class 7math book

dflysf ;a} lrqx¿ eg] lrq xg' \\ . ;dx\" sf ljleGg ks| f/cg;' f/ logLx¿sf eg] lrqx¿ klg km/s km/s xG' 5g\\ . dflysf ljm| ofsnfkx¿sf cfwf/df tnsf kZ| gx¿df 5nkmn u/ M -s_ sg' } klg eg] lrqdf s] s] xG' 5g\\ < -v_ ;jJ{ ofks ;dx\" nfO{ hgfpg s:tf] cfsf/sf] Ifq] ko| fu] ul/Psf] 5 < -u_ ;jJ{ ofks ;dx\" afxs] cGo ;dx\" hgfpgsf nflu s:tf] cfsf/sf] Ifq] ko| fu] ul/Psf] 5 < -ª_ dflysf] 5nkmnsf cfwf/df eg] lrq;u“ ;DalGwt ljleGg tYox¿ kQf nufpm . tL tYox¿ ;fyL;u“ 5nkmn u/ . lgisifn{ fO{ tnsf tYox¿;u“ tn' gf u/L x/] . 1. eg] lrqdf ;jJ{ ofks ;dx\" sf nflu cfotsf/ Ifq] sf] ko| fu] ul/G5 . 2. To:t} cGo ;dx\" sf nflu ufn] fsf/ jf jQ[ fsf/ Ifq] sf] ko| fu] ul/G5 . 3. kT| os] ;dx\" df k/s] f ;b:ox¿nfO{ ;DalGwt Ifq] leq kg{] u/L g} /flvPsf] xG' 5 . 4. eg] lrqdf ;dx\" x¿sf lsl;dcg;' f/ ;femf ;b:ox¿nfO{ vlK6Psf] efudf /flvPsf] xG' 5 . To:t} afs“ L ;b:ox¿nfO{ cfcfkm\\ gf] ;dx\" df /flvPsf] xG' 5 . 5. o;/L ;dx\" jf ;dx\" sf ljleGg ;DaGwx¿nfO{ hgfpg] lrqfTds k:| tl' tnfO{ eg] lrq (venn-diagram) elgG5 . cEof; 11.3 1. tnsf eg] lrqsf cfwf/df kT| os] ;dx\" nfO{ ;\\r\" Ls/0f / JofVof bj' } ljlwaf6 k:| tt' u/L nv] . -s_ -v_ -u_ -3_ U U C D U U 2 B 1D A 4 A 17 28 3 78 9 39 4 10 3 2 8 E9 1 3 4 F 5 11 5 6 9 5 11 10 6 4 10 11 61 56 11 2 7 8 10 7 ca ;jJ{ ofks ;dx\" U kQf nufpm . 2. olb U = {a, b, c, d, e, f, g, h, i, j}, P = {a, b, c, d, e, i} Q = {a, e, i}, R = {b, c, d, j}, S = {i, e, a} / T = {a, b, c, f, g} 5 eg] tnsf ;dx\" x¿nfO{ 56' 6\\ f56' 6\\ } eg] lrqdf k:| tt' u/L bv] fpm M -s_ U, P / Q -v_ U, Q / R -u_ U, Q / S -3_ U, R / T 3. dfly kZ| g g=+ 1 / 2 df lbOP h:t} kZ| gx¿ cfkm“} agfO{ ;dfwfg u/ . ;fyL;u“ ldn/] ;dfwfg u/ . cfkm\\ gf] / ;fyLsf] ;dfwfg tn' gf u//] klg x/] . 96 ul0ft, sIff – &

11.4 cnlUuPsf / vlK6Psf ;dx\" x¿ (Disjoint and Overlapping Sets) tnsf ljm| ofsnfkx¿ cWoog u/ / 5nkmn u/ M -s_ tn rf/cf6] f ;dx\" x¿ lbOPsf 5g\\ M U = {10 ;Ddsf k0\" f{ ;ªV\\ ofx¿} = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} A = { 6 sf u0' fgv08x¿} = {1, 2, 3, 6} B = {8 sf u0' fgv08x¿} = {1, 2, 4, 8} C = {5 sf] 10 ;Ddsf ckjT{ ox¿} = {5,10} dflysf ;dx\" sf cfwf/df tnsf kZ| gx¿df 5nkmn u/ . -s_ s] ;dx\" A sf sg' } ;b:ox¿ ;dx\" C df k5g{ \\ < -v_ s] ;dx\" x¿ B / C df ;femf ;b:ox¿ 5g\\ < -u_ ;dx\" x¿ A / B df ;femf ;b:ox¿ s] s] 5g\\ < oxf“ ;dx\" A sf sg' } klg ;b:o ;dx\" C df k/s] f 5g} g\\ . To;n} ] ;dx\" A / C cnlUuPsf ;dx\" x¿ xg' \\ . To;} u/L ;dx\" x¿ A / B sf ;femf ;b:ox¿ 1 / 2 xg' \\ . To;n} ] ;dx\" x¿ A / B cfk;df vlK6Psf ;dx\" x¿ xg' \\ . -3_ ;dx\" x¿ B / C s:tf ;dx\" x¿ xfn] fg\\ < 5nkmn u/ . -v_ dflysf ;dx\" x¿nfO{ eg] lrqdf bv] fpg] ko| f; u/f“} M – ;aeGbf klxn] ;jJ{ ofks ;dx\" U agfcf“} . AB U C – A / B sf ;femf ;b:ox¿ 1 / 2 nfO{ vlK6Psf] Ifq] df e/f“} . 31 4 5 – A sf afs“ L ;b:ox¿ 3 / 6 nfO{ A sf] afs“ L Ifq] df e/f“} . 0 62 8 – B sf afs“ L ;b:ox¿nfO{ B sf] afs“ L Ifq] df e/f“} . 10 7 – C sf ;a} ;b:ox¿nfO{ C df e/f“} . -u_ dflysf] e]g lrqnfO{ cEof; k'l:tsfdf agfO{ vlK6Psf ;d\"xx¿sf] vlK6Psf] efunfO{ /ªu\\ nufP/ bv] fpm . -3_ dflysf 5nkmnsf cfwf/df vlK6Psf ;dx\" / cnlUuPsf ;dx\" x¿sf] kl/efiff lbg] ko| f; u/ . cfkm\" n] nv] s] f] kl/efiffnfO{ ;fyLn] nv] s] f] kl/efiff;u“ tn' gf u/L 5nkmn u/L x/] . lgisifn{ fO{ tnsf] kl/efiff;u“ tn' gf u/L x/] . 1. olb sg' } bO' { jf bO' e{ Gbf a9L ;dx\" x¿df ;femf ;b:ox¿ 5g\\ eg] To:tf ;dx\" x¿nfO{ vlK6Psf ;dx\" (overlapping sets) elgG5 . 2. olb sg' } bO' { jf bO' e{ Gbf a9L ;dx\" x¿df ;femf ;b:ox¿ 5g} g\\ eg] To:tf ;dx\" x¿nfO{ cnlUuPsf ;dx\" (disjoint sets) elgG5 . ul0ft, sIff – & 97

cEof; 11.4 1. tn lbOPsf ;dx\" sf cfwf/df vlK6Psf / cnlUuPsf ;dx\" x¿ 56' o\\ fpm M A = {2, 4, 6, 8, 10, 12} B = {1, 3, 5, 7, 9, 11} C = {0, 1, 3, 5, 7, 11} D = {5, 6, 7, 8, 10} E = {0, 2, 9, 12} -s_ A / B -v_ A / C -u_ B / C -3_ A / D -ª_ A / E -r_ B / D -5_ B / E -h_ C / D -em_ C / E 2. A = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} eP tnsf ;dx\" x¿nfO{ ;r\" Ls/0f ljlwaf6 nv] . -s_ A sf kT| os] ;b:ox¿df 1 hf8] b\\ f aGg] ;dx\" B -v_ A sf kT| os] ;b:ox¿nfO{ 2 n] u0' fg ubf{ aGg] ;b:o ;dx\" C -u_ A df ePsf lahf/] ;ªV\\ ofx¿sf] ;dx\" D -3_ A df ePsf 10 sf u0' fg v08x¿sf] ;dx\" E -ª_ dflysf ;dx\" x¿ A, B, C, D / E df vlK6Psf / cnlUuPsf ;dx\" x¿ 56' o\\ fpm . 3. sg' } kfr“ kfr“ cf6] f vlK6Psf / cnlUuPsf ;dx\" vfh] ÷agfpm . ;fyL;u“ Pscsfs{ f ;dx\" x¿ 56' o\\ fpg cEof; u/ . 98 ul0ft, sIff – &

11.5 ;dx\" sf] ;o+ fh] g (Union of Sets) Y tnsf ljm| ofsnfk / lrqx¿ cWoog u/L 5nkmn u/ M X dflysf ;dx\" x¿ X / Y sf ;b:ox¿nfO{ lrgfpg] gfd lbP/ ldnfP/ /fvf“} / 5fof kf/L bv] fcf“} . ca tn lbOPsf kZ| gx¿df 5nkmn u/f“} . 1. ;dx\" x¿ X / Y sf ;femf ;b:ox¿ s] s] xg' \\ < la/fnf] v/fof] l;x+ 2. s] ;dx\" X df ;dx\" Y sf ;a} ;b:ox¿ k5g{ \\ < af3 s's'/ aª\\u'/ 3. ;dx\" x¿ X / Y sf ;b:ox¿nfO{ ;r\" Ls/0f ljlwaf6 nv] . :ofn ufO{ xfTtL e}“;L 4. ;dx\" x¿ X / Y sf ;femf ;b:ox¿nfO{ ;r\" Ls/0f ljlwaf6 nv] . 5. ;dx\" x¿ X / Y sf ;a} ;b:ox¿nfO{ gbfx] f¥] ofOsg ;r\" Ls/0f ljlwaf6 nv] . dfly 5fof k/s] f] efunfO{ ;dx\" x¿ X / Y sf] ;o+ fh] g (union of sets) elgG5 . o;nfO{ ;o+ fh] g lrxg\\ ∪ n] hgfOG5 . To;n} ] X U Y = X ;o+ fh] g Y = {la/fnf,] af3, :ofn, e;}“ L, v/fof,] ss' /' , ufO,{ l;x“ , aªu\\ /' , xfQL} xG' 5 . dflysf] 5nkmnsf cfwf/df ;dx\" sf] ;o+ fh] gsf] kl/efiff / tYox¿ nV] g] ko| f; u/ . cfkm\\ gf] nv] fOnfO{ ;fyL;u“ 5nkmn u/ . clGtd lgisifn{ fO{ tnsf] kl/efiff / tYox¿;u“ tn' gf u/L x/] . 1. sg' } bO' { ;dx\" x¿ X jf Y jf bj' } ;dx\" sf ;b:ox¿ X / Y sf ;Dk0\" f{ ;b:ox¿ k/s] f] ;dx\" x¿sf] cj:yfnfO{ ;dx\" x¿ X / Y sf] ;o+ fh] g elgG5 . 2. o;nfO{ X∪Y n] hgfOG5 . 3. o;nfO{ X ;o+ fh] g Y(X union Y) eg/] kl9G5 . ul0ft, sIff – & 99

pbfx/0f 1 ;d:of M olb U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, P = {1, 2, 3, 4, 5}, Q = {1, 4, 5}, R = {3, 4, 5, 6, 7, 8} / S = {2, 3, 6, 7, 10} eP tnsf kT| os] ;dx\" x¿sf] ;o+ fh] g lgdf0{ f u/ / eg] lrqdf klg bv] fpm M -s_ P ∪ Q -v_ Q ∪ R -u_ Q ∪ S U P ;dfwfg 6 2 Q 9 7 1 -s_ oxf“ P ∪ Q = {1, 2, 3, 4, 5} ∪ {1, 4, 5} 4 5 83 10 = {1, 2, 3, 4, 5} lrqdf P ∪ Q hgfpg] ;dx\" nfO{ 5fof kf/L bv] fOPsf] 5 . -v_ Q ∪ R = {1, 4, 5} U { 3, 4, 5, 6, 7, 8} Q U 2 R = {1, 3 ,4 ,5, 6, 7, 8} 91 3 lrqdf Q ∪ R nfO{ 5fof kf/L bv] fOPsf] 5 . 4 6 10 57 8 -u_ Q ∪ S = {1, 4, 5} ∪ {2, 3, 6, 7, 10} Q S U 1 8 = {1, 2, 3, 4, 5, 6, 7, 10} 4 23 9 5 6 lrqdf Q ∪ S nfO{ 5fof kf//] bv] fOPsf] 5 . 7 10 pbfx/0f 2 olb E = {6 sf u0' fgv08x¿} / F = {10 ;Ddsf lahf/] ;ªV\\ ofx¿ } eP -s_ ;dx\" x¿ E / F nfO{ ;r\" Ls/0f ljlwaf6 nv] . -v_ E ∪ F / F ∪ E nfO{ eg] lrqdf bv] fpm . -u_ E ∪ F = F ∪ E xG' 5 egL kd| fl0ft u/ . -3_ F ∪ F = F xG' 5 egL kd| fl0ft u/ . E FU ;dfwfg -s_ E = {1, 2 ,3, 6} 5 2 17 F = { 1, 3, 5, 7, 9} 6 39 -v_ E ∪ F / F ∪ E nfO{ eg] lrqdf bv] fpb“ f, ul0ft, sIff – & 100

-u_ E ∪ F = {1, 2, 3, 6} ∪ {1, 3, 5, 7, 9} = {1, 2, 3, 5, 6, 7, 9} F ∪ E = { 1, 3, 5, 7, 9} ∪ {1, 2, 3, 6} = {1, 2, 3, 5, 6, 7, 9} To;n} ] E ∪ F = F ∪ E xG' 5 . kd| fl0ft eof] . F ∪ F = {1, 3, 5, 7, 9} ∪ {1, 3, 5, 7, 9} = {1, 3, 5, 7, 9} = F ∴ F ∪ F kd| fl0ft eof] . pbfx/0f 3 tnsf ;dx\" sf cfwf/df ;fl] wPsf kZ| gx¿sf] ;dfwfg u/ M U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}, J = {1, 3, 5, 7, 9, 11}, K = {2, 4, 6, 8, 10} / L = {0, 1, 3, 6, 11} 5 . -s_ (J ∪ K) ∪ L = J ∪ (K ∪ L), kd| fl0ft u/ . -v_ (J ∪ K) ∪ L nfO{ eg] lrqdf bv] fpm . -u_ s] (J ∪ K) ∪ L = J ∪ (K ∪ L) = U nV] g ;lsG5 < ;dfwfg -s_ oxf“ (J ∪ K) ∪L = [{1, 3 ,5, 7, 9, 11} ∪ {2, 4, 6, 8, 10}] U {0, 1, 3, 6, 11} = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11} ∪ {0, 1, 3, 6, 11} = {0,1, 2, 3, 4, 5, 6, 7, 8, 9, 10,11} To;n} ,] J ∪ (K ∪ L) = {1, 3, 5, 7, 9, 11} ∪ [{2, 4, 6, 8, 10} U {0, 1, 3, 6, 11}] = {1, 3, 5, 7, 9, 11} ∪ {0, 1, 2, 3, 4, 6, 8, 10, 11} ={0,1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11} J K U To;n} ] (J ∪ K) ∪ L = J ∪ (K ∪ L) kd| fl0ft eof] . 5 28 -v_ oxf“ (J ∪ K) ∪ L nfO{ eg] lrqdf bv] fpb“ f 7 4 10 9 1, 3, 11 6 o L ul0ft, sIff – & 101

-u_ oxf“ (J ∪ K) ∪ L = {0,1,2, 3, 4, 5, 6, 7, 8, 9, 10, 11} 5 . To:t,} J ∪ (K ∪ L) = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11} 5 . / U = {0, 1, 2, 3, 4 5, 6, 7, 8, 9, 10, 11} 5 . To;n} ] (J ∪ K) ∪ L = J ∪ (K ∪ L) = U nV] g ;lsG5 . cEof; 11.5 1. bfofs“ f] h:tf] 56' 6\\ f56' 6\\ } eg] lrq lvr . tn lbPsf ;dx\" sf] ;o+ fh] g hgfpg] efu kQf nufO{ 5fof kf//] bv] fpm M P -s_ P ∪ Q -v_ R ∪ Q -u_ Q ∪ R -3_ P ∪ R -ª_ R ∪ P -5_ (P ∪ Q) ∪ R -h_ P ∪ (Q ∪ R) Q R 2. olb U = {12 ;Ddsf kf| sl[ ts ;ªV\\ ofx¿}, A = {hf/] ;ªVofx¿} / B = {lahf/] ;ªV\\ ofx¿} eP -s_ ;dx\" x¿ U, A / B nfO{ ;r\" Ls/0f ljlwaf6 nv] . -v_ eg] lrq agfO{ lgDgfg;' f/sf ;dx\" kQf nufpm . -c_ A ∪ B -cf_ B ∪ A -O_ B ∪ B -O_{ A∪ A -u_ kd| fl0ft u/ M -c_ A ∪ B = B ∪ A -cf_ B ∪ B = B -O_ A ∪ A = A -O_{ A ∪ B = B ∪ A = U 3. olb U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} / A = {1, 3, 5, 6, 7, 9} eP -s_ U ∪ A nfO{ eg] lrqdf bv] fpm . -v_ A ∪ U = U ∪ A = U kd| fl0ft u/L bv] fpm . 4. olb M = {8 sf u0' fgv08x¿} / N = { } = Ø eP, -s_ M nfO{ ;r\" Ls/0f ljlwaf6 nv] . -v_ M ∪ N nfO{ eg] lrqdf bv] fpm . -u_ N ∪ M nfO{ ;r\" Ls/0f ljlwaf6 bv] fpm . -3_ s] M ∪ N = N ∪ M nV] g ;lsG5 < kd| fl0ft u/L bv] fpm . 5. olb U = {a, b, c, d, e, f, g, h, i, j, k}, E = {a, b, c, d}, F = {e, f, g, h} / G = {a, b, c, e, f, g, i, j} eP, -s_ lgDgfg;' f/sf ;dx\" x¿nfO{ eg] lrqdf bv] fpm M -c_ (E ∪ F) ∪ G -cf_ E ∪ (F ∪ G) -v_ s] (E ∪ F) ∪ G = E ∪ (F ∪ G) nV] g ;lsG5 < kd| fl0ft u/L bv] fpm . 102 ul0ft, sIff – &

11.6 ;dx\" x¿sf] kl| tR5b] g (Intersection of Sets) sIff 7 df k9g\\ ] 10 hgf ljBfyLn{ ] uLt ufpg] / gfRg] kl| tofl] utfdf efu lnPsf 5g\\ . uLt ufpgs] f] ljBfyLx{ ¿sf] ;dx\" nfO{ A / gfRg] ljBfyLs{ f] ;dx\" nfO{ B dflgPsf] 5 . uLt ufpg] (A) = {cK;/f, alatf, l;df, sz] /, /lag, ;lag} gfRg] (B) = {sz] /, ho/fd, alatf, c~h,' k0\" f,{ kl| dnf} ca, uLt ufpg] ljBfyLs{ f] ;dx\" / gfRg] ljBfyLs{ f] ;dx\" nfO{ eg] lrqdf bv] fpb“ f, U AB cK;/f ho/fd l;df s]z/ c~h' /lag alatf k\"0f{ ;lag k|ldnf -s_ bj' } ljm| ofsnfk dg k/fpg] ljBfyL{ sg' sg' /x5] g\\ < eGg ;S5f} < -v_ bj' } ljm| ofsnfk dg k/fpg] ljBfyL{ kg{] Ifq] nfO{ 5fof kf//] bv] fpm . -u_ s] uLt ufpg] dfq ljBfyLs{ f] ;dx\" agfpg ;S5f} < -3_ s] gfRg hfGg] ljBfyL{ dfqsf] ;dx\" agfpg ;S5f} < A / B bO' { cf6] f ;dx\" x¿ 5g\\ . ;dx\" x¿ A / B sf ;femf ;b:ox¿sf ;dx\" nfO{ ;dx\" sf] kl| tR5b] g (Intersection of sets) elgG5 . o;nfO{ ∩ n] hgfOG5 . cyft{ ,\\ A∩B nfO{ A kl| tR5b] g B (A intersection B ) elg k9g\\ k' 5{ . pbfx/0f 1 olb A = {1,2,3,4,5} / B = {2,4,6,8,10} eP A ∩ B ;dx\" lgdf0{ f u/L eg] lrqdf 5fof kf/L bv] fpm . ;dfwfg U oxf,“ A = {1, 2, 3, 4, 5} / B = {2, 4, 6, 8,10} 5g\\ . ca, A ∩ B egs] f] bj' } ;dx\" df kg{] ;femf ;b:ox¿ xg' \\ . AB ctM A ∩ B = {1, 2, 3, 4, 5} ∩ {2, 4, 6, 8, 10} = {2, 4} 16 2 348 5 10 ul0ft, sIff – & 103

pbfxf/0f 2 eg] lrqdf bv] fpb“ f olb A = {1,3,5,7,9} / B = {1, 9} eP A ∩ B nfO{ eg] lrqdf 5fof kf/L bv] fpm . U ;dfwfg M oxf,“ A = {1, 3, 5, 7, 9} / B = {1, 9} 5g\\ . 3 BA A ∩ B = {1, 3, 5, 7, 9} ∩ {1.9} = {1, 9} 5 1 9 7 pbfx/0f 3 eg] lrqdf bv] fpb“ f U olb A = {a, b, c d, e} / B = {x, y, z} eP A∩B nfO{ eg] lrqdf 5fof kf/L bv] fpm . AB ;dfwfg x oxf,“ A = {a, b, c, d, e} / B = {x, y, z} ab y ca, A / B df sg' } klg ;b:o ;femf gePsfn] A∩B vfnL ;dx\" xf] . cd z To;n} ] 5fof kf/L bv] fpg ;lsPg . To;n} ] A∩B = { } jf φ xG' 5 . pbfx/0f 4 e eg] lrqdf bv] fpb“ f U olb ;jJ{ ofks ;dx\" U = {1, 2, 3, 4, 5, 6, 7 , 8, 9}, AB A = {1, 3, 5, 7, 9} / B = {1, 2, 3, 4, 5} eP A ∩ B kQf nufO{ eg] lrqdf bv] fpm . 671 2 4 ;dfwfg 8 9 3 5 A ∩ B = {1, 3, 5, 7, 9} ∩ {1, 2, 3, 4, 5} = {1, 3, 5} cEof; 11.6 AB 1. lbOPsf] eg] lrqaf6 tn lbOPsf ;dx\" x¿ kQf nufpm M a bdg -s_ A ∩ B -v_ B ∩ A -u_ A ∩ A c eh -3_ U ∩ A -ª_ U ∩ B fi 2. olb U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {1, 2, 3, 6} / B = {1, 2, 3, 4, 5} eP A∩B kQf nufpm / eg] lrqdf klg k:| tt' u/ . 3. olb U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, A = {1, 3, 5, 7, 9}, B = {3, 6, 9} / C = {2, 4, 6, 10} eP -s_ A∩B -v_ B∩C -u_ A∩C kQf nufO{ kT| os] nfO{ eg] lrqdf klg bv] fpm . 4. olb P = {a, b, c, d, e} / Q = {a, b, c} eP P∩Q kQf nufO{ eg] lrq klg agfpm . 5. olb U = {10 eGbf ;fgf kf| sl[ ts ;ªV\\ ofx¿}, M = {2 sf ckjTox{ ¿}, N = {8 sf u0' fg v08x¿} / O = {hf/] ;ªVof} eP, -s_ dflysf ;dx\" x¿nfO{ eg] lrqdf bv] fpm . -v_ M∩N kQf nufpm . -u_ s] M∩(N∩O) = (M∩N)∩O nV] g ;lsG5 < kd| fl0ft u/L bv] fpm . 104 ul0ft, sIff – &

PsfO 12 k0\" f{ ;ªV\\ of (Whole Number) 12.1 ;ªV\\ ofsf] ju{ / jud{ n\" (Square and Square Root of the number) 1. ;ªV\\ ofsf] ju{ tnsf ljm| ofsnfkx¿ cWoog u/L 5nkmn u/ M nx/df 4 cf6] f / kªl\\ Stdf 4 cf6] f uR' rfx¿ /fvf“} . hDdf uR' rfx¿ slt eP < oxf,“ bj' } nx/ / kªl\\ Stdf rf/ rf/ cf6] f uR' rfx¿ 5g\\ . To;n} ] 4 x 4 = 16 xG' 5 . To;n} ] 4 sf] ju{ ;ªV\\ of 16 xG' 5 . dflysf] ljm| ofsnfkaf6 jus{ f] kl/efiff nv] . sg' } k0\" f{ ;ªV\\ ofnfO{ cfkm“} ;u“ u0' fg ubf{ cfpg] u0' fgkmnnfO{ g} ju{ ;ªV\\ of elgG5 . ju{ ;ªV\\ of lgsfNbf lbPsf] ;ªV\\ ofnfO{ ToxL ;ªV\\ ofn] u0' fg ugk{' 5{ . h:t} M 4 sf] ju{ ;ªV\\ of 4 x 4 = 16 xG' 5 . 4 sf] jun{ fO{ 42 = 4 x 4 = 16 nV] g ;lsG5 . gf6] M sg' } ;ªV\\ of dfgf“} 5 sf] ju{ eGgfn] 5 PsfO nDafO ePsf] Pp6f jus{ f] Ifq] kmn eGg] a‰' gk' 5{ . ctM 52 = 5 x 5 = 25 ePsf] xf] . pbfx/0f 1 tn lbOPsf ;ªV\\ ofsf] ju{ ;ªV\\ of lgsfn M -s_ 5 -v_ 12 -u_ 1 -3_ 0.04 2 ;dfwfg -s_ 5 sf] ju{ ;ªV\\ of = 52 = 5 x 5 = 25 -v_ 12 sf] ju{ ;ªV\\ of = 122 = 12 x 12 = 144 -u_  1 2 sf] ju{ ;ªV\\ of =  1 2  1  1  1 -3_ (0.04)2 sf] ju{ ;ªV\\ of = (0.04)2 =0.04x0.04=0.0016  2  2 2 2 4 2. ;ªV\\ ofsf] jud{ n\" 105 -s_ jud{ n\" sf] kl/ro tnsf ljm| ofsnfkx¿ cWoog u/L 5nkmn u/ M -c_ Pp6f ljBfnosf] sIff 7 df hDdf 36 hgf ljBfyLx{ ¿ 5g\\ . ltgLx¿nfO{ jufs{ f/ ¿kdf ldnfP/ /fvf“} . kT| os] lsgf/fdf slt slt ljBfyL{ k5g{ \\ < oxf,“ Pp6f lsgf/fdf 6 hgf ljBfyLx{ ¿ k/s] f 5g\\ . 36 = 6 x 6 xG' 5 . To;n} ] x/s] lsgf/fdf 6/6 hgf kg{] u/L ldnfOPsf] /x5] . ca, 36 sf] jud{ n\" 6 xG' 5 . jud{ n\" nfO{ xfdL ( ) lrxg\\ n] hgfp5“ f“} . ul0ft, sIff – &

-cf_ dflysf] ljm| ofsnfksf cfwf/df jud{ n\" sf] kl/efiff nV] g ;S5f“} < ;fyL;u“ 5nkmn u/L nv] s] f] kl/efiffnfO{ tnsf] kl/efiff;u“ tn' gf u//] x/] . sg' } klg ju{ ;ªV\\ ofsf bO' c{ f6] f p:tfp:t} u0' fgv08x¿ xG' 5g\\ eg] tL u0' fgv08x¿dWo] Pp6fnfO{ To; ;ªV\\ ofsf] jud{ n\" elgG5 . jud{ n\" nfO{ ( ) lrxg\\ df klg nl] vG5 . h:t} M 36  62  6 xG' 5 . -v_ u0' fgv08 ljlwaf6 jud{ n\" lgsfNg] tl/sf u0' fgv08 ljlwaf6 jud{ n\" lgsfNbf lgDgfg;' f/sf] kl| jm| of ckgfpgk' 5{ M 1. lbOPsf] ;ªV\\ ofsf] ¿9 u0' fgv08 lgsfNg] 2. ¿9 u0' fgv08nfO{ lrxg\\ leq /fVg] 3. hf8] f hf8] f ;ªV\\ ofnfO{ 3ftfªs\\ sf] ¿kdf nV] g] 4. kT| os] hf8] fsf] Pp6f Pp6f ;ªV\\ of nV] g] / u0' fg ug{] 5. kf| Kt u0' fgkmn g} ;f] ;ªV\\ ofsf] jud{ n\" xG' 5 pbfx/0f 2 ca, 81 sf] jud{ n\" lgsfNbf, 81 sf] jud{ n\" lgsfn M 81 sf] jud{ n\" ;dfwfg M oxf“ 81 sf ¿9 u0' fgv08x¿ lgsfNbf,  81 3 81  3333 3 27 39  32  32  33 3 9 t;y{ 81 = 3 x 3 x 3 x 3 pbfx/0f 3 sg' } Pp6f ljBfnosf sIff 7 df k9g\\ ] ljBfyLx{ ¿n] nl' DagL ed| 0fsf nflu ljBfyL{ ;ªV\\ of hlt 5 Tolt g} ?lkof“ hDdf ubf{ ?= 15,625 hDdf eP5 eg] ;f] sIffsf] ljBfyL{ ;ªV\\ of lgsfn . ;dfwfg M oxf“ cfjZos ljBfyL{ ;ªV\\ of = 15,625 sf] jud{ n\" xG' 5 . To;n} ] ;jk{ y| d 15,625 sf] ¿9 u0' fgv08x¿ lgsfnf“} . 15625 sf] ¿9 u0' fgv08 lgsfNbf, ca, 15,625 sf] jud{ n\" lgsfNbf 5 15625 15625  5  5  5  5  5  5 5 3125  52  52  52 5 625 5 125  555 5 25  125 5 ∴ 15625 = 5 x 5 x 5 x 5 x 5 x 5 ctM pSt ljBfnodf sIff 7 df k9g\\ ] ljBfyL{ 125 hgf /x5] g\\ . 106 ul0ft, sIff – &

-u_ efu ljlwåf/f jud{ n\" lgsfNg] tl/sf tn efu ljlwaf6 jud{ n\" lgsfNg] tl/sf / pbfx/0f cWoog u/L 5nkmn u/ M pbfx/0f 4 1764 sf] efu ljlwåf/f jud{ n\" lgsfNg] ko| f; u/f“} M 42 tl/sf 4 1764 1. ;ªV\\ ofsf] Ps :yfgaf6 hf8] f hf8] f ldnfpb“ } hfgk' 5{ . h:t} M 1764 +4 -16 2. klxnf] hf8] f 17 nfO{ ju{ ;ªV\\ ofdf ljrf/ ubf{ 17 eGbf ;fgf] t/ ;a}eGbf 82 0164 7n' f] ju{ ;ªV\\ of 16 xG' 5 . o;sf] jud{ n\" 4 cfpg] u/L lx;fa ul/G5 . +2 - 164 3. 4 nfO{ tndfly /fv/] u0' fgkmn 17 sf] tnk6l\\ 6 /fvL 36fpgk' 5{ . 84 000 clg cufl8sf] 4 / 4 nfO{ hf8] lrxg\\ /fvL hf8] g\\ k' 5{ . 4. z]if cfPsf] 1 ;“u csf]{ hf]8f ;ª\\Vofx¿ 64 nfO{ tn emfgk{' 5{ . ca efHo 164 x'G5 . 5. ca, 82 sf] b;sf] :yfgsf] ;ªV\\ ofn] efHo 164 sf] b;sf] :yfg / ;o :yfgsf] ;ªV\\ of 16 nfO{ efu hfg] efukmn cgd' fg ugk{' 5{ / glhssf] ;ªV\\ ofn] u0' fg ugk{' 5{ . oxf“ 16 nfO{ 8 n] 2 k6s efu hfG5 . 6. ca, zi] f 0 cfPsfn] 1764 sf] jud{ n\" 42 xG' 5 . pbfx/0f 5 gf6] M 11025 sf] efu ljlwaf6 jud{ n\" lgsfn M 1. oxf“ 11025 nfO{ Pssf] :yfgaf6 ;dfwfg M 11025 sf] efu ljlwaf6 jud{ n\" lgsfNbf, hf8] L ldnfpb“ f 11025 xG' 5 . 105 2. klxnf] k6s 10 emfgk{' 5{ . t/ 2 1 11025 +1 - 1 kl5 sg' } cªs\\ /fVbf 10 nfO{ efu 205 01025 ug{ ldNbg} . To;n} ] Pp6f efugf:tL + 5 -1025 zG\" o (0) ykL 25 kl5 emf//] 1025 210 0000 agfOPsf] 5 . 3. efukmndf zG\" o ykk] l5 efhsdf ∴ 11025 sf] jud{ n\" 105 xG' 5 . klg zG\" o yKgk' 5{ . -3_ leGg ePsf ;ªV\\ ofsf] jud{ n\" lgsfNg] tl/sf tn lbOPsf] leGgsf] jud{ n\" lgsfNg] tl/sf / pbfx/0f leGgsf] jud{ n\" lgsfNg] tl/sf cWoog u/L 5nkmn u/ M 1. ;jk{ y| d x/ / cz+ sf] 56' 6\\ f56' 6\\ } pbfx/0f 6 jud{ n\" lgsfNgk' 5{ . 2. leGgsf] ;/n u/L pTt/ lgsfNg' 49 sf] jud{ n\" lgsfn M 81 k5{ . ul0ft, sIff – & 107

oxf,“ 49 sf] jud{ n\" = 49  77  72  7 7 81 81 3333 32  32 33 9 To;n} ,] 49 sf] jud{ n\" 7 xG' 5 . 81 9 pbfx/0f 6 1399 sf] jud{ n\" lgsfn M 625 ;dfwfg 1 399  1 625  399  1024 sf] jud{ n\"  1024  32 32  32  1 7 625 625 625 625 25 25 25 25 To;n} ,] 1 399 sf] jud{ n\" 1 7 xG' 5 . 625 25 cEof; 12.1 -s_ tn lbOPsf ;ªV\\ ofsf] ¿9 u0' fgv08 ljlwåf/f jud{ n\" lgsfn M 1. 64 2. 196 3. 324 4. 400 5. 1225 6. 2916 7. 5625 8. 11664 9. 121 x 169 10. 343 x 112 11. 144  196 12. 25  625 -v_ tn lbOPsf ;ªV\\ ofx¿sf] ju{ ;ªV\\ of lgsfn M 1. 8 2. 12 3. 15 4. 19 5. 25 6. 77 7. 95 8. 100 9. 205 10. 500 -u_ tn lbOPsf ;ªV\\ ofx¿sf] efu ljlwåf/f jud{ n\" lgsfn M 1. 169 2. 625 3. 2304 4. 8836 5. 9801 10. 1024144 6. 11025 7. 95481 8. 166464 9. 646416 -3_ tn lbOPsf leGgsf] jud{ n\" lgsfn M 144 625 1225 4. 1 91 5. 8 568 1. 169 2. 1024 3. 2916 2025 729 -ª_ tnsf kZ| gx¿ ;dfwfg u/ M 1. Pp6f jufs{ f/ hUufsf] nDafO 20m eP To;sf] Ifq] kmn kQf nufpm . 2. Pp6f jufs{ f/ ;l] dgf/ xnsf] Ifq] kmn 625m2 eP To;sf] nDafO kQf nufpm . 108 ul0ft, sIff – &

12.2 ;ªV\\ ofsf] 3g / 3gdn\" (Cube and Cube Roots) 1. ;ªV\\ ofsf] 3g tnsf ljm| ofsnfkx¿ cWoog u/L 5nkmn u/ M -s_ 2 nfO{ tLg k6s u0' fg u//] x/] f“} . 2 2 2x2x2=8 2 -v_ To:t} 3 / 4 nfO{ klg tLg tLg k6s u0' fg u//] x/] f“} . 3 x 3 x 3 = 27 / 4 x 4 x 4 = 64 xG' 5 . oxf,“ 2 sf] 3g ;ªV\\ of 8 xf] . To:t} 3 / 4 sf 3g ;ªV\\ ofx¿ jm| dzM 27 / 64 xg' \\ . -u_ lrqdf kT| os] eh' f 2 PsfO ePsf] 3gfsf/ j:t' bv] fOPsf] 5 . o; 3gfsf/ j:ts' f] cfotg (V) = 2 x 2 x 2 = 23 = 8 3g PsfO xG' 5 . dflysf] ljm| ofsnfksf cfwf/df 3g ;ªV\\ ofsf] kl/efiff nv] . ;fyL;u“ 5nkmn u/L ltdLn] nv] s] f] kl/efiff tnsf] kl/efiff;u“ bfh“ /] x/] . tLgcf6] f pxL ;ªV\\ ofsf] u0' fgkmnnfO{ 3g ;ªV\\ of elgG5 . h:t} M sg' } ;ªV\\ of 2 eP 2 sf] 3g ;ªV\\ of 23 xG' 5 . To:t} sg' } ;ªV\\ of a eP a sf] 3g ;ªV\\ of a3 xG' 5 . pbfx/0f 1 1, 7 / 10 sf] 3g ;ªV\\ of lgsfn M ;dfwfg 1 sf] 3g ;ªV\\ of = 13 = 1 x 1 x 1 = 1 7 sf] 3g ;ªV\\ of = 73 = 7 x 7 x 7 = 343 10 sf] 3g ;ªV\\ of = 103 = 10 x 10 x 10 = 1000 pbfx/0f 2 ltdf| ] ljBfnodf vfgk] fgLsf nflu Pp6f 5 PsfO nfdf,] 5 PsfO rf8} f / 5 PsfO cUnf] 6o\\ fªs\\ L hldgdl' g lgdf0{ f ug{ slt 3g PsfOsf] vfN8f] cfjZostf knf{ < ;dfwfg oxf,“ 5 sf] 3g ;ªV\\ of g} cfjZos ;dfwfg xf,] lsg < 53 = 5 x 5 x5 = 125 3g PsfO xG' 5 . t;y,{ cfjZos vfN8f] a/fa/ 125 3g PsfO . ul0ft, sIff – & 109

2. ;ªV\\ ofsf] 3gdn\" ljm| ofsnfkx¿ cWoog u/L 5nkmn u/ M 3g ;ªV\\ ofsf] hfgsf/L lnO;sk] l5 ca xfdL tL 3g ;ªV\\ ofsf] ¿9 u0' fgv08 lgsfnL x/] f“} . 1= 1 x 1 x 1 8=2x2x2 27 = 3 x 3 x 3 64 = 2 x 2 x 2 x 2 x 2 x 2 = 4 x 4 x4 oxf,“ xfdL s] eGg ;S5f“} eg,] 1 sf] 3gdn\" 1 xG' 5 . 8 sf] 3gdn\" 2 xG' 5 . To:t,} 27 / 64 sf] 3gdn\" jm| dzM 3 / 4 xG' 5g\\ . ca, s] ltdLx¿n] 3gdn\" sf] kl/efiff nV] g jf eGg ;S5f} < nv] / ;fyL;u“ 5nkmn u/ . sg' } 3g ;ªV\\ ofsf tLgcf6] f p:t} u0' fgv08x¿dWo] Pp6fnfO{ pSt 3g ;ªV\\ ofsf] 3gdn\" elgG5 . 3gdn\" nfO{ 3 n] hgfOG5 . h:t} a3 3g ;ªV\\ of xf] eg] a3 sf] 3gdn\" 3 a3  a xG' 5 . pbfx/0f 3 tnsf 3g ;ªV\\ ofsf] 3gdn\" lgsfn M -s_ 216 -v_ 512 -u_ 1728 ;dfwfg -s_ 216 sf] 3g ;ªV\\ of = 3 216  3 2  2  2  3  3  3  3 23  33  2  3  6 -v_ 512 sf] 3g ;ªV\\ of = 3 512  3 2 2 2 2 2 22 2 2  3 23  23  23  2 2 2  8 -u_ 1728 sf] 3g ;ªV\\ of = 3 1728  3 222222 3 3 3  3 23 23  33  22 3  12 110 ul0ft, sIff – &

cEof; 12.2 1. tn lbOPsf ;ªV\\ ofx¿sf] 3g ;ªV\\ of lgsfn M -s_ 6 -v_ 11 -u_ 13 -3_ 15 -ª_ 18 -r_ 24 -5_ 30 -h_ 45 -em_ 80 -`_ 100 2. tn lbOPsf ;ªV\\ ofsf] 3gdn\" lgsfn M -s_ 8 -v_ 125 -u_ 343 -3_ 1000 -ª_ 3375 3. Pp6f 3gfsf/ afs;sf] nDafO 12 ld6/ 5 eg] ;f] afs;sf] cfotg lgsfn . 4. 45m nDafO ePsf] Pp6f 3gfsf/ 3/sf] cfotg slt xG' 5 xfn] f < 5. vfg]kfgL cfof]hgfn] 25m nDafO ePsf] 3gfsf/ 6\\ofª\\sL lgdf{0f u/]5 eg] Tof] 6o\\ fªs\\ Lsf] Ifdtf slt xfn] f < [ olb 1m3 = 1000l ] 6. Pp6f 3gfsf/ sf7] fdf 4096m3 xfjf c6fp5“ eg] ;f] sf7] fsf] prfO lgsfn . 7. Pp6f 3gfsf/ vfgk] fgL 6o\\ fªs\\ Lsf] hDdf Ifdtf 64,000l 5 eg] ;f] 6o\\ fªs\\ Lsf] nDafO lgsfn . ul0ft, sIff – & 111

12.3 dxQd ;dfkjts{ (Highest Common Factor - HCF) 1. dxQd ;dfkjts{ sf] kl/ro tnsf ljm| ofsnfk cWoog u/L 5nkmn u/ M dfgf,“} bO' { cf6] f ;ªV\\ of 24 / 36 5g\\ . oxf,“ ;ªV\\ of 24 sf u0' fgv08sf] ;dx\" agfcf“} . F24 = {1, 2, 3, 4, 6, 8, 12, 24} To:t} ;ªV\\ of 36 sf u0' fgv08sf] ;dx\" agfcf“} . F36 = { 1, 2, 3, 4, 6, 9, 12, 18, 36} ca, 24 / 36 sf ;femf u0' fgv08sf] ;dx\" agfcf“} . ;femf u0' fgv08sf] ;dx\" {1, 2, 3, 4, 6, 12} 5 . o;df ;ae} Gbf 7n' f] ;femf u0' fgv08 = 12 5 . To;n} ,] dxQd ;dfkjts{ -d=;=_ = 12 xG' 5 . dflysf] ljm| ofsnfksf cfwf/df d=;= sf] kl/efiff nv] / tn lbOPsf] d=;= sf] kl/efiff;u“ tn' gf u/L x/] . lbOPsf kf| sl[ ts ;ªV\\ ofx¿sf] ;femf u0' fgv08x¿dWo] ;ae} Gbf 7n' f] u0' fgv08nfO{ dxTtd ;dfkjts{ (highest common factor) elgG5 . o;nfO{ 5f6] s/Ldf d=;= (H.C.F.) nl] vG5 . 2. efu ljlwaf6 d=;= lgsfNg] tl/sf ;fgf] ;ªV\\ of 24 n] 36 nfO{ efu ubf,{ tnsf ljm| ofsnfk cWoog u/L 5nkmn u/ M 24) 36 (1 - 24 dfly lbOPs} ;ªV\\ ofx¿ 24 / 36 sf] efu ljlwaf6 d=;= lgsfNg] ko| f; u/f“} M 12 1. ;jk{ y| d ;ae} Gbf ;fgf] ;ªV\\ of / ;ae} Gbf zi] f 12 n] efHo 24 nfO{ efu ubf,{ 7n' f] ;ªV\\ of kQf nufpgk' 5{ . oxf“ ;aeGbf ;fgf] ;ªV\\ of 2 / ;ae} Gbf 7n' f] ;ªV\\ of 12) 24 (2 36 5 . x 24 0 2. ;fgf] ;ªV\\ ofn] 7n' f] ;ªV\\ ofnfO{ efu ub{} hfgk' 5{ . -efhseGbf ;fgf] zi] f gcfP;Dd_ oxf,“ lgMzi] f efu nufpg] efhs 12 g} ;ªV\\ of 24 / 36 sf] ;ae} Gbf 7n' f] ;femf u0' fg v08 xf] . 3. z]ifn] efHonfO{ efhs dfgL efu ub}{ To;n} ,] 24 / 36 sf] d=;= = 12 xG' 5 . hfgk' 5{ . 112 ul0ft, sIff – &

dflysf] kl| jm| ofnfO{ Ps} 7fpd“ f /fvL lgDgfg;' f/ d=;= lgsfNg ;lsG5 . 24 / 36 sf] d=;= lgsfNbf, 24) 36 ( 1 -24 12) 24 (2 - 24 0 To;n} ,] d=;= = 12 xG' 5 . pbfx/0f 1 35 / 60 nfO{ lgMzi] f efu hfg] ;ae} Gbf 7n' f] ;ªV\\ of sg' xf] < ;dfwfg oxf,“ cfjZos ;ªV\\ of 35 / 60 sf] d=;= xG' 5 . ca 35 / 60 sf] efu ljlwaf6 d=;= lgsfNbf, 35) 60 (1 -35 25) 35 (1 -25 10) 25 (2 -20 5) 10 (2 10 0 oxf,“ 35 / 60 sf] d=;= = 5 5 . To;n} ,] 35 / 60 nfO{ lgMzi] f efu hfg] ;aeGbf 7n' f] ;ªV\\ of 5 xG' 5 . pbfx/0f 2 40 cf6] f lstfa, 50 cf6] f sfkL / 60 cf6] f sndx¿ a9Ldf slt hgf ljBfyLx{ ¿nfO{ a/fa/ u/L af8“ g\\ ;lsPnf < kT| os] n] slt sltcf6] f kfpnfg\\ < kQf nufpm . ;dfwfg oxf,“ cfjZos ;ªV\\ of 40, 50 / 60 sf] d=;= xG' 5 . kT| os] d=;=n] kT| os] ;ªV\\ ofnfO{ efu ubf{ cfpg] efukmn g} ;a} ljBfyLx{ ¿n] a/fa/ ;ªV\\ ofdf kfpg] xG' 5 . ul0ft, sIff – & 113

40 / 50 sf] d=;= lgsfNbf, ca, efhs 10 n] csf{] ;ªV\\ of 60 nfO{ efu ubf,{ 40) 50 (1 - 40 10) 60 (6 10) 40 (4 - 60 - 40 0 0 ∴ d=;= = 10 xG' 5 . To; sf/0f, 40 lstfa, 50 sfkL / 60 snd a9Ldf 10 hgf ljBfyLx{ ¿nfO{ a/fa/ u/L af8“ g\\ ;lsG5 . kT| os] n] 40 ÷ 10 = 4 cf6] f lstfa, 50 ÷ 10 = 5 cf6] f sfkL / 60 ÷ 10 = 6 cf6] f snd kfp5“ g\\ . cEof; 12.3 1. tn lbOPsf ;ªV\\ ofx¿sf] u0' fgv08sf] ;dx\" agfP/ d=;= kQf nufpm M -s_ 3, 6 -v_ 8, 10 -u_ 15, 18 -3_ 9, 12 -ª_ 12, 18 -r_ 9, 18 -5_ 21, 28, 35 -h_ 16, 20, 28 -em_ 20, 35, 55 -`_ 14, 26, 54 2. tn lbOPsf ;ªV\\ ofx¿sf] efu ljlwaf6 d=;= lgsfn M -s_ 18, 24 -v_ 36, 42 -u_ 40, 50 -3_ 25, 35 -ª_ 48, 64 -r_ 60, 72 -5_ 54, 72 -h_ 12, 15, 18 -em_ 20, 35, 40 3. 72 cf6] f uR' rf / 99 cf6] f rsn6] a9Ldf slt hgfnfO{ a/fa/ xg' ] u/L af8“ g\\ ;lsPnf / kT| os] n] slt sltcf6] f la:s6' / rsn6] kfp5“ g\\ xfn] f < kQf nufpm . 4. 125 cf6] f ;G' tnf, 150 cf6] f df;} d / 225 cf6] f cDaf a9Ldf slt ljBfyLn{ fO{ a/fa/ xg' ] u/L af8“ g\\ ;lsG5 < kT| os] n] x/s] kmnkm\" n slt sltcf6] f kf| Kt u5g{ \\ xfn] f < kQf nufpm . 5. Pp6f j4[ f>ddf 80 cf6] f sDan, 90 cf6] f :j6] / / 120 cf6] f Gofgf] Hofs6] ljt/0fsf nflu Joj:yf ul/P5 . tL sk8fx¿ a9Ldf slt hgfnfO{ a/fa/ efu nfUg] u/L af8“ g\\ ;lsG5 < kT| os] n] x/s] sk8f slt slt ;ªV\\ ofdf kf| Kt u5g{ \\ xfn] f < kQf nufpm . 6. sIff 7 sL 5fqf kg' dn] cfkm\\ gf] hGdlbgsf] cj;/df 60 cf6] f n88\\ ,' 72 cf6] f k8] f / 108 cf6] f akmL{ afl“ 85g\\ . pSt ld7fOx{ ¿ a9Ldf slt hgfnfO{ a/fa/ u/L afl“ 8g\\ xfn] f < kQf nufpm . 114 ul0ft, sIff – &

12.4 n3Q' d ;dfkjTo{ (Lowest Common Multiple - LCM) 1. n3Q' d ;dfkjTos{ f] kl/ro tnsf ljm| ofsnfkx¿ cWoog u/L 5nkmn u/ M -s_ bO' c{ f6] f ;ªV\\ ofx¿ 6 / 8 sf ckjTox{ ¿sf] ;dx\" agfP/ x/] f“} . ;ªV\\ of 6 sf ckjTox{ ¿sf] ;dx\" (M6) = {6, 12, 18, 24, 30, 36, 42, 48, 54, 60 ...} ;ªV\\ of 8 sf ckjTox{ ¿sf] ;dx\" (M ) = {8, 16, 24, 32, 40, 48, 56, 64, 72, 80, ...} 8 ca, ;ªV\\ of 6 / 8 sf ;femf ckjTox{ ¿sf] ;dx\" = {24, 48, ...} oxf“ 6 / 8 sf] ;ae} Gbf ;fgf] ckjTo{ 24 5 . To;n} ] 24 nfO{ ;ªV\\ of 6 / 8 sf] n3Q' d ;dfkjTo{ elgG5 . -v_ ca, dflysf M6 / M8 sf kT| os] ckjTox{ ¿nfO{ 6 / 8 n] 56' 6\\ f56' 6\\ } efu u//] x/] . o;af6 s] lgisif{ lgsfNg ;S5f} < nv] . -u_ s] 6 / 8 n] 24 nfO{ klg 56' 6\\ f56' 6\\ } efu hfG5 < efu u/L x/] . sg' } klg bO' { jf bO' e{ Gbf a9L kf| sl[ ts ;ªV\\ ofn] lgMzi] f efu hfg] ;ae} Gbf ;fgf] kf| sl[ ts ;ªV\\ ofnfO{ tL ;ªV\\ ofx¿sf] n3T' td ;dfjTo{ (Lowest Common Multiple - L.C.M.) elgG5 . 2. efu ljlwaf6 n=;= lgsfNg] tl/sf efu ljlwaf6 n=;= lgsfNbf, ;ªV\\ ofx¿ 48 / 64 sf] efu ljlwaf6 n=;= lgsfNg] 2 48, 64 ko| f; u/f“} M 2 24, 32 2 12, 16 n=;= lgsfNg] tl/sf÷kl| jm| of 2 6, 8 1. lbOPsf ;a} ;ª\\Vofx¿nfO{ kª\\lSt (row) df 3, 4 cwl{ j/fd (,) /fvL ldnfP/ /fVg] oxf“ n=;= = 2 x 2 x 2 x 2 x 3 x 4 = 192 2. ;aeGbf ;fgf] ;femf ¿9 u0' fgv08åf/f efu ub{} ctM ToxL u0' fgkmn 192 lbOPsf hfg] ;ªV\\ ofx¿ 48 / 64 sf] n=;= xG' 5 . 3. lbOPsf ;ªV\\ ofx¿dWo] sDtLdf bO' { cf6] fnfO{ k0\" f{ 115 ¿kdf ¿9 u0' fgv08 gcfP;Dd efu ub{} hfg] 4. ;a} efhs ¿9 u0' fgv08x¿ / clGtd kªl\\ Stsf afs“ L ;ªV\\ ofx¿sf] u0' fgkmn lgsfNg] . oxL u0' fgkmn lbOPsf ;ªV\\ ofx¿sf] n=;= xG' 5 . ul0ft, sIff – &

pbfx/0f 1 9 / 12 sf] n=;= lgsfn M -s_ ckjTox{ ¿sf] ;dx\" agfP/ -v_ efu ljlwaf6 ;dfwfg -s_ ;dx\" agfP/ 9 / 12 sf] n=;= lgsfNbf, 9 sf ckjTox{ ¿sf] ;dx\" (M9) = {9, 18, 27, 36, 45, 54, 63, 72, 81, 90, ...} 12 sf ckjTox{ ¿sf] ;dx\" (M12) = { 12, 24, 36, 48, 60, 72, 84, 96 .....} ca, ckjTox{ ¿sf] ;dx\" df tn' gf u//] xb] f{ 9 / 12 sf ckjTox{ ¿dWo] ;aeGbf ;fgf] ckjTo{ = 36 xf] . To;n} ] ;ªV\\ ofx¿ 9 / 12 sf] n=;= = 36 xG' 5 . -v_ efu ljlwaf6 9 / 12 sf] n=;= lgsfNbf, 3 9, 12 34 ca, n=;= = 3 x 3 x 4 = 36 xG' 5 . pbfx/0f 2 15, 18, 24 / 30 n] l7s efu hfg] ;ae} Gbf ;fgf] ;ªV\\ of kQf nufpm . -efu ljlwaf6_ ;dfwfg oxf“ lbOPsf ;ªV\\ ofx¿n] l7s efu hfg] ;ae} Gbf ;fgf] ;ªV\\ of lbOPsf ;ªV\\ ofx¿sf] n=;= xG' 5 . To;n} ] efu ljlwaf6 15, 18, 24 / 30 sf] n=;= lgsfNbf, 2 15, 18, 24, 30 3 15, 9, 12, 15 5 5, 3, 4, 5 1, 3, 4, 1 ca, n=;= = 2 x 3 x 5 x 3 x 4 = 360 o;y{ 15, 18, 24 / 30 n] l7s efu hfg] ;ae} Gbf ;fgf] ;ªV\\ of 360 xG' 5 . pbfx/0f 3 sIff 7 sf ljBfyLx{ ¿nfO{ klxnf] 5/5 hgfdf, bf;] f| ] 6/6 hgfdf / t;] f| ] 10/10 hgfdf ;dx\" agfP/ ;fdb' flos÷kl/ofh] gf sfo{ ug{ nufOPsf] /x5] . ca ljBfyLx{ ¿ sDtLdf slt hgf ePdf kT| os] sfod{ f ljBfyLx{ ¿ afs“ L gxg' ] u/L ;dx\" agfpg ;lsPnf < kQf nufpm . 116 ul0ft, sIff – &

;dfwfg oxf,“ cfjZos ;dx\" ;ªV\\ of eGgfn] kT| os] ;dx\" sf ljBfyL{ ;ªV\\ of jm| dzM 5, 6 / 10 sf] n=;= xG' 5 . ca, 5, 6 / 10 sf] efu ljlwaf6 n=;= lgsfNbf, 2 5, 6, 10 5 5, 3, 5 1, 3, 1 ca n=;= = 2 x 5 x 3 = 30 To;n} ] sDtLdf 30 hgf ljBfyLx{ ¿ ePdf jm| dzM klxnf] 5/5 hgf, bf;] f| ] 6/6 hgf / t;] f| ] 10/10 hgfsf] ;dx\" agfO{ ;dx\" sfo{ ug{ ;lsG5 . cEof; 12.4 1. tnsf kT| os] ;ªV\\ ofx¿sf] ckjTox{ ¿sf] ;dx\" agfP/ tyf efu ljlw u/L bj' } tl/sfn] n=;= lgsfn M -s_ 18 / 48 -v_ 12 / 30 -u_ 36 / 48 -3_ 49 / 35 -ª_ 15, 20 / 25 -r_ 30, 40 / 50 -5_ 28, 42 / 56 -h_ 36, 54 / 72 -em_ 210, 280, 420 / 530 -`_ 100, 200, 300 / 400 2. 30, 36, 48 / 60 n] l7s efu hfg] ;aeGbf ;fgf] ;ªV\\ of kQf nufpm . 3. ltg cf6] f dh] l/ª 6k] x¿ jm| dzM 24cm, 35cm / 54cm nDafOsf 5g\\ . ca sg' rflx“ ;ae} Gbf 5f6] f] nDafO ;a} 6k] n] l7s efu hfg] u/L -l7Ss xg' ] u/L_ gfKg ;lsPnf < kQf nufpm . 4. Tof] ;ae} Gbf ;fgf] ;ªV\\ of kQf nufpm, h;af6 tLg 36fpb“ f cfpg] 36fp kmnnfO{ 18, 24 / 36 n] lgMzi] f efu hfG5 . 5. Tof] ;ae} Gbf ;fgf] ;ªV\\ of kQf nufpm, h;df 7 hf8] b\\ f cfpg] ofu] kmnnfO{ 32, 64 / 192 n] l7s efu nfU5 . 6. dfly kZ| g 1 bl] v 5 ;Dd lbOP h:t} ;d:ofx¿ agfO{ ;fyL;u“ ;f6/] ;dfwfg u/L lzIfsnfO{ bv] fpm . ul0ft, sIff – & 117

12.5. låcfwf/ / k~rcfwf/ ;ªV\\ of k4lt (Binary and Quinary Number System) 1. låcfwf/ ;ªV\\ of k4ltsf] kl/ro bzdnj ;ªV\\ of k4ltsf af/d] f xfdLn] cl3Nnf sIffx¿df kl9;ss] f 5f“} . bzdnj cyft{ \\ lxGb' c/l] as ;ªV\\ of k4ltdf 0 bl] v 9 ;Ddsf 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 u/L hDdf 10 cªs\\ x¿ xG' 5g\\ . lxGb' c/l] as ;ªV\\ of k4ltdf ;ªV\\ ofnfO{ 10 sf] 3ftfªs\\ sf ¿kdf JoSt ul/G5 . h:t} M 24 = 2 x 10 + 4 = 2 x 101 + 4 x 100 576 = 5 x 100 + 7 x 10 + 6 = 5 x 102 + 7 x 101 + 6 x 100 ca xfdL låcfwf/ ;ªV\\ of k4ltsf af/d] f 5nkmn u/f“} . lj1fg tyf ;r\" gf kl| jlwsf] ljsf;n] sDKo6' / kl| jlwsf dfWodaf6 5f6] f] ;dod} hl6neGbf hl6n ul0ftLo ;d:ofx¿ ;dfwfg ug{ ;lsg] ePsf] 5 . sDKo6' /df ljBt' Lo ;ls6{ ( electrical circuit) vfN] g] / aGb ug{] (on and off ) bO' { cf6] f kl| jm| ofnfO{ jm| dzM ;ªs\\ t] 0 / 1 n] hgfOPsf] xG' 5 . o;/L låcfwf/ ;ªV\\ of k4ltdf 0 / 1 u/L bO' c{ f6] f dfq ;ªV\\ ofx¿ ko| fu] ul/Psf xG' 5g\\ . bzdnj ;ªV\\ of k4wltdf bzcf6] f cªs\\ ko| fu] ePh:t} låcfwf/ ;ªV\\ of k4wltdf bO' c{ f6] f 0 / 1 dfq ko| fu] xG' 5g\\ . låcfwf/ ;ªV\\ of k4ltdf ;ªV\\ ofx¿ 2 sf] 3ftfªs\\ df nl] vG5 . 2. bzdnj ;ªV\\ of k4ltaf6 låcfwf/ ;ªV\\ of k4ltdf ¿kfGt/0f ca xfdL bzdnj ;ªV\\ of k4ltaf6 låcfwf/ ;ªV\\ of k4ltdf abNg] tl/sf af/d] f 5nkmn u/f“} . bzdnjdf ePsf] ;ªV\\ ofnfO{ 2 n] efu ub{} hfg] / efukmndf 0 gcfP;Dd efu ul//xgk' 5{ . clg zi] fnfO{ bfoft“ km{ nV] b} hfgk' 5{ / cGTodf tnk6l\\ 6af6 dflylt/ jm| dzM zi] fnfO{ ldnfP/ nV] gk' 5{ . pbfx/0f 1 25 nfO{ låcfwf/ ;ªV\\ of k4ltdf ¿kfGt/0f u/ . ;dfwfg oxf,“ 25 nfO{ låcfwf/ ;ªV\\ of kbw\\ ltdf ¿kfGt/0f ubf,{ 2 25 2 12 1 26 0 23 0 21 1 01 t;y,{ 25 = 110012 xG' 5 . 118 ul0ft, sIff – &

3. låcfwf/ ;ªV\\ of k4ltaf6 bzdnj ;ªV\\ of k4ltdf ¿kfGt/0f tnsf] ljm| ofsnfk cWoog u/L 5nkmn u/ M 02 = 0 = 0 12 = 20 = 1 10 = 1 x 21 + 0 x 20 = 2 2 11 = 1 x 21 + 1 x 20 = 3 2 1002 = 1 x 22 + 0 x 21 + 0 x 20 = 4 1012 = 1 x 22 + 0 x 21 + 1 x 20 = 5 låcfwf/ ;ªV\\ of k4ltnfO{ 56' o\\ fpg k5fl8 2 /fv/] 112 cyjf 1012 nV] g] ul/G5 . pbfx/0f 2 10012 / 11112 nfO{ bzdnj ;ªV\\ of k4ltdf abn M ;dfwfg 1001 = 1 x 23 + 0 x 22 + 0 x 21 + 1 x 20 2 =8+0+0+1=9 11112 = 1 x 23 + 1 x 22 + 1 x 21 + 1 x 20 =8+4+2+1 = 15 4. k~rcfwf/ ;ªV\\ of k4ltsf] kl/ro bzdnj ;ªV\\ of k4ltdf bzcf6] f cªs\\ x¿, låcfwf/ ;ªV\\ of k4ltdf bO' c{ f6] f cªs\\ eP h:t} k~rcfwf/ ;ªV\\ of k4ltdf 0, 1, 2, 3, 4 u/L kfr“ cf6] f cªs\\ x¿ ko| fu] ul/G5g\\ . k~rcfwf/ ;ªV\\ of kbw\\ ltdf ;ªV\\ ofnfO{ 5 sf] 3ftsf ¿kdf nl] vG5 . tnsf] tflnsf cWoog u/L 5nkmn u/ 5 sf] :yfgdfg 55 54 53 52 51 50 bzcfwf/ 3125 625 125 25 51 5. bzdnj ;ªV\\ of k4ltaf6 k~rcfwf/ ;ªV\\ of k4ltdf ¿kfGt/0f ca xfdL bzdnj ;ªV\\ of k4ltaf6 k~rcfwf/ ;ªV\\ of k4ltdf abNg] tl/sfaf/d] f 5nkmn u/f“} . ;jk{ y| d bzdnjdf ePsf] ;ªV\\ ofnfO{ 5 n] efukmndf 0 gcfP;Dd efu ul//xgk' 5{ . clg zi] fnfO{ bfoft“ km{ nV] b} hfgk' 5{ . cGTodf tnk6l\\ 6af6 dflylt/ jm| dzM zi] fnfO{ ldnfP/ nV] gk' 5{ . pbfx/0f 3 432 nfO{ k~rcfwf/ ;ªV\\ of k4ltdf abn M ul0ft, sIff – & 119

;dfwfg oxf“ 432 nfO{ k~rcfwf/ ;ªV\\ of k4ltdf ¿kfGt/0f ubf,{ 5 432 5 86 2 5 17 1 53 2 03 432 = 32125 6. k~rcfwf/ k4ltaf6 bzdnj ;ªV\\ of k4ltdf ¿kfGt/0f tnsf] ljm| ofsnfk cWoog u/L 5nkmn u/ M 05 =0 = 0 1 = 50 = 1 5 10 = 1 x 51 + 0 x 50 = 5 5 115 = 1 x 51 + 1 x 50 = 6 1005 = 1 x 52 + 0 x 51 + 0 x 50 = 25 1015 = 1 x 52 + 0 x 51 + 1 x 50 = 26 k~rcfwf/ ;ªV\\ of k4ltnfO{ 56' o\\ fpg ;ªV\\ ofsf k5fl8 5 /fv/] 115 nV] g] ul/G5 . pbfx/0f 4 dflysf] :yfgdfg tflnsf x/] /] 2345 nfO{ bzdnj ;ªV\\ of k4ltdf ¿kfGt/0f u/ M ;dfwfg 234 = 2 x 52 + 3 x 51 + 4 x 50 5 = 2 x 25 + 3 x 5 + 4 x 1 = 50 + 15 + 4 = 69 cEof; 12.5 1. tn lbOPsf bzdnj k4ltsf] ;ªV\\ ofnfO{ låcfwf/ ;ªV\\ of k4ltdf abn M -s_ 11 -v_ 25 -u_ 79 -3_ 104 -ª_ 250 -r_ 366 2. tnsf kT| os] låcfwf/ ;ªV\\ ofnfO{ bzdnj ;ªV\\ of k4ltdf abn M -s_ 11 -v_ 101 -u_ 111 -3_ 10101 -ª_ 11001 2 2 2 2 2 -r_ 111112 -5_ 1100112 -h_ 1000002 -em_ 10000112 -`_ 11110012 3. tnsf kT| os] bzdnj k4ltsf ;ªV\\ ofnfO{ k~rcfwf/ ;ªV\\ of k4ltdf abn M -s_ 21 -v_ 55 -u_ 112 -3_ 650 -ª_ 1128 -r_ 3650 4. tnsf kT| os] k~rcfwf/ ;ªV\\ ofnfO{ bzdnj ;ªV\\ of k4ltdf abn M -s_ 215 -v_ 345 -u_ 1235 -3_ 3435 -ª_ 21135 -r_ 12345 120 ul0ft, sIff – &

PsfO 13 k0\" ffª{ s\\ (Integer) 13.1 k0\" ffª{ s\\ sf rf/ ;fwf/0f lgod 1. k0\" ffª{ s\\ sf] kl/ro tnsf ljm| ofsnfk cWoog u/L 5nkmn u/ M -s_ N = {1, 2, 3, 4, 5, ...} nfO{ xfdL sg' ;ªV\\ ofx¿sf] ;dx\" eG5f“} < -v_ W = {0, 1, 2, 3, 4, 5, ....} nfO{ xfdL sg' ;ªV\\ ofx¿sf] ;dx\" eG5f“} < klxnfn] fO{ kf| sl[ ts ;ªV\\ ofx¿ (natural numbers) sf] ;dx\" / bf;] f| n] fO{ k0\" f{ ;ªV\\ ofx¿ (whole numbers) sf] ;dx\" elgG5 . -u_ k0\" f{ ;ªV\\ ofdf ePsf sg' } bO' { cf6] f ;ªV\\ of 5 / 8 lnpm“ . 5 + 8 = 13, s] k0\" f{ ;ªV\\ ofdf k5{ < 8 - 5 = 3, s] k0\" f{ ;ªV\\ ofdf k5{ < 8 x 5 = 40, s] k0\" f{ ;ªV\\ ofdf k5{ < -3_ ca 5 / 8 sf] 36fp kmn (5-8) lgsfNg] sfl] ;; u/f“} . o;nfO{ ;ªV\\ of /v] faf6 36fP/ x/] f“} . 36fp kmn (5-8) hg' 0 eGbf 3 PsfO afof“ k5{ . 0 eGbf afof kg{] ;ªV\\ of t k0\" f{ ;ªV\\ ofdf kbg{} . o;nfO{ -3 nl] vG5 . -3 k0\" ffª{ s\\ df kb5{ . -ª_ ca, (6-8), (2-6), (1-2), (4-5) nfO{ klg ;ªV\\ of /v] fdf bv] fP/ x/] f“} . 1. (6-8) nfO{ ;ªV\\ of /v] fdf bv] fpb“ f, oxf“ (6-8), hg' 0 eGbf 2 PsfO afof“ k¥of] . To;n} ] (6-8) = -2 xG' 5 . 2. (2-6) nfO{ ;ªV\\ of /v] fdf bv] fpb“ f, oxf,“ (2-6) hg' 0 eGbf 4 PsfO afof“ k¥of] . To;n} ] (2-6) = -4 xG' 5 . 121 ul0ft, sIff – &

3. (1-2) nfO{ ;ªV\\ of /v] fdf bv] fpb“ f, oxf“ (1-2) hg' 0 eGbf 1 PsfO afof“ k¥of] . To;n} ] (1-2) = -1 xG' 5 . 4. (4-5) nfO{ ;ªV\\ of /v] fdf bv] fpb“ f, oxf,“ (4-5) hg' 0 eGbf 1 PsfO afof“ k/s] f] 5 . To;n} ] (4-5) = -1 xG' 5 . ca, xfdL eGg ;S5f“} -1, -2, -3, -4, .... cflb 0 eGbf afof“ / +1, +2, +3, +4, ..... cflb 0 eGbf bfof“ xG' 5g\\ . 0 eGbf afofs“ f ;ªV\\ ofx¿ 0 eGbf ;fgf xG' 5g\\ . 0 eGbf bfofs“ f ;ªV\\ ofx¿ 0 eGbf 7n' f xG' 5g\\ . To;n} ] 0 ;lxtsf ;a} wgfTds / C0ffTds ;ªV\\ ofnfO{ k0\" ffª{ s\\ elgG5 . o;nfO{ Z n] hgfOG5 . 2. k0\" ffª{ s\\ x¿sf ks| f/ / tn' gf -s_ k0\" ffª{ s\\ sf ks| f/ tnsf ljleGg ks| f/sf k0\" ffª{ s\\ x¿sf] cWoog u/L 5nkmn u/ . k0\" ffª{ s\\ (Z) ={ ............ -4, -3, -2, -1, 0, +1, +2, +3, +4, .........} wgfTds k0\" ffª{ s\\ (Z+) = {+1, +2, +3, .............} / C0ffTds k0\" ffª{ s\\ (Z-) = {-1, -2, -3, -4, .........} -v_ k0\" ffª{ s\\ x¿larsf] tn' gf dflysf] ;ªV\\ of /v] fsf cfwf/df 5 hf8] L ;ªV\\ ofx¿nfO{ (< / >) lrxg\\ ko| fu] u/L nv] . h:t} M (5<10) / (5>-2) -u_ ljdv' k0\" ffª{ s\\ k0\" ffª{ s\\ +2 / k0\" ffª{ s\\ -2 sf] tn' gf u/L x/] f“} . k0\" ffª{ s\\ +2 n] pbu\\ d laGb' 0 af6 bfoffl“ t/ /xs] f] k0\" ffª{ s\\ nfO{ hgfp5“ . To:t,} -2 n] 0 af6 plTts} b/' Ldf /xs] f] afofl“ t/sf] ;ªV\\ of hgfp5“ . 122 ul0ft, sIff – &

To;n} ] +2 / -2 nfO{ cfk;df ljdv' k0\" ffª{ s\\ x¿ elgG5 . sg' } k0\" ffª{ s\\ ;ªV\\ of /v] fsf] pbu\\ d laGb' zG\" oaf6 hlt b/' Ldf 5 l7s Tolt g} b/' Ldf /xs] f] csf{] ljk/Lt lbzfsf] k0\" ffª{ s\\ nfO{ Tof] k0\" ffª{ s\\ sf] ljdv' elgG5 . –x sf] ljdv' k0\" ffª{ s\\ +x xG' 5 . ca -5 sf] ljdv' k0\" ffª{ s\\ slt xfn] f < 3. k0\" ffª{ s\\ sf] lg/kI] fdfg (Absolute Value of Integer) tnsf] ;ªV\\ of /v] f x/] L 5nkmn u/ M oxf,“ pbu\\ d laGb' a;kfs{ xf] . a;kfsa{ f6 nugvn] 3km klZrd cyft{ \\ afof“ 5 . dxf/fhuGh 3km kj\" { cyft{ \\ bfof“ 5 . ca eg t, dxf/fhuGhbl] v nugvn] sf] b/' L slt 5 < s] (-3km) + (+3km) = 0km xG' 5 < kSs} xb“' g} . cyjf 3km + 3km = 6km xG' 5 . xf,] -3km = 3km / +3km = 3km dfGg] xf] eg] bO' { :yfglarsf] b/' L 3km + 3km = 6km xG' 5 . To;n} ] -3 / +3 bj' s} f] lg/kI] f dfg 3 xG' 5 . sg' } klg k0\" ffª{ s\\ sf] wgfTds ;fªl\\ Vos dfgnfO{ lg/kI] fdfg elgG5 . To;n} ] |+x| = |-x| = x xG' 5 . 3.1 k0\" ffª{ s\\ sf] hf8] / 36fp (Addition and Subtraction of Integers) -s_ k0\" ffª{ s\\ sf] hf8] (Addition of Integers) tn lbOPsf ;ªV\\ of /v] fsf cfwf/df ul/Psf k0\" ffª{ s\\ sf] hf8] cWoog u/L 5nkmn u/ . 1. (+2) +(+3) = ? +2 PsfO bfof“ uP/ +3 PsfO bfof“ g} hfb“ f sxf“ kl' uG5 < xf] +5 PsfO bfof“ kl' uG5 . ctM (+2) +(+3) = +5 xG' 5 . 2. (+6) +(-2) = ? To;n} ] (+6) + (-2) = +4 xG' 5 . oxf,“ pbu\\ d laGba' f6 6 PsfO bfof“ / 2 PsfO afof“ cfPkl5 xfdL 0 af6 4 PsfO bfof“ kU' 5f“} . ul0ft, sIff – & 123

3. (+2) +(-6) = ? 0 af6 2 PsfO bfof“ / +2 af6 6 PsfO afof“ hfbf“ 4 PsfO 0 af6 afof“ kl' uG5 . To;n} ,] (+2) + (-6) = -4 xG' 5 . pbfx/0f 1 ;ªV\\ of /v] f ko| fu] u/L hf8] u/ -s_ (+4) + (+6) -v_ (+4) + (-6) -u_ (-4) + (+6) ;dfwfg -s_ (+4) + (+6) ctM (+4) + (+6) = +10 xG' 5 . -v_ (+4) + (-6) ctM (+4) + (-6) = -2 xG' 5 . -u_ (-4) + (+6) ctM (-4) + (+6) = (+2) xG' 5 . -v_ k0\" ffª{ s\\ sf] hf8] sf lgod k0\" ffª{ s\\ sf] hf8] sf sx] L dxTTjk0\" f{ lgodx¿nfO{ tn ab“' fut ¿kdf pNnv] ul/Psf] 5 . 1. ljlgod lgod (Commutative Law) k0\" ffª{ s\\ x¿sf] hf8] kmn lgsfNbf k0\" ffª{ s\\ x¿nfO{ hg' ;s' } jm| ddf /fv/] klg kl/0ffd Pp6} lg:sg] lgodnfO{ k0\" ffª{ s\\ sf] hf8] sf] ljlgod lgod elgG5 . h:t} M a + b = b + a hxf“ a / b bj' } k0\" ffª{ s\\ x¿ xg' \\ . 124 ul0ft, sIff – &

2. ;ª3\\ Lo lgod (Associative Law) tLgcf6] f k0\" ffª{ s\\ x¿nfO{ hf8] b\\ f klxnf hg' ;s' } 2 cf6] f k0\" ffª{ s\\ hf8] /] cfPsf] hf8] kmndf t;] f| ] k0\" ffª{ s\\ hf8] b\\ f klg hf8] kmn a/fa/ cfp5“ eg] To:tf] lgodnfO{ k0\" ffª{ s\\ sf] hf8] sf] ;ª3\\ Lo lgod elgG5 . h:t} M (a+b)+c = a+(b+c) = (a+c)+b hxf“ M a, b / c k0\" ffª{ s\\ x¿ xg' \\ . 3. ljk/Lt kl/0ffd (Inverse Quantity) olb sg' } bO' c{ f6] f k0\" ffª{ s\\ x¿ hf8] b\\ f hf8] kmn 0 xG' 5 eg] To; lgodnfO{ k0\" ffª{ s\\ sf] hf8] sf] ljk/Lt kl/0ffd elgG5 . h:t} M +a / -a cfk;df ljk/Lt kl/0ffd xg' \\ . hxf“ (+a)+(-a)=0 xG' 5 / a Pp6f k0\" ffª{ s\\ xf] . -u_ k0\" ffª{ s\\ sf] 36fp (Subtraction of Integer) tnsf] ;ªV\\ of /v] fsf cfwf/df bv] fOPsf k0\" ffª{ s\\ sf 36fp;DaGwL ljm| ofsnfkx¿ cWoog u/L 5nkmn u/ M 1. (+4) - (+2) = ? (+4) PsfO bfof“ uP/ -2 PsfO afof“ hfb“ f sxf“ kl' uG5 < 2 PsfO bfof“ kl' uG5 . To;n} ,] (+4) - (+2) = (+2) xG' 5 . 2. (+4) - (+7) = ? 0 af6 (+4) PsfO bfof“ uP/ -7 PsfO afof“ hfb“ f sxf“ kl' uG5 < 3 PsfO afof“ kl' uG5 . To;n} ,] (+4) – (+7) = (-3) xG' 5 . 3. (-3) + (-5) = ? 0 af6 (-3) PsfO afof“ uP/ 5 PsfO afof“ g} hfb“ f sxf“ kl' uG5 < (-8) PsfO afof“ g} kl' uG5 . To;n} ,] (-3) + (-5) = (-8) xG' 5 . ul0ft, sIff – & 125

pbfx/0f 2 ;/n u/ M -s_ (+5) - (+3) -v_ (+2) - (+5) -u_ (-2) - (+5) ;dfwfg -s_ (+5) - (+3) nfO{ ;ªV\\ of /v] fdf bv] fpb“ f, +2 -3 +5 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 t;y{ (+5) - (+3) = (+2) -v_ (+2) - (+5) nfO{ ;ªV\\ of /v] fdf bv] fpb“ f, –5 +2 -3 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 t;y,{ (+2) - (+5) =-3 -u_ (-2) - (+5) nfO{ ;ªV\\ of / /v] fdf bv] fpb“ f, -(+5) -2 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 t;y{ (-2) - (+5) = -7 cEof; 13.1 1. tn lbOPsf k0\" ffª{ s\\ sf ljm| ofx¿nfO{ ;ªV\\ of /v] fdf bv] fpm M -s_ (+3) + (+4) -v_ (+7) + (-4) -u_ (-5) + (+2) -3_ (-4) + (-3) -ª_ (-6) - (+2) -r_ (+6) - (+2) -5_ (+6) - (-2) -h_ (-5) + (+7) -em_ (-4) - (-6) -`_ (+3) + (-5) 2. tn lbOPsf k0\" ffª{ s\\ x¿sf] 10 PsfO bfof“ / 10 PsfO afof“ kg{] k0\" ffª{ s\\ nv] M -s_ (-5) -v_ (-3) -u_ 0 (3) +4) -ª_ (+7) -r_ (+10) 126 ul0ft, sIff – &

3. tn lbOPsf k0\" ffª{ s\\ x¿sf] ljdv' k0\" ffª{ s\\ nv] M -s_ (+4) -v_ (+1) -u_ (-3) -3_ (-5) -ª_ 0 -r_ (+7) 4. tn lbOPsf k0\" ffª{ s\\ x¿sf] lg/kI] f dfg lgsfn M -s_ |+6| -v_ |-4| -u_ |+10| -3_ |-3| -ª_ |-5| -r_ |-7| 5. ;/n u/ M -s_ (+6) + (+4) + (+3) -v_ (+8) + (-4) + (+3) -u_ (-7) + (+6) + (-5) -3_ (-12) - (-10) + (+6) -ª_ (+15) - (+10) - (-3) -r_ (-35) + (+25) + (+10) -5_ (+24) + (-20) + (-15) -h_ (-10) - (+10) -(+10) 6. ljlgdo lgod ko| fu] u/L hf8] u/ M -s_ (+17) + (+12) + (+20) -v_ (+20) + (-10) + (-10) -u_ (+25) + (20) + (-15) -3_ (+35) + (+24) + (-18) -ª_ (-46) + (+58) + (-44) 7. (+25) df slt hf8] b\\ f (-25) xG' 5 < 8. (-35) af6 slt 36fpb“ f (-20) xG' 5 < 9. ;h' gnfO{ ;G' tnf ar] /] ?= 145 gfkmf ePsf] 5 . hg' f/ ar] /] ?= 74 gfS] ;fg eP5 . sn' sf/fa] f/af6 ;h' gnfO{ gfkmf jf gfS] ;fg s] eP5 / slt eP5 < kQf nufpm . 10. bO' c{ f6] f a;x¿ Ps} :yfgaf6 Ps} ;dodf 56' 5] g\\ . Pp6f a;n] 127 km kj\" { ofqf u¥of] / csf{] a;n] 139 km klZrd ofqf u¥of] . tL bO' { a;larsf] b/' L kQf nufpm . 11. sg' } bO' c{ f6] f k0\" ffª{ s\\ x¿sf] ofu] kmn -119 5 . olb 7n' f] k0\" ffª{ s\\ 177 eP ;fgf] k0\" ffª{ s\\ slt xfn] f < kQf nufpm . 12. ;ª3\\ Lo lgod ko| fu] u/L ;/n u/ M -s_ (+7) + (-25) - (-65) -v_ (+45) + (-146) + (+209) 13. jm| d ljlgod ko| fu] u/L ;/n u/ M -s_ (-5) - (+2) -v_ (+2) - (-5) 14. ;/n u/ M -s_ -4 + 14 + 15 + (-52) -v_ (-13) + (+7) - 8 + 14 - 40 ul0ft, sIff – & 127

13.2 k0\" ffª{ s\\ sf] u0' fg / efu (Multiplication and Division of Integers) 1. k0\" ffª{ s\\ sf] u0' fg -s_ tnsf] ljm| ofsnfk cWoog u/L 5nkmn u/ M B O A klZrd gf/fo0fu9 kj\" { 8 306f 40km/hr 8 306f klZrdaf6 kj\" t{ km{ 40km kl| t 306fsf b/n] ul' 8/xs] f] a; gf/fo0fu9af6 8 306f kj\" { / 8 306f klZrdsf] a;sf] l:ylt eGg ;S5f} < -s_ 8 306fsf] klZrdsf] l:ylt = (40km/hr) x (8)hr = (320)km = 320km kj\" { -v_ 8 306f cl3sf] kj\" s{ f] l:ylt = (40km/hr) x (8hr) = (320)km = 320km klZrd gf6] M kj\" s{ f l:yltnfO{ -±_ / klZrdsf] l:yltnfO{ -–_ dfGbf, To;}n], wgfTds k0\" ffª{ s\\ x wgfTds k0\" ffª{ s\\ = wgfTds k0\" ffª{ s\\ -±_ -±_ -±_ wgfTds k0\" ffª{ s\\ x C0ffTds k0\" ffª{ s\\ = C0ffTds k0\" ffª{ s\\ -±_ -—_ -—_ -v_ tn lbOPsf] k0\" ffª{ s\\ sf] u0' fgsf] 9fr“ f cWoog u/ M (+3) x (-3) = - 9 (+2) x (-3) = -6 (+1) x (-3) = -3 (0) x (-3) = 0 (-1) x (-3) = +3 (-2) x (-3) = +6 (-3) x (-3) = +9 To;n} ,] C0ffTds k0\" ffª{ s\\ x C0ffTds k0\" ffª{ s\\ = wgfTds k0\" ffª{ s\\ xG' 5 . pbfx/0f 1 -v_ (+7) x (-6) -u_ (-5) x (+8) -3_ (-7) x (-7) u0' fg u/ M -s_ (+4) x (+6) ;dfwfg -v_ (+7) x (-6) = (-42) -s_ (+4) x (+6) = (+24) -3_ (-7) x (-7) = (+49) -u_ (-5) x (+8) = (-40) 128 ul0ft, sIff – &

2. k0\" ffª{ s\\ sf] u0' fgsf lgodx¿ (Properties of multiplication of Integers) k0\" ffª{ s\\ sf u0' fgsf sx] L dxTTjk0\" f{ lgodx¿nfO{ tn ab“' fut ¿kdf pNnv] ul/Psf] 5 . 1. ljlgdo lgod (Commutative Property) : a x b = b x a sg' } bO' c{ f6] f k0\" ffª{ s\\ x¿sf] u0' fg kmn ltgLx¿sf :yfg abNbf xg' ] u0' fg kmn;u“ a/fa/ xG' 5 . h:t} M (+2) x (+4) = (+4) x (+2) = +8, (-3) x (+7) = (+7) x (-3) = -21 / (-8) x (-6) -= (-6) x (-8) = +48 xG' 5 . ctM olb a / b bO' c{ f6] f k0\" ffª{ s\\ x¿ xg' \\ eg] a x b = b x a xG' 5 . 2. ;ª3\\ Lo lgod (Associative Property): (a x b) x c = a x (b x c) sg' } ltg cf6] f k0\" ffª{ s\\ x¿sf] u0' fgkmn ltgLx¿sf] klxnf bO' { / clGtdsf] u0' fg;u“ / klxnf] / clGtd bO' { u0' fg ubf{ cfpg] u0' fgkmn;u“ a/fa/ xG' 5 . h:t} M [(+2) x (+3)] x (+4) = (+2) x [(+3) x (+4)] xG' 5 . [(+5) x (-2)] x (-7) = (+5) x [(-2) x (-7)] xG' 5 . ctM olb a, b / c tLgcf6] f k0\" ffª{ s\\ ¿ xg' \\ eg] (a x b) x c = a x (b x c) 3. kb ljR5b] g lgod (Distributive Property) : a(b+c) = ab + ac h:t} M (+6) [(+5)+(+3)] = (+6) x (+5)+(+6) x (+3) = (+30) + (+18) = (+48) To:t,} (+6) x [(-5) + (+3)] = (+6) x (-5) + (+6) x (+3) = (-30) + (18) = -12 xG' 5 . olb a, b / c tLgcf6] f k0\" ffª{ s\\ x¿ 5g\\ eg] a(b+c) = ab + ac xG' 5 . 4. 1 sf] u0' fg lgod (Multiplicative Property of 1) h:t,} (+y) x 1 = +y (-5) x 1 = -5 1 x (+9) = 9 xG' 5 . olb a Pp6f k0\" ffª{ s\\ xf] eg] ax(+1)=(+1) x (a)=a xG' 5 . 5. 0 sf] u0' fg lgod (Multiplicative Property of 0) h:t} M 2 x 0 = 0 0x2=0 100 x 0 = 0 olb a Pp6f k0\" ffª{ s\\ xf] eg] a x 0 = 0 x a = 0 xG' 5 . 0 x 0 slt xG' 5 xfn] f < ul0ft, sIff – & 129

pbfx/0f 2 u0' fg u/ M [(-8)x(-5)]x(-4) ;dfwfg [(-8)x(-5)]x(-4) =(+40)x(-4) =-160 pbfx/0f 3 ;ª3\\ Lo lgod ko| fu] u/L u0' fg u/ M (+6)x[(-15)x(12)] ;dfwfg (+6)x[(-15)x(12)] = [(+6)x(-15)] x (12) = (-90) x 12 = - 1080 2. k0\" ffª{ s\\ sf] efu (Division of Integers) efu ljm| of u0' fg ljm| ofsf] ljk/Lt ljm| of (inverse operation) xf] . To;n} ] efu ljm| ofdf klg u0' fg ljm| ofs} lgod nfu' xG' 5g\\ . tnsf pbfx/0fx¿ x/] f“} M -v_ (-8)x(+4)=(-32) -s_ (+12)x (+3) = (+36) (-32)÷(+4)=(-8) (+36) ÷ (3) = (+12) (-32)÷(-8)=(+4) (+36) ÷(12)= (+3) To;}n], wgfTds k0\" ffª{ s\\ nfO{ wgfTds k0\" ffª{ s\\ n] efu ubf{ efukmn wgfTds xG' 5 . wgfTds k0\" ffª{ s\\ nfO{ C0ffTds k0\" ffª{ s\\ nn] efu ubf{ efukmn C0ffTds xG' 5 . C0ffTds k0\" ffª{ s\\ nfO{ C0ffTds k0\" ffª{ s\\ n] efu ubf{ efukmn wgfTds xG' 5 . C0ffTds k0\" ffs{ nfO{ wgfTds k0\" ffª{ s\\ n] efu ubf{ efukmn C0ffTds xG' 5 . cEof; 13.2 1. u0' fg u/ . -s_ (+5) x (+5) -v_ (+5) x (-8) -u_ (-7) x (+8) -3_ (-9) x (-8) -ª_ (+4) x (+6) x (+5) -r_ (+7)x(+8)x(-6) -5_ (+12)x(-8)x(+2) -h- (+7)x(-5)x(+4) 130 ul0ft, sIff – &

-em_ (+6)x(+4)x(-3)x(-2) -`_ (+5)x(-4)x(-8)x(-3) 2. u0' fgsf] ;ª3\\ Lo lgod ko| fu] u/L bj' } tl/sfn] u0' fgkmn lgsfn M -s_ (+5)x(+6)x(+7) -v_ (+7)x(-5)x(-3) -u- (-3)x(-3)x(-3) -3_ (+4)x(+6)x(-5) -ª_ (+8)x(+6)x(-7) 3. u0' fgsf] kbljR5b] g lgod ko| fu] u/L ;/n u/ M -s_ (+6)x[(-18)+(30)] -v_ (-5)x[(+24)-(-6)] -u_ (+7)x [(-12) – (+6)] -3_ (+12)x [(-30)+(+45)] -ª=_ (-16)x[(-13)+(-5)] 4. efukmn lgsfn M -s_ (+30) ÷ (+6) -v_ (-25)÷ (+5) -u_ (+48) ÷ (-6) -3_ (+95) ÷ (-19) -ª_ (-100)÷(-20) -r_ (+120)÷(20) 5. bO' { cf6] f k0\" ffª{ s\\ x¿sf] u0' fgkmn (+49) 5 . Pp6f k0\" ffª{ s\\ (+7) eP csf{] k0\" ffª{ s\\ kQf nufpm . 6. bO' { cf6] f k0\" ffª{ s\\ x¿sf] u0' fgkmn (+60) 5 . Pp6f k0\" ffª{ s\\ (+5) eP csf{] k0\" ffª{ s\\ kQf nufpm . 7. (-5) nfO{ sltn] u0' fg u/] u0' fg kmn (+80) xG' 5 < 8. (+72) nfO{ sltn] efu u/] efu kmn (+9) xG' 5 < 9. u0' fgkmn (-225) agfpg (15) nfO{ slt k6s u0' fg ugk{' 5{ < 10. (-96) nfO{ (-24) n] slt k6s efu ug{ ;lsG5 < ul0ft, sIff – & 131

13.3 rf/ ;fwf/0f lgodsf] ;/nLs/0f (Simplification of Four Operations) hf8] , 36fp, u0' fg / efu oL rf/cf6] f ul0ftsf cfwf/et\" ljm| ofx¿ xg' \\ . logLx¿sf] jm| d ldnfP/ ;/n ugk{' 5{ . ;/nLs/0f;DaGwL tnsf lgodx¿ ofb u/ M -s_ hf8] , 36fp tyf u0' fg ldl>t ;d:ofdf klxnf u0' fgsf] sfd ugk{' 5{ . -v_ hf8] , 36fp tyf efu ljm| of ;dfjz] ePsf ;d:ofsf] ;dfwfg ubf{ ;ae} Gbf klxnf efu ljm| of ugk{' 5{ . -u_ u0' fg / efu ;dfjz] ePsf ;d:ofdf klxnf efu ljm| of ug{] jf afofa“ f6 bfofl“ t/ ;/n ub{} hfb“ f hg' lrxg\\ klxnf cufl8 cfp5“ ToxL ljm| of klxnf ugk{' 5{ . pbfx/0f 1 pbfx/0f 2 ;/n u/ . 15 af6 5 36fO{ 4 n] uG' bf slt xG' 5 < oxf“ lbOPsf] ;d:ofnfO{ ul0ftLo efiffdf JoSt ubf,{ (+10) + (-5) + (+25)÷(-5)-(-6)x(+8) = (+10)+(-5)+(-5)-(-6)x(+8) (15-5)x4 = (+10)+(-5)+(-5)-(-48) = 10x4 = (+5) + (-5) - (-48) = 40 = 0-(-48) = (+48) t;y,{ cfjZos ;ªV\\ of = 40 xG' 5 . cEof; 13.3 1. ;/n u/ M -s_ 20 ÷ 2 + 19 -v_ 45 - 81 x 5 -u_ 88 - 3 x 20 ÷ 6 -3_ 108 x 3 - 55 ÷ 11 + 105 -ª_ (-6)x(-4)÷(+4)+(-3)-(-2) 2. 55 af6 3 sf] 6 u0' ff 36fpb“ f slt xG' 5 < 3. 200 nfO{ 4 n] efu u/L 33 hf8] b\\ f slt xG' 5 < 4. 25 sf] 2 u0' ffsf] 10 efuaf6 2 36fO{ 5 n] u0' fg ubf{ slt xG' 5 < 5. 32 / 20 sf] km/snfO{ 4 n] efu u/L 15 hf8] b\\ f slt xG' 5 < 132 ul0ft, sIff – &

PsfO 14 cfgk' flts ;ªV\\ of (Rational Nnumber) 14.1 cfgk' flts / bzdnj ;ªV\\ of (Rational and Decimal Number) 1. cfgk' flts ;ªV\\ ofx¿ tnsf ljm| ofsnfkx¿ cWoog u/L 5nkmn u/ M -s_ tn ;ªV\\ ofx¿sf ljleGg ;dx\" x¿ lbOPsf 5g\\ M 1. kf| sl[ ts ;ªV\\ ofx¿sf] ;dx\" (N) = {1, 2, 3, 4, 5, ...} 2. k0\" f{ ;ªV\\ ofx¿sf] ;dx\" (W) = {0, 1, 2, 3, 4, ...} 3. k0\" ffª{ s\\ x¿sf] ;dx\" (Z) = {... -3, -2, -1, 0, 1, 2, 3 ...} ca, sg' } bO' c{ f6] f k0\" ffª{ s\\ x¿ (-3) / (+4) lnpm“ . logLx¿lar hf8] , 36fp, u0' fg / efu ljm| of ug{] ko| f; u/f“} M 1. hf8] ljm| of M (-3) / (+4) sf] ofu] kmn (-3)+(+4)= +1 xG' 5 . oxf“ +1 Pp6f k0\" ffª{ s\\ xf] . 2. 36fp ljm| of M oxf“ (-3) / (+4) sf] 36fp kmn (-3) - (+4) = (-3) + (-4)= (-7) xG' 5 . oxf“ -7 klg Pp6f k0\" ffª{ s\\ xf] . 3. u0' fg ljm| of M (-3) / (+4) sf] u0' fgkmn (-3) x (+4) =-12 xG' 5 . oxf“ (-12) klg Pp6f k0\" ffª{ s\\ xf] . 4. efu ljm| of M (-3) / (+4) sf] efukmn (-3)÷(+4) xG' 5 . oxf“ (-3)÷ +4)=- 3 Pp6f k0\" ffª{ s\\ xfO] g . 4 o;y{ hf8] , 36fp / u0' fgsf] ljm| ofaf6 kf| Kt xg' ] ;ªV\\ of k0\" ffª{ s\\ df kb5{ eg] efu kmn jf cgk' ft k0\" ffª{ s\\ df kbg{} . h:t} M 2 ,4 ,1 ,4 cflb k0\" ffª{ s\\ xfO] gg\\ . logLx¿ a sf ¿kdf cfp5“ g\\ . To;n} ] o:tf 3625 b ;ªV\\ ofx¿ cfgk' flts ;ªV\\ ofx¿ xg' \\ . -v_ dflysf] 5nkmnaf6 cfgk' flts ;ªV\\ ofsf] kl/efiff nv] . cfkm\\ gf] nv] fOnfO{ sIffdf 5nkmn u/ . kf| Kt lgisifn{ fO{ tnsf] lgisif;{ u“ tn' gf u//] x/] . a / b bO' c{ f6] f k0\" ffª{ s\\ x¿ xg' \\ / b = 0 eP a sf ¿kdf JoSt ul/g] ;ªV\\ ofnfO{ cfgk' flts b ;ªV\\ of (rational number) elgG5 / o;nfO{ Q n] hgfOG5 . ctM cfgk' flts ;ªV\\ of (Q) = { ........ 1,  1 13 , 0,1, , ....... } 2 24 To;n} ,] ;a} k0\" ffª{ s\\ x¿ cfgk' flts ;ªV\\ ofdf kg{] ePsfn] k0\" ffª{ s\\ sf] ;dx\" cfgk' flts ;ªV\\ ofsf] pk;dx\" xf] . To;n} ,] Z  Q nV] g ;lsG5 . ul0ft, sIff – & 133

2. bzdnj ;ªV\\ ofx¿ tnsf ljm| ofsnfkx¿ cWoog u/L 5nkmn u/ M -s_ tnsf] tflnsf e/ M j|m=;+= cGTo xg' ] bzdnj ;ªV\\ of cGToxLg bzdnj ;ªV\\ of kg' /fjT[ t bzdnj 1 1  0.5 2  0.285714285714 1  0.3333.... 2 7 3 2. 1  .... 22  ....... 2  ....... 4 7 3 3 4. dflysf] tflnsfsf cfwf/df yk 5/5 cf6] f cGTo xg' ,] cGToxLg / kg' /fjQ[ xg' ] bzdnj ;ªV\\ ofx¿ kQf nufP/ nv] . -v_ dflysf] tflnsfsf cfwf/df s] lgisif{ lgsfNg ;S5f} < nv] . cfkm\\ gf] lgisifn{ fO{ sIffdf 5nkmn u/ . kf| Kt lgisifn{ fO{ tnsf] lgisif;{ u“ tn' gf u//] x/] . o;/L cfgk' flts ;ªV\\ ofnfO{ bzdnjdf JoSt ubf{ cGTo xg' ,] cGToxLg / kg' /fjQ[ xg' ] bzdnj ;ªV\\ ofdf JoSt ug{ ;lsg] /x5] . dfly pbfx/0fdf 1 , 1 ..... cflb cGTo xg' ,] 2 , 22 ,..... cGToxLg / 1 , 2 ..... cflb kg' /fjQ[ bzdnj 24 77 33 ;ªV\\ ofx¿ xg' \\ . 3. bzdnj ;ªV\\ ofsf ks| f/ dflysf] pbfx/0fsf cfwf/df bzdnj ;ªV\\ ofx¿sf ks| f/nfO{ tn rrf{ ul/Psf] 5 . -s_ cGTo xg' ] bzdnj ;ªV\\ of (Terminating Decimal) olb cfgk' flts ;ªV\\ ofsf] x/n] cz+ nfO{ efu ubf{ efukmndf bzdnj k5fl8sf ;ªV\\ ofx¿ cGTo xG' 5 eg] To:tf] ;ªV\\ ofnfO{ cGTo xg' ] bzdnj ;ªV\\ of elgG5 . h:t} M 1  0.25 4 -v_ cGToxLg bzdnj ;ªV\\ of (Non-terminating Decimal) olb cfgk' flts ;ªV\\ ofx¿sf] x/n] cz+ nfO{ efu ubf{ efukmndf bzdnj k5fl8sf ;ªV\\ ofx¿ slxn] klg cGTo xb“' g} g\\ eg] To:tf] ;ªV\\ ofnfO{ cGToxLg bzdnj ;ªV\\ of elgG5 . h:t} M 2  0.28571... 7 134 ul0ft, sIff – &

-u_ kg' /fjT[ t bzdnj ;ªV\\ of (Recurring Decimals) olb cfgk' flts ;ªV\\ ofx¿sf] x/n] cz+ nfO{ efu ubf{ efukmndf bzdnj k5fl8sf ;ªV\\ ofx¿df Pp6} ;ªV\\ of bfx] fl] /P/ cfO/xG5g\\ eg] To:tf] ;ªV\\ ofnfO{ kg' /fjQ[ bzdnj ;ªV\\ of elgG5 . h:t} 2  0.66666... 3 gf6] M 1. olb cfgk' flts ;ªV\\ ofsf] x/df 2 cyjf 5 jf ¿9 u0' fg v08 2 / 5 xG' 5g\\ eg] Tof] ;ªV\\ of cGTo xg' ] bzdnj ;ªV\\ of xG' 5 . h:t} M 1 , 1 , 7 , 7 , 17 , ............. 2 5 10 25 30 2. olb cfgk' flts ;ªV\\ ofsf] x/df 2 / 5 afxs] c¿ ;ªV\\ of ePdf To:tf bzdnj ;ªV\\ of cGToxLg jf kg' /fjQ[ xG' 5g\\ . h:t} M 1,2,5,2, ................ 3377 2. cfgk' flts ;ªV\\ ofsf ljzi] ftfx¿ (Properties of Rational Numbers) sIff 6 df leGgsf] hf8] , 36fp, u0' fg / efu ljm| ofsf ;d:ofx¿ ;dfwfg ul/;ss] f 5f“} . dfgf,“} a, b / c sg' } cfgk' flts ;ªV\\ ofx¿ xg' \\ . cfgk' flts ;ªV\\ ofsf] hf8] / u0' fgsf ljzi] ftfx¿ tnsf cfgk' flts ;ªV\\ ofx¿sf ljzi] ftfx¿ cWoog u/L 5nkmn u/ M 1. PsfTds lgod (Identity Property) : a00 a  a / a 11 a  a b bb b bb hf8] sf] PsfTds lgod u0' fgsf] PsfTds lgod 100 1  1 lgod M 1 1  1 1  1 2 22 2 22 2 00 2  2 2 1  1 2  2 3 33 3 33 3 00 3  3 3 1  1 3  3 7 77 7 77 a 00 a  a a 1  1 a  a b bb b bb lgod M sg' } klg cfgk' flts ;ªV\\ ofdf zG\" o sg' } klg cfgk' flts ;ªV\\ ofnfO{ -)_ hf8] b\\ f cfpg] ;ªV\\ of ToxL ;ªV\\ of Ps (1) n] u0' fg ubf{ ToxL g} xG' 5 . ;ªV\\ of cfp5“ . ul0ft, sIff – & 135

2. ljk/Lt u0' f (Inverse Property) hf8] sf] ljk/Lt u0' f u0' fgsf] ljk/Lt u0' f -1 + 1 = 0 1x1=1 11 0 2 1 1 22 2 2 2 0 23 1 33 32 aa 0 ab 1 bb ba lgod M sg' } klg cfgk' flts ;ªV\\ of a lgod M sg' } klg cfgk' flts ;ªV\\ of a df  ljBdfg eO{ b ab a a    a  b df sg' } ;ªV\\ of a eO{ ba b b b  b  1 0 cfp5“ eg] o:tf] u0' fgnfO{ hf8] sf] cfp5“ eg] To:tf] u0' fnfO{ u0' fgsf] ljk/Lt u0' f elgG5 . ljk/Lt u0' f elgG5 . 3. jm| d ljlgdo u0' f (Commutative Property) hf8] sf] jm| d ljlgdo u0' f u0' fgsf] jm| d ljlgod u0' f 2+3=3+2 2x3=3x2 12  2 1 12  21 23 32 23 32 25 52 25 52 36 63 36 63 ac ca ac  ca bd db bd db lgod M ac ca nfO{ hf8] sf] lgod M a  c  c  a nfO{ u0' fgsf] bd db b d d b jm| d ljlgod elgG5 . jm| d ljlgdo elgG5 . 4. ;ª3\\ Lo lgod (Associative Property) hf8] sf] ;ª3\\ Lo lgod u0' fgsf] ;ª3\\ Lo lgod 2 + (5 + 7) = (2 + 5) +7 2 x (5 x 7) = (2 x 5) x 7 1   2  3    1  2   3 1   2  3    1  2   3 2 3 5 2 3 5 2 3 5 2 3 5 a   c  e    a  c   e a   c  e    a  c   e b d f  b d f b d f  b d f 136 ul0ft, sIff – &

lgod M a   c  e    a  c   e lgod M a   c  e    a  c   e nfO{ b d f  b d f b d f  b d f nfO{ hf8] sf] ;ª3\\ Lo lgod elgG5 . u0' fgsf] ;ª3\\ Lo lgod elgG5 . 5. lgs6tfsf] lgod (Closure Property) u0' fgsf] lgs6tfsf] lgod hf8] sf] lgs6tfsf] lgod 1 / 2 cfgk' flts ;ªV\\ of xg' \\ . 1 / 2 cfgk' flts ;ªV\\ of xg' \\ . 2 3 2 3 1  2  34  7 klg cfgk' flts ;ªV\\ of xf] . 12  2 klg cfgk' flts ;ªV\\ of xf] . 23 6 6 23 6 ac e eP e klg cfgk' flts ;ªV\\ of xf] . bd f f lgod M olb ac e eP hf8] kmn e klg lgod M olb ac  e eP e klg bd f f bd f f cfgk' flts ;ªV\\ of xG' 5 . cfgk' flts ;ªV\\ of xG' 5 . cEof; 14.1 1. tnsf kZ| gsf] pQ/ nv] M -s_ s] ;a} kf| sl[ ts ;ªV\\ of cfgk' flts ;ªV\\ of xg' \\ < -v_ s] ;a} k0\" f{ ;ªV\\ of cfgk' flts ;ªV\\ of xg' \\ < -u_ s] ;a} k0\" ffª{ s\\ cfgk' flts ;ªV\\ of xg' \\ < -3_ s] zG\" o (0) cfgk' flts ;ªV\\ of xf] < -ª_ s] ;a} cfgk' flts ;ªV\\ of k0\" ffª{ s\\ xg' \\ < 2. tn lbOPsf sg' sg' ;ªV\\ ofx¿ cGTo xg' ,] cGToxLg / kg' /fjQ[ bzdnj ;ªV\\ of xg' ,\\ 56' o\\ fpm M -s_ 1 -v_ 3 -u_ 2 -3_ 15 -ª_ 17 2 5 7 2 13 -r_ 55 -5_ 37 -h_ 25 -em_ 22 -`_ 12 10 20 17 7 25 3. tn lbOPsf ;ªV\\ ofsf] hf8] sf] ljk/Lt / u0' fgsf] ljk/Lt ;ªV\\ of kQf nufpm M -s_ 2 -v_ 5 -u_ 22 -3_ 12 -ª_  11 5 7 7 7 8 4. 1 / 3 sf] lardf sg' } bO' c{ f6] f cfgk' flts ;ªV\\ of nv] . 2 5 ul0ft, sIff – & 137

PsfO 15 cggk' flts ;ªV\\ of (Irrational Number) cggk' flts ;ªV\\ ofsf] kl/ro (Introduction of Irrational Number) 1. cgfgk' flts ;ªV\\ of -s_ tnsf ljm| ofsnfk cWoog u/L 5nkmn u/ M 4 sf] jud{ n\"  4  2 -cfgk' flts ;ªV\\ of_ 9 sf] jud{ n\" 9  3 -cfgk' flts ;ªV\\ of_ 9 sf] jud{ n\"  9  3 -cfgk' flts ;ªV\\ of_ 16 16 4 2 sf] jud{ n\" 2  1.4421.... of] cfgk' flts ;ªV\\ of xfO] g, lsg < cyjf 2 nfO{ a sf] ¿kdf JoSt ug{ ;lsb“ g} . b -v_dflysf] 5nkmnsf cfwf/df cggk' flts ;ªV\\ ofsf] kl/efiff nv] . a sf] ¿kdf JoSt ug{ g;lsg] ;ªV\\ ofnfO{ cggk' flts ;ªV\\ of elgG5 . b cyft{ ,\\ cggk' flts ;ªV\\ ofdf gkg{] ;ªV\\ ofnfO{ cggk' flts ;ªV\\ of elgG5 . h:t} M ..., 2, 3, 1, 7,3 10, cflb cggk' flts ;ªV\\ ofx¿ xg' \\ . 3 cgfgk' flts ;ªV\\ ofnfO{ Ir n] hgfOG5 . To;n} ] Q / Ir cnlUuPsf ;dx\" xg' \\ . ctM Q∩lr = ∅ xG' 5 . cEof; 15.1 1. tn lbOPsf ;ªV\\ of cfgk' flts ;ªV\\ of jf cgfgk' flts ;ªV\\ of s] xg' \\ < 56' o\\ fpm . -s_ 3 -v_ 2 -u_ 5 -3_ 2 -ª_ 10 4 5 20 -r_ 1 -5_ 25 -h_ 40 -em_ 0.735 3 10 50 2. s] ;a} cfgk' flts ;ªV\\ ofx¿ cgfgk' flts ;ªV\\ of xg' \\ < 3. s] ;a} k0\" ffª{ s\\ cgfgk' flts ;ªV\\ of xg' \\ < 4. s] cfgk' flts ;ªV\\ of / cgfgk' flts ;ªV\\ of cnlUuPsf ;dx\" xg' \\ < 138 ul0ft, sIff – &

PsfO 16 leGg / bzdnj (Fraction and Decimal) 16.1 leGgsf zflAbs ;d:of (Word Problems on Fraction) xfdLn] sIff 6 df leGgsf wf/0ffx¿, leGgsf hf8] / 36fp, u0' fg / ;/nLs/0fsf af/d] f 5nkmn ul/;ss] f 5f“} . ca xfdL leGgsf zflAbs ;d:ofx¿sf af/d] f 5nkmn ub}{ 5f“} . tnsf pbfx/0f cWoog u/L cfkm\" n] klg cEof; u/ . pbfx/0f 1 bLksn] kT| os] dlxgf ? 12,000 sdfp5“ . p;sf] cfDbfgLsf] tLg efusf] Ps efu lzIffdf vr{ u5{ . To:t} rf/ efusf] Ps efu vfgfdf vr{ u5{ . ca p;n] hDdf slt efu vr{ u5{ < hDdf slt /sd vr{ u5{ < ;dfwfg bLkssf] lzIffdf vr{ tLg efusf] Ps efu / vfgfdf vr{ rf/ efusf] Ps efu xG' 5 . ca hDdf vr{ lgsfNbf bj' } vr{ hf8] g\\ k' 5{ . hDdf vr{ = lzIffsf] vr{ ± vfgfsf] vr{ = 1  1 3 4 4x1  3x1 ca, 3 / 4 x/df ePsf ;ªV\\ ofsf] n=;= = 12 = 12  43 clg, kT| os] x/n] n=;= nfO{ efu u/f“} / efukmnn] cz+ nfO{ u0' fg 12 u/f“} / ;/n u/f“} . 7 efu 12 bLksn] hDdf 12 efusf] 7 efu vr{ u/s] f] /x5] . o;y{ p;n] cfkm\\ gf] cfDbfgLsf] 7 efu vr{ u5{ 12 ca bLkssf] hDdf jf:tljs vr{ = ?= 12,000 sf] 7 = ? 12,000x 7 = ?= 7000 . o;y{ bLkssf] 12 12 hDdf vr{ = ?= 7000 xG' 5 . pbfx/0f 2 Pp6f sIffsf 42 hgf ljBfyLx{ ¿dWo] ltg efusf] bO' { efu s6] f / afs“ L s6] L lyP eg] s6] L slt hgf /x5] g\\ < kQf nufpm . ;dfwfg tl/sf 1 oxf,“ s6] fsf] ;ªV\\ of = hDdf ljbo\\ fyLs{ f] 2 efu 5, To;n} ] s6] fsf] jf:tljs ;ªV\\ of = 42 hgfsf] 2 efu = 42 hgf x 2 = 28 hgf . 3 3 3 ul0ft, sIff – & 139

140 ul0ft, sIff – &

ca tLg} hgfsf] leGgsf] x/df ePsf cªs\\ sf] n=;= lgsfnf“} . n=;= = 6 ca ;a} leGgsf] x/ 6 agfcf“} M 1  1x2  3 2 32 6 1  1x2  2 3 32 6 11 66 ca cz+ df ;ae} Gbf 7n' f] 3 ePsfn] ] 1 efu vfg] ;~hn' ] ;aeGbf w/] } ss] vfOG5g\\ . 2 cEof; 16.1 1. hf8f] dlxgfsf] Ps lbg sIff 7 sf hDdf 70 hgf ljBfyLd{ Wo] ltg efusf] Ps efu dfq pkl:yt eP5g\\ eg] slt efu cgk' l:yt eP5g\\ < kQf nufpm . 2. ku| lt lzIff ;bgdf sIff 7 df hDdf 42 hgf ljBfyL{ lyP . 24 hgf s6] f lyP eg] s]6f slt efu lyP < s6] L slt efu lyP < kQf nufpm . 3= ;h' gsf] dfl;s cfGbfgL ?= 9,000 5 . p;n] 1 efu vfhfdf vr{ u5{ . 3 efu sk8fdf vr{ efu oftfoftdf vr{ u5{ . slt efu 5 10 kQf nufpm . 2 u5{ . 5 art u5{ / slt ?lkof“ art u5{ < 4= a6' jn cfB} fl] us dn] fdf sdnfn] cfkm\" ;u“ ePsf] ?lkofs“ f] 1 efu dgf/] ~hgdf vr{ ul/g\\ . 1 5 2 3 efu vfgfsf] nflu vr{ ul/g\\ . 10 efu nTtf sk8fdf vr{ ul/g\\ . ;ae} Gbf w/] } sg' zLifs{ df vr{ u/s] L /lx5g\\ < kQf nufpm . 5= sg' } ;ªV\\ ofsf] 3 efu 90 xG' 5 eg] ;f] ;ªV\\ of kQf nufpm . 5 6= sg' } ?lkofs“ f] 4 efu ?= 600 xG' 5 eg] 3 efu a/fa/ slt xG' 5 < kQf nufpm . 5 4 7= /fli6o« jfl0fHo aª} s\\ df ;G' b/k;| fbn] 10 kl| tzt kl| tjifs{ f b/n] Ps jifs{ f] Aofh ae' mfpb“ f ?= 30000 ltgk{' ¥of] . p;n] hDdf slt /sd C0f lnPsf] /x5] < 8= Pp6f kfgL 6\\ofªs\\ Lsf] 1 efu ebf{ 700 ln6/ kfgL hDdf eP5 eg] 6\\ofªs\\ Lsf] Ifdtf 5 slt /x5] < kQf nufpm . 9= ks| l[ tsL cfdfn] pgnfO{ lbgx“' Pp6f a66\\ fsf] 3 efu xlnS{ ; vj' fpgx' G' 5 . 30 a66\\ f xlnS{ ;n] 10 pgnfO{ hDdf slt lbgnfO{ kU' nf < kQf nufpm . ul0ft, sIff – & 141

16.2 bzdnjsf] ;/nLs/0f / zflAbs ;d:ofx¿ (Simplification and Word Problems on Decimal) pbfx/0f 1 (1) hf8] lrxg\\ , hf8] lrxg\\ / 36fp lrxg\\ , 36fp lrxg\\ ;/n u/ M 5.24+3.01 - 1.92 -5.67 ;dfwfg ldnfP/ Ps} 7fpd“ f /fVg] 5.24+3.01 - 1.92 -5.67 r/0f 1 r/0f 2 = 8.25-7.59 = 0.66 -@_ 5.24 1.92 8.25 pbfx/0f 2 ;/n u/ M + 3.01 + 5.67 -7.59 ;dfwfg 8.25 7.59 0.66 (64.32 - 40.64) x 2.22 = (64.32 - 40.64) x2.22 23.68 = 23.68 x 2.22 = 52.5696 x 2.22 4736 4736 4736 52.5606 pbfx/0f 3 cfotsf/ ?dfnsf] nDafO 5.2 cm / rf8} fO 4.8 cm /x5] eg] ?dfnsf] kl/ldlt lgsfn M ;dfwfg cfotsf/ ?dfnsf] kl/ldlt 2(l+b) xG' 5 . ca, nDafO (l) = 5.2 cm / rf8} fO -b_ = 4.8 cm 5 . To;n} ] ?dfnsf] kl/ldlt M = 2(l+b) = 2(5.2cm+4.8cm) = 20cm pbfx/0f 4 Pp6f 0.45 m cwJ{ of; ePsf] ;fOsnsf] kfªu\\ f| n] 100 rSs/ nufpb“ f slt b/' L kf/ u5{ < kQf nufpm . (π = 3.14) ;dfwfg oxf“ ;fOsnsf] kfªu\\ f| n] Ps rSs/ nufpg' egs] f] kl/lw lgsfNg' xf] . To;n} ,] kfªu\\ f| sf] Ps rSs/ = 2πr = 2π x 0.45m = 2 x 3.14 x 0.45m 142 ul0ft, sIff – &

ul0ft, sIff – & 143

PsfO 17 cgk' ft, ;dfgk' ft / kl| tzt (Ratio, Proportion and Percent) 17.1 kl| tztsf ;/n ;d:ofx¿ (Simple Problems on Percentage) kl| tzt egs] f] kl| t ;odf lx;fa ug{' xf] . ctM x/df 100 ePsf] leGg g} kl| tzt xf] . h:t} M 20% egs] f] 100 df 20 cyft{ 20 xG' 5 . 100 75% egs] f] 100 df 75 cyft{ 75 xG' 5 . 100 To:t,} 12  12 100%  12% 100 100 1  1 100%  25% 44 leGgnfO{ kl| tztdf abNbf 100 n] u0' fg u/L % lrxg\\ /fVgk' 5{ . kl| tztnfO{ leGgdf abNbf 100 n] efu u/L % nfO{ x6fpgk' 5{ . pbfx/0f 1 ?= 150 sf] 20% slt xG' 5 < ;dfwfg oxf“ ?= 150 sf] 20% ?= 150 x 20 -% nfO{ leGgdf abNbf 100 n] efu u/s] f_] 100 = ?= 30 ctM ?= 150 sf] 20% egs] f] ?= 30 xG' 5 . pbfx/0f 2 20 sf] 15 slt % xG' 5 < ;dfwfg oxf“ 20 efudWo] 15 efu 5 . To;n} ] kl| tztdf JoSt ubf{ 15  100 % x'G5 . 20 20 sf] 15 nfO{ leGgdf nV] bf 15 xG' 5 . 20 15 nfO{ % df abNbf = 15 100% = 75 % 20 20 ctM 20 sf] 15 egs] f] 75% xG' 5 . 144 ul0ft, sIff – &

csf{] tl/sf, cfjZos kl| tztnfO{ x% dfGbf 20 sf] x% = 15 cyjf, x 20 × 100 =15 cyjf, x = 15 5 cyjf, x = 15 x 5 x∴ = 75% To;n} ,] 20 sf] 75 % = 15 xG' 5 . pbfx/0f 3 slt ?lkofs“ f] 25% n] ? 350 xG' 5 < ;dfwfg oxf,“ cfjZos ?lkofn“ fO{ x dfgf“} . x sf] 25% = ?= 350 cyjf, x  25 = ?= 350 100 cyjf, x = ?= 350 4 cyjf, x = 4 x ?= 350 x = ?= 1400 ct M ?= 1400 ?lkofs“ f] 25% n] ?= 350 xg' ] /x5] . pbfx/0f 4 kg' dn] 20 k0\" ffª{ s\\ sf] k/LIffdf 16 cªs\\ kf| Kt ul/g\\ eg] pgn] slt % kf| Kt ul/5g\\ < ;dfwfg dflysf] ;d:ofnfO{ ul0ftLo efiffdf nV] bf, 16 xG' 5 . 20 16 nfO{ % df abNbf 16 100%  80% xG' 5 . 20 20 ctM kg' dn] 80% cªs\\ kf| Kt ul/5g\\ . ul0ft, sIff – & 145