संख्यात्मक योग्यता PDF

सभी प्रतियोगी परीक्षाओं के लिए  िवषयसूची S Page Chapter Title No. No. 1 सं या प त 1 2 सरलीकरण 8 3 लघु म समापव य व मह म समापवतक 12 4...

सभी प्रतियोगी परीक्षाओं के लिए  िवषयसूची S Page Chapter Title No. No. 1 सं या प त 1 2 सरलीकरण 8 3 लघु म समापव य व मह म समापवतक 12 4 करणी व घातांक 15 5 तशतता 19 6 लाभ - हािन 23 7 ब ा 28 8 अनुपात व समानुपात 31 9 िम ण एवं एलीगेशन 35 10 औसत 37 11 समय और काय 41 12 चाल, समय और दरू ी 44 13 नाव और धारा 48 14 पाइप और टंक 50 15 साझेदारी 53 16 साधारण याज 56 17 च वृ याज 59 18 बीजग णत 62 19 यािम त 67 20 े िम त 84 21 ि कोणिम त 99 22 डेटा इंटरि टेशन 106  la[;k i)fr 1 (Number System) (Number System) CHAPTER la[;k i)fr %& fdlh Hkh ;kSfxd jkf'k ds ifj.kkeksa dk cks/k djkus ds fy, ftl i)fr dk mi;ksx gksrk gS] la[;k i)fr dgykrh gSA la[;kvksa dks muds xq.kksa vkSj fo'ks"krkvksa ds vk/kkj ij fuEu izdkj ls oxhZd`r fd;k tk ldrk gS & lfEeJ la[;k,¡ (Complex Number) (i) /[email protected] la[;k,¡ % tc izkd`r la[;kvksa ds os lHkh la[;k,¡ tks okLrfod vkSj dkYifud la[;kvksa ls ifjokj esa 0 dks Hkh 'kkfey dj ysrs gS] rc og iw.kZ feydj cuh gksrh gSA la[;k,¡ dgykrh gSA bUgsa (a + ib) ds :i esa fy[kk tkrk gSA tgk¡ a vkSj b W = {0, 1, 2, 3, 4, 5, ….} okLrfod la[;k,¡ gS rFkk i = –1 gSA uksV % pkj yxkrkj izkd`frd la[;kvksa dk xq.kuQy ges'kk 24 ls iw.kZr% foHkkT; gksrk gSA Z = a ¼okLrfod la[;k½ + ib ¼dkYifud la[;k½ 1. okLrfod la[;k,¡ (Real Numbers): ifjes; ,oa A. izkd`r la[;k,¡ % ftu la[;kvksa dk bLrseky oLrqvksa vifjes; la[;kvksa dks lfEefyr :Ik ls okLrfod dks fxuus ds fy, fd;k tkrk gS] izkd`r la[;k dgrs la[;k dgrs gSaA bUgsa la[;k js[kk ij iznf'kZRk fd;k gSA N = {1, 2, 3, 4, 5, ……} tk ldrk gSA I. iw.kkZad la[;k,¡ % la[;kvksa dk ,slk leqPp; ftlesa izFke n izkd`frd la[;kvksa dk ;ksx = n(n + 1) iw.kZ la[;kvksa ds lkFk&lkFk _.kkRed la[;k,¡ Hkh 2 lfEefyr gks] iw.kkZad la[;k,¡ dgykrh gS] bls I ls izFke n izkd`frd la[;kvksa ds oxksZa dk ;ksx lwfpr djrs gSaA n(n + 1)(2n + 1) = I = {–4, –3, –2, –1, 0, 1, 2, 3, 4,…..} 6 1  izFke n izkd`frd la[;kvksa ds ?kuksa dk ;ksx = mnkgj.k & (4,9), (15, 22), (39, 40) HCF = 1  n(n + 1)  2  2  (b) ;kSfxd la[;k,¡ (Composite Numbers) :- os nks yxkrkj izkd`frd la[;kvksa ds oxksZa dk varj muds izkd`r la[;k,¡ tks 1 ;k Lo;a dks NksM+dj fdlh vU; ;ksxQy ds cjkcj gksrk gS A la[;k ls Hkh foHkkT; gks] ;kSfxd la[;k,¡ dgykrh gSA tSls & 4] 6] 8] 9] 10 vkfnA mnkgj.k & 112 = 121 (ii) le la[;k,¡ % la[;k,¡ tks 2 ls iw.kZr% foHkkT; gks 122 = 144 le la[;k dgykrh gSA 11 + 12 → 23 Difference 144 – 121 = 23 n oka in = 2n (a) vHkkT; la[;k,¡ (Prime Numbers) :- ,d la[;k izFke n le la[;kvksa dk ;ksx = n(n+1) ftlds dsoy nks gh xq.kd gksrs gSa] 1 vkSj og la[;k izFke n le la[;kvksa ds oxksZa dk ;ksx = Lo;a] mUgsa vHkkT; la[;k dgrs gSaA 2n(n + 1)(2n + 1) tSls & {2, 3, 5, 7, 11, 13, 17, 19……..} 3 rhu vadks dh lcls NksVh vHkkT; la[;k = 101  vafre in  n =  rhu vadks dh lcls cM+h vHkkT; la[;k = 997  2  tgk¡ 1 Prime Number ugha gSA (iii) fo"ke la[;k,¡ % og la[;k,¡ tks 2 ls foHkkftr u 2 ,dek= le Prime la[;k gSA 3, 5, 7 Øekxr fo"ke vHkkT; la[;k dk bdykSrk tksM+k gks] fo"ke la[;k,¡ gksrh gSA gSA izFke n fo"ke la[;kvksa dk ;ksx = n2 1 ls 25 rd dqy vHkkT; la[;k ¾ 9  vfare in + 1  25 ls 50 rd dqy vHkkT; la[;k ¾ 6 n =   2  1-50 rd dqy 15 Prime Number gSA II. n'keyo 51-100 rd dqy 10 Prime Number gSA n'keyo os la[;k,¡ gS tks nks iw.kZ la[;kvksa ;k iw.kkZadks vr% 1-100 rd dqy 25 Prime Number gSA ds chp vkrh gSA tSls & 3-5 ,d n'keyo la[;k gS 1 ls 200 rd dqy vHkkT; la[;k ¾ 46 tks 3 o 4 ds chp fLFkr gSA 1 ls 300 rd dqy vHkkT; la[;k ¾ 62 izR;sd n'keyo la[;k dks fHkUu ds :i esa fy[kk tk 1 ls 400 rd dqy vHkkT; la[;k ¾ 78 ldrk gS vkSj blds foijhr izR;sd fHkUu dks Hkh 1 ls 500 rd dqy vHkkT; la[;k ¾ 95 n'keyo :i esa fy[kk tk ldrk gSA  vHkkT; la[;kvksa dk ijh{k.k %& nh x;h la[;k ds (i) lkar n'keyo laHkkfor oxZewy ls cM+h dksbZ la[;k yhft,A ekuk ;g Okg la[;k,¡ tks n'keyo ds ckn dqN vadksa ds ckn la[;k x gS] vc x ls NksVh leLr vHkkT; la[;kvksa dh [kRe gks tk;s tSls & 0.25, 0.15, 0.375 bls fHkUu lgk;rk ls nh x;h la[;k dh foHkkT;rk dk ijh{k.k la[;k esa fy[kk tk ldrk gS A dhft,A (ii) vlkar n'keyo ;fn ;g buesa ls fdlh ls Hkh foHkkT; ugha gS rks ;g tks la[;k,¡ n'keyo ds ckn dHkh [kRe ugha gksrh fuf'pr :i ls ,d vHkkT; la[;k gksxhA cfYd iqujko`fÙk djrh gks] vuar rdA mnkgj.k & tSls & 0.3333, 0.7777, 0.183183183……….. D;k 349 ,d vHkkT; la[;k gS ;k ugha \ ;s nks izdkj ds gks ldrs gSa & gy & A. vkorhZ n'keyo fHkUu (Repeating) 349 dk laHkkfor oxZewy 19 gksxk vkSj 19 ls NksVh lHkh og n'keyo fHkUu n'keyo fcanq ds ckn ,d ;k vHkkT; la[;k,¡ % 2] 3] 5] 7] 11] 13] 17 gSA vf/kd vadksa dh iqujko`fÙk gksrh gSA Li"V gS fd 349 bu lHkh vHkkT; la[;kvksa ls foHkkT; ugha gS vr% 349 Hkh ,d vHkkT; la[;k gSA tSls & 1 = 0.333..., 22 = 3.14285714..... 3 7 lg vHkkT; la[;k,¡ (Co-prime Numbers) − og ,slh fHkUuksa dks O;Dr djus ds fy, nksgjk, tkus okys la[;k,¡ ftudk HCF flQZ 1 gksA vad ds Åij ,d js[kk [khap nsrs gaSA 2  bls ckj cksyrs gSA mnkgj.k & 0.333..... = 0.3 2 4 10 7 , , , 22 = 3.14285714.... = 3.142857 3 5 –11 8 7 B. vukorhZ (Non-Repeating) 'kq) vkorhZ n'keyo fHkUu dks fuEu izdkj ls lk/kkj.k tks la[;k,¡ n'keyo ds ckn dHkh [kRe ugha gksrh ij ;s fHkUu esa cnys & viuh la[;kvksa dh fuf'pr iqujko`fÙk (Repeat) ugha P pq pqr djrhA 0.P = 0.pq = 0.pqr = 9 99 999 tSls &  = 3.1415926535897932… fefJr vkorhZ n'keyo fHkUu dks fuEu izdkj ls 2 = 1.41421356237… lk/kkj.k fHkUu esa cnys & vifjes; (Irrational) la[;k,¡ & bUgsa P/Q form esa pq – p pqr – pq iznf'kZr ugha fd;k tk ldrkA 0.pq = 0.pq r = 90 900 mnkgj.k & pqr – p pqrs – pq 2 , 3 , 11 , 19 , 26...... 0.pqr = 0.pq rs = 990 9900 fHkUu (Fraction) :- fHkUu ,d ,slh la[;k gS tks fdlh mnkgj.k & lEiw.kZ pht dk dksbZ Hkkx fu:fir djrh gSA (i) 0.39 = = 39 13 tSls ,d lsc ds pkj Hkkx fd;s tkrs gS] mlesa ls ,d 99 33 1 fgLlk fudky fn;k x;k rks mls ds :i esa iznf'kZr 625 – 6 619 4 (ii) 0.625 = = 990 990 3 fd;k tkrk gSA tcfd 'ks"k cps Hkkx dks ds :i esa 3524 – 35 3489 1163 4 (iii) 0.3524 = = = 9900 9900 3300 iznf'kZr fd;k tk;sxkA fHkUu nks Hkkxksa esa caVk gksrk gS & va'k o gj ifjes; (Rational) la[;k,¡ & og la[;k,¡ ftUgsa p → v'ak P/Q form esa fy[kk tk ldrk gS] ysfdu Q tgk¡ ekuk dksbZ fHkUu = 'kwU; ugha gksuk pkfg,] P o Q iw.kkZad gksus pkfg,A q → gj fHkUuksa ds izdkj 3 2. dkYifud la[;k,¡ (Imaginary Numbers)% ftUgsa 2. Binary dks Decimal esa cnyuk % la[;k js[kk ij iznf'kZr ugha fd;k tk ldrk gSA Binary system esa 1 dk eku tc og gj ckj viuh ckbZ vksj ,d LFkku f[kldrk gS] Lo;a dk nqxquk gks ijQsDV la[;k (Perfect Number) tkrk gS rFkk tgk¡ dgha Hkh 0 vkrk gS mldk eku 0 og la[;k ftlds xq.ku[k.Mksa dk ;ksx ml la[;k ds gksrk gSA cjkcj gks ¼xq.ku[k.Mksa esa Lo;a ml Lka[;k dks NksM+dj½ mnkgj.k & mnkgj.k & 1 0 1 1 0 0 1 6 → 1, 2, 3 → ;gk¡ 1 + 2 + 3 → 6 26 25 24 23 22 21 20 28 → 1, 2, 4, 7, 14 → 1 + 2 + 4 + 7 + 14 → 28 vc iw.kZoxZ la[;k dh igpku (1011001)2 = 1 × 26 + 0 × 25 + 1 × 24 × 1 × 23 + 0 × 22 + ↓ 0 × 21 + 1 × 2 0 bdkbZ vad tks ,d tks ugha gks ldrs = 64 + 0 + 16 + 8 + 8 + 0 + 1 {20 = 1} = 89 iw.kZ oxZ la[;k ds gks ldrs gSA Hkktdksa dh la[;k ;k xq.ku[kaM dh la[;k fudkyuk 0 2 ____ igys la[;k dk vHkkT; xq.ku[kaM djsaxs vkSj mls Power 1 3 ____ ds :Ik esa fy[ksaxs rFkk izR;sd (Power) ?kkr esa ,d tksM+dj ?kkrks dk xq.kk djsxa s rks Hkktdkas dh la[;k izkIr gks tk;sxhA 4 7 ____ mnkgj.k & 5 or 25 8 ____ 2280 dks dqy fdruh la[;kvksa ls iw.kZr% Hkkx fn;k tk 6 ldrk gSA 9 gy & fdlh Hkh la[;k ds oxZ ds vafre nks vad ogh gksaxs tks 2280= 23 × 31 × 51 × 191 1-24 rd dh la[;kvksa ds oxZ ds vafre nks vad gkasxsA Hkktdksa dh la[;k = (3+1) (1+1) (1+1) (1+1) = 4 × 2 × 2 × 2 = 32 uksV & vr% lHkh dks 1-25 ds oxZ vo’; ;kn gksus pkfg,A bdkbZ dk vad Kkr djuk Binary o Decimal esa cnyuk 1- tc la[;k ?kkr (Power) ds :Ik esa gks 1. Decimal la[;k dks Binary esa cnyuk % tc Base dk bdkbZ vad 0, 1, 5 ;k 6 gks] rks dksbZ Hkh fdlh Mslhey ¼nl&vk/kkjh½ la[;k ds lerqY; izkd`frd ?kkr ds fy, ifj.kke dk bdkbZ vad ogh jgsxkA Binary number Kkr djus ds fy, ge iznÙk Mslhey ¼nl&vk/kkjh½ la[;k dks yxkrkj 2 ls rc rd Hkkx tc base dk bdkbZ vad 2, 3, 4, 7, 8, ;k 9 gks] rks Power esa 4 ls Hkkx nasxs vkSj ftruk ‘ks”k izkIr gksxk nsrs gS tc rd fd vafre HkkxQy ds :Ik esa 1 izkIr ugha gksrk gSA mruk gh Base ds bdkbZ vad ij power j[ksaxAs tc vc lHkh 'ks"kQy dks mYVs Øe esa fy[kk tk, rks power, 4 ls iw.kZr% foHkkftr gks tkrk gS rks base ds ifjofrZr ckbujh la[;k izkIr gksrh gSA bdkbZ vad ij 4 power j[kasxsA mnkgj.k & 2- ljyhdj.k ds :Ik esa gks izR;sd la[;k ds bdkbZ ds vad dks fy[kdj fpUg ds vuqlkj ljy djsaxs tks ifj.kke vk;sxk mldk bdkbZ vad mÙkj gksxkA Power okyh la[;kvksa esa Hkkx nsuk ¼Hkktd fudkyuk½ 1- ;fn an + bn fn;k gks rks n fo"ke gksus ij (a+b) bldk Hkktd gksxkA vr% 89 ds lerqY; Binary number = (1011001)2 2- ;fn an – bn fn;k gks rksA 4  n fo"ke gksus ij Hkktd → (a–b) jkseu i)fr ds ladsrd n le gksus ij Hkktd → (a – b) ;k (a + b) ;k nksuksaA 1 → I 20 → XX 2 → II 30 → XXX (i) a ÷ (a – 1) gks] rks 'ks"kQy ges'kk 1 cpsxkA n 3 → III 40 → XL n  ;fn n le gks] rks ges'kk 1 cpsxk 4 → IV 50 → L (ii) a ÷ (a + 1)   ;fn n fo"ke gks] rks 'ks"kQy a gksxk 5 → V 100 → C (iii) (an + a)  (a – 1) gks] rks 'ks"kQy 2 cpsxk 6 → VI 500 → D 7 → VII 1000 → M  ;fn n le gks] rks 'ks"kQy 'kUw; ¼0½ gksxkA (iv) (an + a) ÷ (a+1)  8 → VIII ;fn n fo"ke gk]s rks 'ks"kQy (a – 1) gksxkA 9 → IX 10 → X foHkkT;rk ds fu;e la[;k fu;e 2 ls vfUre vad le la[;k ;k 'kwU; (0) gks tSls & 236] 150] 1000004 3 ls fdlh la[;k es vadksa dk ;ksx 3 ls foHkkftr gksxk rks iw.kZ la[;k 3 ls foHkkftr gksxhA tSls & 729, 12342, 5631 4 ls vfUre nks vad 'kwU; gks ;k 4 ls foHkkftr gks tSls & 1024, 58764, 567800 5 ls vfUre vad 'kwU; ;k 5 gks tSls & 3125, 625, 1250 6 ls dksbZ la[;k vxj 2 rFkk 3 nksuksa ls foHkkftr gks rks og 6 ls Hkh foHkkftr gksxhA tSls & 3060, 42462, 10242 7 ls ;fn nh x;h la[;k ds bdkbZ vad dk nqxquk ckdh la[;k ¼bdkbZ dk vad NksM+dj½ ls ?kVkus ij izkIr la[;k 7 ls foHkkftr gS rks iwjh la[;k 7 ls foHkkftr gks tk,xhA vFkok fdlh la[;k esa vadks dh la[;k 6 ds xq.kt esa gks rks la[;k 7 ls foHkkftr gksxhA tSls & 222222, 44444444444, 7854 8 ls ;fn fdlh la[;k ds vfUre rhu vad 8 ls foHkkT; gks ;k vafre rhu vad ‘000’ ¼'kwU;½ gks A tSls & 9872, 347000 9 ls fdlh la[;k ds vadksa dk ;ksx vxj 9 ls foHkkT; gks rks iw.kZ la[;k 9 ls foHkDr gksxhA 10 ls vafre vad 'kwU; (0) gks rks 11 ls fo"ke LFkkukas ij vadkas dk ;ksx o le LFkkuksa ij vadksa ds ;ksx dk vUrj 'kwU; (0) ;k 11 dk xq.kt gks rks tSls & 1331, 5643, 8172659 12 ls 3 o 4 ds foHkkT; dk la;qDr :Ik 13 ls fdlh la[;k esa ,d gh vad 6 ckj nksgjk, ;k vfUre vad dks 4 ls xq.kk djds 'ks"k la[;k ¼bdkbZ vad NksM+dj½ esa tksM+us ij izkIr la[;k 13 ls foHkkftr gks rks iw.kZ la[;k 13 ls foHkkftr gksxhA tSls & 222222, 17784 5  vH;kl iz'u mnk-2 rhu vHkkT; la[;kvksa dk ;ksx 100 gS ;fn muesa ls ,d la[;k nwljh la[;k ls 36 vfèkd gks rks la[;kvksa ds ;ksx] varj rFkk xq.kuQy ij ,d la[;k D;k gksxk \ vkèkkfjr Hkkx] HkkxQy rFkk 'ks"kQy ij vkèkkfjr mnk-1 ;fn fdlh la[;k dk 3/4 ml la[;k ds 1/6 ls 7 mnk-1 64329 dks tc fdlh l[;k ls Hkkx fn;k tkrk vfèkd gS] rks ml la[;k 5/3 D;k gksxk\ gS] rks 175, 114 rFkk 213 yxkrkj rhu 'ks"kQy (a) 12 (b) 18 vkrs gS rks HkkT; D;k gS \ (c) 15 (d) 20 (a) 184 (b) 224 mÙkj (d) (c) 234 (d) 296 mnk-2 ;fn nks la[;kvksa dk ;ksxQy rFkk mudk mÙkj (c) xq.kuQy a rFkk b, muds O;qRØekas dk ;ksxQy gksxk mnk-2 (325 + 326 + 327 + 328) foHkkftr gSA 1 1 b (a) 11 (b) 16 (a) + (b) (c) 25 (d) 30 a b a a a mÙkj (d) (c) (d) b ab mnk-3 foHkktu ds ,d ;ksxQy esa foHkktd] HkkxQy dk mÙkj (c) 1" 12 xquk rFkk 'ks"kQy dk 5 xquk gSA rnuqlkj] ;fn mnk-3 nks la[;kvksa dk ;ksx 75 gS vkSj mudk varj 25 mlesa 'ks"kQy 36 gks] rks HkkT; fdruk gksxk ? gS] rks mu nksuksa la[;kvksa dk xq.kuQy D;k gksxk\ ¼a) 2706 (a) 1350 (b) 1250 (b) 2796 (c) 1000 (d) 125 (c) 2736 mÙkj (b) (d) 2826 mnk-4 ,d fo|kFkÊ ls fdlh la[;k dk 5/16 Kkr djus mÙkj (c) ds fy;s dgk x;k vkSj xyrh ls ml la[;k dk bdkbZ vad fudkyuk vk/kkfjr 5/6 Kkr dj fy;k vFkkZr~ mldk mÙkj lgh mÙkj ls 250 vfèkd Fkk rks nh gqbZ la[;k Kkr dhft;sA (a) 300 (b) 480 (c) 450 (d) 500 mÙkj (b) mnk-1 416  333 + 2167  118 – 114  133 ds ifj.kke le] fo"ke rFkk vHkkT; la[;kvksa ij vkèkkfjr dk bdkbZ vad Kkr dhft, ? fdruk gS ? (a) 0 (b) 2 (c) 3 (d) 5 mnk-1 ;fn fdUgha rhu Øekxr fo"ke çk—r la[;kvksa dk ;ksx 147 gks] rks chp okyh la[;k gksxh A (a) 47 (b) 48 (c) 49 (d) 51 mÙkj (c) 6  izkd`frd la[;kvksa ds squre/cube ds lcls cM+h rFkk lcls NksVh la[;k@fHkUu ;ksx ,oa varj ij vk/kkfjr Kkr djus ij vk/kkfjr 2 4 mnk-1 fuEu es ls vkSj ds chp mifLFkr fHkUu gSa ? mnk-1 (11 + 12 + 13 + …. + 20 ) = ? 2 2 2 2 5 9 (a) 385 (b) 2485 3 2 (a) (b) (c) 2870 (d) 3255 7 3 4 1 mnk-2 13 + 23 + 33 +... + 103 = ? (c) (d) 5 2 n'keyo la[;k vk/kkfjr mnk-2 fuEu esa ls cM+h la[;k gSaA 1 1 1 ( 3) 3 , (2 ) 2 , 1, ( 6 ) 6 1 (a) ( 2 ) 2 (b) 1 1 1 (c) ( 6 ) 6 (d) ( 3) 3 mnk-1 ,d fo|kFkhZ dks fuEufyf[kr O;atd dks ljy djus dks dgk x;k vkjksgh@vojksgh Øe vk/kkfjr 2 0.0016  0.025 0.1216  0.105  0.002  6 3  +  27 − 6  0.325  0.05 0.08512  0.625  0.039  4 19 mldk mÙkj FkkA mlds mÙkj esa fdrus 10 mnk-1 2, 3 4 , 4 6 dks cyk dks’Bd { }  4  4 2  12   6 3- cM+k dks’Bd [ ] Step 5 – lcls igys NksVk dks’Bd] fQj ea>yk dks’Bd vkSj mlds ckn cM+k dks’Bd gy fd;k tkrk gS A r`rh; LFkku ij “O” gS tks fd “of” ;k “Order” ls   13 5 1 29  13  4  4 − 2  12   6    cuk gS] ftldk eryc ^^xq.kk** ls ;k ^^dk** ls gksrk gSA prqFkZ LFkku ij “D” gS ftldk eryc “Division” 4   13 30 − 29  13 Step 6 –    24  6  gS] fn, x;s O;atu esa fHkUu&fHkUu fØ;kvksa esa lcls Step 7 –  13  1   13 igys Hkkx djrs gS ;fn fn;k gS rks A  4 24  6 iape LFkku ij “M” gS ftldk eryc Step 8 –   24   13 13 “Multiplication” gS] fn;s x, O;atu esa “Division”  4  6 ds ckn “Multiplication” ¼xq.kk½ djsx a sA 8  6 n Step 9 – 13  6  13 ;fn izFke o vafre in Kkr gks rks Sn = a +  2 = 36 Ans. tgk¡ = vafre in a+b chtxf.krh; lw= nks jkf'k;ksa ds e/; lekarj ek/; A = [a, b dk 2 1. (a + b)2 = a2 + 2ab + b2 lekarj ek/; A gS A] 2. (a – b)2 = a2 – 2ab + b2 xq.kksÙkj Js.kh 3. (a + b)2 + (a – b)2 = 2(a2 + b2) ;fn Js.kh ds izR;sd in dk mlls iwoZ in ls vuqikr ,d 4. (a2 – b2)= (a + b) (a – b) fuf'pr jkf'k gksrh gS rks xq.kksÙkj Js.kh gksrh gS A bl 5. a2 + b2 + c2 = (a + b + c)2 – 2(ab + bc + ca) fuf'pr jkf'k dks lkoZvuqikr dgrs gSa A 1  1 2 xq.kksÙkj Js.kh dk n ok¡ in 6. a + 2 2 = a+  −2 a  a Tn = a.rn-1 1 tgk¡ a = izFke in a2 + b2 + c2 − ab − bc − ca = ( a − b ) + (b + c ) + ( c − a)  2 2 2 7.   2 r = lkoZ vuqikr 8. a3 + b3 = (a + b)3 – 3ab(a + b) = (a + b) (a2 – n = inksa dh la[;k ab + b2) xq.kksÙkj Js.kh ds n inkas dk ;ksxQy 9. a3 – b3 = (a – b) 3 + 3ab (a – b) = (a – b) (a2  1 − rn  Sn = a  tc r  1  rn − 1  + ab + b2) ; Sn = a   ; tc r  1  1−r   r −1  10. a3 + b3 + c3 – 3abc = (a + b + c) (a2 + b2 + c2 1. nks jkf'k;ksa ds e/; xq.kksÙkj ek/; G = ab – ab – bc – ca) 2. ;fn nks /kukRed jkf'k;ksa a o b ds e/; lekarj ek/; = 1 ( a + b + c )( a − b )2 + (b − c )2 + ( c − a)2  2 rFkk xq.kksÙkj ek/; A o G gS rks ;fn a + b + c = 0 gks rks a+b A > G,  ab a3 + b3 + c3 = 3abc 2 1  1 3  1 gjkRed Js.kh 11. a + 3 =  a +  − 3 a +  3 fdlh Js.kh ds inkas ds O;qRØe mlh Øe esa fy[kus ij lekarj a  a  a Js.kh esa gks rks mls gjkRed Js.kh dgrs gS A 3 1  1  1 gjkRed Js.kh dk n oka in 12. a − 3 =  a −  + 3  a −  3 a  a  a Tn = 1 lekUrj Js.kh a + (n − 1)d og Js.kh ftldk izR;sd in vius iwoZ in ls dksbZ fu;r 2ab gjkRed ek/; (H) = jkf’k tksM+us vFkok ?kVkus ls izkIr gksrk gS A a+b tSls & 2, 5, 8, 11, ……. lekarj ek/;] xq.kksÙkj ek/; o gjkRed ek/; esa lekarj Js.kh dk n ok¡ in laca/k Tn = a + (n – 1) ekuk A, G rFkk H nks jkf’k;ksa a o b ds e/; Øe'k% lekarj tgk¡ a = izFke in ek/;] xq.kksÙkj ek/; o gjkRed ek/; gS rc d = lkoZ varj ¼f}rh; in & izFke in½ n = inksa dh la[;k G2 = AH rFkk A  G  H n lekarj Js.kh ds n inksa dk ;ksx Sn = 2a + (n − 1)d 2 9  vH;kl iz'u mnk-3 33 - 4 √35 dk oxZewy D;k gS \ VBODMAS – vk/kkfjr (a)  ( 2 7 + 5 ) (b)  ( 7 + 2 5 ) (c)  ( 7 − 2 5 ) (d)  ( 2 7 − 5 ) ?kukUrj rFkk ?kuewy vk/kkfjr mnk-1 24  2  12 + 12  6 of 2  (15  8  4) of (28  7 of 5) dk Ekku gksxk – (a) 4 32 (b) 4 8 75 75 3 2 1 mnk-1 (√43 + 152 ) dk eku D;k gS \ (c) 4 (d) 4 3 6 (a) 4913 (b) 4313 mnk-2 ljy djsa (c) 4193 (d) 3943  1  1 1  1 1 1    1 mÙkj (a) 1 3  1 −  2 − −     of 4  mnk-2 710 esa dkSulh NksVh la[;k tksM+h tkuh pkfg,  4  4 2  2 4 6    2 3 rkfd ;ksx ,d iw.kZ ?ku cu tk, \ mnk-3 ljy djsa A (a) 29 (b) 19 3 5 7 1 1 5 3 3 (c) 11 (d) 21 2 1 × ( + )+  of 8 3 4 7 4 7 4 6 mÙkj (b) 56 (b) 49 (a) 77 80 fHkUu vk/kkfjr (c) 2 (d) 3 2 3 9 oxkZUrj rFkk oxZewy vk/kkfjr mnk-1 fuEufyf[kr dk eku gS & (c) 1 (d) 1 16 32 mÙkj (a) mnk-1 fuEufyf[kr dk eku gS & 1 mnk-2 ;fn 2 = x + gS rks x dk eku Kkr 1 1+ √5 + √11 + √19 + √29 + √49 is 1 3+ 4 djsa A (a) 3 (b) 9 (c) 7 (d) 5 (a) 18 (b) 21 17 17 mÙkj (a) (c) 13 (d) 12 mnk-2 ;fn (102)2 = 10404 gS] rks 17 17 √104.04 + √1.0404 + √O. 010404 mÙkj (b) dk eku fdlds cjkcj gS \ 998 mnk-3 999 x 999 fdlds cjkcj gS \ 999 (a) 0.306 (b) 0.0306 (a) 998999 (b) 999899 (c) 11.122 (d) 11.322 (c) 989999 (d) 999989 mÙkj (d) mÙkj (a) 10 mnk-4 1 + 999 494  99 dk eku Kkr djsa A lehdj.k vk/kkfjr 5 495 (a) 90000 (b) 99000 (c) 90900 (d) 99990 mÙkj (b) mnk-1 ,d i;ZVd izfrfnu mrus gh :i;s [kpZ djrk gS chtxf.krh; lw=ksa ij vk/kkfjr ftrus mlds i;ZVu ds fnuksa dh la[;k gS A mldk dqy [kpZ :i;s 361 gS] rks Kkr djsa fd mldk i;ZVu fdrus fnuksa rd pyk \ (a) 17 days (b) 19 days (c) 21 days (d) 31 days 2 mÙkj (b) mnk-1  2 + 1  ds cjkcj gS \ mnk-2 ;fn nks la[;kvksa dk ;ksx 22 gS] vkSj muds oxksaZ dk ;ksx  2 404 gS] rks mu la[;kvksa dk xq.kuQy Kkr djsa \ (a) 2 1 (b) 3 1 2 2 (a) 40 (b) 44 (c) 80 (d) 89 (c) 4 1 (d) 5 1 2 2 mÙkj (a) mÙkj (c) mnk-3 tc ,d nks vadkas dh la[;k dks mlds vadkas ds ;ksx ls xq.kk fd;k tkrk gS] rks xq.kuQy 424 gksrk gSA 0.51  0.051  0.051 + 0.041  0.041  0.041 mnk-2 dk tc mlds vadkas dks vkil esa cnyus ls izkIr la[;k 0.51  0.051− 0.051 × 0.041+ 0.041  0.041 eku D;k gS \ dks vadkas ds ;ksx ls xq.kk fd;k tkrk gS rks ifj.kke 280 gksrk gS A la[;k ds vadkas dk ;ksx fdruk gS\ (a) 0.92 (b) 0.092 (a) 7 (b) 9 (c) 0.0092 (d) 0.00092 (c) 6 (d) 8 mÙkj (b) mÙkj (d) Js.kh vk/kkfjr ¼lekUrj Js.kh] xq.kksÙkj Js.kh] gjkRed Js.kh½ mnk-1 50 ls de 3 ds lHkh xq.ktksa dk ;ksxQy Kkr djks\ (a) 400 (b) 408 (c) 404 (d) 412 mÙkj (b) mnk-2 fuEufyf[kr lekarj Js.kh esas fdrus in gSa \ 7, 13, 19,.............. , 205 mnk-3 5 ds mu lHkh /kukRed xq.kkadksa dk ;ksx Kkr djsa tks 100 ls de gS \ 11  3 Yk?kqÙke lekioR;Z o egÙke CHAPTER lekiorZd (LCM & HCF) xq.ku[k.M mnk-1 HCF fudkyuk % nks la[;kvksa dk HCF Hkkx fof/k ,d la[;k dks nwljs dk xq.ku[k.M dgk tkrk gS] ;fn }kjk fudkyk tkrk gS] rks HkkxQy Øe'k% 3, 4, ;g nwljs dks iwjh rjg ls foHkkftr dj nsA bl izdkj 3 ,oa 5 izkIr gksrk gSA ;fn nks la[;kvksa dk HCF, o 4, 12 ds xq.ku[k.M gSaA 18 gks rks la[;k,¡ Kkr dhft,A lekiorZd gy nks la[;k,¡ a ,oa b gSa og la[;k tks nks ;k nks ls vf/kd nh gq;h la[;kvksa dks iw.kZr% foHkkftr dj ns] mu la[;kvksa dk lekiorZd dgykrh gSA bl izdkj 9, 18, 21 ,oa 33 dk ,d lekiorZd 3 gSA vfUre Hkktd HCF gksrk gS A d = 18 LCM (Lowest Common Multiple) c = 5 × d = 5 × 18 = 90 ¼y?kqÙke lekioR;Z½ a = (4 × C) + d = (4 × 90) + 18 = 378 og lcls NksVh la[;k tks nh x;h la[;kvksa ls b = 3a + c iw.kZr;k% foHkkT; gks] LCM dgykrh gSA = (3 × 378) + 90 = 1134 + 90 Power okys la[;k dk LCM fudkyuk & vHkkT; = 1224, 378 Ans xq.ku[k.M djus ds ckn Power ds :Ik esa fy[ksxs vkSj ftrus vHkkT; la[;k dk iz;ksx gksxk mls xq.kk Power okyh la[;k dk HCF fudkyuk ds :Ik esa fy[ksxs vkSj ml ij vf/kdre Power igys Base dk vHkkT; xq.ku[k.M djasxs vkSj mls Power j[ksaxAs ds :Ik esa fy[ksx a s vkSj tks lHkh esa Common vHkkT; la[;k mnk-1 (12)16, (18)15, (30)18 dk LCM fudkysA gksxh] mls xq.kk ds :Ik esa fy[kasxs vkSj ml ij U;wure Power j[ksaxAs gy (12)16 = (2 × 2 × 3)16 = (22 × 3)16 = 232 × 316 (18)15 = (2 × 3 × 3)15 = (2 × 32)15 = 215 × 330 mnk-1 (24)8, (36) 12, (18) 16 dk HCF fudkysAa (30)18 = (2 × 3 × 5)18 = 218 × 318 × 518 gy 24 = (23 × 3) 8 = 224 × 38 vr% LCM = 232 × 330 × 518 Ans. 36 = (22 × 32)12 = 224 × 324 18 = (2 × 32)16 = 216 × 332 fHkUuksa dk LCM fudkyuk vr% e-l-i- = 216 × 38 v'akks dk LCM LCM = fHkUu dk HCF fudkyuk gjks dk HCF 1 o 5 dk LCM ? v'ak dk HCF mnk-2 fHkUu dk HCF = 2 8 gj dk LCM 1 o 5 dk LCM 18 12 6 LCM =  5 mnk-1 , , 2 o 8 dk HCF 2 25 7 35 18,12,6 dk HCF 6 gy = HCF (Highest Common Factor) 25,7,35 dk LCM 175 egÙke lekiorZd fdlh nks la[;kvksa dk tksM+ rFkk y-l-Ik- dk e-l-i-] mu og lcls cM+h la[;k ftlls nh x;h lHkh la[;k,¡ la[;kvksa ds e-l- ds cjkcj gksrk gSA iw.kZr% foHkkftr gks] HCF dgykrk gSA ekuk nks la[;k,¡ x rFkk y gS] rFkk mudk e-l- H gSA tSls & 18 ,oa 24 dk e-l-i- 6 gSA vr% x = Ha y = Hb 12 tgk¡ a rFkk b ijLij vHkkT; gSaA mnk-3 N% ?kf.V;k¡ ,d lkFk ctuh vkjEHk gqbZ] ;fn ;s x, y dk LCM = Hab ?kf.V;k¡ Øe'k% 2, 4, 6, 8, 10, 12 lsd.M ds vkSj x + y = H(a + b) varjky ls cts] rks 30 feuV esa fdruh ckj ;s vc ‘a’ rFkk ‘b’ ijLij vHkkT; la[;k,¡ gS] rks (a + b) rFkk ,d lkFk bDdÎh ctsx a h\ ab Hkh ijLij vHkkT; gksxhA blls ;g fu"d"kZ fudyrk (a) 4 ckj gS fd H(a + b) rFkk Hab dk e-l- H gh gksxk] tks x rFkk (b) 10 ckj y dk Hkh e-l- gSA (c) 16 ckj (d) buesa ls dksbZ ugha LCM ,oa HCF esa Relation LCM × HCF = nksukas la[;kvksa dk xq.kuQy fHkUuksa ds y-l-i- rFkk e-l-i- mnk-1 nks la[;kvksa dk LCM ,oa HCF Øe'k% 420 ,oa 28 gSaA ;fn ,d la[;k 84 gS] rks nwljh la[;k Kkr dhft, & 420 × 28 gy nwljh la[;k = = 140 84 14 42 21 mnk-1 , , dk egÙke lekioZrd Kkr dhft, & tc dgk tk;s fd x,y,z ds fy;s og NksVh ls NksVh 33 55 22 la[;k D;k gksxh ftlesa Hkkx nsus ij r 'ks"k cp tk;s] 11 55 33 44 blds fy, mŸkj gksxk x, y, z dk (LCM + r)A mnk-2 , , , dk y?kqÙkee lekioZrd Kkr 14 42 35 63 og NksVh ls NksVh la[;k ftls x,y,z ls Hkkx djus ij dhft, & 'ks"kQy Øekxr a,b,c gksA blds fy;s mÙkj gksxk & (x, y, z) - K dk LCM A mnk-3 rhu O;fDr ,d 11 fdeh- yEcs o`Ùkkdkj iFk ij ,d lkFk ,d gh fn'kk esa pyuk izkjaHk djrs gSA vH;kl ç'u mudh pky Øe'k% 4, 5.5 ,oa 8 fdeh- izfr ?kaVk egÙke lekiorZd vkèkkfjr gSA os rhuksa ,d lkFk fdrus le; ckn izkjafHkd fcUnq ij feysx a s\ y-l-i- rFkk e-l-i- ds eè; lacaèk vkèkkfjr mnk-1 84, 126, 140 dk egÙke lekiorZd fdruk gS \ mnk-2 x6 – 1 vkSj x4 + 2x3 – 2x1 – 1 dk e-l- D;k gksxk \ (a) x2 + 1 (b) x – 1 (c) x2 – 1 (d) x + d mÙkj (c) mnk-1 nks la[;kvksa dk y-l- 225 rFkk e-l- 5 gSA ;fn mlesa ls ,d la[;k 25 gS] rks nwljh la[;k Kkr y?kqÙke lekioR;Z vk/kkfjr djsa \ (a) 5 (b) 25 (c) 45 (d) 225 mÙkj (c) mnk-2 nks la[;kvksa dk ;ksx 36 gS] budk egÙke mnk-1 15, 18, 24, 27, 36 dk y?kqÙke lekioR;Z D;k lekiorZd 3 rFkk y?kqÙke lekioR;Z 105 gS] bu gksxk \ la[;kvksa ds O;qRØeksa dk ;ksx fdruk gksxk \ mnk-2 nks la[;kvksa dk ;ksx 45 gSA mudk varj ;ksx 2 3 1 (a) (b) dk gS] rks mudk y-l- Kkr djsaA 35 25 9 4 2 (a) 200 (b) 250 (c) (d) 35 25 (c) 100 (d) 150 mÙkj (c) mÙkj (c) 13 mnk-3 nks la[;kvksa ds e-l- rFkk y-l- dk ;ksx 680 gS mnk-4 nks la[;kvksa ds egÙke lekiorZd rFkk y?kqÙke mudk y-l-] e-l- dk 84 xq.kk gSA ;fn ,d lekioR;Z Øe'k% 12 rFkk 72 gS] ;fn bu la[;kvksa la[;k 56 gS] rks nwljh la[;k Kkr djsa \ dk ;ksx 60 gks] rks buesa ls NksVh la[;k fuEu esa (a) 84 (b) 12 ls dkSu&lh gS \ (c) 8 (d) 96 (a) 12 (b) 24 mÙkj (d) (c) 60 (d) 72 mÙkj (b) 14