class 7math book

PsfO 6 lgbz{] fªs\\ (Co-ordinate) 6.1. nv] flrqdf lbOPsf laGbx' ¿sf] lgbz{] fªs\\ (Co-ordinate of the Given Points in Graphs) tnsf ljm| ofsnfkx¿ cWoog u/L 5nkmn u/ . 1. rty' fz+{ x¿sf] kl/ro ;u“ s} f nv] flrqdf, bf;] f| ] rty' fz+{ klxnf] rty' fz+{ -—, ±_ -±, ±_ bO' { ;ªV\\ of /v] fx¿ XOX' / YOY' cfk;df nDa xg' ] u/L laGb' O df O rfy} f] rty' fz+{ sfl6Psf 5g\\ . -±, —_ t;] f| ] rt'yf{z+ t;] f{] /v] f X'OX nfO{ X - cIf (X-axis) elgG5 . To:t} 7f8f] /v] f YOY' nfO{ Y - cIf (Y-axis) -—, —_ elgG5 . laGb' O nfO{ pbu\\ d laGb' (origin) elgG5 . laGb' O nfO{ k;| ªu\\ sf] laGb' (point of Reference) elgG5 .  lrqdf slt cf6] f Ifq] x¿ 5g\\ < ltgLx¿sf] gfd nv] . dflysf] lrqdf YOX, YOX', X'OY' / XOY' rf/ Ifq] x¿ -rty' fz{+ _ xg' \\ . Ifq] YOX nfO{ klxnf] rty' fz+{ , YOX' nfO{ bf;] f| ] rty' fz+{ , X'OY' nfO{ t;] f| ] rty' fz+{ / XOY' nfO{ rfy} f] rty' fz+{ elgG5 . 2. nv] flrqdf laGbx' ¿sf] lgbz{] fªs\\ ca xfdL nv] flrqdf rf/cf6] } rty' fz+{ sf laGbx' ¿ jm| dzM A, B, C / D sf lgbz{] fªs\\ x¿ kQf nufpg sfl] ;; u/f“} . -s_ laGb' A sf] lgbz{] fªs\\ tnsf] lrqdf, X- cIfdf pbu\\ d laGbs' f] bfofl“ t/ wgfTds ;ªV\\ ofx¿ nl] vPsf] 5 . To:t} Y- cIfdf pbu\\ d laGba' f6 dflylt/ wgfTds ;ªV\\ ofx¿ nl] vPsf 5g\\ . o;/L YOX Ifq] df X / Y sf bj' } lgbz{] fªs\\ x¿ wgfTds xG' 5g\\ .  laGb' A sg' rt' yfz+{ df k5{ <  laGb' A sf] x- lgbz{] fªs\\ slt xfn] f <  To:t} laGb' A sf] y – lgbz{] fªs\\ slt xfn] f < 46 ul0ft, sIff – &

ca A(x,y) sf] lgbz{] fªs\\ lgsfnf“} M lrqdf laGb' A af6 YY' ;Ddsf] b/' L 3 PsfO 5 . BA To;n} ] laGb' A sf] x - lgbz{] fªs\\ 3 xG' 5 . To:t} u/L laGb' A af6 XX' ;Ddsf] b/' L 5 PsfO O 5 . To;n} ] laGb' A sf] y- lgbz{] fªs\\ 5 5 . To;n} ] CD laGb' A sf] lgbz{] fªs\\ A(x,y) = A(3,5) xG' 5 . -v_ ljm| ofsnfk -s_ sf cfwf/df laGb' B sf] lgbz{] fªs\\ B(x,y) kQf nufpm . -u_ s] laGb' C(x,y) sf] lgbz{] fªs\\ C(-4,-5) xG' 5 < -3_ ca laGb' D sf] lgbz{] fªs\\ slt xfn] f < lrqdf lbOPsf laGbx' ¿sf lgbz{] fªs\\ x¿ lgDgfg;' f/ xG' 5g\\ M A(x,y) = A(3,5), B(x,y) = B(-2,4), C(x,y) = C(-4,-5) / D(x,y) = D(4,-4) xG' 5 . o;/L X – cIf / Y – cIfsf ;ªV\\ of /v] fx¿sf] dfgsf cfwf/df lbOPsf kT| os] laGbx' ¿sf] (x,y) lgbz{] fªs\\ lgsflnG5 . sg' } klg laGbs' f] lgbz{] fªs\\ n] To; laGbs' f] cjl:yltnfO{ hgfpb“ f x - lgbz{] fªs\\ n] pbu\\ d laGbe' Gbf slt PsfO bfof“ afof“ eGg] ae' mfp5“ . To:t} y - lgbz{] fªs\\ n] pbu\\ d laGbe' Gbf slt PsfO dfly jf tn eGg] ae' mfp5“ . pbfx/0f 1 lbOPsf] lrqdf, -s_ laGbx' ¿ P, Q, R / S sf PS lgbz{] fªs\\ x¿ kQf nufpm . -v_ kT| os] laGbx' ¿nfO{ jm| dzM hf8] b\\ } hfpm . ss] f] lrq aGof] nv] . ;dfwfg O -s_ lrqdf laGbx' ¿ P, Q, R / S sf QR lgbz{] fªs\\ x¿ lgDgfg;' f/ 5g\\ M P sf] lgbz{] fªs\\ = P(-4,4) Q sf] lgbz{] fªs\\ = Q(-4,-4) ul0ft, sIff – & 47

R sf] lgbz{] fªs\\ = R(4,-4) / X sf] lgbz{] fªs\\ = S(4,4) -v_ laGbx' ¿ jm| dzM P,Q,R / S hf8] b\\ f Pp6f ju{ PQRS aGof] . h;df kT| os] eh' fsf] nDafO 8 PsfO 5 . s;/L < lzIfs;u“ k/fdz{ u/ . pbfx/0f 2 lbOPsf] nv] flrqdf ;dfgfGt/ rte' h{' ABCD sf kT| os] zLifl{ aGbx' ¿sf lgbz{] fªs\\ kQf nufpm . ;dfwfg oxf,“ laGb' A sf] lgbz{] fªs\\ lgsfNbf, A D B C laGb' A af6 YY' ;Ddsf] b/' L 3 PsfO 5 . laGb' A bf;] f| ] rty' fz+{ df kg{] ePsfn] A sf] x lgbz{] fªs\\ (-3) xG' 5 . To:t} u/L A af6 XX' ;Ddsf] b/' L 4 PsfO 5 . laGb' A bf;] f| ] rty' fz+{ df ePsfn] laGb' A sf] y- lgbz{] fªs\\ 4 xG' 5 . To;n} ] laGb' A sf] lgbz{] fªs\\ = A(x,y) = A (-3, 4) xG' 5 . To;n} ,] laGb' A sf] h:t} kl| jm| ofaf6 jm| dzM laGbx' ¿ B, C / D sf lgbz{] fªs\\ x¿ lgsfNbf, laGb' B sf] lgbz{] fªs\\ = B(x, y) = B (-5, -3) laGb' C sf] lgbz{] fªs\\ = C(x,y) = C(2,-3) laGb' D sf] lgbz{] fªs\\ = D (x,y) = D(4,4) t;y,{ ;dfgfGt/ rte' h{' ABCD sf kT| os] zLifl{ aGbs' f lgbz{] fªs\\ x¿ jm| dzM A(-3,4), B(-5, -3) C(2,-3) / D (4,4) xG' 5g\\ . 48 ul0ft, sIff – &

cEof; 6.1 1. nv] flrqsf cfwf/df lgbz{] fªs\\ x¿ kQf nufpm . jm| dzM laGbx' ¿ F, B, C, D, E, A / B hf8] . ss] f] lrq aGof,] nv] . A D OE C B F 2. nv] flrqdf lbOPsf] kT| os] cfsl[ tsf zLifl{ aGbx' ¿sf] lgbz{] fªs\\ x¿ kQf nufpm . P SA O Q RB C DG EF 3. ltdf| ] ;fyLn] nv] flrqdf bv] fPsf] sg' } laGbs' f] lgbz{] fªs\\ kQf nufpm . 4. kZ| g g=+ 1 / 2 df h:t} u/L Ps Pscf6] f ;d:of agfO{ ;dfwfg u/ . ;fyL;u“ cfk;df ;d:of ;f6/] ;dfwfg u/ . pQ/nfO{ hfr“ /] klg x/] . ul0ft, sIff – & 49

6.2 nv] flrqdf lbOPsf laGbx' ¿sf] cªs\\ g (Plotting the Given Points in the Graph) tnsf ljm| ofsnfk cWoog u/L 5nkmn u/ M nv] flrqdf lbOPsf laGbx' ¿sf] cªs\\ g ug{] tl/sf ;u“ s} f] nv] flrqdf laGbx' ¿ A(6,4), B(-3,4), C(-3,-5) / D(6,-5) nfO{ kl] G;nn] nv] flrqdf cªs\\ g ug{] ko| f; u/ . -s_ ;jk{ y| d sfkLdf lrqdf bv] fOP h:t} X- cIf / B(-3,4) A(6,4) Y- cIfdf ;ªV\\ of /v] fx¿ agfO{ nv] flrq tof/ u/ . C(-3,-5) D(6,-5) -v_ A(6, 4) nfO{ nv] flrqdf cªs\\ g ug{ s] ugk{' nf{ < – X- cIfdf pbu\\ d laGbb' l] v wgfTds lbzfdf -bfof_“ 6 PsfO hfg] – ToxL 6 PsfO af6 Y- cIfsf] wgfTds lbzf -dfly_ 4 PsfO hfg] / laGbn' fO{ ;ª\\st] ug{] – To; laGb' glhs} A(6,4) nV] g] – laGb' A(6, 4) nfO{ nv] flrqdf cªs\\ g ug{] sfd ;lsof] . -u_ ca B(-3,4) nfO{ -s_ s} kl| jm| of ckgfO{ s;/L dflysf] nv] flrqdf cªs\\ g ug{ ;lsG5 xfn] f < ;fyL;u“ klg 5nkmn u/ . -3_ To;} u/L jm| dzM C(-3,-5) / D(6,-5) nfO{ klg nv] flrqdf cªs\\ g u/ . -ª_ ca A, B, C / D laGbx' ¿nfO{ hf8] . ss] f] lrq aGof,] nv] . oxf“ ABCD Pp6f ju{ xf] . ju{ ABCD n] slt Ifq] kmn cfu] 6s] f] 5 < sf7] f ug/] kQf nufpm . lgbz{] fªs\\ vn] +5 +4 sIffdf ;fyLx¿;u“ ldn/] lrqdf bv] fPh:t} u/L sfuhsf 6j' m| fx¿df +3 X- cIf / Y- cIfsf ;ªV\\ of /v] fx¿ agfpm . bO' { ;dx\" df afl“ 8P/ +2 lgbz{] fªs\\ vn] vn] . +1 -5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5 vn] sf] lgod tyf vN] g] tl/sf -1 1. ;j{k|yd ;ª\\Vofx¿nfO{ lrqdf b]vfPh:t} cfk;df nDa -2 -3 xg' ] u/L rf/} df /fv . lgbz{] fªs\\ sf] jl/kl/ bO' { ;dx\" x¿df -4 56' 6\\ f56' 6\\ } nfOg nfu/] a; . -5 50 ul0ft, sIff – &

2. ;dx\" -s_ sf] klxnf] ;fyLn] ;dx\" -v_ sf] ;fdG' gs] f] ;fyLnfO{ sg' } lgbz{] fªs\\ df pleg eGg] . p;n] ;fc] g;' f/ plegk' 5{ . pleg g;sd] f aflx/ uO{ bzs{ aGb5 . t/ g;Sg] ;fyLnfO{ plegk' g{] :yfgsf] hfgsf/L ;fyLx¿af6 l;sfOlbgk' 5{ . 3. ca nv] flrqsf] cfkm\\ gf] lgbz{] fªs\\ df plePsf] ;fyLn] g} ;dx\" -s_ sf] csf{] ;fyLnfO{ g=+ 2 df h:t} u/L sg' } csf{] gof“ lgbz{] fªs\\ eGg] / ;fc] g;' f/ pleg nufpgk' 5{ . 4. bj' } ;dx\" dWo] hg' ;dx\" df w/] } ;fyL l7s lgbz{] fªs\\ df pleg ;s] pxL ;dx\" sf] lht xG' 5 . pbfx/0f 1 laGbx' ¿ E(-5, -5), F(-3, -3), O(0, 0), G(3, 3) / H(6, 6) nfO{ nv] flrqdf cªs\\ g u/ . ;a} laGbn' fO{ hf8] L x/] , s] s] lrq aGof] < nv] . lgbz{] fªs\\ sf] tflnsf klg agfpm . ;dfwfg F(-3,-3) H(6,6) E(-5,-5) G(3,3) laGbx' ¿ E(-5, -5), F(-3, -3), O(0, 0), G(3, 3) / H(6, 6) nfO{ hf8] b\\ f aGg] O(0,0) nv] flrqnfO{ bfofl“ t/ pNnv] ul/Psf] 5 . lgbz{] fªs\\ sf] tflnsfdf /fVbf, x -5 -3 0 3 6 y -5 -3 0 3 6 o;/L dfly lbOPsf laGbx' ¿ hf8] b\\ f Pp6f l;wf /v] f aGof] . pbfx/0f 2 P(-5,3) Q(0,3) S R(0,-2) laGbx' ¿ P(-5,3), Q(0,3), R(0,-2) / S Pp6f jus{ f zLifl{ aGbx' ¿ xg' \\ eg,] -s_ nv] flrqdf lbOPsf laGbx' ¿nfO{ cªs\\ g u/ . -v_ laGb' S sf] lgbz{] fªs\\ kQf nufpm . -u_ pSt jus{ f] Ifq] kmn kQf nufpm . ;dfwfg -s_ lbOPsf laGbx' ¿ P(-5, 3), Q(0, 3), R(0, -2) nfO{ jm| dzM cªs\\ g u/L bfof“ nv] flrqdf bv] fOPsf] 5 . ul0ft, sIff – & 51

-v_ PQRS Pp6f ju{ ePsfn] ;f] jus{ f] nDafO QR = 5 PsfO 5, ca R(0, -2) af6 PQ ;u“ X- cIfsf] C0ffTds lbzflt/ cyjf -5 df S laGb' kQf nufpm . ca laGb' S sf] lgbz{] fªs\\ S(-5, -2) xG' 5 . -u_ ca, pSt ju{ PQRS sf] Ifq] kmn lgsfNbf, jus{ f] Ifq] kmn = l2 ju{ PsfO = 52 ju{ PsfO = 25 ju{ PsfO o;y,{ jun{ ] cfu] 6s] f sf7] fx¿ uGbf klg 25 ju{ PsfO g} xG' 5 . To;n} ] pSt ju{ PQRS sf] Ifq] kmn 25 ju{ PsfO xG' 5 . cEof; 6.2 1. tnsf kT| os] laGbx' ¿nfO{ nv] flrqdf cªs\\ g u/ M A(3, -2), B(7, 0), C(-1, 2), D(1, 1), P(4, 4), Q(-5, 0), R(0, 5), S(7, -2) E(0, 4), F(-3,-3), G(-5,-4) / H(-3,-4) 2. kZ| g g=+ 1 sf sg' sg' laGbx' ¿ X- cIfaf6 a/fa/ b/' Ldf k5g{ ,\\ nv] . 3. kZ| g g=+ 1 sf sg' sg' laGbx' ¿ Y- cIfaf6 a/fa/ b/' Ldf k5g{ ,\\ nv] . 4. tnsf k|To]s laGb'x¿nfO{ n]vflrq agfO{ cª\\sg u/ . k|To]s laGb'nfO{ j|mdzM hf]8\\b} hfpm . o;/L aGg] cfsl[ tsf] gfd klg nv] M -s_ A(-4,2), B(4,3) / C(2,-5) -v_ P(-6,4), Q(0,4), R(0,0) / S(-6,0) -u_ H(-5,-4), I(-2,1), J(1,7), K(4,2), L(1,3), M(4,0) / N(-2,1) 5. laGbx' ¿ P(2, 5) / Q(2, -5) nfO{ nv] flrqdf cªs\\ g u/ / tnsf kZ| gsf] hjfkm nv] M -s_ PQ sf] dWolaGb' R eP R sf] lgbz{] fªs\\ slt xfn] f < -v_ PQ sf] nDafO slt xfn] f < 6. Pp6} nv] flrqdf / pxL pbu\\ dlaGb' lnP/ tnsf bO' { ;dx\" x¿ -s_ / -v_ df lbOPsf laGbx' ¿ cªs\\ g u/L kT| os] ;dx\" sf laGbx' ¿ jm| d;} u“ hf8] b\\ } hfpm . -s_ A(-6, 5), B(-4, 5), C(0, 5), D(2, 5) / E(6, 5) -v_ F(5, -5), G(5, -3), H(5, 0) / I(5, 6) pSt bO' { ;dx\" sf nv] flrqx¿ sfl6Psf] laGb' J sf] lgbz{] fªs\\ kQf nufpm . 52 ul0ft, sIff – &

PsfO 7 kl/ldlt / Ifq] kmn (Perimeter and Area) 7.1. ju{ / cfotsf] kl/ldlt (Perimeter of Square and Rectangle) ju{ / jus{ f] kl/ldlt (Square and Perimeter of Square) ljm| ofsnfk 1 tnsf ljm| ofsnfk cWoog u/L sIffdf 5nkmn u/ .  ;u“ s} f] lrqdf eh' fx¿ AB, BC, CD / DA gfk . s] ;a} A D eh' fx¿ a/fa/ 5g\\ <  To:t} leqL sf0] fx¿ ∠A, ∠B, ∠C / ∠D klg gfk . s] kT| os] sf0] fx¿ Ps ;dsf0] f (=90°) sf] gfksf 5g\\ <  s] rte' h{' ABCD Pp6f ju{ xf] < s;/L < 5nkmn u/ . B C  ju{ ABCD sf] jl/kl/sf] 3/] fsf] nDafOsf] gfk slt xfn] f < oxf,“ ;a} eh' fx¿ a/fa/ 5g\\ . cyft{ \\ AB = BC = CD = DA = 3cm 5g\\ . dfgf,“} ju{ ABCD sf] Pp6f eh' fsf] nDafO l 5 eg] AB = BC = CD = AD = l = 3cm xG' 5 . ca, ju{ ABCD sf] kl/ldlt eGgfn] jl/kl/sf rf/ cf6] } eh' fx¿ AB, BC, CD / AD sf] gfksf] ofu] kmn xG' 5 . t;y,{ ju{ ABCD sf] kl/ldlt (P) = AB + BC + CD + DA xG' 5 . = 3cm + 3cm + 3cm + 3cm = 12cm ca s] 12cm nfO{ 4 x 3cm = 4l nV] g ;lsG5 s;/L < 5nkmn u/ . lsgls l = 3cm 5 .  s] jus{ f] kl/ldlt kQf nufpg] cGo pkfox¿ 5g\\ <  s] o;sf nflu sg' } ;q\" kQf nufpg ;lsG5 < lgisif{ M sg' } klg jus{ f] kl/ldlt To; jus{ f] Pp6f eh' fsf] nDafOsf] rf/ u0' ff xG' 5 . ca dflysf] tYonfO{ ;ªs\\ t] ;q\" df nV] g] ko| f; u/f“} . jus{ f] kl/ldltnfO{ P / nDafOnfO{ l dfGbf, jus{ f] kl/ldlt (P) = (l + l + l + l) PsfO = 4l PsfO xG' 5 . t;y,{ ;q\" M jus{ f] kl/ldlt (P) = 4l xG' 5 . ;a} eh' fx¿ a/fa/ ePsf] / kT| os] leqL sf0] fx¿ Ps ;dsf0] f (=90°) ePsf] rte' h{' nfO{ ju{ elgG5 . kl/ldlt egs] f] jl/kl/sf] 3/] fsf] nDafO xf] . t;y{ ju{ ABCD sf] jl/kl/sf] 3/] f (AB + BC + CD + DA) sf] nDafOnfO{ ju{ ABCD sf] kl/ldlt elgG5 . ul0ft, sIff – & 53

ljm| ofsnfk 2 Pp6f jufs{ f/ sfuh np] m÷agfpm . To;nfO{ l P lrqdf lbP h:t} u/L MNOP gfd bp] m . M l kT| os] eh' f gfk/] lrqdf lbP h:t} u/L l l = ... df jf:tljs gfk /fv . ca ju{ MNOP sf] kl/ldlt gfkL ;q\" ko| fu] gul/sg NO tyf ;q\" ko| fu] u//] kQf nufpm . l ljm| ofsnfk 3 sg' } Pp6f jufs{ f/ ;tx ePsf] j:t' vfh] . sfkLdf ;ªs\\ t] lrq klg agfpm . To; j:ts' f] jufs{ f/ ;txsf] kl/ldlt ;q\" sf] ko| fu] u/L kQf nufpm . cfkm\\ gf] sfo{ ;fyL tyf lzIfsnfO{ bv] fO{ 5nkmn u/ . 2. cfot / cfotsf] kl/lw (Rectangle and Perimeter of Rectangle) ljm| ofsnfk 4 ;u“ s} f] lrqdf eh' fx¿ EF, FG, GH / HE gfk . s] EF = HG eof] < To:t} s] HE = FG eof] <  To:t} u/L leqL sf0] fx¿ ∠E,∠F,∠G / ∠H klg gfk . s] kT| os] sf0] fx¿ Ps ;dsf0] f (= 90°) 5g\\ <  ;u“ s} f] lrqdf ;Ddv' eh' fx¿ EF = HG = 3cm / HE = FG = 3.5 cm 5g\\ . To:t} leqL sf0] fx¿ ∠E = ∠F = ∠G = ∠H = 90° 5g\\ . E H oxf,“ rte' h{' EFGH sf ;Ddv' eh' fx¿ a/fa/ 5g\\ / kT| os] leqL sf0] fx¿ Ps ;dsf0] f (=90°) sf 5g\\ . To;n} ] rte' h{' EFGH Pp6f cfot xf] . ca, cfot EFGH sf] kl/ldlt eGgfn] jl/kl/sf rf/cf6] } eh' fx¿ F G EF, FG, GH / EH sf] ofu] kmn xG' 5 . t;y,{ cfot EFGH sf] kl/ldlt (P) = EF + FG + GH + HE xG' 5 . = 3cm + 3.5cm + 3cm + 3.5cm = 13cm ca s] 13cm nfO{ 2(3.5+3)cm = 2(l+b) PsfO n]Vg ;lsG5, s;/L < 5nkmn u/ . lsgls l = 3.5cm / b = 3cm 5 . 54 ul0ft, sIff – &

 s] cfotsf] kl/ldlt kQf nufpg] cGo pkfox¿ 5g\\ <  s] o;sf nflu sg' } ;q\" kQf nufpg ;lsG5 < lgisif{ M sg' } klg cfotsf] kl/ldlt To; cfotsf] nDafO / rf8} fOsf] ofu] kmnsf] bO' { u0' ff jf bfA] a/ xG' 5 . ca dflysf] tYonfO{ ;ªs\\ t] ;q\" df nV] g] ko| f; u/f“} . cfotsf] kl/ldltnfO{ P, nDafO l, / rf8} fOnfO{ b dfGbf, cfotsf] kl/ldlt (P) = ( l + b + l + b) PsfO = 2l + 2b = 2(l+b) PsfO xG' 5 . t;y,{ ;q\" M cfotsf] kl/ldlt (P) = 2(l+b) xG' 5 . ;Ddv' eh' fx¿ a/fa/ ePsf] / ;a} leqL sf0] fx¿sf] gfk Ps ;dsf0] f (=90°) ePsf] rte' h{' nfO{ cfot (rectangle) elgG5 . sg' } klg cfotsf jl/kl/sf 3/] f cyft{ \\ rf/cf6] } eh' fx¿sf] ofu] nfO{ To; cfotsf] kl/ldlt elgG5 . ljm| ofsnfk 5 l D b Pp6f cfotfsf/ sfuh np] m÷agfpm . To;nfO{ lrqdf lbP A C h:t} u/L gfd bp] m . ;q\" ko| fu] gul/sg tyf ;q\" ko| fu] u//] bj' } tl/sfn] pSt cfotfsf/ sfuhsf] kl/ldlt kQf nufpm . b ljm| ofsnfk 6 B l ltdf| ] 3/ jf ljBfnodf Pp6f cfotfsf/ j:t' vfh] . To;sf] kl/ldlt ;q\" ko| fu] gul/sg tyf ;q\" ko| fu] u//] kQf nufpm . ;ªs\\ t] lrq klg agfpm . cfkm\\ gf] sfo{ ;fyL / lzIfs ;dIf bv] fO{ 5nkmn u/ . pbfx/0f 1 l5l/ª;u“ ePsf] Pp6f jufs{ f/ ?dfnsf] nDafO 15cm eP To; ?dfnsf] kl/ldlt kQf nufpm . ;dfwfg oxf,“ lbPcg;' f/ jufs{ f/ ?dfnsf] nDafO (l) = 15cm kl/ldlt (P) = ? ;q\" cg;' f/, jus{ f] kl/ldlt (P) = 4l = 4 x 15cm = 60 cm ctM To; jufs{ f/ ?dfnsf] kl/ldlt = 60cm xG' 5 . pbfx/0f 2 l;hg{ f;u“ kl/ldlt 120 ld6/ ePsf] Ps 6j' m| f jufs{ f/ hUuf 5 . o;sf] nDafO kQf nufpm . ;dfwfg M oxf,“ lbOPcg;' f/ hUuf -ju_{ sf] kl/ldlt (P) = 120m nDafO (l)= ? ul0ft, sIff – & 55

;q\" cg;' f/, jus{ f] kl/ldlt (P) = 4l cyjf, 120m = 4l cyjf, 4l = 120m cyjf l = 120  30 m 4 t;y,{ pSt hUufsf] nDafO 30 ld6/ 5 . pbfx/0f 3 cfOt] tfdfªn] cfkm\\ gf] 40 m nDafO ePsf] jufs{ f/ t/sf/L af/Lsf] jl/kl/ kvfn{ dfly 7 kmGsf] sf8“ t] f/ agfpgnfO{ slt sf8“ t] f/ rflxG5 < ;dfwfg kZ| gdf lbOPcg;' f/, jufs{ f/ t/sf/L af/Lsf] nDafO (l) = 40 m kl/ldlt (P) = ? 7 kmGsf] af/Lsf] nDafO = ? ;\"qcg';f/, jus{ f Ifq] sf] kl/ldlt (P) = 4l = 4 x 40 m = 160 m t/ 7 kmGsf] sf8“ t] f/ nufpg rflxg] tf/ = 7 x P = 7 x 160 m = 1120 m t;y,{ 7 kmGsf] af/ nufpg 1120 m sf8“ t] f/ rflxG5 . pbfx/0f 4 nvg rfw} /L;u“ 900 m nDafO / 600 m rf8} fO ePsf] Pp6f cfotsf/ cfk“ sf] aur“} f 5 . To; aur“} fsf] jl/kl/sf] 3/] f -kl/ldlt_ slt xfn] f < ;dfwfg oxf“ lbOPcg;' f/, cfk“ sf] aur“} fsf] nDafO (l) = 900m rf8} fO (b) = 600m kl/ldlt (P) = ? ;\"qcg';f/, cfotfsf/ j:t' jf Ifq] sf] kl/ldlt (P) = 2(l+b) =2(900m+600m) = 3000m = 3km t;y,{ pSt aur“} fsf] kl/ldlt 3km 5 . 56 ul0ft, sIff – &

pbfx/0f 5 zldn{ fsf] Pp6f cfotfsf/ g;/{ Lsf] kl/ldlt 200 ld6/ 5 . olb To; g;/{ Lsf] nDafO 60 ld6/ eP rf8} fO slt xfn] f < ;dfwfg oxf“ kZ| gdf lbOPcg;' f/, g;/{ Lsf] kl/ldlt (P) = 200m nDafO (l) = 60m rf8} fO (b) = ? ;\"qfg';f/, cfotfsf/ j:t' jf Ifq] sf] kl/ldlt (P) = 2 (l + b) cyjf, 200m = 2(60m + b) cyjf, 200m = 120m + 2b cyjf, 2b = 200m - 120m cyjf, b  80m 2 cyjf, b = 40m t;y,{ pSt g;/{ Lsf] rf8} fO 40 ld6/ xG' 5 . pbfx/0f 6 snfbj] L /fOn{ ] 20m nDafO / 15m rf8} fO ePsf] t/sf/L af/Lsf] jl/kl/ 5 kmGsf] l;ªu\\ f] lgufnfsf] af/ nufpg] ljrf/ ul/5g\\ . pgn] slt lgufnf] lsGgk' nf{ < ;dfwfg kZ| gfg;' f/, pSt cfotfsf/ t/sf/L af/Lsf] nDafO (l) = 20m rf8} fO (b) = 15m kl/ldlt (P) = ? 5 kmGsf] lgufnfsf] nDafO = ? ;\"qcg';f/, cfotfsf/ Ifq] sf] kl/ldlt (P) = 2(l + b) = 2(20m + 15m) =2(35m) = 70m ca 5 kmGsf] lgufnfsf] af/ nufpgk' g{] ePsfn,] hDdf lgufnfsf] nDafO l = 5 x P = 5 x 70m = 350m t;y,{ pSt t/sf/L af/Ldf af/ nufpg 350 ld6/ lgufnf] rflxG5 . ul0ft, sIff – & 57

cEof; 7.1 1. tn lbOPsf kT| os] cfsl[ tsf] nDafO / rf8} fO gfkL ;q\" ko| fu] u//] kl/ldlt lgsfn M -s_ -v_ E H A D B C F G P V -u_ M -3_ S NO T U 2. tn lbOPsf jufs{ f/ jf cfotsf/ j:ts' f] kl/ldlt lgsfn M -s_ jufs{ f/ sfuh, nDafO = 13cm -v_ jufs{ f/ ?dfn, nDafO = 18cm -u_ cfotfsf/ 6a] n, nDafO = 73cm, rf8} fO = 56cm -3_ cfotfsf/ s/;] faf/L, nDafO = 5m, rf8} fO = 3m 3. Pp6f jufs{ f/ g;/{ Lsf] nDafO 7m 5 eg,] -s_ pSt Ifq] sf] jl/kl/sf] 3/] f slt xG' 5 < -v_ olb ;f] g;/{ Lsf] jl/kl/ 8 kmGsf] sf8“ t] f/ nufpgk' /d] f slt sf8“ t] f/ rflxPnf < 4. Pp6f jufs{ f/ ?dfnsf] kl/ldlt 225cm /x5] eg] To; ?dfnsf] nDafO slt xfn] f < 5. Pp6f cfotfsf/ 6a] nsf] nDafO 78cm / rf8} fO 65cm 5 eg] To; 6a] nsf] kl/ldlt kQf nufpm . 6. sfG5L bgj' f/sf] 650m kl/ldlt ePsf] cfotfsf/ t/sf/L af/Lsf] nDafO 205m /x5] eg] rf8} fO slt xfn] f < 7. Zofd lj=s=n] Pp6f 5m nDafO ePsf] Pp6f jufs{ f/ 38/] L lsg] 5g\\ M -s_ pgsf] ;f] 38/] Lsf] kl/ldlt slt xfn] f < -v_ pgn] 8 kmGsf] 8f/] Ln] ag] k{' bf{ slt 8f/] L rflxPnf < 8. wlgofn“ ] cfkm\\ gf] 23m nDafO / 21m rf8} fO ePsf] 3/sf] jl/kl/sf] sDkfpG8sf] kvfn{ nufpg rflx5g\\ M -s_ pSt sDkfpG8sf] kl/ldlt slt xfn] f < -v_ pgn] sDkfpG8sf] dfly jl/kl/ 4 kmGsf] sf8“ t] f/ nufpgsf nflu slt nfdf] sf8“ t] f/ lsg/] Nofpgk' nf{ < 9. dfly plNnlvt ;ªV\\ of 1 bl] v 8 ;Dd lbOP h:t} u/L cfkm“} kZ| gx¿ lgdf0{ f u/L yk cEof; u/ . cfkm\\ gf] sfon{ fO{ ;fyL tyf lzIfs;u“ 5nkmn u/ . 58 ul0ft, sIff – &

7.2 if8d\\ v' f / 3gsf] k/\" f ;txsf] Ifq] kmn (Total Surface area of Cuboid and Cube) xfdLn] sIff 6 df if8d\\ v' f / 3gsf] kl/ro, vfj] m| f gd'gfx¿ / cfotg;DaGwL ;/n ;d:ofx¿af/] 5nkmn ul/;ss] f 5f“} . ca xfdL o; kf7df if8d\\ v' f / 3gsf] ;txsf] Ifq] kmnsf af/d] f 5nkmn ug{] 5f“} . o;sf ;fy} o;;DaGwL Jofjxfl/s ;d:ofx¿ ;dfwfg ug{] l;k klg xfl;n ug{] 5f“} . tnsf ljm| ofsnfkx¿ cWoog u/L 5nkmn u/ . 1. if8d\\ v' fsf] k/\" f ;txsf] Ifq] kmn if8d\\ v' fsf] hfnL agfO{ lgDgfg;' f/ nDafO -l_, rf8} fO -b_, / prfO -h_, 56' o\\ fP/ nv] . clg tnsf kZ| gdf 5nkmn u/ . b h hb l b l h bh h b b l l -s_ dflysf bO' c{ f6] f lrqdf s] ;DaGw 5 < -hfnL / gdg' fdf_ -v_ if8d\\ v' fsf] hfnL sltcf6] f cfot ldn/] ags] f] /x5] < -u_ if8d\\ v' fsf ;a} cfot a/fa/ 5g\\ ls 5g} g\\ < -3_ sltcf6] f cfotsf] Ifq] kmn l x b 5g\\ < -ª_ sltcf6] f cfotsf] Ifq] kmn l x h 5g\\ < -r_ sltcf6] f cfotsf] Ifq] kmn b x h 5g\\ < -5_ lrqdf 5fof k/s] f] efun] hfnLsf] slt efu hgfp5“ < -h_ lrqdf 5fof k/s] f] efusf] Ifq] kmn slt xfn] f < ?n/n] nDafO -l_, rf8} fO -b_, prfO -h_, gfk/] kT| os] ;txsf] Ifq] kmn lgsfn . ;a} ;txsf] Ifq] kmn hf8] /] lgsfn . oxL sn' Ifq] kmn g} l;ªu\\ f] if8d\\ v' fsf] ;tx Ifq] kmn xG' 5 . ul0ft, sIff – & 59

dflysf bO' c{ f6] f lrqx¿df klxnf] if8d\\ v' fsf] hfnL / gdg' f bf;] f| ] xf] . if8d\\ v' fsf] hfnL hDdf 6 cf6] f cfot ldn/] ags] f] 5 . kT| os] cfdg;] fdgs] f bO' { bO' { hf8] L cfotx¿ ;dfgfGt/ / a/fa/ 5g\\ . ;a} cfot a/fa/ 5g} g\\ . lrqdf 2 cf6] f cfotsf] Ifq] kmn l x b 5 . To:t} 2 cf6] f cfotsf] Ifq] kmn l x h 5 . / 2 cf6] f cfotsf] Ifq] kmn b x h 5 . oxf“ if8d\\ v' fsf] ;tx Ifq] kmn (A) = ;a} cfotsf] Ifq] kmnsf] ofu] kmn xG' 5 . ca, 5fof k/s] f] efusf] Ifq] kmn (A)= (l x b) + (l x h) + (b x h) xG' 5 . To;n} ] k/' } hfnLsf] Ifq] kmn (A)= 2(l x b)+2(l x h) +2(b x h) = 2(lb + lh + bh) xG' 5 . ;q\" M t;y{ sg' } klg if8dv' fsf] ;txsf] Ifq] kmn (TSA)= 2(lb+lh+bh) ju{ PsfO xG' 5 . hxf,“ if8d\\ v' fsf] l = nDafO, b = rf8} fO, / h = prfO . dflysf] ljm| ofsnfkdf agfPsf] if8dv' fsf] nDafO -l_, rf8} fO -b_, / prfO -h_ 56' o\\ fP/ nv] . pSt if8d\\ v' fsf] ;txsf] Ifq] kmn ;q\" ko| fu] u/L lgsfn . pQ/ hfr“ /] klg x/] . 2. 3gsf] k/\" f ;txsf] Ifq] kmn nDafO, rf8} fO / prfO a/fa/ ePsf] if8d\\ v' fnfO{ 3g -Cube_ elgG5 . To;n} ] 3gdf l = b = h xG' 5 . ca ;q\" M if8d\\ v' fsf] ;txsf] Ifq] kmn (A) = 2(lb + lh + bh) df l = b = h = a -dfgf_“} xb“' f M 3gsf] Ifq] kmn (A) = 2(aa+aa+aa) = 2(3a2) = 6a2 ;q\" M 3gsf] ;tx Ifq] kmn (A)=6a2 ju{ PsfO xG' 5, hxf“ 3gsf] Pp6f eh' fsf] nDafO = a PsfO 5. pbfx/0f 1 Pp6f afs;sf] nDafO (l) = 42cm, rf8} fO (b)=39cm / prfO (h) = 28cm 5 . ca To;sf] k/\" f ;txsf] Ifq] kmn kQf nufpm . ;dfwfg oxf“ lbPcg;' f/, pSt afs;sf] nDafO (l)= 42cm rf8} fO (b)= 39cm / prfO (h)= 28cm 5 . afs; if8d\\ v' f xf] . 60 ul0ft, sIff – &

;q\" cg;' f/, if8d\\ v' fsf] k/\" f ;txsf] Ifq] kmn (T.S.A.)=2(lb+lh+bh) ju{ PsfO = 2[(42 x 39) + (42 x 28) + (39 x 28)]cm2 = 2(1638 + 1176 + 1092)cm2 = 2 x 3906cm2 = 7812cm2 ctM pSt afs;sf] k/\" f ;txsf] Ifq] kmn 7812cm2 xG' 5 . pbfx/0f 2 Pp6f ;nfOs{ f] a66\\ fsf] nDafO 4cm, rf8} fO 3cm / k/\" f ;txsf] Ifq] kmn 45cm2 /x5] . ca To;sf] prfO kQf nufpm . ;dfwfg oxF“ kZ| gfg;' f/, pSt ;nfOs{ f] a66\\ fsf] nDafO (l) = 4cm rf8} fO (b) = 3cm k/\" f ;txsf] Ifq] kmn (A) = 45cm2 / prfO (h) = ? oxf,“ ;nfOs{ f] a66\\ f Pp6f if8d\\ v' f xf] . ;q\" fg;' f/, if8d\\ v' fsf] k/\" f ;txsf] Ifq] kmn (A)= 2(lb+lh+bh) ju{ PsfO cyjf, 45cm2 = 2[(4 x 3) + (4 x h) + (3 x h)] cyjf, 45 = 2(12 + 4h + 3h) cyjf, 45 = 2(12 + 7h) cyjf, 45 = 24 + 14h cyjf, 14h = 45 – 24 cyjf, h = 21  1.5 cm 14 ctM pSt ;nfOs{ f] a66\\ fsf] prfO 1.5cm xG' 5 . pbfx/0f 3 kD] afn] Pp6f lsgf/f 15.4cm ePsf] 3gsf] gdg' f agfO5g\\ . pSt 3gsf] k/\" f ;txsf] Ifq] kmn slt xfn] f < ul0ft, sIff – & 61

;dfwfg oxf“ lbPcg;' f/, 3gsf] Pp6f eh' fsf] nDafO (a)= 15.4cm k/\" f ;txsf] Ifq] kmn (A)= ? ;q\" cg;' f/, 3gsf] k/\" f ;txsf] Ifq] kmn (A)= 6a2 = 6x(15.4)2cm2 =(6x237.16)cm2 =1422.96cm2 ct M pSt 3gsf] gdg' fsf] k/\" f ;txsf] Ifq] kmn (A) = 1422.96cm2 xG' 5 . pbfx/0f 4 Pp6f 3gfsf/ j:ts' f] k/\" f ;txsf] Ifq] kmn (A) = 4056cm2 nl] vPsf] /x5] . ca To; j:ts' f] Pp6f eh' fsf] nDafO slt xfn] f < ;dfwfg kZ| gdf lbPcg;' f/, pSt 3gfsf/ j:ts' f] k/\" f ;txsf] Ifq] kmn (A)= 4056cm2 / Pp6f eh' fsf] nDafO (a) = ? ;q\" cg;' f/, 3gsf] k/\" f ;txsf] Ifq] kmn (A)= 6a2 ju{ PsfO cyjf, 4056cm2 = 6a2 cyjf, 4056 cm2 a2= 6 cyjf, a2 = 676cm2 cyjf, a = 676cm2 cyjf, a = 26cm ct M To; 3gfsf/ j:ts' f] Pp6f eh' fsf] nDafO 26cm xG' 5 . 62 ul0ft, sIff – &

cEof; 7.2 1. tn lrqdf lbOPsf kT| os] 7f;] j:ts' f] ;fl] wPsf] s/' f kQf nufpm (A = if8d\\ v' fsf] k/\" f ;txsf] If]qkmn_ -s_ -u_ -v_ l= -3_ A=? -ª_ dfly s / v df lbOP h:t} u/L 2/2 cf6] f ;d:of agfpm . ;fyL;u“ ldn/] k/\" f ;txsf] Ifq] kmn kQf nufpm . -r_ dfly u / 3 df lbOP h:t} u/L 2/2 cf6] f ;d:of agfpm / ;fyL;u“ ldn/] c1ft eh' fsf] nDafO kQf nufpm . 2. tnsf kT| os] if8d\\ v' fsf] k/\" f ;txsf] Ifq] kmn lgsfn M -s_ l = 5cm, b = 3cm / h = 4.5cm -v_ l = 6.2m, b = 3.3m / h = 6.8m -u_ dfly s / v df h:t} u/L bO' { bO' c{ f6] f ;d:of agfpm÷vfh] . ;fyL;u“ ldn/] cfk;df kT| os] if8d\\ v' fsf] Ifq] kmn kQf nufpm . 3. tnsf kT| os] cj:yfdf 3gsf] k/\" f ;txsf] Ifq] kmn kQf nufpm M -s_ a = 6cm -v_ a = 3.5cm -u_ a = 12m -3_ dfly s, v, / u df h:t} 5 cf6] f ;d:ofx¿ agfpm . ;fyL;u“ kT| os] 3gsf] k/\" f ;txsf] Ifq] kmn kQf nufpm . 4. tnsf kT| os] cj:yfdf if8d\\ v' fsf] glbOPsf] eh' f kQf nufpm M -s_ A = 350cm2, l = ?, b = 10cm / h = 2.5cm -v_ A= 136.24cm2, l = 4.2cm, b= 3.6cm / h=? 5. tnsf kT| os] cj:yfdf 3gsf] Pp6f eh' f kQf nufpm M 63 -s_ A= 600cm2, a =? -v_ A = 5400cm2, a =? ul0ft, sIff – &

6. Pp6f if8d\\ v' fsf] k/\" f ;txsf] Ifq] kmn 6.3 m2 nDafO 1.5m / prfO 1.2m 5 eg,] -s_ pSt if8d\\ v' fsf] rf8} fO slt xfn] f < -v_ pSt if8d\\ v' fnfO{ sf7] fdf /fVbf sf7] fsf] slt ;tx cfu] 6n\\ f < 7. Pp6f afs;sf] nDafO 125cm, rf8} fO 85cm / prfO 70cm 5 eg,] -s_ pSt afs;sf] k/\" f ;txsf] Ifq] kmn kQf nufpm . -v_ To; afs;n] sf7] fsf] slt ;tx 9fSnf < 8= Pp6f kf| ylds pkrf/ afs; (First Aid Box) sf] nDafO 18cm, rf8} fO 10cm / prfO 12cm 5 eg,] -s_ pSt afs; hgfpg] hfnL (Net) sf] lrq agfpm . -v_ pSt hfnLsf cfwf/df k/\" f ;txsf] Ifq] kmn kQf nufpm . -u_ pSt afs;nfO{ 6a] ndfly /fVbf 6a] nsf] slt ;tx 9fSnf < -3_ pSt afs;sf] lasf{] RofltP/ x/fP5 . ca lasf{] gePsf] afs;sf] k/\" f ;txsf] Ifq] kmn slt xfn] f < 9. prfO 7cm / rf8} fO 8cm ePsf] Pp6f rssf] a66\\ fnfO{ 6a] ndf /fVbf 6a] n' sf] ;tx 80cm2 9fs5] eg,] -s_ pSt a66\\ fsf] nDafO kQf nufpm . -v_ pSt a66\\ fsf] k/\" f ;txsf] Ifq] kmn kQf nufpm . 10. dfly kZ| g 6 bl] v 9 ;Dd lbOP h:t} u/L 2/2 cf6] f ;d:of agfO{ lx;fa u/ . pSt lx;fa ;fyL ;fyLlar ;f6/] Pscsfs{ f] ;dfwfg u/ . 5nkmn u/L ;fyLsf] pTt/ hfr“ /] klg x/] . 64 ul0ft, sIff – &

PsfO 8 :yfgfGt/0f (Transformation) 8.1 k/fjtg{ (Reflection) tnsf ljm| ofsnfk tyf tYox¿ cWoog u/L 5nkmn u/ . Y -s_ lrqdf s] bV] 5f} < lrqdf k/fjtg{ (reflection) nfO{ bv] fOPsf] 5 . oxf“ 5fqfsf] kl| tlaDa (image) Pg] fdf bv] fOPsf] 5 . ltdLn] klg Pg] fsf] glhs / 6f9f uO{ x/] . kl| tlaDa k/fjtg{ klg 6f9f / glhs xG' 5 . -v_ cªu\\ h]| L j0fd{ fnfsf] 7'nf] cIf/ V nfO{ laGb' /v] f YY' df Y' k6o\\ fpb“ f l7s bO' { a/fa/ efudf k6l\\ 6G5 . oxf“ kT| os] efu Pscsfs{ f] kl| tlaDa jf k/fjtg{ xf] . -u_ lqeh' PQR nfO{ laGb' /v] f YY' df k/fjtg{ u/fpb“ f ΔP'Q'R' ags] f] 5 . oxf“ Y PP', QQ' / RR' /v] f k/fjtg{ cIf YY' df nDa 5g\\ . To:t} PA = AP', QB = BQ' / RC = CR' klg xG' 5 . oxf“ PQR / kl| tlaDa P'Q'R' cg¿' k 5g\\ . dflysf kT| os] ljm| ofsnfkdf kT| os] cfsl[ t / kl| tlaDa k/fjtg{ cIfaf6 Y' a/fa/ b/' Ldf k/s] f 5g\\ . s;/L < dflysf 3 cf6] f ljm| ofsnfksf cfwf/df k/fjtg{ sf sx] L tYox¿ kQf nufpg] sfl] ;; u/f“} . 1. s'g} klg j:t' jf lrqsf] k|ltlaDanfO{ PA P' k/fjtg{ elgG5 . dfly lrqx¿df jm| dzM s6] L, V sf] cfwf efu / P'Q'R' ;a} laGb' /v] f YY' df k/fjtg{ ePsf 5g\\ . 2. h'g /]vfdf k|ltlaDa ag]sf] 5 To;nfO{ Q Q' k/fjtg{ cIf (axis of reflection) elgG5 . B R' lrqx¿df laGb' /v] f YY' k/fjtg{ sf] cIf xf] . RC 3= jf:tljs cfsl[ t k/fjtg{ eO{ kl| tlaDa aG5 . $= jf:tljs cfsl[ t / kl| tlaDacg¿' k xG' 5g\\ . cyft{ \\ jf:tljs cfsl[ t / kl| tlaDasf] Ifq] kmn klg cfk;df a/fa/ xG' 5g\\ . 5. sg' } klg HofldtLo lrq jf cfsl[ tnfO{ k/fjtg{ ubf{ cfsl[ t / kl| tlaDa k/fjtg{ cIfaf6 a/fa/ b/' Ldf k5g{ \\ . ul0ft, sIff – & 65

cEof; 8.1 1. tnsf HofldtLo lrqx¿nfO{ k/fjtg{ cIf XX' laGb/' v] fdf k/fjtg{ ubf{ aGg] kl| tlaDa nv] M -s_ laGb' A -v_ laGb' P -u_ /v] f PQ -3_ ∠PQR -ª_ laGb' E -r_ ∠CGE -5_ /v] f IH -h_ laGb' l -em_ ∠KLM -`_ ∠LKM X X' 2. dfly kZ| g g=+ -s_ bl] v -`_ ;Dd lbP h:t} c¿ ;d:of agfO{ ;fyL;u“ 5nkmn u/L lzIfssf] ;xofu] df cfsl[ tsf] kl| tlaDa lrGg] vn] vn] . 3. tnsf kT| os] cfsl[ tnfO{ k/fjtg{ sf] cIf;u“ k/fjtg{ ubf{ aGg] kl| tlaDa lvr/] bv] fpm M -s_ -v_ -u_ KM PQ L N -h_ dfly lbP h:t} u/L sg' } 5 cf6] f km/s km/s cfsl[ tx¿ / k/fjtg{ cIf agfO{ ;d:of agfpm . ;fyL;u“ 5nkmn u/L cfk;df k/fjtg{ u/L kl| tlaDa lvr/] bv] fpm . gf6] M kT| os] cfsl[ tsf zLifl{ aGbx' ¿bl] v k/fjtg{ sf] cIfdf nDa lvr/] -;6] :Sjfo/n] jf sDkf;n_] kl| tlaDa agfpm . 66 ul0ft, sIff – &

4. uf| km kk] /df sg' } /v] f AB / MN nfO{ k/fjtg{ sf cIfx¿ dfgL tnsf cfsl[ tsf kl| tlaDa agfpm M ul0ft, sIff – & 67

8.2 kl/jm| d0f (Rotation) tnsf ljm| ofsnfkx¿df 5nkmn u/ M 1. kl/jm| d0f -Rotation_ -s_ wf/f vfN] bf / aGb ubf{ s] ugk{' 5{ < -v_ bft“ df‰g] dGhgsf] lasf{] vfN] bf / aGb ubf{ s] ul/G5 < -u_ ;fr“ fn] ] tfNrf vfN] bf / aGb ubf{ s] ul/G5 < -3_ 9fs] f vfN] bf / aGb ubf{ s] ugk{' 5{ < -ª_ 38L ldnfpgsf nflu ;O' x{ ¿nfO{ s] ugk{' 5{ < plNnlvt sfo{x¿ ubf{ j:t'nfO{ lglZrt laGb'df 3'dfpg] sfo{ ul/G5 . of] 3'dfpg] sfd xfl] ;of/Lsf ;fy cfjZos dfqfdf ugk{' 5{ . o;/L j:tx' ¿sf] 3d' fOsf] kl| jm| ofnfO{ g} kl/jm| d0f elgG5 . dflysf 5nkmnsf cfwf/df kl/jm| d0fsf] kl/efiff nv] . ;fyL;u“ 5nkmn u/L lgisif{ lgsfn . lglZrt laGbd' f j:tx' ¿sf] 3d' fOnfO{ kl/jm| d0f elgG5 . 2. kl/jm| d0fsf ks| f/ -s_ lbOPsf 38Lsf lrqx¿ x/] / 5nkmn u/ M  bf;] f| ] 38Ldf slt ahs] f] 5 <  o;nfO{ 38Lsf] lbzfdf 15 ldg6] kl/jm| d0f ubf{ slt xG' 5 <  ca bf;] f| ] 38LnfO{ lrq g+= 1 lrq g+= 2 lrq g+= 3 38Lsf] ljk/Lt lbzfdf 15 ldg6] n] kl/jm| d0f ubf{ slt xG' 5 < dfly bf;] f| ] 38Ldf 2.00 ahs] f] 5 . o;nfO{ 38Lsf] lbzfdf 15 ldg6] -90°_ df kl/jm| d0f ubf{ 2.15 ahs] f] 5 . o:tf] 38Lsf] ;O' s{ f] rfncg;' f/sf] kl/jm| d0fnfO{ C0ffTds kl/jm| d0f -negative rotation_ elgG5 . To;} u/L olb bf;] f| ] 38LnfO{ 38Lsf] ljk/Lt lbzfdf 15 ldg6] -90°_ df kl/jm| d0f ul/of] eg] 1.45 ahs] f] xG' 5 . o:tf] kl/jm| d0fnfO{ wgfTds kl/jm| d0f (positive rotation) elgG5 . 68 ul0ft, sIff – &

-v_ HofldtLo lrqsf] kl/jm| d0f lrq g=+ -4_ df /v] f AB nfO{ laGb' O sf] 38Lsf] ;O' s{ f] ljk/Lt lbzf jf wgfTds lbzfdf 90° sf] kl/jm| d0f ubf{ ags] f] kl| tlaDa A' B' xf] . To:t} /v] f AB nfO{ laGb' O sf] 38Lsf] lbzf jf C0ffTds lbzfdf 90° sf] kl/jm| d0f ubf{ aGg] kl| tlaDa A\" B\" xf] . lrq g+= 4 3. kl/jm| d0fsf tYox¿ dflysf 5nkmnx¿af6 kl/jm| d0fsf af/d] f s] s] tYox¿ kQf nufpg ;S5f} < nv] . tL tYox¿nfO{ ;fyL;u“ 5nkmn u/ . tn lbOPsf tYox¿;u“ tn' gf u/L x/] .  sg' } klg HofldtLo cfsl[ tnfO{ lbOPsf] sf0] f / lbzfdf lbOPsf] laGbs' f] jl/kl/ 3d' fP/ :yfgfGt/0f ugn{' fO{ kl/jm| d0f -rotation_ elgG5 .  38Lsf] ;O' s{ f] lbzfdf ePsf] kl/jm| d0fnfO{ C0ffTds kl/jm| d0f -negative rotation_ elgG5 .  38Lsf] ;O' s{ f] ljk/Lt lbzfdf ePsf] kl/jm| d0fnfO{ wgfTds kl/jm| d0f -positive rotation_ elgG5 .  kl/jm| d0fn] ;dtn ;txdf /xs] f HofldtLo cfsl[ tx¿nfO{ sg' } laGba' f6 Pp6} lbzf / plTts} sf0] fdf :yfgfGt/0f ub5{ . pbfx/0f 1 tn lbOPsf] lrq ABC nfO{ laGb' O sf] jl/kl/ 60° sf] wgfTds lbzfdf kl/jm| d0f ubf{ aGg] kl| tlaDa lvr/] bv] fpm . ;dfwfg tl/sf A 1. A / O hf8] f“} . O nfO{ kl/jm| d0f laGb' dfg/] OA cwJ{ of; lnP/ laGb' A nfO{ 60° sf] wgfTds lbzf -38Lsf] ljk/Lt lbzf_ df kl/jm| d0f O C u/L A sf] kl| tlaDa A' kQf nufcf“} . B ul0ft, sIff – & 69

2. B / O hf8] f“} . O nfO{ sG] b| dfg/] OB cwJ{ of; lnP/ laGb' B nfO{ 60° sf] wgfTds lbzfdf kl/jm| d0f u/L B sf] kl| tlaDa B' kQf nufcf}“ . 3. C / O hf8] f“} . O nfO{ sG] b| dfg/] OC cwJ{ of; lnP/ laGb' C nfO{ 60° wgfTds lbzfdf kl/jm| d0f u/L C sf] kl| tlaDa C' kQf nufcf“} . 4. ?n/sf] ;xfotfn] A'B'C' jm| d;} u“ hf8] f“} . ca aGg] lrq ΔA'B'C' lbOPsf] ΔABC sf] cfjZos wgfTds kl/jm| d0f xf] . pbfx/0f 2 lbOPsf] lrqdf ;dfgfGt/ rte' h{' OACB nfO{ sG] b| laGb' O sf] jl/kl/ 180° sf] C0ffTds lbzfdf kl/j|md0f u/ . o;/L aGg] kl| tlaDasf] lrq lvr/] bv] fpm . ;dfwfg tl/sf 1. laGb' A nfO{ OA cwJ{ of; lnP/ 180° sf] C0ffTds lbzfdf O af6 kl/jm| d0f u/L A' kQf nufcf“} . 2. To:t} u/L jm| dzM laGbx' ¿ B / C nfO{ klg jm| dzM OB / OC cwJ{ of; lnP/ 180° sf] C0ffTds lbzfdf kl/jm| d0f u/L kl| tlaDax¿ B' / C' kQf nufcf“} . 3= ca jm| dzM O / A', O / B', B' / C' tyf C' / O hf8] f“} . o;/L aGg] ;dfgfGt/ rte' h{' OA'C'B' lbOPsf] ;dfgfGt/ rte' h{' OABC sf] cfjZos C0ffTds kl/jm| d0fsf] kl| tlaDa xf] . gf6] M s] 180° sf] C0ffTds / wgfTds bj' } kl/jm| d0fn] Pp6} kl| tlaDa bn] fg\\ < 5nkmn u/ . cEof; 8.2 1. tnsf ;dodf 38Lsf] ldg6] ;O' n{ ] slt k6s kl/jm| d0f u5{ < kQf nufpm M -s_ 1 306f -v_ 1/2 306f -u_ 15 ldg6] -3_ 20 ldg6] -ª_ 45 ldg6] 70 ul0ft, sIff – &

2. 38Lsf ;O' x{ ¿n] lgDgfg;' f/sf kl/jm| d0f ubf{ slt ;do 36L jf a9L xG' 5 < -s_ ldg6] ;O' s{ f] wgfTds lbzfdf Ps rfy} fO kmGsf (90°) sf] kl/jm| d0f -v_ ldg6] ;O' s{ f] C0ffTds lbzfdf bO' { rfy} fO jf cfwf kmGsf (180°) sf] kl/jm| d0f -u_ ldg6] ;O' s{ f] wgfTds lbzfdf Ps k/\" f kmGsf (360°) sf] kl/jm| d0f -3_ 306f ;O' s{ f] C0ffTds lbzfdf Ps rfy} fO (90°) sf] kl/jm| d0f -ª_ ;s] G] 8 ;O' s{ f] wgfTds lbzfdf tLg rfy} fO (270°) sf] kl/jm| d0f 3. tnsf HofldtLo cfsf/x¿nfO{ lbOPsf kl/jm| d0fsf] laGb' O, lbzf / sf0] fdf kl/jm| d0f u/ . kl/jm| d0fsf] tl/sf klg nv] . o;/L aGg] kl| tlaDasf] lrq lvr/] bv] fpm . -s_ -v_ -u_ -3_ AC F 0 H O BO 0 D EG laGb' O df 90° laGb' O df 270° sf] laGb' O df 30° sf] laGb' O df 180° wgfTds lbzfdf C0ffTds lbzfdf C0ffTds lbzfdf sf] C0ffTds lbzfdf gf6] M 270° sf] C0ffTds kl/jm| d0f = 90° sf] wgfTds kl/jm| d0f xG' 5 < s;/L < lzIfs;u“ k/fdz{ u/L kQf nufpm . 4. lbOPsf] lrqnfO{ C0ffTds lbzfdf 90° / 180° df kl/jm| d0f ubf{ aGg] lrq lvr/] bv] fpm . kl/jm| d0f ubf{ sg' sg' kl| jm| of ckgfof} < ab“' fut ¿kdf nv] . ul0ft, sIff – & 71

8.3 lj:yfkg (Displacement) 1. lj:yfkgsf] kl/ro tnsf ljm| ofsnfk cWoog u/ / 5nkmn u/ M -s_ Pp6f k:' tsnfO{ 7n' f] ;t] f] sfuhdfly /fvf“} . To;af6 6;«] u/L Pp6f PQRS rte' h'{ agfcf“} . o; k:' tsnfO{ l;wf /v] fdf cln k/ ;f/f“} . To;af6 klg 6;«] u/L csf{] rte' h{' P'Q'R'S' agfpm . ca PP',QQ',RR' / SS' sf] ;DaGw s] xfn] f 5nkmn u/ . oxf“ k:' ts PQRS nfO{ P'Q'R'S' df lj:yfkg ul/Psf] elgG5 . s] cfsl[ t / kl| tlaDa cg¿' k 5g\\ < ca dflysf] ljm| ofsnfksf cfwf/df lj:yfkgsf] kl/efiff nv] . cfkm\" n] nv] s] f] kl/efiffnfO{ ;fyL;u“ tn' gf u/L x/] / 5nkmn u/ . ca lgisifn{ fO{ tnsf tYox¿;u“ tn' gf u/L x/] . 1. ;dtn ;txdf /xs] f HofldtLo cfsl[ tsf x/s] laGbn' fO{ plTts} b/' L / pxL lbzfdf :yfGt/0f ugn{' fO{ lj:yfkg -displacement_ elgG5 . 2. lj:yfkgnfO{ kl/eflift ugs{ f nflu lj:yfkgsf] kl/df0f jf gfk / lbzf pNnv] ugk{' b5{ . 3. lj:yfkgsf cfsl[ t / kl| tlaDacg¿' k xG' 5g\\ . 4. sg' } klg laGbn' fO{ lj:yfkg ubf{ lbOPsf] kl/df0f / lbzfdf ;dfgfGt/ /v] f lvRgk' 5{ . 5nkmn u/ -s_ Pp6f df6] / 15 ld6/ ;fe] m} cufl8 a9fof] eg] s] To;sf ;a} laGbx' ¿ a/fa/ b/' L / pxL lbzfdf :yfgfGt/0f xG' 5g\\ < s] of] lj:yfkg xf] < -v_ eO' d“ f /fd/| L ldnfP/ la5o\\ fOPsf] sDdn jf 7n' f] sfk6{] sf] sg' } Pp6f dfq 6K' kf] ;dft/] 1 ld6/ cfkm\" lt/ tfGbf s] afs“ L ;a} 6K' kfx¿ pxL lbzf / kl/df0fdf :yfgfGt/0f xfn] fg\\ < 72 ul0ft, sIff – &

pbfx/0f 1 O ;u“ s} f] /v] fv08 AB nfO{ ls/0f /v] f (ray) ON A sf] kl/df0f / lbzfdf lj:yflkt u/ M N ;dfwfg tl/sf B 1. laGb' A af6 ON sf] lbzf / kl/df0f;u“ a/fa/ / O B ;dfgfGt/ xg' ] u/L AA' lvr . A 2. laGb' B af6 ON sf] lbzf / kl/df0f;u“ a/fa/ / ;dfgfGt/ xg' ] u/L BB' lvr . 3. A' / B' hf8] . N o;/L /v] fv08 A'B' g} /v] fv08 AB sf] ON df lbzf / kl/df0fdf lj:yflkt kl| tlaDa xf] . A' pbfx/0f 2 B' P lbOPsf] lrqdf ΔPQR nfO{ lbPsf] ls/0f /v] f OM sf] lbzf / kl/df0fdf lj:yfkg u/ . ;dfwfg tl/sf 1. P af6 OM ;u“ a/fa/ / ;dfgfGt/ xg' ] u/L OM sf] lbzfdf M O R PP' lvr . Q R 2. Q af6 OM ;u“ a/fa/ / ;dfgfGt/ xg' ] u/L OM sf] P lbzfdf QQ' lvr . 3. R af6 OM ;u“ a/fa/ / ;dfgfGt/ xg' ] u/L OM sf] P' lbzfdf RR' lvr . $= ca P', Q' / R' nfO{ jm| d;} u“ hf8] . → O M o;/L ags] f] ΔP'Q'R' g} ΔPQR sf] OM sf] lbzf / Q' Q kl/df0fdf lj:yflkt kl| tlaDa xf] . R' gf6] M ls/0f /v] fsf] gfk lnb“ f bO' c{ f6] f laGbx' ¿sf] larsf] dfq nDafOsf] gfk lngk' 5{ . ;dfgfGt/ /v] f lvRbf ;6] :Sjfo/sf] ko| fu] ugk{' 5{ . ul0ft, sIff – & 73

cEof; 8.3 M 1. lj:yfkgsf] pbfx/0f;lxt kl/ro bp] m . 2. lj:yfkgdf sg' sg' tYox¿ xG' 5g\\ < lrq;lxt nv] . 3. lj:yfkgsf sg' } 3 cf6] f tYox¿nfO{ pbfx/0f / lrq;lxt pNnv] u/ . 4. /v] f XY nfO{ OM sf] gfk / lbzfdf lj:yfkg u//] bv] fpm . XOY 5. Pp6f kl] G;nnfO{ Goh' lkG| 6 jf 8O« ª kk] /sf] lar efudf /fvL 6;]« u/ . To;nfO{ rf/} lbzfdf 5cm sf] kl/df0fdf jm| dzM lj:yfkg u/L cfsl[ tx¿ agfpm . pSt cfsl[ tx¿nfO{ sIffdf 5nkmn u/L ;hfP/ /fv . 6. ;fyLn] lbPsf] lbzf / kl/df0fdf Ps Pscf6] f lrq jf j:tx' ¿nfO{ lj:yfkg u/ . 4. tnsf kT| os] HofldtLo lrqx¿nfO{ lbzfdf lj:yfkg ubf{ aGg] kl| tlaDa lvr/] bv] fpm . -s_ -v_ F A m B DE m -u_ A m m B D 5. dfly kZ| g g=+ 4 dCf lbOP h:t} 5 cf6] f ;d:of agfpm / lj:yfkg u/L kl| tlaDa klg agfpm . cfkm\\ gf kT| os] ;d:of ;fyL;u“ cfk;df ;dfwfg u/ . ;fyLsf] kl| tlaDa / lj:yfkg tl/sfnfO{ ltdf| ;] u“ tn' gf u/L x/] . 74 ul0ft, sIff – &

PsfO 9 ;dldlt / 6;] n;] g (Symmetry and Tessellation) 9.1 /v] f / laGb' ;dldlt (Line and Point Symmetry) tnsf laGb' ;dldltx¿ cWoog u/L 5nkmn u/ M -s_ tnsf lrqx¿ / ljm| ofsnfksf cfwf/df laGb' ;dldlt kQf nufpm . lrq g=+ 9.1 lrq g=+ 9.2 lrq g+= 9.3 lrq g+= 9.4 1. lrq 9.1 sf] cfsl[ tnfO{ Pp6f kftnf] sfuhdf 6;«] u/ . o;/L 6;«] ul/Psf] lrq, lrq 9.1 ;u“ ;jfª{ u\\ ;d cyft{ \\ cg¿' k xG' 5 . 2. o;nfO{ lrq 9.1 df l7s ldNg] u/L jT[ tsf] sG] b| O df kl] G;nsf] 6K' kfn] ] lyr . 3. ca dflysf] lrqnfO{ 3d' fpb“ } hfb“ f lrq 9.1 df k/' } gvK6;] Dd 3d' fpb“ } hfpm . 4. ca Ps kmGsf] 3d' fpb“ f dflysf] lrqsf] slt cz+ 3d' fof} xfn] f < laGb' A sxf“ kU' of] xfn] f < 5. ca o; l:yltaf6 km] l/ gvK6;] Dd 3d' fpm . o; k6s Ps kmGsfs] f] slt k6s 3d' fof} xfn] f . lrq g=+ 9.1 df 6;«] ul/Psf] ;jfª{ u\\ ;d cfsl[ tnfO{ laGb' O df k/\" f Ps kmGsf] 3d' fpb“ f 2 k6s vlK6of] . o:tf lrqdf >0] fL 2 (order -2) sf] laGb' jf kl/jm| lds ;dldlt ePsf] dflgG5 . To;n} ] lrq 9.1 sf] laGb' ;dldltsf] >0] fL 2 eof] . -lrq g=+ 9.2 df 6;]« ul/Psf] ;jfª{ u\\ ;d cfsl[ tnfO{ laGb' O df k/\" f Ps kmGsf] 3d' fpb“ f 3 k6s vlK6of] . To;n} ] o; lrqsf] laGb' ;dldltsf] >0] fL 3 eof] ._ lrq 9.2 sf] ljm| ofsnfk 1. lrq g=+ 9.2 nfO{ kftnf] sfuhdf 6;«] u/ . 2. o;nfO{ lrq g=+ 9.2 sf] O laGbd' f sG] b| kg{] u/L Ps k/\" f kmGsf] 3d' fpm . 3. o;/L 3d' fpb“ f slt k6s lrq vlK6of] < nv] . 4. ca lrq g=+ 9.2 sf] laGb' ;dldltsf] >0] fL kQf nufpm . ul0ft, sIff – & 75

lrq 9.3 sf] ljm| ofsnfk 1. lrq g=+ 9.3 nfO{ kftnf] sfuhdf 6;«] u/ . 2. o;nfO{ lrq g=+ 9.3 dfly /fvL k/\" f Ps kmGsf] 3d' fpb“ f slt k6scg¿' k cfsl[ t vlK6G5 < x/] / nv] . 3. o;sf] laGb' ;dldltsf] >0] fL kQf nufpm . 4. s] ltdf| ] >0] fL 4 cfof,] s;/L < lrq 9.4 sf] ljm| ofnfk 1. lrq g=+ 9.4 nfO{ klg kftnf] sfuhdf 6;«] u/ . 2. o;nfO{ jT[ t ABC dfly /fv/] k/\" f Ps kmGsf] 3d' fpm . 3. sltk6s cfsl[ t vlK6of] < nv] . laGb' O df lrq g=+ 9.4 df 6;«] ul/Psf] ;jfª{ ;\\ d cfsl[ t laGb' O df 3d' fpb“ f cfsl[ t Ps k6s klg vlK6Pg . To;n} ] of] cfsl[ ts laGb' ;dldlt xb“' g} . lrq 9.4 sf] ljm| ofsnfkdf laGb' ;dldlt xb“' g} . 1. sg' } klg cfsl[ tsf] cg¿' k cfsl[ tnfO{ sg' } lglZrt laGbd' f k/\" f Ps kmGsf] 360 l8uL| kl/jm| d0f ubf{ vlK6g] cj:yf cfpgn' fO{ laGb' ;dldlt ePsf] elgG5 . dflysf lrqx¿dWo] lrq g=+ 9.1, lrq g=+ 9.2 / lrq g=+ 9.3 sf cj:yfx¿df laGb' ;dldlt xG' 5 . t/ lrq g=+ 9.4 df laGb' ;dldlt xb“' g} . k/\" f Ps kmGsf] 3d' fpb“ f lbPsf] cfsl[ t hlt k6s vlK6G5 To;nfO{ cfsl[ tsf] >0] fL elgG5 . 2. /v] f ;dldlt (Line of Symmetry) -s_ tnsf lrq / ljm| ofsnfksf cfwf/df 5nkmn u/L /v] Lo ;dldlt kQf nufpg] ko| f; u/f“} . lrq g=+ 9.5 lrq g=+ 9.6 lrq g+= 9.7 dflysf kT| os] lrqdf 86 nfOgn] lrqnfO{ l7s 2 efudf afl“ 8Psf] 5 . To;n} ] dflysf kT| os] lrq ;dldlt xg' ] vfnsf (symmetrical) 5g\\ . kT| os] lrqnfO{ bO' { efudf af8“ g\\ ] 86 /v] f (dot line) nfO{ /v] f jf /v] Lo ;dldlt (line of symmetry) elgG5 . /v] Lo ;dldltnfO{ ;dldltsf] cIf (axis of symmetry) jf Pg] f /v] f (mirror line) klg eGg] ul/G5 . 76 ul0ft, sIff – &

ca tnsf kZ| gx¿df 5nkmn u/f“} M 1. dflysf] lrq g=+ 9.5, lrq g=+ 9.6 / lrq g=+ 9.7 df slt sltcf6] f /v] Lo ;dldltsf cIfx¿ 5g\\ < 2. lrq g=+ 9.5 sf] kl/jm| lds ;dldlt slt >0] fLsf] xfn] f < 3. To:t} lrq 9.6 / 9.7 sf] kl/jm| lds ;dldlt slt slt >0] fLsf 5g\\ < dflysf lrqx¿dWo] lrq g=+ 9.5 / 9.6 df /v] Lo ;dldltsf] cIf Pp6f 5 . To:t} lrq g=+ 9.7 df ;dldltsf cIf 2 cf6] f 5g\\ . lrq g=+ 9.6 nfO{ 86 /v] fdf 3d' fpb“ f Ps k6s dfq lrq vlK6G5 . To;n} ] lrq 9.6 sf] /v] Lo ;dldltsf] kl/jm| lds >0] fL (order) 1 xG' 5 . To:t} lrq g=+ 9.7 sf] kl/jm| ldssf] /v] Lo ;dldlt >0] fL 2 xG' 5 . s;/L < -v_ ca tnsf bO' c{ f6] f lrqsf cfwf/df tnsf ljm| ofsnfkdf 5nkmn u/f“} . lrq g+= 9.8 lrq=+ 9.9  lrq 9.8 sf] /v] Lo ;dldltsf] kl/jm| d0f >0] fL slt xfn] f <  lrq 9.8 sf] /v] Lo ;dldltsf] cIf slt cf6] f 5g\\ <  xf] dflysf] lrq g=+ 9.8 df /v] Lo ;dldlt / /v] Lo ;dldltsf cIfx¿ 2 cf6] f 5g\\ . o;sf] kl/jm| lds ;dldltsf] >0] fL 4 xG' 5 < s;/L < lrq 9.8 sf] ljm| ofsnfk  ;jk{ y| d lrq 9.8 nfO{ cEof; kl' :tsfdf 6l«] ;ª u/ .  lrq 9.9 df k/\" f slt efudWo] slt efunfO{ 5fof kfl/Psf] 5 <  kl/jm| lds ;dldltsf] >0] fL slt xfn] f <  sltcf6] f /v] Lo ;dldltsf cIfx¿ 5g\\ .  o;nfO{ sg' } csf{] efudf 5fof“ kf¥of] eg] 2 cf6] f /v] Lo ;dldltsf] cIf / kl/jm| lds ;dldltsf] >0] fL 2 ePsf] gof“ cfsl[ t aG5 <  o;/L ags] f] lrqdf Pp6} dfq ;dldltsf] cIf / kl/jm| lds ;dldltsf] >0] fL 36g\\ ] u/L csf{] t;] f| ] efunfO{ 5fof kf/ . s] ltdLn] kl/jm| lds ;dldltsf] >0] fL 2 xg' ] u/L rfy} f] efudf 5fof kfg{ ;S5f} < ul0ft, sIff – & 77

 o;sf] /v] Lo ;dldltsf] cIfx¿ sltcf6] f 5g\\ < dflysf ljm| ofsnfkx¿af6 s] lgisif{ lgsfNg ;S5f} < ;fyL;u“ 5nkmn u/L nv] . s] ltdf| ] lgisif{ tnsf] lgisif;{ u“ ldN5 . tn' gf u/L x/] . HofldtLo cfsl[ tx¿df /v] Lo ;dldltsf] kl/jm| lds >0] fL 1 eGbf sd xg' ;Sbg} . lrq / /v] Lo ;dldltsf] cIfcg;' f/ o;sf] >0] fLdf km/s kb5{ . pbfx/0f 1 lrqdf laGb/' v] f (dot line) sf] ko| fu] u/L /v] Lo ;dldlt agfpm . kT| os] lrqdf slt >0] fL eof] < nv] . ;dfwfg M Pp6f /v] Lo ;dldltsf] cIf agfpb“ f, oxf“ ;dldltsf] >0] fL 2 5 . pbfx/0f 2 tn /v] f ;dldltsf] cIf / cfwf lrq lbOPsf] 5 . o;nfO{ k/\" f u/ . /v] Lo ;dldltsf] kl/jm| lds >0] fL klg kQf nufpm . ;dfwfg o;sf] kl/jm| lds /v] Lo ;dldltsf] >0] fL 4 5 . pbfx/0f 3 lbOPsf] lrqnfO{ laGb' 0 df kl/jm| d0f u/ . ca laGb' ;dldltsf cfwf/df O kl/jm| lds >0] fL kQf nufpm . ;dfwfg M kZ| gsf] lrqnfO{ laGb' O df Ps k/\" f kl/jm| d0f ubf{ 3 k6s vlK6G5 . To;n} ] o; laGb' ;dldltsf] kl/jm| d0f >0] fL 3 xG' 5 . cEof; 9.1 1. tnsf lrqx¿ cEof; kl' :tsfdf ;f/ . kT| os] lrqnfO{ 6;]« u/L O laGbd' f kl/jm| d0f u/L x/] / kl/jm| lds >0] fL kQf nufpm . -s_ -v_ -u_ O OO 78 ul0ft, sIff – &

-3_ -ª_ -r_ -5_ O O OO -h_ -em_ -`_ O O O 2. tnsf kT| os] laGb/' v] fnfO{ 6l«] ;ª u/ . /v] Lo ;dldlt lvr . /v] Lo ;dldltsf] kl/jm| d0f >0] fL slt eof] nv] . -s_ -v_ -u_ -3_ -ª_ -r_ 3. tn lrqdf /v] Lo ;dldltsf] cIf / cfwf lrq lbOPsf] 5 . lrq k/\" f u/ / kl/jm| lds >0] fL klg kQf nufpm . -s_ -v_ -u_ ul0ft, sIff – & 79

9.2 axe' h' sf 9fr“ fx¿af6 6;] n] ;] g (Tessellation by using Polygons) 1. 6;] n] ;] gsf] wf/0ff tnsf kZ| g / ljm| ofsnfkdf 5nkmn u/ . -s_ ltdLn] O6“ f jf 9ª' u\\ fx¿ 5fks] f bv] s] f 5f} < ltdf| ] 3/ / ljBfnodf o:tf] sxf“ sxf“ 5g\\ < -v_ O6“ fx¿nfO{ s;/L ldnfP/ /flvPsf] 5 < -u_ O6“ fx¿ sg' sg' cfsf/sf 5g\\ < -3_ s] oL O6“ fx¿nfO{ csf{] tl/sfn] klg 5fKg ;lsGYof] < -ª_ ltdf| ] 3/df ePsf] 8fs] f,] af;“ jf Knfl:6ssf] 8fs] f] tyf sndbfgLdf kT| os] kftfx¿ s;/L /fvs] f 5g\\ < -r_ o:tf cGo pbfx/0fx¿ vfh] L u/ . -h:t} M sfk6{] , sldh, st' f{ ;?' jfn, tGgf cflb_ -5_ tnsf lrqdf slt slt cf6] f p:t} HofldtLo cfsl[ tx¿ ko| fu] ePsf 5g\\ < 5nkmn u/ . -h_ df/} Ln] cfkm\\ gf] 3f/ s;/L agfPsf] xG' 5 < lrq agfP/ nv] . – dflysf kT| os] lrqx¿df ;txx¿nfO{ k/\" f ug{ PseGbf a9L p:t} ks| f/sf HofldtLo cfsl[ tx¿ ko| fu] ePsf 5g\\ . – dflysf ;a} pbfx/0fx¿ 6;] n] ;] gsf pbfx/0fx¿ xg' \\ . -em_ ;dafx' lqeh' , ju,{ cfot / lgoldt axe' h' af6 Ps Ps cf6] f 6;] n] ;] g agfO{ bv] fpm . -`_ ca dflysf pbfx/0fx¿ / lj|mofsnfkx¿sf cfwf/df 6];]n];gsf] cy{ n]v . cfk\\mgf] kl/efiffnfO{ ;fyLsf] kl/efiff;u“ tn' gf u/L x/] . lgisifn{ fO{ sIffdf 5nkmn u/ . s] ltdf| ] lgisif{ tnsf] egfO;u“ ldN5 < tn' gf u/L x/] . PseGbf a9L p:t} ks| f/sf HofldtLo cfsl[ tsf 6fon jf lrqx¿ gvK6fOsg / vfnL 7fp“ g/fvLsg ;dtn ;tx 9fSg] jf 5fK] g] kl| jm| ofnfO{ 6;] n] ;] g jf 6folnª (tessellation or tilling) elgG5 . 80 ul0ft, sIff – &

2. 6;] n] ;] gsf ks| f/ tnsf lrqx¿ cWoog u/L 5nkmn u/ M lrq g=+ 9.11 lrq g=+ 9.12 lrq g=+ 9.13 -s_ lrq g=+ 9.11 df slt ks| f/sf / s:tf HofldtLo lrq ko| fu] ePsf 5g\\ < -v_ s] lrq g=+ 9.12 df ko| fu] ePsf bj' } HofldtLo lrqx¿ lgoldt (regular) 5g\\ < -u_ lrq g=+ 9.13 df ko| fu] ePsf HofldtLo lrq lgoldt jf clgoldt s] xg' \\ < dflysf lrqdf lrq g=+ 9.11 sf] lrq lgoldt 6;] n] ;] g (regular tessellation) xf] . lrq g=+ 9.12 sf] cw{ jf lgoldt 6;] n] ;] g (semiregular tessellation) xf] . To:t} lrq g=+ 9.13 sf] lrq clgoldt 6;] n] g] (irregular tesselation) xf] . ca dflysf 5nkmnsf cfwf/df lgoldt, cw{ lgoldt / clgoldt 6;] n] ;] gsf] kl/efiff nv] . cfkm\\ gf] kl/efiffnO{ ;fyLsf] kl/efiff;u“ tn' gf u/L lgisifn{ fO{ sIffdf 5nkmn u/ . s] ltdf| ] lgisif{ tnsf] egfO;u“ ldN5 < tn' gf u/ . 1. 6;] n] ;] g lgoldt, cw{ lgoldt / clgoldt u/L 3 lsl;dsf xG' 5g\\ . 3. Ps} ks| f/sf lgoldt HofldtLo cfsl[ t ko| fu] eP/ ags] f 6;] n] ;] gnfO{ lgoldlt 6;] n] ;] g (reguar tessellation) elgG5 . 3. bO' { jf bO' e{ Gbf a9L ks| f/sf lgoldt HofldtLo cfsl[ t ko| fu] u/L ags] f 6;] n] ;] gnfO{ cw{ lgoldt 6;] n] ;] g (semiregular tessellation) elgG5 . 4. clgoldt HofldtLo cfsl[ tx¿ ko| fu] u/L ags] f 6;] n] ;] gnfO{ clgoldt 6;] n] ;] g (irregular or non-ragular tessellation) elgG5 . ul0ft, sIff – & 81

3. axe' h' sf 9fr“ fx¿ -s_ tnsf axe' h' sf 9fr“ f cWoog u/L 5nkmn u/ M jm| =;= Hofldlt cfsl[ t÷9fr“ f 1. ;dafx' lqeh' sf] 9fr“ f 2. jus{ f] 9fr“ f 11111111111111122222222222222233333333333333311444444414444411141442252552555255555525552366636636663663636366647777777447744777774475888888585885888855885699699696999999996699677007700700770000000008888881111118111111111222222222222222333333333333333444444444444444555555555555555 3. lgoldt k~reh' sf] 9fr“ f 111111111111111112222222222222222233333333333333333444444444444444441151155551515115555555155562626626226662626226666666737773373337737373777777778884448884448488488488888895995959595995599599999959000060606006060060006606007777777111171111111111111788288882288222222222222282333333333333333334444444444444444455555555555555555666666666666666667777777777777777788888888888888888 11111111111111222222222222223333333333333344444444444444515515511555111551555566222666226226626666667777377337373337773777848448844848888848488899559995555999995999596006600060606600000600171717711717117111171122222222222222333333333333334444444444444455555555555555 4. if8e\\ h' sf] 9fr“ f 11111111111111122222222222222233333333333333311111114444444444444442222222555555555555555333333366666666666666644444447777777777777775555555888888888888888666666699999999999999977777770000000000000008888888111111111111111222222222222222333333333333333444444444444444 -v_ dflysf kT| os] 6;] n] ;] gnfO{ 8O« ª kk] /df agfP/ sIffsf7] fdf ;hfP/ /fvL 5nkmn u/ . cEof; 9.2 1. tnsf 6;] n] ;] gsf 9fr“ fx¿nfO{ 6l«] ;ª u//] uf| km kk] /df ;f//] k/\" f u/ M -s_ -v_ -u_ -3_ 82 ul0ft, sIff – &

2. tnsf kT| os] 6;] n] ;] gdf sg' sg' HofldtLo cfsl[ t ko| fu] ePsf 5g,\\ nv] M -s_ -v_ -u_ -3_ -ª_ 3. kZ| g g=+ 2 df ko| fu] ePsf jf cGo 5/5 cf6] f 6;] n] ;] g agfpm÷vfh] . kT| os] 6;] n] ;] gnfO{ sfuhdf 6;«] u/ / cfkm\" nfO{ dg kg{] /ª nufP/ bv] fpm . 4. kZ| g g=+ 3 sf 6;] n] ;] gx¿dWo] lgoldt, cw{ lgoldt / clgoldlt 6;] n] ;] g 56' o\\ fP/ nv] . 5. 6;] n] ;] gsf] pbfx/0f;lxt kl/efiff nv] . 6. 6;] n] ;] g slt ks| f/sf xG' 5g\\ < kT| os] sf] pbfx/0f;lxt kl/efiff nv] . 7. sg' } 2/2 cf6] f lgoldt, cw{ lgoldt / clgoldt 6;] n] ;] gsf 9fr“ f ko| fu] u/L sfuh tyf sk8fdf 6;] n] ;] g u/L sIffdf kb| zg{ u/ . leTtfdf jf wfufn] l;lnªdf em' G8o\\ fP/ kb| zg{ klg u/ . ul0ft, sIff – & 83

PsfO 10 lbzfl:ylt / :sn] 8O« ª (Bearing and Scale Drawing) 10.1 lbzfl:ylt / gS;fsf] k9fO (Bearing and Map Reading) 1. lbzf l:ylt (Bearing) pQ/ klZrd (NW) pQ/ (N) pQ/ k\"j{ (NE) tnsf ljm| ofsnfkx¿ cWoog u/L 5nkmn u/ . klZrd (W) 45° -s_ lrqdf sltcf6] f lbzfx¿ k\"j{ (E) bv] fOPsf] 5 < ltgLx¿ s] s] xg' \\ < -v_ pTt/ / kj\" { lbzf bv] fpg] /v] fn] blIf0f klZrd (SW) blIf0f (S) blIf0f kj\" { (SE) slt l8uL| sf] sf0] f agfPsf] 5 < nv] . -u_ pQ/ / pQ/ kj\" { lbzf bv] fpg] /v] flar slt l8uL| sf] sf0] f 5 < gfk/] kQf nufpm . -3_ s] pQ/ / klZrd, klZrd / blIf0f tyf blIf0f / kj\" { bv] fpg] /v] fx¿lar klg 90° sf sf]0f ags] f 5g\\ < -ª_ s] pQ/ / pQ/ klZrd, klZrd / blIf0f klZrd tyf blIf0f / blIf0f kj\" { bv] fpg] ;a} /v] fx¿n] cfk;df 45°/45° sf sf0] f agfPsf 5g\\ . -r_ ljBfno sDkfpG8 jf rp/df uP/ dfly lrqdf bv] fPh:t} u/L kT| os] lbzfdf Ps Ps hgf ;fyLx¿ pleP/ lbzf l:ylt kQf nufpg] cEof; u/ . kT| os] ;fyLnfO{ lbzfl:yltsf] gfdn] afn] fpm . 2. gS;fsf] k9fO (Map Reading) www.dos.gov.np ul0ft, sIff – & 84

dflysf] gS;f / gS;fdf lbOPsf :yfgx¿ cWoog u/L 5nkmn u/ . -s_ sf7df8fn“} fO{ sG] b| dfgL tnsf :yfgx¿sf] lbzf l:ylt kQf nufpm M -c_ hgsk'/ -cf_ hD' nf -O_ gk] fnuGh -O_{ Onfd -p_ jL/uGh -p_ ;u/dfyf lxdfn -v_ kfv] /fnfO{ sG] b| dfgL lgDglnlvt :yfgx¿sf] lbzfl:ylt kQf nufpm M -c_ dxG] bg| u/ -cf_ cGgk0\" f{ lxdfn -O_ Onfd -O_{ jL/uGh -pm_ ;u/dfyf lxdfn -pm_ sf7df8f“} -P_ hD' nf sg' } klg gS;fdf efu} fl] ns :yfg, gbLgfnf, lxdfn, jghªu\\ n, af6f3] f6f,] kb| z] 56' o\\ fOPsf] xG' 5 . sg' } klg gS;fdf lbOPsf] :yfg kQf nufpg gS;fnfO{ k9g\\ ] af/d] f 1fg, l;k / Ifdtf xfl;n ugk{' g{] xG' 5 . /fd xl/snf ;Ltf cEof; 10.1 1. ;u“ s} f] lrqsf] cfwf/df tnsf kZ| gx¿sf] pQ/ bp] m M -s_ l5l/ª plePsf] :yfgsf] lbzfl:ylt sg' xf] < l5l/ª ;fl] gof -v_ wlgofsf] lbzfl:ylt sg' xfn] f < -u_ s] wgdfof / kD] afsf] lbzfl:ylt ldN5 < wlgof wgdfof kD] af -3_ dflysf] lrqaf6 kZ| g g=+ -s_, -v_ / -u_ afxs] 3/3 cf6] f yk kZ| gx¿ agfO{ ;fyL;u“ cfk;df lbzfl:ylt kQf nufpg] cEof; u/ . 2. tnsf] lrq k9L ;fl] wsf kZ| gx¿sf] hjfkm nv] M -s_ :yfg O af6 laGb' G sf] lbzfl:ylt kQf nufpm . -v_ :yfg O af6 laGb' C sf] lbzfl:ylt kQf nufpm . -u_ :yfg D af6 laGb' F sf] lbzfl:ylt kQf nufpm . -3_ :yfg O af6 laGb' A sf] lbzfl:ylt kQf nufpm . -ª_ dflysf] lrqaf6 aGg ;Sg] yk ;d:of agfO{ ;dfwfg u/ . ul0ft, sIff – & 85

3. tnsf] gk] fnsf] gS;fsf cfwf/df ;fl] wPsf kZ| gx¿sf] hjfkm nv] M www.dos.gov.np -s_ a6' jnaf6 kfv] /fsf] lbzfl:ylt nv] . -v_ jL/uGhaf6 lj/f6gu/sf] lbzfl:ylt nv] . -u_ jL/G] bg| u/af6 dxG] bg| u/sf] lbzfl:ylt nv] . 4. gk] fnsf] /fhgLlts gS;f lnP/ sg' } Pp6f lhNnf ;b/ds' fdnfO{ sG] b| dfgL 8 cf6] } lbzfdf kg{] Ps Pscf6] f :yfg kQf nufO{ gfd nv] . 86 ul0ft, sIff – &

10.2 :sn] 8O« ª (Scale Drawing) tn lbOPsf ljm| ofsnfkx¿ cWoog u/L 5nkmn u/ . -s_ tnsf] elnan sf6] s{ f] lrqsf] cfwf/df lbOPsf] tflnsf e/ . elnan sf6] { 18cm :sn] 1:200 (1cm = 200cm) jm| =;=+ /v] fsf] gfd gS;fsf] jf:tljs jf:tljs sf6] { / gS;fsf] lgisif{ /v] fsf] gfk sf6] s{ f] gfk gfksf] cgk' ft 1. sf6] s{ f] nDafO 9cm 18m 1:200 2. sf6] s{ f] rf8} fO 3. dWo /v] fbl] v kx| f/ /v] f;Ddsf] b/' L – s] jf:tljs sf6] s{ f] ;a} b/' L / gS;fsf] b/' Lsf cgk' ft 1:200 cfof] < – dflysf] ljm| ofsnfkx¿sf cfwf/df s] lgisif{ lgsfNg ;S5f“} < cfkm\\ gf] lgisifn{ fO{ ;fyL;u“ 5nkmn u/ . -u_ Pp6f xfQL / cfv“ fn] bV] g g;lsg] cldaf cflbnfO{ Pp6} kh] df s;/L nV] g ;lsG5 < nv] / 5nkmn u/ . lgisif{ M 1. o;/L Hofb} 7n' f] / Hofb} ;fgf ;fgf j:tn' fO{ /v] fªs\\ g ugk{' bf{ lglZrt :sn] sf] ko| fu] ul/G5 . 2. o:tf] :sn] df jf:tljs j:t' / lrq lvrL cfjZostfcg;' f/ 7n' f] jf ;fgf] gfk lnP/ lglZrt cgk' ft agfOG5 . h:t} M dfly ljm| ofsnfk -v_ df elnan sf8] s{ f] cgk' ft 1:200cm lnOPsf] 5 . o;sf] cy{ elnan sf6] s{ f] kT| os] 200cm b/' LnfO{ gS;fdf 1cm dfgL :sn] agfOPsf] 5 . ul0ft, sIff – & 87

cEof; 10.2 1. Pp6f aur“} fsf] jf:tljs nDafO 150m / rf8} fO 100m 5 . 1cm:10cm sf] cgk' ft lnP/ aur“} fsf] /v] fªs\\ g u/L bv] fpm . 2. 1cm n] jf:tljs 15m hgfpg] u/L 90m nDafO / 45m rf8} fO ePsf] km' 6an db} fgsf] lrq agfP/ bv] fpm . 3. ;u“ s} f] lrqdf 1cm n] jf:tljs 4m hgfpg] u/L :sn] 8O« ª u/ / jf:tljs ¿kdf w/x/f slt cUnf] /x5] < kQf nufpm . 4. lrqdf lbOPsf] ?v 1:15cm sf] scale df agfOPsf] 5 . ul0ft, sIff – & :sn] n] gfk/] ?vsf] jf:tljs prfO kQf nufpm . 5. 1cm = 1.5m sf] :sn] df lvrs] f] -lrqdf lbOPsf_] uf8Lsf] jf:tljs nDafO slt xfn] f < 6. tn u08sL kb| z] sf] gS;f lbOPsf] 5 . ;f] gS;fdf 1 cm = 50 km sf] :sn] df ljleGg :yfgx¿ lbOPsf] 5 . ca lbOPsf] lhNnfx¿larsf] ;aeGbf 5f6] f] / ;ae} Gbf nfdf] b/' L kQf nufpm . -s_ sf:sLbl] v kfNkf -v_ afunª' bl] v uf/] vf -u_ DofUbLbl] v ndhª' -3_ uf/] vfbl] v :ofªh\\ f 88

PsfO 11 ;dx\" (Set) 11.1 ;jJ{ ofks ;dx\" (Universal Set) tnsf ljm| ofsnfkx¿ cWoog u/L 5nkmn u/ M 1. lxdfno dfWolds ljBfno DofUbLsf sIff 7 sf ljBfyL{x¿n] ;d\"x PsfOdf 5nkmn ubf{ lgDgfg;' f/sf ;dx\" agfP5g,\\ -s_ sIff 7 sf s6] Lx¿sf] ;dx\" , A = { ;Ltf, xl/snf, km' ndfof, wlgof,“ ¿kf, ;/:jtL, sdnf, clDasf, nIdL} -v_ sIff 7 sf s6] fx¿sf] ;dx\" , B = { /fdljnf;, >L ufl] jGb, cfOt,] si[ 0f, /fds[i0f, zflns/fd, pdz] } -u_ sIff 7 sf r:df nufpg] ljBfyLx{ ¿sf] ;dx\" C = {/fdljnf;, pdz] , xl/snf} ca dflysf tLgcf6] } ;dx\" x¿sf u0' f jf ljzi] ftf cfpg ;Sg] sg' } Pp6f lglZrt ;dx\" s] xfn] f < S = {sIff 7 sf ljBfyLx{ ¿} df dflysf tLgcf6] } ;dx\" kb5{ g\\ < ltdLn] klg cfkm\\ gf] sIffsf] S = {sIff 7 sf ljBfyLx{ ¿sf] ;dx\" } df 5nkmndf cfpg ;Sg] sg' } 3 ;dx\" x¿ agfpm . ;fyL;u“ 5nkmn klg u/ . dflysf ;a} ;dx\" x¿sf u0' f jf ljzi] ftf cfpg ;Sg] ;dx\" S = {sIff 7 sf ljBfyLx{ ¿} ;jJ{ ofks ;dx\" xf] . 2. ;ªV\\ ofsf] 1fgaf6 aGg ;Sg] ljleGg ;dx\" x¿ agfpg] af/d] f 5nkmn u/ M -s_ ju{ ;ªV\\ ofx¿sf] ;dx\" S -3_ lahf/] ;ªV\\ ofx¿sf] ;dx\" O -v_ 3g ;ªV\\ ofx¿sf] ;dx\" C -ª_ ¿9 ;ªV\\ ofx¿sf] ;dx\" P -u_ hf/] ;ªV\\ ofx¿sf] ;dx\" E -r_ ;o+ S' t ;ªV\\ ofx¿sf] ;dx\" A ca dflysf ;a} ;dx\" x¿sf u0' f jf ljzi] ftf 5nkmndf cfpg ;Sg] sg' } Pp6f ;dx\" agfpm . s] N = {kf| sl[ ts ;ªV\\ ofx¿} df dflysf ;a} ;dx\" x¿ 5nkmndf cfpg ;S5g\\ < xf] dflysf -s_ bl] v -r_ ;Ddsf ;a} ;ªV\\ ofx¿ N = {kf| sl[ ts ;ªV\\ ofx¿} = {1, 2, 3, 4, 5, 6, 7, 8, ....} df 5nkmndf cfpg ;S5g\\ . o;/L dflysf ;a} ;dx\" x¿sf nflu ;dx\" N ;jJ{ ofks ;dx\" xg' ;S5 . ca dflysf bO' c{ f6] f ljm| ofsnfkx¿ (1 / 2) sf cfwf/df ;jJ{ ofks ;dx\" sf] cy{ nV] g sfl] ;; u/ . cfkm\" n] nv] s] f] cyn{ fO{ ;fyL;u“ tn' gf u/L x/] . s] ltdLx¿n] lgsfns] f] lgisif{ tnsf] tYox¿;u“ ldN5 < tn' gf u/L x/] . ul0ft, sIff – & 89

;jJ{ ofks ;dx\" sf sx] L dxTTjk0\" f{ cfwf/et\" tYox¿ 1. sg' } Pp6f lglZrt ;dx\" df 5nkmnleq cfpg ;Sg] ;a} ks| f/sf ;dx\" x¿ ;dfjz] eP5g\\ eg] Tof] lglZrt ;dx\" nfO{ ;jJ{ ofks ;dx\" (universal set) elgG5 . 2. ;jJ{ ofks ;dx\" nfO{ U n] hgfOG5 . pbfx/0f 1 tnsf ;fyLx¿larsf] s/' fsfgLnfO{ sIffdf clego u/ M lzIfsn] sfnfk] f6Ldf sg' } Pp6f ;dx\" nV] ge' Psf] 5 . /fdljnf; M o;df lahf/] ;ªV\\ ofx¿ {1, 3, 5, 7, 9, 11, 13, 15} dfq 5g\\ . dR} ofª M o;df uGtL ;ªV\\ of 16 t k/g] . km\" ndfof M o;df hf/] ;ªV\\ ofx¿ {2, 4, 6, 8, 10, 12, 14} 5g\\ < >Lsi[ 0f M o;df 0 klg 5 . cfOt] M o;df ¿9 ;ªV\\ ofx¿ klg 5g\\ t < ;fgd' fof M o;df pkoS' t leGg k/g] t . ;lrq M P ! o;df t bzdnj ;ªV\\ of klg 5g} . ca dflysf] 5nkmnsf cfwf/df lzIfsn] sfnfk] f6Ldf nv] ]sf] ;jJ{ ofks ;dx\" kQf nufpm . ;dfwfg oxf“ 15 ;Ddsf k0\" f{ ;ªV\\ ofsf] ;dx\" (W) = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15} df dflysf ;a} ;dx\" x¿ 5nkmndf cfpg ;S5g\\ . To;n} ] o; kZ| gsf nflu ;dx\" W ;jJ{ ofks ;dx\" xG' 5 . pbfx/0f 2 ;jJ{ ofks ;dx\" (U) = {50 eGbf ;fgf k0\" f{ ;ªV\\ ofx¿} xf] . ca tnsf ;d:ofx¿ ;dfwfg u/ . -s_ 4 sf ckjTox{ ¿sf] ;dx\" M4 nfO{ ;r\" Ls/0f ljlwaf6 nv] . -v_ 6 sf ckjTox{ ¿sf] ;dx\" M6 nfO{ ;r\" Ls/0f ljlwaf6 nv] . -u_ ;dx\" U df kg{] yk 2 ;dx\" agfpm . ;dfwfg -s_ 4 sf ckjTox{ ¿sf] ;dx\" (M4) = {4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48} -v_ 6 sf ckjTox{ ¿sf] ;dx\" (M6) = {6, 12, 18, 24, 30, 36, 42, 48} -u_ 2 sf ckjTox{ ¿sf] ;dx\" (M2) = {2, 4, 6, 8, 10, 12, 14, 16, .........48} / 5 n] lgMzi] f efu hfg] ;ªV\\ ofx¿sf] ;dx\" (A) = {5, 10, 15, 20, 25, 30, 35, 40, 45} 90 ul0ft, sIff – &

cEof; 11.1 1. ;jJ{ ofks ;dx\" U = {30 eGbf ;fgf kf| sl[ ts ;ªV\\ ofx¿} 5 . ca lgDglnlvt ;d:ofx¿ ;dfwfg u/ M -s_ ¿9 ;ªV\\ ofx¿sf] ;dx\" P nfO{ ;r\" Ls/0f ljlwaf6 nv] . -v_ 6 sf ckjTox{ ¿sf] ;dx\" M6 nfO{ ;r\" Ls/0f ljlwaf6 nv] . -u_ ;o+ S' t ;ªV\\ ofx¿sf] ;dx\" C nfO{ ;r\" Ls/0f ljlwaf6 nv] . -3_ lahf/] ;ªV\\ ofx¿sf] ;dx\" nfO{ ;r\" Ls/0f ljlwaf6 nv] . -ª_ -s_ / -3_ lardf s] km/s 5 < -r_ ;dx\" -v_ / -u_ larsf] ;DaGw bv] fpm . 2. lgDgfg;' f/sf ;dx\" ;dfjz] ePsf] ;dx\" sf] cfwf/df ;jJ{ ofks ;dx\" (U) kQf nufpm M -s_ ;dx\" P = {30 leqsf ¿9 ;ªV\\ ofx¿} -v_ ;dx\" C = {20 leqsf ;o+ S' t ;ªV\\ ofx¿} -u_ ;dx\" A = {0, 1, 2, 4, 20} -3_ ;dx\" N = {20 ;Ddsf kf| sl[ ts ;ªV\\ ofx¿} 3. Pp6f ljBfnodf sIff 7 sf ljBfyLx{ ¿lar ePsf] tnsf] s/' fsfgLsf cfwf/df ;jJ{ ofks ;dx\" kQf nufpm . ufd] f M o;df M ljrf/ ubf{ M = {4, 8, 12} xG' 5 . 4 4 ;fl] gof M o;df zG\" o 5g} t . ufk] fn M o;df 13 klg 5g} lg . kD] af M P ! o;df cgk' oS' t leGg klg cfpb“ g} . l5l/ª M o;df hf/] ;ªV\\ ofx¿ 2, 4, 6, 8, 10 / 12 k5g{ \\ . >j0f M o;df bzdnj ;ªV\\ of t Pp6f klg 5g} . 4. dfly kZ| g g=+ 1 bl] v 3 ;Dd lbP h:t} yk 2/2 cf6] f ;d:of agfO{ ;dfwfg u/ . pSt ;d:ofnfO{ ;fyLlar ;f6/] ;dfwfg u/ / lzIfsnfO{ bv] fpm . ul0ft, sIff – & 91

11.2 pk;dx\" (Sub Sets) 1. pk;dx\" sf] kl/ro tnsf ljm| ofsnfk cWoog u/L 5nkmn u/ M dfgf“} sg' } Pp6f ;jJ{ ofks ;dx\" U = {4 ;Ddsf kf| sl[ ts ;ªV\\ ofx¿sf] ;dx\" } = {1, 2, 3, 4} 5. -s_ dflysf] ;dx\" sf ;b:ox¿af6 aGg ;Sg] cGo ;dx\" x¿ agfpm . ;dx\" x¿ zI} fl0fs kf6Ldf ;ªs\\ ng u/ / tnsf] tflnsf;u“ tn' gf u/L x/] . jm| =;=+ ;dx\" ;d\"xsf] gfd÷u0fgfTdstf 1. A = {1}, B = {2}, C = {3}, D = {4} Pp6f dfq ;b:o ePsf ;dx\" x¿ n(A) = n(B) = n(C) = n(D) = 1 2. E = {1,2 }, F = {1, 3}, G = {1,4} bO' c{ f6] f ;b:o ePsf ;dx\" x¿ H = {2, 3}, I = {2, 4 }, J = {3, 4} n(E) = n(F) = n(G) = n(H) = n(I) = n(J) = 2 3. K = {1,2,3}, L = {1, 2,4,}, tLgcf6] f ;b:o ePsf ;dx\" x¿ M = { 2, 3, 4} N = {1, 3, 4} n(K) = n(L) = n(M) = n(N) = 3 4. O = {1, 2, 3, 4} 4 ;b:o ePsf] ;dx\" n(O) = 4 5. P = { } vfnL ;dx\" n(P) = 0 dflysf] tflnsfsf cfwf/df tnsf kZ| gx¿df 5nkmn u/ . 1. s] ;dx\" A sf ;b:o ;dx\" U sf klg ;b:o xg' \\ < 2. s] jm| =;= 1 sf ;a} ;dx\" sf ;b:ox¿ ;dx\" U sf klg ;b:o xg' \\ < 3. s] ;dx\" E sf ;a} ;b:ox¿ ;dx\" U sf klg ;b:ox¿ xg' \\ < 4. s] ;dx\" K sf ;a} ;b:ox¿ ;dx\" U sf klg ;b:ox¿ xg' \\ < 5. s] ;dx\" O sf ;a} ;b:ox¿ ;dx\" U sf klg ;b:ox¿ xg' \\ < 6. s] P = { } = vfnL ;dx\" sf ;a} ;b:o cGo ;a} ;dx\" sf ;b:odf klg k5g{ \\ < oxf,“ ;dx\" A sf ;a} ;b:o ;dx\" U sf klg ;b:o xg' \\ . To;n} ] ;dx\" A nfO{ ;dx\" U sf] pk;dx\" elgG5 . o;nfO{ ul0ftLo ;ªs\\ t] df A ⊂ U jf U ⊃ A nl] vG5 . oxf“ ;dx\" B sf ;a} ;b:o ;dx\" E sf klg ;b:o xg' \\ . To;n} ] ;dx\" B ;dx\" E sf] pk;dx\" xf] . o;nfO{ B ⊂ E nl] vG5 . To:t} E⊂K, E ⊂ M klg nV] g ;lsG5 . 92 ul0ft, sIff – &

-v_ dflysf] 5nkmnsf cfwf/df pk;dx\" sf] cy÷{ kl/efiff nv] / ;fyL;u“ 5nkmn u/ . olb Pp6f ;dx\" X df ePsf ;a} ;b:ox¿ csf{] ;dx\" Y sf klg ;b:ox¿ xg' \\ eg] X nfO{ Y sf] pk;dx\" (sub-set) elgG5 . ;ªs\\ t] df X⊂Y cyjf Y⊃X n] hgfpg ;lsG5 . To:t} Y nfO{ X sf] cltl/St ;dx\" (super set) elgG5 . o;nfO{ ul0ftLo ;ªs\\ t] df nV] bf X ⊂Y jf Y ⊃ X nl] vG5 . dflysf] tflnsfsf ;dx\" sf cfwf/df ;DefJo pk;dx\" x¿ 56' o\\ fpm . ;an} fO{ ⊂ / ⊃ ;ªs\\ t] ko| fu] u/L nv] . 2. pkoS' t / cgk' oS' t pk;dx\" (Proper and Improper Subsets) dfly ljm| ofsnfk 1 df lbOPsf] tflnsfsf cfwf/df tnsf tYox¿df 5nkmn u/f“} . -s_ s] pk;dx\" O df ;jJ{ ofks ;dx\" U sf ;a} ;b:ox¿ k/s] f 5g\\ < -v_ s] O afxs] cGo pk;dx\" x¿ A, E, K, P cflbdf ;jJ{ ofks ;dx\" U sf ;a} ;b:ox¿ k/s] f 5g\\ < tn' gf u/L x/] . dflysf] 5nkmnsf cfwf/df pk;dx\" O df U sf ;a} ;b:ox¿ k/s] f 5g\\ . To;n} ] pk;dx\" O ;jJ{ ofks ;dx\" U sf] cgk' oS' t pk;dx\" (improper subset) xf] . cgk' oS' t pk;dx\" nfO{ ⊆ ;ªs\\ t] n] hgfOG5 . ;ªs\\ t] df nV] bf O ⊆ U xG' 5 . O afxs] cGo ;a} pk;dx\" df ;jJ{ ofks ;dx\" U sf ;a} ;b:ox¿ k/s] f 5g} g\\ . To;n} ] O afxs] ;a} pk;dx\" x¿ ;jJ{ ofks ;dx\" U sf pkoS' t pk;dx\" x¿ (proper subsets) xg' \\ . pkoS' t pk;dx\" nfO{ ⊂ ;ªs\\ t] n] hgfOG5 . ;ªs\\ t] df nV] bf A ⊂ U xG' 5 . -u_ ca U sf cGo pkoS' t pk;dx\" x¿ s] s] xg' ;S5g,\\ nv] / 5nkmn u/ . -3_ s] D ⊂ G nV] g ;lsG5 < s;/L < -ª_ ⊂ ;ªs\\ t] ko| fu] u/L cGo 10 cf6] f pkoS' t pk;dx\" x¿ agfO{ ;ªs\\ t] df nv] . ;fyLsf] nv] fO;u“ cfkm\\ gf] nv] fOnfO{ tn' gf u/L x/] . -r_ dflysf] 5nkmnsf cfwf/df pkoS' t / cgk' oS' t ;dx\" sf] cy{ nv] . cfkm\\ gf] cyn{ fO{ ;fyLx¿;u“ 5nkmn u/ . -5_ s] ltdLx¿sf] 5nkmnsf] lgisif{ tnsf tYox¿;u“ ldN5 < tn' gf u/L x/] . ul0ft, sIff – & 93

lgisif{ 1. olb sg' } ;jJ{ ofks ;dx\" U jf cGo ;dx\" x¿ A, B, C ......... cflbaf6 pSt ;dx\" sf ;a} ;b:ox¿ lnP/ pk;dx\" agfOG5 eg] To;nfO{ cgk' oS' t pk;dx\" (improper subset) elgG5 . o;nfO{ ⊆ ;ªs\\ t] n] hgfOG5 . 2. olb sg' } ;jJ{ ofks ;dx\" U jf cGo ;dx\" x¿ A, B, C, ... cflbaf6 sx] L dfq ;b:ox¿ lnP/ sg' } pk;dx\" agfOG5 eg] To;nfO{ pkoS' t pk;dx\" (proper subset) elgG5 . o;nfO{ ⊂ ;ªs\\ t] n] hgfOG5 . 3. vfnL ;dx\" cGo sg' } klg ;dx\" sf] pkoS' t pk;dx\" xG' 5 . 4. a/fa/ ;dx\" x¿ cfk;df cgk' oS' t pk;dx\" x¿ xG' 5g\\ . cEof; 11.2 1. olb W = { 5 ;Ddsf k0\" f{ ;ªV\\ ofx¿sf] ;dx\" } 5 eg,] -s_ ;dx\" W nfO{ ;r\" Ls/0f ljlwaf6 nv] . -v_ ;dx\" W af6 aGg] bO' c{ f6] f Ps ;b:oLo pk;dx\" x¿ agfO{ gfds/0f u/ . -u_ kZ| g -s_ af6 bO' { ;b:oLo yk sltcf6] f ;dx\" x¿ aGg ;Snfg\\ < agfP/ x/] . -3_ ;dx\" W af6 Ps Ps cf6] f, bO' { bO' c{ f6] f, tLg tLgcf6] f, rf/ rf/cf6] f, kfr“ kfr“ cf6] f, 5 5cf6] f / ;b:o gePsf pk;dx\" x¿ lgdf0{ f u/L gfds/0f u/ . -ª_ kZ| g g=+ -3_ sf cfwf/df lbOPsf] ;dx\" W af6 hDdf sltcf6] f pk;;dx\" ag,] nv] . 2. F = {s/] f, :ofp, cªu\\ /' } af6 aGg] ;a} pk;dx\" x¿ nv] . 3. ;dx\" Q = {1} af6 aGg ;Sg] ;a} pk;dx\" x¿ nv] . 4. ;dx\" R = {1, 2} af6 aGg ;Sg] ;a} pk;dx\" x¿ nv] . 5. ;dx\" S = {1, 2, 3} af6 aGg ;Sg] ;a} pk;dx\" x¿ nv] . 6. ;dx\" T = {1, 2, 3, 4} af6 aGg ;Sg] ;a} pk;dx\" x¿ nv] . 7. kZ| g g=+ 3 bl] v 6 ;Ddsf pTt/sf cfwf/df tnsf] tflnsf sfkLdf agfO e/ M jm| =;+ ;d\"x pk;d\"xx¿ ;b:o ;ªV\\ of pk;dx\" sf] ;ªV\\ of 1. {1} ......... ......... ......... 2. {1, 2}. ........ ......... ......... 3. {1,2, 3} ......... ......... ......... 4. {1, 2, 3, 4} ......... ......... ......... dflysf] tflnsfsf] cfwf/df sg' } klg ;dx\" af6 aGg ;Sg] ;DefJo pk;dx\" sf] ;ªV\\ of kQf nufpg] ;q\" lgsfn . 94 ul0ft, sIff – &

11.3 eg] lrq (Venn Diagram) tnsf ljm| ofsnfkx¿ cWoog u/L 5nkmn u/ M dfgf“} sg' } ;jJ{ ofks ;dx\" U = {10 ;Ddsf k0\" f{ ;ªV\\ ofx¿} = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} 5 . ca o;af6 lgDgfg;' f/sf km/s km/s pk;dx\" x¿ agfcf“} . A = {1, 2, 3, 4, 5, 6, 7, 8, 9} B = {2, 4, 6, 8, 10} C = {1, 3, 5, 7, 9} D = {3, 5, 7} E = {3, 5, 7} ca ;jJ{ ofks ;dx\" U af6 ags] f ;dx\" x¿nfO{ o;/L klg bv] fpg ;lsG5 . cWoog u/L 5nkmn u/ . ltdLn] klg cEof; kl' :tsfdf eg] lrq agfpg] ko| f; u/ . -s_ ;dx\" x¿ C / D nfO{ tn' gf u/L x/] f“} M U D sf ;a} ;b:ox¿ ;dx\" C sf klg C ;b:ox¿ xg' \\ . To;n} ] ;dx\" D ;dx\" C sf] pkoS' t pk;dx\" xf] . 0 1 D 6 oxf“ D ⊂ C cyjf C ⊃ D xG' 5 . 2 3, 7, 58 49 10 -v_ ;dx\" x¿ D / E nfO{ tn' gf u/L x/] f“} M U oxf“ ;dx\" D / E sf ;a} ;b:ox¿ Ps cfk;df pxL / plTts} 5g\\ . To;n} ] ;dx\" x¿ 1 E D6 D / E a/fa/ ;dx\" x¿ / cgk' oS' t pk;dx\" bj' } xG' 5g\\ . 2 3, 5, 8 D = E 7 40 9 D ⊂E 10 -u_ ;dx\" x¿ B / C nfO{ cfk;df tn' gf u/L x/] f“} M B C U oxf,“ ;dx\" B / C sf sg' } klg ;b:ox¿ cfk;df ldNbf 5g} g\\ . To;n} ] logLx¿ 28 17 cfk;df cnlUuPsf 5g\\ . B / C 4 10 39 cnlUuPsf ;dx\" (disjoint sets) x'g\\ . 06 5 -3_ ;dx\" x¿ A / B nfO{ cfk;df tn' gf u/L x/] f“} M oxf“ ;dx\" A / B df sx] L ;b:ox¿ 2, 4, 6 / AB U 8 ;femf jf ldNbf 5g\\ . To;n} ] o:tf ;dx\" vlK6Psf ;dx\" x¿ (overlapping sets) xg' \\ . 13 2 0 5 7 4,6 10 98 ul0ft, sIff – & 95