la[;k i)fr v/;k;&1 1- ljy djks ifjek.k ds vojksgh Øe esa fy[kks 4- ljy djsa ljy djsa ljy djsa dk eku Kkr djsaa

Transkript

1 v/;k;&1 la[;k i)fr 1- ljy djks vksj 4 5 esa dksu lh cm+h gs\ 3- ifjek.k ds vojksgh Øe esa fy[kks 3 ] 3 3 ] ljy djsa ljy djsa ljy djsa dk eku Kkr djsa dk eku Kkr djs dks ljy djsa 10- eku Kkr djsa dks ljy djks dks ljyre :i esa fy[ks 13- ljy djsa vksj 1 4 ds chp nks ifjes; la[;k, crkvks

2 15-1 vksj 3 ds chp nks ifjes; la[;k, Kkr djks 16-5 vksj 3 4 ds chp rhu ifjes; la[;k, Kkr djks 17-1 vksj 1 ds chp pkj ifjes; la[;k, Kkr djks dks p q ds :i esa O;r djks dks p q ds :i esa O;r djks dks p q ds :i esa O;r djks 1-3 dk n'keyo ds nks LFkkuksa rd lgh&lgh eku ;k gksxk\ a b dks ljyre :i esa O;r djks a b c dks ljyre :i esa O;r djks dk n'keyo izlkj ;k gksxk\ dk n'keyo izlkj ;k gksxk\ dk n'keyo izlkj ;k gksxk\ 7-0 vksj 1 10 ds chp esa rhu ifjes; la[;k, Kkr djks dk ifjes; gj crkb,

3 9- ;fn gks rks 3 dk eku Kkr djks 30- fueu esa gj dk ifjes;dj.k djks %& esa gj dk ifjes;dj.k djks esa gj dk ifjes;dj.k djks dk eku Kkr djks dk eku Kkr djks 35- ;fn x 1 rks 1 x x dk eku Kkr djks 36- fueu esa ls ifjes; ;k vifjes; la[;k, crkb, %& ¼d½ 3 ¼[k½ ;fn P 3 rks p 1 p dk eku ;k gksxk\ 38- xy 3 xy dks vifjes; la[;k esa O;r djks 39- a 1. ab 3 dks vifjes; la[;k esa O;r djks dk eku Kkr djks 41- ljy :i crkb, %& 7 80

4 4- ;fn rc 1 10 dk eku Kkr djks 43- ;fn x rc x 1 dk eku Kkr djks x 44-0 rfkk 1 ds chp esa rhu ifjes; la[;k, Kkr djks dk eku Kkr djks dk eku Kkr djks dk eku ;k gksxk\ 48- ;fn 1.41 rks 1 dk eku ;k gksxk\ dks ls Hkkx djks dk ifjes;dkjh O;atd crkb, dk eku Kkr djks 5- ;fn rks 5 3 dk eku Kkr djks 53- ;fn 3 x 1 rks x dk eku Kkr djks 54- ;fn 4 49 x 1 rks x dk eku Kkr djks dk eku Kkr djks

5 mùkjekyk , 3, 15 3, ,, ;k dksbz vu; nks ;k dksbz vu; rhu ,,, ;k ;k , 8 16 ;k dksbz vu; nks ab 4a a bc b c

6 ,, ;k dksbz vu; rhu ,, ;k dksbz vu; rhu ¼d½ vifjes; ¼[k½ ifjes; x y a b

7 v/;k;& cgqin 1- cgqin 4 y dh?kkr crkb;s - cgqin 3 5 x 4 x 7 x dh?kkr crkb, 3- dksuls cgqin jsf[kd] f}?kkrh vfkok f=?kkrh gsa\ ¼d½ 3 x x ¼[k½ y y 4 ¼x½ 3t 4- cgqin 4x 7 x 5 dk eku Kkr djks tcfd x 3 gks 3 5- ;fn f x x 3x 1 gks rks f dk eku crkvks 3 6- cgqin 11 y y dh?kkr crkb, 7- ;fn P y y y 1 gks rks P 3 dk eku ;k gksxk\ 8- cgqin Px x 5 dk 'kwu;d ;k gksxk\ 9- x y z esa x dk xq.kkad ;k gs\ 10- x 5 x esa x dk xq.kkad ;k gs\ 11- cgqin 0 dh?kkr crkb;s 4 1- y 6y 9 4y dk ekud :i crkb;s q 4 q q dk ekud :i ;k gksxk\ ;fn Px x ax 6 x a dks x a ls Hkkx djsas rks 'ks"kqy ;k gksxk\

8 15- 'ks"kqy crkb;s tc x dks x 1 ls Hkkx fd;k tk, 16- k dk eku crkb;s] ;fn 3 x ] 3x kx 6 dk,d xq.ku[k.m gs x 60x 150x 15 dks f}in?ku ds :i esa crkvks 18- xq.ku[k.m djks %& 9 p 16q 19- xq.ku[k.m djks %& x x y z z x y z 0- folr`r :i crkb;s x 5y 3 1- folr`r :i crkb;s x folr`r :i crkb;s x 3- folr`r :i crkb;s 3 x 1 4- xq.ku[k.m cukb, 5- xq.ku[k.m dhft, 50 x 7y m 3m 3 6- cgqin x 14x 40 ds 'kwu;d crkvks 7- cgqin x 8 x 10 ds 'kwu;dksa dk xq.kuqy crkvks 8- xq.ku[k.m djks y 9 x cgqin 4 y a b 1a b 6ab y dh?kkr crkvks ds xq.ku[k.m djks x 7 y 36 x y 54 xy ds xq.ku[k.m djks 3-36a 60ab 5b ds xq.ku[k.m djks

9 33- x 5x 4 dk xq.kuqy Kkr djks 34- x 3 x 7 dk xq.kuqy Kkr djks 35- x 6x 8 dk xq.kuqy Kkr djks 36- ;fn x 1 cgqin 3 x x x 1 dk,d xq.ku[k.m gs rks 'ks"kqy crkvks Px x 3x 3x x 1 g x x 1 ;fn Px dks g x ls Hkkx fn;k tk, rks 'ks"kqy ;k gksxk\ 38- ;fn x 3 x x k dk,d xq.ku[k.m x gs rks k dk eku Kkr djks 39- ;fn x 3 kx 9 x 5 dk,d xq.ku[k.m x 1 gs rks k dk eku Kkr djks 40- ;fn x kx 3x 4 dk,d xq.ku[k.m x 1 gs rks k dk eku Kkr djks 41- ;fn 3x x k dk,d xq.ku[k.m x 1 gs rks k dk eku Kkr djks 4- cgqin 3 3 x 15x 10 x esa x dk xq.kkad ;k gs\ 43- Px x 4x 7 Px dk eku Kkr djks ;fn x f 3 7 dk eku Kkr djks ;fn f x 4x 3x 45- ;fn f x x 5 x 14 gks rks cgqin x 15 x 34 ds 'kwu;d Kkr djks f dk eku Kkr djks 47- cgqin x 5x 4 esa ;k tksm+k tk, fd cgqin dk 'kwu;d 3 gks tk,\ 48-3,,, 1 esa ls dksu&lh la[;k, cgqin x 4 dk 'kwu;d gs\ 49- HkkxQy Kkr djks ;fn x 7x 1 dks x 3 ls Hkkx fd;k tk, 50- og cgqin Kkr djks ftlds 'kwu;d rfkk gsa

10 mùkjekyk (i) f=?kkrh, (ii) f}?kkrh, (iii) jsf[kd 4- & & 'kwu; q q q 14-5a y y 6y x P 4q3 P 4q 19- x z x y z x 15y 15x y 75xy x x 94x - x 4x 4 3- x 4-5x 6y5 x 6y x 3 x 9 6x 1 m 6-4, 10 y y a b x 3y 3 3-6a 5b 33- x x x 10x x 14x k 39- k k k ] ] 49- x x

11 v/;k;&3 funsz'kkad T;kferh 1- fdl xf.krk us funsz'kkad T;kferh fodflr dh\ - fdlh js[kk[k.m dks [khapus ds fy, fdrus fcanqvksa dh vko';drk gksrh gs\ 3-,d lery dks nks v{k pkj Hkkxksa esa ck Vrs gsa] bu pkj Hkkxksa dks ;k dgrs gsa\ 4- ewy fcanq ds funsz'kkad ;k gksrs gsa\ 5- nksuksa v{kksa ds izfrpnsnh fcanq dks ;k dgrs gsa\ 6- y & v{k ls fdlh fcanq dh nwjh dks ;k dgk tkrk gs\ 7- x & v{k ls fdlh fcanq dh nwjh dks ;k dgk tkrk gs\ 8- dkrhz; lery esa fdlh fcanq dh flfkfr fu/kkzfjr djus ds fy, tks {ksfrt rfkk m/okz/kj js[kk, cukbz tkrh gsa muds uke crkvks 9- x & v{k dk lehdj.k crkvks 10- x 0 dksu&ls v{k dks n'kkzrk gs\ 11- js[kk y fdl v{k ds lekarj gs\ 1- js[kk x 3 fdl v{k ds lekarj gs\ 13- fcanq P 4, 3 dh x & v{k ls ykafcd nwjh crkvks 14- fcanq Q 5, 7 dh y & v{k ls ykafcd nwjh crkvks 15- fueu esa dksu&ls fcanq pksfks prqfkkza'k esa gsa %& 3, 5,, 1, 4, 4, 7, 5, 6

12 16- ;fn 0, 0, 5, 0, 5, 5 vksj 0, 5 dks dkrhz; ry esa feyk;k tk, rks vkd`fr dk uke crkvks 17- ;fn 0, 0, 5, 0, 5, 3 rfkk 0, 3 dks dkrhz; ry esa feyk;k tk, rks izkir vkd`fr dk uke crkvks 18-5, 0, 0, 5 rfkk 5, 0 dks dkrhz; ry esa feykus okyh vkd`fr dk uke crkvks 19- fcanq 7, 0 dksu&ls v{k ij flfkr gs\ 0- fcanq 0, 9 dksu&ls v{k ij flfkr gs\ 1- ml fcanq ds funsz'kkad crkb, ftldk Hkqt 9 vksj dksfv gks - ml fcanq ds funsz'kkad crkb, ftldh dksfv 3 gks vksj tks y v{k ij flfkr gs 3- ml fcanq ds funsz'kkad crkb, ftldk Hkqt 3 gks vksj tks x v{k ij flfkr gks 4-, 4 dksu&ls prqfkkzaa'k esa gsa\ 5- ;fn fcanq P 1, 0, Q (5, 0), R (5, ) fdlh vk;r PQRS ds rhu 'kh"kksaz ds funsz'kkad gks rks pksfks 'kh"kz S ds funsz'kkad crkvks 6- ;fn fcanq 0, 0, (, 0), (, ) fdlh oxz ds rhu 'kh"kksaz ds funsz'kkad gks rks pksfks 'kh"kz ds funsz'kkad crkvks 7- dksu lk fcanq x & v{k ij flfkr gsa %& (i) 3, 0 (ii), 3 8- ;fn fcanq x, y ] x v{k ij flfkr gks rks y dk eku ;k gksxk\ 9- ;fn fcanq x, y ] y v{k ij flfkr gks rks x dk eku ;k gksxk\

13 30- ml fcanq ds funsz'kkad crkvks ftldk Hkqt 9 vksj dksfv 5 gks 31- ml fcanq ds funsz'kkad crkvks ftldk Hkqt 3 vksj dksfv 7 gks 3- dksu&lk fcanq x & v{k ij flfkr gsa %& (i) (0, ) (ii) (, 0 ) 33- dksu&lk fcanq y & v{k ij flfkr gsa %& (i) (0, 3) (ii) (, 3 ) 34- dksu&lk fcanq x & v{k ij flfkr gsa %& (i) ( 0, 4 ) (ii) (5, 0 ) 35-,d fcanq tks x & v{k ij ewy fcunq ds nkb± vksj flfkr gs rfkk y & v{k ls 6 bdkbz dh nwjh ij gs bl fcanq ds funsz'kkad crkb, 36-,d fcanq tks y & v{k ij flfkr gs vksj x & v{k ls Åij dh vksj 1 bdkbz dh nwjh ij gs bl fcanq ds funsz'kkad crkb, 37- fcanqvksa,,, ds funsz'kkad crkb, Y X X Y

14 38- fcanqvksa,,,, E, F, G rfkk H ds funsz'kkad crkb, Y H X G F X E Y Y 39- vkd`fr igpkfu, 'kh"kksza ds funsz'kkad Hkh crkb, X X Y 40- vkd`fr esa y & v{k ls fn, gq, fcanq dh ykafcd nwjh ;k gksxh] ;fn ;g x & v{k dh /kukred fn'kk esa flfkr gks M P Y X O L X Y

15 41- vkd`fr esa x & v{k ls fn, gq, fcanq dh ykafcd nwjh crkb,] ;fn fcanq y & v{k dh /kukred fn'kk esa flfkr gks Y M P X O L X Y Y 4- vkd`fr ns[kdj crkb, %& (i) fcanq ds funsz'kkad (ii) fcanq ds funsz'kkad (iii) fcanq ds funsz'kkad (iv) fcanq ds funsz'kkad X X Y Y 43- vkd`fr esa crkb, %& (i) fcanq H dh dksfv (ii) fcanq I dh dksfv X I X H Y

16 44- vkd`fr ns[kdj crkb, %& (i) fcanq dk Hkqt (ii) fcanq E dk Hkqt Y X X 45- vkd`fr ns[kdj crkb, %& (i) og fcanq ftlds funsz'kkad (, 3) gksa (ii) og fcanq ftlds funsz'kkad (3, 3) gksa Y Y X X Y Y 46- vkd`fr ns[kdj crkb, %& (i) fcanq L ds funsz'kkad (ii) fcanq M ds funsz'kkad X X 47- og prqfkkza'k crkb, ftlesa fueu fcanq flfkr gs\ (i) ( 7, 5) (ii) (, 9) (iii) (9, 0) (iv) (0, 5) Y

17 1- jsu ndkrsz - nks 3- prqfkkza'k 4- (0, 0) 5- ewy fcanq 6- Hkqt 7- dksfv 8- x & v{k vksj y & v{k 9- y y & v{k 11- x & v{k 1- y & v{k 13-3 ek=d 14-5 ek=d 15-, 1 rfkk 4, oxz 17- vk;r mùkjekyk 18- f=hkqt 19- x & v{k 0- y & v{k 1-9, - 0, 3 3-3, 0 4- III prqfkkza'k 5-1, 6-0, 7-3, 0 8- y 0 9- x , , , 0

18 33-0, , , , ,, 4, 4 5,, ( 3, 4) 38-, 3, 4, 5, 3 5,, E, 4 F 3, 0 G(,0), H (0, 4) 39- f=hkqt 4, 4, 4, 4 0, Hkqt x 41- dksfv y 4- (, ) (, 4) (3, 1) 43- H I E (i) E (ii) 46- L 1, 47- (i) III M 1, (ii) IV (iii) x & v{k (iv) y & v{k ( 3, 0)

19 1- lehdj.k ;k gksrk gs\ v/;k;&4 nks pjksa okys jsf[kd lehdj.k -,d pj esa jsf[kd lehdj.k ;k gksrk gs\ 3-,d pj ds jsf[kd lehdj.k ds fdrus gy gksrs gsa\ 4- nks pj okys jsf[kd lehdj.k ;k gksrs gs\ 5- nks pj okys jsf[kd lehdj.k ds fdrus gy gksrs gsa\ 6- fueu lehdj.k ax by c 0] a 0, b 0 dks y ds :i esa crkb, 7- lehdj.k x y 4 dks y ds :i esa O;r dhft, 8- lehdj.k x 5y 9 dks x ds :i esa O;r dhft, 9- lehdj.k x y 9 x ds :i esa ;k gksxk\ 10- lehdj.k 3 x 8 y vksj ax by c 0 dh rqyuk djks vksj izr;sd flfkfr esa, vksj c dk eku crkb, 11- lehdj.k x 6y vksj ax by c 0 dh rqyuk djks vksj izr;sd flfkfr esa a, b vksj c dk eku crkb, y x dks ax by c 0 ds :i esa O;r djks vksj izr;sd flfkfr esa a, b vksj c dk eku crkb, 13- x 9 jsf[kd lehdj.k dks nks pj pkys jsf[kd lehdj.k ds :i esa O;r djks 14-3 y 7 jsf[kd lehdj.k dks nks pj pkys jsf[kd lehdj.k ds :i esa O;r djks 15- jsf[kd lehdj.k 5y dks nks pj okys jsf[kd lehdj.k ds :i esa O;r djks a b

20 16- tk p djds crkvks fd x ] y 1 jsf[kd lehdj.k 5x 3y 14 dk gy gs ;k ugha 17- tk p djds crkvks fd x 1 ] jsf[kd lehdj.k 0.5x dk gy gs ;k ugha 18- tk p djks fd x 0 lehdj.k x x 3 x 8 dk gy gs x 3 x tk p djks fd x lehdj.k dk gy gs 0- tk p djks fd 3] lehdj.k x 7 3x 8 dk gy gs 1- ml fcanq ds funsz'kkad Kkr djks tgk lehdj.k 3x y 6 dk vkys[k nksuksa v{k ij dkvrk gs - ml fcanq ds funsz'kkad Kkr djks tgk lehdj.k y 3x 9 nksuksa v{k ij dkvrk gs 3- ;k fcanq 4, 1 lehdj.k x 5y 13 }kjk iznf'kzr vkys[k ij flfkr gs\ 4- ml fcanq ds funsz'kkad Kkr djks tgk lehdj.k x 3y 6 ] x & v{k rfkk y & v{k dks dkvrk gs 5- fueu jsf[kd lehdj.k ds dksbz nks gy Kkr djks %& (i) x 5y 13 (ii) x y 4 0 (iii) x 4 y p dk eku Kkr djks ;fn x 1, y 1 lehdj.k 3x py 6 ds gy gsa 7- p dk eku Kkr djks ;fn x 1, y 1 lehdj.k px y 10 ds gy gsa 8- p dk eku Kkr djks ;fn x 1, y 1 lehdj.k 5x py 3a ds gy gsa fueu dfkuksa dks nks pj okys lehdj.k ds :i esa O;r djks %& 9-,d la[;k nwljh la[;k dh nks frgkbz gs 30- HkkbZ vksj cgu dh vk;q dk ;ksx 50 o"kz gs

21 31- :ikyh dh vk;q tk;uk dh vk;q dh 7 xquh gs 3-,d la[;k ds nqxus dks nwljh la[;k esa tksm+us ij 15 izkir gksrk gs 33- fn;s gq, fueu vkys[kksa esa lgh lehdj.k pqfu, %& (i) y x (ii) x y 0 (iii) y x (iv) 3y 7x 34- fueu vkys[k dk laxr lehdj.k crkvks %& (, 3) (, ) 35- fueu vkys[k dk laxr lehdj.k crkvks %&

22 36- lehdj.k 3 x 0 dk xzkq vkys[k fdl v{k ds lekarj gksxk\ 37- lehdj.k y 5 dk xzkq,d js[kk gksxh og fdl v{k ds lekurj gs\ 38- lehdj.k x 3 dk xzkq x & v{k dks fdl fcanq ij dkvrk gs\ 39- x y 0 ds vkys[k es a js[kk nksuks a v{k x & v{k rfkk y & v{k dks fdl fcanq ij dkvrh gs\ 40- y 0] dksu&ls v{k dks iznf'kzr djrk gs\ 41- y esa Hkqt x esa izr;sd eku ds fy, dksfv y dk eku ;k gksxk\ 4- x 1 ls cus xzkq esa dksfv y ds izr;sd eku ds fy, Hkqt dk eku ;k gksxk\ 43- x 1 dk vkys[k fdl v{k ds lekarj gksxk 44- nks pj pkys jsf[kd lehdj.k ds fdrus gy gksaxs\ 45- nsgyh esa vkwvks fj'kk dk fdjk;k bl izdkj gs %& igys,d fdyksehvj dk fdjk;k 10:- gs vksj blds ckn dh nwjh ds fy, izfr fdyksehvj dk HkkM+k 5:- gs ;fn r; dh xbz nwjh x fdyksehvj gks vksj dqy HkkM+k y :- gks rks bldk jsf[kd lehdj.k crkvks 46-,d isu dh dher 10:- rfkk isafly dh dher :- gs og lehdj.k crkb, tks [kjhns x, isu rfkk isaflyksa dh fofhkuu la[;kvksa dks iznf'kzr djrk gks ;fn dqy [kpz 110:- gks 47- tk p djks fd lehdj.k x y 4 dk fueu esa ls dksu&lk gy ugha gs %& (i) x 0, y 4 (ii) x 3, y = (iii) x 1, y 1 (iv) y 0, x 48- lehdj.k 3x 5 y 11 0 esa y dks x ds :i O;r djks og fcanq Kkr djks tgk lehdj.k x y ] y & v{k dks izfrpnsn djrk gs

23 mùkjekyk 1- lehdj.k,d lerk dk dfku gs ftlesa,d ;k vf/kd vkkr jkf'k;k gksrh gs] ftugsa pj dgrs gsa - lehdj.k,d pj okyk jsf[kd lehdj.k dgykrk gs ;fn lehdj.k esa,d pj gks ftldh vf/kdre?kkr,d gks 3- dsoy,d gy 4-,slk jsf[kd lehdj.k tks ax by c 0 ds :i esa gks tgk a, b, c oklrfro la[;k, gks vksj a 0] b 0 rfkk x, y nks pj gks 5- vifjfer :i ls vusd gy 6- y c ax b 1-5 x y 50 0, a 5, b 1, c x 0. y x 3 y x 5 y gy ugha gs 17- gy ugha gs 18- gy ugha gs 19- gy gs 0- gy ugha gs 1- tc x & v{k ij dkvs & (, 0) x 4 y x 9 5y 9 x tc y & v{k ij dkvs & (0, 6) - tc x & v{k ij dkvs & ( 3, 0) tc y & v{k ij dkvs & (0, 9) 3- gk 4- tc x & v{k ij dkvs & (3, 0) 10- a 3, b =, c 8 tc y & v{k ij dkvs & (0, ) 11- a, b = 6, c 0 5- (i) 1 1 (4, 1) (, ) 5

24 6- p 3 (ii) (, ) (1, 5) 7- p 1 (iii) (1, 1) (, ) 37- x v{k 38- ( 3, 0) 39- (0, 0) 40- x v{k P 3 a & x 3 y 30- x y x 7 y 3- x y (ii) x y x 43- y & v{k 44- vuar 45- y 5 x 1 10 y 5x x y (iii) ] tc x [kjhns x, isuksa dh la[;k y [kjhnh xbz isaflyksa dh la[;k 35- y y v{k 48- y 11 3x 5 ] fcanq & 11 0, 5

25 v/;k;&5 ;qfym dh T;kfefr dk ifjp; 1- ;wukuh Hkk"kk ds fdu 'knksa ls ^T;kfefr* 'kn cuk gs\ - bu nks 'knksa dk vfkz ;k gs\ 3- ^T;kferh* 'kn dk ;k vfkz gs\ 4- os dksu&dksu ls yksx Fks ftugksaus T;kfefr dk v/;;u 'kq: fd;k\ 5- os dksu&dksu ls yksx Fks ftugksaus T;kfefr ds Kku dk mi;ksx [ksrksa ds {ks=qy rfkk vuu HkaMkjksa ds vk;ru Kkr djus ds fy, fd;k\ 6- prqhkqzt rfkk f=hkqt tslh vkd`fr;ksa ds {ks=qy ds lw= fdlus fodflr fd,\ 7- O;ofLFkr :i ls T;kfefr dk v/;;u djus dk Js; fdldks fn;k tkrk gs\ 8- ;wuku ds nks lokzf/kd yksdfiz; xf.krkksa dk uke crkb, 9- ^T;kfefr dk firk* fdls ekuk tkrk gs\ 10- ;qfym ds izfl) xzafk dk uke crkb,\ 11- nks Hkkjrh; xf.krkksa ds uke crkb, ftudk T;kfefr ds {ks= esa ;ksxnku ljkguh; gs\ 1- ^vfhkx`ghr* rfkk ^vfhk/kkj.kk* esa ;k varj gs\ 13-,d ljy js[kk ds fy, ;qfym dh vfhk/kkj.kk ;k gs\ 14-,d o`ùk ds fy, ;qfym dh vfhk/kkj.kk ;k gs\ 15- ledks.kksa ij ;qfym dh vfhk/kkj.kk crkb, 16- ;qfym dh ik poh vfhk/kkj.kk ;k gs\

26 17- ;qfym dh ik poh vfhk/kkj.kk dk lerqy; :ikarj.k fdlus fd;k\ 18- ;qfym dh ik poh vfhk/kkj.kk ds lerqy; :ikarj.k dks ;k dgk tkrk gs\ 19- ;qfym dh ik poh vfhk/kkj.kk ds lerqy; :ikarj.k dk dfku crkb, 0- ;qfym ds vuqlkj ^vfhkx`ghr* rfkk ^vfhk/kkj.kk* esa ;k varj gs\ 1- ;qfym dh ik poh vfhk/kkj.kk dk,sfrgkfld egùo crkb, - de ls de fdrus fhkuu fcanqvksa ls,d vf)rh; js[kk [khaph tk ldrh gs\ 3- nks fhkuu js[kk, vf/kdre fdrus fcanqvksa ij izfrpnsn djsaxh 4- ^IysQs;j vfhkx`ghr* dk dfku crkb, 5- ;qfym ds ml izfl) xzafk dk uke crkb, ftlds 13 Hkkx gsa\ 6-,d fcanq ls fdruh js[kk, [khaph tk ldrh gs\ 7- ;k nks fhkuu izfrpnsnh js[kk,,d gh js[kk ds leakrj gks ldrh gs\ dkj.k lfgr mùkj nhft, 8- fn, gq, nks fcunqvksa L vksj M ls fdrus js[kk[kam [khaps tk ldrs gsa\ 9- rhu lajs[k fcanqvksa x, y vksj z }kjk cuus okys js[kk[kamksa ds uke crkb, 30- T;kfefr dh rhu vk/kkjhkwr vo/kkj.kk, ;k gsa\ fjr LFkku %& iz- la[;k 31 ls iz- la[;k os olrq,sa tks,d gh olrq ds cjkcj gksa,d nwljs &&&&&&&&&&&&& ds gksrh gas 3- &&&&&&&&&&&&& vius Hkkx ls cm+k gksrk gs\ 33- &&&&&&&&&&&&& olrqvksa ds nqxus ijlij cjkcj gksrs gsa 34-,d lery esa nks fhkuu fcanqvksa ls gksdj,d &&&&&&&&&&&&& js[kk [khaph tk ldrh gs 35-,d js[kk],d lery dks &&&&&&&&&& Hkkxksa esa fohkkftr djrh gs] &&&&&&&&&&&&& vksj &&&&&&&&&&&&&

27 36- nks fhkuu &&&&&&&&&&&&& dk,d lery esa,d fcanq ls vf/kd mhk;fu" ugha gks ldrk 37- ;fn,d js[kk rfkk,d fcanq ¼tks js[kk ij flfkr u gks½ fn;k gks rks],d vksj dsoy,d &&&&&&&&&&&& gksrk gs tks js[kk rfkk fcanq ls gksdj tkrk gks rfkk js[kk ij &&&&&&&&&&&& gks 38- ;fn fcanq ] fcanq vksj fcanq ds chp flfkr gs] 15] 9 ] ;k gksxk\ 39- ;k,d js[kk dh yackbz gksrh gs\ 40- T;kfefr; lh/kh js[kk dk mnkgj.k nhft, 41- vkd`fr 1 esa fueu ds uke crkb, %& ¼d½ 3 js[kk [kam ¼[k½ 4 lajs[k fcanq ¼x½ vizfrpnsnh js[kk[kamksa dk,d ;qxe 4- izfrpnsnh js[kkvksa rfkk laxkeh js[kkvksa esa ;k varj gs\ vkñfr&1 43- lekarj vfhkx`ghr crkb, 44- ;fn,,, R,d js[kk l ds lekarj gks rks,,, vksj R &&&&&&&& gsa 45- ^e/;lfkrk* ls vki ;k le>rs gsa\ 46- mnkgj.k dh lgk;rk ls crkb, fd,d js[kk dk e/; fcanq ;k gksrk gs\ 47- js[kk[kamksa dh lokzaxlerk ls vki ;k le>rs gsa\ 48- ;qfym ds vuqlkj ^,d fcanq* dh ifjhkk"kk crkb, 49- ;qfym ds vuqlkj ^,d js[kk* dh ifjhkk"kk crkb, 50- uhps nh gqbz vkd`fr dks ns[kdj crkb, fd dksu*&ls dfku lr; gsa vksj dksu&ls dfku vlr; gsa %& ¼d½ PQ QR PR ¼[k½ P R PS PS vkñfr& ¼x½ js[kk[k.m PQ vksj PS laikrh gsa ¼?k½ fcanq R, S, T js[kk PQ ij flfkr gs P Q R S T Q R S

28 1- ^ft;ks* (geo) vksj ^ehvªhu* (metrein) - ft;ks dk vfkz & i`foh ;k Hkwfe ehvªhu dk vfkz & ekiuk mùkjekyk 14- fdlh fcanq dks dsunz ekudj vksj fdlh f=t;k ls,d o`ùk [khapk tk ldrk gs 15- lhkh ledks.k,d&nwljs ds cjkcj gksrs gsa 3- Hkwfe dks ekiuk 4- felz] cschyksu ds yksx 5- felzoklh 6- cschyksuoklh 7- ;wukuh 8- FksYl] ikbfkkxksjl 9- ;qfym 10-,yhesaV~l (Elements) 11- czãxqir] HkkLdj II ] vk;zhkv~v 1- vfhkx`ghr & os dyiuk, ftudk iz;ksx xf.kr esa fujarj fd;k x;k vksj ftudk dsoy T;kfefr ls fo'ks"k laca/k ugha gksrk vfhk/kkj.kk & os dyiuk, tks fof'k"v :i ls T;kfefr ls lacaf/kr Fkha 13-,d fcanq ls,d vu; fcanq rd,d lhèkh js[kk [khaph tk ldrh gs 16- ;fn,d lh/kh js[kk nks lh/kh js[kkvksa ij fxj dj vius,d gh vksj nks var%dks.k bl izdkj cuk, fd bu nksuksa lh/kh js[kk, vfuf'pr :i ls c<+k, tkus ij mlh vksj feyrh gsa ftl vksj ;g ;ksx nks ledks.kksa ls de gksrk gs 17- tkwu IysQs;j] 179 esa 18- IysQs;j vfhkx`ghr 19- nks fhkuu izfrpnsnh js[kk,,d gh js[kk ds lekarj ugha gks ldrh 0- vfhkx`ghr & vk/kkjhkwr rf; ftudh lr;rk dh tk p dh vko';drk ugha gksrh ;s Li"Vr% lozo;kih lr; gksrh gsa izes; & os dfku ftugsa ifjhkk"kkvksa] vfhkx`ghrksa rfkk igys fl) fd, x, dfkuksa vksj fuxefud rdz.k }kjk fl) fd;k tkrk gs 1- vusd iz;klksa ds ckotwn ;g vfhk/kkj.kk,d izes; ds :i esa fl) ugha gks ldh fdarq bu iz;klksa ds ifj.kkelo:i,d ubz T;kfefr dh mùifùk gqbz ftls v;wfymh; T;kfefr ds uke ls tkuk tkrk gs

29 - nks 3-,d 4- izr;sd js[kk l vksj ml ij u flfkr izr;sd fcanq P ls gksdj tkrh gs vksj l ds lekurj gks 5-,fyesUV~l 6- vla[; 7- ugha] IysQs;j vfhkx`ghr 8- dsoy,d 9- XY,YZ,ZX 30- fcanq] js[kk rfkk lery 31- cjkcj 3- iw.kz 33-,d gh 34- vf)rh; 35- rhu( nks v/kzlery rfkk,d js[kk 36- js[kkvksa 37- yec] yec gk ¼pkSMkbZ jfgr½ 40- Ldsy ¼ekid½ dk fdukjk 41- (i), Q, RS ;k vu; dksbz rhu (ii),, Q, S (iii), Q 4- izfrpnsnh & nks js[kkvks a dk,d mhk;fu" fcanq gksrk gs laxkeh js[kk, & nks ls vf/kd js[kkvksa dk,d mhk;fu" fcanq gksrk gs 43- ;fn l,d js[kk rfkk P,d fcanqw gs tks l ij flfkr ugha gs] rks,d vksj dsoy,d js[kk gs tks P ls gksdj tkrh gks rfkk l ds lekarj gks 44- lajs[k 45- e/;lfkrk ls vfhkizk; gs fd,d fcanq ] fcanq rfkk ds chp bl izdkj flfkr gks fd 46-,d fcanq M js[kk dk e/;&fcanq gksxk ;fn M ] ij flfkr gks rfkk M M gks 47- ;fn nks js[kk[kam leku yackbz ds gksa rks] os lokzaxle gksaxs 48- fcanq dk dksbz Hkkx ugha gksrk 49-,d js[kk pksm+kbz jfgr yackbz gksrh gs 50- ¼i½ lr; ¼ii½ vlr; ¼iii½ lr; ¼iv½ lr;

30 v/;k;&6 js[kk, vksj dks.k 1- nh xbz vkd`fr esa nks vkluu dks.kksa ds ;qxeksa ds uke crk, F E - vkd`fr esa O Kkr djsa ;fn O 68 vksj O 4 3- ;fn XOY vksj YOZ nks vkluu dks.k gsa rks XOZ dk eku Kkr djsa ;fn XOY 56 vksj YOZ 34 Z Y O X 4- ;fn,d dks.k vius iwjd dks.k ds cjkcj gks rks dks.k dk eki ;k gksxk\ 5- ;fn fdlh dks.k dk leiwjd dks.k 98 gks rks dks.k dk eki ;k gksxk\ 6- jsf[kd ;qxe ds dks.kksa dk vuqikr 8%1 gs izr;sd dks.k dk eki ;k gksxk\ 7- nh xbz vkd`fr esa x dk eku Kkr djsa 8- ;fn x 58 vksj y 4 rks nh xbz vkd`fr ls 1 z dk eku Kkr djsa

31 9- nh xbz vkd`fr esa x y dk eku Kkr djsa 10- nh xbz vkd`fr esa ;fn OP vksj OQ Øe'k% O vksj O ds lef}hkktd gsa rks POQ dk eku Kkr djsa 11- vkd`fr esa ;fn PE 50 rks x dk eku Kkr djsa 1- vkd`fr esa p q - ;fn 1 70 rks dk eki Kkr djsa 13- vkd`fr esa p q ] r s ;fn 1 80 rks 3 dk eki Kkr djsa 14- vkd`fr esa vksj EF ;fn 30 djs a ] E 10 rks EF Kkr

32 15- vkd`fr esa ;fn O 60 ] O 0 rks O dk eku crk, 16- vkd`fr esa 103 vksj 35 rks x vksj y dk eku Kkr djsa 17- vkd`fr esa PQR 69 rks PRS dk eku crk, ] QPR 5 18-,d f=hkqt ds rhuksa dks.kksa dk vuqikr :3: 4 gs f=hkqt ds rhuksa dks.k crk, 19-,d f=hkqt ds rhuksa dks.kksa dk vuqikr 1:3: 6 gs rhuksa dks.k Kkr djsa 0- vkd`fr esa l m 1: 3: rks 3 dh eki crk, 1- vkd`fr esa O vksj O Øe'k% vksj ds lef}hkktd gsa ;fn 6 vksj 54 gks rks O Kkr djsa

33 - vkd`fr esa ] PQ 50 vksj PR 15 rks x vksj y dk eku Kkr djks 3- vkd`fr esa PQ RS djks ] MXQ 135 ] MYR 40 gks rks XMY dk eku Kkr 4- vkd`fr esa ] rks x dk eku Kkr djsa 5- vkd`fr esa SPR 135 vksj PQT 110 rks PRQ dk eku Kkr djsa 6- fdlh dks.k dk iwjd dks.k blds rhu xqus ls vf/kd gs dks.kkssa dk eki crkvks 7- fdlh dks.k rfkk mlds laiwjd dks.k dk vuqikr :3 gs izr;sd dks.k dk eki Kkr djks 8- vkd`fr esa vksj nks js[kk, gsa tks ijlij O fcanq ij izfrpnsn djrh gs ;fn vksj OE ] O dk lef}hkktd gs] rks dk eku Kkr djks 1 70

34 9- ;fn iwjd dks.k fn, x, dks.k dk nqxuk gks rks dks.k dk eku Kkr djks 30- ;fn iwjd dks.k fn, x, dks.k ls vf/kd gks] rks dks.k dk eku Kkr djks 31- vkd`fr esa OP 40 ] OQ ] Kkr djsa OP dk dks.k lef}hkktd gks rks OQ dk eku 3- vkd`fr esa ] dk nqxuk gs ;fn 50 rks Kkr djsa 33- ;fn l m vksj m n rks x dk eku Kkr djsa 34- vkd`fr esa l m ] 1: 7 :3 rks 1 vksj Kkr djsa 35- vkd`fr esa l m 1: :3 rks 3 Kkr djsa

35 36- vkd`fr esa 80 vksj 40 O vksj O Øe'k% vksj ds dks.k lef}hkktd gsa rks O Kkr djsa 37- vkd`fr esa E ] dk lef}hkktd gs vksj E ;fn 130 rks Kkr djsa E 38- vkd`fr esa RT ] PRS dk dks.k lef}hkktd gs vksj PQ RT PQR dk eku crk, P vksj PRS 110 T rks Q 39- vkd`fr esa O vksj O ckg~; vksj ckg~; ds lef}hkktd gsa ;fn 80 vksj 40 rks O dk eku crk, R S 40- vkd`fr esa ;fn l m dk eku crk, vksj 1 45 gks rks

36 41- vkd`fr esa l m ] M l lef}hkktd gks vksj 1 30 m ;fn M ] P dk vksj N ] Q dk ] N rks Kkr djsa P 1 N Q l 4- vkd`fr esa x dk eku Kkr djsa M m 43- vkd`fr esa x dk eku Kkr djsa 44- vkd`fr esa x dk eku Kkr djsa 45- vkd`fr esa P 100 vksj R 40 rks PQS dk eku Kkr djsa 46- vkd`fr esa 50 ] E 90 rks y dk eku crk,

37 47- vkd`fr esa l m rks x dk eku crk, 48- vkd`fr esa l m ] vksj p q ;fn rks dk eku Kkr djsa 49- nh xbz vkd`fr esa PS QR T vksj PT T Kkr djks ] QTR 30 x ] y vksj z dk eku P Q 30 Z T R y S x 50- nh xbz vkd`fr esa M ] M 35 ] P 40 x dk eku Kkr djks M x P

38 1- vksj ( EF vksj EF - O 6 3- XOZ izr;sd dks.k vksj 0 7- x x y POQ x EF O x 68 ] y PRS 94 mùkjekyk 18-40, 60, , 54, O 11 - x 50 ] y XMY x PRQ ] , OQ X ] O PQR O x x x 30º 45- PQS y x x 10 ] y 10 ] z

39 v/;k;&7 f=hkqt 1- vkd`fr esa fn, x, nksuksa f=hkqt lokzaxle gsa nksuksa f=hkqtksa ds chp esa laxrrk crkb, 3cm X Y 4cm 3cm 5cm 5cm 4cm - lokzaxlerk dk og dksu&lk fu;e gs tks bu nksuksa f=hkqtksa dks lokzaxle djrk gs\ Z 3- bl vkd`fr esa dksu&ls nks f=hkqt lokzaxle gsa\ O 4- dksu&ls fu;e }kjk ;g fl) dj ldrs gsa fd SQ ] PR dks lef}hkkftr djrk gs\ P Q O S R

40 5- vkd`fr esa PQ PR PS rks QRS dk eku crkvks S P Q R 6- vkd`fr esa ;fn rfkk rks x Kkr djks 7- vkd`fr esa ;fn PQ SR rfkk PQ QR ] x y dk eku crkvks P 8- vkd`fr esa ;fn L N LM MN, QM MR LM PQ, MN PR Q 50 rks x dk eku Kkr djks Q 50 M x R 9- ;fn ] rfkk rks x eku Kkr djks 1 P 10- esa] rfkk 1 ] P rfkk P Kkr djks

41 11- ;fn,d leckgq f=hkqt gs rks x y Kkr djks 1- ;fn,d ledks.k f=hkqt esa,d U;wu dks.k nwljs dks.k dk nqxquk gs rks nksuksa dks.k Kkr djks 13-,d f=hkqt esa nks Hkqtkvksa dk ;ksx lnso rhljh Hkqtk ls &&&&&&&&&&& gksrk gs 14- vkd`fr esa x y Kkr djks x P y 15- vkd`fr esa ;fn,d oxz gs Hkqtkvksa rfkk dks P rfkk Q rd bl izdkj c<+k;k x;k fd P Q ;fn P 7cm rks Q Kkr djks Q Q 16- vkd`fr esa x Kkr djks P 17- vkd`fr esa ;fn PQ PR rks x dk eku Kkr djks

42 18- vkd`fr esa rfkk P nks lef}ckgq f=hkqt gsa x dk eku Kkr djks 19- ;fn XYZ rfkk PYZ nks lef}ckgq f=hkqt gsa] rfkk nksuksa,d gh vk/kkj YZ ij cus gsa ;fn P 10 rfkk XYP 40 rks x Kkr djks 0- fdlh Hkh f=hkqt esa cm+h Hkqtk dk leeq[k dks.k lnso &&&&&&&&&&& gksrk gs 1- vkd`fr esa lcls yech Hkqtk dksu*&lh gs\ - XYZ dk dksu&lk dks.k lcls cm+k gs\ 3cm X 6cm 3- esa] rfkk Y 6.5cm ] rks x Kkr djks X 4- fdlh f=hkqt esa cjkcj Hkqtkvksa ds leeq[k dks.k &&&&&&&&&&& gksrs gsa 5- ledks.k f=hkqt esa d.kz &&&&&&&&&&& Hkqtk gksrh gs

43 6- vkd`fr esa x dk eku Kkr djks X 7- nh xbz vkd`fr esa XYZ,d ledks.k f=hkqt gs ;fn XY 8cm rfkk YZ 4cm rks XZ Kkr djks 8cm Y? 4cm Z 8- vkd`fr esa ;fn 30 rks x y Kkr djks z 9- vkd`fr esa x y z Kkr djks x y 30- esa 5cm ] 3cm rfkk 5.5cm dksu&lk dks.k lcls cm+k dks.k gs\ 31- nh xbz vkd`fr esa x dk eku Kkr djks

44 3- vkd`fr esa P 4 S 40 ] Q 46 rfkk x dk eku Kkr djks 33- ] rfkk dks lef}hkkftr djrk gs ;fn 3cm rks Kkr djks 3cm? 34- vkd`fr esa ;fn l m ] rks rfkk Kkr djks 3cm 4cm O L P 35- vkd`fr esa ;fn P Q rks PQ Kkr dhft, M O 4cm Q 36- Kkr djks ;fn 37- vkd`fr 1 esa ;fn E rks x y Kkr djks 38- EF ;fn 4cm rks F Kkr djks 115 4cm F E

45 39- esa ;fn ] 40,oa, vksj Øe'k%, vksj ds lef}hkktd gksa] rks x Kkr djks 40- fdlh f=hkqt esa rhuksa 'kh"kzyacksa dk ;ksx f=hkqt ds ifjeki ls &&&&&&&&&&& gksrk gs 41- fdlh f=hkqt dk ifjeki mldh rhuksa ekf/;dkvksa ds ;ksx ls &&&&&&&&&&& gksrk gs 4- vkd`fr esa PQR,d lef}ckgq f=hkqt gs] PQ PR ] 1 rks x rfkk y esa laca/k crkb, 43- rfkk EF leckgq f=hkqt gsa ] E rfkk F dk eku crkvks 44- nh xbz vkd`fr esa ;fn 48 rks x Kkr djks F 45- nh xbz vkd`fr esa x dk eku Kkr djks ;fn E 90 gks

46 1- XY 3cm XZ 5cm YZ 4cm - Hkqtk dks.k Hkqtk fu;e 3- O O 4- S vfkok S fu;e 5- QRS x x x P 15 P x y , cm+k 14- x y 10 mùkjekyk 15- Q x x x x cm+k 1- - X 3- x leku cm 5- lcls yach 6- x XZ x y 10 cm 9- x y z x x cm 34-6cm 8cm 35- PQ 8 cm x y F 4cm 39- x NksVk 41- NksVk 4- y x F x x 45 ] 60 ]

47 1: : 4:5 1: 4:5 49, x :3: 6 : 7 90 PQRS x

48 x x y PQRS 4 PQRS PR QS S 5cm R P Q 1cm MNOP x y 3cm PQRS 4cm O

49 3:5:9: :5: 7 57 x 3x 5x x y

50 6-,d lekurj prqhkqzt dk ifjeki ;fn 14-6 lseh- gks rfkk cm+h Hkqtk dh yeckbz 4-8 lsehgks rks NksVh Hkqtk dh yeckbz Kkr djks 7-,d lekurj prqhkqzt ds vk/kkj rfkk mlds {ks=qy dk vuqikr 1:8 gks rks 'kh"kzyec dh yeckbz Kkr djks 8- lekurj prqhkqzt ds 'kh"kzyec rfkk mlds {ks=qy dk vuqikr :11 gks rks vk/kkj dh yeckbz Kkr djks 9- vkd`fr esa fn, x, lekurj prqhkqzt esa f=hkqt dk {ks=qy Kkr djks ;fn L lekurj prqhkqzt dk {ks=qy Kkr djks ;fn 5cm P 4cm L o cm, P 90 gks 19cm vkd`fr esa lekurj prqhkqzt dk ifjeki crkb, 8cm 3- vkd`fr esa PQRS,d lekurj prqhkqzt gs rfkk PQ 1cm ] ST 9cm ] QM 6cm dh yeckbz Kkr djks E gks rks SP 33- lekurj prqhkqzt ] ftldk {ks=qy 184cm gks rks f=hkqt E dk {ks=qy Kkr djks P M 6cm S 90 9cm T 1cm 6cm Q R

51 34- vkd`fr esa f=hkqt PTQ o vk;r PQRS ds {ks=qyksa dk vuqikr Kkr djsa S 4cm T R 35- vkd`fr esa PQ SR ] NR PQ vksj NP MQ gsa PQRN dk {ks=qy Kkr djks ;fn PQ 9cm o ST 5 cm SM 3 cm gks P 9cm N S M P T Q Q R 36- vkd`fr esa ] E o E ] dk e/;fcanq gs E ;fn dk {ks=qy 96cm gks rks f=hkqt E dk {ks=qy Kkr djks 37- vkd`fr esa, E, F Øe'k%, o ds e/; fcanq gsa ;fn 50 ] 60 rks EF esa ] E o F Kkr djks o F E,d vk;r o lekurj prqhkqzt EF ds {ks=qy leku gksa rfkk vk/kkj 8cm o Å pkbz 3cm gks rks lekurj prqhkqzt EF dk ifjeki Kkr djks ;fn ] EF dk e/;fcanq gks 8 cm 3 cm E F 39- fn, x, f=hkqt esa] o E Øe'k% o ds e/;fcanq gsa E dh yeckbz rfkk dh eki Kkr djks 40- f=hkqt esa,d ekf/;dk gs] ;fn f=hkqt dk {ks=qy xcm gs rfkk f=hkqt dk {ks=qy ycm gs rks x o y esa laca/k Kkr djks

52 , 60, 10, , , x , 60, 10, , 90, , cm cm 17-6cm mùkjekyk 19-0, 100, , x 6, y = ls-eh- 7-8 ;wfuv ;wfuv 9-38 sq. cm sq. cm 31-3cm 3-18cm 33-9 sq. cm 34-1: sq. cm 36-3 sq. cm 37-70, 50, cm 39-6cm ] y x

53 v/;k;&9 o`ùk 1- rhu vlajs[k fcanqvksa ls fdrus o`ùk xqtjrs gsa\ O - nh xbz vkd`fr esa fcanq thok dk e/; fcanq gs O dk eku crkb, 3-,d pøh; prqhkqzt gs ;fn 95 rks dk eku crkb, 4- vkd`fr esa O 10 ] Kkr djks O nh xbz vkd`fr esa thok dh dsunz ls nwjh 4 lseh- gs ;fn o`ùk dh f=t;k 5 lseh- gks rks thok dh yeckbz crkb, 5cm O 4cm 6-,d o`ùk esa ftldk dsunz O gs] ;fn O 60 gks rks izfrorhz O dk eku Kkr djks 7- ;fn,d pøh; prqhkqzt gks] rks rfkk dk ;ksx ;k gs\ 8- o`ùk ds nh?kz o`ùk[k.m esa fdl izdkj dk dks.k curk gs\ 9- o`ùk ds y?kq o`ùk[k.m esa fdl izdkj dk dks.k curk gs\ 10- ;fn O dsunz okys o`ùk dk O;kl gs rfkk o`ùk ij dksbz fcanq gs rks dk eku ;k gksxk\ 11- vkd`fr esa,d pøh; prqhkqzt gs Hkqtk ] P rd bl izdkj c<+kbz xbz gs fd P 110 ] dk eku crkb, 110 P

54 1- vkd`fr esa,d pøh; prqhkqzt gs] Hkqtk nksuksa rjq bl izdkj c<+kbz xbz gs rfkk Q 95 ] rfkk dk ;ksx ;k gs\ fd P 110 Q ,d pøh; prqhkqzt gs ;fn 110 rfkk 60 ] rfkk dk ;ksx ;k gs\ 14- vkd`fr esa thok dh o`ùk ds dsunz ls nwjh 5 lseh- gs ;fn thok dh yeckbz 4 lseh- gs rks o`ùk dh f=t;k Kkr djks P 4cm? 5cm O 48cm? 15-5cm vkd`fr esa 5 lseh- f=t;k okys o`ùk dh thok dh O yeckbz 48 lseh- gs thok dh dsunz ls nwjh Kkr djks 16-,d o`ùk ftldk dsunz O gs] thok vksj dsunz ls lenwjlfk gs ;fn 8cm rks thok dh yeckbz Kkr djks 17- vkd`fr esa ;fn O 140 rks 18- nh xbz vkd`fr esa O Kkr djks Kkr djks O O 35

55 19- vkd`fr esa,d pøh; prqhkqzt gs ftlesa ;fn Hkqtk nksuksa rjq c<+kbz xbz gks rfkk Q 80 rks P Kkr djks Q 0- PQRS,d pøh; prqhkqzt gs ;fn P rfkk R 1: ds vuqikr esa gs] P rfkk R dk eku Kkr djks 1- vkd`fr esa,d pøh; prqhkqzt gs ftlesa rfkk ] dk eku Kkr djks - vkd`fr esa ;fn 75 rks 80 P Kkr djks vkd`fr esa E,d pøh; prqhkqzt gs ;fn E 80 ] Kkr djks 4- vkd`fr esa x dk eku Kkr djks P O 4 x 80 E 5- o`ùk dh lcls cm+h thok dk uke crkb, Q 6- fdlh o`ùk dh nks thokvksa esa ls dsunz ds ikl okyh thok &&&&&&&&&&&& gksrh gs 7- ;fn thok }kjk vius,d gh vksj flfkr nks vu; fcanqvksa ij cjkcj dks.k varfjr fd, x, gksa rks pkjksa fcanq &&&&&&&&&&&& dgykrs gsa

56 8- vkd`fr esa x Kkr djks x O 9- vkd`fr esa x Kkr djks x 10 O vkd`fr esa PQ RS rfkk O;kl gs x dk eku Kkr djks P O x 30 S Q R 31- vkd`fr esa] pøh; prqhkqzt dk fod.kz dsunz ls xqtjrk gs rfkk ds eku Kkr djks O 3- pøh; prqhkqzt ds fod.kz rfkk dsunz ls xqtjrs gsa fdl izdkj dk pøh; prqhkqzt gs 33- o`ùk ds nks O;kl ijlij yec gsa buds fljksa dks tksm+us ij fdl izdkj dk prqhkqzt curk gs 34- vkd`fr esa,d pøh; prqhkqzt gs ftlesa rfkk 80 ] Kkr djks o`ùk (o, r) esa ;fn rks thok dh yeckbz ;k gksxh\ 36- vkd`fr esa o`ùk dh thok ¾ thok ;fn O 60 ] O Kkr djks O 60

57 37- vkd`fr esas o`ùk dh thok gs ftldk dsunz O gs thok fcanq rd bl izdkj c<+kbz xbz gs fd O ;fn O 30 rks O Kkr djks 38- vkd`fr esa] o`ùk dh nks thok, rfkk leku gsa M dks.k dk lef}hkktd gs crkb, fd o`ùk dk dsunz dgk ij flfkr gs\ 39- vkd`fr esa O ;fn O 3cm rks dk eku crkb, M O 40- vkd`fr esa O ;fn O ] dk ifjdsunz gks rks O dk eku crkb, 41- vkd`fr esa x dk eku crkb, E vkd`fr esa] E v/kzo`ùk esa cuk,d iaphkqt gs E dk eku crkb, x O E 43- vkd`fr esa gs rfkk esa ;k laca/k gs rfkk dk eku crkb, 48 O

58 44- vkd`fr esa] RTQ rfkk RQT Kkr djks P S O T Q R 45- vkd`fr esa] nks ladsunzh; o`ùk ftudk dsunz O gs fn, x, gsa rfkk thok, gsa OM ;fn 18cm rfkk M 8cm ] rks Kkr djks M O 46- vkd`fr esa rfkk o`ùk dh nks leku thok, gsa] O rfkk OE E fdl izdkj dk f=hkqt gs\ 47- vkd`fr esa] nks o`ùk ftuds dsunz O rfkk 1 O gsa],d&nwljs dks rfkk ij izfrpnsn djrs gsa PQ Kkr djks O E O O' 48- PQ o`ùk dh thok gs ftldh f=t;k " r " gs ;fn fcanq o`ùk ij bl izdkj gs fd rks PQ dk eku Kkr djks PQ 90 P 49- vkd`fr esa] O o`ùk dk dsunz gs ftldh f=t;k 5 lseh- gs OP 8cm rfkk 6cm rks PQ Kkr djks P Q ] OQ ;fn 5cm O Q

59 1-,d lseh U;wudks.k 9- vf/kddks.k lseh lseh lseh lsehmùkjekyk 0- P O;kl 6- cm+h 7- o`ùk ij ] R rfkk 3- vk;r 33- oxz thok ds cjkcj js[kk M ij flfkr gksxh 39-6 lseh vksj RTQ 45 rfkk RQT 45-1 lseh lef}ckgq f=hkqt 47- RQT r

60 v/;k;&10 ghjksu dk lw= 1- f=hkqt dk {ks=qy Kkr djus ds fy, ghjksu dk lw= crkb, - ghjksu ds lw= esa a, b vksj c ;k gsa\ 3- ;fn a 1cm ] b 13cm rfkk c 15cm gks rks s dk eku ;k gksxk\ 4- fueu f=hkqt esa dksu&lh Hkqtk, a, b rfkk c dks n'kkzrh gsa\ 5- fjr LFkku Hkjks %&? s 6- ;fn a 5cm ] b 15cm vksj c 0cm rks (s b) dk eku ;k gksxk\ 7- fdlh f=hkqtkdkj IykWV dh Hkqtk, :3: 4 vuqikr esa gsa rfkk mldk ifjeki 36 ls-eh- gs rks mldh Hkqtk, &&&&& ] &&&&& vksj &&&&& gsa 8- s Kkr djsa ;fn 5cm 9- fdlh leckgq f=hkqt dh Hkqtk 6 lsa-eh- gks rks mldk {ks=qy &&&&&&&&&&& gksxk 10-,d f=hkqt esa a 15cm ] b 7cm vksj s 15cm gks rks " c " dk eku Kkr djks 11-,d leckgq f=hkqt esa s 30cm gks rks f=hkqt dh izr;sd Hkqtk dh yeckbz ;k gksxh\ 1- f=hkqt dk {ks=qy tc ghjksu lw= dk iz;ksx fd;k tkrk gs ¾ :i esa crkvks 7cm 6cm ] bls ljyre 13- eku Kkr djks %& 1889

61 14- ;fn,d f=hkqt esa s a 7cm ] s b 8cm vksj s c 6cm gks rks s? 15-,d leckgq f=hkqt ftldh,d Hkqtk x gs mlesa s? x 16-,d f=hkqt esa a 3b 6c rks s? c 17- nh xbz vkd`fr esa] vksj ds fy, s dk eku Kkr djks 30m 15m 35m 15m 30m 18-,d f=hkqt dh Hkqtk, 3:3:5 ds vuqikr esa gsa rfkk mldk ifjeki 55 eh- gks rks mldh Hkqtk, Kkr djks 19- f=hkqt dh Hkqtk, a 1cm ] b 14cm ] c 0cm gks rks fueu dks iwjk djks %& f=hkqt dk {ks=qy ¾ ;fn s a 1cm ] s b 9cm ] s c 4cm gks rks s =? 1-,d f=hkqt esa] 'kh"kz ds leeq[k Hkqtk =? vksj 'kh"kz ds leeq[k Hkqtk =? - ;fn,d leckgq f=hkqt dh Hkqtk 6lsaeh- gks rks leckgq f=hkqt dk {ks=qy crkb, 3- leckgq f=hkqt esa 3 s a ] ¼tgk a f=hkqt dh Hkqtk gs½ f=hkqt dk ifjeki Kkr djks

62 1- s( s a) s bs c - a, b, c f=hkqt dh Hkqtk, gsa mùkjekyk cm 3-0 lseh- 4- c ] a ] 5- a+b+c b 11-0 lseh lseh- 7-8 lseh-] 1 lseh- vksj 16 lseh- 8-9 lseh cm 10-8 lseh unit² 17-40cm each lseh-] 15 lseh-] 5 lseh lseh- 1- ] lsa-eh-² 3-3a

63 v/;k;&11 i`"h; {ks=qy vksj vk;ru 1-,d Hkwfexr ikuh dh Vadh 7 eh- Hkqtk okys?ku ds vkdkj dh gs] bldk vk;ru ;k gksxk\ -,d ckwl dk vk;ru ;k gksxk ftldh yeckbz 16 eh0 ] pksm+kbz 8 eh- rfkk Å pkbz 5 eh- gks\ 3-1 eh- yacs] 10 eh- pksm+s rfkk 9 eh- Å ps dejs dh pkjksa nhokjksa dk {ks=qy ;k gksxk\ 4-,d?ku dk vk;ru 7 a 3 gs mldh dksj dh yeckbz Kkr djks 5-13 eh- yacs] 8 eh- pksm+s rfkk 4 eh- Å ps fmcs dks <du ds lkfk cukus esa fdruh,y;qfefu;e 'khv dh vko';drk gksxh\ 6-,d?ku dk vk;ru 1331 cm 3 gs blds fdukjs dh yackbz Kkr djks 7-,d?ku ds fod.kz dh yackbz 17-3 cm gs?ku dk vk;ru Kkr djks 8-6] 8 rfkk 10 lseh Hkqtk okys leku /kkrq ls cus rhu?kuksa dks fi?kykdj,d?ku cuk;k x;k u,?ku dk vk;ru Kkr djks 9-10 eh- dksj okys nks?kuksa dks fdukjksa ls tksm+k x;k gs bl izdkj cus?kukhk dk i`"h; {ks=qy Kkr djks 10-,d?ku dk vk;ru 51 ls-eh- 3 gs bldh Hkqtk Kkr djks lseh- yacs] 10 lseh- pksm+s rfkk 5 lseh Å ps,d vk;rkdkj fmcs dks cukus ds fy, vko';d xùks dk {ks=qy Kkr dhft, 1-,d yec o`ùkh; csyukdkj VSad dk vk;ru ;k gksxk ftldh f=t;k 7 lseh- rfkk Å pkbz 5 lseh gks\ 13- lseh- f=t;k okys,d ksl xksys esa ls 3 cm f=t;k okys fdrus ksl xksys cuk, tk ldrs gsa\ 14- ;fn,d xksys dk vk;ru vksj i`"h; {ks=qy vkafdd :i ls leku gks rks mldh f=t;k ;k gksxh\

64 15-,d yec o`ùkh; csyu dk vk;ru 3 39 cm rfkk Å pkbz 8 lseh- gs f=t;k Kkr djsa 16-,d xksys dk i`"h; {ks=qy 448 cm ] bldh f=t;k ;k gksxh\ 17-,d?ku dh dksj ¼Hkqtk½ ;k gksxh ;fn bldk oø i`"h; {ks=qy 34 oxz lseh- gks\ 18-,d v/kzxksys dk vk;ru cm ] bldh f=t;k ;k gksxh\ 19-,d 'kadq dk oø i`"h; {ks=qy 140 cm² gs 'kadq dh f=t;k ;k gksxh ;fn bldh frjnh Å pkbz 5 lseh- gs\ 0-,d ksl xksys dh f=t;k 1 lseh- gs blesa ls 6 lseh- f=t;k okys fdrus xksys cuk, tk ldrs gsa\ 1-,d?kukHk dk vk;ru 840?ku lseh- gs ;fn bldh yackbz 14 lseh- vksj pksm+kbz 5 lsehgks rks Å pkbz Kkr djks - pkj cjkcj?ku ftlesa ls izr;sd dh Hkqtk 5 lseh- gs fljs ls fljs ij la;qr fd;s tkrs gsa bl izdkj izkir gksus okys?kukhk dk i`"h; {ks=qy Kkr dhft, 3-,d leprqhkqzt dk {ks=qy 56 oxz lseh- rfkk mldk,d fod.kz 7 lseh- gs leprqhkqzt dk nwljk fod.kz Kkr djks 4- ml cm+h ls cm+h NM+ dh yeckbz ;k gksxh tks ml fmcs esa j[kh tk lds ftldh var% foek, 30 lseh-] 4 lseh] rfkk 18 lseh- gksa\ 5-,d csyu dk oø i`"h; {ks=qy 16 ls-eh- gs ;fn bldh Å pkbz 18 lseh- gks rks bldh f=t;k Kkr dhft, 6-,d o`ùkkdkj csyu dk :i nsus ds fy, 60 o`ùkkdkj IysVksa ftudh f=t;k cjkcj gs] dks,d&nwljs ds Åij j[kk x;k bl izdkj cus csyu dh Å pkbz Kkr djsa ;fn,d IysV dh eksvkbz 3 4 cm gks 7-,d 'kadq dk oø i`"h; {ks=qy nwljs 'kadq ds oø i`"h; {ks=qy ls frxquk gs nwljs dh frjnh Å pkbz igys dh frjnh Å pkbz ls frxquh gs mudh f=t;kvksa dk vuqikr Kkr dhft,

65 8-,d dqvk ftldk O;kl eh- gs] 14 eh- xgjk [kksnk tkrk gs dq, ls fudkyh x;h fev~vh dk vk;ru Kkr djks 9- ;fn,d ksl xksys dk vk;ru 36?ku lseh gks rks bldh f=t;k Kkr dhft, 30-4 lseh- Å pkbz rfkk 7 lseh- f=t;k okys,d 'kadq dks,d cpps us ksl xksys ds :i esa <ky fn;k xksys dh f=t;k Kkr djks mùkjekyk m³. 640m³ m² 4. 3a m² 6. 11cm cm³ cm³ m² 10. 8cm cm² cm units 15. 7cm cm or 4 7cm 17. 9cm 18. 6cm 19. 8cm cm. 450cm² 3. 16cm cm or 30 cm 5. 6cm 6. 45cm 7. 9: m³ 9. 3cm cm

66 v/;k;&1 lkaf[;dh 1- oxz varjky 1&5 dh mpp lhek &&&&&&&&&& gs - oxz varjky 18&6 dk oxz fpug &&&&&&&&&& gs 3- vkadm+s 31] 3-5] 0-3] 7-9] 19-7] 31-7 ds fy, ifjlj &&&&&&&&&& gs 4- ;fn feyku fpug gks rks ckjeckjrk ;k gksxh\ 5-,d Vhe us QqVcky ds 10 espksa esa fueufyf[kr xksy fd, ] 3] 5] 4] 0] 1] 3] 3] 4] 3 bu xksyksa dk ek/; Kkr djks 6- Åij fn, x, iz'u ds vuqlkj xksyksa dk ek/;d Kkr djks 7- ] 3] 4] 5] 0]1] 3] 3] 4] 3 bu vk dm+ksa esa cgqyd Kkr djks 8- vk dm+ksa 14] 5] 14] 8] 18] 17] 14] 3] ] 14] 18 dk cgqyd Kkr djks 9- ;fn 6] 8] 5] 7] x rfkk 4 dk ek/; 7 gs rks x dk eku Kkr djks 10- ;fn 10 izs{k.kksa dk ek/; 0 vksj vu; 15 izs{k.kksa dk ek/; 16 gs rks lhkh 5 izs{k.kksa dk ek/; Kkr djks 11-4] 3] 7] 0] 0] 6] 8 dk ek/; fudkyks 1-7] 9] 1] 13] 7] 1] 15] 7] 1] 7] 5] 18] 7 bu vk dm+ksa dk cgqyd Kkr djks 13- rhu la[;kvksa dk ek/; 6 gs ;fn muesa nks la[;k, Øe'k% 5 vksj 8 gksa rks rhljh la[;k Kkr djks 14- ;fn x 1 ] x dk ek/; 6,oa x 1, x, x 3 dk ek/; 7 gks rks x 3 dk eku Kkr djks

67 15- izfke rhu izkd`frd la[;kvksa dk ek/; Kkr djks 16- ;fn x ] 3] 4] 5 dk ek/; 3 gs rks x dk eku Kkr djks 17- izfke rhu iw.kz la[;kvksa dk ek/; Kkr djks 18- p, q, r dk ek/; ;k gksxk\ 19-4] 4] 3] 5] 6] dk ek/; &&&&&&&&&& gs 0- fdlh fo kfkhz us ik p fo"k;ksa esa fueufyf[kr vad izkir fd, bu vadksa dk ek/; fudkysa 70] 64] 56] 54] p ds fdl eku ds fy, fueu vk dm+ksa dk cgqyd 5 gs %& 1] ] 5] 7] 5] ] 7] 5] 9] ] 3] p ] 11 - fueu vk dm+ksa dks vkjksgh Øe esa fy[kk x;k gs] ekf/;dk Kkr djks 36] 39] 4] 48] 5] 68] 69] 71] 7] 78 3-,d cavu ds oxz fpug gsa %& 47] 5] 57] 6] 67] 7] 77] 8 oxz vkeki Kkr dhft, 4- Åij fn, x, iz'u esa izfke oxz fpug dh oxz lhek Kkr djks 5- fueu vkadm+ksa ds fy, ifjlj Kkr djks %& 40] 4] 80] 69] 56] 47

68 mùkjekyk p q r &

69 v/;k;&13 izkf;drk 1-,d fldk mnkyus ij fpùk vkus dh ;k izkf;drk gs\ - rk'k dh xm~mh esa 5 iùks gsa fp= iùkk vkus dh ;k lahkkouk gs\ 3-,d ikls dks,d ckj mnkyus ij vhkkt; la[;k vkus dh ;k izkf;drk gs\ 4-,d ikls dks mnkyus ij 5 ls de vad vkus dh ;k izkf;drk gs\ 5- rk'k dh xm~mh esa 5 iùks gsa gqdqe dh nqdh vkus dh ;k izkf;drk gs\ 6-,d fødsv esp esa,d cyysckt [ksyh xbz 30 xsanksa esa 5 ckj pksdk ekjrk gs pksdk u ekjs tkus dh ;k lahkkouk gs\ 7-,d FkSys esa 5 lqsn] 4 dkyh] 3 yky ckwysa gsa ;n`pn;k,d ckwy fudkyh xbz dkyh ckwy vkus dh ;k lahkkaouk gs\ 8-,d flds dks 500 ckjs mnkyus ij fueufyf[kr ckjackjrk, izkir gksrh gsa %& fpùk 45] iv 55 fpùk?kvuk dh izkf;drk Kkr djks 9-50 fnuksa esa fd, x, eksle iwokzuqekuksa esa ls 175 ckj iwokzuqeku lgh jgk gs rks fn, gq, fnu ij iwokzuqeku lgh u gksus dh izkf;drk ;k gksxh\ 10-,d uoha d{kk ds 36 cppksa esa 0 cpps ym+ds gsa d{kk esa ym+fd;k gksus dh izkf;drk ;k gksxh\ 11-5 ls 15 la[;kvksa ds chp fo"ke la[;k gksus dh izkf;drk Kkr djks ifjokjksa esa( 814 ifjokjksa esa cpps gsa ckfd ifjokjksa esa 1 cppk gs 1 cppk gksus dh izkf;drk Kkr djks 13- 'India' essa 'I' gksus dh izkf;drk Kkr djks

70 14-,d FkSys esa 56 lsc gsa ftuesa ls 19 lsc [kjkc gks x,,d lsc dks ;kn`pn;k pquk x;k hd lsc vkus dh izkf;drk ;k gksxh\ 15-,d ikls dks,d ckj mnkyus ij vad 7 vkus dh izkf;drk Kkr djks 16-,d FkSys esa 5 lqsn] 6 dkyh vksj 3 gjs dkmz gsa,d dkmz fudkyus ij gjk dkmz u vkus dh izkf;drk Kkr djks 17-,d d{kk esa 50 cppksa esa ls 70 izfr'kr cpps ikl gks x, Qsy gksus okys cppksa dh izkf;drk Kkr djks 18-,d Vhe dh QqVckWy esp esa thrus dh lahkkouk 0-4 gs rks gkjus dh izkf;drk Kkr djks 19- rk'k ds 5 iùkksa esa ls,d yky ckn'kkg vkus dh izkf;drk Kkr djks 0-,d eqgyys esa 67 xkfm+;k gsa ftuesa ls 39 dkyh xkfm+;k gsa dkyh xkm+h u gksus dh lahkkouk ;k gksxh\ 1- lkaf[;dh ds ckjs esa 50 fo kffkz;ksa dk er Kkr fd;k x;k ftlesa 143 fo kfkhz lkaf[;dh ilan djrs gsa lkaf[;dh u ilan djus okys fo kfkhz dh izkf;drk Kkr djks - LSSES esa 'S' vkus dh izkf;drk Kkr djks 3- fdlh d{kk esa 47 cppksa esa ls 9 cpps x`g fokku i<+rs gsa ckfd MªkWbZax ds fo kfkhz gsa MªkWb±x ds cppksa dh izkf;drk ;k gksxh\ 4- Mathematics 'kn esa Loj (vowel) vkus dh izkf;drk Kkr djks 5-1&70 ds chp esa 7 ls fohkkt; la[;kvksa dh izkf;drk Kkr djks

71 mùkjekyk