MATHS GRADE 7 teacher guide

PsfO M 1
/v] f / sf]0fx¿

(Line and Angles)

cgd' flgt 3G6L M 15

k"j{1fg
of] kf7 ;'? ug'{kj" { ljb\ofyL{x¿df lgDgfg';f/sf] 1fgsf] k'g/fjnf]sg u/fpgk' g]{ 5 M
-s_ kf| ]6]S6/sf] k|ofu] u/L sf]0fx¿ lvRg / lbOPsf sf0] fx¿ gfKg
-v_ sDkf;sf] ko| f]u u/L /]vfsf] nDafw{s lvRg

-u_ sDkf;sf] k|of]u u/L lbOPsf] sf]0fs]f cws{ lvRg
-3_ Gog" sf]0f, ;dsf0] f, clws sf0] f, ;/n sf]0f, ax[ tsf]0f 56' o\ fpg

-ª_ ;dfgfGt/ / kl| tR5]lbt /v] fx¿ lvRg .
kf7 kl/ro
Hofldlt cWoog laGb'af6 ;?' u/L /]vflt/ xb' } ;dtnLo cfs[lt tyf 7f;] j:tx' ¿sf af/d] f cWofog ul/G5 .
cfwf/et" tx sIff & df ljb\ofyL{x¿n] sf]0fx¿, hf8] f sf]0fx¿ / ltgLx¿sf u'0fx¿sf] k/LIf0fsf ;fy ;fy}
;dfgfGt/ /]vfx¿nfO{ 5]bsn] sf6b\ f aGg] hf]8f sf0] fx¿ / ltgLx¿sf] ;DaGwsf] af/]df cWoog ug5]{ g\ .
HofldltnfO{ ljbo\ fyL{ {n] u//] l;Sg] ljlwaf6 cWoog u/fpbf k|efjsf/L / cyk{ "0f{ l;sfO x'G5 .

1.1 sDkf;sf] k|ofu] åf/f sf]0fx¿sf] /rgf (Construction of angles using compass)
pbb\ Z] ox¿ M o; kf7sf] cWoog u/L;s]kl5 ljbo\ fyLx{ ¿ lgDg sfo{ ug{ ;Ifd x'g] 5g\ M
1. sDkf;sf] ko| fu] af6 60, 120,90 ,30, 45, 135, 75, 105 sf sf]0fx¿ lvRg
2= lbPsf] sf0] f;u“ a/fa/ x'gu] /L sf0] fsf] /rgf ug{
3. zLiff{ledv' sf]0f, cfzGg sf]0f, ;dk/" s sf]0f, kl/k/" s, PsfGt/, ;ª\ut / j|mdfut leqL sf]0fx¿sf]
kl/ro lbg .
4. zLiff{ledv' sf]0f, ;/nsf0] f, k0" fs{ f0] f ;DaGwL sx] L ;fdfGo HofldtLo tYox¿sf] k/LIf0f ug{ .

lzIf0f ;fdu|Lx¿
sDkf;, k|f6] S] 6/, ?n/ cflb ePsf] HofldtLo cf}hf/ afs;
l;sfO ;xhLs/0f ljm| ofsnfkx¿

1. sDkf; / ?n/sf] k|ofu] af6 60 sf] sf]0fsf] /rgf
DRAFT
-s_ ;a} ljb\ofyL{x¿nfO{ cf–cfkm\ gf] sfkLdf lglZrt gfksf] cwA{ of; lnO{ Ps Ps cf6] f j[Tt lvRg
nufpg'xf];\ .

-v_ jT[ tdf Pp6f Aof; AB lvRg nufpgx' f];\ . jT[ tsf] s]Gb|nfO{ O gfd lbg nufpgx' f];\ .

-u_ j[Ttsf] cwA{ of; a/fa/sf] rfkn] laGb' B af6 kl/wLdf sf6g\ nufpg'xf;] \ . C

-3_ pSt sfl6Psf] laGb'nfO{ C gfd lbg nufP/ laGb' O / C hf]8g nufpgx' f];\ . 60
O
-ª_ COB nfO{ k|f6] ]S6/n] gfKg nufpg'xf;] \ . A B

-r_ pSQ sf]0f slt l8uL| gfksf] x'G5 < 5nkmn ug{ nufpg'xf];\ .

2. sDkf; / ?n/sf] k|of]uaf6 120 sf] sf0] fsf] /rgf DC

-s_ ljb\ofyLx{ ¿nfO{ Pp6f /]vfv08 PQ lvRg nufpg'xf;] \ . 120 B
A Q
P

-v_ PQ df 3 cm gfksf] b'O{ 5]pdf j|mdzM laGb' A / B lng nufpg'xf;] \ .

-u_ AB a/fa/sf] sDkf;df rfk lnP/A nfO{ s]Gb| dflg cwj{ T[ t lvRg nufpg'xf];\ .

-3_ ;fx] L rfkn] laGb' B af6 cw{jT[ tsf] kl/lwdf rfk sf6g\ nufpg] / C gfd lbg nufpg'xf];\ .

-ª_ k'gM laGb' C af6 ;fx] L rfkn] kl/lwdf sf6\g nufP/ D gfd lbg nufpg'xf];\ .

2

-r_ ?n/sf] k|of]u u/L laGb' A /D hf8] g] . kf| ]6S] 6/ Kf|of]u u/L BAD gfKg nufpgx' f];\ / stL l8u|Ldf
sf65\ .

-5_ ;/rgfsf r/0f / sf]0fsf] dfg ;DaGwdf kZ| gf]Tt/ / 5nkmn u/fpgx' f];\ .

csf{] tl/sf,

-s_ Pp6f l;wf /]vf AOB lvRgx' f;] \ . sDkf;sf] k|ofu] u/L laGb' O df 60sf] sf]0f lvRg] . C

-v_ oxf“ BOC =60xG' 5 . caAOC gfk/] xg] {nufpg'xf];\ . slt l8uL| xG5 < 120
k|Zgf]Tt/ / 5nkmn u/fpg'xf];\ . O

3. sDkf; / ?n/sf] k|ofu] af6 75sf] sf0] fsf] /rgf A B

-s_ /]vfv08 XY lvRg] / 4 cm sf] efudf lrqdf h:t}
u/L A / B gfd lbg nufpg'xf;] \ .

-v_ laGb' A df sDkf;sf] ko| f]u u/L 90sf] sf]0f lvRg nufpg'xf;] \ .

-u_ xfdLnfO{ yfxf 5, 90 – 15 = 75 x'G5 .
t;y,{ xfdLn] 60/ 90sf] larsf] sf0] fsf]
cw{s lvRgk' 5{ .

-3_ sDkf;df lgZrLt gfksf] rfk lnO{ E/Caf6 rfk lvrL rfkx¿ sfl6Psf] 7fp“ P nfO{ laGb' A ;Dd
hf8] g] / BAF nfO{ gfKg nufpgx' f;] \ .

-ª_ cfaZos sf0] f BAF=75x'G5 .

gf6] M oxf“ lrqdfPAX sf] gfd 105xG' 5 .
DRAFT
4. sDkf; / ?n/sf] ko| fu] af6 105°sf] /rgf

xfdLnfO{ yfxf 5, 90°+15°=105° xG' 5 . C ED B
-s_ ljb\ofyLx{ nfO{ klxn] 90 sf] sf]0f lvRg nufpg'xf;] \ .
-v_ 90° / 120° sf] sf]0fsf] larsf] sf]0f slt x'G5 < kTtf nufpgx' f;] \ . 105° A
O
-u_ Tof] sf]0f 30° sf] x'G5 . lrqdf COD = 30°x'G5 .
-3_ pSt sf]0fsf] cws{ lvRg nufpgx' f;] \ .

AOE= AOD + DOE= 90°+15°= 105° x'G5 .

To;}u/L, 135° / 150° sf nflu 135° = 90° + 45° / 150° = 120°+ 30° sf sf]0fx¿ lvrL /rgf ug{
nufpgx' f;] \ .

SR Q 3 C
X B

135° E 150°
DBA

cGTodf ljbo\ fyLx{ ¿nfO{ ;d"xdf ljefhg u/L k|To]s ;dx" nfO{ j|mdzM 60 ,120, 75,135, 150, 90
gfksf sf]0fx¿sf] /rgfu/L sIffdf k|:tt' ug{ nufpgx' f];\ .

d"Nofªs\ g

ljb\ofyL{x¿nfO{ km/s km/s gfksf] rfk lnP/ 60 ,120, 75,135, 150, 90 sf sf0] fx¿ /rgf ug{
nufpg'xf];\ / k|To]s ljb\ofyL{x¿sf] sfos{ f] lgl/If0f Ufg'{xf];\ .
lzIfsnfO{ yk hfgsf/L M

3l8sf] ;'Os{ f] ljkl/t lbzfdf jm| dzM 45° / 30° sf] sf]0fsf] /rgf u/L sf0] fsf] gfk k|f6] ]S6/sf] k|of]u u/L
135° / 150° sf sf0] fx¿sf] /rgf ug{ ;lsG5 .

1.2 sDkf;sf] k|of]uaf6 a/fa/ sf0] fsf] /rgf ( Construction of equal angles using compass)
cg'dflgt lzIf0f 3G6L @
pb\oZ] oDRAFT

o; kf7sf] cGTodf ljbo\ fyLx{ ¿n] lbPsf] sf0] f;u“ a/fa/ x'gu] /L sf0] fsf] /rgf ug{ ;Ifd xg' ] 5g\ M

;xhLs/0f lj|mofsnfkx¿

1.s_;a} ljbo\ fyLx{ ¿nfO{ cf–cfkm\gf] skLdf :sn] / l;;fsndsf] ko| fu] B
u/L Ps Ps cf]6f sf0] fx¿ lvRg nufpg'xf;] \ / To;nfO{

AOB gfdfs/0f ug{ nufpg'xf;] \ . Y

v_ laGb' O df sDkf;sf] l;of] /fvL lglZrt gfksf] rfk lvRg O XA
nufppgx' f;] \ .

-u_ OA / OB df sfl6Psf] laGb'nfO{ j|mdzM X / Y gfd lbg nufpg'xf];\ . R

o;/L ags] f] sf0] f AOB ;u“ a/faf/ x'g]csf]{ sf0] f lvRg lgDg cg;' f/ D
sfos{ nfk u/fpg'xf];\ M

To;kl5 Pp6f csf]{ l;wf/]vf PQ lvRg nufpg'xf;] \ . PC Q

 lj|mofsnfk v df pNnv] u/] cg';f/sf] rfkn] laGb' P df klg rfk lvRg nufpg'xf;] \ .

 ca sDkf;sf] l;of]nfO{ laGb' Xdf /fvL Y ;Ddsf] gfk lng] / ;fx] L gfksf] rfkn] C af6 klxns] f]
rfkdf sf6\g] / D gfd lbg] / :sn] n] laGb' D / P hf8] g] .

ca AOB /QPR gfk/] 5nkmn ug{ nufpg'xf];\ .

kl| tlaDa M a/fa/ sf0] fsf] /rgf ubf{ lsg a/fa/ rfk lnOof,] /]vf PQ sf] gfk lsg hltlnPg xG' 5, a/fa/
sf]0fsf] /rgfsf] pkofl] utf s] 5 h:tf s/' fx¿df k|ltljDag u/fpg] .

1.3 sf]0fsf hf] 8fx¿sf] kl/ro (Introduction to pair of angles)

cgd' flgt lzIf0f 3G6L M #

pb\b]Zox¿

4

o; kf7sf] cGTodf ljbo\ fyL{x¿ lgDg sfo{ ug{ ;Ifd xg' ] 5g\ M

-s_ cfzGg sf0] f, zLiffl{ ed'v sf]0f, ;dk/" s sf0] f, kl/k"/s sf0] fx¿sf] kl/efiff atfpg

-v_ cf;Gsf0] f, zLiffl{ edv' sf0] f, ;dsf]0f, kl/k"/s sf0] fx¿ lrGg / 56\ofpg .

lzIf0f ;fduL| x¿

Hofldlt afs; / ;fbf sfuh

;xhLs/0f lj|mofsnfkx¿

-s_ ;a} ljbo\ fyL{x¿nfO{ JolStut ¿kdf Pp6} zLifl{ aGb' ePsf bO' { jf ;f]eGbf w]/} sf0] fx¿ agfpg
nufpgx' f;] \ . s:tf sf]0fx¿ agfp5g\ cjnfs] g ub{} cfjZos ;xofu] ug'{xf];\ .

-v_ ljb\ofyLn{ fO{ 4÷4 hgfsf] ;d"xdf ljefhg ug{x' f];\ pgLx¿nfO{ jm| dzM a, b,
c / d gfd lbgx' f;] \ . ;a}nfO Ps Ps cf]6f ;fbf k]k/x¿ af8g\ x' f];\ . pSt

sfuhsf] kfgfnfO{ bO' { lt/af6 k6o\ fpg nufpgx' f];\ . s:tf] cfsl[ t aG5 <
sflkdf agfpg nufpgx' f;] \ . To;kl5 pSt cfs[ltdf b]lvPsf b'O{ cf]6f
/v] fv08x¿nfO{ jm| dz: AB /CD gfd lbO sfl6Psf] 7fp“nfO{ Ogfdfs/0f
ug{ nufpgx' f];\ . slt cf6] f sf0] fx¿ aG5g\ / j|mdz sf]0fx¿nfO{ a,b,c /
d gfd lbg] . a / b gDa/ ePsf ljb\ofyLx{ ¿n] a, b / c / d gDa/
eP/ ljbo\ fyL{n] c / d sf ;femf efux¿sf] ;fy} ;DaGw klxrfg u/L
n]Vg nufpg'xf];\ .
DRAFT
To;kl5 Pp6f ;femf e'hf / Pp6} zLif{laGb' ePsf cfzGg sf]0fx¿sf] af/]df

:ki6 kfl/lbg'xf;] \ . kT| o]s hf8] LnfO{ cfkmg\ f] efusf sf]0fx¿sf] gfd n]Vg nufpg'xf;] \ . AOC,

COB, BOD, AOD ca ;a} ljb\ofyL{x¿nfO{ lrqdf bv] fOP h:t} lrq lbg] / slt hf]8f cfzGg
sf]0fx¿ 5g\, JoSltut ¿ksf ;r" L tof/ ug{ nufpgx' f;] \ / cfjZos ;xof]u ug'{xf];\ .

gf] 6 M cGgf;Gg sf0] fx¿sf af/d] f 5nkmn u/fg'xf];\ .

2. ljb\ofyLx{ ¿nfO{ 4/ 4 hgfsf] ;d"xdf ljefhg ug{'xf;] \ . ! / # gDa/ ePsf ljbo\ fyL{x¿n] !, # / @
/ $ gDa/ eP/ ljbo\ fyLn{ ] @ / $ sf sf0] fx¿ gfkL ;DaGwsf] Aff/] nV] g nufpg'xf];\ .

o;/L b'O{ /v] fx¿ cfk;df sfl6“bf ljkl/t lbzfdf aGg] hf8] f sf0] fx¿nfO{ zLif{ledv' sf0] fx¿ elgG5

egL k|:6 kfgx'{ f];\ .

To;kl5 k|Tos] ;d"xdf JoSltut ¿kdf b'{O cf6] f l;wf/v] fx¿Ps cfk;df AC

sfl6Psf lrq agfP/ zLif{led'v sf0] fx¿ 5'6\ofpg nufpgx' f];\ . O

lzIfsn] oL hf]8L sf0] fx¿sf] kl/efiff, k|sf/sf af/d] f :ki6 kfgx'{ f];\ .

d"Nofªs\ g DB
lbOPsf] lrqdf s_ rf/ hf8] f cfzGgsf]0fx¿

-v_ b'O{ hf]8f zLifl{ ed'v sf0] fx¿sf ;"lr tof/ kfg{ nufpgx' f];\ .

3. ljbo\ fyLx{ ¿sf ;d"x tof/ kfg'{xf;] / k|Tos ;dx" nfO{ Ps Ps hf]8f x
y
;ªV\ of lbgx' f;] \ tL ;ª\Vofx¿sf] of]ukmn 90 cyjf 180 x'gk' 5{ . h:t} M

30 / 60; 90 / 90; 120 / 60; 20 / 70; 135 / 45 cflb .
o;kl5 kT| os] Ps Ps cf]6f ;fbf sfuh lbgx' f;] \ / cfkm"nfO{ k/s] f]
;ª\Vof a/fa/ gfk ePsf] sf0] f k|f]6S] 6/ k|ofu] u/L lvRg nufpg'xf;] \
;d"xnfO{ pSt b'O{ sf0] fx¿sf] of]ukmn lgsfNgnufpg'xf;] \ . kT| o]s hf8] Lsf]

5
xy

lvrs] f hf8] f sf0] fx¿ slt sf]0fx¿ /x]5g < kfn}kfnf] k|:t't ug{ nufpg'xf];\ .

o;/L, b'O{ cf]6f sf]0fx¿sf] ofu] kmn 90 eof] eg] pSQ hf]8f sf]0fx¿nfO{ ;dk/" s sf0] fx¿ elgG5 . To:t},
bO' { sf]0fx¿sf] of]ukmn 180 eof] eg] pSQ hf8] f sf0] fx¿nfO{ kl/k/" s sf]0fx¿ elgG5 .

cGTodf olb bO' s{ f0] fx¿ x / y 5g\ / x+ y=90 eP pSt hf]8L sf]0fx¿ ;dk'/s sf0] fx¿ xG' 5g\ /
x+y=180eP pSt hf]8L sf]0fx¿ kl/k/" s sf]0fx¿ x'G5g egL ;fdfGoLs/0f ug{'xf;] \ .

5]bsn] /]vfx¿;“u agfpg] sf0] fx¿ (Angles made by transversal with lines)

pbb\ ]Zox¿
o; kf7sf] cGTodf ljb\ofyLx{ ¿ lgDg sfo{x¿ ug{ ;Ifd x'g] 5g\ M

-s_ 5]bssf] kl/ro lbg, 5]bsn] bO' { cf6] f /v] fv08nfO{ sf6b\ f aGg] leqL tyf aflx/L sf0] fx¿ 5'6\ofpg .

-v_ PsfGt/ sf0] fx¿, ;ª\utsf]0fx¿ tyf j|mdfut leqL sf0] fx¿sf] kl/ro lbg .

-u_ lbOPsf] lrqdf PsfGt/ sf0] fx¿, ;ª\ut sf0] fx¿ tyf j|mdfut leqL sf]0fx¿sf hf8] L kTtf nufpg .

;xhLs/0f lj|mofsnfkx¿

1. k|Tos] ljb\ofyLx{ ¿nfO{ Ps Ps cf]6f ;fbf sfuhsf] kfgf lng
nufpgx' f;] \ / ;an} fO{ lrqdf b]vfP h:t} t];f]a{ f6 @ k6s / 7f8fa] f6
58\s] u/L Psk6s k6\ofpg nufpgx' f];\ . To;kl5 k6o\ fP/ vf]n]sf]
sfuhdf ag]sf] cfs[ltnfO{ cf cfkmgf] sfkLdf ptfg'x{[ f];\ / slthf]8f
sf0] fx¿ aGb5g\ ug]/ n]Vg nufpgx' f];\ .
DRAFT
2. ;a} ljbo\ fyL{nfO{ $/$ hgfsf] ;dx" df laefhg ugx{' f;] \ . ;dx ! nfO{ lelq / aflxl/ sf]0fx¿sf] ;"lr tof/ kfg{
k/s] f leqL cGgfzGgsf]0fx¿ ;r" L tof/ kfg{ nufpgx' f;] \ / ;a} hfgx' f;] \ .
;dx" x¿sf] sfon{ fO{ cjnfs] g u/L cfjZostfcg';f/ ;xofu] ug{'xf];\ .
;a} ;d"xnfO{ cfcfkm\ gf] sfos{ f] af/d] f sIffdf k|:t't ug{ nufpgx' f;] \ /

ki[ 7kf]if0f lbb“ }] / :ki6 kfb{}

To;kl5 af8] {df Pp6f lrq lvRg] / klxn] agfOPsf ;dxx¿nfO{ jm| dzM pgLx¿sf] sfd 12
adfl] hd leqL / aflx/L sf]0fx¿ PsfGt/ sf]0fx¿, ;ª\ut sf0] fx¿ / j|mdfut 34
lelqsf0] fx¿sf] hf]8f kTtf nufpg nufpg'xf];\ .
56
;fy} ljbo\ fyLx{ ¿n] ;d"xdf k:| t't ul/;sk] l5 pgLx¿nfO{ JolStut ¿kdf kl/efiff n]Vg / 78
eGg nufpg'xf];\ .

gf]6M w/} ljbo\ fyL{ ePdf aG] rcg';f/sf] ;d"x lgdf0f{ u/L lbOPsf zLifs{ x¿ bfx] f]¥ofP/ cEof; u/fpg
;lsG5 .

d"Nofªs\ g

lbOPsf lrqdf a sf lbOPadf]lhdsf hf] 8L sf]0fx¿ kTtf nufpm . gh
-s_ PsfGt/ sf0] f fa
-v_ ;ª\ut sf]0f
-u_ jm| dfut lelqsf]0f eb
lzIfsnfO{ yk hfgsf/L de

ljb\offyLx{ ¿nfO{ yk ljm| ofsnfkx¿df ;Ung u/fP/ kf7\o k':tssf cEof;sf ;d:ofx¿ ;dfwfg
u/fpg'xf;] \ .

6

sf0] fx¿sf u0' fx¿sf] k/LIf0f ( Experimental Varification of Angles)

;xhLs/0f lj|mofsnfkx¿

1= zLiff{ledv' sf0] fx¿ MR

-s_ ljbo\ fyL{x¿nfO{ cfjZostfg';f/ ;d"xdf laefhg ug{'xf;] \ . !
$@
-v_ k|To]snfO{ cfk;df k|ltR5]lbt bO' { b'O{ cf]6f /v] fv08x¿ lVfRg nufpg'xf];\ /
o;/L ags] f sf]0fx¿nfO{ j|mdzM !, @, #, / $ gfd lbO{ / kf| ]6]S6/sf] k|of]u u/L #
;a}nfO{ JolStut ¿kdf ;a} sf]0fx¿sf] gfKg nufpgx' f;] \ . sg' s'g sf0] fx¿
a/fa/ eP ;d"xsf ;fyLx¿lar 5nkmn ug{ / ;dx" df lgisif{ lgsfNg QN
nufpg'xf];\ . ! =… ,  @ =… = ,  # =… = , /  $ =… =. o;/L
;a}sf] Pp6} lgisif{ s] kfOG5 eg] zLif{fled'v sf]0fx¿ a/fa/ xG' 5g .

-u_ o;}u/L cGo pbfx/0f / ;d:ofx¿ ;dfwfg u/fpg'xf];\ .

2. sg' } laGbs' f] al/kl/ Ps kl/j|md0fdf ag]sf sf]0fx¿df of]ukmn 360 xG' 5 .

-s_ ljb\ofyLx{ ¿nfO{ ;d"xdf ljefhg ug]{ / kT| os] nfO{ j|mdzM b]xfoadf]lhdsf lrqx¿ agfpg nufpg'xf];\ M

DRAFTa ab c
b cd ab

c d

-v_ ca k|To]sn] k|f]6]S6/ sf] k|ofu] u/L cfk"mn] agfPsf] sf0] fx¿sf] gfk kTtf nufpg'xf];\ .

 a = ....... ,b =....... c =........ d =........

-u_ sf]0fx¿sf] gfkkl5 cfPsf dfkgx¿sf] ofu] kmn kTtf nufpg'xf;] \ . h:t,} a + b + c = ...... a +

b + c + d =………

-3_ cfcfk\mgf] glthfnfO{ ;d"xsf ;fyLx¿;u“ 5nkmn u/L / ;femf lgisif{ sIffdf k:| t't ug{ nufpg'xf];\ .

o;sf] glthf sg} laGbs' f] kl/j|md0fdf ags] f sf0] fx¿sf] ofu] kmn 360 xG' 5 elg lzIfsn] :ki6 kfg{] .

gf6] M o;nfO{ kf7\o k:' tsdf lbOPsf tl/sfaf6 klg k/LIf0f ug{ ;lsG5 .

dN" ofªs\ g M ljb\ofyLx{ ¿nfO{ ;dx" df x sf] dfg lgsfNg nufpgx' f];\ .

s_ v_ u_

120 30 90 C
115
150 x 90
x
x

3. ;/n /]vfsf] s'g} laGb'df Ps} lt/ ag]sf sf]0fx¿sf] ofu] kmn 180 xG' 5 . AB
-s ljbo\ fyL{x¿nfO{ # / # hgfsf] ;d"xdf ljefhg u/L !, @ / # gfdfs/0f O

ug]{ . CD

7 A B
O

DE

C

A OB

-v_ ;an} fO{ l;wf/]vf AB lvrL lardf laGb' O nufpg lgb]{z lbg'xf];\ .

-u_ != hltn] O af6 Pp6f /]vf lvRg], @= hltn] O af6 b'O{ /v] f lvRg] /
#= hltn] u/L O af6 ltg cf6] f /]vfx¿ lvRg nufpg'xf];\ .

-3_ ca k|To]sn] cfcfkm\ gf] ;fgf sf]0fx¿sf] gfk k|f]6]S6/ k|of]u u/L kTtf
nufO{ ofu] kmn M AOC ,BOC,COD,
lgsfNg nufpg'xf;] \ . h:t}
BOD,DOE of]ukmn kTtf nufpg
,BOE./ ltgLx¿sf]
nufpgx' f;] \ .

-ª_ ;a}sf] ofu] kmn slt slt cfof] ;dx" df 5nkmn u/L ;fdl" xs lgisif{ lgsfNg / sIffdf k:| tt' ug{
nufpgx' f];\ . ljb\ofyL{sf lj|mofsnfkx¿sf] cjnfs] g ugx{' f];\ / pgLx¿nfO{ k[i7kf]if0f lbg'xf;] \ .
cGTodf lgDgfg';f/ lgisif{ atfpg'xf;] \ .

;/n /v] fsf] s'g} laGb'df Psl} t/ ags] f sf]0fx¿sf] ofu] kmn 180 xG' 5 .

dN" ofª\sg

cEof; 1.4 sf kZ| gx¿df x, y, z sf] dfg kTtf nufpg nufpg'xf];\ .

DRAFT
PsfO M @ cg'dflgt 3G6LM !@
lqe'h, rt'e'h{ / axe' 'h

(Triangle ,Quadrilateral and Polygon)

k"j1{ fg

 eh' f / sf0] fsf cfwf/df lq'eh' sf] juLs{ /0f

 sDkf; / ?n/sf] k|of]uåf/f ju{ -Pp6f eh' f lbOPdf_ / ;dafx' lqe'hsf] /rgf

 ax'eh' sf] kl/ro
kf7 kl/ro
l;wf /v] fv08x¿ cfk;df ldn/] aGg] aGb cfs[ltnfO{ g} ax'e'h elgG5 . # cf6] f $ cf6] f, % cf6] f
e'hfx¿n] ag]sf aGb cfsl[ tx¿nfO{ sd| z lqeh' , rte' {'h k~reh' elgG5 . lqe'h tyf rt'e{'hx¿ klg
w/] } ks| f/sf xG5g\ . ltgLx¿sf u'0fx¿ klg km/s km/s x'G5g\ . ljb\ofyLx{ ¿n] bl} gs hLjgdf ljleGg
lqeh' fsf/, cfoftsf/, juf{sf/ j:tx' ¿sf] ko| fu] ul//x]sf xG' 5g\ .

pbb\ ]Zox¿ M o; kf7 cWoog ul/;s]kl5 ljb\ofyLx{ ¿ lgDg sfo{ ug{ ;Ifd x'g] 5g\ M

8

!= lqe'hsf] /rgf ug{ .
@= rte' '{h - cfoft, ju{, ;dafx' rte' h'{ , ;dfgfGt/ rt'e{'h_ sf lazi] ftfx¿ kTtf nufpg .
#= rte' '{hsf] /rgf ug{ .
$=lgoldt axe' 'hx¿sf leqL tyf aflx/L sf0] fx¿ kTtf nufpg .
kf7zLif{s M lqe'hsf] /rgf

pbo\ Z] ox¿
o; kf7sf] cGTodf ljbo\ fyLx{ ¿ lgDg sfox{ ¿ ug{ ;Ifd xg' ] 5g\ M
s_ lbOPsf] cj:yfdf lqeh' sf] /rgf ug{ ;Sg5] g\ .

lzIf0f ;fdu|Lx¿ XC
HofldtLo cf}hf/ afs; / Go'h lk|G6 k]k/ cflb .

l;sfO ;xhLs/0f lj|mofsnfkx¿ 6 cm

lqe'h egs] f] s] xf] < k|To]s ljbo\ fyL{x¿nfO{ cfcfkmg\ f] 45
sfkLdf nV] g nufO{ k'g/fjnfs] g u/fpgx' f];\ .
A 5 cm B
lj|mofsnfk 1

DRAFT1. Pp6f sf]0f / bO' { cf6] f eh' fsf] nDafO lbPdf

-s_ ?n/sf] k|ofu] u/L ;a}nfO{ Ps Ps cf]6f l;wf/v] f AB lvRg nufpgx' f;] \ . h:t} AB = 5 cm

-v_ sDkf;sf] k|ofu] u/L laGb' B df lbOPsf] sf]0f lvRg nufpg'xf;] \ . h:t}, B=45 /BX /v] f lvRg
nufpgx' f;] \ .

-u_ ca BX df laGb' B af6 sDkf;df 6 cm a/fa/sf] rfk lnO{ lrx\g nufpg / C gfd lbg
nufpg'xf];\ .

-3_ laGb' A / C ?n/sf] ko| f]u u/L hf8] g nufpgx' f];\ .

-ª_ cfjZos lqe'h ABC tof/ eof] .

h;df B = 45, AB = 5 cm / BC = 6cm 5 .

2. Pp6f eh' fsf] gfk / To;df ag]sf b'O{ sf0] fx¿ lbPdf

-s_ ?n/sf] ko| fu] u/L ;a}nfO{ Ps Ps cf6] f l;wf/]vf PQ lvRg nufpgx' f];\ . h:t} PQ = 8 cm laGbx' ¿
P / Q df s:tf k|sf/sf sf]0fx¿
lvRbf lqe'h aG5 xf]nf ;fR] f/] n]Vg YX
R

nufpg'xf;] \ .

-v_ sDkf;sf] ko| fu] u/L laGb' P / Q df 60 45
lbOPsf sf]0fx¿ lvRg nufpg'xf;] \ . P
h:t}, XPQ=60 / YQP=45 . 8 cm Q
/]vfx¿ PX / QY sfl6Psf] laGb'nfO{
R gfd lbgnufpgx' f;] \ .

-u_ cfjZos lqe'h PQR tof/ eof] .
h;df P = 60, PQ = 8 cm / Q = 45cm 5 .

lzIfssf nflu yk hfgsf/LM

ljb\ofyLx{ nfO{ lbOPsf] /]vfsf b'O{ 5p] x¿df km/s km/s sf]0fx¿ lvRbf s:tf] / a/fa/ gfksf sf0] fx¿ lvRbf
s:tf] lqeh' aGb5 < ;fW] g] . sIffdf ;d"x lgdf0{ f u/L /rgf ug{ nufO{ ;xhLs/0f ug{ ;lsG5 .

3= ltg cf]6f eh' fx¿sf] gfk lbPdf (XY = 6 cm, YZ = 5 cm / XZ = 4 cm)

9

-s_ ?n/sf] k|of]u u/L ;a}nfO{ Pp6f cfwf/ e'hf XY lvRg nufpgx' f];\ . Z
h:t} XY = 6 cm.

-v_ sDkf;df 5 cm a/fa/sf] rfk lnO{ laGb' Y af6 nfdf] rfk lvRg 4 cm 5 cm
nufpgx' f;] \ .

-u_ X af6 klg 4 cm sf] rfkn] cl3Nnf] nfdf] rfknfO{ sf6\g] . sfl6Psf]
laGb'nfO{ Z gfd lbg]

-3_ laGb' X / Z tyf laGb' Y / Z hf8] g] X 6cm Y

-ª_ cfjZos lqeh' XYZ tof/ eof] . h;df XY = 6 cm, YZ = 5 cm / ZX= 4 cm 5 .

dN" ofªs\ g

cEof; 2.1 sf /rgfx¿ sIffdf ug{ nufO{ d"Nofª\sg ug{x' f];\ .

DRAFTkf7 zLifs{ M rte' h'{ sf] kl/ro / k|sf/

pb\oZ] ox¿

o; kf7 cWoog ul/;s]kl5 ljb\ofyLx{ ¿ lgDg sfo{x¿ ug{ ;Ifd xg' ] 5g\ M
!= rt'eh{' sf] kl/ro lbg / k|sf/ atfpg
@= ;dfgfGt/ rt'e{'h tyf ;dafx' rte' h{' sf u0' fx¿ vf]hL ug{
#= ju{ / cfotsf u0' fx¿ vf]hL ug{ .

lzIf0f ;fdu|Lx¿

HofldtLo cf}hf/ afs; / Go'hlk|G6 k]k/ cflb .

l;sfO ;xhLs/0f lj|mofsnfkx¿

1. lgDg kZ| gx¿df n5kmn u/fpg'xf];\ M
-s_ ;a}eGbf sd slt cf]6f l;wf /]vfv08x¿ ldn/] aGb cfsl[ t aGg ;S5 < ;a} ljbo\ fyL{x¿nfO{ nV] g
nufpgx' f];\ .
-v_ ;an} fO{ 2, 3, 4,5, 6 .........cf6] f dWo] ;ª\Vofsf /v] fv08x¿ ko| f]u u//] s'g} Ps aGb cfsl[ t
lvRg nufO{ pSt aGb cfs[ltsf] gfd cgd' fg u/L nV] g nufP/ kfnk} fnf] sIffdf k|:t't ug{
nufpg'xf;] \ .
-u_ rt'e'{h Pp6f axe' h' xf] egL lg:sif{ lbg] .

@= rte' {h' sf k|sf/x¿
-s_ 4/4 hgfsf] jf a]Gr cg';f/sf] ;dx" lgdf{0f ug{'xf];\ .
-v_ ;a} ;d"xx¿nfO{ Ps Ps cf6] f rt'e'{hsf lrqx¿ -lj;dafx', ;dafx', cfot, ju{, ;dfgfGt/ rte' '{h
x'g]u/L_ kb| fg ug]{ / pSt rt'e'h{ sf] gfd cgd' fg u/L nV] g nufpg'xf];\ .
-u_ kf| ]6]S6/ / ?n/sf] ko| f]u u/L rt'e'{hsf sf]0fx¿sf] gfk / e'hfx¿sf] nDafO gfKg nufpgx' f;] \ .
-3_ sf0] fx¿sf] gfk / eh' fx¿sf] nDafO s:tf kfOof] < ;d"xdf 5nkmn u/L nV] g nufpg'xf];\ .
ljb\ofyLx{ ¿nfO{ sIffdf k:| t't ug{ nufpgx' f];\ / cfjZos ki[ 7kf]if0f k|bfg ug{'xf];\ .
-ª_ cGTodf ljb\ofyL{ ;d"xx¿nfO{ klxn]sf] eGbf km/s rt'e{h' sf lrq lbO{ sf]0fx¿ tyf eh' fx¿sf]
;DaGwsf af/]df ;fdfGoLs/0f ug{ nufpg'xf;] \ .

ljm| ofsnfk #M
ljifdafx' rt'e'{hsf] cjwf/0ff lbg] . lj|mofsnfk @ df gfkhfr ePsf rt'e'h{ x¿dWo] ;a} e'hfsf] gfk km/s
km/s ePsf] rt'eh{' 5gf6] ug{ nufO{ of] ljifdafx' /xs] f] s/' f :ki6 kfg]{ .

10

kf7 zLif{sM ;dfgfGt/ rt'eh{' sf u'0fx¿

pb\o]Zox¿

o; kf7sf] cGTodf ljb\ofyLx{ ¿ lgDg k/LIf0f ug{ ;Ifd x'g] 5g\ .
!= ;dfgfGt/ rt'eh'{ sf ;Dd'v sf0] fx¿ a/fa/ x'G5g\ .
@= ;dfgfGt/ rte' {h' sf ;Dd'v e'hfx¿ a/fa/ x'G5g\ .
#= ;dfgfGt/ rte' {h' sf las0fx{ ¿ k/:k/ ;dl4efhg x'G5g\ .

lzIf0f ;fdu|Lx¿
HofldtLo cfh} f/ afs; / Go'hlk|G6 kk] / cflb .

;xhLs/0f lj|mofsnfkx¿

1. ;dfgfGt/ rte' {h' sf ;Ddv' sf]0fx¿

-s_ ;dfgfGt/ rte' {'h eg]sf] s] xf] < cfcfkm\gf] sfkLdf nv] L a]Gr ;d"xdf 5nkmn u/L lgisifn{ fO{ sIffdf
k:| t't ug{ nufpgx' f];\ .

-v_ ;a} ljbo\ fyLx{ ¿nfO{ Ps Ps cf6] f km/s km/s gfksf ;dfgfGt/ rt'e{h' lvRg nufpgx' f];\ .

DRAFT
-u_ kT| os] ljb\ofyL{x/n] JolStut ¿kdf cfkm"x¿n] lvr]sf ;dfgfGt/ rte' 'h{ sf sf]0fx¿sf gfkL ltgLx¿sf]
;DaGw nV] g nufpgx' f;] \ .

-3_ aG] r ;d"xn] cfk;df 5nkmn u/L Go'hlk|G6df lrq;lxt ;dx" sf] lgisif{ tof/ u/L sIffdf k|:tt' ug{
nufpgx' f;] \ .

ª_ lzIfsn] ljb\ofyLx{ ¿sf] k|:t'ltdf k[i7kfi] f0f lbb} ;dfgfGt/ rte' 'h{ sf] ;Ddv' sf0] fx¿larsf] ;DaGw ki| 6
kfg'{xf];\ .

dN" ofªs\ g

tnsf lrqx¿df x sf] dfg kTtf lgsfNg nufpgx' f];\ .

-s_ -v_ -u_

x x x

45 150 84 95

2. ;dfgfGt/ rte' {h' sf ;Ddv' eh' fx¿ a
s_ ;dfgfGt/ rt'e'{h eg]sf] s] xf] < cfcfkm\gf] sfkLdf n]vL aG] r ;dx" df 5nkmn u/L lgisifn{ fO{ sIffdf
k|:t't ug{ nufpgx' f];\ .

-v_ ;a} ljbo\ fyL{x¿nfO{ Ps Ps cf6] f km/s km/s gfksf ;dfgfGt/ rte' '{h lvRg nufpgx' f];\ .

-u_ kT| o]s ljb\ofyL{x/n] AolStut ¿kdf cfkmx" ¿n] lvr]sf ;dfgfGt/ rte' h'{ sf eh' fx¿sf gfk lnO{
ltgLx¿sf] ;DaGwsf af/]df nV] g nufpgx' f];\ .

-3_ a]Gr ;d"xn] cfk;df 5nkmn u/L Go'hlk|G6df lrq;lxt ;dx" sf] lgisif{ tof/ u/L sIff sf]7fdf k|:t't
ug{ nufpg'xf];\ .

-ª_ lzIfsn] ljbo\ fyLx{ ¿sf] k|:t'ltdf ki[ 7kf]if0f lb“b} ;dfgfGt/ rte' 'h{ sf ;Ddv' eh' fx¿sf] ;DaGw k|i6
kfg'{xf;] \ .

dN" ofªs\ g

tnsf lrqx¿df x / y sf] dfg lgsfNg nufpg'xf;] \ M

-s_ -v_ -u_ y

y 4 cm

3 cm 4 cm 11 x 4 in

x x

5 cm y 7 in

3. ;dfgfGt/ rte' h'{ sf ljs0fx{ ¿

-s_ ltg ltg hgfsf] ;d"x lgdf0{ f u/L ;a} ljbo\ fyLx{ ¿nfO{ Ps Ps cf6] f km/s km/s gfksf ;dfgfGt/
rte' {'h ABCD df las0f{x¿AC / BD lvRg nufpg'xf];\ . las0fx{ ¿ sfl6Psf] laGb'nfO{ O gfd
lbg] . h:t} M

A BA B A B
O O O

D CD CD C

-v_ kT| os] ljb\ofyL{x¿n] AolStut ¿kdf cfkm"x¿n] lvr]sf ;dfgfGt/ rt'e{h' sf las0f{x¿ / ltgLx¿sf
efux¿ h:t} OA, OC, OB / OD gfKg] / ltgLx¿sf] gfk / k/' f las0fx{ ¿sf] gfksf] larsf]
;DaDwsf af/]df nV] g nufpg'xf;] \ .
DRAFT
-u_ aG] r ;dx" nfO{ cfk;df 5nkmn u/L Goh' lkG| 6df lrq;lxt lgisif{ tof/ ug{ nufpgx' f;] \ .

-3_ ljb\ofyLx{ ¿sf lj|mofsnfkx¿sf] cjnf]sg ub{} cfjZs ;xofu] / ;'emfj lbO{ sIffdf k|:t't ug{
nufpgx' f];\ .

-ª_ ljbo\ fyL{x¿sf] k|:tl'tdf k[i7kf]if0f lbb} ;dfgfGt/ rte' h'{ sf las0f{x¿sf] ;DaGw k|i6 kfgx{' f];\ .

d"Nofª\sg

tnsf lrqx¿df x / y sf] dfg kTtf nufp .

-s_ P S -v_ B -u_ E F
C
x cm A x
O
3 cm Oy Oy

Q R 3 in 3 in H 2.5 cm
D G

lzIfsnfO{ yk hfgsf/L M

lzIfsn] o;sf ;fy;fy} ;dafx' rte' '{h / o;s larsf] ;DaGw atfpb} ;dafx' rte' {h' sf las0fx{ ¿sf af/d] f
klg :ki6 kfl/lbg'xf;] \ .

kf7 zLif{s M cfot / ju{sf u0' fx¿sf] vf]hL

pbo\ Z] ox¿
o; kf7sf] cWoog kl5 ljb\ofyLx{ ¿ lgDg sfox{ ¿ ug{ ;Ifd xg' ] 5g\ M
-s_ cfot / jud{ f km/s 5'6o\ fpg
-v_ cfot / ju{sf u0' fx¿sf] vf]hL ug{
lzIf0f ;fdu|Lx¿
HofldtLo cfh} f/ afs; / Go'hlk|G6 k]k/ cflb . .
l;sfO ;xhLs/0f ljm| ofsnfkx¿
1. cfot / ju{lar km/s

12

-s_ cfot / /ju{sf] kl/efiff eGg nufO{ 5nkmn u/fpg'xf;] \ .

-v_ ljbo\ fyL{x¿nfO{ cfot / jud{ f s] km/s 5 < JolStut ¿kdf n]Vg nufpg'xf];\ / aG] r ;dx" df 5nkmn
u/L ;femf pTt/nfO{ sIffdf k|:tt' ug{ nufpgx' f];\ .

-u_ ljb\ofyL{x¿nfO{ a]Gr ;dx" df ljefhg ug]{ / hf]/ a]Gr gDa/ ePsf] h:t}, 2,4,6,8,..;d"xsf
;b:ox¿nfO{ cfot / lahf]/ h:t} 1, 3, 5, 7,... ePsfnfO{ JoSltut ¿kdf Ps Ps cf6] f km/s km/s
gfksf ju{ lvRg nufpg'xf;] \ .

-3_ ;a} ;d"xnfO{ cfcfkm\ gf] lrqdf las0f{x¿sf gfk ?n/sf] k|ofu] u/L
lng nufpgx' f];\ .

-ª_ k'gM las0fx{ ¿ sfl6Psf 7fpd“ f ags] f rf/ cf]6f sf0] fx¿ gfKg nufpg'xf;] \ .

-ª_ a]Gr ;d"xdf cf cfkm\gf] glthfaf/] 5nkmn ug{ nufpg'xf;] \ .

-r_ ;fdl" xs lgisif{ tof/ u/L sIffdf k|:t't ug{ nufpg'xf];\ .

-5_ ljb\ofyL{x¿sf] k:| t'ltdf k[i7kf]if0f lbb“ } lgisif{ M cfotsf las0fx{ ¿ a//a/ xG' 5g\ / ju{sf las0fx{ ¿
k/:k/ ;dsf0] fx'g]u/L sfl6Psf x'G5g egL :ki6 kfgx{' f;] \ .

2. ju{ / ;dafx' rte' '{h

-s_ sIffsf ;a} ljbo\ fyL{nfO{ ;d"xdf ljefhg ug'x{ f;] \ .

DRAFT
-v_ Pp6f sf]nd ;dx" nfO{ ju{ / csf{] ;dx" nfO{ ;dafx' rte' h'{ s]f af/d] f nV] g nufpg'xf;] \ . ;a}nfO{ Ps Ps
cf6] f ;fbf sfuh kb| fg ugx{' f;] \ .

-u_ ;d"x A df /xs] f pk;dx" x¿nfO{ ju{ / ;dx" B sf nflu
;dafx' rte' 'h{ lvrL ;a}nfO{ bj' } las0fx{ ¿ lvRg
nufpg'xf];\ .

-3_ k|To]s ;d"xn] las0fx{ ¿n] zLifl{ aGbx' ¿df agfPsf sf]0fx¿sf]
gfKg] / ;fyLx¿;“u 5nkmn u/L zLif{sf]0f / ljs0fn{ ]
zLifl{ aGbd' f agfPsf sf]0fx¿sf ;DaGwsf af/] ;femf wf/0ff lgdf0f{ u/L sIff sf7] fdf k:| tt' ug{
nufpg'xf];\ .

-ª_ cGTodf ju{ / ;dafx' rt'{e'hsf las0f{x¿ / zL]if{sf0] fx¿sf] ;DaGwsf af/]df 5nkmn u/fO{ :ki6
kfg'{xf;] \ .

3. ;dafx' rt'eh'{ sf u0' fx¿

s_ sIffdf ;a} ljbo\ fyLx{ ¿nfO{ hf]8Ldf a:g nufpg'xf;] \ .

v_ k|To]s hf]8LnfO{ Ps Ps cf6] f ;dafx' rt'e{'hsf cfs[ltx¿ lbg] . las0f{x¿ lvRg nufpgx' f;] \ .

u_ las0fx{ ¿ sfl6Psf] laGb'nfO{ O gfdfs/0f ug]{ . AD

-3_ To;kl5 hf]8Ldf Ps hgfn] las0f{ AC / csf{n] las0f{ BD sf efux¿
/ sf0] fx¿ gfKg nufpg'xf];\ .

-ª_ hf8] Ldf 5nkmn u/L las0f{sf efux¿sf] nDafO / sf]0fsf af/d] f B
;femf lgisif{ kTtf nufpg ;xofu] ug{'xf;] \ .
C

-ª_ kT| o]s hf8] LnfO{ cfcfkm\gf] lgisif{ sIffsf]7fdf k:| t't ug{ nufpg'xf];\ .

d"Nofª\sg

kf7\o k:' tssf] cEof; 2.2 sf kZ| gx¿;dfwfg ug{ nufP/ d"Nofªs\ g ug{x' f];\ .

kf7 zLif{s M rte' h'{ sf] /rgf (construction of quadrilateral )
pbb\ ]Zox¿

13

zI} fl0fs ;fduL|

l;sfO ;xhLs/0f ljm| ofsnfk

1= cf;Gg e'hf / ltgLx¿larsf] sf0] f lbP/ ;dfgfGt/ rt'eh{ sf] /rgf

cf;Gg e'hfx¿sf] gfk j|mdzM 6cm / 5.4cm tyf ltgLx¿larsf]
sf]0f 120° ePsf] ;dfgfGt/ rt'eh{' sf] /rgf u/ .
lrq lstfaf6 kh] @%pbfx/0f !

lgDgfg;' f/ lgb{]zgcg';f/ /rgf ug{ nufpg'xf];\ M

/rgf ug]{ tl/sf

-s_ Pp6f PQ = 6cm ePsf] /]vfv08 lvRr nufpg'xf;] \ .
-v_ laGb' P df sDkf;sf] ;xfotfn] 120° sf] sf]0f lvRg cEof;

u/fpgx' f;] \ .
-u_ 120° sf] sf0] f agfpg] /]vfdf P af6 5.4cm sf6]/ laGb' R kTtf nufpg nufpgx' f];\ .
-3_ R af6 6cm / Q af6 5.4cm rfk lnO{ cfk;df sf6]/ laGb' S kTtf nufO{ Q / S tyf R / S hf]8g

nufpgx' f;] \ . ca cfjZos ;dfgfGt/ rt'e'h{ PQSR tof/ eof] .
-ª_ oxf“ ckgfOPsf k|Tos] r/0fsf af/]df ljb\ofyLx{ ¿ lardf k|ZgfT] t/ / 5nkmn u/fpgx' f];\ .
2= cf;Gg e'hfx¿sf] gfk lbP/ cfotsf] /rgf
Pp6f cfot EFGH lvr, h;df EF = 4.5cm / EH = 2cm 5 .
lgDgfg;' f/sf] lgb{]zgsf cfwf/df /rgf ug{ nufpg'xf;] \ .
DRAFT
/rgf ug]{ tl/sf lrq lstfaf6 kh] @%pbfx/0f @

-s_ cfwf/ /v] f EF = 4.5cm lvrL sDkf; / ?n/sf]
;xfotfn] laGb' E df 90° sf] sf0] f lvr .

-3_ laGb' E df 90° sf0] f agfpg] /v] fdf 2cm
gfk/] laGb' H kTtf nufpg'xf;] \ .

-u_ laGb' H af6 4.5cm sf] rfk F af6 2cm sf] rfk lnO{
cfk;df sf6]/ laGb' G kTtf nufO{ laGb' G / H tyf F
/ G hf]8g nufpgx' f;] \ . ca, cfjZos cfot EFGH tof/ eof] .

-3_ oxf“ ckgfOPsf k|To]s r/0fsf af/]df k|ZgfT] t/ / 5nkmn u/fpgx' f];\ .

3= e'hf / sf]0fsf] gfk lbP/ ;dafx' rt'e'{hsf] /rgf

e'hfx¿ 4.5cm / cf;Gg eh' fx¿n] agfPsf] Pp6f sf0] f 45° ePsf] Pp6f ;dafx' rte' 'h{ sf]
lgDgfg';f/sf] lgb{]zgsf cfwf/df /rgf ug{ nufpgx' f];\ .

/rgf ug{] tl/sf lrq lstfaf6 k]h @^pbfx/0f #

-s_ cfwf/ /v] f QR = 4.5 cm xg' ] u/L Pp6f /]vfv08 lvrL
-v_ laGb' Q df sDkf; / ?n/sf] ko| f]u u/L 45° sf] sf]0f
/rgf u/fpg'xf];\ .
-u_ laGb' Q sf] 45° sf0] f agfpg] /]vdf 4.5cm gfk]/ laGb'

T kTtf nufO{ laGb' R / T af6 4.5cm s} rfkx¿
lnP/ cfk;df sf6/] laGb' S kTtf nufpg nufpgx' f];\ .

14

-3_ T / S tyf S / R hf]8gx' f;] \ . ca cfjZos ;dafx' rt'e{'h QRST tof/ eof] .
o; /rgfdf ckgfOPsf kT| o]s r/0fsf af/]df k|ZgfT] t/ / 5nkmn u/fpgx' f];\ .
d"Nofªs\ g
kf7\o k:' tssf] cEof; 2.3 sf kZ| gx¿ lbO{ d"Nofª\sg ug]{ ugx{' f];\ .
k|ltlaDa

km/s km/s cj:yfdf rt'e'{hx¿sf] /rgfsf r/0fx¿ t'ngf u/L;r' Ls/0f ug{ nufpgx' f];\ .

lzIfsnfO{ yk hfgsf/L

ljleGg gfksf eh' fx¿/ sf]0fx¿ lnP/ km/s km/s rt'eh{' x¿ /rgf u/fpg ;lsG5 .

kf7 zLif{s M ax'e'hsf leqL tyf aflx/L sf]0fx¿

pb\oZ] ox¿

o; kf7sf] cWoog kl5 ljb\ofyL{x¿ lgDg sfox{ ¿ ug{ ;Ifd x'g] 5g\ M

!= axe' 'hsf leqL tyf aflx/L sf]0fx¿ 5'6o\ fpg

@= ax'eh' sf leqL tyf aflx/L sf0] fx¿sf gfk kTtf nufpg

lzIf0f ;fduL| x¿

ljleGg axe' h' sf cfsl[ tx¿, h;' kfOksf 6j' m| fx¿, 5\jfnL tyf l;Gsfx¿ / wfuf] .

;xhLs/0f ljm| ofsnfkx¿

1. s_ ;a} ljbo\ fyL{x¿nfO{ ax'e'hsf leqL tyf aflx/L sf]0fx¿sf af/d] f yfxf ePsf s/' fx¿ 3 ldg]6df
n]Vg nufpgx' f];\ . ;u}sf ;fyL;“u 5nkmn u/L ;femf pTt/ agfO{ sIffdf k|:t't u/fpgx' f];\ .

-v_ ;d"x ljefhg u/L kT| os] ;d"xnfO{ tnsf dWo] sg' } Ps cf]6f lrq lbO{ leqL / aflx/L sf]0fx¿sf] ;r" L
tof/ ug{ nufpg'xf;] \ .
DRAFT
IH GF V U T A Y XY Q
A E S A
B P Z
D QR XE A R
C
Z
D

P

-u_ kT| o]s ;d"xn] cfkmn" fO{ k/]sf] lrq Go'hlkG| 6df agfO{ leqL tyf aflx/L sf]0fx¿sf] ;r" L tof/ ug{] /
To;nfO{ ;a} ;d"xdf 3'dfpg] / cGo ;dx" n] klg cWoog ug{] lgb{]zg lbgx' f;] \ .

-3_ cGTodf lzIfsn] leqL tyf aflx/L sf]0fx¿sf af/]df :ki6 kfgx'{ f];\ .

d"Nofª\sg

hf8] Ldf PsPscf6] f ax'eh' agfP/ gfdfs/0f u/L cfk;df ;f6f;f6 u/L pSt ax'eh' sf leqL tyf aflx/L
sf0] fx¿sf] ;"rL tof/ ug{ nufpgx' f];\ .

2. -s_ 1, 2, 3, 4, 5, 6, 7 ....ub}{ a9Ldf 5/5 hgfsf] ;dx" lgdf{0f ug{]

-v_ k|Tos] ;dx" nfO{ j|mdzM 3 cf]6f, 4 cf]6f, 5 cf6] f, 6 cf]6f, ......a/fa/ gfksf 5\jfnLsf 6j' |mfx¿ / sl] x
wfuf] lbgx' f];\ .

-u_ ;a} 6j' |mfsf 5]px¿ cfk;df hf8] ]/ axe' h' agfpg] . tn lrqdf b]vfP h:t} Pp6f zLifl{ aGb'af6 wfuf]
ko| fu] u/]/ lqe'hx¿df ljefhg ug{'xf;] \ .

15

-3_ lzIfsn] ljb\ofyLx{ ¿nfO{ sg' } lqeh' sf ltg leqL sf]0fx¿sf] ofu] kmn 180 x'G5 . eg] k|To]s ;dx" n]
cfcfkm\ gf] ax'eh' sf] leqL sf0] fx¿sf] of]ukmn lgDgfg;' f/ tflnsf agfP/ eg{ nufpgx' f;] \ M

ax'eh' sf] gfd eh' fsf] ;ª\Vof lqe'hsf] leqLsf]0fx¿sf] ofu] kmn
;ª\Vof

lqe'h 3 1=3-2 180 (3-2)=180

rte' h{' 4 2=4-2 180 (4-2)=360

k~re'h 5 3=5-2 180 (5-2)=540

ifi7eh' 6 4=6-2 180 (6-2)=720

============DRAFT ======== . …………

.

n n-2 180 (n-2)

-ª_ ;a} ;d"xn] k|:tt' ul/;s]kl5 ;an} fO{ o;sf] lgisif{ cg'dfg u/L nV] g nufpg'xf;] \ / cGTodf
lgDgfg;' f/sf] lgisif{ atfOlbg'xf;] \ .

olb sg' } klg ax'e'hsf eh' fx¿sf] ;ª\Vof n eP To;sf leqL sf]0fx¿sf] ofu] kmn (n -2) x 180 x'G5 .

3.

-s_ lgoldt axe' h' eGgfn] s] al' emG5 < o;df eh' f / sf0] fx¿sf] gfk s] s:tf] xfn] f < cgd' fg u/L n]Vg
nufpg'xf];\ .

-v_ kT| o]s ;d"xnfO{ Ps Ps cf]6f km/s km/s lgoldt axe' 'hsf lrqx¿ lbg'xf];\ / eh' fx¿sf ;ªV\ of uGg
nufpgx' f;] \ .

-u_ ;d"xdf 5nkmn u/L (n -2)  180 ko| f]u u/L leqL sf]0fx¿sf] ofu] kmn kTtf nufpg'xf;] \ .

-3_ of]ukmnnfO{ e'hfsf] ;ªV\ ofn] efu ug{] / n]Vg] .

-ª_ kf| ]6]S6/sf] k|of]u u/L Pp6f leqL sf]0f gfKg nufpg'xf];\ .

-r_ 3 / ª sf] glthf t'ngf ug{ nufpgx' f];\ .

-5_ t;y{, lgoldt axe' h' sf] leqL sf]0fsf] gfk leqL sf0] fx?sf] of]ukmn
(x) = e'hfx?sf] ;ªVof

(n -2)  180
=n

4. lgoldt ax'eh' sf] aflx/L sf]0fsf] gfk

16

-s_ kT| os] ljb\ofyL{x¿nfO{ lbOPsf] lrqnfO{ sfkLdf agfpg nufpgx' f];\ .

-v_ x / y sf] gfk k|f]6]S6/sf] k|of]u u/L x / y sf]0f gfKg nufO{ tL
sf0] fx¿sf] ofu] kmn slt x'G5 < kTtf nufpg nufpg'xf;] \ .

-u_ ca y sf] dfg slt xfn] f <

-3_ leqL sf]0f kTtf nufpg] ;q" s] xf] <

lzIfsn] ljbo\ fyLs{ f lj|mofsnfkx¿ cjnfs] g u/L lgDgfg';f/ :ki6 kfl/lbg] .
;“u}sf] lrq Pp6f lgoldt axe' h' sf] lrq xf] .

o;df e'hf GF nfO{ laGb' P ;Dd nDAofpb“ f aGg] aflx/L sf0] f PFE = y
dfgf“} .

To:t} y sf] cf;Gg sf]0f GFE = x dfgf}“ .

ca, x + y = GFP -l;ªu\ f] 6j' |]m tYocg';f/_
cyjf, x + y = 180° -GFP ;/n sf]0f ePsfn_]
cyjf, y= 180 - x = 180 -180n(n-2)

= 180n-180n+360
n
DRAFT
= 360
n

dflysf lj|mofsnfkaf6 s] lgisif{ lgsfNg ;lsG5 < ljb\ofyLx{ ¿nfO{ n]Vg nufpg'xf];\ .

lgisif{ M lgoldt axe' h' sf] aflx/L sf]0f (y) =36n0 x'G5, .hxf“ n = ax'eh' sf] e'hfx¿sf] ;ªV\ of 5 .

dN" ofªs\ g

cEof; 2.4 sf 3 / 4 sf kZ| gx¿ ;dfwfg ug{ nufP/ d"Nofª\sg ug'x{ f];\ .

PsfO M #
;d¿k / cg'¿k cfs[ltx¿

(Similar and Congruence Shapes)

cg'dflgt lzIf0f 3G6LM #

k"j{1fg
ljleGg HofldtLo cfs[ltx¿, ltgLx¿sf] cfsf/ tyf gfksf] hfgsf/L
kf7 kl/ro
p:t} cfs[lt / km/s km/s cfsf/sf HofldtLo cfs[ltx¿nfO{ ;d¿k cfs[ltx¿ elgG5 . h:t} Pp6f ;dafx'
lqeh' csf]{ ;dafx' lqe'h;“u ;d¿k x'G5 . To:t} p:t} cfsl[ t tyf ;dfg cfsf/ ePsf HofldtLo cfs[ltx¿
cg¿' k cfs[lt x'g\ . h:t} Pp6f % ;]=ld= e'hf ePsf] ;dafx' lqe'h / csf]{ % ;=] ld= e'hf ePsf] ;dafx' lqe'h
cg'¿k x'G5g\ . HOfldtLdf :yfgfGt/0f -kl/j|mmd0f, k/fjtg{ tyf lj:yfkg_ kl5 aGg] kl| tlaDa cfsl[ t / lbOPsf]
cfs[ltx¿ cg'¿k g} x'G5g\ .

DRAFT
kf7 zLif{s M ;d¿k cfs[ltx¿

pb\o]Zox¿

o; kf7sf] cGTodf ljbo\ fyLx{ ¿ lgDg sfo{ ug{ ;Ifd xg' ] 5g\ M
-s_ ;d¿k cfsl[ tsf] kl/ro lbg .
-v_ lbOPsf] hf]8f cfs[ltx¿ ;d¿k eP gePsf] 5'6\ofpg
zI} fl0fs ;fduL| x¿ M sf8a{ f]8,{ HofldtLo cfsl[ tx¿sf df]8n] x¿
;xhLs/0f ljm| ofsnfkx¿

1. ;d¿k cfsl[ tx¿
-s_ cfjZostfcg;' f/ ljbo\ fyL{x¿nfO{ ;d"xdf ljefhg ug{'xf];\ .
-v_ af]8{df km/s km/s gfksf 2 hf8] L Pp6} cfs[ltx¿ kb| z{g UFg{]
-u_ hf8] LnfO{ Ps Ps cf6] f cfsl[ t sfkLdf agfpg nufpgx' f];\ .
-3_ k|Tos] hf]8LnfO{ cfkm'n] agfPsf] cfs[ltx¿df ePsf ;dfgtfx¿ kTtf nufO{ cfsl[ tx¿ tn' gf ug{

nufpg'xf;] \ .
-ª_ To;kl5 pgLx¿n] tof/ u/s] f] lgisif{nfO{ sIffdf k:| tt' ug{ nufpgx' f];\ / ki[ 7kf]if0f lbgx' f];\ .
-r_ ;d¿k cfsl[ tx¿sf] kl/efiff ;DaGwdf 5nkmn u/fpgx' f];\ .
-5_ ljbo\ fyL{x¿nfO{ sx] L ;d¿k / km/s ;dt] cfsl[ tx¿ ld;fO lbg] . tL cfs[ltx¿af6 ;d¿k cfsl[ tx¿
5gf] ug{ nufpg] .
d'Nofª\sg

18

;a} ljb\ofyL{x¿nfO{ sIffsf]7f jf aflx/ uO{ Ps Ps hf]8f ;d¿k cfs[ltx¿ vfl] h gfd nV] g nufpgx' f;] \ .

2. cg¿' k cfsl[ tx¿

-s_ kT| os] hf]8LnfO{ Ps Ps cf]6f ;dfg gfk / cfsf/ ePsf cfsl[ tx¿ kb| fg ug{] / bj' }df ePsf ;dfgtf
jf ;dfg u'0fx¿sf] af/]df n]Vg nufpg'xf];\ .

-v_ kT| o]s hf8] LnfO{ pSt cfsl[ tx¿sf af/]df lgisif{df k'Ug ;xof]u ugx{' f];\ .

-u_ lgisif{nfO{ k|To]s ;dx" sf Ps Ps hgfnfO{ sIffsf]7fdf k:| tt' ug{ nufpg'xf;] \ .

-3_ cGTodf k|:t'ltnfO{ ki[ 7kfi] f0f lb“b} tof/ kf/s] f] lgisif{nfO{ ;xhLs/0f u/L ;fdfGoLs/0f u/fpgx' f;] \ .

d'Nofªs\ g

ljb\ofyL{x¿nfO{ hf]8f cfs[ltx¿ kb| fg u/L s'g s'g hf]8f cfsl[ tx¿ cg¿' k 5g\ sg' sg' hf]8f cfsl[ t
;d¿k x'g jf sg' sg' hf]8f cfsl[ tx¿ ;d¿k jf cg¿' k 5}gg\ eg] nV] g nufpg'xf;] \ . ;fy} ljbo\ fyL{x¿nfO{
cg'¿k cfsl[ t xg' nfO{ cfjZos kg{] lazi] ftfx¿ JolStut ¿kdf n]vL sIffdf k:tt' ug{ nufpg'xf];\ .

3. ;d?k÷cg?' k cfsl[ tx¿ klxrfg

-s_ k|Tos] ljb\ofyL{x¿nfO{ Ps Ps cf]6f cfsl[ tx¿ k|bfg ug{] / ;f] cfsl[ t;u{ cg'¿k tyf ;d¿k x'g]
cfs[ltx¿ 6]a'ndf /fVg] To;kl5 ljb\ofyL{x¿nfO{ cfk\mgf] cfs[lt;“u cg?' k÷;d¿k x'g] vfnsf] cfs[lt
vf]Hg nufpgx' f];\ .

-v_ ;an} ] cfcfk\mgf] cfs[ltsf] cg'?k÷;d?k cfsl[ t vf]h]kl5 pgLx¿ ;d"xdf pSt cfs[ltsf] lrq cfk\mgf]
sfkLdf agfP/ pSt cfs[ltx¿sf ljzi] ftfx¿ n]Vg nufpg'xf;] \ .

-u_ ljb\oflyx{ ¿nfO{ ;d¿k tyf cg¿' k cfsl[ tx¿ lvRg nufpgx' f;] \ .

gf6] M of] v]n ;d¿kdf klg v]Ng÷v]nfpg ;lsG5 .
DRAFT
PsfOM$

jT[ t(Circle)

cg'dflgt 3G6LM #

k"j1{ fg Mj[Tt / jT[ tfsf/ ;tx ePsf j:tx' ¿af/] hfgsf/L
kf7 kl/ro
xfdLn] ljleGg 3/fo;L ;fdfu|L h:t} lunf;sf] ;tx tyf lk“w, r'/f, k;} f cflbsf] 3/] faf6 j[Tt agfpg ;S5f} .
;fy} o; sIffdf xfdL j[Tt / jT[ tdf ljleGg efux¿sf af/d] f cWoog ub{5f} . o;af6 ljb\ofyL{x¿n] bl} gs
hLjgdf bv] ]sf v]ndf k|of]u ePsf tyf cfk"mn] tof/ kf/]sf ljleGg j[Ttfsf/ cfs[lt / ;txx¿sf] hfgsf/L
kf| Kt xG' 5 . s'g} lzif{ laGb' jf sf]0f gePsf] Hofldlto ;dtn cfs[ltnfO{ jT[ t elgG5 .

kf7 zLif{s M j[Tt / o;df ljleGg efux¿
pb\b]Zox¿
o; kf7sf] cWoogkl5 ljb\ofyLx{ ¿ lgDg sfox{ ¿ ug{ ;Ifd xg' ] 5g\ M
!= jT[ tsf] kl/ro lbg
@= jT[ tsf] efux¿sf] kl/ro lbg
;xhLs/0f lj|mofsnfkx¿
1. -s_ sIffdf ljb\ofyLx{ ¿nfO{ HofldtLo cfs[ltx¿ dWo] dfx] 8f / zLifl{ aGb' gxg' ] cfs[ltx¿sf] gfd nV] g

nufpgx' f;] \ .

19

-v_ pgLx¿nfO{ k};f r/' f jf s'g} Pp6f j[Ttfsf/ 3]/f ePsf] j:tx' ¿ kb| fg ugx{' f];\ / ;fx] L cg¿' ksf] cfs[lt
sfkLdf 6;]« ug{ nufpg'xf;] \ .

-u_ rp/df Pp6f lsnf uf8L Pp6f 8f/] L lng'xf;] \ .

-3_ 8f]/Lsf] Ps 5]p lsnfdf v'sn' f] x'g]u/L af“Wgx' f];\ csf{] 5p] Ps hgf ljbo\ fyL{nfO{ ;dfTg nufpg'xf];\ .

-ª_ pSQ ljbo\ fyL{nfO{ 8f]/L tGsg] u/L 3D' g nufpg'xf;] \ / p;n] lx8s] f] af6f]df s:tf] cfs[lt aG5
cjnfs] g u/L n]Vg nufpg'xf];\ .

2.

tn lbOPadfl] hdsf km/s km/s lrqx¿ d]6fsf8d{ f tof/ kfg{ nufpgx' f];\ .

-s_ ljb\ofyLx{ ¿nfO{ kf“r ;d"xdf -a]Gr ;d"xdf ljefhg u/] klg xG' 5_ k|Tos] ;d"xnfO{ dflysf Ps Ps cf6] f
jT[ tx¿sf lrq kb| fg ugx{' f;] \ .
DRAFT
-v_ kT| os] ;d"xdf kT| os] n] cfcfk\mgf] efudf k/]sf lrqdf uf9f lrxg\ nufPsf] efusf] gfd ;dx" df 5nkmn
u/L n]Vg nufpg'xf];\ . cfjZos k/] lzIfsn] ;xofu] ug{'xf];\ .

-u_ ca lzIfsn] Pp6f circle board lng'xf];\ h;df /a/ AofG8 klg xg' 'k5{ .

-3_ ljb\ofyL{ ;d"xsf Ps Ps hgfn] kfn}kfnf] kl/lw sG] b|
cfkm' nfO{ k/s] f] lrqsf] gfd eGGf Jof;
nufpgx' f;] \ / circle board af6 :ki6 cw{Aof;
kfl/lbgx' f];\ . jT[ t lhjf

-ª_ km] l/ -s_ df eGbf km/s cfs[ltx¿nfO{
-klxn]sf] ;d"xdf gbfx] l/g] u/L_ ;a}
;dx" nfO{ af8“ \g] / pSt lrx\g nufPsf]
jf /ª\ufPsf] efusf] gfd n]Vg
nufpg'xf];\ .

-r_ cGTodf ;a} efux¿nfO{ Pp6} j[Ttdf agfpg nufpgx' f;] \ .

d'Nofª\sg

-s_ k|To]s ljb\ofyLx{ ¿nfO{ Ps Ps cf]6f j[Tt agfP/ To;df tn lbOPsf efux¿ lvRg nufpg'xf];\

!_ cwJ{ of; @_ s]Gb| #_ Jof; $_ If]qs %_ hLjf

^_ kl/lw &_ cwj{ [Tt

-v_ ;fy} ljbo\ fyL{x¿nfO{ lgDgfg';f/ k|Zgx¿ ;fw] /] klg dN" ofª\sg ug{ ;lsG5 .

!= lhjf / Jof;df s] km/s 5 <

@= Ifq] s / cwj{ [Ttdf egs] f] s] xf] <

#= Pp6f j[tnfO{ sltj6f cwj{ t[ df laefhg ug{ ;lsG5 <

$= Pp6f Jof;df slt cf]6f cw{Jof; x'G5g\ <

lzIfsnfO{ yk lgb]{zgM

laBfyL{x¿nfO{ rf/} df jT[ tfsf/ ¿kdf pleg nufP/ klg j[Ttsf ljleGg efux¿sf] lzIf0f ug{ ;lsG5 .

20

PsfO M %
7f]; cfs[ltx¿

(Solid figures)

cgd' flgt 3G6LM ^

kj" {1fgM
7f]; cfsf/ / ltgsf ;tx, lsgf/f / sg' fsf] kl/ro, 3g, if8\dv' f, ;f]nL, ufn] f / a]ngfsf] kl/ro
tyf 3g / if8\d'vfsf vf]j|mf gdg' fx¿af/] hfgsf/L
kf7 kl/ro M
nDafO, rf8} fO / prfO tLg cfofdx¿ ePsf] Hofldlto cfsl[ tx¿nfO{ 7f;] cfs[lt elgG5 . h:t} 3g, a]ngf,
if8dv' f lk|Hd cflb . 7f]; cfsl[ tsf hfnLx¿ eGgfn] pSt cfs[ltsf ;a} ;txx¿ bl] vg]u/L ;dtn ;txdf
lvlrPsf] bO' { cfoflds lrq xf] eg] hfnL eGgfn] s'g} df]x8f gePsf] lsgf/f dfq ePsf] 7f;] j:t'sf] 9fFrf xf]
. o;df lsgf/fx¿ h;' kfOk, l;Gsfx¿ tyf 5j\ flnaf6 ags] f xG' 5g\ . sIff & df xfdL ljleGg 7f;] tyf
ltlgx¿sf] vf|]sf gd'gfx¿sf af/]df cWoog ub5f} ;fy} a]ngf, 66] «fx8] g / cS6fx8] g« sf vf]sf| gd'gf tof/
ug{] l;Sg]5f} .
DRAFT
pbb\ ]Zox¿
o; kf7sf] cWoogkl5 ljb\ofyLx{ ¿ lgDg sfo{ ug}{ ;Ifd xg' ] 5g\ M
!= 7f]; cfsl[ t / ;dtnLo cfs[lt lrGg .
@= 7f]; cfsl[ tx¿ 66] «fx8] g, cS6fx]8g« sf, ;f]nL tyf an] gfsf] kl/efiff lbg .
#= 7f;] cfs[ltx¿ 6]6f« x]8g, cS6fx]8«gsf, ;fn] L tyf an] gfsf] vf]j|mf] gdg' f tof/ kfg{ .
lziF0F ;fdu|Lx¿
aleGg 7f;] cfs[ltx¿sf df]8]nx¿
;xhLs/0f ljm| ofsnfkx¿

1.

-s_ k|To]s ljb\ofyL{nfO{ cfk"mn] b]vs] f] HofldtLo cfsl[ tx¿sf] ;"rL tof/ kfg{ nufpg'xf;] \ .
-v_ ;a}nfO{ A]Gr ;d"x jf $/ % hgfsf] ;d"xdf 5nkmn u/FO{ ;d"xsf] Pp6} ;r" L tof/ kfg{ nufpg'xf;] \ .
-u_ k|To]s ;dx" nfO{ gbfx] f]l/g] u/L Ps Ps u/L gfd sIffdf k|:t't ug{ nufpgx' f;] \ / af]8{df tnsf] h:t}

tflnsf agfP/ l6Kb} hfg'xf;] \ .

7f]; cfsl[ tx¿ ;dtnLo cfsl[ tx¿

an] gf, 3g, lkG| d,+======== lqeh' , ju{, rte' {h' ,=======

-3_ kT| o]s ;d"xnfO{ Ps Ps cf]6f 7f]; cfsl[ tx¿ h:t} 3g a]ngf ;fl] n 6]6«fx8] g\, cS6f|x]8g lkHd cflb lbO{
/ pSt cfsl[ tdf /x]sf ;dtnLo cfs[ltx¿ df]xf]8fx¿ zLifl{ aGb'x¿ lsgf/fx¿ ug/] cfcfkm\ gf] sfkLdf
nV] g nufpg'xf];\ . h:t} M olb 3g ePdf V =8, E = 12 /F=6 xG' 5 .

21

DRAFT-ª_ kT| os] ;dx" nfO{ zLifl{ aGbs' f] ;ª\Vof df]xf8fsf] ;ª\Vof hf]8/] hf]8kmnaf6 ls/fgfsf] ;ªV\ of 36fpg
nufpg'xf];\ / slt cfp5“ sfkLdf l6Kg nufO{ / sIffdf k|:tt' ug{ nufpgx' f];\ .

-r_ ;as} f] kfnf] ;lsPkl5 ;as} f] glthfsf] lgisif{ ;f]r/] ljb\ofyL{x¿nfO{ n]Vg nufpgx' f;] \ / cGTodf lgisif{
atfOlbg'xf];\ .

2. 66] «fx]8g« / o;sf] gd'gf lgdf{0f
-s_ JolStut ¿kdf 6]6«fx8] g« egs] f] s] xf]nf / To;sf u0' fx¿ s] s] x'g <

nV] g nufpgx' f];\ .
-v_ ljb\ofyL{x¿nfO{ 4/5 hgf jf a]Gr ;d"xdf ljefhg u/L k|Tos] ;dx" nfO{

a/fa/ gfkdf 6/6 cf6] f l;Gsf jf h'; kfOk 6'j|mf / s]lx wfuf] klg
kb| fg ug'{xf];\ .
-u_ klxnf] ltg cf]6f 6'j|mf k|of]u u/L Pp6f lqe'h agfpg nufpg'xf];\ .
-3_ km/] L kT| o]s zLifl{ aGbd' f Ps Ps cf6] f 6's|f hf]8g nufpg'xf];\ .
-ª_ tL ltg cf6] fnfO{ Ps 7fpd“ f afWg'xf;] \ .
r_ slt cf6] f lsgf/f / slt cf]6f zLifl{ aGb' ePsf] cfs[lt aGof] cfcfk\mgf]
sfkLdf nV] g nufpgx' f];\ .
o;df ;dafx' lqe'h cfwf/ / $ cf]6f zLifl{ aGb' EfPsf] 7f]; cfsl[ tnfO{ 66] f« x]8«g elgG5 egL hfgsf/L
u/fpgx' f];\ .
-5_ ljbo\ ffyL{x¿nfO{ pSt cfs[ltsf zLif{laGb' dfx] 8f -;xt_ / lsgf/fx¿ ug]/ ;DaGW vf]Hg / n]Vg
nufpgx' f];\ .
dN" ofªs\ g
66] «fx8] «gsf ;j} ;txsf cfsf/ s:tf 5g\ / ltgLx¿sf kT| os] lsgf/fsf nDafO gfkL s:tf] cfsl[ t aGb5,
nV] g nufpg'xf];\ .
lzIfsnfO{ yk hfgsf/L
h:t} sfuhsf] ;dafx' lqe'hdf df]8n tof/ kf//] pSt lqeh' k|Tos] ltg cf6] f e'hfsf] dWo] laGb'af6 k6o\ fP/
klg 66] f« x]8g« agfpg ;lsG5 .
3. cS6fx]8«g / o;sf] hfnL
-s_ 12 cf6] f a/fa/ gfksf h;' kfOksf 6j' |mf jf 5\jfnLsf 6'j|mf lngx' f];\ .
-v_ rf/ cf]6f ldnfP/ Pp6f ju{ agfpgx' f;] \ / kT| os] rf/ cf6] f
zLifl{ aGbd' f bO' { bO{ cf]6f 6's|f hf]8g nufpgx' f];\ / tndfly
b'a}lt/ Ps} 7fp“df kgu{] /L af“Wg nufpg'xf;] \ .
-u_ s:tf] cfs[lt aG5 To;df df]x8f zLifl{ aGb' / lsgf/f ug]/ n]Vg
nufpg'xf;] \ .
-3_ cfwf/ Pp6} ;dafx' lqe'h / df]xf]8f ePsf] cfs[ltnfO{ cS6fx]8g«
elgG5 elg elglbg'xf;] \ .
-ª_ V-E+F sf] dfg kTtf nufpg nufpgx' f];\ .
dN" ofª\sg
cEof; %=! sf] kZ| g g= ! b]lv ^ ;Ddsf kZ| gx¿sf pTt/x¿ ;dx" df 5nkmn u/L n]Vg nufpg'xf];\ .
lzIfsnfO{ yk hfgsf/L
sfuhsf] ;dafx' lqe'hx¿af6 klg kf7o\ k:' tssf] tl/sfaf6 cS6fx]8g« agfpg ;lsG5 .
4. an] gf / ;f]nLsf] kl/ro

22

sIffdf b'O{ kª\ltsf ljb\ofyLx¿nfO{ bO' { ;dx" agfpg] / ;dx" A nfO{ ;f]nL / ;d"x B nfO{ a]ngfsf/
cfs[ltx¿ k|bfg u/L tnsf kZ| gx¿sf] pTt/ nV] g nufpg'xf;] \ .

-s_ k|Tos] cfsl[ tdf slt cf]6f cfwf/ 5g\ <

-v_ cfwf/sf cfsl[ tx¿ s:tf 5g <

-u_ zLif{laGb' 5g jf 5]}gg\ / eP slt cf6] f 5g\ <

-3_ ;a} ;tx s:tf cfsl[ tsf 5g\ <

-ª_ ;a} cfs[ltx¿sf ;dfg u0' fx¿ s] s] xfn] fg\ <

-r_ ljb\ofyLx{ ¿nfO{ ;dx" df 5nkmn u/fP/ ;fdl" xs lgisz{ lgsfNg nufO{ sIffdf k:| t't u/fpg'xf;] \ .

5. ;fn] L tyf an] gfsf vf]j|mf gdg' fsf] lgdf{0f

-s_ ljb\ofyLx{ ¿nfO{ a/fa/ bO' { ;d"xdf ljefhg ugx'{ f];\ h:t,} A/B . Pp6f ;d"xnfO{ an] gfsf/ / csf{]
;d"xnfO{ ;f]nL cfsf/sf j:t'x¿sf] ;r" L JolStut ¿kdf agfpg nufpgx' f];\ . kfnk} fnf] sIffdf k|:tt' ug{
nufO{ af]8d{ f l6Kg'xf;] \ .

-v_ lrqdf b]vfP h:t} lrqx¿ Aff8] {df 6f:gx' f;] \ .

;d"x A ;dx" B

DRAFT
-u_ lbOPsf cfsl[ tx¿cWoog u/L / tnsf k|Zgsf] pTt/ AolStut ¿kdf nV] g nufpgx' f;] \ .
1. dfly lbPsf j:tx' ¿ s:tf cfsf/sf 5g\ <
2. oL j:t'x¿sf cfwf/ s:tf cfsf/sf 5g\ <
3. ltgLx¿ lardf s] km/s kfof} <

-3_ lrqx¿cWoog u/L a]Gr ;d"xdf j:tx' ¿sf u0' fx¿ kTtf nufpmg nufpg'xf];\ .
-ª_ dflysf lj|mofsnfkssf cfwf/df an] gf jf an] gfsf/ / ;f]nL cfsf/ j:t'sf] kl/efiff JolStut ¿kdf

nV] g nufpg'xf];\ .
-r_ cfk\mgf] kl/efiffnfO{ ;u“ s} f] ;fyL;u“ 5nkmn u/L ;femf kl/efiff tof/ kf/L sIffsf7] fdf k:| t't ug{

nufpg'xf;] \ .
-5_ cGTodf kl/efiffnfO{ :ki6 kf/Llbgx' f];\ .
6. an] gfsf] vf]j|mf gdg' f lgdf0{ f
-s_ ;a} ljbo\ fyLx{ ¿nfO{ Ps Ps cf6] f cfotfsf/ sfuhsf] kfgf kb| fg ug{x' f];\ .

23

-v_ To;nfO{ lbOPsf lrq tyf gd'gf lgdf0{ fsf r/0fcg';f/ agfp“b} hfgx' f;] \ .
 cfotfsf/ sfuhnfO{ pq} b'O{ cf]6f jT[ tx¿sf] lrq v h:t} x'g] u/L a]g'{xf;] \ .
 sfuhsf wf/x¿nfO{ cfk;df l;wf x'g] u/L 6f:“ g'xf];\ .
 ca s]sf] cfsf/ aG5 x]/f}“ / 5nkmn u/f}“ .

-u_ pSt cfsl[ tnfO{ cf–cfk\mgf] sfkLdf agfP/ k|bz{g u/fpgx' f;] \ .
k|ltlaDag
xfdf| ] bl} gs hLjgdf ko| f]u x'g] / j/k/ kfOg] a]ngfsf] gd'gf x¿ / ltgLx¿sf vfj] |mf gd"gfx¿sf k|of]ux¿sf
af/]df 5nkmn u/fpg'xf;] \ .

7. ;fn] Lsf] vf]jm| f gd'gf lgdf0{ f

tnsf r/0fx¿sf cfwf/df Pp6f ;f]nLsf] lgdf{0f ug{ nufpgx' f;] \ .

1. lrq g=+ s df h:t} u/L sg' } j[Tt sf6L To;df sg' } If]qs AOB lrx\g nufpgx' ]f;\ .
2. If]qs AOB nfO{ lrq g= v df h:t} u/L sf6]/ x6fpg'xf;] \ .
3. lrq g=u df h:t} u/L OA / OB nfO{ hf]8L ss] f] lrq aGof] < nV] g nufpgx' f;] \ .

AA
DRAFT
O B O
lrq g+=s
B lrq g=+ u
lrq g+=v

d¬Nofªs\ g
-s_ ;fn] Lsf] kl/efiff;lxt gdg' f / lrq klg agfpg nufpgx' f];\ .
-v_ a]ngfsf] kl/efiff;lxt gd'gf / lrq klg agfpg nufpg'xf];\ .
-u_ jn] gf / ufn] fsf ;dfg u'0fx¿ / km/sx¿ ;"rL tof/ kfg{ nufpg'xf];\ .

PsfO ^
lgbz{] fªs\

( Co-ordinate)

24

cgd' flgt 3G6LM %

kj' 1{ fg
X-cIf / Y-cIfsf] hfgsf/L
lgb{]zfªs\ sf] kl/ro / rt'yf+{zx¿sf] kl/ro ,
klxnf] rt'yf{+zdf ljGbx' ¿ cªs\ g ug]{ .
kf7 kl/ro

cfwf/et" txdf ljb\ofyLx{ ¿n] cIf/ pb\ud laGbs' f cfwf/df lu|8 kk] /df laGb'x¿ cª\sg ub5\ {g/\ o;sf nflu
xfdLn] laleGg v]n tyf ljm| ofsnfkx¿ u/fpg ;S5f}“ . o;df ljbo\ fyLn{ ] rt'yf+{z / ltgLx¿df kg]{ laGb'sf
lgb]{zfªs\ sf lrx\gsf af/]df ;d]t hfgsf/L kfp5“ g\ . ;fy} lbPsf] laGb'nfO{ uf| kmdf cª\sg ug]{ / u|fkmdf ePsf
laGb'sf] lgb{]zfªs\ kTtf nufOG5 .

DRAFTpbo\ ]Zox¿ M

o; kf7 cWoogkZrft\ ljbo\ fyLx{ ¿ lgDg sfo{ ug{ ;Ifd xg' ] 5g\ M

!= rty' f{zx¿sf] kl/ro lbg / X cIf / Y cIf 5'6\ofpg

@= n]vflrqdf lbOPsf ljGb'x¿sf] lgbz]{ fª\s kTtfnufpg

#= lbOPsf ljGb'nfO{ n]vflrqdf cª\sg ug{ .

zI} fl0fs ;fdu|Lx¿

u|fkm af]8{, lhcf]af8] { cflb, lu|8 sfuh .

;xhLs/0f lj|mofsnfkx¿

1. X cIf / Y cIf

-s_ lbOPsf] lrqnfO{ af8] {df agfpg] jf Y
uf| kmaf]8{sf] ko| fu] u/L cjnf]sg ug{
nufpgx' f;] \ .

-v_ k|Tos] ljb\ofyL{x¿nfO{ tnsf k|Zgx¿sf A
pTt/ nV] g nufpgx' f];\ M B

!= XOX' / YOY' nfO{ s] elgG5 < X’ X

@= laGb' O af6 laGb' A df k'Ug slt O
PsfO bfof“ jf afof“ / tn jf dfly
hfg'k5{ <

#= laGb' O af6 laGb' B df k'Ug slt C
PsfO bfof“ jf afof“ / tn jf dfly D
hfg'k5{ <
Y’

$= laGb' O af6 laGb' C df k'Ug slt PsfO bfof“ jf afof“ / tn jf dfly hfg'k5{ <

% =laGb' O af6 laGb' D df k'Ug slt PsfO bfof“ jf afof“ / tn jf dfly hfgk' 5{ <

-u_ pTt/ nl] v;s]kl5 ;“us} f] ;fyLsf] pTt/;“u t'ngf ug{ nufpg'xf;] \ .

-3_ hf8] Lsf] pTt/nfO{ a]Gr ;d"xdf 5nkmn u/L ;femf pTt/ agfP/ sIffdf k|:tt' ug{ nufpgx' f];\ .

25

-ª_ cGTodf ljb\ofyLx{ ¿nfO{ rf/ rty' f{+z / ltgLx¿sf kg]{ laGbx' ¿sf lgb]{zfª\ssf lrxg\ x¿sf af/d] f lgGg
tYonfO{ k:| tt' ul/lbg'xf];\ .
X cIfaf6dfly wgfTds (+), X cIfaf6 tn C0ffTds (-)
Y cIfaf6 bfofFwgfTds (+), Y cIfaf6 afof“ C0ffTds (-)
;fy}, X cIfaf6 dfly s'g}] laGb'sf] lgb{]zfª\s (4,6) / tn (4,-6) 5 eg] X cIfaf6 pSt bO' { laGb'x¿
a/fa/ b'/Ldf kb{5g\ .

To:t,} Y cIfaf6 bfofFs'g}] laGbs' f] lgb]z{ fªs\ (5,5) / afofF (5,-5) 5 eg] Y cIfaf6 pSt bO' { laGb'x¿
a/fa/ b'/Ldf kb{5g\ .
d"Nofªs\ g
ljb\ofyL{x¿nfO{ kT| o]sn] u|fkm k]k/df Ps Ps cf]6f rte' '{h lvRg nufP/ ;“u}sf] ;fyL;“u pSt cfsl[ t
;f6f;f6 u/L To;sf zLifl{ aGb'x¿sf lgb{]zfªs\ x¿ kTtf nufpg nufpg'xf];\ .

2.

-s_ lgb{]zfª\s / rt'yf{+z tyf ltgLx¿df kg]{ laGb'x¿sf (-,+) (+,+)
lgb]{zfªs\ x¿sf lrx\gsf af/]df kg' /fjnf]sg u/fpg'xf;] \ .

-v_ af]8d{ f s]xL laGb'x¿sf lgb]{zfªs\ x¿ n]v]/ uf| kmdf cªs\ g ug{
nufpg'xf];\ .
DRAFT
-u_ lgb]{zfªd\ f klxnf] cª\sdf ePsf lrx\gsf cfwf/df bfof“ jf afof“

stf k5{ kTtf nufpg nufpgx' f;] \ . (-,-) (+,-)

-3_ bf;] f| ] cªs\ df ePsf] lrx\gsf cfwf/df sg' rty' f+z{ df k5{
lgSof]{n u/fpgx' f;] \ .

-ª_ laGb'nfO{ n]vflrqdf cªs\ g ug]{ laGb'nfO{ ;fgf] sf]i6 k|ofu] u/L n]Vg] hfgsf/L u/fpgx' f;] \ .

dN" ofªs\ g

ljb\ofyLx{ ¿nfO{ lgDgfg;' f/sf laGbx' ¿ n]vflrqdf cªs\ g Y
u/L pSt laGb'x¿nfO{ ?n/n] hf8] ]/ aGg] cfsl[ tsf] gfd
n]Vg nufpgx' f];\ .

-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(1,3), / L(4,0) X’ X
-3_ P(2,4), Q(-2,4), R(-2,-4) / S(2,-4)
O

PsfO & Y’

kl/ldlt / If]qkmn

(Perimeter and area)

26

cgd' flgt lzIf0f3G6LM 10

kj' { 1fg M

eh' fx¿sf] gfkåf/f lqe'hsf] kl/ldlt kTtf nufpg, ju{ uGtL u//] lgoldt tyf clgoldt cfsf/x¿sf]
Ifq] kmn lgsfNg, cfot / jus{ f] If]qkmn -;q' ko| fu] u/]/_ lgsfNg, if8\dv' f / 3gsf] cfotg ;DaGwL ;/n
;d:ofx¿ xn ug{ .

kf7 kl/ro
kl/ldlt eGgfn] s'g} j:t' af cfs[ltsf] jl/kl/ 3/] f eGg] a'lemG5 . h:t} cfotsf] kl/ldlt, ju{sf] kl/ldlt
eg]sf] ltgLx¿sf ;dx' df rf/ cf6] f ehfsf] nDafOsf] of]ukmn xf] . To;}u/L Ifq} kmn eGgfn] s'g} j:tn' ]
;dtnLo ;txdf cf]u6]sf] 7fp“ eGg] a'lemG5 . s'g} 7f]; j:t'sf] k'/f ;txsf] Ifq] kmn eGgfn] k|Tos] ;dtnLo
;txsf] If]qkmnsf] ofu] kmn eGg] al' emG5 . h:t} if8\dv' fsf] k/' f ;txsf] If]qkmn lgsfNg ;a} 5 cf]6f
cfotfsf/ ;txsf] Ifq] kmn lgsfn]/ hf]l8G5 .

kf7 M ju{ / cfotsf] kl/ldlt (Perimeter of Square and Rectangle)
pbo\ ]Zox¿ M o; kf7sf] cWoogkZrft\ ljbo\ fyL{x¿ lgDg sfo{ ug{ ;Ifd x'g] 5g\ M
DRAFT
-s_ ju{ / cfotsf] kl/ldlt lgsfNg-;'q ko| fu] u//] _ . Y

-v_ if8\d'vf / 3gsf] ;txsf] Ifq] kmn -;q' k|of]u u//] _
lgsfNg .

zI} fl0fs ;fdu|Lx¿

ljleGg cfotsf/ jf juf{sf/ cfsl[ tx¿, 3g tyf X’ X
if8\dv' fsf/ j:tx' ¿ O
l;sfO ;xhLs/0f ljm| ofsnfk
1. ju{sf] kl/ldlt

-s_ u|fkm k]k/df lbOPsf] ju{ lvRg] / af8] d{ f 6f“:g] /
ljb\ofyL{x¿nfO{ skL ug{ nufpg'xf;] \ .

-v_ To; jus{ f rf/ cf6] f e'hfx¿ gfKg ;fy} sf]0fx¿ Y’
klg gfKg nufpgx' f];\ .

-u_ ca sf0] fx¿ / e'hfx¿ s:tf /x]5g\ a]Gr ;dx" df 5nkmn u/fpgx' f];\ .

-3_ ;a} ljbo\ fyL{x¿nfO{ pSt ju{sf ;a} eh' fx¿sf] nDafO hf]8g nufpg'xf];\ .

-3_ ljb\ofyLx{ ¿sf lj|mofsnfkx¿sf] cjnfs] g u/L ;a} e'hfx¿ gfksf] of]ukmn pSt ju{sf] kl/ldlt xf] egL
:ki6 kfgx'{ f;] \ .

;fy}, lrqdf rf/ cf6] } e'hfsf] nDafO 8 cm 5 .

of]ukmn, 8cm + 8cm + 8 cm+ 8cm = 32 cm x'G5 .

= 4 x 8cm

= 4 x Pp6f e'hfsf] nDafO

kl/ldlt (P)= 4 x l x'G5 egL ki| 6 kfgx'{ f;] \ .

dN" ofª\sg

-s_ km/s km/s gfksf ju{x¿ lbP/ ltgLx¿sf e'hfx¿sf nDafO gfk]/ kl/ldlt lgsfNg nufpg'xf;] \ .

-v_ s'g} Pp6f eh' fsf] nDafO lbPsf] cj:yfdf jus{ f] kl/ldlt lgsfNg nufpg'xf;] \ .

27

-u_ kl/ldlt lbPsf] cj:yfdf jus{ f] e'hfsf] nDafO kTtf nufpg nufpg'xf;] \ .

2. cfotsf] kl/ldlt
s_ ljb\ofyLx{ ¿sf hf8] L lgdf0{ f u/L ;a}nfO{ Ps Ps cf]6f cfot lvRg nufpgx' f];\ .
-v_ lvrs] f cfotx¿nfO{ hf]8Ldf ;f6f;f6 ug{ nufpgx' f];\ . ?n/sf] k|of]u u/L

;a} eh' f gfKg nufO{ a/fa/ gfk ePsf e'hfx¿ kTtf nufpg eGgx' f;] \ .

-u_ To;kl5 ;a} e'hfx¿sf gfksf] ofu] kmn kTtf nufpg eGgx' f];\ .

-3_ hf]8Ldf 5nkmn u/L lgisif{ lgsfNg nufpg'xf];\ .

-ª_ lzIfsn] ljb\ofyLx{ ¿sf ljm| ofsnfkx¿sf] cjnf]sg u/L ;a} e'hfx¿ gfksf] ofu] kmn pSt cfotsf]
kl/ldlt xf] egL :ki6 kfg'x{ f];\ .

;fy}, rf/ cf]6} e'hfsf] gfksf] ofu] kmn

= l cm + b cm + l cm+ b cm x'G5 .

= 2 l cm +2 b cm

= 2-nDafO + rf8} fO_xG' 5
kl/ldlt (P)= 2(l + b) x'G5 egL k|i6 kfg{x' f;] \ .
kl| tljDag
DRAFT
sg' } klg 7f;] cfs[ltsf] kl/ldlt kTTff nufpgs' f] pkof]lutfsf] af/]df dl:tis dGyg u/fpgx' f;] \ .
dN" ofª\sg
-s_ km/s gfksf cfotx¿ lbP/ ltgLx¿sf eh' fx¿sf nDafO gfk]/ kl/ldlt lgsfNg nufpg'xf];\ .
-v_ bO' { cf]6f eh' fsf] nDafO lbOPsf] cj:yfdf cfotsf] kl/ldlt lgsfNg nufpg'xf;] \ .

-u_ kl/ldlt / nDafO jf rf}8fO lbPsf] cj:yfdf cfotsf] nDafO jf rf}8fO kTtf nufupg lbg'xf];\ .

3. If]qkmn
-s_ Ifq] kmn eg]sf] s] xf] < cfoftsf/ j:t'sf] Ifq] kmn s;/L lgsflnG5 < 5nkmn u/fpgx' f];\ .

-v_ If]qkmn / 3gsf] k/' f ;txsf] Ifq] kmnlar s] km/s 5 < cfcfkm\ gf] sfkLdf n]Vg nufpg'xf;] \ .

-u_ 7f]; j:t'sf] k'/f ;txsf] Ifq] kmn s;/L lgsfNg ;lsG5 < cgd' fg u/L n]Vg nufpg'xf;] \

-3_ ;a} ljb\ofyL{x¿nfO{ 6/6 hgf kg{] u/L ;d"x lgdf0{ f ug'{xf];\ / k|Tos] ;d"xnfO{ Ps Ps cf6] f if8\d'vf
lbg'xf];\ .

-ª_ Ps Ps u/L 6 hgfn] pSt 7f;] sf 6 cf6] f cfotsf/ ;txsf nDafO / rf}8fO gfkL Ifq] kmn /
ltgLx¿sf] ofu] kmn kTtf nufpg nufpg'xf;] \ .

-r_ ljb\ofyL{x¿sf lj|mofsnfkx¿sf] cjnf]sg u/L ;a} cfotsf/ ;txx¿sf] If]qkmnx¿sf] of]ukmn g} pSt
7f]; cfsl[ tsf] k'/f ;txsf] Ifq] kmn xf] egL ki| 6 kf/Llbg'xf;] \ .

if8d\ v' fdf 3 hf]8f cfotsf/ ;txx¿ xG' 5g\ .

Ifq] kmn (A) = l × b + l × b + b × h + b × h + l × h + l × h

=2(l × b) + 2(b × h) + 2( l × h) xG' 5 .

=2(l × b +b × h+ l × h)xG' 5 .

-5_ To:t}, s'g} 7f;] cfs[ltsf 6 cf]6f ;txx¿ jufs{ f/ 5g\ eg] pSt 7f];nfO{ 3g elgG5 egL :ki6
kfl/lbgx' f;] \ .

Ifq] kmn (A) =l×l+l×l+l×l+l×l+l×l+l×l

28

DRAFT = 2(l 2) + 2(l2) + 2( l2) xG' 5 .
= 6 l 2 x'G5 .
dN" ofªs\ g
-s_ if8d\ 'vfsf/ / 3gfsf/ 7f]; cfsl[ tdf slt cf6] f ;dtnLo ;txx¿ x'G5g\ <
-v_ nDafO, rf8} fO, / prfO lbPsf] cj:yfdf if8\dv' fsf] k'/f;txsf] Ifq] kmn kTtf nufpgx' f];\ .
-u_ Pp6f e'hfsf] nDafO lbPsf] ca:yfdf 3gsf] k/' f;txsf] If]qkmn lgsfNg] ;"qx¿ n]Vg nufpgx' f];\ .
lzIfsnfO{ yk hfgsf/L
o;sf ;fy;fy} bl} gs hLjgdf cfwfl/t kZ| gx¿ xn u/fpg] .

29

kf7 8
:yfgfGt/0f (Transformation)

cg'dflgt lzIf0f 3G6L M %

kÒj{ 1fg
:yfgfGt/0fsf] ;fdfGo kl/ro
:yfgfGt/0fsf ;fwf/0f pbfx/0fx¿

kf7 kl/ro
:yfgfGt/0f lzIf0fsf] ;?' jft ;dtn Pg] f, :s]nsf] ko| f]u, 38Lsf] ;O' { tyf 6a] n' sf] lstfa cflbaf6
;?' jft ug{ ;lsG5 . o;af6 ljbo\ fyL{x¿n] cfsl[ t / k/fjt{gkl5sf] kl| tlaDa cfs[lt ;db/' Ldf kg]{,
k/fjtg{ , kl/j|md0f / :yfgfGt/0f -k/fjt{g, kl/j|md0f tyf lj:yfkg_ df j:t' / kl| tlaDa cfs[ltx¿ Ps
csfd{ f cg'¿k xg' ] s'/f;d]t hfgsf/L xfl;n ug{] 5g\ .

pbo\ Z] ox¿
o; kf7sf] cGTodf ljb\ofyL{x¿ lgDg sfo{x¿ ug{ ;Ifd xg' ] 5g\ M
!= k/fjt{gsf] kl/ro lbg
@= sg' } cfsl[ tnfO{ lbOPsf] /v] faf6 k/fjt{g u/L u|L8 k]k/df lgb{]zfª\s kTtf nufpg
DRAFT
#= kl/j|md0fsf] kl/ro lbg A B

$= s'g} cfs[ltnfO{ lbOPsf] sf]0f / laGb'af6 kl/j|md0f u/L uL| 8 k]k/df
lgb{]zfª\s kTtf nufpg .

%= lj:yfkgsf] kl/ro lbg / HofldtLo cfsl[ tx¿nfO{ lj:yfkg u/L k|ltlaDa
cfs[lt lvRg .

z}Ifl0fs ;fdu|Lx¿

;fbf sfuh, u|L8 kk] /, d;L, HofldtLo cfs[ltx¿, lhofa] f]8{ cflb .

l;sfO ;xhLs/0f lj|mofsnfkx¿

1. k/fjtg{ (Reflection)
-s_ ;a} ljbo\ fyL{x¿nfO{ PsPs cf]6f ;fbf sfuh lng nufpgx' f];\ .

-v_ To;df Ps Ps yfk] f lgnf] jf sfnf] dl; /fVg nufpg'xf];\ .

-u_ d;L Psk66\ L dfq kg{u] /L sfuhnfO{ k6\ofpg / xNsf lyr/] dflyaf6 d;LnfO{ km}nfpg eGgx' f];\ .

-3_ k6\ofPsf] sfuhnfO{ vfN] g / s:tf] cfs[lt aGb5 cjnfs] g ug{ nufpgxf];\ .
s] cfs[ltx¿ a/fa/ / p:t} 5g\ < sfuh k6\ofpb“ f ag]sf] /v] fnfO{ AB gfd lbg nufpgx' f;] \ .

-ª_ AB af6 sfuhdf ag]sf] cfsl[ tx¿sf] b'/L slt 5 < gfKg nufpgx' f;] \ .

-r_ ;dx" x¿ lgdf{0f u/L ;a}sf cfs[ltx¿cfk;df ;f6f;f6 ug{ nufpgx' f;] \ .

-5_ ;dx" df 5nkmn u/L ;femf lgisif{ kTtf nufpgx' f;] \ / Ps hgfnfO{ cfs[lt;lxt sIffdf k|:t't ug{
nufpgx¬ f];\ .

-h_ cGTodf, k/fjtg{ , k/fjt{gsf] cIf, k/fjt{g cufl8 / k5fl8 ag]sf cfs[ltx¿sf cfsf/ ks| f/sf tyf
lbOPsf] cfs[lt / k|ltlaDa cfsl[ tsf] AB af6 gfk]sf] b'/Lsf af/d] f :ki6 kf/LbLg'xf];\ .

d"Nofª\sg

k/fjt{gsf cIf / Pp6f cfsl[ t lbOPsf] cj:yfdf k|ltlaDa cfsl[ t lvRg nufpg¬xf;] \ .

30

lzIfsnfO{ yk hfgsf/L

lhcf]af8] d{ f /a/ AofG8sf] k|of]u u//] Pp6f k/fjt{gsf] cIf dfg]/ ljleGg cfsl[ tx¿ lgdf0{ f u/L ltgLx¿sf]
k/fjt{g kl5sf kl| tlaDax¿ agfP/ klg cWoog u/fpg ;lsG5 . o;sf ;fy ;fy} ljb\ofyL{x¿nfO{
vn] x¿af6 klg l;sfpg ;lsG5 .

2. kl/jm| d0f -Rotation)

-s_ sIffdf km/s km/s Ifdtf ePsf Y
ljbo\ fyLx{ ¿ ldnfP/ ;d"xx¿ lgdf{0f O
ugx'{ f;] \ .
Y’
-v_ k|To]s ljb\ofyLx{ ¿nfO{ u|L8 sfuhdf Ps
Ps cf6] f lqe'h lvRg nufpg'xf;] \ .

DRAFTh:t}, A(1,4), B(5,7) / C(6,3) X’ X

-u_ ;f]xL lqe'h a/fa/sf] gfk ePsf] csf{]
lqeh' kf/bzL{ sfuhdf agfpg nufO{
X / Y cIf klg agfpg nufpg'xf];\ .

-3_ laGb' O df 3'Dg] u/L kf/bzL{ sfuhnfO{
u|L8df 6f“:g nufpg'xf;] \ .

-ª_ kf/bzL{ sfuhnfO{ km/s km/s sf0] fdf
wgfTds lbzfdf 3'dfpg eGgx' f];\ .

h:t} M OX nfO{ OY df / OY nfO{ OX' df kg{] u/L 3d' fpg
nufpg'xf;] \ / pSt lqeh' sf zLif{laGb'sf lgb{]zfªs\ x¿ kTtf nufpg
nufpgx' f];\ .

h:t}, A(1,4)A’(-4,1)]

B(5,7)B’(-7,5)/

C(6,3)C’ (-3,6)
-r_ cGTodf ;dx" x¿df Ps hgfn] sIffdf k:| t't ug{] . lzIfsn] lgDg k|Zgx¿ ;fw] ]/ ki| 6 kfl/lbg'xf];\ .

kl/j|md0f eg]sf] s]xf] < wgfTds / C0ffTds kl/j|md0f eGgfn] s] al' emG5, cflb .

dN" ofª\sg

s'g} cfsl[ t uL| 8 sfuhdf lvRg nufP/ pSt cfs[ltnfO{ 90  / 180 df kl/j|md0f u/fpg
nufpg'xf;] \ .

lzIfsnfO{ yk hfgsf/L
4/5 hgf ljbo\ fyLx{ ¿nfO{ cufl8 pEofP/ bfof“ kms{, afof“ kms{, k'/fkms{ h:tf vn] x¿af6 klg kl/j|md0fnfO{
kl/ro u/fpg ;lsG5 .
3. D’ C’ lj:yfkg ( Displacement)

A’ B’
DC

31

DRAFT-s_ lzIfsn] uL| 8 sfuhdf agfPsf] lgDgfg;' f/sf] lrq af]8d{ f 6F;“ g\ x' f];\ . kT| os] ljb\ofyL{nfO{ pSt lrq
lvRg nufpgx' f];\ .

-v_ b'j} lrqsf ;a} e'hfx¿ gfKg nufpgx¬ f;] \ . s] b'j} lrq Pp6} 5g\ < ;fy} bj' } lrqsf zLifl{ aGb'sf
lgb]{zfªs\ x¿ n]Vg nufpg¬xf];\ .

-u_ tL bO' { cfotdf s] km/s 5 < skLdf n]Vg nufpgx' f];\ /;dx" df 5nkmn u/L sIffdf k|:tt' u/fpgx' f];\ .
-3_ lgDg s'/fnfO{ :ki6 kfg'x{ f;] \ .

b'O{ cfot a/fa/ cfsf/sf 5g\ . 7fp“ dfq km/s 5 . t;y{ cfot A'B'C'D' cfot ABCD sf]
lj:yfkg k|ltlaDa xf] .
o;/L s'g} klg cfsl[ tnfO{ To;sf] cfsf/, k|sf/ tyf cj:yfdf s'g} kl/jt{g gxg' ]u/L sg' } lglZrt
lbzfdf / lgZlrt b'/Ldf ;fg'{nfO{ lj:yfkg elgG5 .
k|ltlaDag
k/fjt{g, kl/j|md0f / lj:yfkgsf Ps Ps cf6] f bl} gs hLjgdf k|of]u ePsf pbfx/0fx¿ vf]hL k|:tg'
ug{ nufpg'xf];\ .
dN" ofª\sg M
sg' } HofldtL cfs[lt uL| 8 sfuhdf lvRg nufP/ pSt cfs[ltnfO{ lgZlrt lbzfdf / lgZlrt b'/Ldf lj:yfkg
u/fpg nufpg'xf;] \ .
lzIfsnfO{ yk hfgsf/L
ljb\ofyLx{ ¿nfO{ d6] fsf8d{ f lgb]{zfªs\ x¿ n]v]/ cfk\mgf] 7fp“ ;f//] klg lj:yfkgnfO{ kl/ro u/fpg ;lsG5 .

32

DRAFT PsfO 9
;dldlt / 6l] ;n;] g

Symmetry and Teslation

cgd' flgt lzIf0f 3G6L M 6

k"j{ 1fg
lgoldt / clgoldt lrqx¿, ax'eh' x¿
PsfO kl/ro
oL kf7x¿sf nflu ljleGg cfsf/sf] sfuhx¿nfO{ sf6]/ k|of]ufTds tl/sfn] cWoog ug{k' g{] x'G5 . s'g}lrqsf]
/v] Lo ;dldlt slt cf6] f 5g\, laGb' ;dldltx¿sf] >]0fL slt xG' 5 cflb s'/fx¿sf] hfgsf/L ljbo\ fyL{x¿
:jodn] ko| fu] u/L x]g{ nufpg'kg{] 5 . To;}u/L lgoldt, cwl{ goldt tyf clgoldt sfuhsf 6j' |mfx¿af6 s'g}
7fp“ vfnL gxg' ] u/L 6f:“ g nufO{ 6l] ;n];g u/fpg'kg{] x'G5 . oL kf7x¿sf] k|ofu] sfk{]6, jfn k]k/ tyfcGo
l8hfOgx¿df b]Vg kfOG5 . ul0ftsf oL kf7x¿ ul0ftLo ;'Gb/tfsf cg'kd pbfx/0fx¿ x'g\ .

pb\b]Zox¿
o; kf7sf] cWoogkZrft\ ljbo\ fyLx{ ¿ lgDg sfo{ ug{ ;Ifd x'g] 5g\ M
1= laGb' ;dldltsf] >]0fL kTtf nufpg
2= HofldtLo lrqsf] /]vLo ;dldlt kTtf nufpg
3= 6]l;n;] gsf] kl/efiff eGg / 6]l;n;] g ug{
4= axe' h' sf] 9f“rf agfpg .
z}Ifl0fs ;fdu|L
afSnf] cfotfsf/ sfuhdf j[Tt lvlrPsf] / ;fx] L j[Ttsf] cw{Jof; a/fa/ x'g]u/L cwj{ T[ t, Ps ltxfO efusf]
j[Tt, Ps rf}yfO efusf] Ifq] sx¿sf] s]Gbd| f lkg /fvL k/' f j[Ttsf] s]Gb|df hf]8g] tfls tL j[Tt v08x¿
;lhnl} ;t 3d' fpg ;lsof];\ . ;f]xL cw{Jof; a/fa/ xg' ] Pp6f jT[ t, km/s /ªsf kftnf] sfuhx¿, rf6{ kk] /,
:sn] cflb .
l;sfO ;xhLs/0f ljm| ofsnfk
1=-s_ z}Ifl0fs ;fdu|Ldf lbOPsf] jT[ t v08x¿;lxtsf] j[Tt kb| z{g ugx{' f;] \ . j[Ttv08sf] A k/' f j[Ttsf] A df

k'Ug slt rf]6L 3'dfpg' knf{ < 5nkmn ug{ nufO{ k|:tt' ug{ nufpgx' f;] \ . To;u} /L o;sf] ljGb'
;dldltsf] >]0fL j|mdzM 2, 3, 4 x'g]af/] klg k:| tt' ug{ nufpg'xf;] \ .
-v_ ;f] cfotsf] j[Ttdfly pxL cwJ{ of;sf] j[Ttsf] sG] b|df lkgn] /f]kL dflysf lj|mofsnfkdf e}m“ 3d' fpg
nufpg'xf];\ . ;f] jT[ t slt rf6] L 3d' fp“bf A sf] dfly g} A knf{< 5nkmn u/fpg'xf;] \ . oxf“ k"/f Ps
kmGsf] nufpg'kg{] ePsfn] o;sf] >0] fL gxg' ]af/] 5nkmn ug{ nufpgx' f];\ .
-u_ To:t} ljGb' ;dldltsf] >]0fL 2,3,4 xg' ] lrqx¿ Pp6f Pp6f agfpg nufpg'xf;] \ .
-3_ kf7o\ k':tssf lj|mofsnfkdf 5nkmn ug{ nufpg'xf;] \ .
2=-s_ ljbo\ fyLx{ ¿nfO{ cfkm'v';Lkm/s km/s cfsf/sf lrqx¿ rf6{ kk] /df agfO{ sf6g\ nufpg'xf];\ . tL
sfl6Psf lrqx¿nfO{ l7s cfwf xg' ] u/L k6o\ fpgnufpg'xf;] \ . l7s cfwf xg' ] u/L sfl6Psf] 7fp“df 86\
/]vf sf]g{ nufpg'xf];\ . Tof] g} /]vLo ;dldlt xg' a] f/] 5nkmn ug{ nufpgx' f;] \ .
-v_ /]vf ;dldlt x'g] lrqx¿ s'g} 5 cf]6f agfpg nufpg'xf];\ .
-u_ s'g s'g lrqx¿df bO' { cf6] f jf bO' { cf]6f eGbf a9L /]vf ;dldltx¿ x'G5g\ xf]nf < 5'6o\ fpg
nufpgx' f];\ . Toxf“ gePdf agfpg nufpg'xf];\ .
-3_ kf7\o k:' tssf] /v] f ;dldltsf lj|mofsnfkx¿ 5nkmn ug{ / lgisif{ eGg nufpgx' f;] \ .

33

DRAFT3=-s_ km/s /ªsf kftnf sfhux¿nfO{ ;de'h lqeh' fsf/, jufs{ f/ jf cfotfsf/ ¿kdf sf6\g nufO{ rf6{
k]k/ jf Goh' lk|G6 jf klqsfdf 7fp“ vfnL gx'g] u/L 6f“:g nufpg'xf;] \ . Tof] g} lgoldt 6]l;n];g x'g]af/]
5nkmn ug{ nufpg'xf;] \ .

-v_ km/s km/s cfsf/sf kftnf] sfuhx¿ dWo]af6 s'g} b'O{ cf6] faf6 -;s;] Dd km/s /ªsf_ 6]l;n;] g ug{
nufpg'xf;] \ . Tof] g} cw{ lgoldt ;dldltxg' ]af/] 5nkmn ug{ nufpgx' f;] \ . ljb\ofyL{nfO{ cfk"mv;' L cGo
lrqx¿af6 cw{ lgoldt 6];n;] g ug{ nufpgx' f];\ .

-u_ To;}u/L kf7\o k:' tsdf lbOPsf afxs] cGo clgoldt 6l] ;n;] g ug{ nufpg'xf;] \ .
4= rf6{ kk] /df ;dafx' lqeh' , ju{, lgoldt k~re'h, if8\eh' cflbsf] 9f“rf agfpg] ;d"x sfo{ lbgx' f;] \ /

sIffdf ;hfpg nufpgx' f];\ . /fdf| ] 9f“rfnfO{ k'/:st[ ug'{xf;] \ .
d"Nofª\sg
1= ljb\ofyL{x¿nfO{ cfk"mv;' L lrq agfpg nufO{ /]vf ;dldlt / /]vf ;dldltsf] >]0fL tyf laGb' ;dldltsf]

>0] fL kTtf nufpg lbg'xf;] \ .
2= lgoldt, cw{ lgoldt / clgoldt 6]l;n];gsf] Pp6f / Pp6f pbfx/0f agfpg lbg'xf];\ .
3= ljb\ofyL{nfO{ cfkm" v';L ax'e'hsf] 9f“rf tof/ ug{ nufpgx' f];\ .
lzIfsnfO{ lgb]{zg
oL kf7x¿ k|of]ufTds lj|mofsnfkdf cfwfl/t kf7 ePsfn] o;df ljb\ofyLs{ f] ;lj|mo ;xeflutf ckl/xfo{ 5 .
ctM ljbo\ fyLx{ ¿åf/f cfcfkm'n] tof/ u/]sf ;fduL| x¿ ;f6f;f6 ug{ nufO{ k|:tt' ug{ nufpg] tyf
ljlzi6gd"gfx¿nfO{ sIffsf]7f ;hfpg k|f]T;fxg ug{'xf;] \ .

34

DRAFT PsfO 10M
lbzfl:ylt / :s]n 8«Oª

Bearing and Scale Drawing

cg'dflgt lzIf0f 3G6L M5

k"j{ 1fgM gS;f k7\g, rf/ lbzf 56' \ofpg ;Sg]
PsfO kl/roM
o; kf7cGtu{t ljb\ofyL{nfO{ pgLx¿n] yfxf kfPsf rf/ lbzfafx]s cGo lbzfx¿sf] hfgsf/L Pjd\
gk] fnsf] gSzf x/] L s'g} 7fp“nfO{ cfwf/ dfgL csf]{ 7fps“ f] lbzfl:ylt kTtf nufpg] / s'g} :yfg jf
j:t'sf]:s]n 8«Oª ug{ nufpg] tl/sfaf/] 5nkmn ul/g] 5 . o;sf] nflu cgk' ftsf] Go"gtd kbdf n}hfg
;Sg] l;k ljb\ofyLd{ f x'g'kg{] 5 .

kf7 pb\bZ] o
o; kf7sf] cWoogkZrft\ ljb\ofyL{x¿ lgDg sfo{ ug{ ;Ifd x'g] 5g\ M
1= rf/ lbzfsf ;fy} tL lbzfsf] lardf kg]{ lbzfx¿sf] gfd eGg / bv] fpg
2= gk] fnsf] gSzfdf /xs] f b'O{ :yfgx¿ dWo] Pp6faf6 csf{s] f] lbzfl:ylt atfpg
3= s'g} j:t' jf :yfgsf] :sn] 8«Oª ug{ .
zI} fl0fs ;fduL|
cfotfsf/ sfuhx¿, l:kg/ agfpg afSnf] sfuhsf j[Ttx¿, lkg, afSnf l;Gsfx¿, lbzf b]vfpg]
sDkf; -pknJw ePdf_, lhNnf cªl\ st gk] fnsf] gSzfx¿, :s]n cflb .
l;sfO ;xhLs/0f ljm| ofsnfk
1= -s_ k|To]s ljb\ofyLx{ ¿nfO{ cfotfsf/ sfuh lbO ;Ddv'

nDafO / rf8} fOx¿ k6o\ fpg lbg'xf;] \ . tL b'O{ cf6] f k6\l6Psf
/v] fx¿ hf]8sf] 7fp“df 0 nV] g nufO{ rf/ lbzf klg nV] g
nufpg'xf;] \ .
-v_ To;}u/L N dfly E vlK6g] u/L sf]0f NOE nfO{ cfwf ug{
nufpg'xf];\ . ;f] kfgf vf]Ng nufO{ ;a} lbzfsf gfdx¿ n]Vg
nufpgx' f;] \ / ltgLx¿sf sf]0fx¿ gfKg nufO n]Vg nufpg'xf];\ .
lbzf nV] bf N jf S af6 ;'?
xG' 5 . blIf0f tkm{ eP S af6 / pTt/ tkm{ eP N af6 lbzf nl] vG5 egL :ki6 kfl/lbg] .
-u_ sIffsf ljb\ofyL{x¿nfO{ cfotfsf/ ¿kdf /fvL larlt/ a;]sf Ps hgf ljbo\ fyL{nfO{ ) dfgL
p;sf] k"j{, klZrd, pTt/, blIf0fdf sf] sf] a;s] f 5g\ eGg nufO{ pTt/ k"j,{ pTt/ klZrd, blIf0f
k"j{ / blIf0fklZrddf sf] sf] 5g\ :jod\ ljb\ofyL{x¿nfO{ g} eGg nufpg'xf];\ .
2= -s_ k|Tos] aG] rnfO{ Pp6f÷Pp6f g]kfnsf] gSzf lbO{ cfkm" a;s] f] lhNnfaf6 ljleGg lbzfdf /x]sf

b'O{ b'O{ :yfgsf] gfd eGg÷n]Vg nufpg'xf];\ .
-v_ vn] vN] g nufpgM] ljbo\ fyL{x¿nfO{ bO' { ;dx" df ljefhg ugx{ f];\ . Pp6f ;d"xsf] Ps hgfnfO o:tf

k|Zgx¿ ;f]Wg nufpgx' f;] \ - lj/f6gu/af6 pTt/ klZrddf /xs] f] Pp6f :yfg eGg nufpg'xf;] \ .
eGg;s]df 1 cªs\ lbg] g;sd] f cªs\ glbg] . To:t} csf]{ ;dx" sf ljb\ofyLa{ f6 c? kZ| gx¿
kfnk} fnf] ;f]Wg nufpgx' f];\ . cj:yf / ;do x]/L v]n 6'ª\UofO{ xf/lhtsf] 3f]if0ff ug'x{ f];\ .

35

DRAFT-u_ kT| o]s a]Grdf afSnf] j[Ttfsf/ sfuhsf] sG] bd| f O n]vL lbzfx¿sf] gfd n]Vg nufpg'xf];\ . 5]:sf
/ lkgsf] ;xof]uåf/f :kLg/ (spinner) agfpg nufpg'xf];\ . ljb\ofyLx{ ¿ lar cfkmn" ] eg]sf]

lbzfdf ;O' /f] k/d] f lhTg] u/L v]n v]Ng nufpg'xf];\ .
3= ljb\ofno ejgn] rr{s] f] efusf] gfk ;lxt gSzf agfpg ;d"x sfo{ lbg'xf];\ . n]lvPsf] gfk cg;' f/
:sn] dfgSzf nv] ] gnv] ]sf] ljb\ofyL{nfO{ g} hfR+ g nufpg'xf;] \ / :s]ncg';f/ nV] g cEof; u/fpg'xf;] \ .
dN" ofª\sg

1= cfk"m a;]sf] 3/sf] ;a} lbzfdf /xs] f] 7fp“ jf dlGb/sf] gfd n]Vg nufpg'xf;] \ .
2= ljbo\ fnosf] pTt/ klZrd lbzfdf s'g :yfg /xs] f] 5<
3= 1m eGbf nfdf] lgoldt j:t'sf] :s]n 8O« ª ug{ nufpgx' f];\ .
lzIfsnfO{ lgb]{zgM
dflysf wf/0ffx¿ lbO;s]kl5 kf7o\ k':tssf ljm| ofsnfkx¿ klg ;fy;fy} u/fpg' pko'St x'G5 .
ljZjsf] gSzf x/] L ljleGg dxfb]z tyf b]zx¿sf lbzfl:ylt atfpg / :s]n x]/L jf:tljs b'/L ;d]t
lgsfNg kf| T] ;fxg ug{' pko'St xfn] f .

36

PsfO 11
;d"x(Sets)

cg'dfgLt lzIf0f 30l6 M12

k"j{ 1fg
;d"xsf] kl/ro, ;d"xsf ks| f/, ;d"x ;ª\st]
PsfO kl/ro
o; PsfOcGtut{ ljb\ofyLx{ ¿nfO{ lbOPsf ;dx" x¿sf] ;j{Jofks ;dx" , s'g} klg ;d"xsf pk;d"xx¿ Pj+;d"xx¿
ljr ;o+ fh] g / kl| tR5]bg ;DaGwL wf/0ff tyf e]g lrqsf] k|ofu] ug]{ af/] 5nkmn ul/g5] .

pb\b]ZoDRAFT
o; kf7sf] cWoogkZrft\ ljbo\ fyLx{ ¿ lgDg sfo{ ug{ ;Ifd x'g] 5g\ M
1= lbOPsf ;dx" x¿sf] ;j{Jofks ;dx" agfpg
2= lbOPsf ;jJ{ ofks ;dx" sf] pk;dx" x¿ agfpg
3= pkoS' t / cgk' o'St ;d"xx¿ 5'6o\ fpg / kl/efiff lbg
4= ;d"xx¿nfO{ e]g lrqdf b]vfpg .
z}Ifl0fs ;fduL|
a6\6fx¿ -4÷5_ cf\6f, a66\ fdf c6fpg] sfuhsf cfotfsf/ 6j' |mfx¿ kl| tljb\ofyL{ Pp6f
l;sfO ;xhLs/0f lj|mofsnfk

1= ;jJ{ ofks ;dx"

-s_ k|Tos] ljbo\ fyLn{ fO{ Ps Ps 6j' |mf sfuh lbO{ To;df Pp6f vfbo\ \o j:t'sf] gfd nV] g nufpg] /
sIffsf] cue| fudf/fVg nufpg'xf];\ .

-v_ 3÷4 hgf ljb\ofyLn{ fO{ sfuhdf nl] vPsf vfb\o\o j:t'x¿ k9g\ nufO{ juLs{ /0f ug{ nufpg'xf;] \ .
h:tM} vfgfsf gfdx¿, ko] kbfy{sf gfdx¿, t/sf/Lsf gfdx¿, kmnkm' nsf gfdx¿ cflb .

-u_ juL{s/0f ul/Psf vfbo\ o\ j:t'x¿nfO{ 56' \6f 5'66\ } a66\ fdf /fVg nufpgx' f;] \ .
-3_ of] a66\ fdf s:tf vfbo\ o\ j:t'sf] gfdx¿ 5g\ < 5nkmn ub}{ a66\ fdf ;DalGwt ;dx" sf] gfd nV] g

nufpg'xf];\ .
h:t}M t/sf/Lx¿sf] ;dx" , k]o kbfy{x¿sf] ;dx" cflb . Tof] g} tL a6\6fleq /x]sf ;b:ox¿sf]
;j{Jofks ;dx" xg' a] f/] lgisif{ lgsfNg nufpg'xf];\ .
-ª_ kf7\o k':tssf] pbfx/0f 1 5nkmn ug{ nufpgx' f;] \ .
2=pk;dx" sf] kl/ro
-s_ Ps hgf ljb\ofyLn{ fO{ dfly pNn]lvt sg' } Pp6f a66\ fsf sfuhdf n]lvPsf 3 cf]6f 6'j|mfx¿
lgsfNg nufpgx' f;] .\ To;df n]lvPsf gfdx¿ csf{] ljbo\ fyL{nfO{ n]vgkf6Ldf n]Vg nufpg'xf];\ . h:tM}
;j{Jofks;d"x U={t/sf/Lx¿sf] ;dx" } a66\ faf6 A ={ cfn,' Kofh, ;u“ } agfpg nufpg'xf;\ .
To:t} c? ljbo\ fyLn{ fO{ klg Pp6f, b'O{ cf6] f ;b:o ePsf ;dx" x¿ agfpg nufO{ nv] gkf6Ldf nV] g
nufpg'xf;] \ .
h:tM} {k/j/}, C={sfpnL, aGbfsfe] L} cflb . oL ;d"xx¿U sf pk;d"x xg' a] f/] 5nkmn ug{
nufpg'xf];\ .

37

-v_ kf7o\ k:' tssf lj|mofsnfk pk;dx" sf] kl/ro 5nkmn ug{ nufpg'xf];\ .

3. pkoS' t ;d"x / cg'ko'St ;d"x

-s_ Ps hgf ljb\ofyL{nfO{ dfly pNnl] vt csf]{ a6\6fsf ;b:ox¿ Ps Ps ub}{ ;a} gfdx¿ lgsfNg
nufpgx' f];\ .csf{] Ps hgf ljbo\ fyL{nfO{ gfd l6Kg nufpb}+ hfg'xf];\ . ;a} ;b:ox¿ ePsf] Pp6f
;d"x agfpg nufpg'xf;] \ .

of] ;d"x cg'ko'St ;d"x x'g] / cGo ;d"xx¿ pko'St x'ga] f/] o;sf ;dx" ;ª\s]t ;dt] 5nkmn
ug{ nufOkf7o\ k:' tssf] lj|mofsnfk 5nkmn ug{ nufpgx' f];\ .

-v_ ljbo\ fyLx{ ¿nfO{ b'O{ ;d"xdf ljefhg ug'{xf;] \ . Pp6f ;d"xsf] Ps hgfnfO{ sg' } Pp6f
;j{Jofks;d"x eGg nufpgx' f;] \ . csf{] ;d"xsf] Ps hgfnfO{ To;sf] pko'St ;dx" Pp6f /
cg'ko'St ;dx" eGg nufpg'xf];\ . ldnd] f 1 cª\s lbgx' f];\ . To:t} bf]>f] ;dx" sf] Ps hgfnfO{
;j{Jofks ;dx" eGg nufpg'xf;] \ / klxnf] ;d"xsf] csf{] Ps hgf ljbo\ fyLn{ fO{ Pp6f pkoS' t /
cgk' o'St ;d"x eGg nufpgx' f];\ . ;a}sf] kfnf] ;lsPkl5 a9L cª\s kfpg] ;d"xnfO{ ljhoL
u/fpgx' f;] \ .

4= e]g lrq

-s_ Ps hgf ljb\ofyL{nfO{ n]vg kf6Ldf ;dx" ;ª\st] k|of]u u/L sg' } Pp6f ;jJ{ ofks ;d"x nV] g
nufpg'xf];\ .

-v_ To;sf] ;b:ox¿nfO{ Pp6f cfot lvRg nufO{ To;df tL ;b:ox¿ 5l/Psf] ¿kdf n]Vg
nufpg'xf];\ . csf]{
DRAFT
lstfj, skL, lstfj, skL,
kG] l;n, O/h] /, kG] l;n, O/h] /,
rs, dfs/{ ,
rs, dfs/{ ,
HofdLlt afs;
HofdLlt afs;

-u_ Ps hgf ljbo\ fyLn{ fO{ To;sf 5 cf6] f ;b:ox¿ nV] g nufpg'xf];\ . cfot leq tL 5 cf]6f
;b:ox¿nfO{ ufn] f] 3/] fleq /fVg nufpg'xf;] \ . of] lrq g} eg] lrq ePsf] / ufn] f] 3]/f leqsf]
pk;d"x xg' a] f/] 5nkmn ub}{ kf7o\ k':tssf] lj|mofsnfk 5nkmn ug{ nufpgx' f];\ .

d"Nofªs\ g

1= A={elnjn, km"6an}, B ={af:s]6an}, C ={Jof8\ldG6g}, D ={ljm| s]6} sf] ;jJ{ ofks ;dx" n]v .

2= s'g} Pp6f ;j{Jofks ;dx" agfp / To;sf pk;dx" x¿ ;a} n]v . tL dWo] pkoS' t ;dx" /
cgk' o'St;d"x sg' s'g x'g\ < sg' } Pp6f pk;dx" nfO{ eg] lrqdf bv] fpm .

lzIfsnfO{ lgb]z{ g

;jJ{ ofks ;d"x, pk;d"xx¿, pko'St tyf cg'ko'St pk;d"xx¿ / eg] lrqsf] wf/0ff lbg yk
ljm| ofsnfkx¿ ljbo\ fyL{x¿nfO{ u[xsfo{sf] ¿kdf lbg nufpg' pko'St x'G5 . o; sIffdf e]g lrqsf]
k|ofu] sf] ;'?jft ePsf] xb'“ f o;df pgLx¿nfO{ a9L pbfx/0fx¿ agfpg nufO{ cEof; u/fpg' pkoS' t
xG' 5 .

38

kf7M cnlUuPsf / vlK6Psf] ;d"x

pbb\ ]ZoM o; kf7sf] cWoogkZrft\ ljb\ofyLx{ ¿ lbOPsf] ;d"xx¿ dWo] cnUuLPsf jf vlK6Psf] ;dx"
56' o\ fpg;Sg]5g\ .

zI} fl0fs ;fduL|

1 b]lv 10 ;Dd n]lvPsf kTtLx¿,

;xhLs/0f lj|mofsnfk

1= -s_ sIffsf] cu| efudf 1 b]lv 10 ;Dd nl] vPsf kTtLx¿ /fVgx' f];\ .

b'O{ hgf ljbo\ fyL{x¿nfO{ cufl8 8fs/] Ps hgf ljbo\ fyLn{ fO{ Pp6f ;6] sf] 5 cf6] f kTtLx¿
kfnk} fnf] xg] { nufpgx' f];\ .

csf]{ ljb\ofyL{nfO{ nv] g kf6Ldf ;d"x ;ª#st] k|ofu] u/L tL 5 cf]6f kTtLx¿sf] ;dx" nV] g
nufpgx' f];\ . To;} u/L csf{] bO' { hgf ljb\ofyLx{ ¿nfO{ cufl8 8fs]/ dfly pNnv] u/] em}+ n]vg
kf6Ldf ;d"x nV] g nufpg'xf];\ . h:t,} A = {2,3,6,7,9}, B = {1, 3, 4,5,7,9}

e]g lrqdf k|:t't ubf,{

AB

23 1
67 4

95

-v_ tL b'O{ cf]6f ;dx" x¿ k9g\ nufpgx' f];\ / kZ| g ugx{' f];\ . oL bO' { ;dx" x¿ dWo] s'g} ;femf ;b:ox¿
5g\< e]g lrqdf bv] fpg nufpg'xf];\ . oL ;d"xx¿ Ps cfk;df 5fP] sf 5g\ < 5nkmn ug{'xf;] \ / To;}
u/L cnUuLPsf ;dx" x¿ af/] klg 5nkmn ugx{ f;] \ .
h:Tf,} A ={a,e,i,o,u} B= {b,f,g,k,l,p}
eg] lrqdf k|:t't ubf{
DRAFT
A B

aei bf
ou gk
lp

-u_ kT| o]s ljbo\ fyL{nfO{ cfk"mv';L Pp6f ;jJ{ ofks ;d"x agfpg nufpgx' f];\ . To;sf cnlUuPsf /
vlK6Psf 3/3 cf6] f ;dx" x¿ agfpg nufpgx' f];\ / k:| tt' ug{ nufpgx' f];\ .

d"Nofª\sgM
s'g} Pp6f ;j{Jofks ;d"x nv] L o;sf 5 j6f pk;dx" x¿ nv] . tL ;d"xx¿sf] cnUuLPsf] /
vlK6Psf ;dx" x¿ 2/2 cf6] f n]v / e]g lrqdf b]vfpm .

kf7M ;d"xsf] ;o+ fh] g / ;d"xsf] k|ltR5]bg
kf7 pb\bZ] o M o; kf7sf] cWoogkZrft\ ljbo\ fyLx{ ¿ lgDg sfo{ ug{ ;Ifd xg' ] 5g\M
1= bO' { ;dx" x¿lar ;+ofh] g ug{
2= bO' { ;dx" x¿lar kl| tR5]bg ug{

39

DRAFTz}Ifl0fs ;fduL|
jT[ tfsf/ sfhux¿ kl| t a]GrnfO{ b'O{ cff] sf b/n], 6]k, 1 b]lv 10 ;Dd nl] vPsf cfotfsf/ kTtLx¿,
l;sfO ;xhLs/0f ljm| ofsnfkx¿

1. ;d"xsf] ;o+ f]hg
-s_ kT| os] aG] rnfO{ j[Ttfsf/ sfuhx¿ lbg] / s'g} ;d"xsf ;b:ox¿ nV] g nufpg'xf;] \ . nl] vPsf

j[Ttx¿ n]vg kf6Ldf 6]kåf/f 6f:“ g nufpgx' f;] \ . tL jT[ tx¿ cnUuLPsf ;dx" x¿ jf vlK6Psf s]
xg' \ < vlK6Psf ;dx" x¿ eP vK6o\ fpg nufpg'xf;] \ .
-v_ cnUuLPsf / vlK6Psf ltg ltg cf6] f ;d"xx¿ cfcfkmg\ f] sfkLdf nV] g nufpg'xf;] \ .
-u_ bO' { hgf ljb\ofyL{x¿nfO{ sIffsf] cu|efudf /fVg'xf];\ . Ps hgfnfO{ 3fK] 6o\ fP/ /fvs] f cfotfsf/
kTtLx¿ Pp6f/Pp6f u/L 5 cf]6f kTtLx¿ kN6fpg nufpgx' f;] \ . oL ;dx" x¿ s:tf ;d"xx¿ x'g\ <
5nkmn u/L ;sk] l5 ba' } ;d"xnfO{ Pp6} ;dx" dfnV] g' kbf{ s;/L n]Vg] egL 5nkmn ugx{ f;] \ .
-3_ b'a} ;d"xnfO{ Pp6} ;dx" df nV] bf ;o+ f]hg x'g] / ∪ lrxg\ k|of]u ug]{ af/] ;dx" ;ª\s]t ko| fu] u/L
ljbo\ fyLn{ ] n]vs] f cnUuLPsf ;d"xx¿sf] ;+ofh] g nV] g nufpg'xf;] \ .
-ª_ k|To]s aG] rnfO{ Pp6f Pp6f /i] 6'/]G6sf] dg] ' Go'hlkG| 6df n]Vg nufpgx' f];\ . nv] ]sf d]g'x¿ nv] g
kf6Ldf 6f:“ g nufpg'xf;] \ . s'g} bO' { dg] 'x¿ lnO To;sf vfbo\ ;fdu|Lx¿nfO{ Pp6} dg] ' agfpg
cyf{t\ ;+ofh] g ug{ nufpgx' f];\ .
-r_ Ps hgf ljb\ofyLn{ fO{ cfkm"n] n]vs] f ;o+ f]hg n]vg kf6Ldf n]Vg nufpgx' f;] \ .
-5_ olb sg' } b'O{ vfbo\ j:t' bf]xfl] /P/ nv] ]df lsg b'O{ rf]6L nV] of} < h:t} M s] cfn' / lrof nV] g'
cfjZos 5 < h:tf k|Zgx¿ u/L ;b:ox¿ gbf]xfl] /g] u/L nV] g'kg{] atfp+b} ;+ofh] g ug{
nufpgx' f];\ .
-h_ ljbo\ fyLn{ ] n]vs] f vlK6Psf ;dx" x¿sf] ;o+ f]hg ug{ nufpgx' f];\ .
-em_ kf7o\ k:' tssf] pbfx/0f 1,2,3 5nkmn ug{ nufO{ lgrf]8 eGg / n]Vg nufpg'xf];\ .
2. ;d"xsf] ;+ofh] g
-s_ dfly vlK6Psf ;d"xx¿sf ;b:ox¿ dfq lnb“ f aGg] ;dx" k|ltR5l] bt x'g]af/] 5nkmn ug{] / ;d"x
;ªs\ t] df nV] g nufpgx' f;] \ .
-v_ kf7o\ k:' tsdf ePsf pbfx/0f 1, 2, 3, 4 5nkmn ug{ nufO{ k|:t't ug{ nufpgx' f;] \ .
dN" ofª\sg
1= b'O{ cf]6f cnUuLPsf ;dx" x¿ nv] / ;o+ f]hg u/
2= bO' { cf6] f vlK6Psf ;d"xx¿ n]v / ;+of]hg u/
3= b'O{ cf6] f cnlUuPsf ;d"xx¿ n]v / kl| tR5b] g u/
4= bO' { cf]6f vlK6Psf ;dx" x¿ nv] / k|ltR5]bg u/ .
lzIfsnfO{ lgb{z] g
ljb\ofyLx{ ¿nfO{ ;dx" df sfo{ u/fpb“ f k|To]sn] efu lng} kg{] cj:yfsf] jftfj/0f agfpgsf ;fy} cfkm\gf]
ts{ k|:t't ug{ nufpg} k5{ h:n] ubf{ ljbo\ fyLn{ ] ljifo j:t'sf] unt wf/0ff /fv]sf eP ;Rofpg] dfs} f
kfp“5 .

40

Psf OM 12
k"0f{ ;ª\Vof(Whole numbers )

cgd' flgt lzIf0f 3G6L M 15

k"j{ 1fgM ;ª\Vofsf] :yfgdfg

PsfO kl/ro

o; PsfOdf s'g} ;ªV\ ofsf] ju{ tyf 3g lgsfNg,] k"0f{ ju{ ;ª\Vofsf] ?9 u'0fg v08 tyf efu ljlwåf/f ju{dn"
lgsfNg] / ?9 u'0fg v08 ljlwaf6 3gd"n lgsfNg cWoog ug{] tl/sfx¿af/] 5nkmn ug]{5f}+ . jus{ f] k|of]u
bl} gs hLjgdf 6fon jf wall paper, eO"“ + cflb 7fpx“ ¿df k|i6 bl] vg] / If]qkmn lgsfNg o:sf] k|of]u eO/xs] f]
kfOG5 eg] 3gfsf/ afs;x¿, uf6] Lx¿ cflb bl] v /xs] f x'G5g\ eg] of] cfotg lgsfNg ko| f]u ub{5f+} . To;}u/L
;ª\Vofx¿sf] d=;= / n=;= lgsfNg] tl/sf af/] 5nkmn ug'{sf ;fy} låcfwf/ tyf k~rcfwf/ ;ªV\ of
kb\wltsf ¿kfGt/ ug{] tl/sfx¿af/] 5nkmn ug]{5f}+ .

kf7 M ;ª\Vofsf] ju{ / jud{ "n tyf ;ªV\ ofsf] 3g / 3gdn"
pbb\ ]Zo M o; kf7sf] cWoogkZrft\ ljbo\ fyL{x¿ lgDg sfo{ ug{ ;Ifd xg' ] 5gM\

1= sg' } ;ª\Vofsf] ju{ lgsfNg / k0" f{ ju{ ;ªV\ ofsf] ?9 v08Ls/0f tyf efu ljlwaf6 jud{ n" lgsfNg
2= s'g} ;ªV\ ofsf] 3g lgsfNg / k"0f{ 3g;ª\Vofsf] ?9 v08Ls/0f ljlwaf6 3gd"n lgsfNg

z}Ifl0fs ;fduL|
DRAFT
2, 3, 4, 5, 6, 0.1, 0.7, 8, =======cflb nl] vPsf kTtLx¿, r];af8] {, k|lt a]GrnfO{ Pp6f Pp6f rf6{ k]k/,

s}r+ L, Un" l:6s cflb .

l;sfO ;xhLs/0f lj|mofsnfk

1. ;ª\Vofsf] ju{ / jud{ "n

dfly z}Ifl0fs ;fduL| df pNnl] vt kTtLx¿ sIffsf] cue| fudf 3fK] 6o\ f/ /fVg'xf];\ .

-s_ v]n vn] fpg] M ljb\ofyL{x¿nfO{ bO' { ;d"xdf ljefhg ug'{xf];\ . klxnf] ;dx" sf] Ps hgfnfO{ Pp6f kTtL
kN6fpg nufpg'xf];\ . To;sf] ju{ csf]{ ;d"xsf] Ps hgfnfO{ eGg nufpg'xf;] \ . -;'?df Pp6f kTtLsf] ju{
lzIfsn] egL lbg] ._ ldnd] f Ps cª\s lbgx' f];\ . kfn}kfnf] klN6Psf kTtLx¿sf] ju{ eGg ;s]kl5 a9L
cª\s kfpg] ;dx" ljhoL 3fl] ift ug{xf];\ .

-v_ k|Tos] a]Grdf ljb\ofyL{x¿ lar kf7o\ k':tsdf pNn]lvt u0' fg v08 ljlwaf6 ju{d"n lgsfNg] tl/sf
5nkmn ug{ nufO{ k:| tt' ug{ nufpg'xf;] \ . r;] sf] kftfsf jux{ ¿ slt cf]6f 5g\ < nDafO / rf}8fOdf
slt slt cf6] f jux{ ¿ 5g\ < 5nkmn ug{'xf];\ .

-u_ efuljlwaf6 ju{dn" lgsfNg] tl/sf 5nkmn ug{ nufO{ PP6f pbfx/0f ;lxt k:| t't ug{ nufpg'xf;] \ .

-3_ v]n v]nfpg]M ljbo\ fyL{x¿nfO{ b'O{ ;dx" df ljefhg ug{] . 441 nfO{ n]vg kf6Ldf n]vL bj' } ;dx" sf] Ps
Ps hgfnfO{ 5'66\ f 56' \6} ju{d"n lgsfNg nufpgx' f;] \ . h:n] l56f] jud{ "n lgsfN5 ;f] ;dx" nfO{ Ps cª\s
lbgx' f;] \ . s]xL ;do;Dd bj' } ;dx" sf a/fa/ ljbo\ fyLx{ ¿ lar v]nfO ;sk] l5 j9L cª\s kfpg] ;d"x l
jhoL 3f]lift ug'x{ f;] \ .

2. 3g / 3gdn"

-s_ k|Tos] a]GrnfO{ rf6{ k]k/ lbO{ To;df 10 ;]=ld= nDafOsf] 3gsf] hfnL tof/ kfg{ nufO{ 3gsf] gd"gf
agfpg lbgx' f];\ .

o;sf] cfotg g} o;sf] nDafOsf] 3g xg' ]af/] 5nkmn ugx{' f;] \ . 10 ;=] ld=sf] 3g 1000 3=;]=ld= g} 1
ln6/ xg' ]af/] 5nkmn ug'{xf];\ .

-v_ dfly pNn]lvt kTtLx¿sf] 3g ;ª\Vof eGg nufpg] / l6Kg nufpg'xf;] \ . k"0f{ 3g ;ªV\ ofsf] u'0fg v08
ljlwaf6 3gd"nlgsfNg tl/sf pbfx/0f ;lxt k|bzg{ ug{ nufpg'xf;] \ .

41

DRAFTd"Nofªs\ g
1= s'g} 5 cf]6f k0" f{ ;ª\Vofx¿ nv] L To;sf] ju{ lgsfn / To; ju{ ;ªV\ ofsf] ?9 u0' fg v08Ls/0f / efu

ljlwaf6 ju{d"n lgsfn .
2= sg' } 5 cf]6f k0" f{ ;ªV\ ofx¿ nv] L To;sf] 3g lgsfn / To; 3g ;ª\Vofsf] ?9 v08Ls/0f ljlwaf6

3gd"n lgsfn .
lzIfs lgb]{zgM
b}lgs hLjgdf cfO/xg] ju{ / 3g k|ofu] x'g] If]qx¿;+u ;DalGwt u/L 5nkmn ug{ nufpb+ f o; ;DaGwL
wf/0ff lbg ;lhnf] x'g]5 .

kf7M d=;= / n=;=

pbb\ Z] oM o; kf7sf] cWoogkZrft\ ljbo\ fyLx{ ¿ lgDg sfo{ ug{ ;Ifd x'g] 5g\ M
1= d=;= / n=;= sf] k/Lefiff lbg
2= efu ljlwaf6 ;ªV\ ofx¿sf] d=;= / n=;= lgsfNg
zI} fl0fs ;fduL|
6 OGrL / 9 OGrL nfdf] lni6L jf 5]:sf, 1 OGrL, 2 OGrL = = = = = = 9 OGrL ;Ddsf 6'jm| f 5]:sfx¿, 12
6j' m| f pq} cfotfsf/ sfuh / 18 j6f l;dLx¿, 6k] -12 sfkL / 18 snd a9Ldf slt hgfnfO{ a/fa/ u/L
af8“ \g ;lsPnf < k|To]sn] slt slt cf6] f tL ;fdfgx¿ kfpnf <_ n]lvPsf] rf6{ kk] /, 1 b]lv 16 ;Dd
n]lvPsf] kfgf sfuhx¿,
l;sfO ;xhLs/0f ljm| ofsnfk

1. d=;=

-s_ sIffsf] cue| fudf 6 OGr / 9 OGr nfdf] 5]:sfsf 6j' |mfx¿ /fvL 1 OGrsf] 5]:sfn] gfKg nufpgx' f];\ .
To:t} 9 OGrsf] 5]:sfnfO{ klg gfKg nufO{ . b'j} 5]:sfnfO{ 1 OGrsf] 5:] sfn] gfKg ;ls“bf] /x]5 eGg]
s/' fsf] k|bz{g ugx{' f];\ .

-v_ s] 2 OGrsf] 5]:sfn] tL b'j} 5]:sfnfO{ gfKg ;lsPnf < ljb\ofyL{nfO{ gfKg nufpg'xf];\ . 2 OGrn] 6
OGrnfO{ gfKg ;lsg] t/ 9 OGrnfO{ gfKg g;lsg] af/] 5nkmn ug{ nufpg'xf;] \ . To;u} /L 3 OGr 5:] sfn]
tL bj' }nfO{ gfKg ;lsG5 jf ;lsb“ }g 5nkmn ug{ nufpg'xf];\ .

-u_ Pjd\ k|sf/ 4 OGrn] <5 OGrn] < = = = 9 OGrn] < 5nkmn ug{ nufpg] . of] 5nkmnsf] cfwf/df 6 / 9
nfO{ l7s efu nfUg] ;ª\Vofx¿ 1 / 3 xf] . oL b'O{ ;ª\Vofx¿ dWo] ;ae} Gbf 7n' f] 3 ePsfn] of] g} 6 / 9
sf] d=;= xg' ]af/] k:| t't ug{ nufpgx' f];\ . To;} u/L 16 / 20 sf] d=;= slt xfn] f < 5nkmn ug{
nufpgx' f;] \ .

-3_ sIffsf] cue| fudf 12 cf]6f cfotfsf/ sfuh / 18 cf]6f l;dLx¿ /fVgx' f];\ . b'O{ hgf ljb\ofyL{x¿nfO{
8fs]/ Ps hgfn] csf{n] fO{ 12 cf]6f sfuh / 18 cf]6f l;dL lbg nufpgx' f];\ . kml] / csf]{ bO' { hgfnfO{
8fs/] tL ;fdfgx¿a/fa/ x'g]u/L af“8g\ nufpg'xf;] \ . bO' { hgfn] sltcf]6f s'g ;fdfgx¿ kfP5g\ <
sfuh 6 6j' |mf / l;dL 9 cf6] f kfpga] f/]5nkmn ug{ nufpg'xf;] \ . km]l/ 3 hgfnfO{ tL ;fdfgx¿ a/fa/
xg' u] /L af8“ b\ f kT| os] n] slt slt cf]6f sg' ;fdfg kfP5g\ < k|of]u ug{ nufO lgisif{ eGg nufpg'xf];\ .

To;u} /L 4 hgf ljbo\ fyLn{ fO{ bj' } ;fdfgx¿ a/fa/ xg' ] u/L af8“ \g ;lsPnf < lgisif{ eGg nufpg'xf];\ .
6 hgfnfO{ ljbo\ fyLn{ fO{ b'a} ;fdfg a/fa/ x'gu] /L af“8\g nufpg], k|To]sn] slt slt cf6] f s'g ;fdfg
kfP5g\ < pTt/ eGg nufpg'xf];\ . To:t} 7, 8, 9, …, 12 hgfnfO<{ 13 hgfnfO{ ;fdfgx¿ af8+ \g sfuhsf]
6j' |mfsf] ;ª\Vof kU' nf < 5nkmn ugx{' f;] \ . To;kl5 dfly z}Ifl0fs ;fdu|Ldf n]lvPsf rf6{ kk] / nv] g kf6Ldf
6f“:gx' f;] \ .

To;sf] pTt/ dflysf] ko| fu] sf] cfwf/df ljb\ofyL{nfO{ eGg nufpg'xf;] \ .

42

DRAFT-v_ ;f] k|Zgsf] pTt/ lgsfNg efu ljlwsf] kb| z{g ub}{ ljb\ofyLx{ ¿nfO{ klg ug{ nufpg'xf;] \ .
-u_ kf7\o k':tssf] pbfx/0fx¿ 5nkmn ug{ nufO{ k:| t't ug{ nufpg'xf;] \ .

2. n=;=

-s_ /f]n g+= 1 bl] v 16 ;Ddsf ljb\ofyLx{ ¿nfO{ sIffsf] cufl8 /fv]/ k|Tos] nfO{ sfuhdf nl] vPsf] cfcfkm\gf]
/f]n g=+ sldhdf lkgn] c8o\ fpgnufpg'xf;] \ . c¿ Ps hgf ljb\ofyLn{ fO{ /f]n g= 2, 4, 6, … nfO{ Pp6f
Pp6f 5:] sf afF8g\ nufpgx' f;] \ . To;kl5 /f]n g+= 3, 6, 9, …. nfO{ Pp6f / Pp6f cfotfsf/ sfuhsf]
6j' m| f af8“ \g nufpg'xf];\ . s;n] ba' } kfP5g\ < 5nkmn ug{ nufpg'xf;] \ .
/fn] g+= 6, 12 n] b'O{ cf]6} ;fdfgx¿ kfP5g\ eGg] lgisif{ lgsfNg nufpgx' f];\ . ;a}eGbf klxn] s;n]
kfP5g\ < 5nkmn ub{} Tof] g} b'j} ;ªV\ ofsf] n=;= xg' ]af/] 5nkmn ugx{ f];\ .

-v_ efuljlwaf6 2, 3, 4 sf] n=;= lgsfNg] tl/sf 5nkmn ug{xf;] \ .
-u_ kf7o\ k:' tssf pbfx/0fx¿ 5nkmn ug{ nufO{ k|:t't ug{ nufpg'xf];\ .
d"Nofª\sg
1. 16, 24 / 32 sf] d=;= slt xfn] f < kTtf nufpg'xf;] \ .
2= Ps hgf bftf 120 cf]6f sfkL / 150 cf6] f sndx¿ ljb\ofyLx{ ¿nfO{ ljt/0f ug{ cfP5 . p;n] s]xL

ljbo\ fyLx{ ¿nfO{ tL ;fdfgx¿ a/fa/ x'g] u/L af8“ g\ rfx5] g\ eg] ;a}eGbf a9L slt hgf a/fa/L kfpg]
/x]5g\ < o;/L af“8b\ f sg' ;fdfgx¿ slt slt cf]6f kfpg] /x]5 <
3. 12, 18 / 24 sf] n=;= lgsfn .
lzIfs lgbz{] g
;ªV\ ofsf] d=;=/ n=;= lgsfNg ;Sg] ePkl5 dfq zflAbs ;d:ofsf] ;dfwfg ug{ nufpg' pkoS' t xG' 5 .
n=;= ;ªV\ ofx¿ dWo] s'g} ;ªV\ of hf]8g' jf 36fpg' kg{] ;d:ofx¿sf] ;dfwfg ug]{ tl/sfsf] r/0fj4 ¿kdf
5nkmn ug'x{ f]nf . h:t}M s'g} ;ªV\ ofdf 5 36fpFbf km/snfO{ 20, 25 / 40 n] lgMz]if efu hfnf<

kf7M k~r cfwf/ / låcfwf/ ;ªV\ ofsg kb\wlt

pbb\ ]Zo M o; kf7sf] cWoogkZrft\ ljbo\ fyLx{ ¿ lgDg sfo{ ug{ ;Ifd xg' ] 5g\ M
1= bzdnj ;ª\Vofaf6 låcfwf/ ;ªV\ of kbw\ ltdf ¿kfGt/ ug{
2= låcfwf/ ;ªV\ ofaf6 bzdnj ;ª\Vofdf ¿kfGt/0f ug{
3= bzdnj ;ª\Vofaf6 k~rcfwf/ ;ªV\ of kbw\ ltdf ¿kfGt/ ug{
4= k~rcfwf/ ;ªV\ ofaf6 bzdnj ;ªV\ ofdf ¿kfGt/0f ug{
zI} fl0fs ;fduL| M
1 b]lv 20 ;Ddsf låcfwf/ ;ª\Vofx¿ nl] vPsf] rf6{, 1 bl] v 20 ;Ddsf k~rcfwf/ ;ªV\ ofx¿ n]lvPsf] rf6{,
;xhLs/0f lj|mofsnfk M
1= -s_ n]vg kf6Ldf 1 bl] v 10/10 sf] column df 50 ;Dd n]vg kf6Ldf n]Vgx' f];\ / ljbo\ fyLx{ ¿nfO{ klg

n]Vg nufpgx' f];\ . of] s'g cfwf/ kbw\ ltsf] ;ªV\ of xf] / o;df sg' sg' cª\sx¿ 5g\ < 5nkmn
ugx{' f];\ .
-v_ To;/L g} låcfwf/ ;ªV\ of kb\wltdf s'g s'g cª\sx¿ x'G5g\ xfn] f < ljb\ofyL{af6 0 / 1 eGg] pTt/
cfPdf 2 bl] v 9 cªl\ st ;a} sf6Llbg] / ljb\ofyL{nfO{ klg ;fx] L ;ªV\ ofx¿ sf6\g nufpg'xf;] \ . afs“ L
cªs\ x¿nfO{ lgDgfg;' f/ n]Vg nufpg'xf;] M\

00

11

2 10

43

DRAFT 3 11

4 100

5 101

6 110

7 111 cflb .
-u_ låcfwf/ ;ªV\ of kb\wltsf] rf6{ k|bz{g u/L 20 ;Ddsf ;f] kb\wltsf ;ª\Vofx¿ 5nkmn ugx{ f];\ .
-3_ bzdnj ;ª\Vof kb\wltaf6 låcfwf/ ;ªV\ of kb\wltdf ¿kfGt/0f ug]{ tl/sf kf7\o k':tssf cfwf/df

5nkmn u/fO{ k:| t't ug{ nufpgx' f];\ .
2= låcfwf/ ;ª\Vof kb\wltaf6 bzdnj ;ª\Vofdf ¿kfGt/ ug{ bzdnj kb\wltsf] :yfgdfg tflnsfsf]

k'g/fjl[ Tt ug{ nufpg'xf;] \ .
h:t} M 323=3x102+ 2x101+ 3x100 oxf“ 10 sf] 3ftfª\s :yfgcg;' f/ a9\b} hfG5 .
To;} cg';f/, 110102=1x24 + 1x23 + 0 x22 + 1x21 + 0x20

=16+ 8 + 0+ 2 + 0

=26 xG' 5 .
3= -s_ låcfwf/ ;ªV\ of kb\wltdf u/s] f] lj|mofsnfk 1 -s_ em}+ u/L k~r cfwf/ kb\wltdf s'g s'g cªs\ x¿

xfn] fg\ < 5nkmn ug{nufpg'xf;] \ . 5 bl] v 9 ;Dd cfpg] cª\sx¿ sf6\b} hf“bf aGg] ;ª\Vofx¿nfO{
j|mdabw\ ¿kdf n]Vg nufO{ To;sf] dfg klg nV] g nufpg'xf];\ .

00

11

22

33

44

5 10

6 11

7 12 cflb .
-v_ k~r cfwf/ ;ª\Vof kb\wltsf] rf6{ k|bzg{ u/L 20 ;Ddsf ;f] kbw\ ltsf ;ªV\ ofx¿ 5nkmn ug{

nufpgx' f];\ .
-u_ bzdnj ;ªV\ of kbw\ ltaf6 k~r cfwf/ ;ª\Vof kbw\ ltadf ¿kfGt/ ug{] tl/sfsf nflu kf7o\ k':tsdf

lbOPsf pbfx/0f x]g{ nufO{ k|:t't ug{ nufpg'xf];\ .
4= -s_ låcfwf/ ;ªV\ of kb\wltaf6 bzdnj ;ª\Vof kb\wltdf ¿kfGt/ u/] em+} k~r cfwf/ ;ª\Vof kb\wltaf6

bzdnj ;ªV\ of kb\wltdf :yfgdfgsf] ;xfotfn] ¿kfGt/ ug{ nufpg'xf];\ .
h:t} M 43235=4x53 + 3 x52 + 2 x51 + 3x30

=500+ 75 + 10+ 3

=500 xg' ]af/] k:| t't ug{ nufpgx' f];\ .
-v_ v]n vn] fpg]M ljbo\ fyL{nfO{ 4 ;d"xdf ljefhg ug{xf];\ . Pp6f ;dx" nfO sg' } Pp6f bzdnj ;ªV\ of

kb\wltaf6 låcfwf/ ;ªV\ of kbw\ ltdf jf k~r cfwf/ ;ª\Vof kbw\ ltdf jf låcfwf/ ;ªV\ of kb\wltaf6
bzdnj ;ª\Vof kbw\ ltdf jf k~r cfwf/ ;ªV\ of kb\wltaf6 bzdnj ;ª\Vofkb\wltdf ¿kfGt/0f ug]{
k|Zg ;f]Wg nufpg'xf];\ . klxnf] ;dx" n] k|Zg ;fW] g], bf];|f] ;dx" nfO{ pTt/ lbg] kfnf] lbg], ldnfPdf 1cªs\
lbg], gldnfPdf t];|f] ;d"xnfO{ kfnf] lbg] / ldnfPdf 1 cªs\ lbg] gldnfPdf rf}“yf] ;dx" nfO{ pTt/ lbg]
kfnf] lbg], ldnfPdf 1 cª\s lbg'xf;] \, gldnfPdf klxnf] ;dx" nfO{ n]vg kf6Ldf pTt/ lgsfNg
nufpg'xf;] \ . kfn}kfnf] vN] g nufO{ a9L cª\s kfpg;] d"xnfO{ ljhoL u/fpg'xf;] \ .

44

DRAFTdN" ofª\sg
cfk"mv;' L sg' } bzdnj ;ªV\ of lnO{ To;nfO{ låcfwf/ ;ªV\ of kbw\ ltdf / k~r cfwf/ ;ª\Vof kbw\ ltdf
¿kfGt/0f ug{nufpg'xf];\ / ;f] ;ª\Vof ldn] gldns] f] csf{] ljbo\ fyLn{ fO{ hfR+ g nufpg'xf];\ .
lzIfs lgbz]{ g
bzdnj ;ªV\ of kbw\ ltsf] :yfgdfg tflnsfdf 10 sf] 3ftf+sdf /xs] f] tYo :ki6 ePdf låcfwf/ ;ªV\ of
kb\wlt tyf k~r cfwf/ ;ª\Vof kb\wltaf6 bzdnj ;ªV\ of kb\wltdf ¿kfGt/ ug{] tl/sfdf ljb\ofyLx{ ¿
;lhn} efu lng ;S5g\ .

45

PsfOM 13
k"0ffª{ \s (Integers)

cg'dflgt lzIf0f 3G6L M 5

k"j1{ fg

o; kf7sf] nflu kf| s[lts ;ª\Vof, ;ªV\ of /]vfdf laGb' cªs\ g ug{ ;Sg] tyf ;ª\Vof /]vfdf 3gfTds df
pbu\ d ljGb' 0 eGbf bfof / C0ffTds ;ª\Vof pbu\ d ljGb'jf6 afof“lt/ cªs\ g ug]{ l;ksf] cfjZostf /xG5 .

PsfO kl/ro

o; PsfOdf k"0ffª{ s\ sf] kl/ro, k0" ff+ª{ \ssf k|sf/ tyf lar t'ngf / k0" ffª{ \ssf] lg/k]If dfg ;DaGwL 5nkmn
ul/g]5 . o;sf ;fy} k"0ffª+{ s\ x¿sf] hf8] , 36fp, u'0fg / efu ug]{ tl/sfaf/] 5nkmn ul/g5] .

z}Ifl0fs ;fdu|L

+1 +2 +3 +4 +5 +6 +7

+8 +9 +10 -1 -2 -3 -4 -5
DRAFT
-5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

ln:6sf] ;ªV\ of /]vf, km\Nof; sf8x{ ¿M -2 3, 5 -5, 3 9, -2 1, -2 -3

pb\b]Zo

o; kf7sf] cWoogkZrft\ ljbo\ fyLx{ ¿ lgDg sfo{ ug{ ;Ifd x'g] 5g\ M

1= k0" ffª{ \ssf] kl/ro lbg tyf k0" ff{ª\sx¿sf k|sf/ eGg / ltgLx¿ lar t'ngf ug{ .

2= k0" ff{ªs\ sf] lg/k]If dfg lgsfNg .

3= k0" ff{ªs\ sf] hf]8, 36fp, u'0fg tyf efu ug{ / tT ;DaGwL ;/nLs/0f ug{ .

4= k0" ff{ª\ssf zflAbs ;d:ofx¿ xn ug{ .

l;sfO ;xhLs/0f ljm| ofsnfk

1. k"0fffª{ s\ x¿sf kl/ro

-s_ bO' { hgf ljbo\ fyLx{ ¿nfO{ sIffsf] cu|efudf /fv]/ Ps hgfnfO{ 3f]K6o\ fP/ /fvs] f ;ªV\ of kTtLx¿ Ps Ps
ub}{ ;a} kN6fpgnufpg'xf;] \ . csf{] ljb\ofyL{nfO{ j|md};“u n]vg kf6Ldf nV] g nufpgx' f;] \ . h:t}M
{1,2,3,4,5,6,7,8,9,10 }, {-5,-4,-3,-2,-1} / 0 nfO{ cnUu} /fVg nufpgx' f;] \ .
{1,2,3,4,5,6,7,8,9,10} nfO{ k|fsl[ ts ;ª\Vofsf] ;dx" / 0 ;lxtsf] kf| s[lts ;ª\VofnfO{ k0" f{
;ª\Vofx¿sf] ;dx" elgg] af/] 5nkmn ug{'xf;] \ .

-v_ Ps hgf ljb\ofyLn{ fO{ 1 b]lv 10 ;Ddsf sg' } bO' { cf]6f ;ª\VofkTtLx¿ lgsfNg nufpg'xf];\ . klxnf]
;ªV\ ofaf6 bf;f| ] ;ªV\ of 36fpg nufpgx' f];\ . olb klxnf] ;ª\Vof 7n" f] / bf];f| ] ;ªV\ of ;fgf] ePdf pTt/
slt x'G5 < olb klxnf] ;ªV\ of ;fgf] eof] / bf];|f] ;ª\Vof 7n' f] ePdf pTt/ slt xf]nf< pbfx/0f ;lxt
5nkmn ug{ nufpgx' f];\ . ;f]sf] nflu ;ªV\ of /]vfsf] k|of]u ug{ nufpg'xf;] \ . 5 hgf ljbo\ fyL{nfO{ olx
lj|mofsnfk u/fpb“ } hfg'xf;] \ / ;ª\Vof /]vfdf klg b]vfp“b} hfgx' f;] \ .

46

DRAFT-u_ ctM 0 ;lxt;a} wgfTds / C0ffTds ;ªV\ ofnfO{ k"0ff{ª\s elgg]af/] o;sf] ;ª\s]t ;lxt 5nkmn ug{
nufpgx' f;] \ / k"0ffª{ \ssf wgfTds / C0ffTds k"0ff{ªs\ x¿nfO{ j|mdzM z+ / z- n]lvg]af/]
ljbo\ fyL{x¿nfO{ ;ª\Vof /]vf sf]g{ nufO{ b]vfpg nufpgx' f];\ .

-3_ sIffsf7] f ;ªV\ of /v] fsf] cEof; u/fpg'xf];\ . o;sf nflu Ps hgf ljb\ofyL{nfO{ 0 df pEofO{ cufl8
a9\g nufpg'xf;] ,\ pm hlt cufl8 a9\5 pTtLs} 3gfTds x'G5 . olb 0 eGbf k5fl8 pN6f] lx“8fpgx' f;] ,\
o;f] ubf{ C0ffTds x'g] hfgsf/L lbg'xf;] \ / +3 jf -4 eg]df 0 eGbf cufl8 uof] ls k5fl8 < egL kZ| g
ugx{' f];\ . o;/L 0 eGbf afof“ hf“bf dfg 36\b} hfg] / bfof“ hf“bf a9\b} hfg] s'/f ljbo\ fyL{x¿nfO{ k:| tt' ug{
nufpgx' f];\ .

-ª_ k0" ffª{ \sx¿sf] t'ngf ug{ z}Ifl0fs ;fdu|Ldf lbOPsf km\nf; sf8x{ ¿sf] ko| f]u u/L > jf < dWo] sg'
lrx\g /fVg] 5nkmnug{ nufpgx' f];\ . 4 sf] ljd'v -4 s;/L eof] ;ªV\ of /]vfdf b]vfpg nufpg'xf];\ .

2. k0" ffª{ \sx¿sf] lg/kI] f dfg
ljbo\ fnosf] vn] db} fgdf Pp6f n6\7L uf8]/ 0 nl] vPsf] sfuh em'08\ofpgx' f];\ . Ps hgf ljbo\ fyL{ A nfO{
0 sf] kj" {lt/ 6 kfOnf cufl8 lx8“ fpg nufpgx' f;] \ . B nfO{ 0 sf] klZrdlt/ 4 kfOnf lx“8\g
nufpg'xf];\ . A / B sf] larsf] b'/L slt xfn] f < s;/L < 5nkmn ug]{ . |-4|+ |6| x'“f B b]lv A kU' g
slt kfOnf lx+8\g' knf{< 5nkmn ug{xf];\ .
ctM . -4 . + . 6 . = 4 + 6 = 10 x'g]af/] k:| t't ug{ nufpgx' f];\ . To:t} c? pbfx/0fx¿ klg eGg
nufpg'xf];\ . kf7\o k':tssf pbfx/0fx¿ ;dx" df 5nkmn ug{ nufpg'xf];\ .

3. k0" ffª{ s\ sf ;/nLs/0f
-s_ kf7o\ k:' tsdf ePsf] k0" ff{ª\ssf hf8] sf lgodx¿ ko| fu] u//] k0" ffª{ \ssf] hf]8 / 36fpsf
pbfx/0fx¿ agfpg nufO{ ;d"xdf 5nkmn u/fO{ k:| t't ug{ nufpg'xf;] \ / To:t} c? pbfx/0fx¿
k:| t't ug{ nufpg'xf];\ .
-v_ kf7\o k:' tsdf ePsf] k0" ffª{ s\ sf u'0fgsf lgodx¿ ko| fu] u//] k0" ff{ªs\ sf u0' fgsf pbfx/0fx¿
;d"xdf agfpg nufO k:| tt' ug{ nufpgx' f];\ / To:t} cGo pbfx/0fx¿ lbg nufpgx' f];\ . 3gfTds /
C0ffTds lrxg\ u'0fg / efu ubf{ x'g] lrxg\ x¿sf] af/]df ljb\ofyLx{ ¿nfO{ bO' { ;dx" df ljefhg u/L
Pp6f ;d"xsf] Ps hgfnfO{ k|Zg ;f]Wg nufpg'xf;] \ / csf{] ;dx" sf] Ps hgfnfO{ pTt/ eGg nufpg]
vn] vn] fpgx' f];\ . ldnfPdf 1 cª\s lbg] u/L cGtdf a9L Nofpg] ;d"x ljhoL 3f]lift ug{] u/L v]n
v]nfpg'xf];\ . h:t}M -5 u0' ff 8 slt x'G5 <

4. k"0ffª{ \ssf zflAbs ;d:ofx¿sf] ;dfwfg

rf/ ;fwf/0f lgodx¿ lglxt k"0ff{ªs\ sf ;d:ofx¿sf] ;dfwfg ug{ lgDgfg';f/ syfx¿ agfpg nufO{
ul0ftLo efiffdf nV] g nufpg'xf;] \ .

a'afn] ljb\ofno hfg lbbL / efO k|Tos] nfO{ ?= 40 lbg' eP5 . k|To]sn] ? 10 sf] vfhf vfP5 . bj' }n] 3/
kmsb{ f af“sL ePsf] /sd 5 hgfnfO{ efu nufP5g\ . ca k|Tos] n] slt slt kfP xfn] fg\ <
o;sf nflu ul0ftLo jfSodf n]Vg k|fT] ;fxg ug'{xf;] \M a'afn] kT| o]snfO{ lbg' ePsf] = ? 40

k|Tos] n] vfhf vfPsf] = ? 10
k|To]s;u“ af“sL = ?= 40 - ?= 10

bO' { hgf;“u afs“ L /sd =2(40 – 10)
5 hgfnfO{ efu nufpb“ f =2(40-10) ÷5

o;sf] ;/n ubf{, klxn] sf]i7sf] lj|mof ug{' kg{] ePsf]n] = 2(30) ÷ 5
oxf“ u'0fg / efu lj|mof cfPsfn] ] klxn] efu ubf{ =2 x 30= 2x6 = 12 x'G5 .

5

cyjf, afof“lt/af6 u0' fg / efu ljm| of dWo] klxn] u0' fg ljm| of cfPsfn] ] klxn] u'0fg ug{ ;lsG5 .
= 2 x 30 ÷ 5=60 / 5 = 12 xG' 5 .

47

dN" ofª\sgM 1= k"0ffª{ \ssf ks| f/x¿ n]v .
2= -2 / -1 df s'g 7"nf] xG' 5<
3= . -5 . + . 5 . slt x'G5<
4= ;/n u/M (+3)+(-2), (+3)-(-2), (+3) x (-2), (+15) ÷(-3)
5= ;/n u/M (+12) + (-3) + ( +23) + (-3) - ( -3) x (-8)

lzIfsnfO{ lgbz{] g
3gfTds k0" ff{ªs\ x“'bf 0 af6 bfofl“ t/ jf cufl8 / C0ffTds xb'“ f afof“lt/ jf k5fl8 hfg'kg{] wf/0ff lbg
k|To]s ljbo\ fyL{nfO{dflysf] lj|mofsnfkdf kfnk} fnf] ;j|mLo efu lng nufpg' cfjZos 5 .

PsfO 14=
cfg'kflts ;ª\Vof

(Rational numbers)
DRAFT
cgd' flgt 3G6L= 3

k"j1{ fg

o; kf7sf] nflu ljb\ofyL{x¿df k"0ff{ª\ssf] hf]8, 36fp, u'0fg / efu ug{ hfg]sf], k0" ffª{ \ssf hf]8 / u'0fgsf
lgodx¿ pbfx/0f ;lxt eGg ;Sg] x'gk' 5{ .

PsfO kl/ro

o; PsfOdf cfgk' flts ;ªV\ ofx¿sf] kl/ro ljzi] ftf, o;sf efukmnsf ks| f/x¿af/] lzIf0f tl/sfx¿sf]
5nkmn ugs{' f ;fy} cfg'kflts ;ª\Vofsf ljz]iftfx¿ af/] 5nkmn ul/g] 5 .

kf7 pbb\ Z] oM o; kf7sf] cWoogkZrft\ ljbo\ fyLx{ ¿ lgDg sfo{ ug{ ;Ifd xg' ] 5g\ M

1= cfgk' flts ;ª\Vofsf] kl/ro lbg

2= cfgk' flts ;ªV\ ofsf x/n] cz+ nfO{ efu u/L xg' ] efukmnsf ks| f/ atfpg .

3= cfgk' flts ;ªV\ ofsf ljz]iftfx¿ atfpg .

z}Ifl0fs ;fduL|

1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 ( ÷ , x, +, - ) n]lvPsf kmn\ f;sf8x{ ¿ b'O{ ;]6

l;sfO ;xhLs/0f ljm| ofsnfk

1= cfg'kflts ;ª\Vofsf] kl/ro

1=-s_ zI} fl0fs ;fduL| df pNnl] vt kTtLx¿ dWo] ;ªV\ ofx¿ ePsf] / lrx\gx¿ ePsf] 56' 6\ f 56' \6} sIffsf] cu|
efudf 3fK] 6\ofP/ /fVg'xf;] \ .

-v_ ljbo\ fyL{x¿nfO{ b'O{ ;d"xdf ljefhg ug{xf];\ . klxnf] ;d"xsf] 3 hgf ljb\ofyLx{ ¿nfO{ cufl8 8fs]/ b'O{
hgfnfO{ ;ª\Vof kTtLx¿ Pp6f÷Pp6f / csf{] ljbo\ fyLn{ fO{ lrxg\ kTtL Pp6f lng nufpg'xf;] \ . To;
cg'¿ksf] pTt/ bf];|f] ;d"xsf sg' } Ps hgf ljbo\ fyL{nfO{ cufl8 cfP/ eGg / nv] g kf6Ldf nV] g
nufpgx' f;] \ . s] of] k0" ff{ª\s xf] < pTt/ ldnd] f 1 cª\s lbg], gldn]df cªs\ glbg] . To:t} u/L bf;] f| ]
;dx" sf 3 hgfnfO{ cufl8 8fs/] dfly s} k|lj|mof cg;' f/ vn] v]nfpgx' f];\ . kT| o]s ;d"xsf 5/5 k6s
kfnf] ;lsP kl5 xf/lhtsf] 3f]if0ff ug{xf];\ .

-u_ kg' ZrM o; v]ndf b'O{ k0" ff{ª\sx¿ ljr hf]8, 36fp, u0' fg ljm| of ubf{ k"0ffª{ s\ g} x'g] t/ efu lj|mof ubf{
k"0ffª{ \s gx'g] s/' fsf]ljZjf; ;ªV\ of /v] fdf /x]sf ;ª\Vofx¿ bv] fO lbnfpgx' f];\ .

48

-3_ cGo efu ljm| of ePsf k"0ff{ª\sx¿ 5/5 cf6] f n]Vg kf6Ldf n]Vg nufpg'xf;] \ .
oL ;ªV\ ofx¿nfO{ cfg'kflts ;ªV\ of eGga] f/] 5nkmn ug{ nufpgx' f];\ . t/ x/ 0 gx'g] s/' f ljb\ofyL{af6
lngx' f;] \ .

2= cfgk' flts ;ª\Vofsf] x/n] cz+ nfO{ efu

-s_ n]vg kf6Ldf , / n]Vgx' f];\ . ltg hgf ljb\ofyL{x¿nfO{ nv] g kf6Ldf nl] vPsf tL cfgk' flts
;ªV\ ofx¿nfO{ bzdnjdf ¿kfGt/ ug{ nufpgx' f];\ . klxnf] leGgsf] bzdnjsf] :yfg kl5 s] eof] < bf;] f| ]
leGgsf] bzdnj kl5sf cªs\ x¿ s] eP < To:t} u/L t];|f] leGgsf] bzdnj kl5sf cª\sx¿ s] eP <
5nkmn ub{} cGTo xg' ], cGToxLg tyf kg' /fjT[ t bzdnj x'g] lgisif{ lgsfNg nufpg'xf];\ .

-v_ kT| os] ljb\ofyL{nfO{ 3/3 cf]6f cGTo x'g,] cGToxLg tyf kg' /fj[Tt bzdnj x'g] cfgk' flts ;ª\Vofx¿ n]Vg
nufpgx' f;] \ .

-u_ kf7\o k:' tsdf lbOPsf cfg'kflts ;ªV\ ofx¿sf] hf8] / u0' fgsf] ljz]iftfx¿cGtu{tsf lgodx¿ 5nkmn
ug{ nufO pbfx/0f k:| t't ug{ nufpg'xf;] \ .

d"Nofªs\ g
!= cfgk' flts ;ª\Vofsf] pbfx/0f ;lxt kl/ro bp] .
@= cGTo xg' ], cGToxLg tyf k'g/fj[Tt bzdnj x'g] cfg'kflts ;ª\Vofx¿ 2/2 cf6] f lx;fa u/]/ b]vfpm .
#= cfgk' flts ;ª\Vofsf ljz]iftfx¿ cGtu{t o;sf lgodx¿ pbfx/0f ;lxt nv] .
lzIfsnfO{ lgb{]zg
k"0ff{ªs\ sf hf]8 / u0' fgsf lgodx¿ tyf cfgk' flts ;ª\Vofsf ljzi] ftflarsf ;DaGwaf/] pbfx/0f ;lxt
5nkmnu/L k:| t't ug{ nufp+bf ljb\ofyL{x¿df o;sf] wf/0ff :ki6 xG' 5 .
DRAFT
PsfO 15=
cgfg'kflts ;ª\Vof

(Irrational Numbers)

cgd' flgt 3G6L M 3

kj" {1fgM cfgk' flts ;ªV\ ofsf] kl/ro
PsfO kl/ro
o; kf7df ;ªV\ ofx¿ dWo] cfgk' flts ;ªV\ ofafx]s cgfg'kflts ;ªV\ ofx¿ klg xg' a] f/] 5nkmn ul/Psf] 5 .

pbb\ Z] o

o; kf7sf] cWoogkZrft\ ljb\ofyLx{ ¿ cgfgk' flts ;ªV\ ofsf ljz]iftfx¿ atfpg tyf cg'kflts /
cgfgk' flts

;ªV\ ofx¿ 56' \ofpg ;Ifd xg' ] 5g\ .

z}Ifl0fs ;fduL| M cfgk' flts / cgfgk' flts ;ª\Vofx¿ nl] vPsf kTtLx¿, √2, √3, √5, √7, ∛3, 6k]

l;sfO ;xhLs/0f ljm| ofsnfk

1=-s_ zI} fl0fs ;fdu|Ldf pNn]lvt kTtLx¿ sIffsf] cue| fudf ;a} ld;fP/ /fVgx' f];\ . k|To]s ljbo\ fyLn{ fO{
kfnk} fnf] Pp6f/Pp6f kTtL l6Kg nufpg] / cfgk' flts ;ª\Vof xf] egL nv] g kf6Ldf nV] g nufpg'xf;] \ .

49

olb x}g eg] 6k] åf/f tL kTtLx¿ n]vg kf6Ldf 6f“:g nufpgx' f;] \ . tL ;ª\Vofx¿ cfgk' flts ;ª\Vof
gePsf] sf/0f eGg nufpg'xf;] \ . kf7o\ k':tssf] pbfx/0fx¿ x]g{ nufO{ 5nkmn u//] k:| t't ug{
nufpgx' f];\ . o:tf ;ªV\ ofnfO{ cgfgk' flts ;ª\Vof elgg]af/] 5nkmn u/fpg'xf;] \ .

-v_ ;dx" df tnsf] rf6{ tof/ ug{ nufpg'xf];\ M

;ª\Vof

cfgk' flts ;ªV\ of cgfg'kflts ;ª\Vof

k"0ffª{ s\

C0ffTds z"Go wgfTds
d"Nofªs\ g
1= cfg'kflts ;ªV\ of / cgfg'kflts ;ª\Voflar s] km/s 5 < pbfx/0f;lxt nv] .
2= ltg cf6] f cgfg'kflts ;ª\Vofx¿ nv] .
3= cfgk' flts / cgfgk' flts ;ª\Vofx¿nfO{ eg] lrqdf b]vfpm .
lzIfsnfO{ lgb]{zg
k|To]s ljbo\ fyL{nfO{ kf| s[lts ;ª\Vof, k"0f{ ;ªV\ of, k"0ff{ª\s ;ªV\ of, cfg'kflts tyf cgfg'kflts ;ªV\ ofsf]
pbfx/0f ;lxt rf6{ tof/ kfg{ lb+bf tL ;ª\Vofx¿ larsf] ;DaGwaf/] hfgsf/L kfpg] xG' 5 . h;af6
o; ;DaGwL wf/0ff rL/:yfoL x'G5 .
DRAFT
PsfO 16=
leGg / bzdnj

(Fraction and Decimals)

cgd' flgt 3G6L M 8

kj" {1fg

leGgx¿sf] hf]8, 36fp, u'0fg, efu tyf ;/nLs/0f ug{ ;Sg,] k"0ffª{ s\ sf] ;/nLs/0f ug{ ;Sg] tyf
bzdnjsf] hf]8, 36fp, u'0fg / efu ug{ ;Sg] .lqe'hsf] Ifq] kmn, cfotfsf/ j:t'sf] kl/ldlt / Ifq] kmn,
jT[ tsf] kl/lw.

PsfO kl/ro

o; kf7df leGg;u“ ;DalGwt zflAbs ;d:ofx¿sf] ;dfwfg ug{] lzIf0f tl/sfaf/] 5nkmn ug{] 5f} .
ljbo\ fyL{sf] b}lgs hLjgdf k|ofu] eO/xg] x'“bf o;sf] lzIf0fdf pgLx¿sf b}lgs hLjg;“u ;DalGwt ug{ ;s]df
Hofb} ke| fjsf/L xG' 5 . To;u} /L o; PsfOdf bzdnj ;ªV\ ofx¿sf] ;/nLs/0f ug]{ tyf tT ;DaGwL zflAbs
;d:ofx¿ xn ug{] tl/sfsf] lzIf0f gd"gfaf/] 5nkmn ul/g] 5 .

zI} fl0fs ;fdu|L
rfdnsf] vfnL af]/f -pknJw ePdf_ gePdf sg' } emf]nf 1 cf6] f, dfs{/ 1, ljb]zL ljlgdo b/ cª\lst
klqsfsf ;"rgfx¿ -k|lt a]GrnfO{ Pp6fsf b/n_]
pbb\ ]Zo
o; kf7sf] cWoogkZrft\ ljbo\ fyLx{ ¿ lgDg sfo{ ug{ ;Ifd xg' ] 5g\ M
1= leGg;u“ ;DalGwt zflAbs ;d:ofx¿ xn ug{

50

DRAFT2= bzdnj ;ª\Vofsf] ;/nLs/0f ug{
3= bzdnj lglxt zflAbs ;d:ofx¿ xn ug{
l;sfO{ ;xhLs/0f lj|Mofsnfk
1. leGgsf] zflAbs ;d:ofx¿ ;dfwfg
-s_ k|Tos] ljbo\ fyL{nfO{ leGg;lxtsf] Pp6f Pp6f ;d:of nV] g nufpg'xf];\ . h:t}M s[i0fn] 3/df 30 s]=hL=

rfdn lsgs] f]df cfh efu vfP/ ;lsP5 eg] ca slt rfdn af“sL xfn] f <

-v_ o;sf] ;dfwfg ug{ oxf“ lbOPsf s/' fx¿ s] s] x'g\ < lgsfNg'kg{] s/' f s] xf] < 5nkmn ugx{' f];\ .
-u_ o; k|Zgsf] lzIf0fsf nflu rfdnsf] vfnL af/] f -30 s]hL nv] ]sf_] k|bz{g ub{} To;df slt tfn} sf] rfdn

lyof] xfn] f <
-3_ To;nfO{ 5 efu nufpb“ f Ps efudf slt tfn} sf] rfdn xfn] f < kZ| g ub}{ af]/fnfO{ 5 efudf dfs{/åf/f

lrxg\ nufpg Ps hgf ljb\ofyLn{ fO{ eGgx' f]nf . Ps efudf xg' } tfn} ljb\ofyLn{ fO{ ;f]Wg'xf;] \ . bO' { efudf
slt tf}n xf]nf < 5nkmn ug{ nufpgx' f;] \ .
-ª_ of] ;d:ofnfO{ lgDgfg';f/ ;dfwfg ug{ ;xofu] ug{x' f];\ M
hDdf rfdn = 30 kg

vr{ ePsf] rfdn = 30 kg x = 6 x 2 = 12 kg x'G5 .

-r_ ca slt rfdn af“sL xf]nf < kTtf nufpg lbgx' f;] \ . o;;“u ;DalGwt as} lNks tl/sf klg 5nkmn
ugx{' f;] \ .

-5_ o;;} “u ;DalGwt cGo kZ| gx¿ klg lbg'xf];\ . h:tM} efu rfdn vfOP5 eg] slt rfdn afs“ L xf]nf <

-h_ kf7o\ k:' tssf pbfx/0fx¿ 5nkmn ug{ nufO{ k|:tt' ug{ nufpg' xf;\ .
2. ;/nLs/0f
-s_ k|To]s a]GrnfO{ bzdnj tyf rf/ ;fwf/0f lj|mofx¿ lglxt ;d:of n]Vg lbg'xf;] \ . To;dWo] sg' } Pp6fnfO{

nv] g kf6Ldfn]Vg nufO{ s;/L ;/n ug{] xf]nf egL k|Zg ugx{' f];\ . o;sfnflu k"0ff{ª\ssf] ;/n ug{] tl/sf
cg';f/ g} bzdnj ;ª\Vofsf] ;/n ug'{kg]{ s/' f 5nkmn u/fpgx' f;] \ . ;/nLs/0f ubf{ ;a} kbx¿df
bzdnjeGbf k5fl8sf cª\sx¿ plTtg} x'gk' g{] s/' fdf hf]8 lbgx' f;] \ / s'g} Ps hgf ljbo\ fyL{nfO{ ;f]
n]lvPsf] ;d:of ;dfwfg ug{ nufpg'xf];\ . To;c} g'¿k cfcfk"mn] agfPsf ;d:ofx¿ ;dfwfg ug{
nufpgx' f];\ . k|To]s ljb\ofyL{nfO{ Pp6f Pp6f ;d:of agfpg nufO{ ;dfwfg ug{ nufpg'xf];\ .
-v_ kf7o\ k':tssf] pbfx/0f 1 / 2 5nkmn ug{ nufO{ k|:tt' ug{ nufpg'xf];\ .
3. bzdnj ;DaGwL zflAbs ;d:ofx¿ ;dfwfg
-s_ klqsfsf ljb]zL ljlgdo b/ nl] vPsf] ;"rgf k|Tos] aG] rnfO{ ljt/0f u/L k|Zgx¿ agfpg lbgx' f];\ / xn
ug{ klg nufpg'xf;] \ . h:t}M 5 $ / 3£ lsg]/ g]=?= 500.25 vr{ ubf{ hDdf slt ?= af“sL xfn] f <
-v_ leGgsf] zflAbs ;d:ofx¿ ;dfwfg ubf{ em+} oxf“ klg lbOPsf s/' fx¿ s] s] xg' \ < kTtf nufpgk' g{]
s/' fx¿ s] xf] < cflb ;d:ofcg;' f/ n]Vg nufpg'xf];\ . dflysf] ;d:ofdf lbOPsf] 5 $ / 3£ nfO{ g]=
?= ¿kfGt/ ug'{kg]{ / tL /sdx¿ hf8] \g' kg]{ tyf To;af6 lbOPsf] g]= ?= 36fpg' kg]{ s'/fx¿
ljbo\ fyLx{ ¿af6 lngx' f];\ . To;sf] nflu ul0ftLo efiffdf nV] g nufO ;dfwfg ug{ nufpgx' f];\ .
-u_ k|To]s ljbo\ fyL{nfO{ o; ;DaGwL Pp6f Pp6f kZ| g agfpg nufO{ ;dfwfg ug{ nufpgx' f];\ .
d"Nofª\sgM

1= ltdLnfO{ a'afn] ?= 1500 lbg'eof] . To;dWo] ltdLn] cfdfnfO{ lbof}“ eg] cfdfnfO{ slt ?= lbof}“ <
ca ltdL;“u slt ?= af“sL xfn] f <

2= ;/n u/ M 124.45 + 34.0012 – 121.302

51