10th Grade Mathematics Textbook PDF

vè;k; 1 la[;k i¼fr 1.1 Hkwfedk fiNyh d{kkvksa esa] vki la[;k js[kk osQ ckjs esa i 0 gSA vuqPNsn 1-2 esa] geus ;g ns[kk gS fd fdl izdkj la[;k js[kk ij n dks] tgk¡ n ,d /ukRed iw.kk±d gS] fu:fir fd;k tkrk gSA vc ge ;g fn[kk,¡xs...

vè;k; 1 la[;k i¼fr 1.1 Hkwfedk fiNyh d{kkvksa esa] vki la[;k js[kk osQ ckjs esa i 0 gSA vuqPNsn 1-2 esa] geus ;g ns[kk gS fd fdl izdkj la[;k js[kk ij n dks] tgk¡ n ,d /ukRed iw.kk±d gS] fu:fir fd;k tkrk gSA vc ge ;g fn[kk,¡xs fd fdl izdkj x dks] tgk¡ x ,d nh gqbZ /ukRed okLrfod la[;k gS] T;kferh; (geometrically) :i ls Kkr fd;k tkrk gSA mnkgj.k ds fy,] vkb, ge bls x = 3.5 ds fy, izkIr djsaA vFkkZr~ ge 3.5 dks vko`Qfr 1.11 T;kehrh; :i ls izkIr djsaxsA ,d nh gqbZ js[kk ij ,d fLFkj fcUnq A ls 3.5 ,dd dh nwjh ij fpÉ yxkus ij ,d ,slk fcUnq B izkIr gksrk gS] ftlls fd AB = 3.5 ,dd (nsf[k, vkÑfr 1.11)A B ls 1 ,dd dh nwjh ij fpÉ yxkb, vkSj bl u, fcUnq dks C eku yhft,A AC dk eè;&fcUnq Kkr 2024-25 20 xf.kr dhft, vkSj ml fcanq dks O eku yhft,A O dks osQUnz vkSj OC dks f=kT;k ekudj ,d v/Zo`Ùk cukb,A AC ij yac ,d ,slh js[kk [khafp, tks B ls gksdj tkrh gks vkSj v/Zo`Ùk dks D ij dkVrh gksA rc BD = 3.5 gSA vf/d O;kid :i esa] x dk eku Kkr djus osQ fy,] tgk¡ x ,d /ukRed okLrfod la[;k gS] ,d ,slk fcanq B ysrs gSa] ftlls fd AB = x ,dd gks vkSj tSlk fd vko`Qfr 1-16 esa fn[kk;k x;k gS] ,d ,slk fcanq C yhft, ftlls fd BC = 1 ,dd gksA rc] tSlk fd geus fLFkfr x = 3.5 osQ fy, fd;k vko`Qfr 1.12 gS] gesa BD = x izkIr gksxk (vko`Qfr 1.12)A ge bl ifj.kke dks ikbFkkxksjl izes; dh lgk;rk ls fl¼ dj ldrs gSaA x +1 è;ku nhft, fd vko`Qfr 1.12 esa] ∆ OBD ,d ledks.k f=kHkqt gSA o`Ùk dh f=kT;k 2 ,dd gSA x +1 vr%, OC = OD = OA = ,dd 2  x + 1 x − 1 vc] OB = x− = ⋅  2  2 vr%] ikbFkkxksjl izes; ykxw djus ij] gesa ;g izkIr gksrk gS% 2 2  x + 1  x − 1 4x BD = OD – OB =  2 2 2  −  = =x  2   2  4 blls ;g irk pyrk gS fd BD = x gSA bl jpuk ls ;g n'kkZus dh ,d fp=kh; vkSj T;kferh; fof/ izkIr gks tkrh gS fd lHkh okLrfod la[;kvksa x > 0 osQ fy,] x dk vfLrRo gSA ;fn ge la[;k js[kk ij x dh fLFkfr tkuuk pkgrs gSa] rks vkb, ge js[kk BC dks la[;k js[kk eku ysa] B dks 'kwU; eku ysa vkSj C dks 1 eku ysa] vkfn&vkfnA B dks osQUnz vkSj BD dks f=kT;k ekudj ,d pki [khafp, tks la[;k js[kk dks E ij dkVrk gS (nsf[k, vkÑfr 1-13)A rc E, x fu:fir djrk gSA 2024-25 la[;k i¼fr 21 vko`Qfr 1.13 vc ge oxZewy dh vo/kj.kk dks ?kuewyksa] prqFkZewyksa vkSj O;kid :i ls nosa ewyksa] tgk¡ n ,d /ukRed iw.kk±d gS] ij ykxw djuk pkgsaxsA vkidks ;kn gksxk fd fiNyh d{kkvksa esa vki oxZewyksa vkSj ?kuewyksa dk vè;;u dj pqosQ gSaA 3 8 D;k gS\ ge tkurs gSa fd ;g ,d /ukRed la[;k gS ftldk ?ku 8 gS] vkSj vkius ;g vo'; vuqeku yxk fy;k gksxk fd 3 8 = 2 gSA vkb, ge 5 243 dk eku Kkr djsaA D;k vki ,d ,slh la[;k b tkurs gSa ftlls fd b5 = 243 gks\ mÙkj gS 3] vr%] 5 243 = 3 gqvkA bu mnkgj.kksa ls D;k vki n a ifjHkkf"kr dj ldrs gSa] tgk¡ a > 0 ,d okLrfod la[;k gS vkSj n ,d /ukRed iw.kk±d gS\ eku yhft, a > 0 ,d okLrfod la[;k gS vkSj n ,d /ukRed iw.kk±d gSA rc n a = b, tcfd bn = a vkSj b > 0A è;ku nhft, fd 2 , 3 8 , n a vkfn esa iz;qDr izrhd ^^ ** dks dj.kh fpÉ (radical sign) dgk tkrk gSA vc ge ;gk¡ oxZewyksa ls lacaf/r oqQN loZlfedk,¡ (identities) ns jgs gSa tks fofHkUu fof/;ksa ls mi;ksxh gksrh gSaA fiNyh d{kkvksa esa vki buesa ls oqQN loZlfedkvksa ls ifjfpr gks pqosQ gSaA 'ks"k loZlfedk,¡ okLrfod la[;kvksa osQ ;ksx ij xq.ku osQ caVu fu;e ls vkSj loZlfedk (x + y) (x – y) = x2 – y2 ls] tgk¡ x vkSj y okLrfod la[;k,¡ gSa] izkIr gksrh gSAa eku yhft, a vkSj b /ukRed okLrfod la[;k,¡ gSaA rc] a a (i) ab = a b (ii) = b b (iii) ( a+ b )( ) a − b =a−b ( (iv) a + b ) (a − b ) = a 2 −b 2024-25 22 xf.kr (v) ( a+ b )( ) c + d = ac + ad + bc + bd ( ) 2 (vi) a + b = a + 2 ab + b vkb, ge bu loZlfedkvksa dh oqQN fo'ks"k fLFkfr;ksa ij fopkj djsaA mnkgj.k 15 : fuEufyf[kr O;atdksa dks ljy dhft,% (i) (5 + 7 ) ( 2 + 5 ) (ii) ( 5 + 5 ) ( 5 − 5 ) (iii) ( 3 + 7 ) (iv) ( 11 − 7 ) ( 11 + 7 ) 2 gy : (i) ( 5 + 7 ) ( 2 + 5 ) = 10 + 5 5 + 2 7 + 35 (ii) ( 5 + 5 ) ( 5 − 5 ) = 5 − ( 5 ) = 25 – 5 = 20 2 2 (iii) ( 3 + 7 ) = ( 3 ) + 2 3 7 + ( 7 ) = 3 + 2 21 + 7 = 10 + 2 2 2 2 21 (iv) ( 11 − 7 ) ( 11 + 7 ) = ( 11 ) − ( 7 ) = 11 − 7 = 4 2 2 fVIi.kh : è;ku nhft, fd Åij osQ mnkgj.k esa fn, x, 'kCn ¶ljy djuk¸ dk vFkZ ;g gS fd O;atd dks ifjes; la[;kvksa vkSj vifjes; la[;kvksa osQ ;ksx osQ :i esa fy[kuk pkfg,A 1 ge bl leL;k ij fopkj djrs gq, fd 2 la[;k js[kk ij dgk¡ fLFkr gS] bl vuqPNsn dks ;gha lekIr djrs gSaA ge tkurs gSa fd ;g ,d vifjes; gSA ;fn gj ,d ifjes; la[;k gks] rks bls ljyrk ls gy fd;k tk ldrk gSA vkb, ge ns[ksa fd D;k ge blds gj dk ifjes;dj.k (rationalise) dj ldrs gSa] vFkkZr~ D;k gj dks ,d ifjes; la[;k esa ifjo£rr dj ldrs gSaA blosQ fy, gesa oxZewyksa ls lacaf/r loZlfedkvksa dh vko';drk gksrh gSA vkb, ge ns[ksa fd bls oSQls fd;k tk ldrk gSA 1 mnkgj.k 16 : 2 osQ gj dk ifjes;dj.k dhft,A 1 gy : ge 2 dks ,d ,sls rqY; O;atd osQ :i esa fy[kuk pkgrs gSa] ftlesa gj ,d ifjes; la[;k 2024-25 la[;k i¼fr 23 1 2 gksA ge tkurs gSa fd 2. 2 ifjes; gSA ge ;g Hkh tkurs gSa fd dks ls xq.kk djus 2 2 2 ij gesa ,d rqY; O;atd izkIr gksrk gS] D;ksafd = 1 gSA vr% bu nks rF;ksa dks ,d lkFk ysus 2 ij] gesa ;g izkIr gksrk gS% 1 1 2 2 = × = 2 2 2 2 1 bl :i esa dks la[;k js[kk ij LFkku fu/kZj.k ljy gks tkrk gSA ;g 0 vkSj 2 osQ eè; fLFkr 2 gSA 1 mnkgj.k 17 : osQ gj dk ifjes;dj.k dhft,A 2+ 3 1 gy : blosQ fy, ge Åij nh xbZ loZlfedk (iv) dk iz;ksx djrs gSaA 2 + 3 dks 2 − 3 ls xq.kk djus vkSj Hkkx nsus ij] gesa ;g izkIr gksrk gS% 1 2− 3 2− 3 × = =2− 3 2+ 3 2− 3 4−3 5 mnkgj.k 18 : osQ gj dk ifjes;dj.k dhft,A 3− 5 gy : ;gk¡ ge Åij nh xbZ loZlfedk (iii) dk iz;ksx djrs gSaA 5 3+ 5 5 3+ 5  −5  ( ) vr%] 3− 5 = 5 3− 5 × 3+ 5 = 3−5 =   2  ( 3+ 5 ) 1 mnkgj.k 19 : osQ gj dk ifjes;dj.k dhft,A 7+3 2 1 1 7 −3 2 7 −3 2 7 −3 2 gy : = ×  = = 7 + 3 2 7 + 3 2  7 − 3 2  49 − 18 31 2024-25 24 xf.kr vr% tc ,d O;atd osQ gj esa oxZewy okyk ,d in gksrk gS (;k dksbZ la[;k dj.kh fpÉ ds vanj gks)] rc bls ,d ,sls rqY; O;atd esa] ftldk gj ,d ifjes; la[;k gS] :ikarfjr djus dh fØ;kfof/ dks gj dk ifjes;dj.k (rationalising the denominator) dgk tkrk gSA iz'ukoyh 1.4 1. crkb, uhps nh xbZ la[;kvksa esa dkSu&dkSu ifjes; gSa vkSj dkSu&dkSu vifjes; gSa% (i) 2− 5 ( (ii) 3 + ) 23 − 23 (iii) 2 7 7 7 1 (iv) (v) 2π 2 2. fuEufyf[kr O;atdksa esa ls izR;sd O;atd dks ljy dhft,% (i) (3 + 3 ) ( 2 + 2 ) ( (ii) 3 + 3 ) (3 − 3 ) ( ) (iv) ( 2) ( 5 + 2) 2 (iii) 5+ 2 5− 3. vkidks ;kn gksxk fd π dks ,d o`Ùk dh ifjf/ (eku yhft, c) vkSj mlosQ O;kl (eku c yhft, d) osQ vuqikr ls ifjHkkf"kr fd;k tkrk gS] vFkkZr~ π= gSA ;g bl rF; dk d var£ojks/ djrk gqvk izrhr gksrk gS fd π vifjes; gSA bl var£ojks/ dk fujkdj.k vki fdl izdkj djsaxs\ 4. la[;k js[kk ij 9.3 dks fu:fir dhft,A 5. fuEufyf[kr osQ gjksa dk ifjes;dj.k dhft,% 1 1 1 1 (i) (ii) (iii) (iv) 7 7− 6 5+ 2 7 −2 1.5 okLrfod la[;kvksa osQ fy, ?kkrkad&fu;e D;k vkidks ;kn gS fd fuEufyf[kr dk ljyhdj.k fdl izdkj djrs gSa\ (i) 172. 175 = (ii) (52)7 = 2310 (iii) = (iv) 73. 93 = 237 2024-25 la[;k i¼fr 25 D;k vkius fuEufyf[kr mÙkj izkIr fd, Fks\ (i) 172. 175 = 177 (ii) (52)7 = 514 2310 (iii) = 233 (iv) 73. 93 = 633 237 bu mÙkjksa dks izkIr djus osQ fy,] vkius fuEufyf[kr ?kkrkad&fu;eksa (laws of exponents) dk iz;ksx vo'; fd;k gksxk] [;gk¡ a, n vkSj m izko`Qr la[;k,¡ gSaA vkidks ;kn gksxk fd a dks vk/kj (base) vkSj m vkSj n dks ?kkrkad (exponents) dgk tkrk gS A] ftudk vè;;u vki fiNyh d{kkvksa esa dj pqosQ gSa% (i) am. an = am + n (ii) (am)n = amn am (iii) = am − n , m > n (iv) ambm = (ab)m an (a)0 D;k gS\ bldk eku 1 gSA vki ;g vè;;u igys gh dj pqosQ gSa fd (a)0 = 1 1 gksrk gSA vr%] (iii) dks ykxw djosQ] vki n = a − n izkIr dj ldrs gSaA vc ge bu fu;eksa a dks ½.kkRed ?kkrkadksa ij Hkh ykxw dj ldrs gSaA vr%] mnkgj.k osQ fy, % 1 (i) 172 ⋅ 17 –5 = 17 –3 = (ii) (52 ) –7 = 5–14 173 23–10 (iii) = 23–17 (iv) (7) –3 ⋅ (9) –3 = (63) –3 237 eku yhft, ge fuEufyf[kr vfHkdyu djuk pkgrs gSa % 2 1 4  1  (i) 2 3 ⋅ 23 (ii)  5  3  1 75 1 1 (iii) 1 (iv) 135 ⋅ 17 5 3 7 ge ;s vfHkdyu fdl izdkj djsaxs\ ;g ns[kk x;k gS fd os ?kkrkad&fu;e] ftudk vè;;u ge igys dj pqosQ gSa] ml fLFkfr esa Hkh ykxw gks ldrs gSa] tcfd vk/kj /ukRed okLrfod la[;k gks vkSj ?kkrkad ifjes; la[;k gks (vkxs vè;;u djus ij ge ;g ns[ksaxs 2024-25 26 xf.kr fd ;s fu;e ogk¡ Hkh ykxw gks ldrs gSa] tgk¡ ?kkrkad okLrfod la[;k gksA)A ijUrq] bu fu;eksa dk dFku nsus ls igys vkSj bu fu;eksa dks ykxw djus ls igys] ;g le> ysuk 3 vko';d gS fd] mnkgj.k osQ fy,] 4 2 D;k gSA vr%] bl laca/ esa gesa oqQN djuk gksxkA n a dks bl izdkj ifjHkkf"kr fd;k tkrk gS] tgk¡ a > 0 ,d okLrfod la[;k gS% eku yhft, a > 0 ,d okLrfod la[;k gS vkSj n ,d /ukRed iw.kk±d gSA rc n a = b gksrk gS, tcfd bn = a vkSj b > 0 gksA 1 ?kkrkadksa dh Hkk"kk esa] ge n a = a n osQ :i esa ifjHkkf"kr djrs gSaA mnkgj.k osQ fy,] 1 3 3 2 = 2 3 gSA vc ge 4 2 dks nks fof/;ksa ls ns[k ldrs gSaA 3  1 3 4 =  42  = 2 = 8 2 3   3 1 1 4 2 = ( 4 ) 2 = ( 64 ) 2 = 8 3 vr%] gesa ;g ifjHkk"kk izkIr gksrh gS% eku yhft, a > 0 ,d okLrfod la[;k gS rFkk m vkSj n ,sls iw.kk±d gSa fd 1 osQ vfrfjDr budk dksbZ vU; mHk;fu"B xq.ku[kaM ugha gS vkSj n > 0 gSA rc] m ( a) m an = n = n am vr% okafNr foLr`r ?kkrkad fu;e ;s gSa% eku yhft, a > 0 ,d okLrfod la[;k gS vkSj p vkSj q ifjes; la[;k,¡ gSaA rc] p q p+q p q pq (i) a.a =a (ii) (a ) = a p a (iii) = ap −q (iv) apbp = (ab)p aq vc vki igys iwNs x, iz'uksa dk mÙkj Kkr djus osQ fy, bu fu;eksa dk iz;ksx dj ldrs gSAa 2 1 4  1  mnkgj.k 20 : ljy dhft,% (i) 2 3 ⋅ 2 3 (ii)  5  3  1 75 1 1 (iii) 1 (iv) 135 ⋅ 17 5 73 2024-25 la[;k i¼fr 27 gy :  2 1 4 2 1  +  3  1 4 (i) 2 ⋅2 =2 3 3  3 3 =2 =2 =2 3 1 (ii)  35  = 35   1 1 1 3−5 −2 1 1 1 1 75  −  (iii) 1 =7  5 3 =7 15 =7 15 (iv) 135 ⋅ 17 5 = (13 × 17) 5 = 2215 3 7 iz'ukoyh 1.5 1 1 1 1. Kkr dhft,% (i) 64 2 (ii) 32 5 (iii) 125 3 3 2 3 −1 2. Kkr dhft,% (i) 9 2 (ii) 32 5 (iii) 16 4 (iv) 125 3 1 7 1 2 1 1 1 112 3. ljy dhft,% (i) 2 ⋅2 3 5 (ii)  3  (iii) 1 (iv) 7 2 ⋅ 8 2 3  4 11 1.6 lkjka'k bl vè;k; esa] vkius fuEufyf[kr fcUnqvksa dk vè;;u fd;k gS% p 1. la[;k r dks ifjes; la[;k dgk tkrk gS] ;fn bls q osQ :i esa fy[kk tk ldrk gks] tgk¡ p vkSj q iw.kk±d gSa vkSj q ≠ 0 gSA p 2. la[;k s dks vifjes; la[;k dgk tkrk gS] ;fn bls osQ :i esa u fy[kk tk ldrk q gks] tgk¡ p vkSj q iw.kk±d gSa vkSj q ≠ 0 gSA 3. ,d ifjes; la[;k dk n'keyo izlkj ;k rks lkar gksrk gS ;k vuolkuh vkorhZ gksrk gSA lkFk gh] og la[;k] ftldk n'keyo izlkj lkar ;k vuolkuh vkorhZ gS] ifjes; gksrh gSA 4. ,d vifjes; la[;k dk n'keyo izlkj vuolkuh vukorhZ gksrk gSA lkFk gh] og la[;k ftldk n'keyo izlkj vuolkuh vukorhZ gS] vifjes; gksrh gSA 5. lHkh ifjes; vkSj vifjes; la[;kvksa dks ,d lkFk ysus ij okLrfod la[;kvksa dk laxzg izkIr gksrk gSA 2024-25 28 xf.kr 6. ;fn r ifjes; gS vkSj s vifjes; gS] rc r + s vkSj r – s vifjes; la[;k,¡ gksrh gSa rFkk r rs vkSj vifjes; la[;k,¡ gksrh gSa ;fn r≠0 gSA s 7. /ukRed okLrfod la[;kvksa a vkSj b osQ laca/ esa fuEufyf[kr loZlfedk,¡ ykxw gksrh gSa% a a (i) ab = a b (ii) = b b (iii) ( a + b )( ) a − b =a−b (iv) (a + b ) (a − b ) = a 2 −b ( ) 2 (v) a + b = a + 2 ab + b 1 a −b 8. osQ gj dk ifjes;dj.k djus osQ fy,] bls ge ls xq.kk djrs gSa] tgk¡ a + b a −b a vkSj b iw.kk±d gSaA 9. eku yhft, a>0 ,d okLrfod la[;k gS vkSj p vkSj q ifjes; la[;k,¡ gSaA rc] (i) ap. aq = ap + q (ii) (ap)q = apq ap (iii) = ap − q (iv) apbp = (ab)p aq 2024-25