dθ

mnkgj.k 35 ;fn x = at2, y = 2at gS rks dy Kkr dhft,A

dx
gy fn;k gS fd 2
x = at , y = 2at
dx dy
blfy, = 2at rFkk = 2a
dt dt
dy
dy dt = 2a = 1
vr% =
dx
dx 2at t
dt

mnkgj.k 36 ;fn x = a (θ + sin θ), y = a (1 – cos θ) gS rks dy Kkr dhft, A

dx
dx dy
gy ;gk¡ = a(1 + cos θ), = a (sin θ)
dθ dθ
dy
dy d θ = a sin θ = tan θ
vr% =
dx a (1 + cos θ)
dx 2
dθ
dy
A fVIi.kh ;gk¡] ;g è;ku nhft, fd dks eq[; pj jkf'k;ksa x vkSj y dks lfEefyr fd,
dx
fcuk gh] osQoy izkpy osQ inksa esa O;Dr djrs gSaA
2 2 2
dy
mnkgj.k 37 ;fn x3 + y3 = a3 gS rks Kkr dhft,A
dx
gy eku yhft, fd x = a cos3 θ, y = a sin3 θ gS rc
2 3 2 2
x3 + y2 = (a cos
3
θ) 3 + ( a sin 3
θ) 3
2 2

= a 3 (cos θ + (sin θ) = a 3
2 2

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 lkarR; rFkk vodyuh;rk 197

2 2 2
vr% x = a cos3 θ, y = a sin3 θ, x 3 + y 3 = a 3 dk izkpfyd lehdj.k gSA
dx dy
bl izdkj] = – 3a cos2 θ sin θ vkSj = 3a sin2 θ cos θ
dθ dθ

dy
d θ = 3a sin θ cos θ = − tan θ = − 3 y
2
dy
blfy,] =
dx − 3a cos 2 θ sin θ
dx x
dθ

iz'ukoyh 5-6
;fn iz'u la[;k 1 ls 10 rd esa x rFkk y fn, lehdj.kksa }kjk] ,d nwljs ls izkpfyd :i esa
dy
lacaf/r gksa] rks izkpyksa dk foyksiu fd, fcuk] Kkr dhft,%
dx
1. x = 2at2, y = at4 2. x = a cos θ, y = b cos θ
4
3. x = sin t, y = cos 2t 4. x = 4t, y =
t
5. x = cos θ – cos 2θ, y = sin θ – sin 2θ
sin 3 t cos3 t
6. x = a (θ – sin θ), y = a (1 + cos θ) 7. x = , y=
cos 2t cos 2t

 t
8.x = a  cos t + log tan  y = a sin t 9. x = a sec θ, y = b tan θ
 2
10. x = a (cos θ + θ sin θ), y = a (sin θ – θ cos θ)
−1 −1 dy y
11. ;fn x = asin t , y = a cos t , rks n'kkZb, fd =−
dx x
5.7 f}rh; dksfV dk vodyt (Second Order Derivative)
eku yhft, fd y = f (x) gS rks
dy
= f ′(x) ... (1)
dx

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;fn f ′(x) vodyuh; gS rks ge x osQ lkis{k (1) dk iqu% vodyu dj ldrs gSaA bl izdkj
d  dy 
ck;k¡ i{k   gks tkrk gS] ftls f}rh; dksfV dk vodyt (Second Order Derviative)
dx  dx 
d2y
dgrs gSa vkSj ls fu:fir djrs gSaA f (x) osQ f}rh; dksfV osQ vodyt dks f ″(x) ls Hkh
dx 2
fu:fir djrs gSaA ;fn y = f (x) gks rks bls D2(y) ;k y″ ;k y2 ls Hkh fu:fir djrs gSaA ge fVIi.kh
djrs gSa fd mPp Øe osQ vodyu Hkh blh izdkj fd, tkrs gSaA
d2y
mnkgj.k 38 ;fn y = x3 + tan x gS rks Kkr dhft,A
dx 2
gy fn;k gS fd y = x3 + tan x gSA vc
dy
= 3x2 + sec2 x
dx
d2y d ( 2
blfy, 2 =
3x + sec 2 x )
dx dx
= 6x + 2 sec x . sec x tan x = 6x + 2 sec2 x tan x
d2y
mnkgj.k 39 ;fn y = A sin x + B cos x gS rks fl¼ dhft, fd + y = 0 gSA
dx 2
gy ;gk¡ ij
dy
= A cos x – B sin x
dx
d2y d
vkSj 2 = (A cos x – B sin x)
dx dx
= – A sin x – B cos x = – y
d2y
bl izdkj +y=0
dx 2
d2y dy
mnkgj.k 40 ;fn y = 3e2x + 2e3x gS rks fl¼ dhft, fd 2
−5 + 6y = 0
dx dx
gy ;gk¡ y = 3e2x + 2e3x gSA vc
dy
= 6e2x + 6e3x = 6 (e2x + e3x)
dx

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 lkarR; rFkk vodyuh;rk 199

d2y
blfy, = 12e2x + 18e3x = 6 (2e2x + 3e3x)
dx 2
d2y dy
vr% 2
−5 + 6y = 6 (2e2x + 3e3x)
dx dx
– 30 (e2x + e3x) + 6 (3e2x + 2e3x) = 0
d2y dy
mnkgj.k 41 ;fn y = sin–1 x gS rks n'kkZb, fd (1 – x2) 2
− x = 0 gSA
dx dx
gy ;gk¡ y = sin–1 x gS rks
dy 1
=
dx (1 − x 2 )
dy
;k (1 − x 2 ) =1
dx
d  dy 
 (1 − x ) ⋅  = 0
2
;k dx dx

;k (1 − x 2 ) ⋅
d 2 y dy d
+ ⋅
dx 2 dx dx
( )
(1 − x 2 ) = 0

d 2 y dy 2x
;k (1 − x 2 ) ⋅ 2
− ⋅ =0
dx dx 2 1 − x 2

d2y dy
vr% (1 − x 2 ) 2
−x =0
dx dx
fodYir% fn;k gS fd y = sin–1 x gS rks
1
y1 = , vFkkZr~ (1 − x 2 ) y 2 = 1
1 − x2 1

vr,o (1 − x 2 ) ⋅ 2 y1 y2 + y12 (0 − 2 x ) = 0
vr% (1 – x2) y2 – xy1 = 0

iz'ukoyh 5-7
iz'u la[;k 1 ls 10 rd esa fn, iQyuksa osQ f}rh; dksfV osQ vodyt Kkr dhft,%
1. x2 + 3x + 2 2. x 20 3. x . cos x
4. log x 5. x3 log x 6. ex sin 5x

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200 xf.kr

7. e6x cos 3x 8. tan–1 x 9. log (log x)

d2y
10. sin (log x) 11. ;fn y = 5 cos x – 3 sin x gS rks fl¼ dhft, fd + y=0
dx 2
d2y
12. ;fn y = cos–1 x gS rks dks osQoy y osQ inksa esa Kkr dhft,A
dx 2
13. ;fn y = 3 cos (log x) + 4 sin (log x) gS rks n'kkZb, fd x2 y2 + xy1 + y = 0
d2y dy
14. ;fn y = Aemx + Benx gS rks n'kkZb, fd 2
− (m + n) + mny = 0
dx dx
d2y
15. ;fn y = 500e7x + 600e– 7x gS rks n'kkZb, fd = 49 y gSA
dx2
2
d 2 y  dy 
16. ;fn e (x + 1) = 1 gS rks n'kkZb, fd
y =   gSA
dx 2  dx 
17. ;fn y = (tan–1 x)2 gS rks n'kkZb, fd (x2 + 1)2 y2 + 2x (x2 + 1) y1 = 2 gSA
5.8 ekè;eku izes; (Mean Value Theorem)
bl vuqPNsn esa ge vody xf.kr osQ nks vk/kjHkwr ifj.kkeksa dks] fcuk fl¼ fd,] O;Dr djsaxsA
ge bu izes;ksa dh T;kferh; O;k[;k (geometric interpretation) dk Hkh Kku izkIr djsaxsA
izes; 6 jksys dk izes; (Rolle's Theorem) eku yhft, fd f : [a, b] → R lao`r varjky
[a, b] esa larr rFkk foo`r varjky (a, b) esa vodyuh; gS vkSj f(a) = f(b) gS tgk¡ a vkSj b dksbZ
okLrfod la[;k,¡ gSaA rc foo`r varjky (a, b) esa fdlh ,sls c dk vfLrRo gS fd f ′(c) = 0 gSA
vko`Qfr 5-12 vkSj 5-13 esa oqQN ,sls fof'k"V iQyuksa osQ vkys[k fn, x, gSa] tks jksys osQ izes;
dh ifjdYiuk dks larq"V djrs gSaA

vko`Qfr 5-12 vko`Qfr 5-13

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 lkarR; rFkk vodyuh;rk 201

è;ku nhft, fd a vkSj b osQ eè; fLFkr oØ osQ ¯cnqvksa ij Li'kZ js[kk dh izo.krk ij D;k
?kfVr gksrk gSA buesa ls izR;sd vkys[k esa de ls de ,d ¯cnq ij izo.krk 'kwU; gks tkrh gSA
jksys osQ izes; dk ;FkkrF; ;gh nkok gS] D;ksafd y = f (x) osQ vkys[k osQ fdlh ¯cnq ij Li'kZ
js[kk dh izo.krk oqQN vU; ugha vfirq ml ¯cnq ij f (x) dk vodyt gksrk gSA
izes; 7 ekè;eku izes; (Mean Value Theorem) eku yhft, fd f : [a, b] → R varjky
[a, b] esa larr rFkk varjky (a, b) esa vodyuh; gSA rc varjky (a, b) esa fdlh ,sls c dk
vfLrRo gS fd
f (b) − f (a )
f ′(c) = gSA
b−a
è;ku nhft, fd ekè;eku izes; (MVT), jksys osQ izes; dk ,d foLrkj.k (extension) gSA
vkb, vc ge ekè;eku izes; dh T;kferh; O;k[;k le>saA iQyu y = f (x) dk vkys[k vko`Qfr
5-13 esa fn;k gSA ge igys gh f ′(c) dh O;k[;k oØ y = f (x) osQ ¯cnq (c, f (c)) ij [khaph xbZ
Li'kZ js[kk dh izo.krk osQ :i esa dj pqosQ gSaA vko`Qfr 5-14 ls Li"V gS fd f (b) − f (a) ¯cnqvksa
b−a
(a, f (a)) vkSj (b, f (b)) osQ eè; [khaph xbZ Nsnd js[kk (Secant) dh izo.krk gSA ekè;eku izes;
esa dgk x;k gS fd varjky (a, b) esa fLFkr ,d ¯cnq c bl izdkj gS ¯cnq (c, f(c)) ij [khaph xbZ
Li'kZ js[kk] (a, f (a)) rFkk (b, f (b)) ¯cnqvksa osQ chp [khaph xbZ Nsnd js[kk osQ lekarj gksrh gSA nwljs
'kCnksa esa] (a, b) esa ,d ¯cnq c ,slk gS tks (c, f (c)) ij Li'kZ js[kk] (a, f (a)) rFkk (b, f (b))
dks feykus okyh js[kk [kaM osQ leakrj gSA

vko`Qfr 5-14
mnkgj.k 42 iQyu y = x2 + 2 osQ fy, jksys osQ izes; dks lR;kfir dhft,] tc a = – 2 rFkk
b = 2 gSA

2018-19
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gy iQyu y = x2 + 2, varjky [– 2, 2] esa larr rFkk varjky (– 2, 2) esa vodyuh; gSA lkFk gh
f (– 2) = f ( 2) = 6 gS vr,o f (x) dk eku – 2 rFkk 2 ij leku gSaA jksys osQ izes; osQ vuqlkj ,d
¯cnq c ∈ (– 2, 2) dk vfLrRo gksxk] tgk¡ f′ (c) = 0 gSA pw¡fd f′ (x) = 2x gS blfy, c = 0 ij
f ′ (c) = 0 vkSj c = 0 ∈ (– 2, 2)
mnkgj.k 43 varjky [2, 4] esa iQyu f (x) = x2 osQ fy, ekè;eku izes; dks lR;kfir dhft,A
gy iQyu f (x) = x2 varjky [2, 4] esa larr vkSj varjky (2, 4) esa vodyuh; gS] D;ksafd bldk
vodyt f ′ (x) = 2x varjky (2, 4) esa ifjHkkf"kr gSA
vc f (2) = 4 vkSj f (4) = 16 gSaA blfy,
f (b) − f (a ) 16 − 4
= =6
b−a 4−2
ekè;eku izes; osQ vuqlkj ,d ¯cnq c ∈ (2, 4) ,slk gksuk pkfg, rkfd f ′ (c) = 6 gksA ;gk¡
f ′ (x) = 2x vr,o c = 3 gSA vr% c = 3 ∈ (2, 4), ij f ′ (c) = 6 gSA

iz'ukoyh 5-8
1. iQyu f (x) = x2 + 2x – 8, x ∈ [– 4, 2] osQ fy, jksys osQ izes; dks lR;kfir dhft,A
2. tk¡p dhft, fd D;k jksys dk izes; fuEufyf[kr iQyuksa esa ls fdu&fdu ij ykxw gksrk gSA
bu mnkgj.kksa ls D;k vki jksys osQ izes; osQ foykse osQ ckjs esa oqQN dg ldrs gSa\
(i) f (x) = [x] osQ fy, x ∈ [5, 9] (ii) f (x) = [x] osQ fy, x ∈ [– 2, 2]
(iii) f (x) = x2 – 1 osQ fy, x ∈ [1, 2]
3. ;fn f : [– 5, 5] → R ,d larr iQyu gS vkSj ;fn f ′(x) fdlh Hkh ¯cnq ij 'kwU; ugha gksrk
gS rks fl¼ dhft, fd f (– 5) ≠ f (5)
4. ekè;eku izes; lR;kfir dhft,] ;fn varjky [a, b] esa f (x) = x2 – 4x – 3, tgk¡ a = 1
vkSj b = 4 gSA
5. ekè;eku izes; lR;kfir dhft, ;fn varjky [a, b] esa f (x) = x3– 5x2 – 3x, tgk¡ a = 1
vkSj b = 3 gSA f ′(c) = 0 osQ fy, c ∈ (1, 3) dks Kkr dhft,A
6. iz'u la[;k 2 esa mijksDr fn, rhuksa iQyuksa osQ fy, ekè;eku ize;s dh vuqi;ksfxrk dh tk¡p dhft,A

fofo/ mnkgj.k
mnkgj.k 44 x osQ lkis{k fuEufyf[kr dk vodyu dhft,%
1 2
(i) 3x + 2 + (ii) esec x + 3cos –1 x (iii) log7 (log x)
2 x2 + 4

2018-19
 lkarR; rFkk vodyuh;rk 203

gy
1 1
1 −
(i) eku yhft, fd y = 3x + 2 + = (3x + 2) 2 + (2 x + 4)
2 2
gSA
2 x2 + 4

2
è;ku nhft, fd ;g iQyu lHkh okLrfod la[;kvksa x > − osQ fy, ifjHkkf"kr gSA blfy,
3
1 1
dy 1 −1 d  1 − −1 d
= (3x + 2) 2 ⋅ (3x + 2) +  −  ( 2 x + 4) 2 ⋅ ( 2 x 2 + 4)
2
dx 2 dx 2 dx
1 3
1 −  1 −
= (3x + 2) 2 ⋅ (3) −   (2 x + 4) 2 ⋅ 4 x
2
2 2
3 2x
= −
2 3x + 2 3
( 2x2 + 4) 2
2
;g lHkh okLrfod la[;kvksa x > − osQ fy, ifjHkkf"kr gSA
3
+ 3cos−1 x gSA ;g [ −1, 1] osQ izR;sd ¯cnq osQ fy, ifjHkkf"kr
2
(ii) eku yhft, fd y = e
sec x

gSA blfy,
dy sec2 x d  1 
= e ⋅ (sec 2 x ) + 3  −
dx dx  1 − x 2 

sec2 x  d  3
= e ⋅  2 sec x (sec x ) −
 dx  1 − x2
2 3
= 2sec x (sec x tan x ) e
sec x
−
1 − x2
2 3
2
= 2sec x tan x e
sec x
−
1 − x2
è;ku nhft, fd iznÙk iQyu dk vodyt osQoy [ −1, 1] esa gh ekU; gS] D;ksafd
cos – 1 x osQ vodyt dk vfLrRo osQoy (– 1, 1) esa gSA

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204 xf.kr

log (log x)
(iii) eku yhft, fd y = log 7 (log x) = (vk/kj ifjorZu osQ lw=k }kjk)
log 7
leLr okLrfod la[;kvksa x > 1 osQ fy, iQyu ifjHkkf"kr gSA blfy,
dy 1 d
= (log (log x))
dx log 7 dx
1 1 d
= ⋅ (log x )
log 7 log x dx
1
=
x log 7 log x
mnkgj.k 45 x osQ lkis{k fuEufyf[kr dk vodyu dhft,%
 sin x   2 x +1 
(i) cos – 1 (sin x) (ii) tan −1   (iii) sin −1  
 1 + cos x   1 + 4x 
gy
(i) eku yhft, fd f (x) = cos – 1 (sin x) gSA è;ku nhft, fd ;g iQyu lHkh okLrfod
la[;kvksa osQ fy, ifjHkkf"kr gSA ge bls fuEufyf[kr :i esa fy[k ldrs gSaA
f (x) = cos–1 (sin x)
−1  π  π
= cos  cos  − x  , since − x ∈ [0.π]
 2  2
π
= −x
2
vr% f ′(x) = – 1 gSA
 sin x 
(ii) eku yhft, fd f (x) = tan – 1   gSA è;ku nhft, fd ;g iQyu mu lHkh
 1 + cos x 
okLrfod la[;kvksa osQ fy, ifjHkkf"kr gS ftuosQ fy, cos x ≠ – 1, vFkkZr~ π osQ leLr
fo"ke xq.ktksa osQ vfrfjDr vU; lHkh okLrfod la[;kvksa osQ fy, ge bl iQyu dks
fuEufyf[kr izdkj ls iqu% O;Dr dj ldrs gSa%
−1  sin x 
f (x) = tan  
 1 + cos x 
 x  x 
 2 sin   cos   
−1 2 2 −1   x  x
= tan   = tan  tan    =
 2cos 2
x    2  2
 2 


2018-19
 lkarR; rFkk vodyuh;rk 205

x
è;ku nhft, fd ge va'k rFkk gj esa cos   dks dkV losQ] D;ksafd ;g 'kwU; osQ cjkcj
2
1
ugha gSA vr% f ′(x) = gSA
2
 2x + 1  .
(iii) eku yhft, fd f (x) = sin – 1   gSA bl iQyu dk izkar Kkr djus osQ fy, gesa mu
 1 + 4x 

2 x +1 2 x +1
lHkh x dks Kkr djus dh vko';drk gS ftuosQ fy, −1 ≤ ≤ 1 gS
A D;ks
f
a d lnSo
1 + 4x 1 + 4x

2 x +1
/u jkf'k gS] blfy, gesa mu lHkh x dks Kkr djuk gS ftuosQ fy, ≤ 1 , vFkkZr~ os
1 + 4x
1
lHkh x ftuosQ fy, 2x + 1 ≤ 1 + 4x gSA ge bldks 2 ≤ + 2x izdkj Hkh fy[k ldrs gSa]
2x
tks lHkh x osQ fy, lR; gSA vr% iQyu izR;sd okLrfod la[;k osQ fy, ifjHkkf"kr gSA vc
2x = tan θ j[kus ij ;g iQyu fuEufyf[kr izdkj ls iqu% fy[kk tk ldrk gS%
x +1
−1  2 
f (x) = sin  x 
1 + 4 

−1  2 ⋅ 2 
x

= sin  2
1 + ( 2 x ) 

−1  2 tan θ 
= sin 
1 + tan 2 θ 
= sin –1 [sin 2θ] = 2θ = 2 tan – 1 (2x)
1 d
vr% f ′(x) = 2 ⋅ ⋅ (2 x )
1 + (2 )
x 2 dx

2
= ⋅ (2 x )log 2
1 + 4x

2 x + 1 log 2
=
1 + 4x

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mnkgj.k 46 ;fn lHkh 0 < x < π osQ fy, f (x) = (sin x)sin x gS rks f ′(x) Kkr dhft,A
gy ;gk¡ iQyu y = (sin x)sin x lHkh /u okLrfod la[;kvksa osQ fy, ifjHkkf"kr gSA y?kqx.kd
ysus ij
log y = log (sin x)sin x = sin x log (sin x)
1 dy d
vc = (sin x log (sin x))
y dx dx
1 d
= cos x log (sin x) + sin x . ⋅ (sin x)
sin x dx
= cos x log (sin x) + cos x
= (1 + log (sin x)) cos x
dy
vc = y((1 + log (sin x)) cos x) = (1 + log (sin x)) ( sin x)sin x cos x
dx
dy
mnkgj.k 47 /ukRed vpj a osQ fy, , Kkr dhft,] tgk¡
dx
1 a
rFkk x =  t +  gSA
t+ 1
y=a t,
t
gy è;ku nhft, fd nksuksa y rFkk x, leLr okLrfod la[;k t ≠ 0 osQ fy, ifjHkkf"kr gSaA Li"Vr%

( )
1
dy d t +1 t+ d  1
= a t = a t  t +  . log a
dt dt dt  t 
1
t+  1
 1 − 2  log a
t
= a
 t 
a −1
dx  1 d  1
blh izdkj = a t +  ⋅ t + 
dt  t dt  t 
a −1
 1  1
= a t +  ⋅ 1 − 2 
 t  t 
dx
≠ 0 osQoy ;fn t ≠ ± 1 gSA vr% t ≠ ± 1 osQ fy,
dt
1
dy t+  1 1
a t  1 − 2  log a t+
dy dt  t  a t log a
= = a −1 = a −1
dx dx  1  1  1
a t +  ⋅  1 − 2  at + 
dt  t  t   t

2018-19
 lkarR; rFkk vodyuh;rk 207

mnkgj.k 48 e cos x osQ lkis{k sin2 x dk vodyu dhft,A

gy eku yhft, fd u (x) = sin2 x rFkk v (x) = e cos x gSA ;gk¡ gesa du = du / dx Kkr djuk gSA Li"Vr%
dv dv / dx
du dv
= 2 sin x cos x vkSj = e cos x (– sin x) = – (sin x) e cos x gSA
dx dx
du 2sin x cos x 2cos x
vr% = = − cos x
dv − sin x e cos x
e

vè;k; 5 ij fofo/ iz'ukoyh

iz'u la[;k 1 ls 11 rd iznÙk iQyuksa dk] x osQ lkis{k vodyu dhft,%
1. (3x2 – 9x + 5)9 2. sin3 x + cos6 x
3. (5x)3 cos x 2x 4. sin–1(x x ), 0 ≤ x ≤ 1.
x
cos −1
5. 2 , – 2 < x < 2.
2x + 7
 1 + sin x + 1 − sin x  π
6. cot −1  ,0<x<
 1 + sin x − 1 − sin x  2
log x
7. (log x) , x > 1
8. cos (a cos x + b sin x), fdUghaa vpj a rFkk b osQ fy,
π 3π
9. (sin x – cos x) <x<
(sin x – cos x)
,
4 4
10. x + x + a + a , fdlh fu;r a > 0 rFkk x > 0 osQ fy,
x a x a

x2
−3
+ ( x − 3) , x > 3 osQ fy,
2
11. xx
π π
12. ;fn y = 12 (1 – cos t), x = 10 (t – sin t), − < t < rks dy Kkr dhft,A
2 2 dx
dy
13. ;fn y = sin–1 x + sin–1 1 − x 2 , 0 < x < 1 gS rks Kkr dhft,A
dx
14. ;fn – 1 < x < 1 osQ fy, x 1 + y + y 1 + x = 0 gS rks fl¼ dhft, fd
dy 1
=−
dx (1 + x )2

2018-19
208 xf.kr

15. ;fn fdlh c > 0 osQ fy, (x – a)2 + (y – b)2 = c2 gS rks fl¼ dhft, fd

3
  dy 2  2
1 +   
  dx   , a vkSj b ls Lora=k ,d fLFkj jkf'k gSA
d2y
dx 2
dy cos 2 ( a + y )
16. ;fn cos y = x cos (a + y), rFkk cos a ≠ ± 1, rks fl¼ dhft, fd =
dx sin a
d2y
17. ;fn x = a (cos t + t sin t) vkSj y = a (sin t – t cos t), rks Kkr dhft,A
dx 2
18. ;fn f (x) = | x |3, rks izekf.kr dhft, fd f ″(x) dk vfLrRo gS vkSj bls Kkr Hkh dhft,A
19. xf.krh; vkxeu osQ fl¼kar osQ iz;ksx }kjk] fl¼ dhft, fd lHkh /u iw.kk±d n osQ fy,
d ( n)
x = nx n−1 gSA
dx
20. sin (A + B) = sin A cos B + cos A sin B dk iz;ksx djrs gq, vodyu }kjk cosines
osQ fy, ;ksx lw=k Kkr dhft,A
21. D;k ,d ,sls iQyu dk vfLrRo gS] tks izR;sd ¯cnq ij larr gks fdarq osQoy nks ¯cnqvksa ij
vodyuh; u gks\ vius mÙkj dk vkSfpR; Hkh crykb,A
f ( x ) g ( x) h( x ) f ′( x ) g ′( x) h′( x)
dy
22. ;fn y = l m n gS rks fl¼ dhft, fd = l m n
dx
a b c a b c
23. ;fn y = ea cos , – 1 ≤ x ≤ 1, rks n'kkZb, fd
−1
x

2
(1 − x2 ) d 2y − x dy − a 2 y = 0
dx dx

lkjka'k
® ,d okLrfod ekuh; iQyu vius izkar osQ fdlh ¯cnq ij larr gksrk gS ;fn ml ¯cnq
ij iQyu dh lhek] ml ¯cnq ij iQyu osQ eku osQ cjkcj gksrh gSA
® larr iQyuksa osQ ;ksx] varj] xq.kuiQy vkSj HkkxiQy larr gksrs gSa] vFkkZr~] ;fn f rFkk
g larr iQyu gSa] rks
(f ± g) (x) = f (x) ± g (x) larr gksrk gSA

2018-19
 lkarR; rFkk vodyuh;rk 209

(f . g) (x) = f (x) . g (x) larr gksrk gSA

f  f ( x)
 g  ( x) = g ( x) (tgk¡ g (x) ≠ 0) larr gksrk gSA
 
® izR;sd vodyuh; iQyu larr gksrk gS fdarq bldk foykse lR; ugha gSA
® Ük`a[kyk&fu;e iQyuksa osQ la;kstu dk vodyu djus osQ fy, ,d fu;e gSA ;fn
dt dv
f = v o u, t = u (x) vkSj ;fn rFkk dk vfLrRo gS rks
dx dt
df dv dt
= ⋅
dx dt dx
® oqQN ekud vodyt (ifjHkkf"kr izkarksa esa) fuEufyf[kr gSa%
d ( −1 ) 1 d ( −1
sin x = cos − 1 x ) =
dx 1 − x2 dx 1 − x2
d ( −1 ) 1 d ( −1 ) −1
tan x = cot x =
dx 1 + x2 dx 1 + x2
d ( −1 ) 1 d ( −1
sec x = cosec −1 x ) =
dx x x 2 −1 dx x x 2 −1
d ( x) x d 1
e =e ( log x ) =
dx dx x
® y?kqx.kdh; vodyu] f (x) = [u (x)]v (x) osQ :i osQ iQyuksa osQ vodyu djus osQ fy,
,d l'kDr rduhd gSA bl rduhd osQ vFkZiw.kZ gksus osQ fy, vko';d gS fd f (x)
rFkk u (x) nksuksa gh /ukRed gksaA
® jksys dk izes;% ;fn f : [a, b] → R varjky [a, b] esa larr rFkk varjky (a, b) esa
vodyuh; gks] rFkk f (a) = f (b) gks rks (a, b) esa ,d ,sls c dk vfLrRo gS ftlosQ
fy, f ′(c) = 0.
® ekè;eku izes;% ;fn f : [a, b] → R varjky [a, b] esa larr rFkk varjky (a, b) esa
vodyuh; gks rks varjky (a, b) esa ,d ,sls c dk vfLrRo gS ftlosQ fy,
f (b) − f (a )
f ′(c) =
b−a

—v—

2018-19
210 xf.kr

vè;k; 6
vodyt osQ vuqiz;ksx
(Application of Derivatives)

vWith the Calculus as a key, Mathematics can be successfully applied

to the explanation of the course of Nature — WHITEHEAD v

6.1 Hkwfedk (Introduction)

vè;k; 5 esa geus la;Dq r iQyuks]a izfrykse f=kdks.kferh; iQyuks]a vLi"V iQyuks]a pj?kkrkadh; iQyuksa
vkSj y?kq?kkrkadh; iQyuksa dk vodyt Kkr djuk lh[kk gSA izLrqr vè;k; esa] ge xf.kr dh
fofHkUu 'kk[kkvksa esa vodyt osQ vuqiz;ksx dk vè;;u djsaxs ;Fkk baftfu;fjax] foKku] lkekftd
foKku vkSj dbZ nwljs {ks=kA mnkgj.k osQ fy, ge lh[ksaxs fd fdl izdkj vodyt dk mi;ksx
(i) jkf'k;ksa osQ ifjorZu dh nj Kkr djus esa] (ii) fdlh ¯cnq ij Li'kZ js[kk rFkk vfHkyac dh
lehdj.k Kkr djus esa] (iii) ,d iQyu osQ vkys[k ij orZu ¯cnq Kkr djus esa] tks gesa mu ¯cnqvksa
dks Kkr djus esa lgk;d gksrk gS ftu ij iQyu dk vf/dre ;k U;wure eku gksrk gSA ge mu
varjkyksa dks Kkr djus esa Hkh vodyt dk mi;ksx djsaxs] ftuesa ,d iQyu o/Zeku ;k ßkleku
gksrk gSA varr% ge oqQN jkf'k;ksa osQ lfUudV eku izkIr djus esa vodyt iz;qDr djsaxsA
6.2 jkf'k;ksa osQ ifjorZu dh nj (Rate of Change of Quantities)
ds
iqu% Lej.k dhft, fd vodyt ls gekjk rkRi;Z le; varjky t osQ lkis{k nwjh s osQ ifjorZu
dt
dh nj ls gSA blh izdkj] ;fn ,d jkf'k y ,d nwljh jkf'k x osQ lkis{k fdlh fu;e y = f ( x)
dy
dks larq"V djrs gq, ifjofrZr gksrh gS rks (;k f ′(x)), x osQ lkis{k y osQ ifjorZu dh nj dks
dx
dy 
iznf'kZr djrk gS vkSj (;k f ′(x0)) x = x0 ij) x osQ lkis{k y dh ifjorZu dh nj dks
dx  x = x0
iznf'kZr djrk gSA

2018-19
 vodyt osQ vuqiz;ksx 211

blosQ vfrfjDr] ;fn nks jkf'k;k¡ x vkSj y, t osQ lkis{k ifjofrZr gks jgh gksa vFkkZr~
x = f (t ) vkSj y = g (t ) gS rc Ük`a[kyk fu;e ls
dy dy dx dx
= , ;fn ≠ 0 izkIr gksrk gSA
dx dt dt dt
bl izdkj] x osQ lkis{k y osQ ifjorZu dh nj dk ifjdyu t osQ lkis{k y vkSj x osQ ifjorZu
dh nj dk iz;ksx djosQ fd;k tk ldrk gS A
vkb, ge oqQN mnkgj.kksa ij fopkj djsaA
mnkgj.k 1 o`Ùk osQ {ks=kiQy osQ ifjorZu dh nj bldh f=kT;k r osQ lkis{k Kkr dhft, tc
r = 5 cm gSA
gy f=kT;k r okys o`Ùk dk {ks=kiQy A = π r2 ls fn;k tkrk gSA blfy,] r osQ lkis{k A osQ ifjorZu
dA d dA
dh nj = ( π r 2 ) = 2π r ls izkIr gSA tc r = 5 cm rks = 10π gSA vr% o`Ùk dk
dr dr dr
{ks=kiQy 10π cm2/cm dh nj ls cny jgk gSA
mnkgj.k 2 ,d ?ku dk vk;ru 9 cm3/s dh nj ls c<+ jgk gSA ;fn blosQ dksj dh yack;ha
10 cm gS rks blosQ i`"B dk {ks=kiQy fdl nj ls c<+ jgk gSA
gy eku yhft, fd ?ku dh ,d dksj dh yack;ha x cm gSA ?ku dk vk;ru V rFkk ?ku osQ i`"B
dk {ks=kiQy S gSA rc] V = x3 vkSj S = 6x2, tgk¡ x le; t dk iQyu gSA
dV
vc = 9 cm3/s (fn;k gS)
dt
dV d 3 d dx
blfy, 9= = ( x ) = ( x3 ) ⋅ ( Ük`a[kyk fu;e ls)
dt dt dx dt
2 dx
= 3x ⋅
dt
dx 3
;k = 2 ... (1)
dt x
dS d d dx
vc = (6 x 2 ) = (6 x 2 ) ⋅ ( Ük`a[kyk fu;e ls)
dt dt dx dt
 3  36
= 12x ⋅  2  = ((1) osQ iz;ksx ls)
x  x
dS
vr%] tc x = 10 cm, = 3.6 cm2/s
dt

2018-19
212 xf.kr

mnkgj.k 3 ,d fLFkj >hy esa ,d iRFkj Mkyk tkrk gS vkSj rjaxsa o`Ùkksa esa 4 cm/s dh xfr ls
pyrh gSaA tc o`Ùkkdkj rjax dh f=kT;k 10 cm gS] rks ml {k.k] f?kjk gqvk {ks=kiQy fdruh rsth
ls c<+ jgk gS\
gy f=kT;k r okys o`Ùk dk {ks=kiQy A = πr2 ls fn;k tkrk gSA blfy, le; t osQ lkis{k {ks=kiQy
A osQ ifjorZu dh nj gS
dA d d dr dr
= (π r 2 ) = (π r 2 ) ⋅ = 2π r ( Üka`[kyk fu;e ls)
dt dt dr dt dt
dr
;g fn;k x;k gS fd = 4 cm
dt
blfy, tc r = 10 cm
dA
= 2π (10) (4) = 80π
dt
vr% tc r = 10 cm rc o`Ùk ls f?kjs {ks=k dk {ks=kiQy 80π cm2/s dh nj ls c<+ jgk gSA
dy
A fVIi.kh x dk eku c<+us ls ;fn y dk eku c<+rk gS rks /ukRed gksrk gS vkSj x
dx
dy
dk eku c<+us ls ;fn y dk eku ?kVrk gS] rks Í.kkRed gksrk gSA
dx

mnkgj.k 4 fdlh vk;r dh yack;ha x, 3 cm/min dh nj ls ?kV jgh gS vkSj pkSMk+ bZ

y, 2 cm/min dh nj ls c<+ jgh gSA tc x = 10 cm vkSj y = 6 cm gS rc vk;r osQ (a) ifjeki vkSj
(b) {ks=kiQy esa ifjorZu dh nj Kkr dhft,A
gy D;ksafd le; osQ lkis{k yack;ha x ?kV jgh gS vkSj pkSM+kbZ y c<+ jgh gS rks ge ikrs gSa fd
dx dy
= – 3 cm/min vkSj = 2 cm/min
dt dt
(a) vk;r dk ifjeki P ls iznÙk gS] vFkkZr~
P = 2(x + y)

dP  dx dy 
blfy, = 2  +  = 2( −3 + 2) = −2 cm/min
dt  dt dt 
(b) vk;r dk {ks=kiQy A ls iznÙk gS ;Fkk
A=x . y

2018-19
 vodyt osQ vuqiz;ksx 213

dA dx dy
blfy, = ⋅ y + x⋅
dt dt dt
= – 3(6) + 10(2) (D;ksfa d x = 10 cm vkSj y = 6 cm)
= 2 cm2/min
mnkgj.k 5 fdlh oLrq dh x bdkb;ksa osQ mRiknu esa oqQy ykxr C(x) #i;s esa
C (x) = 0.005 x3 – 0.02 x2 + 30x + 5000
ls iznÙk gSA lhekar ykxr Kkr dhft, tc 3 bdkbZ mRikfnr dh tkrh gSA tgk¡ lhekar ykxr
(marginal cost ;k MC) ls gekjk vfHkizk; fdlh Lrj ij mRiknu osQ laiw.kZ ykxr esa rkRdkfyd
ifjorZu dh nj ls gSA
gy D;ksafd lhekar ykxr mRiknu osQ fdlh Lrj ij x bdkbZ osQ lkis{k laiw.kZ ykxr osQ ifjorZu
dh nj gSA ge ikrs gSa fd
dC
lhekar ykxr MC = = 0.005(3x 2 ) − 0.02(2 x) + 30
dx
MC = 0.015(3 ) − 0.04(3) + 30
2
tc x = 3 gS rc
= 0.135 – 0.12 + 30 = 30.015
vr% vHkh"V lhekar ykxr vFkkZr ykxr izfr bdkbZ Rs 30.02 (yxHkx) gSA
mnkgj.k 6 fdlh mRikn dh x bdkb;ksa osQ foØ; ls izkIr oqQy vk; #i;s esa R(x) = 3x2 + 36x
+ 5 ls iznÙk gSA tc x = 5 gks rks lhekar vk; Kkr dhft,A tgk¡ lhekar vk; (marginal revenue
or MR) ls gekjk vfHkizk; fdlh {k.k foØ; dh xbZ oLrqvksa osQ lkis{k laiw.kZ vk; osQ ifjorZu
dh nj ls gSA
gy D;ksafd lhekar vk; fdlh {k.k foØ; dh xbZ oLrqvksa osQ lkis{k vk; ifjorZu dh nj gksrh
gSA ge tkurs gaS fd
dR
lhekar vk; MR = = 6 x + 36
dx
tc x = 5 gS rc MR = 6(5) + 36 = 66
vr% vHkh"V lhekar vk; vFkkZr vk; izfr bdkbZ Rs 66 gSA
iz'ukoyh 6-1
1. o`Ùk osQ {ks=kiQy osQ ifjorZu dh nj bldh f=kT;k r osQ lkis{k Kkr dhft, tcfd
(a) r = 3 cm gSA (b) r = 4 cm gSA

2018-19
214 xf.kr

2. ,d ?ku dk vk;ru 8 cm3/s dh nj ls c<+ jgk gSA i`"B {ks=kiQy fdl nj ls c<+ jgk gS
tcfd blosQ fdukjs dh yack;ha 12 cm gSA
3. ,d o`Ùk dh f=kT;k leku :i ls 3 cm/s dh nj ls c<+ jgh gSA Kkr dhft, fd o`Ùk dk
{ks=kiQy fdl nj ls c<+ jgk gS tc f=kT;k 10 cm gSA
4. ,d ifjorZu'khy ?ku dk fdukjk 3 cm/s dh nj ls c<+ jgk gSA ?ku dk vk;ru fdl nj
ls c<+ jgk gS tcfd fdukjk 10 cm yack gS\
5. ,d fLFkj >hy esa ,d iRFkj Mkyk tkrk gS vksj rjaxsa o`Ùkksa esa 5 cm/s dh xfr ls pyrh
gSaA tc o`Ùkkdkj rjax dh f=kT;k 8 cm gS rks ml {k.k] f?kjk gqvk {ks=kiQy fdl nj ls c<+
jgk gS\
6. ,d o`Ùk dh f=kT;k 0-7 cm/s dh nj ls c<+ jgh gSA bldh ifjf/ dh o`f¼ dh nj D;k
gS tc r = 4.9 cm gS\
7. ,d vk;r dh yack;ha x, 5 cm/min dh nj ls ?kV jgh gS vkSj pkSM+kbZ y, 4 cm/min dh
nj ls c<+ jgh gSA tc x = 8 cm vkSj y = 6 cm gSa rc vk;r osQ (a) ifjeki (b) {ks=kiQy
osQ ifjorZu dh nj Kkr dhft,A
8. ,d xqCckjk tks lnSo xksykdkj jgrk gS] ,d iai }kjk 900 cm3 xSl izfr lsdaM Hkj dj
iqQyk;k tkrk gSA xqCckjs dh f=kT;k osQ ifjorZu dh nj Kkr dhft, tc f=kT;k 15 cm gSA
9. ,d xqCckjk tks lnSo xksykdkj jgrk gS] dh f=kT;k ifjorZu'khy gSA f=kT;k osQ lkis{k vk;ru
osQ ifjorZu dh nj Kkr dhft, tc f=kT;k 10 cm gSA
10. ,d 5 m yach lh<+h nhokj osQ lgkjs >qdh gSA lh<+h dk uhps dk fljk] tehu osQ vuqfn'k]
nhokj ls nwj 2 cm/s dh nj ls [khapk tkrk gSA nhokj ij bldh Å¡pkbZ fdl nj ls ?kV
jgh gS tcfd lh<+h osQ uhps dk fljk nhokj ls 4 m nwj gS?
11. ,d d.k oØ 6y = x3 +2 osQ vuqxr xfr dj jgk gSaA oØ ij mu ¯cnqvksa dks Kkr dhft,
tcfd x-funsZ'kkad dh rqyuk esa y-funsZ'kkad 8 xquk rhozrk ls cny jgk gSA
1
12. gok osQ ,d cqycqys dh f=kT;k cm/s dh nj ls c<+ jgh gSA cqycqys dk vk;ru fdl
2
nj ls c<+ jgk gS tcfd f=kT;k 1 cm gS\
3
13. ,d xqCckjk] tks lnSo xksykdkj jgrk gS] dk ifjorZu'khy O;kl (2 x + 1) gSA x osQ lkis{k
2
vk;ru osQ ifjorZu dh nj Kkr dhft,A
14. ,d ikbi ls jsr 12 cm3/s dh nj ls fxj jgh gSA fxjrh jsr tehu ij ,d ,slk 'kaoqQ cukrh
gS ftldh Å¡pkbZ lnSo vk/kj dh f=kT;k dk NBk Hkkx gSA jsr ls cus osQ 'kaoqQ dh Å¡pkbZ
fdl nj ls c<+ jgh gS tcfd Å¡pkbZ 4 cm gS\

2018-19
 vodyt osQ vuqiz;ksx 215

15. ,d oLrq dh x bdkb;ksa osQ mRiknu ls laca/ oqQy ykxr C (x) (#i;s esa)
C (x) = 0.007x3 – 0.003x2 + 15x + 4000
ls iznÙk gSA lhekar ykxr Kkr dhft, tcfd 17 bdkb;ksa dk mRiknu fd;k x;k gSA
16. fdlh mRikn dh x bdkb;ksa osQ foØ; ls izkIr oqQy vk; R (x) #i;ksa esa
R (x) = 13x2 + 26x + 15
ls iznÙk gSA lhekar vk; Kkr dhft, tc x = 7 gSA
iz'u 17 rFkk 18 esa lgh mÙkj dk p;u dhft,%
17. ,d o`Ùk dh f=kT;k r = 6 cm ij r osQ lkis{k {ks=kiQy esa ifjorZu dh nj gS%
(A) 10π (B) 12π (C) 8π (D) 11π
18. ,d mRikn dh x bdkb;ksa osQ foØ; ls izkIr oqQy vk; #i;ksa esa
R(x) = 3x2 + 36x + 5 ls iznÙk gSA tc x = 15 gS rks lhekar vk; gS%
(A) 116 (B) 96 (C) 90 (D) 126
6.3 o/Zeku (Increasing) vkSj ßkleku (Decreasing ) iQyu
bl vuqPNsn esa ge vodyu dk iz;ksx djosQ ;g Kkr djsaxs fd iQyu o/Zeku gS ;k ßkleku ;k
buesa ls dksbZ ugha gSA
f (x) = x2, x ∈ R }kjk iznÙk iQyu f ij fopkj dhft,A bl iQyu dk vkys[k vko`Qfr
6-1 esa fn;k x;k gSA

ewy ¯cnq osQ ck;ha vksj dk eku ewy ¯cnq osQ nk;ha vksj dk eku
x f (x) = x2 x f (x) = x2

–2 4 0 0
3 9 1 1
−
2 4 2 4
–1 1 1 1
1 1 3 9
−
2 4 2 4
0 0 2 4
tSls tSls ge ck¡, ls nk¡, vksj c<+rs vko`Qfr 6-1 tSls tSls ge ck¡, ls nk¡, vksj c<+rs
tkrs gaS rks vkys[k dh Å¡pkbZ ?kVrh tkrs gS rks vkys[k dh Å¡pkbZ c<+rh
tkrh gSA tkrh gSA

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loZizFke ewy ¯cnq osQ nk;ha vksj osQ vkys[k (vko`Qfr 6-1) ij fopkj djrs gSaA ;g nsf[k, fd
vkys[k osQ vuqfn'k tSls tSls ck,¡ ls nk,¡ vksj tkrs gSa] vkys[k dh Å¡pkbZ yxkrkj c<+rh tkrh gSA
blh dkj.k okLrfod la[;kvksa x > 0 osQ fy, iQyu o/Zeku dgykrk gSA
vc ewy ¯cnq osQ ck;ha vksj osQ vkys[k ij fopkj djrs gSaA ;gk¡ ge ns[krs gSa fd tSls tSls
vkys[k osQ vuqfn'k ck,¡ ls nk,¡ dh vksj tkrs gSa] vkys[k dh Å¡pkbZ yxkrkj ?kVrh tkrh gSA
iQyLo:i okLrfod la[;kvksa x < 0 osQ fy, iQyu ßkleku dgykrk gSA
ge vc ,d varjky esa o/Zeku ;k ßkleku iQyuksa dh fuEufyf[kr fo'ys"k.kkRed ifjHkk"kk nsaxsA
ifjHkk"kk 1 eku yhft, okLrfod eku iQyu f osQ izkar esa I ,d varjky gSA rc f
(i) varjky I esa o/Zeku gS] ;fn I esa x1 < x2 ⇒ f (x1) < f (x2) lHkh x1, x2 ∈ I osQ fy,
(ii) varjky I esa ßkleku gS] ;fn I esa x1 < x2 ⇒ f (x1) > f (x2) lHkh x1, x2 ∈ I osQ fy,
(iii) varjky I esa vpj gS] ;fn f (x) = c, x ∈ I tgk¡ c ,d vpj gSA
bl izdkj osQ iQyuksa dk vkys[kh; fu:i.k vko`Qfr 6-2 esa nsf[k,A

vko`Qfr 6-2

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 vodyt osQ vuqiz;ksx 217

vc ge ,d ¯cnq ij o/Zeku ;k ßkleku iQyu dks ifjHkkf"kr djsaxsA

ifjHkk"kk 2 eku yhft, fd okLrfod ekuksa osQ ifjHkkf"kr iQyu f osQ izkar esa ,d ¯cnq x0 gS rc
x0 ij f o/Zeku vkSj ßkleku dgykrk gS ;fn x0 dks varfoZ"V djus okys ,d ,sls foo`Ùk varjky
I dk vfLrRo bl izdkj gS fd I esa] f Øe'k% o/Zeku vkSj ßkleku gS
vkb, bl ifjHkk"kk dks o/Zeku iQyu osQ fy, Li"V djrs gSaA
mnkgj.k 7 fn[kkb, fd iznÙk iQyu f (x) = 7x – 3, R ij ,d o/Zeku iQyu gSA
gy eku yhft, R esa x1 vkSj x2 dksbZ nks la[;k,¡ gSa] rc
x1 < x2 ⇒ 7x1 < 7x2
⇒ 7x1 – 3 < 7x2 – 3
⇒ f (x1) < f (x2)
bl izdkj] ifjHkk"kk 1 ls ifj.kke fudyrk gS fd R ij f ,d o/Zeku iQyu gSA
vc ge o/Zeku vkSj ßkleku iQyuksa osQ fy, izFke vodyt ijh{k.k izLrqr djsxa As bl ijh{k.k
dh miifÙk esa vè;k; 5 esa vè;;u dh xbZ eè;eku izes; dk iz;ksx djrs gSaA
izes; 1 eku yhft, fd f varjky [a,b] ij larr vkSj foo`Ùk varjky (a,b) ij vodyuh; gSA rc
(a) [a,b] esa f o/Zeku gS ;fn izR;sd x ∈ (a, b) osQ fy, f ′(x) > 0 gSA
(b) [a,b] esa f ßkleku gS ;fn izR;sd x ∈ (a, b) osQ fy, f ′(x) < 0 gSA
(c) [a,b] esa f ,d vpj iQyu gS ;fn izR;sd x ∈ (a, b) osQ fy, f ′(x) = 0 gSA
miifÙk (a) eku yhft, x1, x2 ∈ [a, b] bl izdkj gSa fd x1 < x2 rc eè; eku izes; ls x1 vkSj
x2 osQ eè; ,d ¯cnq c dk vfLrRo bl izdkj gS fd
f (x2) – f (x1) = f ′(c) (x2 – x1)
vFkkZr~ f (x2) – f (x1) > 0 ( D;ksafd f ′(c) > 0 )
vFkkZr~ f (x2) > f (x 1)
bl izdkj] ge ns[krs gSa] fd
[ a, b] oQs lHkh x1 , x2 oQs fy, x1 < x2 f ( x1 ) f ( x2 )
vr% [a,b] esa f ,d o/Zeku iQyu gSA
Hkkx (b) vkSj (c) dh miifÙk blh izdkj gSA ikBdksa osQ fy, bls vH;kl gsrq NksM+k tkrk gSA

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fVIi.kh
bl lnaHkZ esa ,d vU; lkekU; izes; osQ vuqlkj ;fn fdlh varjky osQ vaR; fcanqvksa osQ
vfrfjDr f ' (x) > 0 tgk¡ x, varjky esa dksbZ vo;o gS vkSj f ml varjky esa larr gS rc
f dks oèkZeku dgrs gSaA blh izdkj ;fn fdlh varjky osQ vaR; fcanqvksa osQ flok; f1 (x)
< 0 tgk¡ x varjky dk dksbZ vo;o gS vkSj f ml varjky esa larr gS rc f dks ßkleku
dgrs gSaA
mnkgj.k 8 fn[kkb, fd iznÙk iQyu f ,
f (x) = x3 – 3x2 + 4x, x ∈ R
R ij o/Zeku iQyu gSA
gy è;ku nhft, fd
f ′(x) = 3x2 – 6x + 4
= 3(x2 – 2x + 1) + 1
= 3(x – 1)2 + 1 > 0, lHkh x ∈ R osQ fy,
blfy, iQyu f , R ij o/Zeku gSA
mnkgj.k 9 fl¼ dhft, fd iznÙk iQyu f (x) = cos x
(a) (0, π) esa ßkleku gS
(b) (π, 2π), esa o/Zeku gS
(c) (0, 2π) esa u rks o/Zeku vkSj u gh ßkleku gSA
gy è;ku nhft, fd f ′(x) = – sin x
(a) pw¡fd izR;sd x ∈ (0, π) osQ fy, sin x > 0, ge ikrs gSa fd f ′(x) < 0 vkSj blfy,
(0, π) esa f ßkleku gSA
(b) pw¡fd izR;sd x ∈ (π, 2π) osQ fy, sin x < 0, ge ikrs gSa fd f ′(x) > 0 vkSj blfy,
(π, 2π) esa f o/Zeku gSA
(c) mijksDr (a) vkSj (b) ls Li"V gS fd (0, 2π) esa f u rks o/Zeku gS vkSj u gh ßkleku gSA
mnkgj.k 10 varjky Kkr dhft, ftuesa f (x) = x2 – 4x + 6 ls iznÙk iQyu f
(a) o/Zeku gS (b) ßkleku gS

2018-19
 vodyt osQ vuqiz;ksx 219

gy ;gk¡
f (x) = x2 – 4x + 6

;k f ′(x) = 2x – 4

blfy,] f ′(x) = 0 ls x = 2 izkIr gksrk gSA vc

¯cnq x = 2 okLrfod js[kk dks nks vla;qDr varjkyksa]
uker% (– ∞, 2) vkSj (2, ∞) (vko`Qfr 6-3) esa foHkDr vko`Qfr 6-3
djrk gSA varjky (– ∞, 2) esa f ′(x) = 2x – 4 < 0 gSA
blfy,] bl varjky esa] f ßkleku gSA varjky (2, ∞) , esa f ′ ( x) > 0 gS] blfy, bl varjky
esa iQyu f o/Zeku gSA
mnkgj.k 11 os varjky Kkr dhft, ftuesa f (x) = 4x3 – 6x2 – 72x + 30 }kjk iznÙk iQyu f,
(a) o/Zeku (b) ßkleku gSA
gy ;gk¡
f (x) = 4x3 – 6x2 – 72x + 30
;k f ′(x) = 12x2 – 12x – 72
= 12(x2 – x – 6) vko`Qfr 6-4
= 12(x – 3) (x + 2)
blfy, f ′(x) = 0 ls x = – 2, 3 izkIr gksrs gSaA x = – 2 vkSj x = 3 okLrfod js[kk dks rhu
vla;qDr varjkyksa] uker% (– ∞, – 2), (– 2, 3) vkSj (3, ∞) esa foHkDr djrk gS (vko`Qfr 6-4)A
varjkyksa (– ∞, – 2) vkSj (3, ∞) esa f ′(x) /ukRed gS tcfd varjky (– 2, 3) esa f ′(x)
½.kkRed gSA iQyLo:i iQyu f varjkyksa (– ∞, – 2) vkSj (3, ∞) esa o/Zeku gS tcfd varjky
(– 2, 3) esa iQyu ßkleku gSA rFkkfi f, R ij u rks o/Zeku gS vkSj u gh ßkleku gSA

varjky f ′(x) dk fpÉ iQyu f dh izo`Qfr

(– ∞, – 2) (–) (–) > 0 f o/Zeku gS

(– 2, 3) (–) (+) < 0 f ßkleku gS

(3, ∞) (+) (+) > 0 f o/Zeku gS

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220 xf.kr

  π
mnkgj.k 12 varjky Kkr dhft, ftuesa iznÙk iQyu f (x) = sin 3x, x ∈ 0,  esa (a) o/Zeku
 2
gSA (b) ßkleku gSA
gy Kkr gS fd
f (x) = sin 3x vko`Qfr 6-5
;k f ′(x) = 3cos 3x

π 3π  π
blfy,, f ′(x) = 0 ls feyrk gS cos 3x = 0 ftlls 3 x = , (D;ksafd x ∈  0, 
2 2  2

 3π  π π π  π
⇒ 3 x ∈ 0,  ) izkIr gksrk gSA blfy,] x = vkSj gSA vc ¯cnq x = , varjky  0, 2 
 2 6 2 6

 π  π π
dks nks vla;qDr varjkyksa 0, 6  vkSj  6 , 2  esa foHkkftr djrk gSA
  

 π  π π
iqu% lHkh x ∈ 0, 6  osQ fy, f ′ ( x) > 0 D;ksafd 0 ≤ x < ⇒ 0 ≤ 3x < vkSj lHkh
  6 2
 π π
x ∈  ,  osQ fy, f ′ ( x) < 0 D;ksafd π < x ≤ π ⇒ π < 3x ≤ 3π
6 2 6 2 2 2
 π  π π
blfy,] varjky 0, 6  esa f o/Zeku gS vkSj varjky  ,  esa ßkleku gSA blosQ vfrfjDr
 6 2
π  π
fn;k x;k iQyu x = 0 rFkk x = ij larr Hkh gSA blfy, izes; 1 osQ }kjk, f, 0, 6  esa
6  
π π
o/Zeku vkSj  6 , 2  esa ßkleku gSA
 
mnkgj.k 13 varjky Kkr dhft, ftuesa f (x) = sin x + cos x, 0 ≤ x ≤ 2π }kjk iznÙk iQyu f,
o/Zeku ;k ßkleku gSA
gy Kkr gS fd
f (x) = sin x + cos x, 0 ≤ x ≤ 2π
;k f ′(x) = cos x – sin x
π 5π
vc f ′ ( x ) = 0 ls sin x = cos x ftlls gesa x = , izkIr gksrs gSaA D;ksafd 0 ≤ x ≤ 2π ,
4 4

2018-19
 vodyt osQ vuqiz;ksx 221

π 5π  π
¯cnq x = vkSj x = varjky [0, 2π] dks rhu vla;qDr varjkyksa] uker% 0, 4  ,
4 4 

 π 5π   5π 
 ,  vkSj  , 2π  esa foHkDr djrs gSaA
4 4 4 
vko`Qfr 6-6
 π   5π 
è;ku nhft, fd f ′ ( x ) > 0 ;fn x ∈ 0,  ∪  , 2π 
 4  4 

 π  5π
 
vr% varjkyksa  0, 4  vkSj  4 , 2π  esa iQyu f o/Zeku gSA
  

 π 5π 
vkSj f ′ ( x ) < 0, ;fn x ∈ , 
4 4

 π 5π 
vr% f varjky  ,  esa ßkleku gSA
4 4

varjky f ′ ( x) dk fpÉ iQyu dh izo`Qfr

 π
 0, 4  >0 f o/Zeku gS

 π 5π 
 ,  <0 f ßkleku gS
4 4

 5π 
 , 2π  >0 f o/Zeku gS
4 

iz'ukoyh 6-2
1. fl¼ dhft, R ij f (x) = 3x + 17 ls iznÙk iQyu o/Zeku gSA
2. fl¼ dhft, fd R ij f (x) = e2x ls iznÙk iQyu o/Zeku gSA
3. fl¼ dhft, f (x) = sin x ls iznÙk iQyu
 π π 
(a)  0,  esa o/Zeku gS (b)  , π  esa ßkleku gS
 2 2 
(c) (0, π) esa u rks o/Zeku gS vkSj u gh ßkleku gSA

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222 xf.kr

4. varjky Kkr dhft, ftuesa f (x) = 2x2 – 3x ls iznÙk iQyu f

(a) o/Zeku (b) ßkleku
5. varjky Kkr dhft, ftuesa f (x) = 2x3 – 3x2 – 36x + 7 ls iznÙk iQyu f
(a) o/Zeku (b) ßkleku
6. varjky Kkr dhft, ftuesa fuEufyf[kr iQyu f o/Zeku ;k ßkleku gS%
(a) f (x) x2 + 2x + 5 (b) f (x)10 – 6x – 2x2
(c) f (x) –2x3 – 9x2 – 12x + 1 (d) f (x) 6 – 9x – x2
(e) f (x) (x + 1)3 (x – 3)3
2x
7. fl¼ dhft, fd y = log(1 + x ) − , x > – 1, vius lai.w kZ izkra esa ,d o/Zeku iQyu gSA
2+ x
8. x osQ mu ekuksa dks Kkr dhft, ftuosQ fy, y = [x(x – 2)]2 ,d o/Zeku iQyu gSA
 π 4sin θ
9. fl¼ dhft, fd  0,  esa y = − θ , θ dk ,d o/Zeku iQyu gSA
 2 (2 + cos θ)
10. fl¼ dhft, fd y?kqx.kdh; iQyu (0, ∞) esa o/Zeku iQyu gSA
11. fl¼ dhft, fd (– 1, 1) esa f (x) = x2 – x + 1 ls iznÙk iQyu u rks o/Zeku gS vkSj u gh
ßkleku gSA
π
12. fuEufyf[kr esa dkSu ls iQyu  0,  esa ßkleku gS ?
 2
(A) cos x (B) cos 2x (C) cos 3x (D) tan x
13. fuEufyf[kr varjkyksa esa ls fdl varjky esa f (x) = x100 + sin x –1 }kjk iznÙk iQyu f
ßkleku gS\
π   π
(A) (0,1) (B)  , π (C)  0,  (D) buesa ls dksbZ ugh
2   2
14. a dk og U;wure eku Kkr dhft, ftlosQ fy, varjky [1, 2] esa f (x) = x2 + ax + 1 ls
iznÙk iQyu o/Zeku gSA
1
15. eku yhft, [–1, 1] ls vla;qDr ,d varjky I gks rks fl¼ dhft, fd I esa f ( x ) = x +
x
ls iznÙk iQyu f] o/Zeku gSA
 π π 
16. fl¼ dhft, fd iQyu f (x) = log sin x ,  0,  esa o/Zeku vkSj  , π esa ßkleku gSA
 2 2 

2018-19
 vodyt osQ vuqiz;ksx 223

 π  3π 
17. fl¼ dhft, fd iQyu f (x) = log cos x  0,  esa o/Zeku vkSj  , 2π  esa
 2  2 
ßkleku gSA
18. fl¼ dhft, fd R esa fn;k x;k iQyu f (x) = x3 – 3x2 + 3x – 100 o/Zeku gSA
19. fuEufyf[kr esa ls fdl varjky esa y = x2 e–x o/Zeku gS\
(A) (– ∞, ∞) (B) (– 2, 0) (C) (2, ∞) (D) (0, 2)
6.4 Li'kZ js[kk,¡ vkSj vfHkyac (Tangents and Normals)
bl vuqPNsn esa ge vodyu osQ iz;ksx ls fdlh oØ osQ ,d fn, gq, ¯cnq ij Li'kZ js[kk vkSj
vfHkyac osQ lehdj.k Kkr djsaxsA
Lej.k dhft, fd ,d fn, gq, ¯cnq (x0, y0) ls tkus okyh rFkk ifjfer izo.krk (slope) m
okyh js[kk dk lehdj.k
y – y0 = m (x – x0) ls izkIr gksrk gSA
è;ku nhft, fd oØ y = f (x) osQ ¯cnq (x0, y0) ij Li'kZ js[kk dh

dy
izo.krk dx  [ = f ′ ( x0 )] ls n'kkZbZ tkrh gSA blfy,
(x , y )
0 0

(x0, y0) ij oØ y = f (x) dh Li'kZ js[kk dk lehdj.k

y – y0 = f ′(x0)(x – x0) gksrk gSA
blosQ vfrfjDr] D;ksafd vfHkyac Li'kZ js[kk ij yac gksrk gS
−1
blfy, y = f (x) osQ (x0, y0) ij vfHkyac dh izo.krk gSA
f ′ ( x0 ) vko`Qfr 6-7
pw¡fd f ′ ( x0 ) ≠ 0 gS] blfy, oØ y = f (x) osQ ¯cnq (x0, y0) ij vfHkyac dk lehdj.k
fuEufyf[kr gS%
−1
y – y0 = ( x − x0 )
f ′ ( x0 )
vFkkZr~ ( y − y0 ) f ′( x0 ) + ( x − x0 ) = 0

AfVIi.kh ;fn y = f (x) dh dksbZ Li'kZ js[kk x-v{k dh /u fn'kk ls θ dks.k cuk,¡] rc
dy
= Li'kZ js[kk dh io
z .krk = tan θ
dx

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224 xf.kr

fo'ks"k fLFkfr;k¡ (Particular cases)

(i) ;fn Li'kZ js[kk dh izo.krk 'kwU; gS] rc tanθ = 0 vkSj bl izdkj θ = 0 ftldk vFkZ gS fd
Li'kZ js[kk x-v{k osQ lekarj gSA bl fLFkfr esa] (x0, y0) ij Li'kZ js[kk dk lehdj.k y = y0
gks tkrk gSA
π
(ii) ;fn θ → , rc tanθ → ∞, ftldk vFkZ gS fd Li'kZ js[kk x-v{k ij yac gS vFkkZr~ y-v{k
2
osQ lekarj gSA bl fLFkfr esa (x0, y0) ij Li'kZ js[kk dk lehdj.k x = x0 gksrk gS (D;ks\a )A
mnkgj.k 14 x = 2 ij oØ y = x3 – x dh Li'kZ js[kk dh izo.krk Kkr dhft,A
gy fn, oØ dh x = 2 ij Li'kZ js[kk dh izo.krk
dy 
= 3 x − 1 x = 2 = 11 gSA
2

dx  x = 2

mnkgj.k 15 oØ y = 4 x − 3 − 1 ij mu ¯cnqvksa dks Kkr dhft, ftu ij Li'kZ js[kk dh

2
izo.krk gSA
3
gy fn, x, oØ osQ fdlh ¯cnq (x, y) ij Li'kZ js[kk dh izo.krk
−1
dy 1 2
= (4 x − 3) 2 ⋅ 4 = gSA
dx 2 4x − 3
2
D;ksafd izo.krk fn;k gSA blfy,
3
2 2
=
4x − 3 3
;k 4x – 3 = 9
;k x=3
vc y = 4 x − 3 − 1 gSA blfy, tc x = 3, y = 4(3) − 3 − 1 = 2 gSA
blfy,] vfHk"V ¯cnq (3, 2) gSA
2
mnkgj.k 16 izo.krk 2 okyh lHkh js[kkvksa dk lehdj.k Kkr dhft, tks oØ y + ( x − 3) = 0
dks Li'kZ djrh gSA
gy fn, oØ osQ ¯cnq (x,y) ij Li'kZ js[kk dh izo.krk
dy 2
= gSA
dx ( x − 3) 2
D;ksafd izo.krk 2 fn;k x;k gS blfy,]

2018-19
 vodyt osQ vuqiz;ksx 225

2
=2
( x − 3) 2
;k (x – 3)2 = 1
;k x –3=±1
;k x = 2, 4
vc x = 2 ls y = 2 vkSj x = 4 ls y = – 2 izkIr gksrk gSA bl izdkj] fn, oØ dh izo.krk
2 okyh nks Li'kZ js[kk,¡ gSa tks Øe'k% ¯cnqvksa (2] 2) vkSj (4] &2) ls tkrh gSA vr% (2] 2)
ls tkus okyh Li'kZ js[kk dk lehdj.k%
y – 2 = 2(x – 2) gSA
;k y – 2x + 2 = 0
rFkk (4] &2) ls tkus okyh Li'kZ js[kk dk lehdj.k
y – (– 2) = 2(x – 4)
;k y – 2x + 10 = 0 gSA

x2 y2
mnkgj.k 17 oØ + = 1 ij mu ¯cnqvksa dks Kkr dhft, ftu ij Li'kZ js[kk,¡ (i) x-v{k
4 25
osQ lekarj gksa (ii) y-v{k osQ lekarj gksaA

x2 y2
gy + = 1 dk x, osQ lkis{k vodyu djus ij ge izkIr djrs gSa%
4 25

x 2 y dy
+ =0
2 25 dx

dy −25 x
;k =
dx 4 y
(i) vc] Li'kZ js [ kk x-v{k os Q leka r j gS ;fn mldh iz o .krk 'kw U ; gS ] ftlls
dy −25 x x2 y2
=0⇒ = 0 izkIr gksrk gSA ;g rHkh laHko gS tc x = 0 gksA rc + =1
dx 4 y 4 25
ls x = 0 ij y2 = 25, vFkkZr~ y = ± 5 feyrk gSA vr% ¯cnq (0, 5) vkSj (0, – 5) ,sls gSa tgk¡
ij Li'kZ js[kk,¡ x&v{k osQ lekarj gSaA

2018-19
226 xf.kr

4y
(ii) Li'kZ js[kk y-v{k osQ lekarj gS ;fn blosQ vfHkyac dh izo.krk 'kwU; gS ftlls = 0,
25 x
x2 y2
;k y = 0 feyrk gSA bl izdkj] + = 1 ls y = 0 ij x = ± 2 feyrk gSA vr% os ¯cnq
4 25
(2] 0) vkSj (&2] 0) gSa] tgk¡ ij Li'kZ js[kk,¡ y-v{k osQ lekarj gSaA
x−7
mnkgj.k 18 oØ y = osQ mu ¯cnqvksa ij Li'kZ js[kk,¡ Kkr dhft, tgk¡ ;g
( x − 2)( x − 3)
x-v{k dks dkVrh gSA

gy è;ku nhft, fd x-v{k ij y = 0 gksrk gSA blfy, tc y = 0 rc oØ osQ lehdj.k ls

x = 7 izkIr gksrk gSA bl izdkj oØ x-v{k dks (7, 0) ij dkVrk gSA vc oØ osQ lehdj.k dks x

osQ lkis{k vodyu djus ij

dy 1 − y (2 x − 5)
= ( x − 2)( x − 3) (D;ksa)
dx

dy  1− 0 1
;k =

dx  (7,0)
=
(5) (4) 20 izkIr gksrk gSA

1
blfy,] Li'kZ js[kk dh (7, 0) ij izo.krk gSA vr% (7, 0) ij Li'kZ js[kk dk lehdj.k gS%
20
1
y−0= ( x − 7) ;k 20 y − x + 7 = 0 gSA
20
2 2
mnkgj.k 19 oØ x 3 + y 3 = 2 osQ ¯cnq (1, 1) ij Li'kZ js[kk rFkk vfHkyac osQ lehdj.k Kkr dhft,A
2 2
gy x 3 + y 3 = 2 dk x, osQ lkis{k vodyu djus ij]
−1 −1
2 3 2 3 dy
x + y =0
3 3 dx
1
dy  y 3
;k = − 
dx  x

2018-19
 vodyt osQ vuqiz;ksx 227

dy 
blfy,] (1, 1) ij Li'kZ js[kk dh izo.krk = −1 gSA
dx  (1, 1)
blfy, (1]1) ij Li'kZ js[kk dk lehdj.k
y – 1 = – 1 (x – 1) ;k y + x – 2 = 0 gS
rFkk (1] 1) ij vfHkyac dh izo.krk
−1
= 1 gSA
(1]1)ij Li'khZ dh ioz.krk
blfy,] (1] 1) ij vfHkyac dk lehdj.k
y – 1 = 1 (x – 1) ;k y – x = 0 gSA
mnkgj.k 20 fn, x, oØ
x = a sin3 t , y = b cos3 t ... (1)
π
osQ ,d ¯cnq] tgk¡ t = gS] ij Li'kZ js[kk dk lehdj.k Kkr dhft,A
2
gy (1) dk t osQ lkis{k vodyu djus ij
dx dy
= 3a sin 2 t cos t rFkk = −3b cos2 t sin t
dt dt
dy
dy dt −3b cos 2 t sin t −b cos t
;k = = =
dx dx 3a sin 2 t cos t a sin t
dt
π
dy  −b cos
π 2 =0
tc t = rc  π =
2 dx  t= π
2 a sin
2
π
vkSj tc t = , rc x = a rFkk y = 0 gS vr% t = π ij vFkkZr~ (a, 0) ij fn, x, oØ dh Li'kZ
2 2
js[kk dk lehdj.k y – 0 = 0 (x – a) vFkkZr~ y = 0 gSA

iz'ukoyh 6-3
1. oØ y = 3x4 – 4x osQ x = 4 ij Li'kZ js[kk dh izo.krk Kkr dhft,A
x −1
2. oØ y = , x ≠ 2 osQ x = 10 ij Li'kZ js[kk dh izo.krk Kkr dhft,A
x−2

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228 xf.kr

3. oØ y = x3 – x + 1 dh Li'kZ js[kk dh izo.krk ml ¯cnq ij Kkr dhft, ftldk

x-funsZ'kkad 2 gSA
4. oØ y = x3 –3x +2 dh Li'kZ js[kk dh izo.krk ml ¯cnq ij Kkr dhft, ftldk
x-funsZ'kkad 3 gSA
π
5. oØ x = a cos3 θ, y = a sin 3 θ osQ θ = ij vfHkyac dh izo.krk Kkr dhft,A
4
π
6. oØ x = 1 − a sin θ, y = b cos 2 θ osQ θ =
ij vfHkyac dh izo.krk Kkr dhft,A
2
7. oØ y = x3 – 3x2 – 9x + 7 ij mu ¯cnqvksa dks Kkr dhft, ftu ij Li'kZ js[kk,¡ x&v{k
osQ lekarj gSA
8. oØ y = (x – 2)2 ij ,d ¯cnq Kkr dhft, ftl ij Li'kZ js[kk] ¯cnqvksa (2] 0) vkSj
(4] 4) dks feykus okyh js[kk osQ lekarj gSA
9. oØ y = x3 – 11x + 5 ij ml ¯cnq dks Kkr dhft, ftl ij Li'kZ js[kk y = x – 11 gSA
1
10. izo.krk – 1 okyh lHkh js[kkvksa dk lehdj.k Kkr dhft, tks oØ y = , x ≠ – 1 dks
x −1
Li'kZ djrh gSA
1
11. izo.krk 2 okyh lHkh js[kkvksa dk lehdj.k Kkr dhft, tks oØ y = , x ≠ 3 dks Li'kZ
x−3
djrh gSA
1
12. izo.krk 0 okyh lHkh js[kkvksa dk lehdj.k Kkr dhft, tks oØ y = dks Li'kZ
x − 2x + 3
2

djrh gSA
x2 y2
13. oØ + = 1 ij mu ¯cnqvksa dks Kkr dhft, ftu ij Li'kZ js[kk,¡
9 16
(i) x-v{k osQ lekarj gS (ii) y-v{k osQ lekarj gS
14. fn, oØksa ij fufnZ"V ¯cnqvksa ij Li'kZ js[kk vkSj vfHkyac osQ lehdj.k Kkr dhft,%
(i) y = x4 – 6x3 + 13x2 – 10x + 5 osQ (0, 5) ij
(ii) y = x4 – 6x3 + 13x2 – 10x + 5 osQ (1, 3) ij
(iii) y = x3 osQ (1, 1) ij
(iv) y = x2 osQ (0, 0) ij
π
(v) x = cos t, y = sin t osQ t = ij
4

2018-19
 vodyt osQ vuqiz;ksx 229

15. oØ y = x2 – 2x +7 dh Li'kZ js[kk dk lehdj.k Kkr dhft, tks

(a) js[kk 2x – y + 9 = 0 osQ lekarj gSA
(b) js[kk 5y – 15x = 13 ij yac gSA
16. fl¼ dhft, fd oØ y = 7x3 + 11 osQ mu ¯cnqvksa ij Li'kZ js[kk,¡ lekarj gS tgk¡ x = 2
rFkk x = – 2 gSA
17. oØ y = x3 ij mu ¯cnqvksa dks Kkr dhft, ftu ij Li'kZ js[kk dh izo.krk ¯cnq osQ
y-funsZ'kkad osQ cjkcj gSA
18. oØ y = 4x3 – 2x5, ij mu ¯cnqvksa dks Kkr dhft, ftu ij Li'kZ js[kk,¡ ewy ¯cnq ls gksdj
tkrh gSaA
19. oØ x2 + y2 – 2x – 3 = 0 osQ mu ¯cnqvksa ij Li'kZ js[kkvksa osQ lehdj.k Kkr dhft, tgk¡
ij os x-v{k osQ lekarj gSaA
20. oØ ay2 = x3 osQ ¯cnq (am2, am3) ij vfHkyac dk lehdj.k Kkr dhft,A
21. oØ y = x3 + 2x + 6 osQ mu vfHkyacks osQ lehdj.k Kkr dhft, tks js[kk x + 14y + 4 = 0
osQ lekarj gSA
22. ijoy; y2 = 4ax osQ ¯cnq (at2, 2at) ij Li'kZ js[kk vkSj vfHkyac osQ lehdj.k Kkr dhft,A
23. fl¼ dhft, fd oØ x = y2 vkSj xy = k ,d nwljs dks ledks.k* ij dkVrh gS] ;fn
8k2 = 1gSA

x2 y 2
24. vfrijoy; − = 1 osQ ¯cnq (x0, y0) ij Li'kZ js[kk rFkk vfHkyac osQ lehdj.k Kkr dhft,A
a 2 b2
25. oØ y = 3 x − 2 dh mu Li'kZ js[kkvksa osQ lehdj.k Kkr dhft, tks js[kk 4 x − 2 y + 5 = 0 osQ
lekarj gSA
iz'u 26 vkSj 27 esa lgh mÙkj dk pquko dhft,
26. oØ y = 2x2 + 3 sin x osQ x = 0 ij vfHkyac dh izo.krk gS%
1 1
(A) 3 (B) (C) –3 (D) −
3 3
27. fdl ¯cnq ij y = x + 1, oØ y2 = 4x dh Li'kZ js[kk gS\
(A) (1, 2) (B) (2, 1) (C) (1, – 2) (D) (– 1, 2) gSA
6.5 lfUudVu (Approximation)
bl vuqPNsn esa ge oqQN jkf'k;ksa osQ lfUudV eku dks Kkr djus osQ fy, vodyksa dk iz;ksx djsaxsA
* nks oØ ijLij ledks.k ij dkVrs gSa ;fn muosQ izfrPNsnu ¯cnq ij Li'kZ js[kk,¡ ijLij yac gksaA

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eku yhft, f : D → R, D ⊂ R, ,d iznÙk iQyu gS vkSj y = f (x) nh xbZ oØ gSA

eku yhft, x esa gksus okyh fdlh vYi o`f¼ dks izrhd ∆x ls izdV djrs gSaA Lej.k
dhft, fd x esa gqbZ vYi o`f¼ ∆x osQ laxr y esa gqbZ o`f¼ dks ∆y ls izdV djrs gS tgk¡
∆y = f (x + ∆x) – f (x) gSA ge vc fuEufyf[kr dks ifjHkkf"kr djrs gSa%
(i) x osQ vody dks dx ls izdV djrs gSa rFkk
dx = ∆x ls ifjHkkf"kr djrs gSA
(ii) y osQ vody dks dy ls izdV djrs gSa rFkk
 dy 
dy = f ′(x) dx vFkok dy =   ∆x ls
 dx 
ifjHkkf"kr djrs gSaA
bl n'kk esa x dh rqyuk esa dx = ∆x vis{kko`Qr
NksVk gksrk gS rFkk ∆y dk ,d mi;qDr lfUudVu dy
gksrk gS vkSj bl ckr dks ge dy ≈ ∆y }kjk izdV
djrs gSaA
vko`Qfr 6-8
∆x, ∆y, dx vkSj dy osQ T;kferh; O;k[;k osQ
fy, vko`Qfr 6-8 nsf[k,A

AfVIi.kh mi;ZqDr ifjppkZ rFkk vko`Qfr dks è;ku esa j[krs gq, ge ns[krs gSa fd ijra=k pj
(Dependent variable) dk vody pj dh o`f¼ osQ leku ugha gS tc fd Lora=k pj
(Independent variable) dk vody pj dh o`f¼ osQ leku gSA

mnkgj.k 21 36.6 dk lfUudVu djus osQ fy, vody dk iz;ksx dhft,A

gy y = x yhft, tgk¡ x = 36 vkSj eku yhft, ∆x = 0.6 gSA

rc ∆y = x + ∆x − x = 36.6 − 36 = 36.6 − 6

36.6 = 6 + ∆y
vc ∆y lfUudVr% dy osQ cjkcj gS vkSj fuEufyf[kr ls iznÙk gS%
 dy  1
dy =   ∆x = (0.6) ( D;ksafd y = x )
dx 2 x
1
= (0.6) = 0.05
2 36
bl izdkj] 36.6 dk lfUudV eku 6 + 0.05 = 6.05 gSA

2018-19
 vodyt osQ vuqiz;ksx 231

mnkgj.k 22 (25) 3 dk lfUudVu djus osQ fy, vody dk iz;ksx dhft,A

1
gy eku yhft, y = x3 tgk¡ x = 27 vkSj ∆x = −2 gSA
1 1
rc ∆y = ( x + ∆x) 3 − x3
1 1 1
= (25) 3 − (27) 3 = (25) 3 −3
1
;k = 3 + ∆y
(25) 3
vc ∆y lfUudVr% dy osQ cjkcj gS vkSj
 dy 
dy =   ∆x
 dx 
1
1
= 2
( −2) (D;ksafd y= x3 )
3x 3
1 −2
= 1
( −2) = = − 0.074
27
3((27) 3 ) 2
1
bl izdkj] (25) 3 dk lfUudV eku gS%
3 + (– 0. 074) = 2.926
mnkgj.k 23 f (3.02) dk lfUudV eku Kkr dhft, tgk¡ f (x) = 3x2 + 5x + 3 gSA
gy eku yhft, x = 3 vkSj ∆x = 0.02 gSA
f (3. 02) = f (x + ∆x) = 3 (x + ∆x)2 + 5(x + ∆x) + 3
è;ku nhft, fd ∆y = f (x + ∆x) – f (x) gSA
blfy, f (x + ∆x) = f (x) + ∆y
≈ f (x) + f ′(x) ∆x (D;ksafd dx = ∆x)
≈ (3x + 5x + 3) + (6x + 5) ∆x
2

f (3.02) = (3(3)2 + 5(3) + 3) + (6(3) + 5) (0.02) (D;ksafd x =3, ∆x = 0.02)

= (27 + 15 + 3) + (18 + 5) (0.02)
= 45 + 0.46 = 45.46
vr% f (3.02) dk lfUudV eku 45-46 gSA

2018-19
232 xf.kr

mnkgj.k 24 x ehVj Hkqtk okys ?ku dh Hkqtk esa 2% dh o`f¼ osQ dkj.k ls ?ku osQ vk;ru esa
lfUudV ifjorZu Kkr dhft,A
gy è;ku nhft, fd
V = x3
 dV 
;k dV =   ∆x = (3x2) ∆x
 dx 
= (3x2) (0.02x) (D;ksafd x dk 2% = .02x)
3 3
= 0.06x m
bl izdkj] vk;ru esa lfUudV ifjorZu 0.06 x3 m3 gS
mnkgj.k 25 ,d xksys dh f=kT;k 9 cm ekih tkrh gS ftlesa 0-03 cm dh =kqfV gSA blosQ vk;ru
osQ ifjdyu esa lfUudV =kqfV Kkr dhft,A
gy eku yhft, fd xksys dh f=kT;k r gS vkSj blosQ ekiu esa =kqfV ∆r gSA bl izdkj r = 9 cm
vkSj ∆r = 0.03 cmgSA vc xksys dk vk;ru V
4 3
V= π r ls iznÙk gSA
3
dV
;k = 4π r2
dr
 dV 
blfy, dV =  ∆r = (4π r 2 ) ∆r
 dr 
= [4π(9)2] (0.03) = 9.72π cm3
vr% vk;ru osQ ifjdyu esa lfUudV =kqfV 9.72π cm3 gSA

iz'ukoyh 6-4
1. vody dk iz;ksx djosQ fuEufyf[kr esa ls izR;sd dk lfUudV eku n'keyo osQ rhu LFkkuksa
rd Kkr dhft,%
(i) 25.3 (ii) 49.5 (iii) 0.6
1 1 1
(iv) (0.009) 3 (v) (0.999)10 (vi) (15) 4
1 1 1
(vii) (26) 3 (viii) (255) 4 (ix) (82) 4

2018-19
 vodyt osQ vuqiz;ksx 233

1 1 1
(x) (401) 4 (xi) (0.0037) 2 (xii) (26.57) 3
1 3 1
(xiii) (81.5) 4 (xiv) (3.968) 2 (xv) (32.15) 5
2. f (2.01) dk lfUudV eku Kkr dhft, tgk¡ f (x) = 4x2 + 5x + 2 gSA
3. f (5.001) dk lfUudV eku Kkr dhft, tgk¡ f (x) = x3 – 7x2 + 15 gSA
4. x m Hkqtk okys ?ku dh Hkqtk esa 1% o`f¼ osQ dkj.k ?ku osQ vk;ru esa gksus okyk lfUudV
ifjorZu Kkr dhft,A
5. x m Hkqtk okys ?ku dh Hkqtk esa 1% âkl osQ dkj.k ?ku osQ i`"B {ks=kiQy esa gksus okys lfUudV
ifjorZu Kkr dhft,A
6. ,d xksys dh f=kT;k 7 m ekih tkrh gS ftlesa 0-02 m dh =kqfV gSA blosQ vk;ru osQ
ifjdyu esa lfUudV =kqfV Kkr dhft,A
7. ,d xksys dh f=kT;k 9 m ekih tkrh gS ftlesa 0.03 cm dh =kqfV gSA blosQ i`"B {ks=kiQy
osQ ifjdyu esa lfUudV =kqfV Kkr dhft,A
8. ;fn f (x) = 3x2 + 15x + 5 gks] rks f (3.02) dk lfUudV eku gS%
(A) 47.66 (B) 57.66 (C) 67.66 (D) 77.66
9. Hkqtk esa 3% o`f¼ osQ dkj.k Hkqtk x osQ ?ku osQ vk;ru esa lfUudV ifjorZu gS%
(A) 0.06 x3 m3 (B) 0.6 x3 m3 (C) 0.09 x3 m3 (D) 0.9 x3 m3
6.6 mPpre vkSj fuEure (Maxima and Minima)
bl vuqPNsn esa] ge fofHkUu iQyuksa osQ mPpre vkSj fuEure ekuksa dh x.kuk djus esa vodyt
dh ladYiuk dk iz;ksx djsaxsA okLro esa ge ,d iQyu osQ vkys[k osQ orZu ¯cnqvksa (Turning
points) dks Kkr djsaxs vkSj bl izdkj mu ¯cnqvksa dks Kkr djsaxs ftu ij vkys[k LFkkuh; vf/dre
(;k U;wure) ij igq¡prk gSA bl izdkj osQ ¯cnqvksa dk Kku ,d iQyu dk vkys[k [khapus esa cgqr
mi;ksxh gksrk gSA blosQ vfrfjDr ge ,d iQyu dk fujis{k mPpre eku (Absolute maximum
value) vksj fujis{k U;wure eku (Absolute minimum value) Hkh Kkr djsaxs tks dbZ vuqiz;qDr
leL;kvksa osQ gy osQ fy, vko';d gSaA
vkb, ge nSfud thou dh fuEufyf[kr leL;kvksa ij fopkj djsa
(i) larjksa osQ o`{kksa osQ ,d ckx ls gksus okyk ykHk iQyu P(x) = ax + bx2 }kjk iznÙk gS tgk¡
a,b vpj gSa vkSj x izfr ,dM+ esa larjs osQ o`{kksa dh la[;k gSA izfr ,dM+ fdrus o`{k
vf/dre ykHk nsxsa\

2018-19
234 xf.kr

x2
(ii) ,d 60 m Å¡ps Hkou ls gok esa isaQdh xbZ ,d xsan h( x ) = 60 + x − osQ }kjk
60
fu/kZfjr iFk osQ vuqfn'k pyrh gS] tgk¡ x Hkou ls xsan dh {kSfrt nwjh vkSj h(x) mldh
Å¡pkbZ gSA xsan fdruh vf/dre Å¡pkbZ rd igq¡psxh\
(iii) 'k=kq dk ,d vikps gsfydkWIVj oØ f (x) = x2 + 7 }kjk iznÙk iFk osQ vuqfn'k mM+ jgk gSA
¯cnq (1] 2) ij fLFkr ,d lSfud ml gsfydkWIVj dks xksyh ekjuk pkgrk gS tc gsfydkWIVj
mlosQ fudVre gksA ;g fudVre nwjh fdruh gS\
mi;qZDr leL;kvksa esa oqQN loZlkekU; gS vFkkZr~ ge iznÙk iQyuksa osQ mPpre vFkok fuEure
eku Kkr djuk pkgrs gSaA bu leL;kvksa dks lqy>kus osQ fy, ge fof/or ,d iQyu dk
vf/dre eku ;k U;wure eku o LFkkuh; mPpre o LFkkuh; fuEure osQ ¯cnqvksa vkSj bu
¯cnqvksa dks fu/kZfjr djus osQ ijh{k.k dks ifjHkkf"kr djsaxsA
ifjHkk"kk 3 eku yhft, ,d varjky I esa ,d iQyu f ifjHkkf"kr gS] rc
(a) f dk mPpre eku I esa gksrk gS] ;fn I esa ,d ¯cnq c dk vfLrRo bl izdkj gS fd
f (c) ≥ f ( x ) , ∀ x ∈ I
la[;k f (c) dks I esa f dk mPpre eku dgrs gSa vkSj ¯cnq c dks I esa f osQ mPpre eku
okyk ¯cnq dgk tkrk gSA
(b) f dk fuEure eku I esa gksrk gS ;fn I esa ,d ¯cnq c dk vfLrRo gS bl izdkj fd
f (c) ≤ f (x), ∀ x ∈ I
la[;k f (c) dks I esa f dk fuEure eku dgrs gSa vkSj ¯cnq c dks I esa f osQ fuEure eku
okyk ¯cnq dgk tkrk gSA
(c) I esa f ,d pje eku (extreme value) j[kus okyk iQyu dgykrk gS ;fn I esa ,d ,sls ¯cnq
c dk vfLrRo bl izdkj gS fd f (c), f dk mPpre eku vFkok fuEure eku gSA
bl fLFkfr esa f (c), I esa f dk pje eku dgykrk gS vkSj ¯cnq c ,d pje ¯cnq dgykrk gSA

vko`Qfr 6-9

2018-19
 vodyt osQ vuqiz;ksx 235

AfVIi.kh vko`Qfr 6.9 (a), (b) vkSj (c) esa geus oqQN fof'k"V iQyuksa osQ vkys[k iznf'kZr
fd, gSa ftuls gesa ,d ¯cnq ij mPpre eku vkSj fuEure eku Kkr djus esa lgk;rk feyrh
gSA okLro esa vkys[kksa ls ge mu iQyuksa osQ tks vodfyr ugha gksrs gSaA mPpre @ fuEure eku
Hkh Kkr dj ldrs gSa] (mnkgj.k 27)A
mnkgj.k 26 f (x) = x2, x ∈ R ls iznÙk iQyu f osQ mPpre
vkSj fuEure eku] ;fn dksbZ gksa rks] Kkr dhft,A
gy fn, x, iQyu osQ vkys[k (vko`Qfr 6-10) ls ge dg ldrs
gSa fd f (x) = 0 ;fn x = 0 gS vkSj
f (x) ≥ 0, lHkh x ∈ R osQ fy,A

blfy,] f dk fuEure eku 0 gS vkSj f osQ fuEure eku

dk ¯cnq x = 0 gSA blosQ vfrfjDr vkys[k ls ;g Hkh ns[kk tk
ldrk gS fd iQyu f dk dksbZ mPpre eku ugha gS] vr% R
vko`Qfr 6-10
esa f osQ mPpre eku dk ¯cnq ugha gSA

AfVIi.kh ;fn ge iQyu osQ izkar dks osQoy [– 2, 1] rd lhfer djsa rc x = – 2 ij f

dk mPpre eku (– 2)2 = 4 gSA
mnkgj.k 27 f (x) = | x |, x ∈ R }kjk iznÙk iQyu f osQ
mPpre vkSj fuEure eku] ;fn dksbZ gks rks] Kkr dhft,A
gy fn, x, iQyu osQ vkys[k (vko`Qfr 6-11) ls
f (x) ≥ 0, lHkh x ∈ R vkSj f (x) = 0 ;fn x = 0 gSA
blfy,] f dk fuEure eku 0 gS vkSj f osQ fuEure
eku dk ¯cnq x = 0 gSA vkSj vkys[k ls ;g Hkh Li"V gS R
esa f dk dksbZ mPpre eku ugha gSA vr% R esa dksbZ mPpre
eku dk ¯cnq ugha gSA
vko`Qfr 6-11
fVIi.kh
(i) ;fn ge iQyu osQ izkar dks osQoy [– 2, 1] rd lhfer djsa] rks f dk mPpre eku
| – 2| = 2 gksxkA
(ii) mnkgj.k 27 esa è;ku nsa fd iQyu f , x = 0 ij vodyuh; ugha gSA

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mnkgj.k 28 f (x) = x, x ∈ (0, 1) }kjk iznÙk iQyu osQ mPpre vkSj fuEure eku] ;fn dksbZ gks
rks] Kkr dhft,A
gy fn, varjky (0] 1) esa fn;k iQyu ,d fujarj o/Zeku
iQyu gSA iQyu f osQ vkys[k (vko`Qfr 6-12) ls ,slk
izrhr gksrk gS fd iQyu dk fuEure eku 0 osQ nk;ha vksj
osQ fudVre ¯cnq vkSj mPpre eku 1 osQ ck;ha vksj osQ
fudVre ¯cnq ij gksuk pkfg,A D;k ,sls ¯cnq miyC/ gSa\
,sls ¯cnqvksa dks vafdr djuk laHko ugha gSA okLro esa] ;fn
x0
0 dk fudVre ¯cnq x0 gks rks < x0 lHkh x0 ∈ (0,1)
2 vko`Qfr 6-12
x1 + 1
osQ fy, vkSj ;fn 1 dk fudVre ¯cnq x1 gks rks lHkh x1 ∈ (0,1) osQ fy, > x1 gSA
2
blfy, fn, x, iQyu dk varjky (0] 1) esa u rks dksbZ mPpre eku gS vkSj u gh dksbZ fuEure
eku gSA
fVIi.kh ikBd ns[k ldrs gSa fd mnkgj.k 28 esa ;fn f osQ izkar esa 0 vkSj 1 dks lfEefyr dj
fy;k tk, vFkkZr f osQ izkar dks c<+kdj [0, 1] dj fn;k tk, rks iQyu dk fuEure eku
x = 0 ij 0 vkSj mPpre eku x = 1 ij 1 gSA okLro esa ge fuEufyf[kr ifj.kke ikrs gSa (bu
ifj.kkeksa dh miifÙk bl iqLrd osQ {ks=k ls ckgj gS)A
izR;sd ,dfn"V (monotonic) iQyu vius ifjHkkf"kr izkra osQ vaR; ¯cnqvksa ij mPpre@fuEure
xzg.k djrk gSA
bl ifj.kke dk vf/d O;kid :i ;g gS fd laoÙ` k varjky ij izR;sd larr iQyu osQ mPpre
vkSj fuEu"B eku gksrs gSaA

AfVIi.kh fdlh varjky I esa ,dfn"V iQyu ls gekjk vfHkizk; gS fd I esa iQyu ;k rks
o/Zeku gS ;k ßkleku gSA
bl vuqPNsn esa ,d lao`Ùk varjky ij ifjHkkf"kr iQyu osQ mPpre vkSj fuEure ekuksa osQ ckjs
esa ckn esa fopkj djsaxsA
vkb, vc vko`Qfr 6-13 esa n'kkZ, x, fdlh iQyu osQ vkys[k dk vè;;u djsaA nsf[k, fd
iQyu dk vkys[k ¯cnqvksa A, B, C rFkk D ij o/Zeku ls ßkleku ;k foykser% ßkleku ls o/Zeku
gksrk gSA bu ¯cnqvksa dks iQyu osQ orZu ¯cnq dgrs gSaA iqu% è;ku nhft, fd orZu ¯cnqvksa ij vkys[k
esa ,d NksVh igkM+h ;k NksVh ?kkVh curh gSA eksVs rkSj ij ¯cnqvksa A rFkk C esa ls izR;sd osQ lkehI;
(Neighbourhood)esa iQyu dk fuEure eku gS] tks mudh viuh&viuh ?kkfV;ksa osQ v/ksHkkxksa

2018-19
 vodyt osQ vuqiz;ksx 237

vko`Qfr 6-13

(Bottom) ij gSA blh izdkj ¯cnqvksa B rFkk D esa ls izR;sd osQ lkehI; esa iQyu dk mPpre eku
gS] tks mudh viuh&viuh igkfM+;ksa osQ 'kh"kks± ij gSA bl dkj.k ls ¯cnqvksa A rFkk C dks LFkkuh;
fuEure eku (;k lkis{k fuEure eku) dk ¯cnq rFkk B vkSj D dks LFkkuh; mPpre eku (;k lkis{k
mPpre eku) osQ ¯cnq le>k tk ldrk gSA iQyu osQ LFkkuh; mPpre eku vkSj LFkkuh; fuEure
ekuksa dks Øe'k% iQyu dk LFkkuh; mPpre vkSj LFkkuh; fuEure dgk tkrk gSA
vc ge vkSipkfjd :i ls fuEufyf[kr ifjHkk"kk nsrs gSaA
ifjHkk"kk 4 eku yhft, f ,d okLrfod ekuh; iQyu gS vkSj c iQyu f osQ izkar esa ,d vkarfjd
¯cnq gSA rc
(a) c dks LFkkuh; mPpre dk ¯cnq dgk tkrk gS ;fn ,d ,s l k h > 0 gS fd
(c – h, c + h) esa lHkh x osQ fy, f (c) ≥ f (x) gksA rc f (c), iQyu f dk LFkkuh; mPpre
eku dgykrk gSA
(b) c dks LFkkuh; fuEure dk ¯cnq dgk tkrk gS ;fn ,d ,slk h > 0 gS fd (c – h, c + h) esa lHkh
x osQ fy, f (c) ≤ f (x) gksA rc f (c), iQyu f dk LFkkuh; fuEure eku dgykrk gSA
T;kferh; n`f"Vdks.k ls] mi;qZDr ifjHkk"kk dk vFkZ gS fd ;fn x = c, iQyu f dk LFkkuh;
mPpre dk ¯cnq gS] rks c osQ vklikl dk vkys[k vko`Qfr 6-14(a) osQ vuqlkj gksxkA è;ku nhft,
fd varjky (c – h, c) esa iQyu f o/Zeku (vFkkZr~ f ′(x) > 0) vkSj varjky (c, c + h) esa iQyu
ßkleku (vFkkZr~ f ′(x) < 0) gSA
blls ;g fu"d"kZ fudyrk gS fd f ′(c) vo'; gh 'kwU; gksuk pkfg,A

vko`Qfr 6-14

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238 xf.kr

blh izdkj] ;fn c , iQyu f dk LFkkuh; fuEure ¯cnq gS rks c osQ vklikl dk vkys[k
vko`Qfr 6-14(b) osQ vuqlkj gksxkA ;gk¡ varjky (c – h, c) esa f ßkleku (vFkkZr~ f ′(x) < 0) gS
vkSj varjky (c, c + h) esa f o/Zeku (vFkkZr] f ′(x) > 0) gSA ;g iqu% lq>ko nsrk gS fd f ′(c)
vo'; gh 'kwU; gksuk pkfg,A
mi;qZDr ifjppkZ ls gesa fuEufyf[kr ifjHkk"kk izkIr gksrh gS (fcuk miifÙk)A
izes; 2 eku yhft, ,d foo`Ùk varjky I esa f ,d ifjHkkf"kr iQyu gSA eku yhft, c ∈ I dksbZ
¯cnq gSA ;fn f dk x = c ij ,d LFkkuh; mPpre ;k ,d LFkkuh; fuEure dk ¯cnq gS rks f ′(c)
= 0 gS ;k f ¯cnq c ij vodyuh; ugha gSA
fVIi.kh mijksDr izes; dk foykse vko';d ugha gS fd lR; gks tSls fd ,d ¯cnq ftl ij
vodyt 'kwU; gks tkrk gS rks ;g vko';d ugha gS fd og LFkkuh; mPpre ;k LFkkuh; fuEure
dk ¯cnq gSA mnkgj.kr;k ;fn f (x) = x3 gks rks f ′(x) = 3x2 vkSj blfy, f ′(0) = 0 gSA ijUrq 0
u rks LFkkuh; mPpre vkSj u gh LFkkuh; fuEure ¯cnq gSA vko`Qfr 6-15

AfVIi.kh iQyu f osQ izkar esa ,d ¯cnq c, ftl ij ;k rks f ′(c) = 0 gS ;k f vodyuh;
ugha gS] f dk Økafrd ¯cnq (Critical Point) dgykrk gSA è;ku nhft, fd ;fn f ¯cnq c ij
larr gS vkSj f ′(c) = 0 gS rks ;gk¡ ,d ,sls h > 0 dk vfLrRo gS fd varjky (c – h, c + h)
esa f vodyuh; gSA
vc ge osQoy izFke vodytksa dk iz;ksx djosQ LFkkuh; mPpre ¯cnq ;k LFkkuh; fuEure
¯cnqvksa dks Kkr djus dh fØ;kfof/ izLrqr djsaxsA
izes; 3 (izFke vodyt ijh{k.k) eku yhft, fd ,d iQyu f fdlh foo`Ùk varjky I ij
ifjHkkf"kr gSA eku yhft, fd f varjky I esa fLFkr Økafrd ¯cnq c ij larr gSA rc
(i) x osQ ¯cnq c ls gks dj c<+us osQ lkFk&lkFk] ;fn
f ′(x) dk fpß /u ls ½.k esa ifjofrZr gksrk gS
vFkkZr~ ;fn ¯cnq c osQ ck;ha vksj vkSj mlosQ i;kZIr
fudV osQ izR;sd ¯cnq ij f ′(x) > 0 rFkk c osQ
nk;ha vksj vkSj i;kZIr fudV osQ izR;sd ¯cnq ij
f ′(x) < 0 gks rks c LFkkuh; mPpre ,d ¯cnq gSA
(ii) x osQ ¯cnq c ls gks dj c<+us osQ lkFk&lkFk ;fn
f ′(x) dk fpÉ ½.k ls /u esa ifjofrZr gksrk gS]
vFkkZr~ ;fn ¯cnq c osQ ck;ha vksj vkSj mlosQ i;kZIr
fudV osQ izR;sd ¯cnq ij f ′(x) < 0 rFkk c osQ
nk;haa vksj vkSj mlosQ i;kZIr fudV osQ izR;sd ¯cnq
ij f ′(x) >0 gks rks c LFkkuh; fuEure ¯cnq gSA vko`Qfr 6-15

2018-19
 vodyt osQ vuqiz;ksx 239

(iii) x osQ ¯cnq c ls gks dj c<+us osQ lkFk ;fn f ′(x) dk fpÉ ifjofrZr ugha gksrk gS] rks c u
rks LFkkuh; mPpre ¯cnq gS vkSj u LFkkuh; fuEure ¯cnqA okLro esa] bl izdkj osQ ¯cnq dks
ufr ifjorZu ¯cnq (Point of Inflection) (vko`Qfr 6-15) dgrs gSaA

AfVIi.kh ;fn c iQyu f dk ,d LFkkuh; mPpre ¯cnq gS rks f (c) iQyu f dk LFkkuh;
mPpre eku gSA blh izdkj] ;fn c iQyu f dk ,d LFkkuh; fuEure ¯cnq gS] rks f (c) iQyu f
dk LFkkuh; fuEure eku gSA vko`Qfr;k¡ 6-15 vkSj 6-16 ize;s 3 dh T;kferh; O;k[;k djrh gSA

vko`Qfr 6-16
mnkgj.k 29 f (x) = x3 – 3x + 3 }kjk iznÙk iQyu osQ fy, LFkkuh; mPpre vkSj LFkkuh; fuEure
osQ lHkh ¯cnqvksa dks Kkr dhft,A
gy ;gk¡ f (x) = x3 – 3x + 3

;k f ′(x) = 3x2 – 3 = 3 (x – 1) (x + 1)
;k f ′(x) = 0 ⇒ x = 1 vkSj x = – 1
bl izdkj] osQoy x = ± 1 gh ,sls Økafrd ¯cnq gSa tks f osQ LFkkuh; mPpre vkSj@;k LFkkuh;
fuEure laHkkfor ¯cnq gks ldrs gSaA igys ge x = 1 ij ijh{k.k djrs gSaA
è;ku nhft, fd 1 osQ fudV vkSj 1 osQ nk;haa vksj f ′(x) > 0 gS vkSj 1 osQ fudV vkSj 1 osQ
ck;haa vksj f ′(x) < 0 gSA blfy, izFke vodyt ijh{k.k }kjk x = 1] LFkkuh; fuEure ¯cnq gS vkSj
LFkkuh; fuEure eku f (1) = 1 gSA
x = – 1 dh n'kk esa] –1 osQ fudV vkSj –1 osQ ck;ha vksj f ′(x) > 0 vkSj &1 osQ fudV vkSj
&1 osQ nk;ha vksj f ′(x) < 0 gSA blfy, izFke vodyt ijh{k.k }kjk x = –1 LFkkuh; mPpre dk
¯cnq gS vkSj LFkkuh; mPpre eku f (–1) = 5 gSA

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x osQ eku f ′ ( x ) = 3( x − 1)( x + 1) dk fpÉ

nk;ha vksj (ekuk1-1) >0
1 osQ fudV
ck;ha vkjs (ekuk 0 - 9) <0

nk;ha vksj (ekuk – 0 - 9) <0

–1 osQ fudV
ck;ha vkjs (ekuk – 1-1) >0

mnkgj.k 30 f (x) = 2x3 – 6x2 + 6x +5 }kjk iznÙk iQyu f osQ LFkkuh; mPpre vkSj LFkkuh;
fuEure ¯cnq Kkr dhft,A
gy ;gk¡
f (x) = 2x3 – 6x2 + 6x + 5
;k f ′(x) = 6x2 – 12x + 6 = 6 (x – 1)2
;k f ′(x) = 0 ⇒ x = 1
bl izdkj osQoy x = 1 gh f dk Økafrd ¯cnq gSA vc ge bl ¯cnq ij f osQ LFkkuh; mPpre
;k LFkkuh; fuEure osQ fy, ijh{k.k djsaxsA nsf[k, fd lHkh x ∈ R osQ fy, f ′(x) ≥ 0 vkSj fo'ks"k
:i ls 1 osQ lehi vkSj 1 osQ ck;haa vksj vkSj nk;ha vksj osQ ekuksa osQ fy, f ′(x) > 0 gSA blfy,
izFke vodyt ijh{k.k ls ¯cnq x = 1 u rks LFkkuh; mPpre dk ¯cnq gS vkSj u gh LFkkuh; fuEure
dk ¯cnq gSA vr% x = 1 ,d ufr ifjorZu (inflection) ¯cnq gSA

AfVIi.kh è;ku nhft, fd mnkgj.k 30 esa f ′(x) dk fpÉ varjky R esa dHkh Hkh ugha
cnyrkA vr% f osQ vkys[k esa dksbZ Hkh orZu ¯cnq ugha gS vkSj blfy, LFkkuh; mPpre ;k
LFkkuh; fuEure dk dksbZ Hkh ¯cnq ugha gSA
vc ge fdlh iznÙk iQyu osQ LFkkuh; mPpre vkSj LFkkuh; fuEure osQ ijh{k.k osQ fy, ,d
nwljh fØ;kfof/ izLrqr djsaxsA ;g ijh{k.k izFke vodyt ijh{k.k dh rqyuk esa izk;% ljy gSA
izes; 4 eku yhft, fd f, fdlh varjky I esa ifjHkkf"kr ,d iQyu gS rFkk c ∈ I gSA eku yhft,
fd f, c ij nks ckj yxkrkj vodyuh; gSA rc
(i) ;fn f ′(c) = 0 vkSj f ″(c) < 0 rks x = c LFkkuh; mPpre dk ,d ¯cnq gSA
bl n'kk esa f dk LFkkuh; mPpre eku f (c) gSA
(ii) ;fn f ′ (c ) = 0 vkSj f ″(c) > 0 rks x = c LFkkuh; fuEure dk ,d ¯cnq gSA
bl n'kk esa f dk LFkkuh; fuEure eku f (c) gSA
(iii) ;fn f ′(c) = 0 vkSj f ″(c) = 0 gS rks ;g ijh{k.k vliQy gks tkrk gSA
bl fLFkfr esa ge iqu% izFke vodyt ijh{k.k ij okil tkdj ;g Kkr djrs gSa fd c
mPpre] fuEure ;k ufr ifjorZu dk ¯cnq gSA

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 vodyt osQ vuqiz;ksx 241

AfVIi.kh ¯cnq c ij f nks ckj yxkrkj vodyuh; gS blls gekjk rkRi;Z fd c ij f osQ
f}rh; vodyt dk vfLrRo gSA
mnkgj.k 31 f (x) = 3 + | x |, x ∈ R }kjk iznÙk iQyu f dk LFkkuh; fuEure eku Kkr dhft,A
gy è;ku nhft, fd fn;k x;k x = 0 ij vodyuh; ugha gSA bl izdkj f}rh; vodyt ijh{k.k
vliQy gks tkrk gSA vc ge izFke vodyt ijh{k.k djrs gSaA uksV dhft, fd 0 iQyu f dk ,d
Økafrd ¯cnq gSA vc 0 osQ ck;ha vksj] f (x) = 3 – x vkSj blfy, f ′(x) = – 1 < 0 gS lkFk gh 0
osQ nk;ha vksj] f (x) = 3 + x gS vkSj blfy, f ′(x) = 1 > 0 gSA vr,o] izFke vodyt ijh{k.k
}kjk x = 0, f dk LFkkuh; fuEure ¯cnq gS rFkk f dk LFkkuh; U;wure eku f (0) = 3 gSA
mnkgj.k 32 f (x) = 3x4 + 4x3 – 12x2 + 12 }kjk iznÙk iQyu f osQ LFkkuh; mPpre vkSj LFkkuh;
fuEure eku Kkr dhft,A
gy ;gk¡
f (x) = 3x4 + 4x3 – 12x2 + 12
;k f ′(x) = 12x3 + 12x2 – 24x = 12x (x – 1) (x + 2)
;k x = 0, x = 1 vkSj x = – 2 ij f ′(x) = 0 gSA
vc f ″(x) = 36x2 + 24x – 24 = 12 (3x2 + 2x – 2)

 f ′′(0) = −24 < 0

 vko`Qfr 6-17
vr%  f ′′(1) = 36 > 0
 f ′′( −2) = 72 > 0

blfy,] f}rh; vodyt ijh{k.k }kjk x = 0 LFkkuh; mPpre ¯cnq gS vkSj f dk LFkkuh; mPpre
eku f (0) = 12 gSA tcfd x = 1 vkSj x = – 2 LFkkuh; fuEure ¯cnq gS vkSj LFkkuh; fuEure eku
f (1) = 7 vkSj f (–2) = –20 gSA
mnkgj.k 33 f (x) = 2x3 – 6x2 + 6x +5 }kjk iznÙk iQyu f osQ LFkkuh; mPpre vkSj LFkkuh;
fuEure osQ lHkh ¯cnq Kkr dhft,A
gy ;gk¡ ij
f (x) = 2x3 – 6x2 + 6x +5
 f ′( x) = 6 x 2 − 12 x + 6 = 6( x − 1) 2
;k 
 f ′′( x) = 12( x − 1)
vc f ′(x) = 0 ls x = –1 izkIr gksrk gSA rFkk f ″(1) = 0 gSA blfy, ;gk¡ f}rh; vodyt ijh{k.k
vliQy gSA vr% ge izFke vodyt ijh{k.k dh vksj okil tk,¡xsA

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geus igys gh (mnkgj.k 30) esa ns[kk gS fd izFke vodyt ijh{k.k dh n`f"V ls x =1 u rks LFkkuh;
mPpre dk ¯cnq gS vkSj u gh LFkkuh; fuEure dk ¯cnq gS vfirq ;g ufr ifjorZu dk ¯cnq gSA
mnkgj.k 34 ,slh nks /u la[;k,¡ Kkr dhft, ftudk ;ksx 15 gS vkSj ftuosQ oxks± dk ;ksx
U;wure gksA
gy eku yhft, igyh la[;k x gS rc nwljh la[;k 15 – x gSA eku yhft, bu la[;kvksa osQ oxks±
dk ;ksx S(x) ls O;Dr gksrk gSA rc
S(x) = x2 + (15 – x)2 = 2x2 – 30x + 225

S′( x ) = 4 x − 30
;k 
S′′( x) = 4

izkIr gksrk gS rFkk S′′   = 4 > 0 gSA blfy, f}rh; vodyt

15 15
vc S′(x) = 0 ls x =
2 2
15 15 15 15
ijh{k.k }kjk S osQ LFkkuh; fuEure dk ¯cnq x = gSA vr% tc la[;k,¡ vkSj 15 − = gks
2 2 2 2
rks la[;kvksa osQ oxks± dk ;ksx fuEure gksxkA
fVIi.kh mnkgj.k 34 dh Hkk¡fr ;g fl¼ fd;k tk ldrk gS fd ,slh nks ?ku la[;k,¡ ftudk ;ksx
k k
k gS vkSj ftuosQ oxks± dk ;ksx U;wure gks rks ;s la[;k,¡ , gksaxhA
2 2

1
mnkgj.k 35 ¯cnq (0, c) ls ijoy; y = x2 dh U;wure nwjh Kkr dhft, tgk¡ ≤ c ≤ 5 gSA
2

gy eku yhft, ijoy; y = x2 ij (h, k) dksbZ ¯cnq gSA eku yhft, (h, k) vkSj (0, c) osQ chp
nwjh D gSA rc
D = (h − 0) 2 + ( k − c) 2 = h 2 + ( k − c ) 2 ... (1)
D;ksafd (h, k) ijoy; y = x ij fLFkr gS vr% k = h gSA blfy, (1) ls
2 2

D ≡ D(k) = k + ( k − c) 2
1 + 2( k − c)
;k D′(k) =
k + ( k − c )2

2c − 1
vc D′(k) = 0 ls k = izkIr gksrk gS
2

2018-19
 vodyt osQ vuqiz;ksx 243

2c − 1 2c − 1
è;ku nhft, fd tc k < , rc 2(k − c) + 1 < 0 , vFkkZr~ D′( k ) < 0 gS rFkk tc k >
2 2
2c − 1
rc 2(k − c) + 1 > 0 gS vFkkZr~ D′( k ) > 0 (bl izdkj izFke vodyt ijh{k.k ls k = ij
2
k fuEure gSA vr% vHkh"V U;wure nwjh

2
 2c − 1  2c − 1  2c − 1  4c − 1
D = + − c = gSA
 2  2  2  2

AfVIi.kh ikBd è;ku nsa fd mnkgj.k 35 esa geus f}rh; vodyt ijh{k.k osQ LFkku ij
izFke vodyt ijh{k.k dk iz;ksx fd;k gS D;ksafd ;g ljy ,oa NksVk gSA
mnkgj.k 36 eku yhft, ¯cnq A vkSj B ij Øe'k% AP rFkk BQ nks mèokZ/j LraHk gSA ;fn
AP = 16 m, BQ = 22 m vkSj AB = 20 m gksa rks AB ij ,d ,slk ¯cnq R Kkr dhft, rkfd
RP2 + RQ2 fuEure gksA

gy eku yhft, AB ij ,d ¯cnq R bl izdkj gS fd

AR = x m gSA rc RB = (20 – x) m (D;ksfa d AB = 20 m)
vko`Qfr 6-18 ls
RP2 = AR2 + AP2
vkSj RQ2 = RB2 + BQ2
blfy, RP2 + RQ2 = AR2 + AP2 + RB2 + BQ2
= x2 + (16)2 + (20 – x)2 + (22)2 vko`Qfr 6-18
2
= 2x – 40x + 1140
eku yhft, fd S ≡ S(x) = RP2 + RQ2 = 2x2 – 40x + 1140 gSA
vr% S′(x) = 4x – 40 gSA
vc S′(x) = 0 ls x = 10 izkIr gksrk gS vkSj lHkh x osQ fy, S″(x) = 4 > 0 gS vkSj blfy,
S″(10) > 0 gSA blfy, f}rh; vodyt ijh{k.k ls x = 10, S dk LFkkuh; fuEure dk ¯cnq gSA
vr% AB ij R dh A ls nwjh AR = x = 10 m gSA
mnkgj.k 37 ;fn ,d leyac prqHkqZt osQ vk/kj osQ vfrfjDr rhuksa Hkqtkvksa dh yack;ha 10 cm
gS rc leyac prqHkqZt dk vf/dre {ks=kiQy Kkr dhft,A

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gy vHkh"V leyac dks vko`Qfr 6.19 esa

n'kkZ;k x;k gSA AB ij DP rFkk CQ yac
[khafp,A eku yhft, AP = x cm gSA è;ku
nhft, fd ∆APD ≅ ∆BQC gS blfy,
QB = x cm gSA vkSj ikbFkkxksjl izes; ls]
DP = QC = 100 − x 2 gSA eku yhft,
leyac prqHkqZt dk {ks=kiQy A gSA vko`Qfr 6-19
vr% A ≡ A(x)
1
= (lekarj Hkqtkvksa dk ;ksx) (Å¡pkbZ)
2

(2 x + 10 + 10) ( 100 − x 2 )
1
=
2
= ( x + 10) ( 100 − x )
2

+ ( 100 − x 2 )
( −2 x )
;k A′(x) = ( x + 10)
100 − x 2

−2 x 2 − 10 x + 100
=
100 − x 2
vc A′(x) = 0 ls 2x2 + 10x – 100 = 0, ftlls x = 5 vkSj x = –10 izkIr gksrk gSA
D;ksafd x nwjh dks fu:fir djrk gS blfy, ;g ½.k ugha gks ldrk gSA blfy, x = 5 gSA vc
( −2 x)
100 − x 2 (−4 x − 10) − (−2 x 2 − 10 x + 100)
2
100 − x 2
A″(x) =
100 − x 2
2 x 3 − 300 x − 1000
= 3 (ljy djus ij)
(100 − 2 2
x )

2(5)3 − 300(5) − 1000 −2250 −30

vr% A″(5) = 3
= = <0
75 75 75
(100 − (5) 2 ) 2
bl izdkj] x = 5 ij leyac dk {ks=kiQy vf/dre gS vkSj vf/dre {ks=kiQy
A (5) = (5 + 10) 100 − (5) 2 = 15 75 = 75 3 cm2 gSA

2018-19
 vodyt osQ vuqiz;ksx 245

mnkgj.k 38 fl¼ dhft, fd ,d 'kaoqQ osQ varxZr egÙke oØi`"B okys yac o`Ùkh; csyu dh
f=kT;k 'kaoqQ dh f=kT;k dh vk/h gksrh gSA
gy eku yhft, 'kaoqQ osQ vk/kj dh f=kT;k OC = r vkSj Å¡pkbZ
OA = h gSA eku yhft, fd fn, gq, 'kaoqQ osQ varxZr csyu osQ
vk/kj osQ o`Ùk dh f=kT;k OE = x gS (vko`Qfr 6-20)A csyu dh
Å¡pkbZ QE osQ fy,%
QE EC
= (D;ksafd ∆QEC ~∆AOC)
OA OC
QE r−x
;k =
h r
h (r − x)
;k QE =
r
eku yhft, csyu dk oØi`"B S gS A rc vko`Qfr 6-20
2πxh (r − x) 2πh
S ≡ S (x) = = ( rx − x 2 )
r r
 2πh
S′( x) = r ( r − 2 x )
;k 
S′′( x) = − 4πh
 r
r
vc S′(x) = 0 ls x = izkIr gksrk gSA D;ksafd lHkh x osQ fy, S″(x) < 0 gSA vr%
2
r r
S′′   < 0 gSA blfy, x = , S dk mPpre ¯cnq gSA vr% fn, 'kaoqQ osQ varxZr egÙke oØ i`"B
2 2
osQ csyu dh f=kT;k 'kaoqQ dh f=kT;k dh vk/h gksrh gSA
6.6.1 ,d lao`Ùk varjky esa fdlh iQyu dk mPpre vkSj fuEure eku (Maximum and
Minimum Values of a Function in a Closed Interval)
eku yhft, f (x) = x + 2, x ∈ (0, 1) }kjk iznÙk ,d izyu f gSA
è;ku nhft, fd (0] 1) ij iQyu larr gS vkSj bl varjky esa u rks bldk dksbZ mPpre eku
gS vkSj u gh bldk dksbZ fuEure eku gSA
rFkkfi] ;fn ge f osQ izkar dks lao`Ùk varjky [0, 1] rd c<+k nsa rc Hkh f dk 'kk;n dksbZ
LFkkuh; mPpre (fuEure) eku ugha gksxk ijarq bldk fuf'pr gh mPpre eku 3 = f (1) vkSj

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fuEure eku 2 = f (0) gSaA x = 1 ij f dk mPpre eku 3] [0, 1] ij f dk fujis{k mPpre

eku (egÙke eku) (absolute maximum value) ;k lkoZf=kd vf/dre eku (global
maximum or greatest value) dgykrk gSA blh izdkj] x = 0 ij f dk fuEure eku 2] [0, 1]
ij f dk fujis{k fuEure eku (U;wure eku) (absolute minimum value) ;k lkoZf=kd U;wure
eku (global minimum or least value) dgykrk gSA
,d lao`Ùk varjky [a, b] ij ifjHkkf"kr fdlh larr iQyu f osQ laxr vko`Qfr 6.21 esa iznf'kZr
vkys[k ij fopkj dhft, fd x = b ij iQyu f dk LFkkuh; fuEure gS rFkk LFkkuh; fuEure
eku f (b) gSA iQyu dk x = c ij LFkkuh; mPpre ¯cnq gS rFkk LFkkuh; mPpre eku f (c) gSA

vko`Qfr 6-21
lkFk gh vkys[k ls ;g Hkh Li"V gS fd f dk fujis{k mPpre eku f (a) rFkk fujis{k fuEure
eku f (d) gSA blosQ vfrfjDr è;ku nhft, fd f dk fujis{k mPpre (fuEure) eku LFkkuh;
mPpre (fuEure) eku ls fHkUu gSA
vc ge ,d lao`Ùk varjky I esa ,d iQyu osQ fujis{k mPpre vkSj fujis{k fuEure osQ fo"k;
esa nks ifj.kkeksa (fcuk miifÙk) osQ dFku crk,¡xsA
izes; 5 eku yhft, ,d varjky I = [a, b] ij f ,d larr iQyu gSA rc f dk fujis{k mPpre
eku gksrk gS vkSj I esa de ls de ,d ckj f ;g eku izkIr djrk gS rFkk f dk fujis{k fuEure
eku gksrk gS vkSj I esa de ls de ,d ckj f ;g eku izkIr djrk gSA
izes; 6 eku yhft, lao`Ùk varjky I ij f ,d vodyuh; iQyu gS vkSj eku yhft, fd I dk
dksbZ vkarfjd ¯cnq c gSA rc
(i) ;fn c ij f fujis{k mPpre eku izkIr djrk gS] rks f ′(c) = 0
(ii) ;fn c ij f fujis{k fuEure eku izkIr djrk gS] rks f ′(c) = 0
mi;ZqDr izes;ksa osQ fopkj ls] fn, x, lao`Ùk varjky esa fdlh iQyu osQ fujis{k mPpre eku
vkSj fujis{k fuEure eku Kkr djus osQ fy, fof/ fuEufyf[kr gSaA

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 vodyt osQ vuqiz;ksx 247

O;kogkfjd fof/ (Working Rule)

pj.k 1: fn, x, varjky esa f osQ lHkh Økafrd ¯cnq Kkr dhft, vFkkZr~ x osQ og lHkh eku Kkr
dhft, tgk¡ ;k rks f ′ ( x ) = 0 ;k f vodyuh; ugha gSA
pj.k 2: varjky osQ vaR; ¯cnq yhft,A
pj.k 3: bu lHkh ¯cnqvksa ij (pj.k 1 o 2 esa lwphc¼) f osQ ekuksa dh x.kuk dhft,A
pj.k 4: pj.k 3 esa x.kuk ls izkIr f osQ ekuksa esa ls mPpre vkSj fuEure ekuksa dks yhft,A ;gh
mPpre eku] f dk fujis{k mPpre eku vkSj fuEure eku] f dk fujis{k fuEure eku gksaxsA
mnkgj.k 39 varjky [1, 5] esa f (x) = 2x3 – 15x2 + 36x +1 }kjk iznÙk iQyu osQ fujis{k mPpre
vkSj fujis{k fuEure ekuksa dks Kkr dhft,A
gy gesa Kkr gS
f (x) = 2x3 – 15x2 + 36x + 1
;k f ′(x) = 6x2 – 30x + 36 = 6 (x – 3) (x – 2)
è;ku nhft, f ′(x) = 0 ls x = 2 vkSj x = 3 izkIr gksrs gSaA
vc ge bu ¯cnqvksa vkSj varjky [1, 5] osQ vaR; ¯cnqvksa vFkkZr~ x = 1, x = 2, x = 3 vkSj
x = 5 ij f osQ eku dk ifjdyu djsaxsA vc%
f (1) = 2 (13) – 15 (12) + 36 (1) + 1 = 24
f (2) = 2 (23) – 15 (22) + 36 (2) + 1 = 29
f (3) = 2 (33) – 15 (32) + 36 (3) + 1 = 28
f (5) = 2 (53) – 15 (52) + 36 (5) + 1 = 56
bl izdkj] ge bl fu"d"kZ ij igq¡prs gSa fd varjky [1, 5] ij iQyu f osQ fy, x =5 ij fujis{k
mPpre eku 56 vkSj x = 1 ij fujis{k fuEure eku 24 gSA
4 1
mnkgj.k 40 f ( x) = 12 x 3 − 6x3 ,
x ∈ [−1, 1] }kjk iznÙk ,d iQyu f osQ fujis{k mPpre vkSj
fujis{k fuEure eku Kkr dhft,A
gy gesa Kkr gS fd
4 1
f (x) = 12 x 3 − 6x 3
1
2 2(8 x − 1)
;k f ′(x) = 16 x 3 − 2
= 2
x3 x3

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1
bl izdkj f ′(x) = 0 ls x = izkIr gksrk gSA vkSj è;ku nhft, fd x = 0 ij f ′(x) ifjHkkf"kr
8
1 1
ugha gSA blfy, Økafrd ¯cnq x = 0 vkSj x = gSaA vc Økafrd ¯cnqvksa x = 0, vkSj varjky osQ
8 8
vaR; ¯cnqvksa x = –1 o x = 1 ij iQyu f osQ eku dk ifjdyu djus ls
4 1
f (–1) = 12(−13 ) − 6( −13 ) = 18
f (0) = 12 (0) – 6 (0) = 0
4 1
1
f   = 12  1  − 6  1  = − 9
3 3

8  8  8 4
4 1
f (1) = 12(13 ) − 6(13 ) = 6
izkIr gksrs gSaA bl izdkj ge bl fu"d"kZ ij igq¡prs gS fd x = – 1 ij f dk fujis{k mPpre
1 −9
eku 18 gS vkSj x = ij f dk fujis{k fuEure eku gSA
8 4
mnkgj.k 41 'k=kq dk ,d vikps gsfydkWIVj oØ y = x + 7 osQ vuqfn'k iznÙk iFk ij mM+ jgk gSA
2

¯cnq (3] 7) ij fLFkr ,d lSfud viuh fLFkfr ls U;wure nwjh ij ml gsfydkWIVj dks xksyh ekjuk
pkgrk gSA U;wure nwjh Kkr dhft,A
gy x osQ izR;sd eku osQ fy, gsfydkWIVj dh fLFkfr ¯cnq (x, x2 + 7) gSA blfy, (3, 7) ij fLFkr
lSfud vkSj gsfydkWIVj osQ chp nwjh ( x − 3) 2 + ( x 2 + 7 − 7)2 , vFkkZr~ ( x − 3) 2 + x 4 gSA
eku yhft, fd f (x) = (x – 3)2 + x4
;k f ′(x) = 2(x – 3) + 4x3 = 2 (x – 1) (2x2 + 2x + 3)
blfy, f ′(x) = 0 ls x = 1 izkIr gksrk gS rFkk 2x2 + 2x + 3 = 0 ls dksbZ okLrfod ewy izkIr ugha
gksrk gSA iqu% varjky osQ vaR; ¯cnq Hkh ugha gS] ftUgsa ml leqPp; esa tksM+k tk, ftuosQ fy, f ′
dk eku 'kwU; gS vFkkZr~ osQoy ,d ¯cnq] uker% x = 1 gh ,slk gSA bl ¯cnq ij f dk eku
f (1) = (1 – 3)2 + (1)4 = 5 ls iznÙk gSA bl izdkj] lSfud ,oa gsfydkWIVj osQ chp dh nwjh
f (1) = 5 gSA
è;ku nhft, fd 5 ;k rks mPpre eku ;k fuEure eku gSA D;ksafd
f (0) = (0 − 3) 2 + (0) 4 = 3 > 5 gSA
blls ;g fu"d"kZ fudyk fd f ( x ) dk fuEure eku 5 gSA vr% lSfud vkSj gsfydkWIVj osQ
chp dh fuEure nwjh 5 gSA

2018-19
 vodyt osQ vuqiz;ksx 249

iz'ukoyh 6-5
1. fuEufyf[kr fn, x, iQyuksa osQ mPpre ;k fuEure eku] ;fn dksbZ rks] Kkr dhft,%
(i) f (x) = (2x – 1)2 + 3 (ii) f (x) = 9x2 + 12x + 2
(iii) f (x) = – (x – 1)2 + 10 (iv) g (x) = x3 + 1
2. fuEufyf[kr fn, x, iQyuksa osQ mPpre ;k fuEure eku] ;fn dksbZ gksa] rks Kkr dhft,%
(i) f (x) = | x + 2 | – 1 (ii) g (x) = – | x + 1| + 3
(iii) h (x) = sin (2x) + 5 (iv) f (x) = | sin 4x + 3|
(v) h (x) = x + 1, x ∈ (– 1, 1)
3. fuEufyf[kr iQyuksa osQ LFkkuh; mPpre ;k fuEure] ;fn dksbZ gksa rks] Kkr dhft, rFkk
LFkkuh; mPpre ;k LFkkuh; fuEure eku] tSlh fLFkfr gks] Hkh Kkr dhft,A
(i) f (x) = x2 (ii) g (x) = x3 – 3x
π
(iii) h (x) = sin x + cos x, 0 < x <
2
(iv) f (x) = sin x – cos x, 0 < x < 2 π
x 2
(v) f (x) = x3 – 6x2 + 9x + 15 (vi) g ( x ) = + , x>0
2 x
1
(vii) g ( x) = (viii) f ( x ) = x 1 − x , 0 < x < 1
x +2
2

4. fl¼ dhft, fd fuEufyf[kr iQyuksa dk mPpre ;k fuEure eku ugha gS%

(i) f (x) = ex (ii) g (x) = log x
(iii) h (x) = x3 + x2 + x +1
5. iznÙk varjkyksa esa fuEufyf[kr iQyuksa osQ fujis{k mPpre eku vkSj fujis{k fuEure eku Kkr dhft,A
(i) f (x) = x3, x ∈ [– 2, 2] (ii) f (x) = sin x + cos x , x ∈ [0, π]
1  9
(iii) f (x) = 4 x − x 2 , x ∈  −2,  (iv) f ( x ) = ( x − 1) 2 + 3, x ∈[ −3,1]
2  2
6. ;fn ykHk iQyu p (x) =41 – 72x – 18x2 ls iznÙk gS rks fdlh daiuh }kjk vftZr mPpre
ykHk Kkr dhft,A
7. varjky [0, 3] ij 3x4 – 8x3 + 12x2 – 48x + 25 osQ mPpre eku vksj fuEure eku Kkr dhft,A
8. varjky [0, 2π] osQ fdu ¯cnqvksa ij iQyu sin 2x viuk mPpre eku izkIr djrk gS\
9. iQyu sin x + cos x dk mPpre eku D;k gS?

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10. varjky [1, 3] esa 2x3 – 24x + 107 dk egÙke eku Kkr dhft,A blh iQyu dk varjky
[–3, –1] esa Hkh egÙke eku Kkr dhft,A
11. ;fn fn;k gS fd varjky [0, 2] esa x = 1 ij iQyu x4 – 62x2 + ax + 9 mPpre eku izkIr
djrk gS] rks a dk eku Kkr dhft,A
12. [0, 2π] ij x + sin 2x dk mPpre vkSj fuEure eku Kkr dhft,A
13. ,slh nks la[;k,¡ Kkr dhft, ftudk ;ksx 24 gS vkSj ftudk xq.kuiQy mPpre gksA
14. ,slh nks /u la[;k,¡ x vkSj y Kkr dhft, rkfd x + y = 60 vkSj xy3 mPpre gksA
15. ,slh nks /u la[;k,¡ x vkSj y Kkr dhft, ftudk ;ksx 35 gks vkSj xq.kuiQy x2 y5 mPpre gksA
16. ,slh nks /u la[;k,¡ Kkr dhft, ftudk ;ksx 16 gks vkSj ftuosQ ?kuksa dk ;ksx fuEure gksA
17. 18 cm Hkqtk osQ fVu osQ fdlh oxkZdkj VqdM+s ls izR;sd dksus ij ,d oxZ dkVdj rFkk bl
izdkj cusa fVu osQ iQydksa dks eksM+ dj <Ddu jfgr ,d lanwd cukuk gSA dkVs tkus okys
oxZ dh Hkqtk fdruh gksxh ftlls lanwd dk vk;ru mPpre gks\
18. 45 cm × 24 cm dh fVu dh vk;rkdkj pknj osQ dksuksa ij oxZ dkVdj rFkk bl izdkj
cusa fVu osQ iQydksa dks eksM+dj <Ddu jfgr ,d lanwd cukuk gSA dkVs tkus okys oxZ dh
Hkqtk fdruh gksxh ftlls lanwd dk vk;ru mPpre gksA
19. fl¼ fdft, fd ,d fn, o`Ùk osQ varxZr lHkh vk;rksa esa oxZ dk {ks=kiQy mPpre gksrk gSA
20. fl¼ fdft, fd iznÙk i`"B ,oa egÙke vk;ru osQ csyu dh Å¡pkbZ] vk/kj osQ O;kl osQ
cjkcj gksrh gSA
21. 100 cm3 vk;ru okys fMCcs lHkh can csyukdkj (yac o`Ùkh;) fMCcksa esa ls U;wure i`"B
{ks=kiQy okys fMCcs dh foek,¡ Kkr fdft,A
22. ,d 28 cm yacs rkj dks nks VqdM+ksa esa foHkDr fd;k tkuk gSA ,d VqdM+s ls oxZ rFkk nwljs
os o`Ùk cuk;k tkuk gSA nksuksa VqdM+ksa dh yack;ha fdruh gksuh pkfg, ftlls oxZ ,oa o`Ùk dk
lfEefyr {ks=kiQy U;wure gks\
23. fl¼ dhft, fd R f=kT;k osQ xksys osQ varxZr fo'kkyre 'kaoqQ dk vk;ru] xksys osQ vk;ru
8
dk gksrk gSA
27
24. fl¼ dhft, fd U;wure i`"B dk fn, vk;ru osQ yac o`Ùkh; 'kaoqQ dh Å¡pkbZ] vk/kj dh
f=kT;k dh 2 xquh gksrh gSA
25. fl¼ dhft, fd nh gqbZ fr;Zd Å¡pkbZ vkSj egÙke vk;ru okys 'kaoqQ dk v/Z 'kh"kZ dks.k
tan −1 2 gksrk gSA

2018-19
 vodyt osQ vuqiz;ksx 251

26. fl¼ dhft, fd fn, gq, i`"B vkSj egÙke vk;ru okys yac o`Ùkh; 'kaoqQ dk v/Z 'kh"kZ dks.k

 1
sin −1   gksrk gSA
 3

iz'u la[;k 27 ls 29 esa lgh mÙkj dk pquko dhft,A

27. oØ x2 = 2y ij (0, 5) ls U;wure nwjh ij fLFkr ¯cnq gS%
(A) (2 2,4) (B) (2 2,0) (C) (0, 0) (D) (2, 2)

1 − x + x2
28. x, osQ lHkh okLrfod ekuksa osQ fy, dk U;wure eku gS%
1 + x + x2
1
(A) 0 (B) 1 (C) 3 (D)
3
1

29. [ x( x − 1) + 1]3 , 0 ≤ x ≤ 1 dk mPpre eku gS%

1
 1 3 1
(A)   (B) (C) 1 (D) 0
 3 2

fofo/ mnkgj.k
mnkgj.k 42 ,d dkj le; t = 0 ij ¯cnq P ls pyuk izkjaHk djosQ ¯cnq Q ij #d tkrh gSA
dkj }kjk t lsoaQM esa r; dh nwjh] x ehVj esa
2 t
x = t  2 −  }kjk iznÙk gSA
 3

dkj dks Q rd ig¡qpus esa yxk le; Kkr dhft, vkSj P rFkk Q osQ chp dh nwjh Hkh Kkr dhft,A
gy eku yhft, t lsdaM esa dkj dk osx v gSA
 t
vc x = t2  2 − 
 3
dx
;k v= = 4t – t2 = t (4 – t)
dt
bl izdkj v = 0 ls t = 0 ;k t = 4 izkIr gksrs gSaA

2018-19
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vc P vkSj Q ij dkj dk osx v = 0 gSA blfy, Q ij dkj 4 lsoaQMksa esa igq¡psxhA vc 4 lsoaQMksa
esa dkj }kjk r; dh xbZ nwjh fuEufyf[kr gS%
 4  2  32
x] t = 4 = 42  2 −  = 16   = m
 3  3 3
mnkgj.k 43 ikuh dh ,d Vadh dk vkdkj] mèokZ/j v{k okys ,d mYVs yac o`Ùkh; 'kaoQq gS ftldk
'kh"kZ uhps gSA bldk v¼Z 'kh"kZ dks.k tan–1 (0.5) gSA blesa 5 m3/min dh nj ls ikuh Hkjk tkrk gSA
ikuh osQ Lrj osQ c<+us dh nj ml {k.k Kkr dhft, tc Vadh esa ikuh
dh Å¡pkbZ 10 m gSA
gy eku yhft, fd r, h vkSj α vko`Qfr 6-22 osQ vuqlkj gSA rc
r
tan α = gSA
h

−1  r 
blfy, α = tan   = tan–1 (0.5) (fn;k gS)
 h
r h
vr% = 0.5 ;k r =
h 2 vko`Qfr 6-22
eku yhft, 'kaoqQ dk vk;ru V gSA rc
2
1 2 1  h πh3
V = π r h = π  h =
3 3  2 12

dV d  πh3  dh
vr% =  ⋅ ( Ük`a[kyk fu;e }kjk)
dt dh  12  dt
π 2 dh
= h
4 dt

vc vk;ru osQ ifjorZu dh nj vFkkZr~ dV = 5 cm3/min vkSj h = 4 m gSA

dt
π 2 dh
blfy, 5= (4) ⋅
4 dt
dh 5 35  22 
;k = = m/min  π = 
dt 4π 88  7
35
vr% ikuh osQ Lrj osQ mBus dh nj m/min gSA
88

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 vodyt osQ vuqiz;ksx 253

mnkgj.k 44 2 m Å¡pkbZ dk vkneh 6 m Å¡ps fctyh osQ [kaHks ls nwj 5 km/h dh leku pky
ls pyrk gSA mldh Nk;k dh yack;ha dh o`f¼ nj Kkr dhft,A
gy vko`Qfr 6-23 esa] eku yhft,] AB ,d fctyh
dk [kaHkk gSA B ¯cnq ij cYc gS vkSj eku yhft, fd
,d fo'ks"k le; t ij vkneh MN gSA eku yhft,
AM = l m vkSj O;fDr dh Nk;k MS gSA vkSj eku
yhft, MS = s m gSA
è;ku nhft, fd ∆ASB ~ ∆MSN
MS MN
;k =
AS AB
vko`Qfr 6-23
;k AS = 3s
[(D;ksafd MN = 2 m vkSj AB = 6 m (fn;k gS)]
bl izdkj AM = 3s – s = 2s gSA ijUrq AM = l ehVj gSA
blfy, l = 2s
dl ds
vr% = 2
dt dt
dl 5
D;ksafd = 5 km/h gSA vr% Nk;k dh yack;ha esa o`f¼ km/h dh nj ls gksrh gSA
dt 2
mnkgj.k 45 oØ x2 = 4y osQ fdlh ¯cnq ij vfHkyac dk lehdj.k Kkr dhft, tks ¯cnq (1] 2) ls
gksdj tkrk gSA
gy x2 = 4y dk] x osQ lkis{k vodyu djus ij%
dy x
=
dx 2
eku yhft, oØ x2 = 4y osQ vfHkyac osQ laioZQ ¯cnq osQ funsZ'kkad (h, k) gSaA vc (h, k) ij Li'kZ
js[kk dh izo.krk
dy  h

dx  ( h, k ) =
2
−2
⇒ (h, k) ij vfHkyac dh izo.krk = gSA
h

2018-19
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blfy, (h, k) ij vfHkyac dk lehdj.k gS

−2
y–k= ( x − h) ... (1)
h
ijarq ;g ¯cnq (1, 2) ls xqtjrk gSA ge ikrs gSa fd
−2 2
2−k = (1 − h) ;k k = 2 + (1 − h) ... (2)
h h
D;ksafd (h, k) oØ x2 = 4y ij fLFkr gSA blfy,
h2 = 4k ... (3)
vc (2) o (3), ls h = 2 vkSj k = 1 izkIr gksrk gSA h vkSj k osQ bu ekuksa dks (1) esa j[kus ij vfHkyac
dk vHkh"V lehdj.k fuEufyf[kr izkIr gksrk gSA
−2
y −1 = ( x − 2) ;k x + y = 3
2
mnkgj.k 46 oØ y = cos(x + y), – 2π ≤ x ≤ 2π dh Li'kZ js[kkvksa osQ lehdj.k Kkr dhft,
tks js[kk x + 2y = 0 osQ lekarj gS
gy y = cos(x + y) dk x, osQ lkis{k vodyu djus ij]
dy − sin ( x + y )
=
dx 1 + sin ( x + y )

− sin ( x + y )
;k (x, y) ij Li'kZ js[kk dh izo.krk =
1 + sin ( x + y )

−1
pw¡fd fn, x, oØ dh Li'kZ js[kk x + 2y = 0 osQ lekarj gS ftldh izo.krk gSA vr%
2
− sin( x + y ) −1
=
1 + sin( x + y ) 2
;k sin (x + y) = 1
π,
;k x + y = nπ + (– 1)n n ∈ Z,
2
 n π
rc y = cos (x + y) = cos  nπ + ( −1)  , n ∈ Z,
2
= 0 lHkh n ∈ Z osQ fy,

2018-19
 vodyt osQ vuqiz;ksx 255

3π π
iqu% D;ksafd − 2π ≤ x ≤ 2π , blfy, x = − vkSj x = gSA vr% fn, x, oØ osQ osQoy
2 2
 3π  π 
¯cnqvksa  − 2 , 0 vkSj  2 , 0 ij Li'kZ js[kk,¡] js[kk x + 2y = 0 osQ lekarj gaSA blfy, vHkh"V
Li'kZ js[kkvksa osQ lehdj.k
−1  3π 
y–0= x+  ;k 2 x + 4 y + 3π = 0
2 2 
−1  π
vkSj y–0= x−  ;k 2 x + 4 y − π = 0 gSA
2 2
mnkgj.k 47 mu varjkyksa dks Kkr dhft, ftuesa iQyu
3 4 4 3 36
f (x) = x − x − 3x 2 + x + 11
10 5 5
(a) o/Zeku (b) ßkleku gSA
gy gesa Kkr gS fd
3 4 4 3 36
f (x) = x − x − 3x 2 + x + 11
10 5 5
3 4 36
;k f ′(x) = (4 x 3 ) − (3x 2 ) − 3(2 x) +
10 5 5
6
= ( x − 1) ( x + 2) ( x − 3) (ljy djus ij)
5
vc f ′(x) = 0 ls x = 1, x = – 2, vkSj x = 3 izkIr gksrs
gSaA x = 1, – 2, vkSj 3 okLrfod js[kk dks pkj vla;qDr
varjkyksa uker% (– ∞, – 2), (– 2, 1), (1, 3) vkSj (3, ∞) esa vko`Qfr 6-24
foHkDr djrk gSA (vko`Qfr 6.24)
varjky (– ∞, – 2) dks yhft, vFkkZr~ tc – ∞ < x < – 2 gSA
bl fLFkfr esa ge x – 1 < 0, x + 2 < 0 vkSj x – 3 < 0 izkIr djrs gSaA
(fo'ks"k :i ls x = –3 osQ fy, nsf[k, fd] f ′(x) = (x – 1) (x + 2) (x – 3)
= (– 4) (– 1) (– 6) < 0) blfy,] tc – ∞ < x < – 2 gS] rc f ′(x) < 0 gSA
vr% (– ∞, – 2) esa iQyu f ßkleku gSA
varjky (–2, 1), dks yhft, vFkkZr~ tc – 2 < x < 1 gSA
bl n'kk esa x – 1 < 0, x + 2 > 0 vkSj x – 3 < 0 gSA

2018-19
256 xf.kr

(fo'ks"k :i ls x = 0, osQ fy, è;ku nhft, fd] f ′(x) = (x – 1) (x + 2) (x – 3) = (–1)

(2) (–3) = 6 > 0)
blfy, tc – 2 < x < 1 gS] rc f ′(x) > 0 gSA
vr% (&2] 1) esa iQyu f o/Zeku gSA
vc varjky (1,3) dks yhft, vFkkZr~ tc 1 < x < 3 gSA bl n'kk esa fd x – 1 > 0, x + 2 > 0
vkSj x – 3 < 0 gSA
blfy,] tc 1 < x < 3 gS] rc f ′(x) < 0 gSA
vr% (1, 3) esa iQyu f ßkleku gSA var esa varjky (3, ∞), dks yhft, vFkkZr~ tc
3 < x < ∞ gSA bl n'kk esa x – 1 > 0, x + 2 > 0 vkSj x – 3 > 0 gSA blfy, tc
x > 3 gS rks f ′(x) > 0 gSA
vr% varjky (3, ∞) esa iQyu f o/Zeku gSA

mnkgj.k 48 fl¼ dhft, fd f (x) = tan–1(sin x + cos x), x > 0 ls iznÙk iQyu f ,  0, π  esa
 4
fujarj o/Zeku iQyu gSA
gy ;gk¡
f (x) = tan–1(sin x + cos x), x > 0
1
;k f ′(x) = (cos x − sin x )
1 + (sin x + cos x) 2
cos x − sin x
= (ljy djus ij)
2 + sin 2 x
 π
è;ku nhft, fd  0, 4  esa lHkh x osQ fy, 2 + sin 2x > 0 gSA
blfy, f ′(x) > 0 ;fn cos x – sin x > 0
;k f ′(x) > 0 ;fn cos x > sin x ;k cot x > 1
π
vc cot x > 1 ;fn tan x < 1, vFkkZr~] ;fn 0 < x <
4
  π
blfy, varjky  0, 4  esa f ′(x) > 0 gSA
π
vr%  0,  esa f ,d o/Zeku iQyu gSA
 4 

2018-19
 vodyt osQ vuqiz;ksx 257

mnkgj.k 49 3 cm f=kT;k dh ,d o`Ùkkdkj fMLd dks xeZ fd;k tkrk gSA izlkj osQ dkj.k bldh
f=kT;k 0.05 cm/s dh nj ls c<+ jgh gSA og nj Kkr dhft, ftlls bldk {ks=kiQy c<+ jgk gS
tc bldh f=kT;k 3.2 cm gSA
gy eku yhft, fd nh xbZ r'rjh dh f=kT;k r vkSj bldk {ks=kiQy A gSA
rc A = π r2
dA dr
;k = 2πr ( Üak`[kyk fu;e }kjk)
dt dt
dr
vc f=kT;k dh o`f¼ dh lfUudV nj = dr = ∆t = 0.05 cm/s gSA
dt
blfy, {ks=kiQy esa o`f¼ dh lfUudV nj fuEukafdr gS
dA
dA = ( ∆t )
dt
 dr 
= 2πr  ∆t  = 2πr (dr)
 dt 
= 2π (3.2) (0.05) (r = 3.2 cm)
= 0.320π cm2/s
mnkgj.k 50 ,sY;wfefu;e dh 3 m × 8 m dh vk;rkdkj pknj osQ izR;sd dksus ls leku oxZ dkVus
ij cus ,Y;wfefu;e osQ iQydksa dks eksM+dj <Ddu jfgr ,d lanwd cukuk gSA bl izdkj cus lanwd
dk vf/dre vk;ru Kkr dhft,A
gy eku yhft, fd vyx fd, x, oxZ dh Hkqtk dh yack;ha x m gS] rc ckDl dh Å¡pkbZ x,
yack;ha 8 – 2x vkSj pkSM+kbZ 3 – 2x (vko`Qfr 6.25) gSA ;fn lanwd dk vk;ru V(x) gS rc

vko`Qfr 6-25
V (x) = x (3 2x) (8 – 2x)

V′( x ) = 12 x 2 − 44 x + 24 = 4( x − 3)(3x − 2)
= 4x3 – 22x2 + 24x,vr% 
V′′( x) = 24 x − 44

2018-19
258 xf.kr

2
vc V′(x) = 0 ls x = vkSj x = 3 izkIr gksrk gSA ijUrq x ≠ 3 (D;ksa?)
3
2
blfy, x=
3
 2  2
vc V ′′   = 24   − 44 = −28 < 0
 3  3
2 2
blfy, x = mPpre dk ¯cnq gS vFkkZr~ ;fn ge pknj osQ izR;sd fdukjs ls m Hkqtk osQ oxZ
3 3
gVk nsa vkSj 'ks"k pknj ls ,d land
w cuk, rks land
w dk vk;ru vf/dre gksxk tks fuEufyf[kr gS%
3 2
2
V   = 4   − 22   + 24   =
2 2 2 200 3
m
3 3 3 3 27
 x 
mnkgj.k 51 ,d fuekZrk Rs  5 − 100  izfr bdkbZ dh nj ls x bdkb;k¡ csp ldrk gSA
x 
x bdkb;ksa dk mRikn ewY; Rs  + 500 gSA bdkb;ksa dh og la[;k Kkr dhft, tks mls
5
vf/dre ykHk vftZr djus osQ fy, cspuh pkfg,A
gy eku yhft, x bdkb;ksa dk foØ; ewY; S (x) gS vkSj x bdkb;ksa dk mRikn ewY; C (x) gSA
rc ge ikrs gSa
 x  x2
S(x) =  5 −  x = 5x −
 100  100
x
vkSj C (x) = + 500
5
bl izdkj] ykHk iQyu P (x) fuEukafdr }kjk iznÙk gSA
x2 x
P(x) = S( x) − C( x ) = 5 x − − − 500
100 5
24 x2
vFkkZr~ x−
P(x) = − 500
5 100
24 x
;k P′(x) = −
5 50
−1 −1
vc P′(x) = 0 ls x = 240 izkIr gksrk gS vkSj P′′( x) = . blfy, P′′(240) = < 0 gSA
50 50

2018-19
 vodyt osQ vuqiz;ksx 259

bl izdkj x = 240 mPpre dk ¯cnq gSA vr% fuekZrk vf/dre ykHk vftZr dj ldrk gS ;fn
og 240 bdkb;k¡ csprk gSA

vè;k; 6 ij fofo/ iz'ukoyh

1. vodyt dk iz;ksx djosQ fuEufyf[kr esa ls izR;sd dk lfUudV eku Kkr dhft,%
1
1
(a)  17  4 (b) (33)− 5
 
81

log x
2. fl¼ dhft, fd f ( x) = }kjk iznÙk iQyu x = e ij mPpre gSA
x
3. fdlh fuf'pr vk/kj b osQ ,d lef}ckgq f=kHkqt dh leku Hkqtk,¡ 3 cm/s dh nj ls ?kV
jgha gSA ml le; tc f=kHkqt dh leku Hkqtk,¡ vk/kj osQ cjkcj gSa] mldk {ks=kiQy fdruh
rsth ls ?kV jgk gSA
4. oØ x2 = 4y osQ ¯cnq (1] 2) ij vfHkyac dk lehdj.k Kkr dhft,A
5. fl¼ dhft, fd oØ x = a cosθ + aθ sinθ, y = a sinθ – aθ cosθ osQ fdlh ¯cnq θ
ij vfHkyac ewy ¯cnq ls vpj nwjh ij gSA
6. varjky Kkr dhft, ftu ij
4sin x − 2 x − x cos x
f ( x) =
2 + cos x
ls iznÙk iQyu f (i) fujarj o/Zeku (ii) fujarj ßkleku gSA
1
7. varjky Kkr dhft, ftu ij f ( x) = x3 + , x ≠ 0 ls iznÙk iQyu
x3

(i) o/Zeku (ii)ßkleku gSA

x2 y 2
8. nh?kZo`Ùk + = 1 osQ varxZr ml lef}ckgq f=kHkqt dk egÙke {ks=kiQy Kkr dhft,
a 2 b2
ftldk 'kh"kZ nh?kZ v{k dk ,d fljk gSA
9. vk;rkdkj vk/kj o vk;rkdkj nhokjksa dh 2 m xgjh vkSj 8 m3 vk;ru dh ,d fcuk
<Ddu dh Vadh dk fuekZ.k djuk gSA ;fn Vadh osQ fuekZ.k esa vk/kj osQ fy, Rs 70/m2
vkSj nhokjksa ij Rs 45/m2 O;; vkrk gS rks fuEure [kpZ ls cuh Vadh dh ykxr D;k gS\

2018-19
260 xf.kr

10. ,d o`Ùk vkSj ,d oxZ osQ ifjekiksa dk ;ksx k gS] tgk¡ k ,d vpj gSA fl¼ dhft, fd muosQ
{ks=kiQyksa dk ;ksx fuEure gS] tc oxZ dh Hkqtk o`Ùk dh f=kT;k dh nqxquh gSA
11. fdlh vk;r osQ Åij cus v/Zo`Ùk osQ vkdkj okyh f[kM+dh gSA f[kM+dh dk laiw.kZ ifjeki
10 m gSA iw.kZr;k [kqyh f[kM+dh ls vf/dre izdk'k vkus osQ fy, f[kM+dh dh foek,¡ Kkr
dhft,A
12. f=kHkqt dh Hkqtkvksa ls a vkSj b nwjh ij f=kHkqt osQ d.kZ ij fLFkr ,d ¯cnq gSA fl¼ dhft,
2 2 3
fd d.kZ dh U;wure yackbZ (a 3 + b 3 ) 2 gSA
13. mu ¯cnqvksa dks Kkr dhft, ftu ij f (x) = (x – 2)4 (x + 1)3 }kjk iznÙk iQyu f dk]
(i) LFkkuh; mPpre ¯cnq gS (ii) LFkkuh; fuEure ¯cnq gS
(iii) ur ifjorZu ¯cnq gSA
14. f (x) = cos2 x + sin x, x ∈ [0, π] }kjk iznÙk iQyu f dk fujis{k mPpre vkSj fuEure eku
Kkr dhft,A
15. fl¼ dhft, fd ,d r f=kT;k osQ xksys osQ varxZr mPpre vk;ru osQ yac o`Ùkh; 'kaoqQ dh
4r
Å¡pkbZ gSA
3
16. eku yhft, [a, b] ij ifjHkkf"kr ,d iQyu f gS bl izdkj fd lHkh x ∈ (a, b) osQ fy,
f ′(x) > 0 gS rks fl¼ dhft, fd (a, b) ij f ,d o/Zeku iQyu gSA
17. fl¼ dhft, fd ,d R f=kT;k osQ xksys osQ varxZr vf/dre vk;ru osQ csyu dh Å¡pkbZ
2R
gSA vf/dre vk;ru Hkh Kkr dhft,A
3
18. fl¼ dhft, fd v¼Z'kh"kZ dks.k α vkSj Å¡pkbZ h osQ yac o`Ùkh; 'kaoqQ osQ varxZr vf/dre
vk;ru osQ csyu dh Å¡pkbZ] 'kaoqQ osQ Å¡pkbZ dh ,d frgkbZ gS vkSj csyu dk vf/dre
4 3 2
vk;ru πh tan α gSA
27
19 ls 24 rd osQ iz'uksa osQ lgh mÙkj pqfu,A
19. ,d 10 m f=kT;k osQ csyukdkj Vadh esa 314 m3/h dh nj ls xsgw¡ Hkjk tkrk gSA Hkjs x,
xsgw¡ dh xgjkbZ dh o`f¼ nj gS%
(A) 1 m/h (B) 0.1 m/h
(C) 1.1 m/h (D) 0.5 m/h

2018-19
 vodyt osQ vuqiz;ksx 261

20. oØ x = t2 + 3t – 8, y = 2t2 – 2t – 5 osQ ¯cnq (2,– 1) ij Li'kZ js[kk dh izo.krk gS%

22 6 7 −6
(A) (B) (C) (D)
7 7 6 7
21. js[kk y = mx + 1, oØ y = 4x dh ,d Li'kZ js[kk gS ;fn m dk eku gS%
2

1
(A) 1 (B) 2 (C) 3 (D)
2
22. oØ 2y + x2 = 3 osQ ¯cnq (1,1) ij vfHkyac dk lehdj.k gS%
(A) x + y = 0 (B) x – y = 0
(C) x + y +1 = 0 (D) x – y = 1
23. oØ x = 4y dk ¯cnq (1,2) ls gks dj tkus okyk vfHkyac gS%
2

(A) x + y = 3 (B) x – y = 3
(C) x + y = 1 (D) x – y = 1
24. oØ 9y2 = x3 ij os ¯cnq tgk¡ ij oØ dk vfHkyac v{kksa ls leku var% [kaM cukrk gS%
 8  −8 
(A)  4, ±  (B)  4, 
 3 3

 3  3
(C)  4, ±  (D)  ± 4, 
 8  8

lkjka'k
® ;fn ,d jkf'k y ,d nwljh jkf'k x osQ lkis{k fdlh fu;e y = f ( x ) dks larq"V djrs
dy
gq, ifjofrZr gksrh gS rks (;k f ′ ( x ) ) x osQ lkis{k y osQ ifjorZu dh nj dks fu:fir
dx
dy 
djrk gS vkSj dx  (;k f ′(x0 ) ) x = x0 ij) x osQ lkis{k y osQ fu:fir dh nj dks
x = x0

fu:fir djrk gSA

® ;fn nks jkf'k;k¡ x vkSj y, t osQ lkis{k ifjofrZr gks jgh gksa vFkkZr~ x = f (t ) vkSj y = g (t ) ,
rc Üka`[kyk fu;e ls
dy dy dx dx
= , ;fn ≠0
dx dt dt dt

2018-19
262 xf.kr

® ,d iQyu f
(a) varjky [a, b] eas o/Zeku gS ;fn
[a, b] eas x1 < x2 ⇒ f (x1) ≤ f (x2), lHkh x1, x2 ∈ (a, b) osQ fy,
fodYir% ;fn izR;sd x ∈ [a, b] osQ fy, f ′(x) ≥ 0, gSA
(b) varjky [a, b] esa ßkleku gS ;fn
[a, b] esa x1 < x2 ⇒ f (x1) ≥ f (x2), lHkh x1, x2 ∈ (a, b) osQ fy,
fodYir% ;fn izR;sd x ∈ [a, b] osQ fy, f ′(x) ≤ 0 gSA
® oØ y = f (x) osQ ¯cnq (x0, y0) ij Li'kZ js[kk dk lehdj.k
dy 
y − y0 = ( x − x0 ) gSA
dx  ( x0 , y0 )

dy
® ;fn ¯cnq ( x0 , y0 ) ij
dx
dk vfLrRo ugha gS] rks bl ¯cnq ij Li'kZ js[kk y-v{k osQ
lekarj gS vkSj bldk lehdj.k x = x0 gSA
dy 
® ;fn oØ y = f (x) dh Li'kZ js[kk x = x0 ij] x&v{k osQ lekarj gS] rks dx  = 0 gSA
x = x0

® oØ y = f (x) osQ ¯cnq ( x0 , y0 ) ij vfHkyac dk lehdj.k

−1
y − y0 = ( x − x0 ) gSA
dy 
dx  ( x0 , y0 )

dy
® ;fn ¯cnq ( x0 , y0 ) ij
dx
= 0 rc vfHkyac dk lehdj.k x = x0 gSA

dy
® ;fn ¯cnq ( x0 , y0 ) ij
dx
dk vfLrRo ugha gS rc bl ¯cnq ij vfHkyac x-v{k osQ
lekarj gS vkSj bldk lehdj.k y = y0 gSA
® eku yhft, y = f (x) vkSj ∆x, x esa NksVh o`f¼ gS vkSj x dh o`f¼ osQ laxr y esa o`f¼
∆y gS vFkkZr~ ∆y = f (x + ∆x) – f (x) rc
 dy 
dy = f ′ ( x )dx ;k dy =   ∆x
 dx 

2018-19
 vodyt osQ vuqiz;ksx 263

tc dx = ∆x vis{kko`Qr cgqr NksVk gS rks ;g ∆y dk ,d vPNk lfUudVu gSA bls ge

dy ≈ ∆y osQ }kjk fu:fir djrs gSaA

® iQyu f osQ izkar esa ,d ¯cnq c ftl ij ;k rks f ′(c) = 0 ;k f vodyuh; ugha gS]
f dk Økafrd ¯cnq dgykrk gSA

® izFke vodyt ijh{k.k eku yhft, ,d foo`Ùk varjky I ij iQyu f ifjHkkf"kr gSA
eku yhft, I esa ,d Økafrd ¯cnq c ij iQyu f larr gS rc
(i) tc x ¯cnq c osQ ck;ha vksj ls nk;ha vksj c<+rk gS rc f ′(x) dk fpÉ /u ls ½.k
esa ifjofrZr gksrk gS vFkkZr~ c osQ ck;haa vksj vkSj i;kZIr fudV izR;sd ¯cnq ij ;fn
f ′(x) > 0 rFkk c osQ nk;haa vksj vkSj i;kZIr fudV izR;sd ¯cnq ij ;fn f ′(x) <
0 rc c LFkkuh; mPpre dk ,d ¯cnq gSA
(ii) tc x ¯cnq c osQ ck;ha vksj ls nk;ha vksj c<+rk gS rc f ′(x) dk fpÉ ½.k ls
/u esa ifjofrZr gksrk gS vFkkZr~ c osQ ck;haa vksj vkSj i;kZIr fudV izR;sd ¯cnq ij
;fn f ′(x) < 0 rFkk c osQ nk;haa vksj vkSj i;kZIr fudV izR;sd ¯cnq ij ;fn f ′(x) >
0 rc c LFkkuh; fuEure dk ,d ¯cnq gSA
(iii) tc x ¯cnq c osQ ck;ha vksj ls nk;ha vksj c<+rk gS rc f ′(x) ifjofrZr ugha gksrk
gS rc c u rks LFkkuh; mPpre dk ¯cnq gS vkSj u gh LFkkuh; fuEure dk ¯cnqA
okLro esa bl izdkj dk ¯cnq ,d ufr ifjorZu ¯cnq gSA
® f}rh; vodyt ijh{k.k eku yhft, ,d varjky I ij f ,d ifjHkkf"kr iQyu gS
vkSj c ∈ I gSA eku yhft, f, c ij yxkrkj nks ckj vodyuh; gSA rc
(i) ;fn f ′(c) = 0 vkSj f ″(c) < 0 rc x = c LFkkuh; mPpre dk ,d ¯cnq gSA
f dk LFkkuh; mPpre eku f (c) gSA
(ii) ;fn f ′(c) = 0 vkSj f ″(c) > 0 rc x = c LFkkuh; fuEure dk ,d ¯cnq gSA bl
fLFkfr esa f dk LFkkuh; fuEure eku f (c) gSA
(iii) ;fn f ′(c) = 0 vkSj f ″(c) = 0, rc ;g ijh{k.k vliQy jgrk gSA
bl fLFkfr esa ge iqu% okil izFke vodyt ijh{k.k dk iz;ksx djrs gSa vkSj ;g
Kkr djrs gSa fd c mPpre] fuEure ;k ufr ifjorZu dk ¯cnq gSA
® fujis{k mPpre vkSj fujis{k fuEure ekuksa dks Kkr djus dh O;kogkfjd fof/ gS%
pj.k 1: varjky esa f osQ lHkh Økafrd ¯cnq Kkr dhft, vFkkZr~ x osQ os lHkh eku Kkr
dhft, tgk¡ ;k rks f ′(x) = 0 ;k f vodyuh; ugha gSA

2018-19
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pj.k 2: varjky osQ vaR; ¯cnq yhft,A

pj.k 3: (pj.k 1 o 2 ls izkIr) lHkh ¯cnqvksa ij f osQ ekuksa dh x.kuk dhft,A
pj.k 4: pj.k 3 esa x.kuk ls izkIr f osQ lHkh ekuksa esa ls mPpre vkSj fuEure ekuksa
dks yhft,A ;gh mPpre eku] f dk fujis{k mPpre eku vkSj fuEure eku] f dk
fujis{k fuEure eku gksaxsA

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2018-19