mpp xf.kr d{kk 11 oha vad fohkktu

Transcription

1 mpp xf.kr d{kk oha le; % 3?k.Vs iw.kk±d % 00,dy iz'ui vad fohkktu bdkbz fo"k; olrq vad dky[k.m lfeej la[;k, & ¼v½ rhu pjksa ds fo'ks"k ;qxir lehdj.k,oa mudk gy ¼c½ oxkzred lehdj.k ds fl)kar 3 lekurj Js.kh,oa gjkred Js.kh xq.kksùkj Js.kh,oa fo'ks"k Js.kh 05 5 lkjf.kd vko;wg fcunqvksa ds dkrhz; funsz'kkad 09 8 ljy js[kk js[kk;qxe 09 0 o`ùk 'kadq ifjpnsn fdks.kferh; Qyu fdks.kferh; lozlfedk,a] xzkq o lehdj.k 09 4 fhkqt ds xq.k o fhkqt ds gy Å pkbz vksj nwjh lkaf[;dh Øep; lap; xf.krh; vkxeu,oa f}in izes; (A) js[kh; vlerk,a (B) yhfu;j izksxkfeax pj?kkrkadh,oa y?kqx.kdh; Js.kh iqujko`fùk 0 ;ksx 00 80

2 fo"k;&mpp xf.kr d{kk X;kjgoha bdkbz okj vad fohkktu bdkbz- - lfeej la[;k,a 05 bdkbz- - a+ib ds :i esa lfeej la[;kvksa dk lery ij fcunq :i esa fu:i.k vkjxam fp ¼lfEeJ la[;kvksa dk ;ksx o xq.kuqy T;kferh ls½ lfeej la[;kvksa dk cht xf.kr] lfeej la[;kvksa dk oklrfod o dkyifud Hkkx] ekikad,oa dks.kkad] lfeej la[;kvksa dk la;qxeh] lfeej la[;kvksa dk oxzewy] bdkbz dk?kuewy vksj prqfkz ewya] fhkqth; vlekurk Z + Z Z + Z,oa lekurk Z + Z Z. Z lfeej la[;kvksa dk /kzqoh; izn'kzu cht xf.krh; vk/kkj Hkwr izes; dk dfku lfeej la[;k iz.kkyh esa oxz lehdj.k dk gya ¼v½ rhu pjksa ds fo'ks"k ;qxir lehdj.k,oa mudk gya bdkbz- - ¼c½ oxkzred lehdj.k ds fl)kar 05 oxz lehdj.k ds vf/kdre nks eqy gks ldrs gs fofodrdj,oa ewyksa dh izo`fr] ewyksa vksj xq.kkadks ds chp laca/k ewyksa ds lefer Qyu tsls ± aaaaα β, /α /β, α 3 β 3, /α 3 /β 3 vkfn dk eku,oa fn, x, eqyksa ds fy, oxz lehdj.k jpuk oxz vlehdj.k dk ifjp; ax + bx +c 0 bldks la[;k js[kk,oa fpug i)fr ls gy djuk osfnd xf.kr fof/k ls Hkh gy djuka bdkbz- 3- lekurj Js.kh,oa gjkred Js.kh ¼bdkbZ 3 o 4&0 vad½ bdkbz- 4- ifjhkk"kk] izfke in] lekurj Js.kh ls lec/k gjkerd Js.kh dk n ink dk ;ksxqy. gjkred Js.kh dk lekurj Js.kh ls leca/k Js.k dk n okw ina xq.kksùkj Js.kh,oa fo'ks"k Jsf.k;k & xq.kksùkj Js.kh dh ifjhkk"kk] izfke in] lokzuqikr xq.kksùkj Js.kh dk nokw in] vifjfer xq.kksùkj Js.kh] n inks ks ;ksx] vifjfer Js.kh dk ;ksxqy] xq.kksùkj ek/;] nks la[;kvksa ds chp n xq.kksùkj Js.kh ds :i esa vkorz n'keyo dk eku fo'ks"k Js.kh dk eku,oa lekurjh; xq.kksùkj Js.khA bdkbz- 5- lkjf.kd & 05 lkjf.kd] milkjf.kd vksj lg[k.m] lkjf.kd dk izlkj lkjf.kd ds xq.k,oa :ikarj] fhkqt dk {ksqy Kkr djus esa lkjf.kd dk vuqiz;ksx ¼uksV Øsej dk fu;e lfeefyr ugha gsa½

3 bdkbz- 6- vko;wg & 05 vko;wg la[;kvksa ds vk;rkdkj fou;kl ds :i esa vko;wgksa ds izdkj] vko;wgksa dh lekurk] vko;wgksa dk ;ksxqy] vko;wg,oa vkfn'k dk xq.kuqy] vko;wgksa dk xq.kuqy,oa jsf[kd la;kstu] lkgp;z cavu ds fu;e] Ø fofuesa; fu;e dk ikyu u djuk] vko;wg ds ifjoùkz] lg[k.m,oa izfrykse rfkk mudk chtxf.kr o muds xq.k/kez] rhu vpjksa okys jsf[kd lehdj.k dk vko;wg }kjk gya bdkbz- 7- fcunqvksa ds drhz; funsz'kkad & ¼bdkbZ 7] 8 o 9 & 5 vad½ bdkbz- 8- bdkbz- 9- ledksf.k; dkrhz; funsz'kkad] funsz'kkad ds vuqlkj fcunq dk ry ij fu/kkzj.k xzkq] nks fcunqvksa ds chp nwjh] fhkqt dk {ksqy] rhu fcunqvksa ds js[kh; gksus dk izfrca/k] js[kk[kam dk vuqikfrd fohkktu] fhkqt dk dsuæd,oa vur% dsuæ fcunq ifk,oa blds lehdj.k] ewy fcunq,oa v{kksa dk folfkkiua ljy js[kk& ljy js[kk dh izo.krk] fofhkuu izdkj dh js[kkvksa ds lehdj.k ¼½ izo.krk o mlds Y &v{k ls vur% [k.m ds :i esa ¼½,d fcunq izo.krk ds :i esa ¼3½ nks fcunqvksa ls tkus okyh js[kk ds :i esa] ¼4½ nksuksa v{kksa ls dkavs x;s var% [k.m ds :i esa ¼5½ nwjh ds :i esa ¼izkpyu lehdj.k½ ¼6½ ewy fcunq ls vfhkyec ds :i esa] ljy js[kk dk lkeku; lehdj.k] nks lehdj.k] nks js[kkvksa dk izfrpnsnu] nks js[kkvksa ds chp dk dks.k] js[kkvksa ds lekurj vksj yecor gksus ds izfrca/k rhu js[kkvsa ds laxkeh gksus dk izfrca/k fdlh fcunq ls js[kk dh nwjh] fhkqy ds ykfecd dsuæ o ifjdsuæ nks js[kkvksa ds chp ds dks.k v)zd dk lehdj.ka js[kk;qxe & js[kkvksa dk fudk;] nks js[kkvksa ds dvu fcunq ls tkus okyh js[kkvksa ds lehdj.k] f}?kkrh; le?kkr lehjd.k nks pjksa esa] ewy fcunq ls tkus okyh js[kk ;qqxe ds chp ds dks.k ds lef}hkktdksa dk lfeefyr lehdj.k] nks pjks es lkeku; f}?kkr lehdj.k }kjk js[kk ;qxe n'kkzus dk izfrca/k buds izfrpnsn fcunq ds funsz'kkad,oa muds chp dk dks.k] js[kkvksa ds lekurj vksj yecor~ gksus ds izfrca/ka bdkbz- 0- o`ùk & 05 o`ùk dk eku lehdj.k] o`ùk dk dsuæ rfkk ft;k] o`ùk dk izkpfyd lehdj.k] O;kl ds fljksa ds funsz'kkad Kkr gksus ij o`ùk dk lehdj.k ljy js[kk o`ùk dk izfrpnsnu] fdlh js[kk ds Li'khZ gksus dk izfrca/k fdlh fcunq ls tkus okyh Li'kkhZ dk lehdj.k ¼v½ tc fcunq o`ùk ij gks ¼c½ tc fcunq o`ùk ds ckgj gksa o`ùk ds lkis{k fcunq dh flfkfra

4 bdkbz- - 'kadq ifjpnsn ls izkir fofhkuu oø tsls o`ùk] ijoy;] nh?kz o`ùk] vfr ijoy; rfkk ljy js[kk ;qxe,oa fcunq dk ifjp;] nh?kz o`ùk] ijoy;] vfrijoy; ds ekud lehdj.k buds ukfhk o 'kh"kz fcunq ds funsz'kkad fu;rk,oa ukfhk yec ds lehdj.k] ukfhk nwjh] fdlh fcunq dh ijoy;] nh?kz o`ùk] vfrijoy; ds lkis"k flfkfra bdkbz- - fdks.kferh; Qyu ¼bdkbZ o 3 & 0 vad½ dks.k dk va'kks esa ekiu] jsfm;u esa ekiu jfm;u,oa va'kksa esa leca/k] fdks.kferh; vuqikrksa ds eku 30, 45, 60 ds fdks.kferh; vuqikr Kkr djuka vu; dks.kksa ds fdksf.kferh; vuqikrksa dk eku lkfj.kh ls i<+uk ¼0 l 90 rd½ fdks.kferh; lozlehdk,a Sin Cos θ Sec θ + tan θ Coses θ Cot θ dh mrifùk] dks.k ¼ θ½ ds fdks.kferh; vuqikr dks dks.k θ ds fdks.kferh; vuqikr ds :i esa fy[kuk tsls Sin ¼ θ½ Sin θ vkfn fueufyf[kr dks.kksa ds fdks.kferh; vuikrksa dk dks.k θ ds vuqikr ds :i esa O;ä djuk& (π/ ± θ), (π ± θ), nπ ± θ fueu lwksa dh mrifùk,oa mi;ksx& sin (A + B) Sin A. Cos B + Cos A sin B (Cos A + B) Cos A. Cos B + Sin A. sin B tan (A + B) Atan + A B ± tan B µ tan A.tan B lw Sin A + B Sin Cos Sin A Sin B Cos Sin Cos A + Cos B Cos Sin Cos A Cos B Sin Sin

5 Sin A Sin A.Cos A Cos A Cos A Sin A Cos A Sin A Sin 3A 3 Sin A 4 Sin 3 A Cos 3A 4 Cos 3 A 3 Cos A Cos A Sin A/ + Cos A Cos A/ tan A tan 3A bdkbz 3- fdks.kferh; lozlfedk,a] xzkq o lehdj.k & bdkbz 4 fdks.kferh; lozlfedk,a,oa izfrca/kkred loz lfedk,a ¼fHkqt ds dks.kksa ls lacaf/kr½ fdks.kferh; Qyuksa ds vkorz dh vo/kkj.kk] fdks.kferh; Qyuksa ds xzkq tsls & y Sin x y a Cos x, y a. tan (bx) y Sin (bx + c) vkfna 3 SinA 3tan tana A SinB tan A atan 3tan A ba fdks.kferh; lehdj.kksa dk O;kid gy & tsls& Sin θ Sin α θ n π + ( ) n α Cos θ Cos α θ nπ + α tan θ tan α θ nπ + α fhkqt ds xq.k o fhkqt ds gy & fhkqt ds xq.k & lkbu fu;e dks lkbzu fu;e a b + c abc Cos A vkfn usfi;j dh lekurk tan B C b c Cot A/ b + c SinC c

6 iz{ksi lw a b. Cos C + c Cos B fhkqt ds v)z dks.kkasa dk Hkqtkvksa ds :i esa O;Dr djuk tsls& Sin A/ vkfna fhkqt ds dks.kksa dk eku Hkqtkvksa ds inksa ls O;Dr djuka tsls & Sin A /bc s( s a)( s b)( s c) fhkqt ds {ksqy gsjks ds lw }kjk ifjo`ùk dh ft;k R a sina vkfna fhkqt ds {ksqy o ifjo`ùk dh ft;k dk laca/k R fhkqtksa dk gy & ¼v½ tc nks dks.k o,d Hkqtk Kkr gksa ¼c½ tc nksuksa Hkqtk, vksj muds chp dk dks.k Kkr gksa ¼l½ tc rhuksa Hkqtk, nh xbz gksaa abc ( a b) ( s c) 4 bc ¼n½ tc nks Hkqtk vksj muesa ls,d ds lkeus dk dks.k fn;k gksa bdkbz- 5- Å pkbz vksj nwjh & 05 Å pkb o nwjh ls lacaf/kr iz'u¼f}foeh;,oa ffoeh;½ bdkbz- 6- lkaf[;dh & 05 dsuæh; izo`fùk dh eki % ek/; ekf/;dk,oa cgqyd ¼O;fäxr] oxhzñr ¼v½ leothz]¼c½ viothz Js.kh ds fy,½ ek/; dh izr;{k fof/k o y?kqfof/k ¼dfYir ek/; fof/k½ ek/; fopyu] fo{ksi.k] ekud fopuy] fopj.k ekikad] fopj.k xq.kkad ¼O;fäxr [kafmr] oxhzñr½ ¼lrr vksj vlrr~½ ds fy, izr;{k o y?k q fof/k ¼dfYir ek/; fof/k½ bdkbz- 7- Øep; vksj lap;] Jh egkohjkpk;z dk thou ifjp;,oa ;ksxnku 05 x.kuk ds fu;e n vfkz fou;kl ds vfkz esa Øep; ^^p r vksj p;u ds vfkz esa lap; n c r dk vfkz buds vuqiz;ksx o`ùkrh; Øep; vkfna

7 bdkbz- 8- xf.krh; vkxeu,oa f}in izes; % 05 izkñr la[;kvksa ds lanhkz esa xf.krh; vkxeu dk fl)kar vksj mlds ljy vuqiz;ksx] xf.krh; vkxeu dh miifùk }kjk f}in izes; ¼/kuiw.kkZd?kkrkad½ ds fy, lkeku; rfkk fo'ks"k in Kkr djuk fdlh Hkh?kkr ds fy, f}in izesa; ¼miifÙk ugha½] f}in izes; dk lfuudv vuqeku esa vuqiz;ksxa f}in xq.kkadksa ds izxq.ka bdbz- 9- ¼A½ js[kh; vlerk,a & 05 js[kh; mlerk;sa] js[kh; vlerkvksa ds chp xf.krh; gy vksj,d pjh; la[;k js[kk ij mudk izn'kzua ¼B½ yhfu;j izksxzkfeax & nks pj okys] jsf[kd vlehdj.k vksj muds gy] leqpp; ds ys[kk fp] nks pj okys js[kh; vlehdj.k ds fudk;ksa ds gyksa ds leqpp; ds ys[kkfp] yhfu;j izksxzkfeax dk vfkz mldk egro] olrqf"b Qyu] vkivh ekbts'ku] leykhk js[kk vkfn mùky cgqhkqy ¼leqPp;½ mlds xq.k/kez dk Li"Vhdj.k] yhfu;j izksxzkfeax lel;k dk xf.krh; :ikarj,oa ys[kk fp fof/k }kjk gya bdbz- 0- pj/kkrkadh,oa y?kqx.kdh; Js.kh % 05 e ds fy, vaur Js.kh] e dk eku < e < 3 ds chp esa fl) djuk] e x dk folrkj log e ¼+x½ ds fy, vuur Js.kh ds izlkj] mi;qä y?kq xq.kd Js.kh }kjk la[;kvksa ds y?kqx.kd Kkr djuka uksv&;fkk izlax osfnd xf.kr fof/k;sa dk mi;ksx Hkh iz'uksa dks gy djus esa lgk;d fof/k;ksa ds :i esa fd;k tk;sa

8 vuqøef.kdk ¼½ dfbu va'k &88 ¼bdkbZ ls bdkbz 0 rd½ ¼½ Cyw fiz.v 89 ¼3½ iz'u&i 90&98 ¼4½ vkn'kz mùkj 99&3

9 funsz'k %& - iz'u Ø ls 5 rd olrqfu"b iz'u gksaxs ftlds vurxzr tksm+h cukuk],d 'kcn okys iz'u cgqfodyih; iz'u] fjdr LFkkuksa dh iwfrz vkfn ds iz'u gksaxsa izr;sd iz'u esa 5 vad fu/kkzfjr gsaa - olrqfu"b iz'u dks NksM+dj lhkh iz'uksa esa fodyi dk izko/kku j[kk tk;s ;g fodyi leku bdkbz ls rfkk ;FkklaHko leku dfbukbz Lrj okys gksus pkfg,a 3- ljy 50 izfr'kr] lkeku; 35 izfr'kr] dfbu 5 izfr'kra

10 dfbu va'k bdkbz lfeej la[;k (Complex Number) izr;sd oklrfod la[;k dk oxz /kukred gksrk gs pkgs og /kukred] _.kkred dqn Hkh gks fdurq,slh dksbz oklrfod la[;k ugha gs ftldk oxz _.kkred gksa mnkgj.k % x x 5 dk gy oklrfod la[;k leqpp; esa izkir ugha fd;k tk ldrka,slh lel;kvksa ds gy ds fy;s ubz la[;k vfkkzr la[;k i)fr dk folrkj vko';d gsa,,... vkfn dk dksbz xf.krh; vfkz ugha gs],slh la[;kvksa dks vf/kdfyir la[;k, (Imaginary numbers) dgrs gsa,sls lehdj.kksa dk gy lozizfke xf.krk vk;yj us fn;k vksj bldk ladsr i vk;ksx fn;ka oklrfod la[;k,oa vf/kdfyir la[;kvksa dk la;ksx a + ib dks lfeej la[;k uke fn;k x;k gs bls z ls fu:fir djrs gsa,oa blds leqpp; dks c ls fu:fir djrs gsaa blesa a,oa b oklrfod la[;k, gsaa bls Øfer ;qxe ds :i esa (a, b) ls iznf'kzr djrs gsaa z a + ib dk la;qxeh blh izdkj a ib gksrk gs % z i i i 3 i.i i, i 4 i.i. i 5 i 4.i i i (4 ) + i, i 6 i 4.i.i i i (4 ) +. i 7 i 4.i 3.( i) i i (4 ) + 3 i. i 4m + i, i 4m + i 4m + 3 i. i 4m i (4m + ) i. i (4m + ) i (4m + 3) i.

11 mnkgj.k % i (i + i 3 + i 4 + i 5 ) i 3 + i 4 + i 5 + i 6 i i i i 4 i i + 0. lfeej la[;k dk oxzewy Kkr djuk % a + ib ekuk x + iy (a + ib) a b + i.ab oklrfod,oa dkyifud la[;kvksa dks vyx&vyx djus ij] x a b...() y ab...() leh- (),oa () dks gy dj a,oa b dk eku Kkr djrs gsaa mnkgj.k % 0 i dk oxzewy Kkr dhft;sa 0i a ib ekuk oxz djus ij 0i (a ib) a b iab. 84 x + iy oklrfod,oa dkyifud la[;kvksa dks vyx&vyx djus ij] a b ab 0...(i) (a + b ) (a b ) + 4a b...(ii) () + (0) a + b (i) o (iii) dks tksm+us ij % 9....(iii) a 50 a 5 a ± 5. leh- (i) esa a dk eku j[kus ij 5 b

12 b 4 b ± vhkh"v oxzewy a ib ± 5 (± ) i, bdkbz ds?kuewy,oa xq.k mùkj ± (5 i). mùkj () bdkbz ds rhu?kuewy gsa ftlesa ls,d oklrfod,oa nks dkyifud ¼lfEeJ½ ewy gsaa budk eku,,oa 3 i () nks dkyifud ¼lfEeJ½ ewyksa esa ls,d ;fn ω gs rks nwljk ω vfkkzr,d ewy nwljs dk oxz gksrk gsa (3) bdkbz ds rhuksa?kuewyksa dk ;ksx 'kwu; gksrk gsa vfkkzr + ω + ω 0. (4) bdkbz ds nksuksa dkyifud ¼lfEeJ½?kuewyksa dk xq.kuqy gksrk gs ω.ω ω 3. (5) bdkbz ds nks lfeej?kuewyksa esa ls izr;sd nwljs dk O;qRØe gksrk gs ω + 3i ω ω ω. mnk-. ( ω)( ω )( ω 3 )( ω 4 ) 0 fl) dhft;sa gy % ( ω)( ω )( )( ω) 0. ( ω )( ω ) 0 0. mùkj mnk-. fl) dhft;s fd... ω vksj ω. gy % ekuk fd x... x x x x. x + x + 0.

13 x + 3 i 3i, ω ;k ω. mnk- 3. ;fn ω bdkbz dk,d?kuewy gks] rks fl) dhft, fd ( ω + ω ) 3 + ( + ω ω ) 3 6. gy % L.H.S. ( + ω ω) 3 + ( + ω ω ) 3 ( ω ω) 3 + ( ω ω ) 3 ( ω) 3 + ( ω ) 3 8ω 3 8ω 3.ω 3 6ω 3 6 (Θ ω 3 ) R.H.S. ;gh fl) djuk FkkA mnk- 4. dks a + ib ds :i esa iznf'kzr dhft,a gy % 4 ± 3 8z (5) i + 36 i ( ii ) + i6 i i 9 9 i 3i 6 i i. mùkj mnk- 5. lfeej la[;k 5 + i dh la;qxeh dk ekikad Kkr dhft,a gy % ekuk z 5 + i rc z 5 i mùkj mnk i dks /kzqoh; :i esa iznf'kzr dhft, rfkk mldk ekikad crkb;sa gy % dh x + iy ls rqyuk djus ij] x, y

14 ekikad r dks.kkad θ tan ( ) 4. mùkj 3 Fy HG I xk J tan vr% + r (cosθ + i sinθ) 4 mùkj ;gh vhkh"v /kzqoh; :i gsa mnk- 7. bdkbz ds?kuewy Kkr dhft,a gy % ekuk fd bdkbz dk?kuewy x gsa rc x () /3 Þ x 3 x 3 0 (x )(x + x + ) 0 x 0 ;k x + x + 0 x ;k x ± F HG i 3 π 3i x + π y π cos + i sin x, ± 3 i 3 3 i 3, vr% bdkbz ds rhu?kuewy, + i 3 rfkk i 3 I K J gsaa mùkj mnk- 8. fl) dhft, fd bdkbz ds rhuksa?kuewyksa dk ;ksx 'kwu; gksrk gsa 3 gy % iwoz iz'u ls] bdkbz ds rhu?kuewy, + i rfkk vr% bdkbz ds rhuksa?kuewyksa dk ;ksx i 3 gsaa + + i 3 + i 3 + ( + i 3 ) + ( i 3 ) 0. ;gh fl) djuk FkkA

15 bdkbz ;qxir lehdj.k (Simultaneous Equation) xf.kr esa lehdj.kksa dk xf.kr ds yxhkx lhkh 'kk[kkvksa esa Hkjiwj mi;ksx gsa vr% ;qxir lehdj.kksa dks gy djuk Nkksa ds fy;s dkqh mi;ksxh fl) gksxka HkkSfrdh esa Hkh bldk dkqh mi;ksx gsa [kkldj tc dksbz?kvuk,d ls vf/kd pj }kjk izhkkfor gksrh gs rks bls gy djus esa Nk dfbukbz u eglwl djsa bl ckr dks n`f"vxr j[krs gq, bl v/;k; dks ikb~;øe esa lekfgr djuk mfpr izrhr gksrk gsa Nk nloha d{kk esa nks pj okys lehdj.kksa dks gy djuk lh[k pqds gsaa ;gk bl v/;k; esa ge rhu pj okys ;qxir lehdj.kksa dks gy djuk fl[kkus dk iz;kl djsaxsa izfke izdkj&tc rhuksa,d?kkrh; gksa rhuksa lehdj.kksa esa ls nks&nks lehdj.kks dks ysdj fdlh,d pj dks foyksfir dj nks pj okys lehdj.k izkir djrs gsaa fqj bl izdkj ls izkir nks pj okys nks lehdj.kksa esa ls,d vu; pj dks foyksfir dj rhljs pj dk eku Kkr djrs gsaa fqj foyksfir pjksa dk eku Kkr dj ysrs gsa % mnkgj.k % x + y + z...() gy % () () ls x + y + 3z...() x + 4y + 9z 4....(3) y z y + z...(3) leh- () (3) ls y 6z y + 3z....(4) vc leh- (3),oa (4) nks pjksa okys ;qxir lehdj.k gsaa leh- (3) (3) ls z 0 leh- (3) esa z dk eku j[kus ij] z 0 y +.(0) y.

16 leh- () esa y,oa z dk eku j[kus ij] x x 0. x 0 y z mùkj f}lrjh; izdkj ds ;qxir lehdj.k ftlesa rhu ;qxir lehdj.kksa esa ls nks lehdj.k js[kh; gksa rfkk vpj in 'kwu; gksa %,sls ;qxir lehdj.k fudk; ds nksuksa js[kh; lehdj.kksa dks otz xq.ku fof/k ls gy djrs gsa,oa pjksa ds vkuqikfrd eku Kkr dj rhljs lehdj.k esa j[krs gsa blls lekuqikr dk eku Kkr dj pj x, y, z dk eku Kkr djrs gsaa mnkgj.k. lehdj.k gy dhft;s % gy % fn;s x;s lehdj.k leh- () o () ls x + y z 0 7x + 6y 9z 0 x y 3 + z xy z y z x + y z 0...() 7x + 6y 9z 0...() x 3 + y 3 + z (3) x ( 9) ( 6)( ) y ( )( 9) ( 7)( ) z ( )( 6) ( 7)( ) x 9 + U V W k ¼ekuk½ leh- (3) esa x, y, z ds eku j[kus ij] x 3k, y 4k, z 5k. (3k) 3 + (4k) 3 + (5k) 3 78.

17 6k k k mnkgj.k. lehdj.k gy dhft, % xy x 3 x 6, y 4 x 8, z 5 x 0 gy % fn;s x;s lehdj.k gsa % yz 3 zx 6 leh- (), (), (3) dks vkil esa xq.kk djus ij xy...() yz 3...() zx 6...(3) leh- () vksj (4) ls] x y z x 3 x 6 x y z 36 ±6 xyz xy zx 6yz xyz ±...(4) ±6 leh- () vksj (4) ls] leh- (3) vksj (4) ls] ±6 3 x ± z ± 3

18 y ± mùkj x ±, y ±, z ± 3. mùkj mnkgj.k 3. lehdj.k gy dhft, % gy % fn;s x;s lehdj.k gsa % x + y 3 y + z 5 z + x 4. leh- (), () vksj (3) dks tksm+us ij] leh- () vksj (4) ls] x + y 3...() y + z 5...() z + x 4...(3) x + y + z (x + y + z) x + y + z 6...(4) 3 + z 6 leh- (),oa (4) ls] z 3 x x. leh- (3) vksj (4) ls] 4 + y 6 y mùkj x, y, z 3. mùkj mnkgj.k 4. lehdj.k gy dhft, % (x + 3)(y + 5) 4 (y + 5)(z + 7) 48 (z + 7)(x + 3) 3. gy % fn;s x;s lehdj.k gsa % (x + 3)(y + 5) 4...()

19 (y + 5)(z + 7) 48...() (z + 7)(x + 3) 3...(3) leh- () o () vksj (3) dk vkil esa xq.kk djus ij] leh- () vksj (4) ls] (x + 3) (y + 5) (z + 7) 4 x 48 x 3 (x + 3) (y + 5) (z + 7) 4 x 4 x x 6 x (x + 3) (y + 5) (z + 7) 4 x 4 x 6 x 4 (x + 3) (y + 5) (z + 7) ± 4 x 4 x...(4) ± fpug ysus ij] z + 7 ± 8 z fpug ysus ij] z z leh- () vksj (4) ls] z ( x )( y + 5 )( z + 7 ) ( x y )( z y+ 75 ) ± fpug ysus ij] x + 3 ± 4 x fpug ysus ij] x x leh- (3) vksj (4) ls] x 7 y + 5 ± 6 ± 4 4 3

20 + fpug ysus ij] fpug ysus ij] y y y y vr% x, y, z rfkk x 7, y, z 5. mùkj.. oxz lehdj.k ds fl)kur... lkeku; lehdj.k ax + bx + c 0 tgk a, b, c vpj gsa x vr% x ds nks ewy gsa] ekuk fd ;s α,oa β gsaa α + β ewyksa ds ;ksx b/a. αβ ewyksa ds xq.kuqy c/a b ± + b 4 ac b b 4ac, a a... ewyksa dh iz fr oxz lehdj.k ds fofdrdj Discriminant D b 4ac ij fuhkzj djrk gsa (i) (ii) ;fn D b 4ac > 0 rks oxz lehdj.k ds ewy oklrfod,oa vleku gksaxs ;fn D 0, ν b 4ac 0 rks oxz lehdj.k ds ewy ifjes;,oa leku gksaxsa (iii) ;fn D b 4ac < 0 rks oxz lehdj.k ds ewy oklrfod ugha gksaxs o vleku gksaxsa.3.3. ;fn oxz lehdj.k ds ewy α,oa β fn;s gksa rks oxz lehdj.k x (α + β) x + αβ nks oxzlehdj.kksa ds,d mhk;fu"b ewy gksus dh 'krz % ;fn fn;k x;k oxz lehdj.k ekuk fd mhk;fu"b ewy α gsa a x + b x + c 0 a x + b x + c 0

21 a α + b α + c 0 a α + b α + c 0. b c α b c α a c a c a b a b α b c b c a b a b vksj α a c a c a b a b F HG a c a b a c a b IKJ b c b c a b a b (a c a c ) (b c b c )(a b a b ) fl) gqvka..5. og izfrca/k Kkr dhft;s tcfd nks oxz lehdj.kksa esa nksuksa ewy mhk;fu"b gksaa gy % ekuk fd nks oxz lehdj.k a x + b x + c 0...(),oa a x + b x + c 0....() ekuk fd nksuksa oxz lehdj.kksa ds mhk;fu"b ewy α,oa β gsaa () ds ewyksa dk ;ksxqy α + β b /a...(3) () α + β b /a...(4) (3),oa (4) ls b a b a a b a b a a b b (A) leh- () ds ewyksa dk xq.kuqy αβ c a " () " " " αβ c a lehdj.k (5),oa (6) ls...(5)...(6) c a c a c c a a...(b)

22 (A),oa (B) ls a a b b c c. ;gh vhkh"v izfrca/k gsa mnkgj.k 5. ;fn lehdj.k ( + m ) x + cmx + c a 0 ds ewy cjkcj gksa rks fl) dhft;s fd gy % fn;k x;k lehdj.k c a ( + m ). leh- () dh rqyuk Ax + Bx + C 0 ls djus ij] Θ fn;s x;s lehdj.k ds ewy leku gsa % ( + m ) x + cmx + c a 0...() A ( + m ), B cm, C c a B 4AC 0 (m) 4( + m )(c a ) 0 4c m 4( + m )(c a ) 0 4[c m ( + m )(c a )] 0 c m c + a m c + m a 0 c a a + m p r c a ( + m ). fl) gqvka mnkgj.k 6. ;fn lehdj.k p (q r) x + q (r p) x + r (p q) 0 ds ewy leku gsa] rks fl) dhft, fd gy % fn;k x;k lehdj.k gs % lehdj.k () esa x j[kus ij] q. p (q r) x + q (r p) x + r (p q) 0...() p (q r)() + q (r p)() + r (p q) 0 pq pr qr qp + rp rq x lehdj.k () dks larq"v djrk gs blfy, lehdj.k 0 dk,d ewy gksxka lehdj.k () ds ewy cjkcj gsa blfy, bldk nwljk ewy Hkh gksxka

23 ewyksa dk xq.kuqy αβ r ( p q ) p( q r) rp rq pq pr pq pr rp rq pr pq + rq pr pqr pq pqr rq + pqr q + r p + p r q. fl) gqvka mnkgj.k 7. ;fn lehdj.k x px + q ds ewyksa dk vurj gks] rks fl) dhft, fd p + 4q ( + q). gy % fn;k x;k lehdj.k gs % x px + q 0 ekuk lehdj.k () ds ewy α, β gsa] rc ewyksa dk ;ksxiqy α + β p ewyksa dk xq.kuqy αβ q lehdj.k () ls] α + β p (α + β) p α + β + αβ p lehdj.k () ls] α + β + q p α + β p q fn;k x;k gs % α β

24 nksuksa i{kksa dk oxz djus ij] (α β) α + β αβ p q q, p 4q p + 4q nksuksa i{kksa esa 4q tksm+us ij p + 4q + 4q + 4q p + 4q + (q) + (q) p + 4q ( + q). fl) gqvka mnkgj.k 8. ;fn lehdj.k x bx + c 0 rfkk x cx + b 0 ds ewyksa dk vurj leku gks rfkk b c gksa] rks fl) dhft, fd b + c gy % fn;s x;s lehdj.k gsa % x bx + c 0...() rfkk x cx + b 0...() ekuk lehdj.k () ds ewy α, β gsa] rc ewyksa dk ;ksxqy α + β b...(3) ewyksa dk xq.kuqy αβ c...(4) lehdj.k (3) o (4) ls] (α β) (α + β) 4αβ (α β) b 4c ewyksa dk vurj α β b 4c...(5) ekuk lehdj.k () ds ewy α, β gsa ewyksa dk ;ksxqy α + β c...(6) ewyksa dk xq.kuqy α β β...(7) lehdj.k (6) o (7) ls] (α β ) (α + β ) 4α β α β c 4b...(8)

25 fn;k x;k gs] lehdj.k () vksj () ds ewyksa dk vurj leku gsaa α β α β lehdj.k (5) o (8) ls] fd nksuksa i{kksa dk oxz djus ij] lehdj.k () o () ls] b 4c c 4b b 4c c 4b b c 4c + 4b 0 (b c)(b + c) + 4(b c) 0 (b c)(4 + b + c) 0 vc (b c) 0, [ fn;k gs b c] b + c fl) gqvka mnkgj.k 9. ;fn lehdj.k ax + bx + c 0 dk,d ewy nwljs dk oxz gks] rks fl) dhft, gy % fn;k x;k lehdj.k gs % ekuk lehdj.k () ds ewy α, α gsa] rc b 3 + a c b + ac 3abc. a a ax + bx + c 0...() ewyksa dk ;ksxqy α + α b a...() ewyksa dk ;ksxqy α α α 3...(3) lehdj.k () ls] α + α

26 (α + α ) 3 α 3 + (α ) 3 + 3αα (α + α ) b a 3 3 α 3 + α 3 α 3 + 3α 3 (α + α ) b a lehdj.k () o (3) ls] c a c a c a 3 c b a a b a + + F H G I K J c a c 3bc + a a ac + c 3bc a b a b a ac + c 3abc b a HG F bi K J 3 p ( r + a) a c + ac 3abc b 3 b 3 + a c + ac 3abc. fl) gqvka mnkgj.k 0. ;fn lehdj.k x + px + q 0 ds vuqikr r : gks] rks fl) dhft, fd gy % fn;k x;k lehdj.k gs % ekuk lehdj.k () ds ewy rα vksj α gsa] rc r + p q. r q ewyksa dk ;ksxqy rα + α p α (r + ) p 3 x + px + q 0...() α

27 nksuksa i{kksa dk oxz djus ij] α p ( r + )...() ewyksa dk xq.kuqy rα α q α...(3) lehdj.k () o (3) ls] q r p q r r + + r p r q q( r p+ ) + + r p ( rr + r ) r q r + + r r r p q q r + p q r q. fl) gqvka mnkgj.k. ;fn lehdj.k ax + bx + c 0 ds ewyksa esa m : n dk vuqikr gks] rks fl) dhft, fd m n gy % fn;k x;k lehdj.k gs % n + m b ac ekuk lehdj.k () ds ewy mα vksj nα gsa] rc. ax + bx + c 0...()

28 ewyksa dk ;ksxqy mα + nα b a α (m + n) α nksuksa i{kksa dk oxz djus ij] α b a ( m + n)...() ewyksa dk xq.kuqy mα x nα c a α...(3) lehdj.k () o (3) ls] c b bb c amna aac (( m + n )) a( mn) b a a c ( m + n) mn ( m + n) mn m + n mn b ac m mn b + mn b ac m n n + m b ac. fl) gqvka

29 bdkbz 3 lekukurj Js.kh,oa gjkred Js.kh (Arithmatic Progression and Harmonic Progressions) 3.. og vuqøe ftlds fdlh nks yxkrkj inksa dk vurj fu;r jgrk gsa mls l- Js- dgrs gsaa mnk- %, 4, 6, 8,... 3, 5, 7, 9, l- Js.kh dk nok in t n ;k l a + (n ) d. tgk a Js.kh dk izfke in] n inksa dh la[;k] d lkozvurj 3.3. l- Js- ds n inksa dk ;ksx S n n a + n d ( ). l- ek/; % ;fn a, b l- Js- ds nks in gsa,oa x mlds chp,d e/; in gs rks x a + b. ;fn a,oa b l- Js- ds nks inksa ds chp n l- ek- x, x,..., x n gksa rks x a + x a + b N ( ( b a ca) ),,... n + n x N a gjkred Js.kh % ;fn fdlh Js.kh ds inksa dk O;qRØe l- Js- esa gksa rks og Js.kh gjkred Js.kh ¼g- Js-½ esa gksxka ` mnk- %, g- Js- ds nosa in dk eku T n. a + ( n ) d gjkred ek/; % ekuk fd a,oa b g- Js- esa gsa,oa H mudk g- ek- gs rks H ab a + b.

30 nks jkf'k;ksa a,oa b ds chp n g- ek H H ( n + ) ab a + ( n ) b H n ab ( n + ) na + b mnkgj.k. ;fn fdlh l- Js- dk poka in q vksj qoka in p gks rks fl) dhft;s fd p + qoka in 'kwu; gksxka gy % ekuk fd lekurj Js<+h dk izfke in a rfkk lkozvurj d gsa rc] pok in a + (p )d q...() rfkk qok in a + (q )d p...() vc leh- () esa ls leh- () dks?kvkus ij] (p )d (q )d q p ( ac n+ c ) ab (p q)d (q p) dbn + a d d dk eku leh- () esa j[kus ij] a + (p )( ) q < a p + q vr% (p + q)ok in a + (p + q )d p + q + (p + q )( ) 0. ;gh fl) djuk FkkA mnkgj.k. ;fn a, b, c lekurj Js<+h rfkk b, c, d gjkred Js<+h esa gksa] rks fl) dhft, fd a b gy % a, b, c lekurj Js<+h esa gsaa. b...(i)

31 b, c, d gjkred Js.kh esa gsaa c (ii) leh- () dk leh- () ls xq.kk djus ij] bc c ( a + c) d b + d c(b + d) (a + c)d cb + cd ad + cd ad bc a / b c / d. ;gh fl) djuk FkkA mnkgj.k 3. la[;k, 6 vksj 0 dk gjkred ek/; fudkfy;sa gy % pw fd H H H F+ 3 bd ab a 6c 0 bd HG I b d K J 0 60 a b+ 0( b 9+ d). mnkgj.k 4. vksj 0 ds e/; nks lekurj ek/; Kkr dhft,a gy %, A, A, 0 lekrj Js<+h esa gsaa 0 + (4 )d 9 3d d 3 vr% A a + d rfkk A a + d mùkj mnkgj.k 5. Js<+h 7, 4,, 8... dk vfure in 0 gsa inksa dh la[;k Kkr dhft,a gy % ;gk a 7, d vksj l 0 ekuk fd inksa dh la[;k n gsa rc] l a + (n ) d

32 (n ) x ( 3) 3(n ) 7 n n 0 vr% vhkh"v inksa dh la[;k 0 gsa mnkgj.k 6. Js<+h dk vfure in 0 gsa inksa dh la[;k Kkr dhft,a gy % 4, 3,, 4,..., gjkred Js<+h esa gsa ,,,..., gjkred Js<+h esa gksaxsa ;gk lekurj Js<+h dk igyk in a 9, lkozvurj d Θ lekurj Js<+h dk nok in a + (n )d lekurj Js<+h dk 8ok in vr% gjkred Js<+h dk 8ok in. 4, 3,, 4,..., (8 9 ) mùkj mnkgj.k 7. b + c a c + a b a + b c,, lekurj Js<+h esa gksa] rks fl) dhft, fd,, Hkh lekurj Js<+h esa gksaxsa a b c a gy % b + c a c + a b a + b c,, l- Js- esa gsaa a b b + c a c + a b a + b c a b c b + c + a c + a + b a + b + c a b c b c c,, Hkh l- Js- esa gksaxsa,, Hkh l- Js- esa gksaxsa vr% izr;sd in esa (a + b + c) ls Hkkx nsus ij] a b c,, Hkh l- Js- esa gksaxsa ;gh fl) djuk FkkA

33 mnkgj.k 8. ;fn a, b, c lekurj Js<+h esa gksa] rks fl) dhft, fd, rfkk lekurj Js<+h esa gksaxsa gy % a, b, c lekurj Js<+h esa gsaa bc ca ab Hkh Hkh lekurj Js<+h esa gksaxsa fd bc ca ab,, Hkh lekurj Js<+h esa gksaxsa ;gh fl) djuk FkkA mnkgj.k 9. ;fn a vksj b dk lekurj ek/; A vksj gjkred ek/; H gks] rks fl) dhft, a a A b A H b H A. H gy % a vksj b dk lekurj ek/; A vksj gjkred ek/; H gsa rc] L.H.S. A a + b H. ab a + b a (( aa ( ab + b )( b a) ca a,, + b) ) ( a + ) ( ) abc ba Habc b b abc H ab a ( b b a)( b + a) + a+ ( a b ) b( a b) ab ab a b a + b a + b a a b a + ab ab (a + b) ( a b)( a + b) ( b a)( b + a) a ab b ab a + b b + a a b

34 R.H.S. ;gh fl) djuk FkkA mnkgj.k 0. ;fn a, b, c gjkred Js<+h esa gksa] rks fl) dhft, fd b lekurj Js<+h esa gksaxsa gy % a, b, c g- Js- esa gsaa,, l- Js- esa gsaa a b c a + b + c a + b + c a + b + c a b c,, l- Js- esa gsaa + b + c c + + a a + + b a b c b + c c + a a + b a b c,, l- Js- esa gsaa T a9 8+ db+ 8d T + A 9 8d 8d ab H a + b + c c + a a + b,, a b c,, l- Js- esa gsaa ;gh fl) djuk FkkA mnkgj.k. ;fn fdlh lekurj Js<+h dk 9ok in 'kwu; gks] rks mlds 9osa rfkk 9osa inksa dk vuqikr Kkr dhft,a gy % ekuk lekurj Js<+h dk izfke in a rfkk lkozvurj d gsa rc T n a + (n )d T 9 a + (9 )d T 9 a + 8d 0...() rfkk T 9 a + (9 )d a + 8d T 9 a + (9 )d a + 8d T 9 T9 a + 8 d a + 8d [leh- () ls] a 8d] 0 d 0d

35 vr% T 9 : T 9 :. mùkj mnkgj.k. ;fn nks la[;kvksa dk lekurj ek/; 7 rfkk mudk xq.kuqy 45 gks] rks mu la[;kvksa dks Kkr dhft,a gy % fn;k gs % 7 a + b 4...() rfkk ab 45 (a b) (a + b) 4ab (a b) (4) (a b) 6 a b 4...() leh- () o () dks gy djus ij] mnkgj.k dhft, fd S 3 3(S S ). a 9, Ta 3 n b n9 + b 5. mùkj T 9 mnkgj.k 3. fdlh l- Js- ds n, n rfkk 3n inksa ds ;ksxqy Øe'k% S, S rfkk S 3 gsaa fl) gy % ekuk l- Js- dk izfke in a rfkk lkozvurj d gsa rc] S {a + (n )d} S {a + (n )d} rfkk S 3 3 n {a + (3n )d} L NM 3(S S ) 3 n N { a + ( n ) d } { a + ( n ) d} O QP [{4a + (4n )d} {a + (n )d}]

36 n [(4a a) + (4n n + )d] 3 [a + (3n )d] S3. ;gh fl) djuk FkkA n mnkgj.k 4. Js<+h ds fdrus inksa dk ;ksx 7 gksxk \ gy % ekuk Js<+h ds n inksa dk ;ksxqy 7 gsa S n [a + (n )d] 7 [ 4 + (n ) ( 4)] 44 n [48 4n + 4] 44 n [5 4n] 36 n [3 n] 3n n 36 n 3n n 9n - 4n n (n 9) 4(n 9) 0 (n 4)(n 9) 0 n 4 ;k n 9. mùkj mnkgj.k 5. rhu la[;k;sa lekurj Js<+h esa gsaa mudk ;ksx 5 rfkk xq.kuqy 0 gsa la[;k;sa Kkr dhft,a gy % ekuk rhu la[;k;sa a d, a, a, + d gsaa rc iz'ukuqlkj] (a d) + a + (a + d) 5 3a 5 a 5 rfkk (a d).a.(a + d) 0 a (a d ) 0 5 (5 d ) 0 5 d 4 d ± n

37 vr% rhu la[;k;sa vfkkzr~ 4, 5, 6 vfkok 6, 5, 4 gsaa mùkj mnkgj.k 6. rhu lekurj Jsf<+;ksa ds n inksa ds ;ksxqy Øe'k% S, S, S 3 gsaa ;fn izr;sd Js<+h dk izfke in rfkk muds lkozvurj Øe'k%,, 3 gksa] rks fl) dhft, fd S + S 3 S. gy % izfke Js<+h ds fy,] a, d S n( n + ) nwljh Js<+h ds fy,] a, d...() S { + (n ) } n S n...() rhljh Js<+h ds fy,] a, d 3 S 3 { + (n ) 3} 5 n 3+ ( 35 + n, 5, ) + ± ) 7 + ( 3... ) n () o () ls S + S 3 n n n n [n + + 3n ] n mnkgj.k 7. ;fn S + S 3 S. fl) gqvka n dk eku Kkr dhft,a gy % n inks rda ;gk a 3, d 5 3 inksa rd inksa rd 7 gks] rks S n [a + (n )d] S n [ 3 + (n ) ]

38 S n [3 + n ] S n n(n + ) S n n + n inksa rd ;gk a 5, d 8 5 3, n 0 S n [a + (n )d] S 0 [ 5 + (0 ) 3] S 0 5 [0 + 7] S S 0 85 fn;k x;k gs % 7 inksa rd inksa rd 0 n3 + 5 n n n + n 7 85 n + n 95 0 n + 37n 35n 95 0 n(n + 37) 35(n + 37) 0 (n 35)(n + 37) 0 n 35. mnkgj.k 8. ;fn,d lekurj Js<+h dk izfke vafre in 50 vksj ;ksxqy 04 gs] rks Js<+h dk lkozvurj Kkr dhft,a gy % fn;k gs % a, l 50, S n 04 ge tkurs gsa fd S n n [a + l]

39 [ + 50] 04 5n n n 8 l a + (n )d 50 + (8 )d 49 7d d 7. lehdj.k () esa j[kus ij] ( 4) a 8 a 9 S n S 3 [a + (n )d] 04 3 n 5 [ 9 + (3 )( 4)] S 3 [9 3 ] S 3 3 [9 6] S 3 3 ( 53) mùkj S mùkj mnkgj.k 9. ;fn fdlh lekurj Js<+h ds n inksa dk ;ksx 3n + 3n gsa Js<+h dk rok in Kkr dhft,a gy % fn;k gs % S n 3n + 3n,d in dk ;ksx S 3() + 3() 6,d in dk ;ksx a S 6 nks inksa dk ;ksx S 3() + 3()

40 nks inks dk ;ksx a + a + d 8 a + d 8 (6) + d 8 d 6 T n a + (n )d T r 6 + (r ) 6 T r 6 + 6r 6 T r 6r. mùkj mnkgj.k 0. rhu lekurj Jsf<+;ksa ds n inksa ds ;ksxqy Øe'k% S, S, S 3 gsaa ;fn izr;sd Js<+h dk izfke in rfkk mldk lkozvarj Øe'k%,, 3 gksa] rks fl) dhft, fd S + S 3 S. gy % S n inksa rd S L N [ + (n ) ], n( n + ) n Θ S n [ a + n d] M ( ) a a,, d O Q P S n [ + n ] S...() S n inksa rd S [ x + (n ) ], S n[ + n ] S n...()

41 S n inksa rd S 3 n [ + (n ) x 3] S 3 [ + 3n 3] S 3 [3n ]...(3) S + S 3 (3n ) [n + + 3n ] [4n] S + S 3 n S + S 3 S, [leh- () ls] ( Ta np a( n+ ( + qp T( p q ) ) d n+ qa + ( ) dq ) d + ;gh fl) djuk FkkA mnkgj.k. fdlh lekurj Js<+h ds posa qosa inksa dk ek/; rosa vksj sosa inksa ds ek/; ds cjkcj gsa] rks fl) dhft, fd p + q r + s. gy % ekuk lekurj Js<+h dk izfke in a rfkk lkozvarj d gsa rc T p vksj T q dk lekurj ek/; T p a + (p )d T q a + (q )d T p vksj T q dk lekurj ek/; a +...()

42 T r a + (r )d T s a + (s )d T r vksj T s dk lekurj ek/; T r vksj T s dk lekurj ek/; fn;k x;k gs % a + ( r + s ) d...() a + a + a T ( pr r ap ( r + + sq st ( s r q ) ) + sa + ) ( ds ) d d d p + q r + s.

43 bdkbz 4 xq.kksùkj Js<+h,oa fo'ks"k Jsf<+;k og Js.kh ftlds fdlh Hkh nks yxkrkj inksa dk vuqikr leku gksrk gs xq.kksùkj Js<+h dgykrk gsa bls r ls O;Dr djrs gsaa ;fn izfke in a gks rks Js<+h xq.kksùkj Js.kh dk nok in xq.kksùkj Js.kh dk var ls nok in xq.kksùkj Js<+h ds n inksa dk ;ksxqy a, ar, ar, ar 3,... t n ar n S n a r n ( ) r tgk r <.,oa S n ;fn Js<+h ¼xq.kksÙkj½ dk vfure in l fn;s gksa rks tgk r >, n n n lr a ( a r l ar r ) ar ar r r n r r r S n tgk r <,,oa S n tgk r >, vuur xq.kksùkj Js<+h dk ;ksxqy S n tgk r < tc n rks r n 0. S n

44 xq.kksùkj ek/; % (i) ;fn a, b dk xq- ek- la- gs rks G ab G ab (ii) ;fn a,oa b ds chp n xq- ek- G, G,..., G n gksa rks fo'ks"k Js<+h % G a, G a F n b a HG I K J+,... Gn a n F n b a HG I K J+. () izfke n izk r la[;kvksa dk ;ksxqy S n( n +) S n () izfke n izk r la[;kvksa ds oxks± dk ;ksxqy S n n L F n (3) izfke n izk r la[;kvksa ds?kuksa dk ;ksxqy a G b HG I nn( n( n+ + ) ) NM G6 b ak J+ S n n 3 (n + )(n + ). O QP (4) lekukrjh; xq.kksùkj Js.kh ds n inks a dk ;ksxqy (5) lekurjh; xq- Js<+h ds vuar inksa dk ;ksxqy S a dr r + ( r). n a dr r a n d r r + ( ) [ + ( ) ] ( r) r mnkgj.k. ;fn a x b y c z vksj a, b, c xq.kksùkj Js<+h esa gksa] rks fl) dhft, fd x, y, z gjkred Js<+h esa gksaxsa gy % ekuk fd a x b y c z k a k /x, b k /y, c k /z Θ a, b, c xq.kksùkj Js<h esa gsa] b ac n

45 (k /y ) k /x.k /z k /y k /x+/z + x z vr%,, lekurj Js<+h esa gsaa x y z x, y, z gjkred Js<+h esa gsa mnkgj.k. fueufyf[kr Js<+h dk ;ksxqy Kkr dhft, % + x + 3x + 4x (x < ). gy % ekuk S + x + 3x + 4x vc] S.x x + x + 3x S ( x) + x + x + x ;gh fl) djuk FkkA S ( x) S 4 x a ( y r x). mùkj mnkgj.k 3. Js.kh ds vuur inksa dk ;ksxqy Kkr dhft,a gy % ;gk a vksj r vhkh"v ;ksxqy vr% Js<+h ds vuur inksa dk ;ksx gsa mùkj mnkgj.k 4. Js<+h 4,, dk dksu&lk in 8 gs \

46 gy % ;gk a 4, r, T n 8 T n ar n 4 F HG I K J n F HG I K J n F 7 HG I K J F 7 9 4HG I K J F H GI K J n 9 n 0 inksa rd vr% nh gqbz Js<+h dk 0ok in 8 gsa mùkj mnkgj.k 5. gy % ekuk Js<+h dk ;ksxqy S gsa rc] Θ n F HG I 3 F 8 KHG I ab F 4 n J+ nk J+ n HG I K J F S 3 F F n n n HG I KHG I J+ KHG I J+ K J+ ( n inksa rd½ L NM n n + n n { ( ) ( ) } n + n n ( ) ( ). inksa rd ;ksxqy Kkr dhft,a HG I K J+... n inksa rd F n n n n HG O QP mùkj...n I K J mnkgj.k 6. gy % rfkk 64 ds e/; xq.kksùkj ek/; Kkr dhft,a G

47 G fd x G 6 mùkj G 4. mùkj mnkgj.k 7. ;fn y x + x + x rfkk x /kukred,oa bdkbz ls de gks rks fl) dhft, gy % y x + x + x y y xy x x x x + xy y x ( + y) y x ;gh fl) djuk FkkA y + x mnkgj.k + 4 y x Js<+h + 3x + 5x 64 x + 7x ( ds vuur x) x. inksa dk ;ksxqy Kkr dhft,a gy % ekuk fd S + 3x + 5 x + 7 x () rfkk Sx x + 3x + 5x () leh- () esa ls leh- () dks?kvkus ij] S Sx + x + x + x S ( x) + (x + x + x ) S ( x) + F x HG I xk J S ( x) S mùkj

48 fd mnkgj.k 9. ;fn a, b, c lekurj Js<+h esa gsa rfkk x, y, z xq.kksùkj Js<+h esa gksa] rks fl) dhft, gy % Θ a, b, c lekurj Js<+h esa gsa] x b c y c a z a b. b a + c...() x, y, z xq.kksùkj Js<+h esa gsa] y zx...() x b c y c a z a b x b c..z a b, [leh- () ls] b c + c a c a a b x. z. z x 0.z 0 Θ ( xzx ) b b a+ b c c c c + + aa c a aa a + + cc b aa b + c c a b+ c a.. z. z [leh- () ls] x b c y c a z a b. ;gh fl) djuk FkkA mnkgj.k 0. ;fn x < rks Js<+h 5x + 9x 3x rks vuur rd tksfm+,a gy % ekuk fd S 5x + 9x 3x () nksuksa i{kksa esa x dk xq.kk djus ij] leh- () esa ls leh- () dks?kvkus ij] xs x + 5x 9x 3 + 3x () S 5x + 9x 3x xs x + 5x 9x 3 + 3x S ( + x) 4x + 4x 4x

49 x [ x + x... ] 4x vr% S + x 4 x + x 3 + x x mùkj mnkgj.k. Js<+h dk n inksa rd ;ksxqy Kkr dhft,a gy % S n inksa rd 6 9 [ n inksa rd] [(0 ) + (0 ) + (0 3 ) +... n inksa rd] [( n inksa rd½ ( n inksa rd½] L NM O QP QP 6 3x NM O n 0 QP + ( 9 + x ). n a 0( + n [ ) n9 n] 7 3 r 09. mùkj mnkgj.k. fdlh xq.kksùkj Js<+h ds vuur inksa dk ;ksxqy 5 rfkk muds oxks± dk ;ksxqy 45 gsa bl Js<+h dks Kkr dhft,a gy % ekuk fd xq.kksùkj Js<+h dk izfke in a rfkk inkuqikr r gs tcfd r <. rc] Js<+h ds vuar inksa dk ;ksxqy a r iz'ukuqlkj] 5...() xq.kksùkj Js<+h ds inksa dk oxz djus ij izkir xq.kksùkj Js<+h gs % a, a r, a r 4, a r 6,...

50 bl Js<+h ds vuur inksa dk ;ksxqy iz'ukuqlkj] 45...() leh- () ds nksuksa i{kksa dk oxz djds izkir ifj.kke esa leh- () ls Hkkx nsus ij] r 5 5r 6r 4 r vc r dk eku leh- () esa j[kus ij] a 5 + a ar r 3 F L HG I (. 3 K J n 3NM 0 rr) 00 a 000 3a 5 a 5 O QP vr% vhkh"v Js<+h 5, 5 3, 5 mnkgj.k 3. n inksa dk ;ksx dhft, gy % vhkh"v ;ksxqy vfkkzr~ 5, n inksa rd 3 [ n inksa rd], gqbza mùkj 3 9 [ n inksa rd] inksa rd

51 L NM L n 3 NM ( n inksa rd½ F 0 H 0 0 n I O K QP inksa rd mùkj mnkgj.k 4. xq.kksùkj Js<+h ds rhu Øekxr inksa dk xq.kuqy 6 rfkk ;ksxqy 9 gsa rhuksa in Kkr dhft,a L F O F a 6 n3 I F K J+ HG I F NM + F 00K 0 0 J+ + 3 HG +...n 9 n n NM HG I 0 J n 7 HG I 3 r 0 K J. QP 000.a.ar 6 gy % ekuk fd xq.kksùkj Js<+h ds rhu Øekxr in a r, a, ar gsaa rc iz'ukuqlkj] xq.kuqy] a 3 6 a 6 I HG I O K J OJ+...n inksa Q P rd Q P + a + ar r 9 + 6r r 3r 6r 3r r 9r 4r (3r )(r 3) 0 r ;k

52 vr% la[;k,, 6, 6 ;k , 6, 6 vhkh"v in 4, 6, 9 gsa ;k 9, 6, 4 gsaa mùkj mnkgj.k 5. ;fn fdlh xq.kksùkj Js<+h ds n inksa dk ;ksxqy S, n inksa dk xq.kuqy P rfkk inksa ds O;qRØeksa dk n inksa rd ;ksxqy R gks rks fl) dhft, P gy % ekuk fd xq.kksùkj Js<+h dk izfke in a rfkk lkozvuqikr r gsa rc] S a r n ( ) r rfkk P a.ar.ar... ar n gsaa...() a n r (n ) a n r n(n )/ P a n r n(n )...() inksa ds O;qRØeksa l izkir xq.kksùkj Js<+h gs % R F n 3 S6 HG I 3 R / 3 K J n,,.,..., a ar ar ar n Fn ah I n a( r ) ar ( r ) n r n r r F r n K a r r r r HG I K J R vc leh- () o (3) ls] S R vr% leh- () o (4) ls fl) gksrk gs fd n r n ar ( r ) a r n...(3) n FS HG I RK J an r n(n )...(4) P n FS HG I RK J. ;gh fl) djuk FkkA

53 bdkbz 5 lkjf.kd (Determinant) lkjf.kd ds izxq.k (Properties of Determinant) () ;fn fdlh lkjf.kd dh iafdr;ksa dks LrEHkksa esa rfkk LrEHkksa dks iafdr;ksa esa cny fn;k tk;s rks lkjf.kd dk eku vifjofrzr jgrk gsa () ;fn lkjf.kd dh fdugha nks iafdr;ksa ;k LrEHkksa dks vkil esa cny fn;k tk;s rks lkjf.kd dk la[;kred eku ogh jgrk gs ijurq fpug cny tkrk gsa (3) ;fn fdlh lkjf.kd dh dksbz nks iafdr;k ;k LrEHk leku (Identical) gksa rks lkjf.kd dk eku 'kwu; gksrk gsa (4) ;fn fdlh lkjf.kd ds fdlh iafdr ;k LrEHk ds lhkh vo;o 'kwu; gksa rks lkjf.kd dk eku 'kwu; gksrk gsa (5) ;fn fdlh lkjf.kd esa fdlh iafr ;k LrEHk dk izr;sd vo;o ¼?kVd½ nks jkf'k;ksa dk ;ksx ;k vurj gks rks lkjf.kd dks mlh dksfv (order) ds nks lkjf.kdksa ds ;ksx ;k vurj ds :i esa fy[kk tk ldrk gsa ;Fkk] a + α b + β c + g a b c a b c a b c a b c a b c α β g a b c a b c (6) ;fn fdlh lkjf.kd dh fdlh iafdr ;k LrEHk ds lhkh?kvdksa ¼vo;oksa½ dks fdlh la[;k ls xq.kk dj fn;k tk;s rks lkjf.kd ds eku esa Hkh ml la[;k ls xq.kk gks tkrk gsa (7) ;fn fdlh lkjf.kd dh fdlh iafdr ¼vFkok LrEHk½ ds izr;sd vo;o esa fdlh nwljh iafdr ¼vFkok LrEHk½ ds laxr vo;oksa dks fdlh vpj jkf'k ls xq.kk djds tksm+ ;k?kvk fn;k tk;s rks lkjf.kd ds eku esa dksbz ifjorzu ugha gksrk gsa (8) ;fn fdlh lkjf.kd D ds pj x okys vo;o x esa cgqin gksa vksj ;fn x ds LFkku ij dksbz jkf'k a j[kus ls lkjf.kd dk eku 'kwu; gks tkrk gs rks (x a), lkjf.kd D dk,d xq.ku[k.m gksxka (9) fdlh lkjf.kd esa fdlh iafdr ¼vFkok LrEHk½ ds vo;o vksj fdlh vu; iafdr ¼vFkok LrEHk½ ds lg[k.m ds xq.kuqy dk ;ksxqy 'kwu; gksrk gsa

54 mnkgj.k. fn[kkvks fd gy % ekuk D lafø;k R R rfkk R 3 R ls D ¼pw fd R, R 3 lozle gs½ bfrfl)e~ mnkgj.k. fn[kkb;s fd w w w w w w 0 ¼tgk w bdkbz dk vf/kdfyir?kuewy gs½ gy % lafø;k R + R + R 3 ls] L.H.S. + w + w w w w + w + w w + w + w 0 w w 0 w 0 w ¼pw fd + w + w 0 tcfd w bdkbz dk,d?kuewy gs½ 0 ¼pw fd C ds lhkh?kvd 'kwu; gsa C izfke LrEHk dks izdv djrk gs½ R.H.S. bfrfl)e~

55 mnkgj.k 3. fl) djks fd x + y xy gy % lafø;k C C 3 ls] L.H.S x y + y izfke LrEHk C ds inksa esa folrkj ysus ij] ( y) + x mnkgj.k 4. fn[kkb;s fd ( y) ( x) bfrfl)e~ xy R.H.S. bfrfl)e~ b + c c + a a + b q + r r + p p + q y + z z + x x + y a b c p q r x y z gy % ekuk] D b + c c + a a + b q + r r + p p + q y + z z + x x + y b c + a a + b c c + a a + b rks D q r + p p + q + r r + p p + q y z + x x + y z z + x x + y b c a + b b a a + b c c a + b c a a + b ;k] D q r p + q + q p p + q + r r p + q + r p p + q y z x + y y x x + y z z x + y z x x + y

56 vc pw fd r`rh; lkjf.kd esa C vksj C lozle gsa vr% bl lkjf.kd dk eku 'kwu; gksxka b c a b c b b a a b a b c a a c a b vr% D q r p y z x + q r q + y z y q p p y x x + q p q y x y + r p p + z x x r p q z x y b c a c a b ;k] D q r p y z x r p q z x y a b c a b c ;k] D p q r x y z + p q r x y z ;k] D a b c p q r x y z bfrfl)e~ mnkgj.k 5. fl) djks fd a b c a a b b c a b c c c a b (a + b + c) 3. gy % ekuk] D a b c a a b b c a b c c c a b lafø;k C C o C C 3 ls] D ( a + b + c) 0 a a + b + c ( a + b + c) b 0 a + b + c c a b C o C ls (a + b + c) lozfu"b ysus ij] D (a + b + c) 0 a b 0 c a b

57 lafø;k R + R + R 3 ls D (a + b + c) 0 0 a + b + c b 0 c a b izfke iafdr ds inksa esa folrkj ysus ij] D (a + b + c) + (a + b + c) 0 ;k] D (a + b + c) ( + 0) ;k] D (a + b + c) 3 bfrfl)e~ mnkgj.k 6. fl) djks fd a bc ac + c a + ab b ac ab b + bc c 4a b c. gy % ekuk] D a bc ac + c a + ab b ac ab b + bc c C, C o C 3 esa ls Øe'k% a, b o c lozfu"b ysus ij] D abc a c a + c a + b b a b b + c c R ds inksa esa folrkj djus ij] D abc [a(bc ab ac) c(ac + bc ab) + (a + c)(ab + ac + bc)] ;k] D abc (4abc) ;k] D 4a b c. bfrfl)e~

58 cgqfodyih; iz'u lgh fodyi pqudj fyf[k;s % iz'u. fueufyf[kr esa ls fdl lkjf.kd dk eku 'kwu; gs \ (a) a b c a b c a b c (b) a b c a b c a b c (c) a c b b a a b c a (d) c c a b a b a c c 0 iz'u. lkjf.kd dk eku gs % (a) 4 (b) 0 (c) 3 (d) iz'u dk eku gs % (a) 0 (b) (c) (d) buesa ls dksbz ugha iz'u 4. ;fn 0 x a x b x + a 0 x c x + b x + c 0 0 gks rks x dk eku gs % (a) 0 (b) (c) (d) 3 iz'u 5. + x + y cjkcj gs % (a) + x + y (b) 3 + x + y (c) x + y (d) xy iz'u 6. lehdj.k x + a b c b x + c a c a x + b 0 dk,d ewy gs % (a) (a + b) (b) (b + c) (c) a (d) (a + b + c) mùkj %. (a). (b) 3. (b) 4. (a) 5. (d) 6. (d).

59 bdkbz 6 vko;wg (Matrix) Nkksa dks lozizfke fofhkuu izdkj ds vko;wgksa dks Li"V djsa fqj vyx&vyx izdkj ds mnkgj.k nsaa vk;rkdkj vko;wg (Rectangular Matrix) : og vko;wg ftlesa iafdr;ksa dh la[;k LrEHkksa dh la[;k ls fhké gksrh gsa oxz vko;wg (Square Matrix) : og vko;wg ftlesa iafdr;ksa dh la[;k] LrEHkksa dh la[;k ds cjkcj gksrh gsa iafdr vko;wg (Row Matrix) : og vko;wg ftlesa dsoy,d iafdr gksrh gsa LrEHk vko;wg (Column Matrix) : og vko;wg ftlesa dsoy,d LreHk gksrk gsa 'kwu; vko;wg (Zero Matrix) : og vko;wg ftldk izr;sd vo;o 'kwu; gksrk gsa bls 0 ;k 0 mn ls fu:fir djrs gsaa gksaa vfn'k vko;wg (Diagonal Matrix) : og fod.kz vko;wg ftlds lhkh fod.kz vo;o cjkcj,dd ;k ekd vko;wg (Unit Matrix) : og 0 vfn'k 3vkO;wg ftldk izr;sd fod.kz vo;o gksa bls I ;k I n ls fu:fir djrs gsaa fhkqth; vko;wg (Triangular Matrix) : og oxz vko;wg ftlesa eq[; fod.kz ds Åij ds ;k uhps ds lhkh vo;o 'kwu; gksaa vifj fhkqth; vko;wg (Upper Triangular Matrix) : og oxz vko;wg ftlesa eq[; fod.kz ds uhps ds lhkh vo;o 'kwu; gksaa fueu fhkqth; vko;wg (Lower Triangular Matrix) : og oxz vko;wg ftlesa eq[; fod.kz ds Åij ds lhkh vo;o 'kwu; gksaa,dy vo;o vko;wg (Single Element Matrix) : og vko;wg ftlesa,d iafdr vksj,d LrEHk gksa leku vko;wg (Equal Matrix) : ;fn nks vko;wgksa ds Øe (order) rfkk laxr vo;o leku gksa rks os leku vko;wg dgykrs gsaa mnkgj.k. ;fn A L 3 NM O 0 5Q PrFkk B L NM 6 O Q P gks rks 3A 4B dks Kkr djksa

60 gy % 3A 4B mnkgj.k. ;fn] L NM O Q PmÙkj A rks A dk ;kst; izfrykse Kkr djksa gy % A dk ;kst; izfrykse A ( ) A ( ) L 3 ( 4) 0 3 Q P NM 0 0 O P ( NM 40 ) QP L O NM 0 0 QP mùkj mnkgj.k 3. vko;wg A rfkk B dks Kkr djks ;fn A + B L NM gy % nksuksa vko;wgksa dks tksm+us ij 0 O P Q P rfkk A B

61 A ;k] A ;k] A ;k] A mùkj (i) nksuksa vko;wgksa dks?kvkus ij] B L M NM O P O P O QP QP QP M P P ;k] B ;k] B ;k] B mùkj mùkj (ii)

62 mnkgj.k 4. ;fn] A, B, C rks fl) djks fd A (BC) (AB) C gy % AB vc (AB) C L NM O L O LO O O QP Q PN M QP O NM QP + +. P. 3 M M0 P P NM 0 P. + P QP QP O QP...() iqu% BC

63 rfkk A (BC) () vksj () ls Li"V gs fd A (BC) (AB) C L NM...() O O QP QP bfrfl)e~ O QP mnkgj.k 5. ;fn A rfkk I r`rh; Øe dk bdkbz vko;wg gks rks fl) djks fd A 3A + 9I gy % A L NM O QP

64 vc 3A 3 L NM O QP vksj 9I 9 L NM O O QP QP O QP vr% A 3A + 9I L NM O P Q P + L NM 6 M O QP bfrfl)e~

65 mnkgj.k 6. ;fn] A L 0 NM α tan α tan 0 I + A (I A) O QP Lcos NM sin vksj I,d bdkbz vko;wg dks rks fl) dhft;s fd α α sinα cosα O QP gy % I + A + L NM α tan α tan O QP...() vc] I A L 0 NM O 0 Q P L0 L NM α tan O cos α0 O 0 0 Q P sinα α α tan α 0 NM cosα + tan.sin α sinα + tan.cosα sinα cosα QP tan α O tan α α tan.cosα + P QP sinα tan.sinα + cosα NM QP O QP vr% (I A).

66 L NM α α α cos α.cos + sin.sinα sin α.cos + sin.cosα α α cos cos α α α α sin cosα + sin α.cos sin sinα + cos α.cos α α cos cos O QP L NM α tan lehdj.k () vksj () ls] I + A (I A) mnkgj.k 7. ;fn A O P α tan L F4 cos α α 5 QP α cos 0 5 F cosα sinαosin α α sinα cosα QP M α cos L NM H I K H I K NM rks A' Kkr djksa F H I O K H I K QP sin α α α cos F cos α α α cos...() bfrfl)e~ gy % A' ;k] A' L NM O QP mùkj

67 mnkgj.k 8. ;fn A rfkk B rks fl) djks fd (AB)' B'A'. gy % AB (AB)' NM L NM iqu% A' rfkk B' O O Q QP O QP O QP vr% B'A'...()

68 bl izdkj () vksj () ls] (AB)' B'A' bfrfl)e~ mnkgj.k 9. ;fn] A rks adj A Kkr djksa gy % ;gk ] A a () dk lg[k.m ( ) + A a (0) dk lg[k.m ( ) A 3 a 3 () dk lg[k.m ( ) +3 3 L NM O QP A a () dk lg[k.m ( ) A a () dk lg[k.m ( ) A 3 a 3 (0) dk lg[k.m ( ) A 3 a 3 (3) dk lg[k.m ( ) A 3 a 3 () dk lg[k.m ( ) A 33 a 33 () dk lg[k.m ( ) 3+3 0

69 adj A L NM A A A 3 A A A 3 A A A O QP ;k] adj A mùkj mnkgj.k 0. Adj Kkr dhft;sa gy % Adj mùkj mnkgj.k. vko;wg gy % ekuk] A rks A dk O;qRØe Kkr djks ;fn a + b + c + d. L NM NM AdjA cos a + αib ib 4 sin c + α id c sin A c sin + αid id α 5 cos a + 4 αib O QP QP vfkkzr~ A 0. vr% A dk vflrro gsa vc Adj A (a + ib)(a ib) (c + id)( c + id) a + b + c + d ¼iz'ukuqlkj½ A Adj A (Θ A )

70 vr% A mùkj mnkgj.k. vko;wg dk O;qRØe Kkr dhft;sa gy % ekuk A rks A ;k] A ¼lafØ;k C C vksj C 3C ls½ ;k] A ;k] A 3 + O M L O NM PQP NM O Q P QP a4 0 ib c id c id a + ib MNM O 3Q P P (R ds inksa es folrkj ysus ij½ vr% A...() vc] A A 3 A 3 A 3

71 A 3 A 3 A 3 A 3 A 33 0 Adj A ;k] Adj A vc A adja A ;k] A mnkgj.k 3. fl) djks fd L NM M L N cos A θ O 4Q P 3A sin A θ A3 A 3 A NM tan sinθ cosθ N θ MA tan A 0 A L NM O P Q O QP QP O QP P mùkj L NM θ tan θ tan O QP

72 L NM gy % cos θ sinθ sin θ cosθ O QP vc ekuk L NM θ tan L NM θ tan θ tan θ tan O QP O QP A rks mijksdr ifj.kke dks fueukuqlkj fy[k ldrs gsa % A NM L NM QP O P Q cosθ sin θ θ θ cos + sin θ.tan cosθ tan sinθ sinθ cosθ θ θ θ θ sin tan cos θ.tan tan P sinθ tan + cosθ O QP ;k] L NM cosθ sinθ sinθ cosθ O QP AA A ;k] ;k] L NM cosθ sinθ L NM cosθ sinθ O sinθ cosθ QP I sinθ cosθ O QP L NM θ tan L NM θ tan θ tan O QP θ tan O QP bfrfl)e~

73 mnkgj.k 4. fueu lehdj.k fudk; dks esfvªdl fof/k ls gy djks % x + y + z 4 x y + z 5 x + 3y z. gy % fn;s x;s lehdj.k fudk; dks fueukuqlkj fy[k ldrs gsa % tgk AX B...() L NM O P Q 3 P, X, B...() vc A ;k] A ( 3) ( 3) + ( + ) 9 A 0 vksj Adj A x 4 y O 5 L0 / O 9 93 / NM z 5Q P / 93 QP 3 O QP A AdjA A 9...(3) (), () o (3) ls vhkh"v gy X A B ;k] Lx y M zp O NQ L4 5 M P O NQ vr% x, y vksj z. mùkj

74 cgq fodyih; iz'u lgh fodyi pqudj fyf[k;s iz'u.,d vko;wg A [a ij ] m n dks LreHk vko;wg dgsaxs ;fn (a) m (b) n (c) m n (d) m > n iz'u. ;fn A (a) iz'u 3. ;fn A L rks A dk eku gs % (b) 0 NM O 0 Q P gks rks A dk eku gksxk % L NM O Q P (c) (d) (a) (b) (c) (d) M L iz'u 4. ;fn A [ 3] rfkk B (a) L NM M O QP O Q P O L O QP NM QP NM 4 0 O Q P NM (b) rks AB dk eku gksxk % (c) [ 4] (d) iz'u 5. vko;wg L NM O P Q P gs % (a) 'kwu; vko;wg (b) bdkbz vko;wg (c) LrEHk vko;wg (d) buesa ls dksbz ugha

75 iz'u ;fn vko;wg A, B, C esa A dk Øe m n gks] C dk Øe m l gks rfkk A B C vfkkzr~ AB C gks] rks vko;wg B dk Øe gksxk % (a) m l (b) n l (c) m n (d) l n iz'u 7. vko;wg [ 0] fueu izdkj dk gs % (a) LrEHk vko;wg (b) fod.kz vko;wg (c) iafdr vko;wg (d) ekd vko;wg iz'u 8. dk eku gksxk % (a) (b) (c) (d) buesa ls dksbz ugha iz'u 9. dk eku gksxk % (a) (b) (c) (d) buesa ls dksbz ugha iz'u 0. vko;wg dk Øe gs % L NM O Q P O Q P P+L 0 OO N MO 0Q P Q 3Q P 0 0 NM 3 (a) 0 x (b) 0 x 3 (c) 3 x (d) x 3 mùkj. (b),. (b), 3. (a), 4. (c), 5. (b), 6. (a), 7. (c), 8. (d), 9. (c), 0. (d)a lgh fodyi pqudj fyf[k;s % iz'u. ;fn A' A rks A gksxk % (a) lefer (b) fo"ke lefer (c) ekd (d) 'kwu; iz'u. ;fn A dk ifjorz A' gks rks A' dk ifjorz gksxk % (a) A' (b) A (c) adj A (d) buesa ls dksbz ugha iz'u 3. ;fn A, B vksj C rhuksa n n Øe ds vko;wg gksa rks (ABC) dk ifjorz gksxk % (a) A'B'C (b) C'B'A' (c) B'C'A' (d) B'A'C'

76 iz'u 4. ;fn] A, B rfkk C gks rks vifjhkkf"kr O;atd gksxk % (a) A + 3B C (b) CC' (c) (AB)' C (d) AB iz'u 5.,d fo"ke lefer vko;wg esa fod.kz vo;o gksrs gsa % (a) lhkh (b) bdkbz (c) vfuf'pr (d) buesa ls dksbz ugha iz'u 6. ;fn AA' AA' I gks rks vko;wg dgykrk gs % (a) 'kwu;hkkoh (b) vurozfyr (c) oxzle (d) ykfecd iz'u 7. ;fn A iafdr vko;wg gks rks A' gksxk % (a) iafdr vko;wg (b) LrEHk vko;wg (c) lefer vko;wg (d) oxz vko;wg mùkj. (a),. (b), 3. (b), 4. (a), 5. (c), 6. (d), 7. (b) lgh fodyi pqudj fyf[k;s % iz'u 8. ;fn vko;wg O L d cb 3 c NM QP b a P bc NM b 3QP O c a Q P kp P 5Q foy{k.k vko;wg gks rks k dk eku gs % iz'u 9. ;fn A (a) (b) (c) (d) La b c d NM O Q P rks Adj A gksxk % (a) (b) (c) (d) iz'u 0. ;fn A,d r`rh; Øe ds oxz vko;wg A ds laxr lkjf.kd dks iznf'kzr djrk gs rks [ A] cjkcj gksxk % (a) 8 A (b) 8 A (c) A (d) buesa ls dksbz ugha

77 iz'u. ;fn A,d n n Øe dk oxz vko;wg gks rks Adj A dk eku gs % (a) A n (b) A n (c) A r.i (d) A n I iz'u. vko;wg gs % (a) vo;qrøe.kh; (b) O;qRØe.kh; (c) 'kwu; (d) iafdr vko;wg iz'u 3.,d 'kwu; oxz vko;wg dk lg[k.mt gksrk gs % (a) 'kwu; vko;wg (b) bdkbz vko;wg (c) fod.kz vko;wg (d) iafdr vko;wg mùkj 8. (c), 9. (d), 0. (a),. (a),. (b), 3. (a)a lgh fodyi pqudj fyf[k;s % iz'u 4. iz'u 5. x ds fdl eku ds fy, vko;wg dk O;qRØe ugha gksxk % L NM NM O Q P O Q P 0 0 QP (a) 6 (b) (c) 3x 3 06 (d) 3 M x 0P vko;wg dk O;qRØe gs % (a) (b) (c) (d) iz'u 6. fcunq (5, 0) dk X-v{k esa ijkorzu gksxk % (a) (b) (c) (d) iz'u 7. ewy fcunq ds ifjr% 90 ij?kw.kzu esfvªdl gs % (a) (b) (c) (d) mùkj % 4. (d), 5. (c), 6. (c), 7. (b)a

78 bdkbz 7 fcunqvksa ds dkùkhz; funsz'kkad funsz'k % f'k{kd] Nkksa dks dkrhz; funsz'kkad ds fofhkuu lwksa dks layxu mnkgj.k }kjk Li"V djk;saa. nks fcunqvksa P (x y ), Q (x y ) ds chp dh nwjh PQ. ewy fcunq ls (x, y ) dh nwjh ( x 0) ( y 0) 3. rhu fcunq (x y ) (x y ) (x 3 y 3 ) lejs[k gksaxs ;fn + x + y x x x y y y ;k] [x (y y 3 ) + x (y 3 y ) + x 3 (y y )] ;fn A (x ( x x + ( y y) y ),oa B (x y ) dks feykus okyh ( mx my js[kk 3+, nx ny 3dks ) fcunq P (x, y), m : n esa vur% fohkdr m + n djrk gs rks x, y ;fn cká fohkdr djrk gs rks x, y mnkgj.k. fl) dhft, fd fcunq (, ), (, ) rfkk ds 'kh"kz gsaa,d leckgq fhkqt gy % eku yks fd A (, ), B (, ) rfkk C ( 3, 3 ) fn, gq, fcunq gsaa AB [ ( )] + [ ( )] ( + ) + ( + )

79 A (, ) B (, ) BC [ ( 3 )] + [ C ] ( ) + ( + ) CA ( ) + ( ) ( + ) + ( ) 3 + vr% AB BC CA vfkkzr~ AB BC CA ( 3 3 +, 3 + ) ABC leckgq gsa mnkgj.k.,d oxz ds nks leeq[k 'kh"kz Øe'k% (4, 3) rfkk (0, 3) gsa] vu; 'kh"kks± dks Kkr dhft,a gy % eku yks A (0, 3) rfkk C (4, 3) oxz ABCD ds leeq[k 'kh"kz gsa rfkk B ds funsz'kkad (x, y) gsaa B (x, y) A (0, 3) C (4, 3) D

80 AB BC AB BC (x 0) + (y 3) (4 x) + (3 y) x + y 6y x + x + 9 6y + y 8x 6 x ledks.k ABC esa] AC AB + BC (4 0) + (3 3) (x 0) + (y 3) + (4 x) + (3 y) ;k x + y 6y x + x + 9 6y + y ;k y 6y (x izfrlfkkfir djus ij½ ;k (y 5) (y ) 0 y ;k y 5. vu; 'kh"kks± ds funsz'kkad (, ) rfkk (, 5). mnkgj.k 3.,d fhkqt ds 'kh"kz Øe'k% (4, 5 5), (5, 6) rfkk (3, ) gsa( fhkqt dk {ksqy Kkr dhft,a gy % lw [y (x x 3 ) + y (x 3 x ) + y 3 (x x )] ;gk ij x 4, y 5, x 5, ;s eku lw esa izfrlfkkfir djus ij] y 6, x 3 3, y 3 [ 5 (5 3) 6 (3 4) + (4 5)] [ ] vhkh"v {ksqy oxz bdkbz