MATHS GRADE 6

गिणत

क ा६

 

 

नेपाल सरकार  

िश ा ,िव ान तथा िविध म ालय

पा म िवकास के    

सानोिठिम, भ पुर     

ul0ft

sIff ^

k|sfzs
gk] fn ;/sf/
lzIff, lj1fg tyf k|ljlw dGqfno

kf7o\ jm| d ljsf; sG] b|

;fgf]l7dL, eStk/'

k|sfzs
g]kfn ;/sf/
lzIff, lj1fg tyf k|ljlw dGqfno

kf7\oj|md ljsf; sG] b|

;fgfl] 7dL, eStk/'

© ;jf{lwsf/ kf7o\ jm| d ljsf; s]Gb|

o; kf7o\ k':ts;DaGwL ;Dk0" f{ clwsf/ kf7o\ jm| d ljsf; sG] b| ;fgf]l7dL, eStk/' df lglxt /xs] f] 5 . kf7o\ jm| d
ljsf; s]Gb|sf] lnlvt :jLsl[ tlagf Jofkfl/s k|ofh] gsf nflu o;sf] k/' } jf cf+lzs efu x'ax' k|sfzg ug,{
kl/jtg{ u//] k|sfzg ug,{ s'g} ljBt' Lo ;fwg jf cGo k|ljlwaf6 /]s8{ ug{ / kl| tlnlk lgsfNg kfOg] 5}g .

ky| d ;+:s/0f M lj=;=+ @)&)

kl/dflht{ bf];|f] ;+:s/0f M lj=;+ @)&^

d'b0| fM hgs lzIff ;fduL| s]Gb| ln=
;fgf]l7dL, eStk'/ .

dN" o ?=

kf7\ok:' ts;DaGwL kf7sx¿sf sg' } klg k|sf/sf ;´¬ fjx¿ ePdf kf7o\ j|md ljsf;
s]Gb,| ;dGjo tyf k|sfzg zfvfdf k7fOlbg'xg' cg'/fw] 5 . kf7sx¿af6 cfpg]
;¬´fjx¿nfO{ s]Gb| xflb{s :jfut ub5{ . tkfOn“ ] lsg]sf] kf7\ok':tsdf sg' } ql' 6
ePdf glhssf] ljt/saf6 pSt k':ts ;f6g\ ;Sgx' 'g] 5 .

xfd|f] egfO

lzIffnfO{ p2]Zodn" s, Jofjxfl/s / ;d;fdlos agfpg kf7o\ j|md ljsf; s]Gb|n] ljBfno txsf
kf7o\ j|mdtyf kf7o\ k':ts ljsf; tyf kl/dfhg{ ug{] sfo{nfO{ lg/Gt/tf lbFb} cfPsf] 5 . ljBfyLd{ f
/fi6k« d|] , /fli6o« tf kl| tsf] ;dk0{ f / nfs] tflGqs k4ltnfO{ cfTd;ft\ ug]{ efjgfsf] ljsf; u/fO{ gl} tsjfg,\
cgz' fl;t, :jfjnDaL tyf l;hg{ zLn eO{ ;dfjz] L ;dfh lgdf0{ fdf ofu] bfg lbg ;Sg] Ifdtf ljsf; xg'
cfjZos 5 .pgLx¿df eflifs tyf ul0ftLo l;ksf ;fy} lj1fg, ;r" gf tyf ;~rf/ kl| jlw, jftfj/0f,
:jf:Yo tyf hg;ªV\ of;DaGwL cfwf/et" 1fg tyf hLjgfk] ofu] L l;ksf] ljsf; xg' h?/L 5 . To;} u/L
ljBfyL{x¿df snf tyf ;f}Gbo{ kl| tsf] cg'/fu / dfgjLo dN" o dfGotf, cfbz{ tyf j}lzi6\ox¿kl| tsf]
;r]ttf ;lxt ltgsf] ;/+ If0f, ;+jw{g ug{] efjgfsf] ljsf; cfjZos 5 . ;dtf d"ns ;dfhsf]
lgdf0{ fdf ;xofu] k'¥ofpg pgLx¿df ljleGg hfthflt, lnª\u, efiff, wd{, ;+:s[lt / If]qnufotsf
ljljwtfx¿sf] ;Ddfg ug{] / dfgj clwsf/ tyf ;dflhs d"No dfGotf k|lt ;r]t eO{
lhDdj] f/L jxg ug]{ efjgfsf] ljsf; u/fpg' cfjZos 5 . plNnlvt cfjZostfnfO{ bl[ i6ut u/L
cfwf/et" lzIff kf7\oj|md -sIff ^–*_, @)^( nfO{ d"n cfwf/dfgL lzIff;DaGwL ljleGg cfof]usf
;'emfj, lzIfs, ljBfyL{ tyf cleefjsnufot lzIff;u“ ;Da4 ljleGg JolSt ;lDdlnt ufi] 7L /
cGt/lj|mofsf lgisif{ / ljleGg ljBfnodf k/LIf0f u/L k|fKt k[i7kf]if0f ;d]tnfO{ ;d]6L of]
kf7o\ k:' ts tof/ kfl/Psf] xf] .

o; k':tsdf xfd|f] b}lgs hLjg;“u ;DalGwt lj|mofsnfk, k|of]u tyf pbfx/0f ;dfj]z ug]{
sfl] ;; ul/Psf] 5 . o;df ;dfjz] ul/Psf clwsfz+ lj|mofsnfkx¿ :yfgLo ;fdu|Laf6 ug{ u/fpg
;lsg] vfnsf 5g\ . lj=;+= @)%! df 8f= ;Gtf]ifdfg df:s] / xl/gf/fo0f pkfWofon] nV] ge' Psf]
kf7\ok':tsnfO{ cfwf/et" tx -sIff ^–*_ sf] kf7\ojm| d @)^( adf]lhd lbgz] s'df/ >i] 7, 808kfl0f
zdf,{ Zofdl;x+ wfdL, j?0fk|;fb j}B, /dz] k|;fb cj:yL / wj'| gf/fo0f rf}w/L ;lDdlnt sfob{ naf6
kl/dfhg{ ul/Psf] xf] . o;sf] kl/dfh{g sfod{ f sfo{sf/L lgb{]zs vu/fh a/fnsf ;fy} lrqk|;fb
b]jsf6] f, k|f=8f= l;l4 sf]O/fnf, kf| = 8f= lzj/fd Gof}kfg], k|f=8f= lx/faxfb/' dxh{g, 8f= n]vgfy
kf}8]n, ;ª' d\ f t'nfw/, x]d/fh kf]v/]n, d's'Gb/fh zdf,{ lgdn{ f uft} d, hLj/fh cfrfo{, /fhG] b|
b]jsf6] f, d}gf clwsf/L, ;w' L/ emf / OZ{ j/ >i] 7sf] ljz]if of]ubfg /xs] f] 5 . o;sf] efiff
;Dkfbg xl/k|;fb lg/fn} f, snf ;Dkfbg >Lxl/ >i] 7 tyf n]cfp6 l8hfOg ho/fd s'Os“ n] af6
ePsf] xf] . kf7\ok:' tsnfO{ cWofjlws tyf kl/dfhg{ u/L ks| flzt ug{ sfo{df o; sG] bs| f
dxflgb]{zs 8f= n]vgfy kf}8]n, >L u0f]zk;| fb e6\6/fO{ / >L lrgfs'df/L lg/f}nfsf] of]ubfg
/x]sf] 5 . o; kf7\ok:' tssf] ljsf; tyf kl/dfhg{ sfod{ f ;n+ Ug ;a}k|lt kf7o\ j|md ljsf; sG] b|
wGojfb ks| 6 ub{5 .

kf7\ok':tsnfO{ lzIf0f l;sfOsf] dxŒjk0" f{ ;fwgsf ¿kdf lnOG5 . o; kf7o\ k':tssf] k|ofu] af6
kf7o\ j|mdåf/f nlIft ;Ifdtf xfl;n ug{ ljBfyLn{ fO{ ;xof]u k'Ug] ckI] ff ul/Psf] 5 . kf7o\ k:' tsnfO{
;s;] Dd lj|mofsnfkdv' L / ?lrs/ agfpg] k|oTg ul/Psf] 5 . o; kf7\ok:' tsnfO{ ce}m kl/is[t
kfg{sf nflu lzIfs, ljBfyL{, cleefjs, al' 4hLjL Pjd\ ;Dk0" f{ kf7sx¿sf] ;d]t dxŒjk0" f{ el" dsf

/xg] x'“bf ;Da4 ;as} f] /rgfTds ;'emfjsf nflu kf7\ojm| d ljsf; sG] b| xflbs{ cg'/fw] ub{5 .

g]kfn ;/sf/
lzIff, lj1fg tyf kl| jlw dGqfno
lj= ;=+ @)&^
kf7o\ j|md ljsf; sG] b|

ljifo;r" L k[i7;ªV\ of

PsfO PsfO zLifs{ 1
1. /]vf / sf]0f 15
2. lqeh' , rt'e'h{ / ax'e'hx¿ 21
3. 7f]; cfsl[ tx¿ 25
4. lgbz]{ fªs\ x¿ 28
5. kl/ldlt, If]qkmn / cfotg 41
6. :yfgfGt/0f 43
7. ;dldlt / 6l] ;n;] g 48
8. ;d"x 66
9. k"0f{ ;ª\Vofx¿ 92
10. k0" ffª{ s\ x¿ 94
11. cgk' flts ;ªV\ ofx¿ 96
12. leGg / bzdnj 130
13. cg'kft, ;dfg'kft / k|ltzt 138
14. gfkmf / gfS] ;fg 143
15. Pl] ss lgod 149
16. ;fwf/0f Aofh 152
17. tYofªs\ zf:q 158
18. aLhLo cleJo~hs 172
19. ;dLs/0f, c;dfgtf / nv] flrq 186
pQ/dfnf

PsfO 1 /v] f / sf0] f (Line and Angle)

1.1 kl| tR5l] bt / ;dfgfGt/ /v] fx¿ (Intersecting and parallel lines)

tn lbOPsf pbfx/0fx¿ 5nkmn u/ M

A

OD

lrq g=+ 1.1 lrq g=+ 1.2

/v] fx¿ AB / CD n] Pscsfn{ fO{ laGb' 38Lsf] nfdf] ;O' { (minute hand) OD / 5f6] f]
O df e6] 5\ g\ . s] AB / CD n] ;O' { (hour hand) OA n] laGb' O df Pscsfn{ fO{
Pscsfn{ fO{ Psk} 6s laGb' O afxs] e6] s] f 5g\ . s] ToxL ;dodf OA / OD n]
cGo laGbd' f klg e6] 5\ g\ xfn] f < csf{] laGbd' f klg e6] g\ ;S5g\ <

X AB
D DC

P lrq 1.4

RE lbOPsf] lrqdf dr] sf bO' { hf8] L lsgf/fx¿ jm| dzM
(AD, BC) / (AB, DC) xg' \ . s] AD / BC Ps
AF cfk;df kl| tR5b] g xG' 5g\ xfn] f < s] AB / DC
Pscfk;df kl| tR5b] g xG' 5g\ xfn] f <
BY

lrq 1.3 CQ
S

lbOPsf] lrqdf uf8Lx¿ A, B / C ul' 8/xs] f]
;8ssf] lsgf/f RS / uf8Lx¿ D, E / F
ul' 8/xs] f] ;8ssf] lsgf/f XY larsf] b/' L
;dfg 5 ls 5g} <

1. k|ltR5]lbt /]vfx¿ M cfk;df sfl6g] b'O{cf]6f /]vfx¿nfO{ k|ltR5]lbt /]vfx¿ elgG5 .
dflysf] lrq 1.1 df /v] fx¿ AB / DC laGb' O df kl| tR5l] bt 5g\ . To;u} /L lrq g= 1.2 df
nfdf] ;O' { OD / 5f6] f] ;O' { OA klg laGb' O df kl| tR5l] bt 5g\ .

2. ;dfgfGt/ /v] fx¿ M Pp6} ;dtn ;txsf /v] fx¿nfO{ bj' l} t/ hlt nDAofpb“ f klg cfk;df
kl| tR5b] g xb“' g} g\ eg] To:tf /v] fx¿ ;dfgfGt/ xG' 5g\ . lrq g= 1.3 df ;8ssf 5p] x¿ (XY
/ RS) Pscfk;df ;dfgfGt/ 5g\ . o;nfO{ ul0ftLo lrxg\ "//" åf/f hgfOG5 . To;n} ]
lrq g= 1.4 df AD//BC / AB//DC nV] g ;lsG5 .

ul0ft, sIff ^ 1

sx] L pbfx/0fx¿
(i) ltdf| ] cEof; kl' :tsfsf ;Ddv' lsgf/fx¿ ;dfgfGt/ 5g\ .
(ii) ?n/sf ;Ddv' lsgf/fx¿ ;dfgfGt/ 5g\ .
(iii) sfnfk] f6Lsf ;Ddv' lsgf/fx¿ ;dfgfGt/ xG' 5g\ .
(iv) ;dfgfGt/ /v] fx¿ hgfpg] cGo pbfx/0fx¿ sIffsf7] faf6 ;ªs\ ng u/ .

cEof; 1.1

1. tn lbOPsf kT| os] lrqx¿af6 2–2 hf8] f kl| tR5l] bt /v] fv08x¿ nv] .

D

ED E G

AB

A B H F
C C
F
(a) (b)

2. tn lbOPsf lrqx¿df ;dfgfGt/ /v] fv08x¿sf] hf8] f nv] .

XD CK
LP

P Q A BM NO
Y Z

3. olb AB//CD, CD//EF eP s] AB//EF xG' 5, nv] .

AB
CD

EF

4. tn lbOPsf dWo] sg' sg' egfOx¿ l7s 5g\ nv] M
(a) b'O{cf]6f /]vfx¿nfO{ b'j}lt/ a9fp“bf klg Pscfk;df e]6\b}gg\ eg] tL /]vfx¿
;dfgfGt/ xG' 5g\ .
(b) bO' c{ f6] f ;dfgfGt/ /v] fv08x¿larsf] b/' L Ps;dfg xG' 5 .
(c) bO' c{ f6] f ;dfgfGt/ /v] fv08x¿ kl| tR5l] bt xG' 5g\ .

2 ul0ft, sIff ^

1.2 nDa /v] f (Perpendicular lines) A D

(i) lrqdf bv] fP h:t} Pp6f cfotfsf/ sfuhsf] B C
6j' m| f ABCD lnp“ / D nfO{ A df tyf C nfO{ B df AD L
kg{] u/L k6o\ fpm .

(ii) B nfO{ A df tyf C nfO{ D df kg{] u/L sfuh k6o\ fpm .

(iii) lrqdf bv] fP h:t} L, M, P, O / Q BC M
gfds/0f u/ . AB L

P OQ

M

(iv) k6o\ fPsf] sfuh vfn] . /v] fv08x¿ LM / PQ sf] A LD
kl| tR5b] g, laGb' O df ePsf] 5 . s] laGb' O df P
ags] f] kT| os] sf0] fsf] gfk 900 xG' 5 < kf| 6] S] 6/sf] B OQ
;xfotfn] gfk/] kQf nufpm . C

M

cfk;df ;dsf0] f eO{ kl| tR5b] g ePsf /v] fx¿nfO{ nDa/v] fx¿ elgG5 . dflysf] ljm| ofsnfkaf6
kf| Kt /v] fv08x¿ LM / PQ Pscfk;df nDa 5g\ .

bi| 6Jo M bO' c{ f6] f /v] fx¿ (lines), ls/0fx¿ (rays), /v] fv08x¿ (line segments) Pscfk;df nDa
xg' ;S5g\ . bO' c{ f6] f /v] fv08x¿ nDa xg' ' egs] } /v] fx¿ klg nDa xg' ' xf] . " " lrxg\ n] bO' c{ f6] f
/v] fx¿ cyjf /v] fv08x¿ nDa 5g\ eGg] bv] fp5“ . dflysf] lrqdf LM PQ xG' 5 . cfotfsf/
sfuh ABCD sf sg' sg' eh' fx¿ cfk;df nDa xfn] fg\ < gfk/] hfr“ .

cEof; 1.2

1. ltdf| ] Hofldlt afs;df ePsf ;6] :Sjfo/x¿nfO{ sfkLdf km/s km/s 7fpd“ f /fv / cfkm“}
gfds/0f u/L nDa /v] fx¿sf] gfd nv] .

2. sIffsf7] f / cfkm\ gf] jl/kl/ nDa xg' ;Sg] /v] fv08sf sg' } tLg cf6] f pbfx/0f nv] .

3. E, F, H, L, N, T, V, X

dflysf sg' sg' cIf/x¿n] nDa /v] fv08 agfPsf 5g\ < nv] .

ul0ft, sIff ^ 3

4. tn lbOPsf] lrqdf AB / CD bj' } /v] f PQ df nDa 5g\ . s] AB//CD xG' 5 <

AC

P BD Q

5. tnsf k|To]s lrqdf nDa x'g] / ;dfgfGt/ x'g] /]vfv08x¿sf hf]8f 5'6\ofP/ n]v .

A LJ I A H AL J I
K MH B CF G
DE K
D F
B CF G
E

DE BC GH

6. (a) tnsf] lrqdf QR ;“u ;dfgfGt/ x'g] u/L P af6 sltcf]6f /]vfx¿ lvRg
;lsPnf <

(b) QR df nDa xg' ] u/L P af6 sltcf6] f nDa lvRg ;lsPnf <

P

QR
4 ul0ft, sIff ^

1.3 ;dfgfGt/ / nDa/v] fx¿sf] /rgf -;6] :Sjfo/ ko| fu] u//] _

(Construction of parallel and perpendicular lines using a set-square)

(a) ;dfgfGt/ /v] fx¿sf] /rgf
Pp6f laGb' P /v] f AB aflx/ 5 . AB ;u“ ;dfgfGt/ xg' ] / P eP/ hfg] /v] f CD lvRg] .
(i) ;6] :Sjfo/sf] ;dfs] f0] fL eh' fnfO{ AB ;u“ ldNgu] /L /fvf“} .
(ii) ?n/nfO{ ;6] :Sjfo/sf] csf{] ;dsf0] fL eh' f;u“ l;wf xg' ] u/L /fvf“} .
(iii) lrqdf bv] fP h:t} ;6] :Sjfo/nfO{ ?n/ grNg] u/L laGb' P ;Dd nuf“} / CD lvrf“} .
(iv) ;6] :Sjfo/nfO{ x6fcf“} . o;/L CD//AB sf] /rgf eof] .

(b) nDa /v] fx¿sf] /rgf
ljGb' P af6 /v] f AB df nDa PQ lvrf“} hxf“ laGb' P /v] f AB eGbf aflx/ 5 .
(i) /v] f AB df kg{] u/L ?n/nfO{ /fvf“} .
(ii) ;6] :Sjfo/sf] 90° ags] f] eh' fnfO{ ?n/df ldNg] u/L /fvf“} .
(iii) ;6] :Sjfo/sf] 90° ags] f] csf{] eh' fnfO{ laGb' P df ldnfcf“} .
(iv) lrqdf bv] fOP h:t} /v] fv08 PQ lvrf“} / ;6] :Sjfo/nfO{ x6fcf“} . o;/L PQ AB
/rgf eof] .

ul0ft, sIff ^ 5

cEof; 1.3
1. cEof; kl' :tsfdf tn lbOP h:t} u/L /v] fv08 lvrL laGb' cªs\ g u/ / kT| os] /v] fv08;u“

;dfgfGt/ xg' ] u/L lbOPsf] laGba' f6 hfg] /v] fv08sf] /rgf u/ . -;6] :SjfP/sf] ko| fu] u//] _

OR

(a) (b)

A BP Q
(c)
(d) O
S
T

X CD

2. cEof; kl' :tsfdf tn lbOP h:t} cfsl[ t agfO{ kT| os] /v] fv08df lbOPsf] laGba' f6 hfg]

nDasf] /rgf u/ . -;6] :Sjfo/sf] ko| fu] u//] _

P A
(a) (b)

A BC D

(c) (d) D
C DC

P Q

3. cfkmg\ f] sfkLdf /v] fv08 PQ lvr / To;sf] laGb' P / Q df nDa xg' ] u/L 3/3 ;=] ld= nfdf
nDax¿ SP / RQ lvr . RS nfO{ hf8] b\ f ss] f] lrq aG5 <

PQ

4. P af6 QR ;“u ;dfgfGt/ x'g] u/L Pp6f /]vfv08 lvr . R af6 PQ ;u“ ;dfgfGt/ xg' ] u/L
csf{] /v] f lvr . o;/L lvrs] f] bO' { cf6] f /v] fx¿ sfl6Psf] laGbn' fO{ S gfds/0f u/ . s:tf]
cfsl[ t aGof] <

6 ul0ft, sIff ^

1.4 sDkf;sf] ko| fu] af6 /v] fv08sf] nDafws{ sf] /rgf

(Construction of Perpendicular Bisector of a Line Segment using Compass)

r/0f (i): lbOPsf] gfksf] /v] fv08 AB A B
?n/sf] ;xfotfn] lvr .

r/0f (ii): lbOPsf] /v] fv08sf] laGb' A cyjf B A B
af6 /v] fv08sf] cfwfeGbf a9L nDafOsf]
rfk sf6, h:t} M laGb' A af6 .

r/0f (iii): A af6 lnOPsf] rfksf] nDafO
a/fa/ xg' u] /L laGb' B af6 klg np] m .
bj' } rfkx¿n] Pscsfn{ fO{ laGbx' ¿ L /
N df e6] 5\ g\ . ?n/sf] ;xfotfn] L / N
nfO{ hf8] h;n] /v] fv08 AB nfO{
M df e6] 5\ .

AM, BM ∠AML / ∠BML gfk .

To;n} ,] LN /v] fv08 AB sf] nDafws{ xf] .

dflysf] lrqsf cfwf/df lgDg lnlvt kZ| gx¿df 5nkmn u/ M

 s] AM / BM a/fa/ 5g\ <
 s] laGbx' ¿ A / B af6 lvlrPsf rfkx¿ laGb' M df dfq e6] s] f eP laGbx' ¿ L / N kfpg

;Dej lyof] <
 s] AL / BL tyf AN / BN Ps cfk;df a/fa/ xG' 5g\ xfn] f <

∠AML, ∠BML, ∠AMN / ∠BMN df kT| os] sf] gfk slt xG' 5 < gfk/] x/] .

 s] ltdf| ] ljBfnosf sIffsf7] fx¿ tyf 3/sf ‰ofnx¿df nDafws{ xg' ] u//] sf7sf jf
kmnfdsf 58x¿ /flvPsf 5g\ <

sg' } /v] fv08sf] dWo laGba' f6 900 sf] sf0] f agfP/ uPsf] /v] fv08nfO{ pSt /v] fv08sf]
nDafws{ elgG5 . dflysf lrqdf AB sf] nDafws{ LN xf] .

sg' } klg /v] fv08nfO{ cfwf xg' ] u/L uPsf] /v] fv08nfO{ cws{ elgG5 .

ul0ft, sIff ^ 7

cEof; 1.4
1. lbOPsf] lrqaf6,

(a) eh' f BC sf] nDafws{ sf] gfd nv] .
(b) eh' f AB sf] nDafws{ sf] gfd nv] .
(c) s] AD / CD a/fa/ 5g\ <

2. lbOPsf] lrqdf,
(a) s] AB /v] fv08 MN sf] nDafws{ xf] <
(b) s] MN /v] fv08 AB sf] klg nDafws{ xf] <

3. tn lbOPsf gfksf /v] fv08x¿ lvr / pSt /v] fv08sf] nDafws{ sf] /rgf u/ .

(a) AB = 6 ;=] ld= (b) CD = 8 ;=] ld=

(c) PQ = 9 ;=] ld= (d) EF = 9 ;=] ld=

(e) MN = 7.5 ;=] ld= (f) GM = 8.5 ;=] ld=

(g) RS = 51 ;=] ld= (h) KL = 7 1 ;=] ld=
2 2

8 ul0ft, sIff ^

1.5 sf0] fx¿sf] ks| f/ (Types of Angles) X

-s_ Gog" sf0] f (Acute angle) Z

00 eGbf 7n' f] / ;dsf0] feGbf ;fgf] (900 eGbf P
;fgf_] sf0] fnfO{ Gog" sf0] f elgG5 . lrqdf OQ
∠XYZ ;dsf0] feGbf ;fgf] ePsfn] Gog" sf0] f
xf] . Z

Y

-v_ ;dsf0] f (Right angle)

900 gfk ePsf] sf0] fnfO{ ;dsf0] f elgG5 . lrqdf
∠POQ = 900 ePsfn] sf0] f POQ ;dsf]0f xf] .

X

-u_ clwssf0] f (Obtuse angle)

900 eGbf 7'nf] t/ 1800 eGbf ;fgf] sf]0fnfO{
clwssf0] f elgG5 . lrqdf ∠XYZ sf0] f 900 eGbf
7n' f] ePsfn] ∠XYZ clws sf]0f xf] .

Y

-3_ ;/n sf0] f (Straight angle) 180O
1800 gfk ePsf] sf]0fnfO{ ;/n sf]0f P QR
elgG5 . lrqdf ∠PQR sf] gfk 1800
ePsfn] of] Pp6f ;/n sf]0f xf] . Q P
R
-ª_ ax[ ts\ f0] f (Reflex angle) 9

1800 eGbf 7n' f] / 3600 eGbf ;fgf] sf0] fnfO{
a[xt\sf]0f elgG5 . ∠PQR sf] gfk 1800 eGbf
7'nf] ePsfn] of] Ps a[xt\sf]0f xf] .

ul0ft, sIff ^

cEof; 1.5

1. tn lbOPsf kT| os] sf0] fx¿ Gog" sf0] f, ;dsf0] f, clws sf0] f, ;/n sf0] f jf ax[ ts\ f0] f s]
s] xg' ,\ 56' o\ fpm / nv] .

(a) (b) (c)
B M

O

O A CN O
(d) D (f) T

Q (e) BR
R

A
O
PS

2. lrqdf ePsf clwssf0] f, Gog" sf0] f / ;/nsf0] fsf] gfd nv] .

R

PO Q

3. tnsf egfO l7s jf al] 7s s] xg' ,\ 56' o\ fpm M
-s_ ∠X, 00 eGbf 7n' f] / 900 eGbf ;fgf] 5 . X Go"gsf]0f xf] .
-v_ ∠Y, 00 eGbf 7n' f] / 900 eGbf ;fgf] 5 . Y sf] Pp6f dfq dfg x'G5 .

10 ul0ft, sIff ^

-u_ ∠Z, 900 / 1800 sf lardf k5{ . Z n] clwssf]0f hgfp“5 .
-3_ ∠P, 900 ;“u a/fa/ 5 . P n] ;dsf]0f hgfp“5 .
-ª_ ∠L, 1800 ;“u a/fa/ 5 . ∠L clwssf]0f xf] .
4. lrqdf ePsf clwssf]0fx¿sf] gfd n]v .

AD

O

BC

5. gk] fnsf] emG8fsf] /v] fªs\ gdf ePsf clwssf0] f, Gog" sf0] f, ;dsf0] f, ;/nsf0] f / ax[ ts\ f0] f

5'6\ofpm . A

C B
E D

F

6. bO' c{ f6] f l;Gsfx¿sf] ;xofu] af6 Gog" sf0] f, ;dsf0] f, clwssf0] f / ;/nsf0] f agfpm .

ul0ft, sIff ^ 11

1.6 sf0] fsf] /rgf / gfk (Construction of Angle of given Measurement)

1. ;6] :Sjfo/sf] ko| fu] åf/f 30°, 45°, 60° / 90° sf sf0] fx¿sf] /rgf

lrq g= (i) df bv] fOP h:t} ;6] :Sjfo/ (set-square) nfO{ /fvf“} . ?n/sf] ;xfotfn] /v] f
AC / AB lvrf,“} o;/L ∠BAC = 30° kf| Kt xG' 5 .

B

AC B

30° C
A
(i)

lrq g= (ii) df (i) sf] h:t} ljm| ofsnfk bfx] f¥] ofcf“} / ∠CBA = 60° sf] /rgf u/f“} .

C

C

BA A

60°

(ii) B

lrq g= (iii) df (i) sf] h:t} ljm| ofsnfk bfx] f¥] ofcf“} / ∠BAC = 45° sf] /rgf u/f“} .

B

A C B
D (iii)
45° C
A

lrq g= (iv) df (i) sf] h:t} ljm| ofsnfk
bfx] f¥] ofcf“} / ∠DCB = 90° sf] /rgf u/f“} .

D

C B
12
(iv) 90° B
AC
ul0ft, sIff ^

2. sf0] fsf] cws{ sf] /rgf (Construction of bisecter of the Angle)

O af6 OA / OB sfl6g] u/L rfk XY lvr . X / A
Y af6 ;fx] L rfk lnP/ Z df sf6 . O / Z hf8] L
C ;Dd nDAofpm . X ZC
B
lrqdf ∠AOB gfk . To;}u/L ∠AOC / ∠BOC O
klg gfk . /]vf OC n] ∠AOB nfO{ a/fa/ bO' { Y
efudf af8“ s] f] 5 . o;/L Pp6f sf0] fnfO{ bO' { a/fa/
efudf af“8\g] /]vfnfO{ sf]0fsf] cw{s elgG5 .
lrqdf ∠AOB sf] cw{s OC xf] . sDkf;sf]
;xfotfn] sg' } klg sf0] fsf] cws{ lvRg] tl/sf
lrqdf b]vfOPsf] 5 .

3. sDkf;sf] ko| fu] åf/f sf0] fsf] /rgf E B
D
600 sf] sf0] fsf] /rgf AC

Pp6f /v] fv08 AB lvr . A df sDkf;sf]
;xofu] n] Pp6f rfk lvr . ;f] rfkn] /v] f AB
sf] C df sf6\5 . C af6 cl3s} rfk lnO{
klxn] lvlrPsf] rfknfO{ D df sf6 . A / D
hf8] L E ;Dd nDAofpm .
∠EAB = 600 x'G5 .

300 sf] sf0] fsf] /rgf

600 sf] sf]0fsf] /rgf u/ . ;f]xL rfkn] D / E C G
af6 F df sf6g\ ] . sfl6Psf] laGb' F / A hf8] L EF
G ;Dd nDAofpm . oxf“ ∠CAG = ∠GAB =

1  60 = 300 x'G5 .
2

AD B

ul0ft, sIff ^ 13

900 sf0] fsf] /rgf F

Pp6f /v] fv08 AB lvr . laGb' A af6 sg' } rfk lnO{ E
60° sf] lrxg\ D nufpm / km] l/ D af6 60° sf] rfk D
E sf6L 1200 sf]0fsf] /rgf u/ . D / E af6
lvlrPsf a/fa/L rfk sfl6Psf] laGb' / A hf8] L F AC
;Dd nDAofpm . oxf“ ∠FAB = 900 x'G5 .

B

450 sf] sf0] fsf] /rgf F G
EH D
900 sf] sf]0fsf] /rgf u/ . laGb'x¿ H / A af6
a/fa/L rfkn] sfl6Psf] ljGb' D / C hf8] L G ;Dd
nDAofpm . oxf“ ∠BCG = 450 x'G5 .

CA B

cEof; 1.6

1. sDkf; / ;6] :Sjfo/sf] ko| fu] u/L tn lbOPsf sf0] fx¿sf] /rgf u/ M

(a) 60° (b) 30° (c) 90° (d) 45°

2. /v] fv08 AB sf] ljGb' A df sDkf;sf] ;xfotfn] 60° sf] sf0] f /rgf u/ / o;nfO{ cfwf u/ .

AB

3. Pp6f /v] fv08 AB sf] laGb' A df 30° / laGb' B df 90° sf] sf0] f agfpm . sf0] fx¿ agfpg]
/v] fx¿ sfl6Psf] laGbn' fO{ C gfd bp] m / sf0] f C gfk .

14 ul0ft, sIff ^

PsfO 2 lqeh' , rte' h{' / axe' h'

(Triangle, Quadrilateral and Polygon)

2.1 eh' f / sf0] fsf cfwf/df lqeh' sf] juLs{ /0f

(Classification of triangles by sides and angles)

a/fa/ jf km/s km/s gfksf ltg cf6] f uxs“' f] 5j\ fnL jf af;“ sf] kfOk jf h;' kfOk jf cGo
sf7sf 6j' m| f jf 86kg] sf vfj] m| fx¿ np] m / ljleGg ks| f/sf lqeh' x¿ agfpm .

eh' fx¿sf cfwf/df lqeh' sf] juLs{ /0f 15

-s_ ;dafx' lqeh' (Equilateral triangle)
sg' } lqeh' sf tLgcf6] f eh' fx¿sf] nDafO a/fa/ 5 eg] Tof]
lqe'hnfO{ ;dafx' lqe'h elgG5 . lqe'h ABC df
AB = BC = AC ePsfn] of] ;dafx' lqe'h xf] .

-v_ ;dlåafx' lqeh' (Isosceles triangle)
lqeh' sf sg' } bO' c{ f6] f eh' fsf] nDafO a/fa/ 5 eg] To:tf]

lqe'hnfO{ ;dlåafx' lqe'h elgG5 . lrqdf ΔABC df

AB = AC ePsfn] of] lqe'h ;dlåafx' lqe'h xf] .

ul0ft, sIff ^

-u_ ljifdafx' lqeh' (Scalene triangle)

A

sg' } lqeh' sf tLgcf6] f eh' fx¿ km/s km/s gfksf 5g\ eg]

To:tf] lqeh' nfO{ ljifdafx' lqeh' elgG5 .

ΔABC df sg' } klg eh' f a/fa/ 5g} g\ . B C

To;n} ] ΔABC ljifdafx' lqe'h xf] .

sf0] fx¿sf cfwf/df lqeh' sf] juLs{ /0f A

-s_ Gog" sf0] fL lqeh' (Acute-angled triangle) B C
A C
sg' } lqeh' sf tLgcf6] } sf0] fx¿ 900 eGbf ;fgf
5g\ cyft{ \ Gog" sf0] f 5g\ eg] To:tf] lqeh' nfO{ B C
Go"gsf]0fL lqe'h elgG5 . ΔABC df ∠A, ∠B A

/ ∠C ;a} 900 eGbf ;fgf 5g\ . To;n} ] ΔABC

Pp6f Go"gsf]0fL lqe'h xf] .

-v_ ;dsf0] fL lqeh' (Right-angled triangle)
sg' } lqeh' sf] Pp6f sf0] f ;dsf0] f 5 eg] Tof]
lqe'h ;dsf]0fL lqe'h x'G5 . ΔABC df
∠B = 900 ePsfn] pSt lqeh' ;dsf0] fL lqeh'
xf] .

-u_ clwssf0] fL lqeh' (Obtuse angled triangle)
lqeh' sf tLgcf6] f sf0] fdWo] Pp6f sf0] f 90O eGbf

7n' f] 5 eg] Tof] lqeh' clwssf0] fL lqeh' xG' 5 .

B

ΔABC df ∠B, 900 eGbf 7n' f] ePsfn] ΔABC clwssf]0fL lqe'h xf] .

16 ul0ft, sIff ^

cEof; 2.1

1. tn lbOPsf kT| os] lqeh' sf eh' fx¿ gfk / eh' fsf cfwf/df lqeh' sf] juLs{ /0f u/ M

A P (c) L
(a) (b)

Q RM N
CB (f) R
Q
(d) (e)
P A

Q RC BP

(g) (h) (i)
L A

M NB Q P
C R

2. tn lbOPsf lqeh' nfO{ sf0] fsf cfwf/df juLs{ /0f u/ M C

(a) (b) (c)
A A A

B CB B
C
ul0ft, sIff ^
17

(d) (e) (f)
A A A

B CB CB C

(g) A (h) B

BA

C C
A
3. lbOPsf] lrqdf sltcf6] f lqeh' x¿ 5g\ < PQ
BRC
4. lbOPsf] lrqaf6 Ps Pscf6] f A
;dsf0] fL, Gog" sf0] fL / D
clwssf0] fL lqeh' sf] gfd E

nv] .

BC

18 ul0ft, sIff ^

2.2 axe' h' (Polygons)

tnsf] tflnsfdf sx] L axe' h' x¿, ltgLx¿sf eh' fsf] ;ªV\ of / gfd lbPsf] 5 M

lrq eh' fsf] ;ªV\ of gfd

3 lqeh' (Triangle)

4 rte' h{' (Quadrilateral)

5 k~reh' (Pentagon)

6 if8e\ h' (Hexagon)

7 ;Kteh' (Heptagon)

8 ci6eh' (Octagon)

tLg jf tLgeGbf a9L eh' fx¿n] ags] f] ;/n aGb ;dtnLo cfsl[ tnfO{ axe' h' elgG5 .

olb axe' h' sf ;a} eh' fx¿ a/fa/ 5g\ / leqL sf0] fx¿ klg a/fa/ 5g\ eg] To:tf] axe' h' nfO{ lgoldt
axe' h' (Regular Polygon) elgG5 . 5 cf6] f eh' fn] ags] f lgoldt / clgoldt axe' h' sf] lrq x/] .

lgoldt clgoldt clgoldt

bi| 6Jo M lgoldt lqeh' eGg' g} ;dafx' lqeh' xf] . To:t} lgoldt rte' h{' eGg' g} ju{ xf] .

cEof; 2.2
1. tn lbOPsf lrqx¿dWo] sg' axe' h' xfO] g, nv] .

-s_ -v_ -u_ -3_ D
EC
2. lbOPsf] axe' h' sf] gfd nv] / o;leq aGg ;Sg]
2 cf6] f lqeh' / 2 cf6] f rte' h{' sf] gfd nv] . AB
19
ul0ft, sIff ^

3. tn lbOPsf axe' h' x¿sf eh' fx¿sf] ;ªV\ of / axe' h' x¿sf] gfd nv] .

-s_ -v_ -u_ -3_
4. ltdf| ] jl/kl/sf axe' h' cfsf/sf] sg' } kfr“ cf6] f j:tx' ¿sf] gfd nv] .

2.3 sDkf; / ?n/sf] k|of]uåf/f ;dafx' lqe'h / ju{ -Pp6f e'hfsf] nDafO
C
lbOPdf_ sf] /rgf

sDkf; / ?n/ ko| fu] u//] ;dafx' lqeh' sf] /rgf M A B
4 ;=] ld= sf] Pp6f /v] f v08 AB lvr . ;f] /]vf v08df
sDkf; ldnfO{ A af6 dflylt/ 4 ;=] ld= nDafOsf] Pp6f
rfk lvr . To:t} u/L plQs} rfk B af6 klg sf6/] C
laGb' gfds/0f u/ . A / C tyf B / C hf]8 . ABC

Pp6f ;dafx' lqeh' xf] .

sDkf; / ?n/ ko| fu] u//] jus{ f] /rgf M C D
A B
4 ;=] ld= sf] Pp6f /v] fv08 AB lvr . A df 900 sf]
sf]0f /rgf u/ . AC = 4 ;]=ld= lrx\g nufpm . C
af6 sDkf;sf] d2tn] 4 ;]=ld= sf] rfk lvr .
To;u} /L B af6 4 ;=] ld= sf] rfk lvrL C af6
lvlrPsf] rfknfO{ sf6 . sfl6Psf] laGbn' fO{ D gfd
b]pm . C / D tyf B / D hf]8 . ABDC Pp6f ju{
xf] .

cEof; 2.3

1. lgDglnlvt gfksf eh' fx¿ ePsf] ;dafx' lqeh' sf] /rgf u/ M

-s_ eh' f = 3 ;=] ld= -v_ eh' f = 4.5 ;=] ld=

-u_ eh' f = 5 ;=] ld= -3_ eh' f = 6 ;=] ld=

2. lgDglnlvt gfksf eh' fx¿ ePsf] jus{ f] /rgf u/ M

-s_ eh' f = 3 ;=] ld= -v_ eh' f = 4 ;=] ld=
-u_ eh' f = 4.5 ;=] ld= -3_ eh' f = 6 ;=] ld=

20 ul0ft, sIff ^

PsfO 3 7f;] cfsl[ tx¿ (Solid Figures)

tnsf] tflnsfdf sx] L HofldtLo 7f;] cfsl[ t / ltgLx¿sf gdg' f bv] fOPsf] 5 . HofldtLo 7f;]
cfsl[ t;u“ ldNg] o:t} 3/3 cf]6f gd'gfsf] ;"rL tof/ kf/ .

HofldtLo 7f;] cfsl[ t gfd pbfx/0f

ufn] f (Sphere) km' 6an Unfa]
if8d\ v' f (Cuboid) ;nfOs{ f] a66\ f

3g (Cube) uf]6L

an] gf (Cylinder)

bw' sf] a66\ f / kl] G;n

;fn] L (Cone)

cfO;ljm| d sfg] wfgsf] /f;

7f;] cfsl[ tsf sg' fx¿ (Vertices), lsgf/fx¿ (Edges) / dfx] 8fx¿ (Faces)

tnsf] lrqsf] cWoog u/L lgDg lnlvt kZ| gx¿sf cfwf/df 5nkmn u/ M

o;df sltcf6] f cfotfsf/ dfx] 8fx¿ 5g\ < 21

ul0ft, sIff ^

bO' { cfotfsf/ ;tx ldns] f] 7fpn“ fO{ lsgf/f (Edge) elgG5 . if8d\ v' fdf sltcf6] f o:tf lsgf/fx¿
5g\ < tLgcf6] f lsgf/fx¿ ldns] f] 7fpn“ fO{ sg' f (Vertex) elgG5 . cfotfsf/ j:t'df sltcf]6f
sg' fx¿ 5g\ < s] sg' fx¿ aGg 3 cf6] f dfq lsgf/f ldNgk' 5{ jf Tofe] Gbf a9L klg xg' ;S5g\ <

Pp6f if8d\ v' fnfO{ x/] . if8d\ v' fdf sltcf6] f sg' fx¿
5g\ < if8d\ v' fsf] ;ae} Gbf dflysf] sg' fdf sltcf6] f
lsgf/fx¿ hfl] 8Psf 5g\ < To;n} ] 7f;] j:ts' f sg' f
aGgsf nflu 2 eGbf a9L lsgf/fx¿ hfl] 8Psf]
x'g'kb{5 .

sx] L 7f;] j:tx' ¿sf gdg' fx¿ lgdf0{ f (Construction of Some Models of Solids)

1. sfuh k6o\ fP/ (by paper folding) ljleGg cfsl[ tsf gdg' fx¿ agfpg ;lsG5 . lrqdf
bv] fOP h:t} afSnf] sfuhdf cfsl[ tx¿ lvr / jl/kl/sf] 3/] fdf sf6 . oxf“ 86n\ fOg nv] s] f]
7fpd“ f k6o\ fP/ lsgf/fx¿nfO{ ud nufP/ jf ;n] f6] k] ko| fu] u/L hf8] . s] sf] cfsl[ t aG5
k|To]ssf] gfd atfpm .

-s_ -v_ -u_

○○○○○○
○○ ○
○ ○
○
○ ○
○ ○
○

-3_ -ª_ -r_

○○○ ○○○○
○
○
○
○

○
○
○
○

○
○
○
○
○
○
○
○

○
○
○
○

dfly h:t} km/s km/s cfsl[ tsf 7f;] j:tx' ¿ lgdf0{ f u//] ljleGg /ª e/L sIff sf7] fdf ;hfP/ /fVg
;lsG5 . o:tf j:tx' ¿ lunf] df6f] ko| fu] u//] klg lgdf0{ f ug{ ;lsG5 . df6f] ko| fu] u/L 7f;]
cfs[ltsf] lgdf{0f u/]/ lzIfsnfO{ bv] fpm .

22 ul0ft, sIff ^

2. h;' vfg] kfOk jf lgufnf] jf 5j\ fnL ko| fu] u//] 7f;] j:ts' f] vfj] m| f] cfsl[ t (skeleton models)
lgdf0{ f ug{ ;lsG5 . o;sf nflu h'; vfg] kfOk, l;of] / wfuf] cfjZos kb{5 . pbfx/0fsf
nflu klxn] Pp6f lqe'h agfO{ x]/f}“ M

tL cf6] f kfOkdf wfuf] l5/fpg] k6o\ fP/ afW“ bf lqeh' aG5 .

ca, h;' vfg] kfOk ko| fu] u/L lgdf0{ f u/s] f cfsl[ t x/] / o:t} cfsl[ t lgdf0{ f u/L cfkmg\ f]
sIff sf]7f ;hfpm .
-s_ -v_

○○○○○○○○
○
○
○

-u_ -3_

-ª_

ul0ft, sIff ^ 23

cEof; 3
1. tn lbOPsf 7f;] cfsf/x¿sf] lsgf/f, dfx] 8f -;tx_ / sg' fx¿ ug/] nv] .

-s_ -v_

-u_

2. ltdf| ] jl/kl/ /xs] f rssf] a66\ f, 8:6/, lstfa cflb ;ªs\ ng u/ / ltgsf lsgf/f, dfx] 8f
-;tx_ / sg' fx¿ ug/] ;r" L agfpm .

3. pbfx/0fdf lbOPcg;' f/ sfuh tyf h;' vfg] kfOk, 5j\ fnL jf o:t} cGo j:tx' ¿af6 tnsf
gdg' fx¿ tof/ kf/ .

-s_ 3g -v_ if8d\ v' f

24 ul0ft, sIff ^

PsfO 4 lgbz{] fªs\ x¿ (Co-ordinates)

tnsf lrq x/] f+} M

5

4 dlGb/ wf/f
dv}b]nfg
c:ktfn
3

2 lkª 3/ ?v
x'nfs ljBfno

1

zfr} fno

0 1 23 4 5 6 7 8 9

lrqdf '0 bl] v 5 PsfO bfof“ uP/ 2 PsfO dfly ?v 5 eGgsf nflu klxnf t;] f{] To;kl5 7f8f] k9/]

¿v (5, 2) df 5 eGg'k5{ . of] egfOcg';f/ dlGb/ (3, 3) df 5 / xn' fs (0, 1) df 5 eGg ldN5 .

o:t} u/L, lrqdf c¿ j:tx' ¿ s;/L hgfOG5 < ljrf/ u/f“} . Y Y cIf

bfof“tkm{sf] lrq x]/f}“ M 6
5

oxf“ bO' c{ f6] f ;ªV\ of /v] fx¿ O df nDa xg' ] u/L 4
hfl] 8Psf 5g\ . o;df t;] f{] ;ªV\ of /v] fnfO{ x – cIf 3
(x-axis), 7f8f] ;ªV\ of /v] fnfO{ y - cIf (y-axis) / 2
sfl6Psf] laGb' O nfO{ pbu\ d laGb' (Origin) elgG5 .
1 X cIf

0 1 23 4 56 X

bfofs“ f] lrqdf laGb' P sf] :yfg hgfpg P af6 Y pbu\ d laGb'

x – cIf / y – cIfdf nDa lvRbf x – cIf / y – cIf 6
jm| dzM 3 / 4 df sfl6of] . o;nfO{ (3, 4) nl] vG5 / 5
oxf“ 3 nfO{ laGb' P sf] x – lgbz{] fªs\ (x-co-ordinate) 4P
/ 4 nfO{ laGbs' f] y – lgbz{] fªs\ (y-co-ordinate) 3
elgG5 . clg laGb' P nfO{ P (3, 4) klg n]lvG5 . 2
P (3, 4) n] pbu\ d laGb' O af6 3 PsfO bfof“ / 1

01 2 3 4 5 6 X

4 PsfO dfly hfb“ f kg{] laGb' P hgfp“5 . laGb' P sf lgbz{] fªs\ x¿

P (3, 4)

x- lgbz{] fªs\ y- lgbz{] fªs\

ul0ft, sIff – ^ 25

o;/L cIfx¿ (axes) sf] dbtaf6 ;dtn ;tx (plane) df /xs] f laGbx' ¿sf] l:ylt (position) ;xh}
kQf nufpg ;lsG5 .

cEof; 4

1. tn lbOPsf jufª{ l\ st sfuh (squared paper) df lbPsf laGbx' ¿ A, B, C, D, E, F, G,
H / I sf lgb]{zfª\sx¿ kQf nufpm M

Y

12 E
11

10 D F
9

8 C
7
G
6

5 H

4 B
3

2 AI
1

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 X

2. jufª{ l\ st sfuhdf lgDg lnlvt laGbx' ¿ e/ / kT| os] laGbn' fO{ jm| dzM hf8] b\ } hfpm . ss] f]
lrq aG5 <

(a) (2,6); (3,2) / (5,4)
(b) (3,1); (6,1) / (6,4)
(c) (4,4); (7,2) / (7,6)
(d) (0,0); (4,0); (6,4) / (3,5)
(e) (3,3); (7,3); (7,7) / (3,7)
(f) (4,6); (8,2); (7,6) / (8,9)
(g) (4,4); (4,10); (8,7); (6,7) / (8,4)

26 ul0ft, sIff – ^

3. lrqdf lbOPsf kT| os] cfsl[ tsf zLifl{ aGbx' ¿sf] lgbz{] fªs\ nv] M

Y

12 D C

11
10 (i)

9

8A B
7
6 N E
5M (iii)
4 Q
3 (ii)

2 P F GH
1

0 1 2 3 4 5 6 7 8 9 10 11 12 13 X

4. P(6,6), Q (6,10) / R (10,10) ju{sf tLg zLif{laGb'x¿ x'g\ . laGb' S jus{ f] rfy} f] zLifl{ aGb' xf]
eg] laGbx' ¿ cªs\ g u/L S sf] lgb]{zfª\s n]v .

5. laGb' P(3,2) / Q(7,6) hf8] g\ ] /v] fv08sf] dWolaGbs' f] lgbz{] fªs\ slt xG' 5 < jufª{ l\ st
sfuh k|of]u u/L cª\sg u/ .

ul0ft, sIff – ^ 27

PsfO 5 kl/ldlt (Perimeter), Ifq] kmn (Area) / cfotg (Volume)

5.1 lqeh' sf] kl/ldlt (Perimeter of triangle)

tn lbOPsf] lqeh' ABC sf] kl/ldlt s;/L yfxf kfpg ;lsG5 < 5nkmn u/ .

A

BC

oxf“ ΔABC sf] kl/ldlt yfxf kfpg tLgcf6] } eh' fsf] gfk yfxf kfpgk' 5{ . tLgcf6] f eh' f tyf To;sf]
gfk lgDgfg;' f/ 5 M

e'hf gfk

AB = c cm
BC = a cm

AC = b cm

ca ΔABC sf] kl/ldlt = BC + AC + AB

= (a + b + c)cm

ΔABC sf] kl/ldlt = (a + b+ c)cm xG' 5 .

ct M lqeh' sf] kl/ldlt egs] f] tLgcf6] f eh' fsf] nDafOsf] ofu] kmn xf] .

pbfx/0f 1 A
lbOPsf] Δ ABC sf] kl/ldlt slt xfn] f <

oxf“ ΔABC sf tLgcf6] } eh' fsf] gfk ?n/sf ;xfotfn] gfk . B C

eh' fsf] gfk

AB = 3 cm
BC = 4cm

AC = 2cm

ca, ΔABC sf] kl/ldlt = AB + BC + AC

= 3cm + 4cm + 2cm = 9cm

lqeh' ABC sf] kl/ldlt = 9 cm.

28 ul0ft, sIff ^

pbfx/0f 2
Pp6f ;dafx' lqeh' sf] Pp6f eh' f 3 cm 5 eg] To; lqeh' sf] kl/ldlt lgsfn <
oxf“, X

3cm 3cm

YZ
3cm

ΔXYZ sf] kl/ldlt = XY + YZ + ZX

= 3cm + 3cm + 3cm
= 9cm

lqeh' sf] kl/ldlt = 9 cm.
pbfx/0f 3
Pp6f ;dafx' lqeh' sf] kl/ldlt 150cm 5 eg] To; lqeh' sf] Pp6f eh' fsf] nDafO lgsfn .
oxf,“ lqeh' sf] kl/ldlt = 150 cm, eh' fsf] nDafO (l) = ?
lqeh' sf] kl/ldlt = ltg cf6] f eh' fsf] ofu] kmn
cyjf, 150cm = 3l [ ;dafx' lqeh' sf] kl/ldlt = 3l ]

cyjf,
∴ ;dafx' lqeh' sf] Pp6f eh' fsf] gfk = 50 cm 5 .
cEof; 5.1
1. tn lbOPsf] gfksf] lqeh' sf] kl/ldlt lgsfn M

(i) AB = 5cm, BC = 3cm, AC = 4cm

(ii) AB = BC = AC = 6cm
(iii) AB = 2.5cm, BC = 1.5cm, AC = 2.5cm
(iv) XY = 10m, YZ = 10m, ZX = 5cm
(v) AB = 3.5cm, BC = 2.5cm, AC = 1.5cm

ul0ft, sIff ^ 29

2. tnsf lqeh' x¿sf] kl/ldlt gfk/] lgsfn M P

AX

B CY ZQ R

3. Pp6f ;dafx' lqeh' sf] kl/ldlt 18cm 5 eg] lqeh' sf eh' fx?sf] nDafO slt xfn] f <

4. Pp6f ;dafx' lqeh' sf] kl/ldlt 42cm 5 eg] lqeh' sf eh' fx?sf] nDafO slt xG' 5 <

5. Pp6f ;dafx' lqeh' sf] kl/ldlt 60 ;=] ld= eP pSt lqeh' sf] kT| os] eh' fsf] nDafO nv] .

6. Pp6f lqeh' sf] kl/ldlt 20cm / To;sf bO' c{ f6] f eh' fx¿sf] nDafOsf] ofu] kmn 12cm 5
eg] afs“ L eh' fsf] nDafO slt xfn] f <

7. lbOPsf] lrqdf ags] f ABD / ADC sf] kl/ldlt lgsfn M

A 8cm C

6cm 10cm 6cm

BD

8. Pp6f ;dafx' lqeh' sf] Pp6f eh' fsf] nDafO 5.4cm 5 eg] pSt lqeh' sf] kl/ldlt slt
xG' 5 <

9. ;dlåafx' lqeh' df a/fa/ nDafO ePsf bO' { eh' fdWo] Pp6f eh' fsf] gfk 4.6cm / lqeh' sf]
kl/ldlt 15.2cm 5 eg] afs“ L eh' fsf] nDafO slt xfn] f <

10. Pp6f lqeh' ABC df eh' fx¿ AB / AC kT| os] sf] nDafO 6 ;=] ld= / pSt lqeh' sf]
kl/ldlt 20 ;=] ld= 5 . eh' f BC eh' f AB eGbf slt nfdf] 5, nv] .

30 ul0ft, sIff ^

5.2 Ifq] kmn (Area)

-s_ lgoldt tyf clgoldt cfsf/sf] Ifq] kmn (Area of regular and irregular shapes)

tn lbOPsf cfsl[ tx¿df sg' n] ;dtn ;txdf a9L 7fp“ lnPsf] 5 <

-s_ -v_

/ /

(a) (b) (a)
(b)
-u_

/

(a) (b)

oxf“ -s_ df lbPsf bO' { cfsl[ tx¿sf] cfsf/ p:t} eP klg -a_ n] eGbf -b_ n] ;dtndf a9L 7fp“
cfu] 6s] f] :ki6 bV] g ;lsG5 t/ -v_ / -u_ df km/s km/s cfsl[ t ePsfn] sg' n] a9L 7fp“ lnPsf]
5 ;xh} eGg sl7g 5 . sg' cfsl[ tn] a9L 7fp“ lnPsf] 5 eGg] s;/L yfxf kfOG5 eGg] ljifodf
5nkmn u/f“} .
tn lbOPsf cfsl[ tsf hf8] f x/] / kT| os] hf8] fdf sg' lrqn] a9L 7fp“ lnPsf] 5, cgd' fg u/ M

-s_ -v_ /
/
(i)
(i) (ii) (ii)

-u_ / -3_
/

(i) (ii) (i)
(ii)

oxf“ ;a} cj:yfdf tn' gf ugk'{ g]{ j:tx' ¿df p:t} PsfO / cfwf/ ko| fu] ePsfn] kT| os] hf8] fdf sg'
7n' f] 5 ;xh} eGg ;lsG5, s;/L <

ul0ft, sIff ^ 31

(I) (II)

dfly lbOPsf cfsl[ tx¿sf] Ifq] kmn sg' sf] a9L xG' 5 <
7n' f] cfsl[ t 5fKg sltcf6] f ;fgf cfsl[ t rflxPnf <

43

12

o;/L 4 cf6] f ;fgf] cfsl[ tn] 7n' f] cfsl[ t 5fKg ;lsG5 .
t;y,{ cfsl[ t (i) sf] Ifq] kmn cfsl[ t (ii) sf] Ifq] kmneGbf 4 u0' ff a9L 5 .
To;u} /L n] pSt cfsl[ t (i) 5fKg slt cf6] f rflxG5 <

76
85

1 4
2 3

hDdf 8 cf6] f n] cfsl[ t (i) 5fKg ;lsG5 .

;u“ s} f] lrq 1 ;=] ld= x 1 ;]=ld=sf] ju{ xf] . o;nfO{ s'g} klg 1 cm
j:tn' ] ;dtndf slt 7fp“ lnPsf] 5 eg/] u0fgf ug{ jf yfxf 1 cm
kfpg ko| fu] ul/G5 . o;nfO{ Ps ju{ ;=] ld= (1cm2)
n]lvG5 . 111111222222333333444444555555666666777777888888

;dtndf j:tn' ] lnPsf] 7fpn“ fO{ To; j:ts' f] Ifq] kmn
elgG5 . If]qkmnnfO{ ju{ PsfOdf gflkG5 . lrqdf lbOPsf]
cfotdf 1,1 cm2 sf jufs{ f/ sf7] f slt 5g\ <

hDdf ju{ sf7] f ;ªV\ of 12 cf6] f 5g\ . To;n} ] of] cfotsf]
Ifq] kmn 12 ju{ ;=] ld= (12 cm2) xG' 5 .

32 ul0ft, sIff ^

pbfx/0f 1

tnsf cfsl[ tsf] Ifq] kmn PsfO ug/] lgsfn M 111111111111111111111111111111111222222222222222222222222222222222333333333333333333333333333333333444444444444444444444444444444444555555555555555555555555555555555666666666666666666666666666666666777777777777777777777777777777777888888888888888888888888888888888999999999999999999999999999999999000000000000000000000000000000000111111111111111111111111111111111222222222222222222222222222222222333333333333333333333333333333333444444444444444444444444444444444555555555555555555555555555555555666666666666666666666666666666666777777777777777777777777777777777888888888888888888888888888888888999999999999999999999999999999999000000000000000000000000000000000111111111111111111111111111111111222222222222222222222222222222222333333333333333333333333333333333444444444444444444444444444444444555555555555555555555555555555555666666666666666666666666666666666777777777777777777777777777777777888888888888888888888888888888888999999999999999999999999999999999000000000000000000000000000000000111111111111111111111111111111111222222222222222222222222222222222111111111111111111111111111111111222222222222222222222222222222222
-s_ -v_

pQ/ M
-s_ l;ªu\ f] jus{ f] ;ªV\ of = 4

cfwf jus{ f] ;ªV\ of = 4
4 cf6] f cfwf jus{ f] slt cf6] f l;ªu\ f] xG' 5 <
4 cfwf = 2 l;ªu\ f]
To;n} ,] Ifq] kmn = (4 + 2) = 6
To;sf/0f, Ifq] kmn = 6 ju{ ;=] ld=

-v_ l;ªu\ f] jus{ f] ;ªV\ of = 5
cfwfeGbf a9L jus{ f] ;ªV\ of = 4
cfwfeGbf a9LnfO{ 1 ug/] cfwfeGbf sdnfO{ 5f8] /] uGbf
cgd' flgt Ifq] kmn =(5+4) = 9 ju{ PsfO
To;sf/0f, Ifq] kmn = 9 ju{ ;=] ld=

o;/L sf7] f uGg] tl/sfn] lgoldt tyf clgoldt cfsf/x¿sf] Ifq] kmn lgsfNg ;lsG5 .

cEof; 5.2

1. tnsf kT| os] cfsl[ tsf] Ifq] kmn sf7] f uGg] tl/sfaf6 lgsfn M

(a) (b) (c) (d)

(e) (f) (g) (h)

(i) (j) (k)

ul0ft, sIff ^ 33

2. tnsf kT| os] cfsl[ tsf] cgd' flgt Ifq] kmn lgsfn -cfwfeGbf a9LnfO{ 1 ug . cfwfeGbf
sdnfO{ 5f]l8b]pm ._ M

(b)
(a)

(c) (d)

(e)

-v_ cfotsf] Ifq] kmn (Area of a Rectangle) 3 cm

lrqdf nDafO 4cm / rf8} fO 3cm ePsf] Pp6f b
cfot lbOPsf] 5 . o;df 1 cm2 sf sltcf6] f
jux{ ¿ aG5g\ < sf7] fx¿ agfpm . l

t;] fd{] f sltcf6] f sf7] f ag] < 4 cf6] f -nDafOlt/_ 4 cm

7f8fd] f sltcf6] f sf7] fdf ag] < 3 cf6] f -rf8} fOlt/_ l
To;n} ,] Ifq] kmn = n= x rf=}
cfot
= 4cm x 3cm = 12cm2
ju{
cfot tyf jus{ f] Ifq] kmn lgsfNg] ;q" ,
cfotsf] Ifq] kmn (A) = nDafO (l) x rf8} fO (b) xG' 5 . l
ju{ cfsl[ tsf] Ifq] kmn lgsfNbf nDafO / rf8} fO a/fa/

xG' 5g\ . To;n} ] nDafO (l) = rf8} fO (b) = l xb“' f
jus{ f] Ifq] kmn (A) = l x l = l2
To;n} ,] jus{ f] Ifq] kmn (A) = l2 = -eh' f_2 xG' 5 .

To;n} ,] cfotsf] Ifq] kmn (A) = l x b / jus{ f] Ifq] kmn (A) = l2 xG' 5 .

34 ul0ft, sIff ^

pbfx/0f 1

nDafO 5 cm / rf8} fO 4 cm ePsf] cfotsf] Ifq] kmn slt xG' 5 <

oxf“,

nDafO (l) = 5 cm 4cm

rf8} fO (b) = 4 cm

Ifq] kmn (A) = ? 5cm

ca cfotsf] Ifq] kmnsf] ;q" ko| fu] ubf,{ A l= × b = 5 cm × 4 cm

∴ cfotsf] Ifq] kmn = 20 cm2

pbfx/0f 2

rf8} fOeGbf nDafO bfA] a/ ePsf] cfotsf] Ifq] kmn 18m2 5 eg] pSt cfotsf] nDafO / rf8} fO slt
xfn] f <

oxf,“ rf8} fO = bm. / nDafO = 2bm. -rf8} fOeGbf nDafO bfA] a/ ePsfn_] / Ifq] kmn A= 18m2

ca, A = l × b

cyjf, 18 = 2b × b
18
cyjf, b2 = 2 b

cyjf, b2 = 9

∴ b = 3 (b2 = 9 ePsfn_] 2b

To;n} ,] nDafO l = 2b = 2 × 3 = 6

∴ nDafO = 6m, rf8} fO = 3m

pbfx/0f 3

Pp6f jus{ f] Ifq] kmn 9cm2 5 eg] To;sf] nDafO slt xfn] f <

oxf,“ A = l2

9 = l2

3=l l

∴ nDafO = 3cm xG' 5 . l

ul0ft, sIff ^ 35

pbfx/0f 4

Pp6f jus{ f] eh' fsf] gfk 4cm 5 . 2cm rf8} fO ePsf] cfotsf] Ifq] kmn ;fx] L ju;{ u“ a/fa/ 5 eg]
cfotsf] nDafO slt xfn] f <

4 cm 2cm

?

4 cm

oxf,“ jus{ f] eh' f (l)= 4cm
jus{ f] Ifq] kmn (A) = l 2 = (4)2 = 16 cm2

kml] /, cfotsf] rf8} fO (b) = 2

nDafO (l) = ?
ca, A = l × b
cyjf 16 = l × 2

cfotsf] nDafO = 8 cm
cEof; 5.3

1. tn lbOPsf kT| os] cfsl[ tsf] Ifq] kmn lgsfn M

(a) (b)

2 cm 3 cm
3 cm 4 cm

36 ul0ft, sIff ^

(c) (d)

2 cm
4 cm

2 cm
6 cm

2. tn lbOPsf kT| os] cfsl[ tsf] yfxf gePsf] eh' fsf] dfg lgsfn M

(a) (b) (c) ?
? 5 cm A = 15 cm2 12 cm
A = 16 cm2
8 cm ? A = 12 cm2

2 cm (e) (f)
(d)

A= A = 21 cm2 ? A = 25 cm2 ?
10 ? 7 cm 5 cm
cm2

3. tn lbOPsf lrqx¿df 5fof k/s] f] efusf] Ifq] kmn slt xfn] f <

8 cm
(a) (b)

3 cm 1 cm 2 cm 5 cm
1 cm 3 cm 10 cm
37
5 cm

(c) (d) 6 cm
1 cm 5 cm
1 cm
1 cm 12 cm

1.5 cm 1 cm 1.5 cm

ul0ft, sIff ^

4. Pp6f jus{ f] Pp6f eh' fsf] gfk 6 ;=] ld= 5 eg] o;sf] Ifq] kmn / kl/ldlt lgsfn .

5. Pp6f cfotsf] nDafO rf8} fOsf] tA] a/ 5 . Ifq] kmn 12cm2 eP To;sf] nDafO / rf8} fO
slt xfn] f <

6. Pp6f ju{ / cfotsf] Ifq] kmn a/fa/ 5 . jus{ f] Ifq] kmn 16cm2 / jus{ f] eh' f, cfotsf]
nDafOsf] cfwf 5 eg] cfotsf] rf8} fO slt /x5] <

7. lrqdf Pp6f 3/sf] lgdf0{ f ofh] gf lbOPsf] 5 M

(a) a7} s sf7] f, efG5f, ;T' g] sf7] f 1 / ;T' g] sf7] f 2 sf] 5'6\6f5'6\6} If]qkmn lgsfn .

(b) 3/n] hDdf slt Ifq] kmnsf] hUuf cfu] 6s] f] /x5] <

4m 3m

3 m ;T' g] sf7] f 1 efG5f 3 m

2 m gx' fpg] sf7] f a7} s sf7] f 5m
3 m ;T' g] sf7] f 2

3m 4m

38 ul0ft, sIff ^

5.3 if8d\ v' f / 3gsf] cfotg (Volume of Cubiods and Cubes)

lrq x]/ / 5nkmn u/ M

o; ef8“ fdf slt rfdn c6fp5“ <
Pp6f rssf] a66\ fn] slt 7fp“ cfu] 65\ <
sg' } klg j:tn' ] cfu] 6s] f] 7fp“nfO{ To; j:ts' f]
cfotg elgG5 .

lrqdf bv] fOP h:t} 7n' f] Ansdf /xs] f
;fgf ;fgf Ansx¿sf] ;ª\Vof ug .

7n' f] Ans = 12 cf6] f ;fgf Ansx¿ xG' 5g\ . 2 cm
;fgf] Anssf] nDafO, rf8} fO / prfO gfk .
3 cm 2 cm
;fgf] Anssf] nDafO = 1 ;=] ld=, rf8} fO = 1
;=] ld= / prfO = 1 ;=] ld= l
l
;fgf] Anssf] cfotg = 1 3g ;]=ld= x'G5 . 7'nf]
Ansdf 12 cf6] f ;fgf Ansx¿ xG' 5g\ . To;n} ,] 39
7n' f] Anssf] cfotg = 12 3g ;]=ld= x'G5 .
ca, 7n' f] Anssf] nDafO, rf8} fO / prfO gfk .

7n' f] Anssf] nDafO = 3 ;=] ld=, rf8} fO = 2 ;=] ld=,
prfO = 2 ;]=ld= /x]sf] 5 .

To;n} ,] if8d\ v' fsf] cfotg = nDafO x rf8} fO x prfO

= 3 ;=] ld= x 2 ;=] ld= x 2 ;=] ld=

= 12 3g ;=] ld=

∴ if8d\ v' fsf] cfotg = nDafO x rf8} fO x prfO x'G5 .

3gsf] cfotg s;/L lgsfNg ;lsG5, ljrf/ u/ . l
3gsf] nDafO, rf8} fO / prfO a/fa/ xG' 5 .
∴ cfotg = nDafO x rf8} fO x prfO = n= x n= x n=

= -nDafO_3
= l3

ul0ft, sIff ^

cEof; 5.4 (b)

1. tn lbPsf j:ts' f] cfotg slt xfn] f <

(a)

(c) (d)

2. tn lbOPcg;' f/sf gfk ePsf 7f;] x¿sf] cfotg lgsfn M
(a) nDafO = 3 ;=] ld=
rf8} fO = 2 ;=] ld=
prfO = 5 ;=] ld= ePsf] if8d\ v' f

(b) eh' f = 4 ;=] ld= ePsf] 3g
3. nDafO 4 ;=] ld=, rf8} fO 3 ;=] ld= / prfO 2 ;=] ld= ePsf] if8d\ v' fsf] cfotg lgsfn .
4. 5 ;]=ld= e'hf ePsf] 3gsf] cfotg lgsfn .
5. 512 3g ;=] ld= cfotg ePsf] Pp6f 3gfsf/ a66\ fsf] nDafO slt xfn] f <
6. rf8} fOsf] bO' { u0' ff nDafO / 5 ;=] ld= prfO ePsf] if8d\ v' fsf] cfotg 250 3g ;=] ld= eP

if8\d'vfsf] nDafO / rf}8fO lgsfn .
7. Pp6f 3gfsf/ j:ts' f] cfotg 1331 3g ;=] ld= 5 eg] o;sf] Pp6f eh' fsf] nDafO slt

xfn] f <

40 ul0ft, sIff ^

PsfO 6 :yfgfGt/0f (Transformation)

tnsf lrqx¿ cjnfs] g u/ / tn lbOPcg;' f/sf] kl/jtg{ bl] vG5 ls bl] vb“ g} 5nkmn u/ .

B

0 B A
A
lrq g= 2
P]gf

lrq g= 1

B

A lrq g= 4

lrq g= 3

 lrq g= 1 df dg} aQLnfO{ Pg] f cufl8 /fVbfsf] cj:yf A / Pg] f k5fl8sf] cj:yf B 5 .
Pg] faf6 A / B a/fa/ b/' Ldf xG' 5g\ cyjf OA = OB 5 .

 lrq g= 2 df rp/df Pp6f lsnfdf 8f/] Ln] afw“ /] /fvs] f] ss' /' 8f/] L tGsg] u/L 3D' bfsf]
cj:yfx¿ xg' \ .

 lrq g= 3 df Pp6f uf8L :yfg A af6 :yfg B df kU' bf lglZrt lbzfdf b/' L kf/ u/s] f] cj:yf
xf] .

 lrq g= 4 df jux{ ¿ OABC, ODEF / OGHI sf] Pp6} ;femf zLifl{ aGb' O 5 . ju{ ODEF,
ju{ OGHI sf] ;fgf] (reduced) / ju{ OABC sf] 7n' f] (enlarged) cj:yf xf] .

ul0ft, sIff ^ 41

o;/L dflysf lrqx¿af6 xfdL of] lgisifd{ f kU' 5f“} .

sg' } lglZrt lgoddf /xL sg' } j:ts' f] l:ylt (Position) jf cfsf/ (Size) df kl/jtg{ xg' n' fO{ pSt
j:ts' f] :yfgfGt/0f (Transformation) eGb5g\ .

ltdf| ] jftfj/0f jl/kl/ eO/xs] f o:t} kl/jtg{ x¿sf af/d] f 5nkmn u/ .

cEof; 6

1. tn lbOPsf lrqx¿ Pg] f cufl8sf] cj:yf A xf] eg] Pg] fleq bl] vg] cj:yf (B) s:tf] xG' 5
lvr <

(a)

AB AB

P]gf P]gf

(c)

AB

P]gf

2. tn lbOPsf lrqx¿df cj:yf A af6 cj:yf B df hfb“ f l:ylt, :yfg jf cfsf/dWo]
sg' sg' df kl/jtg{ eof] nv] .

(a) B (c)
A
B

(b) B A
A

3. ;fOsnsf] kfªu\ f| 3d' fpb“ f, /f6] ] lkª vN] bf, hft“ f] 3d' fpb“ f, 9ª' u\ f u8' fpb“ fsf cj:yfx¿df
j:td' f s] s] kl/jtg{ bV] 5f,} 5f6] s/Ldf nv] .

42 ul0ft, sIff ^

PsfO 7 ;dldlt / 6l] ;n;] g (Symmetry and Tessellation)

7.1. ;dldtLo lrqx¿ (Symmetrical figures):

tn lbOPsf lrqx¿ cjnfs] g u/ / s] ljzi] ftf bV] of} cfk;df 5nkmn u/ .

To:t,} rf/ cf6] f a/fa/ jufs{ f/ sfuhsf 6j' m| fx¿nfO{ tn lbOPcg;' f/ /v] f AB af6 k6o\ fpm .

s] tL k6o\ fPsf 6j' m| fx¿ Pscfk;df a/fa/ 5g\ < AA

A

AB

B B
B

dflysf lrqx¿df kT| os] jufs{ f/ sfuhsf 6j' m| fnfO{ /v] f AB af6 bO' { a/fa/ efudf k6o\ fpg
;lsG5 . o;/L bO' { a/fa/ efudf k6o\ fpg ;lsg] lrqx¿nfO{ ;dldtLo lrqx¿ elgG5 . /v] f AB
nfO{ ;dldltsf] cIf (Axis of symmetry) elgG5 . Pp6f jufs{ f/ sfuhnfO{ rf/ tl/sfn] bO' {
a/fa/ efudf ljefhg ubf{ rf/cf6] f ;dldtLo cIfx¿ xG' 5g\ . o:t} Ps cyjf PseGbf a9L
;dldtLo cIfx¿ ePsf j:tx' ¿ xfdf| ] jftfj/0f jl/kl/ kfOG5g\ ls kfOb“ g} g\ 5nkmn u/ .

s] ;dldlt cIf gePsf lrqx¿sf klg pbfx/0f lbg ;S5f} <

○○○○○○○○○○○○○ ○○○○○○○○○○○○○○ ○○○○○○○○○○○○○

Pp6f ;dldlt cIf bO' c{ f6] f ;dldlt cIf ;dldlt cIf 5g}

ul0ft, sIff ^ 43

cEof; 7.1
1. tn lbOPsf ;dldtLo lrqx¿sf ;dldtLo cIfx¿sf] gfd nv] .

(a) (b) (c)

2. ltdf| ] 3/ cyjf :sn' df ePsf sg' } tLgcf6] f ;dldtLo j:tx' ¿sf] gfd nv] .

3. ltdf| ] :sn' cyjf 3/ jl/kl/ ePsf af:s6] an sf6] ,{ km' 6an uf| pG8 / elnan sf6] { cflb
;dldtLo lrqsf pbfx/0f xg' \ jf xfO] gg\ nv] .

4. tn lbOPsf kT| os] lrqdf 86n\ fOg ko| fu] u/L ;dldltsf] cIf/v] f lvr . sg' sg' lrqdf
;dldlt cIf/v] f Pp6feGbf a9L xG' 5 <

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)
(j) (k) (l)

44 ul0ft, sIff ^

5. tn lbOPsf kT| os] HofldtLo cfsl[ tsf] 86n\ fOg ko| fu] u/L ;dldltsf] cIf lvr M

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

6. tnsf cªu\ h]| L cIf/ k/" f u/ -s] s] zAb aG5g\ nv] _ M
-s_ -v_

-u_ -3_

ul0ft, sIff ^ 45