RS593_Children Reference Materials No. 41

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ISBN: 978-9937-8247-0-5

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klxnf] ;:+ s/0f M lj=;=+ @)%#
kl/dflh{t bf;] |f] ;:+ s/0f M lj=;=+ @)^^
k'gd'b{ |0f M lj=;=+ @)&#

d"No ? M &@÷—

kf7\oj|md ljsf; s]Gb|sf] lnlvt :jLs[ltlagf Jofkfl/s k|of]hgsf nflu o;sf]
k/' } jf cfl+ zs efu xa' x' ks| fzg ug{, kl/jt{g u/]/ k|sfzg ug{, s'g} ljb\o'tLo ;fwg jf cGo
kl| jlwaf6 clen]vab\w ug{ / kl| tlnlk lgsfNg kfOg] 5}g .

xfdf| ] egfO

lzIffnfO{ på]Zod"ns, Jofjxfl/s, ;d;fdlos / /f]huf/d"ns agfpg ljleGg ;dodf
kf7o\ j|md, kf7o\ k':ts ljsf; tyf kl/dfhg{ ug{] sfo{nfO{ lg/Gt/tf lbOb“ } cfPsf] 5 . ljBfyLd{ f
/fi6,« /fli6«o Pstf / nf]stflGqs ;:+ sf/sf] efjgf k}bf u/fO{ gl} tstf, cgz' f;g / :jfjnDag,
l;hg{ zLntf h:tf ;fdflhs Pjd\ rfl/lqs u0' f tyf cfwf/e"t eflifs tyf ul0ftLo l;ksf ;fy}
lj1fg, k;] f Joj;fo, ;r" gf tyf ;~rf/ kl| jlw, jftfj/0f / :jf:Yo ;DaGwL cfwf/et" 1fg / hLjgf]
kofu] L l;ksf] ljsf; u/fpg' h?/L 5 . pgLx¿df snf / ;f}Gbo,{ dfgjLo dN" o dfGotf, cfbz{ /
j}lzi6\ox¿sf] ;/+ If0f tyf ;j+ wg{ kl| tsf] efj hufpg cfjZos 5 . ;dfj]zL ;dfhsf] lgdf{0fdf
;xof]u k'¥ofpg pgLx¿df ljleGg hfthflt, lnªu\ , ckfª\utf, efiff, wd{, ;:+ s[lt / Ifq] kl| t ;defj
hufpg' / dfgj clwsf/ tyf ;fdflhs dN" o dfGotfk|lt ;r]t eO{ lhDdj] f/Lk0" f{ cfr/0fsf] ljsf;
u/fpg' klg cfhsf] cfjZostf ag]sf] 5 . kf| ylds lzIff kf7o\ jm| d @)^% nfO{ dn" cfwf/ dfgL
lzIff ;DaGwL ljleGg cfof]usf ;'emfj, lzIfs, ljBfyL{ tyf cleefjsnufot lzIff;u“ ;Daå
ljleGg JolSt ;lDdlnt ufi] 7L / cGtlj{|mofsf lgisif{ / ljleGg ljBfnodf k/LIf0f u/L k|fKt
k[i7kfi] f0f;dt] nfO{ ;d6] L of] kf7o\ k':ts tof/ kfl/Psf] xf ] .

zDeg' f/fo0f j}B tyf xl/gf/fo0f pkfWofoåf/f nl] vPsf] o; kf7\ok':tsnfO{ lzIfs, ljzi] f1,
cleefjs, ljBfyL{ Pjd\ cGo ;/fs] f/jfnfx¿sf] ;´' fjcg;' f/ kg' nv]{ g tyf kl/dfhg{ ul/Psf] xf] . o; kf7\
ok':tsnfO{ lrqfTds, lj|mofsnfkd'vL / ljb\ofyL{ s]lGbt| agfpg] ko| f; ul/Psf] 5 . k|:t't k':tssf]
kl/dfhg{ lbgz] sd' f/ >i] 7, lrqk;| fb bj] sf6] f,j?0fk;| fb jB} , xl/gf/fo0f pkfWofo, l8NnLZj/ kw| fg,
808kfl0f zdf{, /fdrGb| kf}8n] / Zofdl;x+ wfdLåf/f ul/Psf] xf] . o; sfod{ f xl/af]n vgfn, 8f=
l;l4k;| fb sfO] /fnf, 8f= lzj/fd Gof}kfg], d'sG' b/fhzdf{, lgd{nf uf}tdnufotsf dxfge' fjx¿sf]
;d]t ;n+ Ugtf /x]sf] 5 . o;sf] efiff;Dkfbg lji0fk' |;fb clwsf/L tyf nfs] ks| fz kl08taf6 ePsf]

xf] . o; k':tsnfO{ rf/ /ªdf ljsf; ug{ / o;sf] ¿k ljGof; Pjd\ snf ;Dkfbg sfo{ >Lxl/

>]i7n] uge{' Psf] xf] . pxf“x¿nufot o; kf7\o k:' tssf] ljsf; tyf kl/dfhg{ sfo{df ;n+ Ug ;a}
kl| t kf7o\ j|md ljsf; s]Gb| wGojfb k|s6 ub{5 .

kf7\ok':tsnfO{ lzIf0f l;sfOsf] dxTTjk"0f{ ;fwgsf ¿kdf lnOG5 . o;af6 ljBfyL{n]

kf7o\ jm| dåf/f nlIft ;Ifdtf xfl;n ug{ dbb\ t k'Ug] ck]Iff ul/Psf] 5 . o; kf7\ok':tsnfO{ ;s];Dd

lj|mofsnfkd'vL / ?lrs/ agfpg] ko| Tg ul/Psf] 5 . kf7o\ k:' tsnfO{ cem} kl/is[t kfgs{ f nflu

lzIfs, ljBfyL{, cleefjs, al' åhLjL Pjd\ ;Dk"0f{ kf7sx¿sf] ;d]t dxTTjk"0f{ e"ldsf /xg] x'b“ f

;Da¢ ;as} f] /rgfTds ;'emfjsf nflu kf7o\ j|md ljsf; sG] b| xflb{s cg'/fw] ub5{  .

g]kfn ;/sf/

lj= ;=+ @)&$ lzIff dGqfno

kf7o\ j|md ljsf; s]Gb|

kf7 jm| d ljifo ;r" L k[i7 ;ªV\ of
kf7 1 M
kf7 2 M kf7zLif{s 1
kf7 3 M Hofldlt 18
kf7 4 M ;ª\Vofsf] 1fg 34
kf7 5 M ul0ftsf cfwf/e"t ljm| ofx¿ 39
kf7 6 M ;do 44
kf7 7 M d'b|f 51
kf7 8 M b'/L 55
kf7 9 M kl/ldlt 59
kf7 10 M If]qkmn 63
kf7 11 M Ifdtf 67
kf7 12 M cfotg 72
kf7 13 M tfn} 77
kf7 14 M leGg / bzdnj 96
kf7 15 kl| tzt 102
kf7 16 P]lss lgod / ;fwf/0f Aofh 106
kf7 17 lan / ah]6 110
kf7 18 tYofªs\ zf:q 118
;d"x 120
aLh ul0ft

!kf7
Hofldlt (Geometry)

1.1 sf]0fx¿sf] gfk O A
B
of] sf]0fsf] gfk slt xf]nf <
s] k|of]u u/]/ gfKg ;lsPnf <

sf]0f gfKg xfdL kf| ]6\ofS6/ ko| f]u u5f{}“ .

tn lbOPsf] lrq x]/ M

lrqdf kf| 6] o\ fS6/sf] 3/] f 180 a/fa/ efudf af“l8Psf] 5 . kT| os] efunfO{ l8u|L
elgG5 . l8uL| (°) nfO{ sf]0f gfKg] PsfOsf ¿kdf lnOG5 . k|f]6\ofS6/df b'O{ cf]6f :s]
n nl] vPsf] xG' 5 . aflx/L :s]n afof“lt/ 0 l8u|Laf6 ;?' eO{ bfof“lt/ a9b\ } uP/ 180°
k'u]sf] 5 eg] leqL :sn] bfofl“ t/ 0 l8uL| af6 ;?' eO{ afofl“ t/ a9b\ } 180° ku' ]sf] 5 .

o;/L Pp6} k|f6] \ofS6/df bO' { cf]6f :s]n lsg nv] ]sf] xfn] f <

xfdL kf| 6] o\ fS6/sf] ko| f]u bO' { sfdsf nflu u5f“{} M

(1) sf]0f gfKgsf nflu

(2) lbOPsf] gfksf] sf]0f lvRgsf nflu

d]/f] ul0ft M sIff % 1

lj|mofsnfk 1

sf0] f ABC sf] gfk lng' 5 . ca, sf0] fsf]
eh' f BC nfO{ kf| 6] \ofS6/sf cfwf/ /]vfdf
l7Ss ldnfP/ o;/L /fvf“} ls laGb' B kf| 6] \
ofS6/sf] s]Gb|df k/f;] \ / BC kf| ]6\ofS6/sf]
00 b]vfpg] /]vf;“u l7Ss ldnf];\ . ca AB
n] kf| ]6o\ fS6/sf] leqL :sn] df sxfg“ /] 5fP] /
uPsf] 5, x]/f}“ . lrqdf AB n] k|f6] o\ fS6/
sf] leqL :s]ndf 300 nfO{ 5fP] / uPsf] 5 .
To;n} ] sf0] f ABC sf] gfk 300 -tL; l8uL| _
eof] . oxf“ xfdLn] leqL :sn] ko| fu] u¥of}“,
lsg <

ljm| ofsnfk 2

sf]0f ABC sf] gfk lng' 5 . ca sf0] f ABC
sf] e'hf BC nfO{ kf| 6] \ofS6/sf] cfwf/

/v] fdf l7Ss ldnfP/ o;/L /fvf“} ls laGb' B
k|f6] \ofS6/sf] s]Gbd| f k/f];\ / BC kf| 6] \ofS6/
sf] afofl“ t/ 00 hgfpg] /v] f;u“ l7Ss ldnf;] \
. ca AB n] k|f6] o\ fS6/sf] aflx/L :s]ndf
sxf“g/] 5f]P/ uPsf] 5 x]/f“} . lrqdf AB
n] k|f6] \ofS6/sf] :sn] df 600 nv] s] f] 7fpd“ f
5f]P/ uPsf] 5 . To;}n] sf]0f ABC sf] gfk
600 eof] . oxf“ xfdLn] aflx/L :sn] k|of]u
ugk{' ¥of], lsg <

ltdLx¿n] lbPsf] sf]0fnfO{ leqL / aflx/L :s]n k|of]u u/L gfKg l;Sof} . ca lbPsf]
gfk a/fa/sf] sf]0f lvRg] tl/sf yfxf kfpg tnsf] pbfx/0f x/] M

2 d]/f] ul0ft M sIff %

pbfx/0f 1
kf| 6] \ofS6/sf] ;xfotfn] 400 gfk ePsf] sf]0f leqL / aflx/L b'j} :sn] ko| fu] u/L lvr .
cfwf/ /v] f AB lvr]/ laGb' A nfO{ kf| 6] o\ fS6/sf] s]Gbd| f kf//] AB nfO{ bfofl“ t/
k|f]6\ofS6/sf] cfwf/ /]vfdf l7Ss ldnfpm . k|f]6o\ fS6/sf] leqL :s]ndf 400 x]//] laGb'
P lrxg\ nufpm .

P

AB
P

k|f6] o\ fS6/ x6fP/ ?n/ / k]lG;nsf]
;xfotfn] A / P nfO{ hf]8 . ca
A B rflxPsf] ∠PAB = 400 aGof] .
cfwf/ /]vf AB lvr]/ laGb' A nfO{ kf| ]6\ofS6/sf] s]Gbd| f kf//] AB nfO{ afofl“ t/sf]
kf| 6] \ofS6/sf] cfwf/ /v] f;u“ ldnfP/ /fv . ca kf| 6] \ofS6/sf] aflx/L :sn] df 400 x]//]
laGb' P lrx\g nufpm .

P

B P
A
B A
ca k|f]6\ofS6/ x6fP/ ?n/ / k]lG;nsf] 3
;xfotfaf6 laGb' A / P hf8] . rflxPsf]
∠PAB = 400 aGof] . o;/L kf| 6] \ofS6/sf] leqL
cyjf aflx/L :sn] ko| fu] u/L pxL gfksf] sf0] f
lvRg ;lsG5 .

d/] f] ul0ft M sIff %

pbfx/0f 2
kf| 6] o\ fS6/sf] ;xfotfn] lrqdf lbOPsf] sf0] f ABC ;u“ a/fa/ xg' ] u/L sf0] f PQR lvr .

A

lbOPsf] sf]0f ABC xf] . sf]0f ABC sf]
gfk a/fa/ xg' ] csf{] sf0] f PQR lvRg'
5 . o;sf nflu sf0] f ABC slt 7"nf]
B C 5 . kf| 6] o\ fS6/ k|of]u u/L gfk . sf]0f
∠ABC = 300 /x5] .

P

QR

ca cfwf/ /v] f QR lvr/] laGb' Q df
P k|f]6o\ fS6/sf] sG] bl| aGb' ldnfO{ QR nfO{

;Lwf 00 b]vfpg] /]vf;u“ ldnfP/ /fv]/
k|f6] o\ fS6/sf] kl/lwdf leqL :s]nsf] 300
agfPsf] 7fpd“ f kg{] u/L laGb' P lrxg\
Q R nufpm .

P

ca k|f6] o\ fS6/ x6fP/ ?n/ / kl] G;nsf]
;xfotfn] P / Q nfO{ hf]8\bf ags] f] sf]0f
Q R PQR sf0] f ABC ;u“ a/fa/ xG' 5 .

4 d]/f] ul0ft M sIff %

cEof; 1.1

(1) tn lbOPsf kT| os] sf]0fsf] gfk cg'dfg u/ . ltd|f] cg'dfg ldNof] cyjf

ldn]g, k|f6] o\ fS6/n] gfk/] x/] M F

-s_ -v_

A

B C D
E

-u_ L -3_ O

M N
JK W

-ª_ S U -r_

V X
T A

(2) bfofs“ f] lrqdf sltcf6] f sf0] fx¿ 5g\, B
k|To]s sf]0fsf] gfk l8uL| df nv] M O C

d]/f] ul0ft M sIff % 5

(3) tnsf kT| o]s lrqdf ∠AOC / ∠BOC sf] gfk l8uL| df n]v . ∠AOC /
∠BOC sf] ofu] kmn slt l8uL| 5 <

CC C

AO BA O BA O B

-s_ -v_ -u_

(4) tnsf kT| os] lrqdf ∠AOC, ∠COB, ∠BOD / ∠DOA gfk/] l8u|Ldf n]v M

C

A C B
C D

A O BO BA O
DD

-s_ -v_ -u_
(5) kZ| g g=+ 4 sf lrqx¿df sg' sg' b'O{ hf8] f sf0] fx¿sf] gfk Pp6} 5 <
(6) tn lbOPsf] lrqdf sf]0f ag]sf 7fp“x¿ kQf nufO{ kT| os] sf]0f gfk .

AB

FE C
D

G H
6 d]/f] ul0ft M sIff %

(7) lrqdf bv] fOPsf] v'nf vfddf sltcf6] f sf0] fx¿ 5g\, kQf nufO{ k|To]snfO{
gfk/] bv] fpm M

P

QR

O

TS

(8) kf| ]6o\ fS6/sf] ;xfotfn] tn lbOPsf] gfksf sf]0fx¿ lvr M
-s_ 200 -v_ 300 -u_ 400 -3_ 700 -ª_ 1100
-r_ 1200 -5_ 1800 -h_ 800 -´_ 900 -`_ 500
(9) k|f6] o\ fS6/sf] ;xfotfn] tn lbOPsf k|Tos] sf0] f;“u a/fa/ x'g] sf]0fsf] /rgf
u/ M

AP

O BQ R

-s_ -v_

X
J

ZY L
K 7

-u_ -3_

d/] f] ul0ft M sIff %

ca, xfdL cfk}“m k|f]6o\ fS6/ agfcf“} -rf}8fOsf] bfA] a/ nDafO_ M

Pp6f sfuhsf] cfotfsf/ 6j' m| f lncf“} .

o;nfO{ lgDgcg;' f/
k6\ofpb“ } hfcf“} .

ca kT| o]s efunfO{
3 a/fa/ efudf k6\ofP/
sfuh vf]Nbf o:tf] bl] vG5 .

o;nfO{ g} 150/150 cGt/sf] sf0] f gfKg
kf| 6] o\ fS6/sf ¿kdf ko| f]u ug{ klg
;lsG5 . o;df hDdf sltcf]6f sf0] fx¿
lvlrPsf 5g\ . kT| o]ssf] gfk nV] g ;S5f} <

lzIf0f lgbz{] g M ul0ftsf] lzIf0f l;sfOsf j|mddf dfly pNnv] u/h] :t} kf7;“u
;DalGwt yk l;hg{ fTds ljm| ofsnfk u/fpg nufpgx' f;] \ .

8 d/] f] ul0ft M sIff %

1.2 lqeh' sf] juLs{ /0f

-s_ eh' fsf cfwf/df lqe'hsf] juL{s/0f

lj|mofsnfk 1
lgDgfg';f/sf 3 ;6] k]lG;n cyjf l;Gsfx¿ hDdf u/ .

;]6 - 1 ;]6 - 2 ;]6 - 3

-tLgcf]6} a/fa/ gfksf k]lG;nx¿_ -bO' c{ f6] f a/fa/ gfksf kl] G;nx¿_ -tLgcf6] } km/skm/s gfksf kl] G;nx¿_

ca tLgcf]6} ;6] af6 km/s km/s lqe'hx¿ agfp“b} hfpm .

;]6 - 1 af6 ag]sf] lqe'h ;]6 - 2 af6 ag]sf] lqeh' ;]6 - 3 af6 ag]sf] lqe'h
-;a} eh' fx¿ a/fa/ ePsf_] -b'O{cf6] f eh' fx¿ a/fa/ ePsf_] -sg' } klg eh' fx¿ a/fa/ gePsf_

o;/L eh' fsf] gfksf cfwf/df lgDglnlvt tLg lsl;dsf lqeh' agfpg ;lsG5 M

(1) tLgcf]6} e'hf a/fa/ ePsf] lqeh' nfO{ ;dafx' lqe'h elgG5 .
(2) b'O{ cf6] f e'hf a/fa/ ePsf] lqe'hnfO{ ;dlåafx' lqe'h elgG5 .
(3) s'g} klg e'hf a/fa/ gePsf] lqe'hnfO{ lj;dafx' lqe'h elgG5 .

d/] f] ul0ft M sIff % 9

-v_ sf]0fx¿sf cfwf/df lqeh' sf] juLs{ /0f

ljm| ofsnfk 1

lrqdf b]vfOPsf] ;]6:Sjfo/ x/] . ;6] :Sjfo/sf] sg' rflx“
sf0] f ;dsf]0f 5 < ;]6 :Sjfo/sf] Pp6f sf0] f ;dsf]0f
ePsfn] o;sf] ko| fu] af6 sg' } sf0] f ;dsf0] feGbf ;fgf] cyjf
7"nf] cyjf a/fa/ s] 5, yfxf kfpg ;lsG5 . tnsf]
lrqdf x]/ M

X

AP

B CQ RY Z

1. ∠ABC sf bj' } eh' fx¿ ;6] :Sjfo/leq k/s] f 5g\ . To;n} ] ∠ABC ;dsf0] feGbf
;fgf] ePsfn] of] Gog" sf0] f xf] .
2. ∠PQR sf] eh' f PQ ;6] :Sjfo/ aflx/ k/s] fn] sf0] f PQR ;dsf0] feGbf 7n" f] xg' fn]
of] clwssf]0f xf] .
3. ∠XYZ sf] bj' } eh' f ;6] :Sjfo/df l7Ss ldns] fn] ∠XYZ ;dsf0] f xf] . ca ;6] :Sjfo/
ko| fu] u/L tnsf kT| os] lqeh' sf leqL sf0] fx¿ Gog" sf0] f, clwssf0] f cyjf ;dsf0] f
s] x'g\, kQf nufpm .

AP

X

B C Q R Y Z
lqeh' - 3
lqeh' - 1 lqeh' - 2

10 d/] f] ul0ft M sIff %

(1) s'g} lqe'hsf tLgcf]6} sf0] fx¿ ;dsf0] feGbf ;fgf 5g\ eg] To:tf] lqe'hnfO{
Go"gsf0] fL lqe'h elgG5 . dflysf] ABC Go"gsf]0fL lqe'h xf] .

(2) sg' } lqe'hsf tLgcf]6f sf]0fdWo] Pp6f sf0] f ;dsf0] f 5 eg] To:tf] lqeh' nfO{
;dsf]0fL lqe'h elgG5 . dflysf] PQR ;dsf0] fL lqe'h xf] .

(3) s'g} lqeh' sf tLgcf6] f sf0] fdWo] Pp6f sf]0f ;dsf]0feGbf 7"nf] 5 eg] To:tf]
lqeh' nfO{ clwssf0] fL lqeh' elgG5 . dflysf] XYZ clwssf0] fL lqeh' xf] .

cEof; 1.2

(1) tn lbOPsf kT| os] lqe'hsf eh' fx¿ gfk / sg' s'g lqeh' x¿ ;dafx,' ;dlåafx'

/ lj;dafx' x'g,\ 56' o\ fpm M

-s_ A -v_ E G

-u_

B D H I
C F P
-3_ J
M

-ª_ -r_

K LN OQ R

(2) tn lbOPsf kT| o]s lqe'hx¿ Gog" sf]0fL, clwssf0] fL cyjf ;dsf0] fL s] x'g,\
56' \ofpm M

-s_ -v_ M -u_ P
J

K LN OQ R
11
d]/f] ul0ft M sIff %

-3_ -ª_ -r_
S
VX
X Y

T UW
Z

(3) kZ| g g=+ 2 sf lqeh' x¿dWo] sg' sg' lqeh' x¿ ;dlåafx,' ;dafx' / lj;dafx'
xg' \ <

(4) tnsf] lrqdf hDdf sltcf6] f lqeh' x¿ 5g\ < tLdWo] ;dafx',;dlåafx' cyjf
lj;dafx' ss] ] x'g,\ nv] .

A

B DC

(5) kZ| g g+= 4 sf lqe'hx¿dWo] Gog" sf]0fL, ;dsf0] fL cyjf clwssf]0fL ss] ] xg' \,
nv] .

lzIf0f lgb{z] g M dflysf] cEof;df lbOPh:t} yk ;d:ofx¿ agfO{ ljBfyLn{ fO{ yk cEof; ug{
nufpgx' f;] \ .

12 d/] f] ul0ft M sIff %

1.3 lqe'hsf sf0] fx¿sf] gfk A

Pp6f lqeh' ABC lvr . ltdLn] lvr]sf] lqeh' BC
ltdf| ] glhssf ;fyLn] lvr]sf] lqeh' ;u“ tn' gf ∠BAC + ∠ ABC + ∠ACB
u/ . s] ltdLn] lvrs] f] lqe'h / ;fyLn] lvrs] f
lqeh' p:t} cfsf/sf 5g\ < ltdLn] lvrs] f] lqeh' sf
tLgcf6] } sf0] fx¿ gfk / tnsf] tflnsf sfkLdf
;f//] e/ M

sf0] f ∠BAC ∠ABC ∠ACB

gfk

ca sf0] fx¿sf] gfksf] of]ukmn slt 5 < ltdLn] lgsfn]sf] of]ukmn glhssf] ;fyLnfO{
b]vfpm . s] of]ukmn Pp6} 5 < c¿ ;fyL;u“ klg ;f]wL x]/, o;af6 s] lgisif{ lgsfNg
;lsG5 <

lgisif{ M lqe'hsf sf]0fx¿sf] ofu] kmn 1800 xG' 5 .

ljm| ofsnfk 1 A

Pp6f 7n" f] ;fOhsf] -kfgf el/g] u/L_ lqeh' lvr . B C
lqeh' sf] zLifl{ aGbs' f] gfd lrqdf bv] fOPem“} u/L nv] .
ca lqeh' sf] jl/kl/ 3/] fdf sr“} Ln] sf6/] lqeh' nfO{
lgsfn . ca B / C n]v]sf] zLifl{ aGbx' ¿nfO{ BC
df ldNg] u/L k6\ofpm .

ca laGb'x¿ B / C ldns] f] 7fpd“ f l7Ss ldNg] A
u/L zLif{laGb' A nl] vPsf] sg' fnfO{ klg k6\ofP/
ldnfpm . A
BC
d/] f] ul0ft M sIff %
13

ca, tLgcf6] } zLif{laGbx' ¿ A, B / C Pp6} laGb' A
O df ldn]sf 5g\ . oxf“ sf0] fx¿ A, B / C ldn/] BC
Pp6f ;/n sf]0f agfPsf 5g\ . t;y{ ∠A + ∠B
+ ∠C = 1800 eof] . O

o:t} km/skm/s ;fOhsf lqe'hx¿ k6o\ fP/ dflysf] k|of]u bf]xf¥] ofO{ x/] . s] ;a
lqeh' nfO{ o;/L g} k6o\ fp“bf tLgcf]6} sf]0fx¿sf] of]u 180° g} x'G5 <

cEof; 1.3

(1) tn lbOPsf kT| o]s lqeh' sf ;a} sf0] fx¿ gfk / ltgLx¿sf] of]ukmn lgsfn M

-s_ A -v_ D

B CE F
J R
L -3_
-u_

Q

KP

(2) tnsf k|To]s lqe'hsf af“sL sf]0fx¿sf] gfk kQf nufpm . -kf| ]6o\ fS6/ k|of]u

gu/Lsg_ -v_ D -u_ G 40° H
-s_ A
20°

I

B CE F

lzIf0f lgb]z{ g M ljBfyLx{ ¿nfO{ km/s km/s lqe'hx¿ lvRg nufO{ tL lqeh' x¿sf sf]0fx¿sf]
ofu] kmn lgsfNg nufpgx' f;] \ . ;fy} dfly k|Zg 2 df lbOPh:t} yk ;d:of agfO{ ;dfwfg lgsfNg]
cEof; ug{ nufpg'xf];\ .

14 d/] f] ul0ft M sIff %

1.4 rte' 'h{ sf sf0] fx¿sf] gfk

ljm| ofsnfk 1 D

Pp6f rt'e'{h ABCD lvr . ltdLn] lvrs] f] A
rt'e'{h ltd|f] glhssf ;fyLnfO{ b]vfpm . s]
ltdLn] lvr]sf] rteh{' / ltd|f ;fyLn] lvrs] f] C
rt'e'{h p:t} cfsf/sf 5g\ <

ca ltdLn] lvr]sf] rte' {'hsf kT| os] sf0] fx¿
gfk / tnsf] tflnsf sfkLdf ;f//] e/ M

B

sf]0f ∠ABC ∠BCD ∠CDA ∠DAB ∠A + ∠B + ∠C + ∠D

gfk
ltdLn] lgsfn]sf sf0] fx¿sf] gfksf] of]ukmn slt 5 < ltdLn] lgsfn]sf] ofu] kmn
glhssf] ;fyLnfO{ bv] fpm . s] of]ukmn Pp6} 5 < c¿ ;fyLx¿;“u klg ;f]wL x/] , s]
;a}n] lgsfn]sf] ofu] kmn 3600 g} 5 < o; ljm| ofsnfkaf6 s] lgisif{ lgsfNg ;lsG5 <

lgisif{ M rt'eh'{ sf sf]0fx¿sf] ofu] kmn 3600 xG' 5 .

lj|mofsnfk 2

P, zLnf ¤ rte' h{' sf sf0] fx¿sf] of] xf] t ¤ o;sf] dtna, rte' '{hnfO{
ukmn t lqeh' sf sf0] fx¿sf] of]uk- 2 cf6] f lqe'hdf ljefhg ug{
mnsf] 7Ls 2 ug' f x'b“ f] /x5] , xlu < ;Sg'k¥of], xg} / <

d]/f] ul0ft M sIff % 15

Pp6f rte' h{' ABCD lvr . B / A A
D nfO{ ?n/ / k]lG;nsf] ;xfotfn] C

hf8] . ca, rte' h{' ABCD sltcf6] f D

lqeh' df ljefhg eof,] x/] . ca, B DB

lqe'h ABD sf sf]0fsf] of]ukmn

1800 5, lqeh' BCD sf] sf0] fsf] of] C

ukmn klg 1800 x'G5 . b'j} lqeh' sf

sf]0fx¿ ldnfpb“ f rt'eh'{ ABCD sf] sf]0fsf] ofu] kmn 3600 xG' 5 .

km] l/, klxnf s} rt'e'{h ABCD sf laGbx' ¿ A / C nfO{ hf8] . rt'e'h{ sltcf]6f
lqeh' df ljefhg x'G5 < s] oxf“ klg rt'e'h{ ABCD sf] sf0] fx¿sf] ofu] kmn a/fa/
b'Oc{ f]6f lqe'hx¿ ABC / ADC sf] sf]0fx¿sf] ofu] kmnsf] hf8] ;“u a/fa/ xG' 5 <

oxf“ /v] fx¿ BD / AC nfO{ rt'eh'{ sf ljs0f{ elgG5 . Pp6f rt'eh{' df slt cf]6f
ljs0f{ xG' 5g\, 5nkmn u/L kQf nufpm .

ljm| ofsnfk 3 D C
sfkLsf] kfgf el/g] u/L Pp6f 7"nf] ju{ ABCD

lvr . ca s}“rLsf] ;xfotfn] ju{sf] jl/kl/sf] O
3]/fdf sf6/] jun{ fO{ cnUofpm / lrqdf b]vfOPh:t}

u/L rf/cf6] } sf]0fsf zLifl{ aGb'x¿ jus{ f] leqL Pp6} A B
laGb' 0 df ldNg] u/L k6o\ fpm . o;/L k6\ofpb“ f rf/
cf6] } sf]0fx¿ A, B, C / D Pp6} k"0f{ sf]0f ags] f] 5 .
xfdLnfO{ yfxf 5 – Pp6f k"0f{sf]0fsf] gfk 3600
xf] . To;n} ] ju{ ABCD sf sf0] fx¿ A, B, C / D
sf] ofu] kmn 3600 eof] .

∠A+∠B+∠C+∠D = 3600

dflysf ljm| ofsnfkx¿af6 cGo rte' {'hsf sf]0fx¿sf] of]ukmn 3600 xG' 5 eGg] lgisif{
lgsfNg ;lsG5 <

16 d/] f] ul0ft M sIff %

cEof; 1.4

(1) tn lbOPsf k|Tos] rte' {'hsf sf0] fx¿ gfk / ltgLx¿sf] ofu] kmn lgsfn M

A DE HP S

B FG R

-s_ C -v_ Q -u_

(2) tnsf kT| os] rt'e{h' ABCD df A / C hf8] / bj' } lqe'hx¿sf sf0] fx¿ gfk/]
ofu] kmn lgsfn . s] bO' { lqe'hsf sf]0fsf] of]ukmn 3600 5 <

A DA DA D

B CB CB C

-s_ -v_ -u_

(3) rt'e'{hsf ;a} sf]0fsf] of]ukmn 3600 x'G5, tn lbOPsf k|To]s rt'e'{hsf]
yfxf gePsf] sf]0f kQf nufpm M

AD E H I 600 L

1200 1000 1100 1200

B 700 F 700 600 J 600

-s_ C K

G -u_

-v_

lzIf0f lgbz{] g M ljBfyLx{ ¿nfO{ yk cEof;sf nflu o:t} ;d:ofx¿ agfO{ ;dfwfg ug{ nufpgx' f;] \ .

d/] f] ul0ft M sIff % 17

@kf7 ;ªV\ ofsf] 1fg
(Concept of Numbers)

2.1 b; s/f8] ;Ddsf ;ª\Vofsf] uGtL / :yfgdfg

lxGb–' c/]las cª\sx¿ b;cf6] f 5g\, tL 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 x'g\ .

oL cª\sx¿sf] k|ofu] u/L Pp6f ;ªV\ of 36582292 agfcf“} . To;nfO{ :yfgdfg
tflnsfdf /fvf}“ / 5 sf] :yfgdfg slt xG' 5 x/] f}“ .

s/f]8 b; nfv nfv b; xhf/ xhf/ ;o b; Ps

365 8 22 9 2

nfvsf] :yfgdf cª\s 5 5 . To;n} ] 5 sf] :yfgdfg 5 nfv = 5,00,000 x'G5 .
:yfgdfg tflnsfdf cª\s 3 s/f8] sf] :yfgdf 5 . To;}n] 3 sf] :yfgdfg 3 s/f8]
= 3,00,00,000 x'G5 .

3,65,82,292 nfO{ ltg s/f8] k;“} 67\ L nfv aof;L xhf/ bO' { ;o aofgAa] egL kl9G5 .

tnsf] tflnsfdf 10 s/f8] ;Ddsf uGtL ;ª\Vof / ltgLx¿nfO{ k9\g] tl/sf jf
;fªl\ Vos gfd lbOPsf] 5 . tflnsf /fdf| ;] u“ x/] / s] yfxf kfof,“} 5nkmn u/ M

slt cª\sn] ag]sf] < ;ªV\ of ;fªl\ Vos gfd

Ps cªs\ n] ags] f] ;aeGbf ;fgf] ;ªV\ of 1 Ps (One)

bO' { cªs\ n] ags] f] ;aeGbf ;fgf] ;ª\Vof 10 b; (Ten)

tLg cª\sn] ag]sf] ;aeGbf ;fgf] ;ª\Vof 100 ;o (Hundred)

rf/ cªs\ n] ag]sf] ;aeGbf ;fgf] ;ª\Vof 1000 xhf/ (Thousand)

kf“r cª\sn] ags] f] ;aeGbf ;fgf] ;ªV\ of 10000 b; xhf/ (Ten thousand)

5 cª\sn] ags] f] ;aeGbf ;fgf] ;ªV\ of 100000 nfv (Lakh)

;ft cªs\ n] ags] f] ;aeGbf ;fgf] ;ªV\ of 1000000 b; nfv (Ten Lakh)

cf7 cª\sn] ags] f] ;aeGbf ;fgf] ;ª\Vof 10000000 s/f]8 (Crore)

gf} cª\sn] ags] f] ;aeGbf ;fgf] ;ª\Vof 100000000 b; s/f8] (Ten Crore)

18 d]/f] ul0ft M sIff %

;ªV\ ofnfO{ cIf/df nV] g] tl/sf

;ªV\ of 106739862 nfO{ :yfgdfg tflnsfdf o;/L n]Vg ;lsG5 M

:yfgdfg

;ª\Vof s/f]8 nfv xhf/ PsfOx¿

b; s/f8] s/f]8 b; nfv nfv b; xhf/ xhf/ ;o b; Ps

106739862 10 6 73 986 2

oxL :yfgdfg hgfpg] u/L ;ªV\ ofnfO{ k9\g ;lhnf] xf;] \ egL cNklj/fd -,_ sf] k|ofu]
ul/G5 . 106739862 nfO{ cNklj/fd k|of]u u/L nV] bf 10,67,39,862 u//]
nl] vG5 . k9b\ f b; s/f8] ;t;\ 67\ L nfv pgfGrfnL; xhf/ cf7 ;o a;} 67\ L egL kl9G5 .

pbfx/0f 1

cIf/df nv] M 19,56,02,183

19,56,02,183 = pGgfO; s/f]8 5kGg nfv b'O{ xhf/ Ps ;o lqof;L
cIf/df lbOPsf ;ª\VofnfO{ cª\sdf n]Vg] tl/sf
b; s/f]8 ;f]x| nfv cf7 xhf/ gf} ;o 5QL;nfO{ cªs\ df nV] bf M
10 s/f8] 16 nfv 8 xhf/ 9 ;o 36 = 10,16,08,936 x'G5 .
To:t} b; s/f8] aof;L nfv kGw| xhf/ kGw|nfO{ cªs\ df n]Vbf M
10,82,15,015 xG' 5 .

pbfx/0f 2

cªs\ df nv] M c7f/ s/f]8 P3f/ nfv rf/ xhf/ ;f7L

c7f/ s/f]8 P3f/ nfv rf/ xhf/ ;f7L = 18,11, 04,060

d/] f] ul0ft M sIff % 19

cEof; 2.1

(1) tnsf kT| o]s ;ªV\ ofdf 2 sf] :yfgdfg slt 5, nv] M

-s_ 13524 -v_ 1235497

-u_ 2305343 -3_ 4263578

(2) tnsf k|Tos] ;ª\Vofdf sltcf6] f nfvx¿ 5g\ <

-s_ 1234567 -v_ 50031247 -u_ 67853479

(3) Ps s/f8] df sltcf6] f xhf/x¿ x'G5g\ <

(4) tnsf k|To]s ;ª\Vofdf cNklj/fd -,_ /fv M

-s_ 1350739 -v_ 2476893 -u_ 275403016

-3_ 75003052 -ª_ 105300274 -r_ 592070593

(5) k|Zg g+= 4 sf kT| os] ;ª\VofnfO{ cIf/df n]v .

(6) cª\sdf n]v M

-s_ Ps s/f]8 aof;L nfv kGw|

-v_ ;q s/f8] krxQ/ nfv tLg xhf/ 5 ;o kfr“

-u_ afO; s/f]8 k};“ 67\ L xhf/ ;ft ;o ;Q/L

-3_ 5QL; s/f8] krf; nfv 5 ;o gAa]

-ª_ pgfG;o s/f]8 kfr“ ;o kRrL;

20 d]/f] ul0ft M sIff %

2.2 ;ªV\ ofnfO{ cGt/f{li6«o k4ltdf nV] g] tl/sf M

;ªV\ of 57363542 nfO{ cNklj/fd /fVg] gk] fnL k4lteGbf cGt/f{li6«o k4lt km/s 5 .
cGt/f{li6«o k4ltdf bfof“af6 x/]s tLg cªs\ df cNklj/fd /fVb} hfgk' 5{ h;cg;' f/
dflysf] ;ªV\ ofnfO{ 57,363,542 n]lvG5 .

o;nfO{ :yfgdfg tflnsfdf /fVbf M

Number Million Thousand Unit
57363542 Hundred Ten
One Hundred Ten One Hundred Ten One
—5 2
7 3 63 5 4

k9\bf, 57 million, 363 thousand and 542

jf Fifty-seven million, three hundred sixty-three thousand five hundred and forty-
two xG' 5 .

pbfx/0f 1

cIf/df nv] M 123,430,316

123,430,316 = 123 million, 430 thousand and 316
= One hundred twenty-three million four hundred thirty thousand three hun
dred and sixteen.

cGt/fl{ i6«o k4ltdf cIf/nfO{ cªs\ df nV] g] tl/sf

Sixty million three hundred forty-two thousand and eighty-two
= 60 million, 342 thousand and 082 = 60,342,082

pbfx/0f 2

cªs\ df nv] M

One hundred seven million two hundred ninety-three thousand six hundred and fifty.
= 107,293,650

d]/f] ul0ft M sIff % 21

cEof; 2.2

1. tnsf] kT| o]s ;ªV\ ofdf cGt/f{li6«o k4ltcg;' f/ cNklj/fd -,_ /fv M

-s_ 35768229 -v_ 38962352

-u_ 158632932 -3_ 628293563

2. cGt/fl{ i6«o k4ltcg';f/ :yfgdfg tflnsfdf /fvL cIf/df n]v M

-s_ 32,567,832 -v_ 178,625,123

-u_ 595,207,257 -3_ 185,090,159

3. cªs\ df nv] M

-s_ Fifty-seven million nine hundred twenty-six thousand one hundred
and thirty-three.

-v_ One hundred twenty-four million sixty-six thousand and three hun

dred four.

-u_ Two hundred fifty-seven million, four hundred twenty thousand and

seventy-four.

-3_ Seven hundred thirty-four million, thirty-seven thousand two hun

dred and sixty-three.

lzIf0f lgb{z] g M dflysf cEof;df lbPsf h:tf k|Zgx¿ ljBfyLx{ ¿af6} agfpg
nufP/ Pp6f ;dx" n] csf]{ ;d"xnfO{ ;fW] g nufO{ xflh/L hafkm k|ltof]lutf h:t}
v]nx¿ v]nfpg'xf];\ .

22 d/] f] ul0ft M sIff %

'u0fg v08x¿2.3 100 ;Ddsf ¿9 (Prime) / ;+o'St (Composite) ;ªV\ of

tnsf] lrqdf 1-20 ;Ddsf ¿9 / ;+o'St ;ªV\ of bv] fOPsf] 5 . lrqdf ko| f]u ePsf
yf]Knfx¿n] Tof] laGb'sf] ;Lwf tn /xs] f] ;ªV\ ofsf] u0' fg v08 atfp“5 . lrqdf x/] /]
1 bl] v 20 ;Ddsf ;ª\Vofx¿df sg' ;ª\Vofsf sltcf]6f u'0fg v08x¿ /x]5g\, ;"rL
tof/ kf/ .

20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2

1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

;ªV\ ofx¿

ltdLn] tof/ kf/]sf] ;r" L x]/ / tnsf kZ| gsf] pQ/ bp] m M
-s_ ToxL ;ª\Vof / 1 dfq u'0fg v08 ePsf ;ªV\ ofx¿ sg' s'g x'g\ <
-v_ lrqdf ufn] f] 3/] f nufOPsf ;ªV\ ofx¿ af/d] f s] eGg ;S5f} <

d]/f] ul0ft M sIff % 23

-u_ ToxL ;ªV\ of / 1 afxs] c¿ ;ª\Vof klg u0' fg v08 ePsf ;ª\Vofx¿ 1–20
;Dd s'gs'g /x5] g,\ 5nkmn u/ .

ToxL ;ª\Vof / 1 dfq u0' fg v08 ePsf ;ªV\ ofx¿nfO{ ¿9 ;ª\Vof elgG5 . ToxL
;ª\Vof / 1 afxs] c¿ klg u'0fg v08 xg' ] ;ªV\ ofnfO{ ;o+ S' t ;ª\Vof elgG5 cyjf
¿9 ;ª\VofnfO{ 1 / ToxL ;ªV\ ofafx]s c¿n] lgMzi] f efu hfb“ g} . 1-20 ;Ddsf
o:tf ¿9 ;ªV\ ofx¿ 2, 3, 5, 7,11, 13, 17, 19 u/L hDdf 8 cf6] f 5g\ . 1 g t ¿9 g
t ;o+ S' t xf] eg] 2 Pp6f dfq hf/] ¿9;ªV\ of xf] . c¿ ;a} ¿9 ;ªV\ of lahf/] xG' 5g\ .

cEof; 2.3

(1) -s_ 10 x 10 sf7] f ePsf] jufª{ l\ st sfuhdf 1-100 ;Ddsf ;a} ;ªV\ ofx¿ nv] .
-v_ ;ªV\ of 1 nfO{ sf6 .
-u_ ;ªV\ of 2 nfO{ 5f8]/ 2 n] efu hfg] ;a} ;ª\VofnfO{ sf6 .
-3_ ;ªV\ of 3 nfO{ 5f8]/ 3 n] efu hfg] ;a} ;ªV\ ofnfO{ sf6 .
-ª_ ;ª\Vof 5 5f8]/ 5 n] efu hfg] ;a} ;ªV\ ofnfO{ sf6 .
-r_ ;ªV\ of 7 nfO{ 5f8]/ 7 n] efu hfg] ;a} ;ª\VofnfO{ sf6 .
-5_ o;/L sf6]kl5 af“sL /xs] f ;a} ;ª\Vof ¿9 jf ;+oS' t s] x'g\ <
-h_ 1-100 ;Dd hDdf slt cf]6f ¿9 ;ªV\ of 5g\ <
-´_ 50 eGbf 7"nf t/ 60 eGbf ;fgf ¿9 ;ªV\ of slt cf6] f 5g\ <

gf6] M k=| g=+ 1 sf] tl/sfn] ¿9 ;ªV\ of kQf nufpg] ljlwnfO{ O/f6fl] :ygLsf] rfNgf]
elgG5 . of] ljlw ;aeGbf klxnf O/f6f]l:ygLn] kQf nufPsfn] o;f] elgPsf] xf] .

(2) tn lbOPsf ;ª\Vofx¿ sg' ¿9 x'g\ / s'g xf]Ogg,\ kQf nufpm M
-s_ 2 -v_ 5 -u_ 6 -3_ 7 -ª_ 13
-r_ 23 -5_ 33 -h_ 41 -´_ 53 -`_ 72
-6_ 75 -7_ 79 -8_ 81 -9_ 83 -0f_ 99
(3) 75 b]lv 85 ;Ddsf ¿9 ;ªV\ ofx¿ nv] .
(4) 11 x 11 = 121, 11 ¿9 ;ª\Vof xf] . 121 s] xf] lg <

24 d/] f] ul0ft M sIff %

2.4 ;ªV\ ofx¿sf] zG" ofGt (Rounding off the Numbers)

x]/ / 5nkmn u/ M 6f9f

glhs

800 810 820 830 840 850 860 870 880 890 900

;ª\Vof /v] fdf 830 ;ªV\ of 800 sf] glhs 5 t/ 900 af6 w]/} 6f9f k/]sf] 5 .
To;}n] 830 nfO{ glhssf] ;odf zG" ofGt ubf{ 800 x'G5 .

900 910 920 930 940 950 960 970 980 990 1000

;ªV\ of /v] fdf 980 ;ª\Vof 1000 sf] glhs 5 t/ 900 af6 w/] } 6f9f k/]sf] 5 .
To;n} ] 980 nfO{ glhssf] ;odf z"GofGt ubf{ 1000 x'G5 . To:t} 950 ;ª\Vof 900 /
1000 af6 a/fa/ b/' Ldf kg]{ xb“' f glhssf] ;odf z"GofGt ubf{ 1000 xG' 5 .

pbfx/0f 1
1200 nfO{ glhssf] xhf/df zG" ofGt u/ M

1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000

1200, 1000 sf] glhs k5{ . To;n} ] 1200 nfO{ glhssf] xhf/df zG" ofGt ubf{ 1000 xG' 5 .

pbfx/0f 2
3657 nfO{ glhssf] xhf/df zG" ofGt u/ M

3657

3000 3100 3200 3300 3400 3500 3600 3700 3800 3900 4000

3657, 4000 sf] glhsdf k5{ . To;n} ] 3657 nfO{ glhssf] xhf/df z"GofGt ubf{
4000 xG' 5 .

d/] f] ul0ft M sIff % 25

pbfx/0f 3
36500 nfO{ glhssf] xhf/df z"GofGt u/ M

36000 36100 36200 36300 36400 36500 36600 36700 36800 36900 37000

36500, 36000 / 37000 sf] 7Ls aLrdf k5{ . 36500 nfO{ glhssf] xhf/df
z"GofGt ubf{ 37000 xG' 5 .

lzIf0f lgbz{] g M ;ª\Vof /]vfsf] k|ofu] u/L glhssf] ;o / xhf/df z"GofGt ug]{
tl/sfaf/] 5nkmn u/fpg'xf;] \ .

cEof; 2.4

1. lbOPsf] ;ªV\ ofnfO{ glhssf] ;odf z"GofGt u/ M

-s_ 2684 2684

2000 2100 2200 2300 2400 2500 2600 2700 2800 2900 3000

-v_ 3824 -u_ 5613 -3_ 6123 -ª_ 12345

2. lbOPsf] ;ªV\ ofnfO{ glhssf] xhf/df z"GofGt u/ M

-s_ 5692 5692

5000 5100 5200 5300 5400 5500 5600 5700 5800 5900 6000

-v_ 24287 24287

24000 24100 24200 24300 24400 24500 24600 24700 24800 24900 25000

-u_ 24500

24000 24100 24200 24300 24400 24500 24600 24700 24800 24900 25000

-3_ 1697 -ª_ 15287 -r_ 58999 -5_ 93902

26 d/] f] ul0ft M sIff %

2.5 ju{ / 3g ;ªV\ ofx¿
2.5.1 ju{ ;ª\Vofx¿

tnsf] tflnsf cWoog u/L 5nkmn u/ M

x 1 2 3 4 5 6 7 8 9 10

1 1 2 3 4 5 6 7 8 9 10

2 2 4 6 8 10 12 14 16 18 20

3 3 6 9 12 15 18 21 24 27 30

44 8 12 16 20 24 28 32 36 40

5 5 10 15 20 25 30 35 40 45 50

6 6 12 18 24 30 36 42 48 54 60

7 7 14 21 28 35 42 49 56 63 70

8 8 16 24 32 40 48 56 64 72 80

9 9 18 27 36 45 54 63 72 81 90

10 10 20 30 40 50 60 70 80 90 100

tflnsfdf ufn] f] 3]/f nufOPsf ;ª\Vofx¿af/] s]xL eGg ;S5f} <

1 x 1 = 1
2x2= 4
3 x 3 = 9

s] ufn] f] 3/] f nufOPsf kT| os] ;ªV\ ofx¿ bO' c{ f6] f p:tp} :t} ;ªV\ ofx¿sf u0' fgkmn xg' \ <

s] dflysf] tflnsfdf ufn] f] 3]/f nufOPsf ;a} ;ªV\ ofnfO{ o;/L g} JoSt ug{ ;lsG5
xf]nf < u'0fg u/L x/] .

d/] f] ul0ft M sIff % 27

ca 3/] f nufOPsf tL ;ªV\ ofnfO{ yfK] nfx¿sf] Pp6f 9fr“ f agfP/ x/] f“} t, s] aGbf] /x5] <

1 4 9 16 25 36 49

oxf“ k|Tos] yf]Knfsf] 9fr“ fn] jufs{ f/ cfs[lt agfPsf 5g\ . o;/L ;ª\VofnfO{
laGbn' ] hgfp“bf jufs{ f/ cfsl[ t agfpg] ;a} ;ªV\ ofnfO{ ju{;ª\Vof elgG5 .

o:tf ju{;ªV\ ofx¿ s'g} ;ªV\ ofnfO{ ToxL ;ªV\ ofn] u0' fg u//] kQf nufpg ;lsG5 .
tnsf] tflnsf x/] M

;ªV\ of u0' fg -;ª\Vof x ;ª\Vof_ ju{ ;ªV\ of

1 1x1 1
2 2x2 4
3 3x3 9
4 4x4 16
5 5x5 25

o;/L s'g} Ps ;ªV\ ofnfO{ ToxL ;ª\Vofn] u'0ff u/]/ cfpg] u'0fgkmn g} Tof] ;ªV\ ofsf]
ju;{ ªV\ of x'G5 . t;y{ lbOPsf] ;ªV\ of ju;{ ªV\ of xf] jf xfO] g eGg] yfxf kfpg Tof]
;ª\VofnfO{ b'O{ cf]6f pxL ;ªV\ ofsf] u0' fgkmndf JoSt ug{ ;lsG5 jf ;ls“bg} eg/]

x]g'k{ 5{ . h:t} M 64 = 8 x 8 x'g] ePsfn] 64 ju{;ªV\ of xf] t/ 15 = 3 x 5 xg' ] ePsfn]
15 ju{;ª\Vof xf]Og .

oxf“, ;ªV\ of 64 sf b'O{ cf]6f p:t} p:t} u'0fgv08x¿ 8 x 8 = 64 df 8 nfO{ 64 sf]
jud{ n" / 64 nfO{ 8 sf] ju{;ª\Vof elgG5 .

28 d/] f] ul0ft M sIff %

ju;{ ª\Vof / ju{d"n ae' mfpg] tnsf] Çnfr] f6{ x]/ M

ju;{ ªV\ of 1 4 9 16 25 ....... 100

u0' fgv08 1 x 1 2 x 2 3 x 3 4 x 4 5 x 5 ....... 10 x 10

ju{d"n 1 2 3 4 5 ....... 10

of] Çnf]rf6{df tnaf6 dfly x]bf{ lbOPsf] ;ªV\ ofsf] ju{;ªV\ of cfp“5 . 3 sf] ju{ 9 xf] .
dflyaf6 tn xb] f{ lbOPsf] ;ªV\ ofsf] ju{dn" cfp“5 . 16 sf] jud{ "n dflyaf6 tn x]bf{ 4
xG' 5 . To:t} 100 sf] ju{d"n 10 xf] .

pbfx/0f 1

;ªV\ of 9 sf] ju{;ª\Vof lgsfn M

oxf“,
lbOPsf] ;ªV\ of = 9
To;}n] 9 sf] ju;{ ª\Vof = 9 x 9 = 81

pbfx/0f 2

r;] af]8{ juf{sf/ xG' 5 / To;sf 64 sf7] f x'G5g\ eg] kT| os] 7f8f] kªl\ St jf t];f{] nx/df
slt slt sf7] f xG' 5g\ xf]nf <

oxf,“
juf{sf/ r];af8] d{ f sf]7fsf] ;ª\Vof = 64

64 = 8 x 8

To;n} ] k|Tos] 7f8f] kªl\ St jf t;] f{] nx/df sf]7fsf] ;ª\Vof = 8

pbfx/0f 3 29
;ª\Vof 25 sf] jud{ n" lgsfn .
oxf,“ 25 = 5 x 5
To;n} ] 25 sf] jud{ n" = 5 xf] .

d/] f] ul0ft M sIff %

cEof; 2.5.1

(1) tn lbOPsf k|To]s ;ªV\ ofsf] ju;{ ªV\ of lgsfn M
-s_ 2 -v_ 5 -u_ 6 -3_ 7 -ª_ 8 -r_ 9
-5_ 10 -h_ 11 -em_ 12 -`_ 13 -6_ 14 -7_ 15
(2) lbOPsf ;ªV\ of ju{;ª\Vof sg' sg' xg' \, kQf nufpm M
-s_ 16 -v_ 25 -u_ 42 -3_ 49 -ª_ 60
-r_ 81 -5_ 100 -h_ 120 -em_ 121 -`_ 225
(3) tnsf kT| o]s ju{;ªV\ ofsf] jud{ "n kQf nufpm M
-s_ 36 -v_ 49 -u_ 64 -3_ 81 -ª_ 100
-r_ 121 -5_ 144 -h_ 169 -em_ 196 -`_ 225
(4) Ps nfOgdf 20 cf]6f la?jf kg]{ u/L jufs{ f/ vt] df sfpnLsf ag] f{ ;fg'k{ bf{
slt a]gf{ rflxG5g\ <
(5) Pp6f sIffdf 36 hgf ljBfyL{nfO{ jufs{ f/ ¿kdf /flvPsf] /x]5 eg] k|To]s
nfOgdf slt ljBfyL{ /flvPsf /x]5g\ .
(6) 144 hgf l;kfxLnfO{ jufs{ f/ ¿kdf kªl\ Sta4 u/L sjfh v]nfpb“ f 1
kªl\ Stdf slt l;kfxL /fVgk' 5{ <
(7) Ps nfOgdf 45 hgf ljBfyL{ /fv/] juf{sf/ ¿kdf ldnfp“bf 5 hgf ljBfyL{
a9L xg' cfp“5g\ eg] hDdf ljBfyL{ slt /x5] g\ <
(8) 1-100 ;Dddf sltcf6] f ju;{ ªV\ of x'G5g\ <

30 d]/f] ul0ft M sIff %

2.5.2 3g ;ªV\ ofx¿ (Cubic Numbers) 2 cm

Pp6f 8fO;sf] nDafO, rf8} fO / prfO k|Tos] 2/2 2 cm 2 cm
cm sf 5g\ . of] Pp6f 3gfsf/ j:t' xf] . o;sf]
nDafO, rf}8fO / prfO hgfpg] ;ªV\ of u'0fg
ubf{ 2 x 2 x 2 = 8 cfp5“ . oxf“ 8 sf p:t}p:t}
tLgcf6] f u0' fgv08x¿ 5g\ . o:tf] ;ªV\ ofnfO{
3g ;ª\Vof elgG5 . tflnsfdf 1 b]lv 5 ;Ddsf
;ª\Vofx¿sf] 3g ;ª\Vof bv] fOPsf] 5 .

;ª\Vof ltg k6s u'0fg ubf{ cfpg] u0' fgkmn 3g ;ªV\ of

1 1x1x1 1

2 2x2x2 8

3 3x3x3 27

4 4x4x4 65

5 5x5x5 125

t;y{, lbOPsf] ;ª\Vofsf] 3g ;ª\Vof lgsfNbf Tof] ;ªV\ ofn] ToxL c¿ b'O{ ;ªV\ ofnfO{
u0' fg ug'{k5{ .

pbfx/0f 1 pbfx/0f 2

6 sf] 3g ;ª\Vof lgsfn M 64 sf] 3gdn" lgsfn M
oxf,“ 6 sf] 3g ;ªV\ of oxf“, 64 = 4 x 4 x 4
To;}n] 64 sf] 3gdn" 4 x'G5 .
=6x6x6

= 36 X 6 = 216

cEof; 2.5.2 -ª_ 10

1. tnsf k|Tos] ;ªV\ ofsf] 3g ;ª\Vof lgsfn M
-s_ 2 -v_ 5 -u_ 7 -3_ 8

d/] f] ul0ft M sIff % 31

2. of] Pp6f 3gfsf/ 7f;] sf] nDafO, rf8} fO / prfO kT| o]s 3cm
sf] 5 . nDafO, rf}8fO / prfO 1/1 cm ePsf] 3gfsf/
agfP/ sf6\bf hDdf slt 6j' |mf xG' 5g\ xf]nf <
3. Pp6f ;ªV\ ofsf 3 cf6] f p:tfp:t} u0' fgv08dWo] Pp6f 5 5 eg] Tof] ;ªV\ of slt /x5] <

4. tnsf ;ª\Vofx¿df s'gsg' 3g ;ªV\ ofx¿ x'g\ <

-s_ 64 -v_ 125 -u_ 16 -3_ 100 -ª_ 27

2.6 ;ªV\ ofx¿sf] ¿9 v08Ls/0f (Prime Factorization of Numbers)

tnsf] pbfx/0f x/] M bO' { cf6] f u'0fg v08x¿ (12 ;o+ 'St ;ªV\ of xf]_
b'O{ cf]6f u'0fg v08x¿ (6 ;+o'St ;ªV\ of xf_]
12 = 12 x 1 b'O{ cf]6f u'0fg v08x¿ (4 ;o+ S' t ;ª\Vof xf_]
= 6 x 2 tLg cf]6f u0' fg v08x¿ (;a} ¿9 ;ªV\ ofx¿ xg' \_
= 4 x3
= 2 x 2 x 3

cem} yk u'0fgv08x¿df 6j' m| ofpg ;lsG5 < 5nkmn u/ .

o;/L Pp6f ;ª\VofnfO{ 2 jf 2 eGbf a9L ;ªV\ ofsf] u0' fgsf ¿kdf JoSt ug]{

tl/sfnfO{ u0' fgv08 lgsfNg' jf v08Ls/0f elgG5 . sg' } ;ªV\ ofnfO{ ;a} u'0fgv08x¿
¿9 ;ªV\ of cfpg] u/L JoSt ugn{' fO{ ¿9 v08Ls/0f elgG5 .

¿9 v08Ls/0f bO' { tl/sfn] ug{ ;lsG5 M

-s_ nuftf/ efu ljm| ofåf/f

pbfx/0f 1 o;/L efu ubf{ ;a}eGbf

;ªV\ of 12 nfO{ ¿9 v08Ls/0f u/ M ;fgf] ¿9 ;ª\Vofx¿n]
efu ug{' kb{5 .
oxf“, 2 12 12 hf]/ 5, 2 n] efu u/L x/] f}“ .

2 6 6 hf/] 5, 2 n] efu u/L x]/f}“ .

3 3 ¿9 ;ªV\ of ePsfn] efu ug{ aGb u/f“} .

To;}n], 12 = 2 x 2 x 3

32 d/] f] ul0ft M sIff %

pbfx/0f 2

;ª\Vof 675 sf] ¿9 v08Ls/0f u/ M

oxf“, 3 675 675 nfO{ 3 n] efu u/s] f] .
3 225 225 nfO{ 3 n] efu u/]sf] .
3 75 75 nfO{ 3 n] efu u/s] f] .
5 25 25 nfO{ 5 n] efu u/s] f] .
5 To;n} ], 675 = 3 x 3 x 3 x 5 x 5

-v_ u0' fgv08sf] jI[ flrq agfP/

pbfx/0f 3
;ª\Vof 675 sf] u0' fg v08sf] jI[ flrq agfP/ b]vfpm M

oxf,“ 675
3 x 225
3 x 3 x 75
3 x 3 x 3 x 25
3x3x3x5x5

To;}n,] 675 = 3 x 3 x 3 x 5 x 5

cEof; 2.6

1. nuftf/ efu ug]{ ljlwaf6 tn lbOPsf k|Tos] ;ª\Vofsf] ¿9 u0' fgv08x¿
lgsfn M

-s_ 8 -v_ 12 -u_ 18 -3_ 24 -ª_ 42
-r_ 62 -5_ 64 -h_ 85 -´_ 121 -`_ 144

2. tnsf k|Tos] ;ªV\ ofsf] ¿9 u'0fgv08sf] j[Iflrq agfP/ b]vfpm M
-s_ 16 -v_ 30 -u_ 44 -3_ 70
-ª_ 162 -r_ 252 -5_ 676 -h_ 625
-´_ 750 -`_ 925

d]/f] ul0ft M sIff % 33

#kf7 ul0ftsf cfwf/e"t lj|mofx¿
(Fundamental Operation in Mathematics)

3.1 cfwf/e"t ljm| ofx¿ ;dfjz] ePsf ;d:ofsf] ;/nLs/0f

k9, 5nkmn u/ / l;s M

ul0ftsf rf/ ;fwf/0f lgodx¿ hf]8 -±_, 36fp -–_, u0' fg -x_ / efu -÷_ lrx\gx¿
;dfj]z ePsf ;d:ofx¿sf] ;dfwfg ugn'{ fO{ ;/nLs/0f elgG5 . ;/nLs/0f ubf{ efu
/ u'0fgdWo] h'g lj|mof klxn] cfp5“ Tof] ljm| of ug{ ;lsG5 . To;kl5 hf]8 / 36fp
ljm| ofdWo] h'g klxn] cfp5“ ToxL lj|mof ug{ ;lsG5 .

pbfx/0f 1
;/n u/ M 15 ÷ 5 x 3 + 7 - 15

of] ;d:ofnfO{ zAbdf JoSt ubf{ 15 sf] 5 v08sf] Ps v08sf] 3 u'0ffdf 7 hf8] ]/ 15
36fpb“ f slt xG' 5 eGg] cy{ a'emfpg] x'“bf ;/n ubf{ j|md};u“ efu, u'0fg, hf]8 / 36fp
ub{} hfgk' 5{ .

To;}n], 15 ÷ 5 x 3 + 7 - 15 [ 15 ÷ 5 = 3, efu lj|mof ]
[ 3 x 3 = 9, u0' fg lj|mof ]
= 3 x 3 + 7 - 15 [ 9 + 7 = 16, hf8] ljm| of ]
= 9 + 7 - 15 [ 16-15 = 1, 36fp lj|mof ]
= 16 - 15
= 1

pbfx/0f 2

;/n u/ M 55 - 576 ÷ 12 + 11 x 3
oxf“, 55 - 576 ÷ 12 + 11 x 3
= 55 - 48 + 11 x 3 [ efu ljm| ofaf6 ]
= 55 - 48 + 33 [ u'0fg ljm| ofaf6 ]
= 7 + 33 [ 36fp lj|mofaf6 ]

= 40 [ hf]8 lj|mofaf6 ]

34 d/] f] ul0ft M sIff %

pbfx/0f 3

16 sf] bO' { u0' ffaf6 20 36fP/ 13 hf]8\bf slt x'G5 <
of] zflAbs ;d:ofnfO{ ul0ftLo efiffdf JoSt ubf{ 16 x 2 - 20 + 13 x'G5 .

To;}n,] 16 x 2 - 20 + 13 cyjf,
= 32 - 20 + 13 [ u0' fgsf] sfd u/]sf] ]
= 12 + 13 [ 36fpsf] sfd u/s] f] ] 16 x 2 - 20 + 13
= 25 [ hf]8sf] sfd u/]sf] ] = 32 - 20 + 13
= 32 + 13 - 20
cEof; 3.1 = 45 - 20
= 25
;/n u/ M

(1) 44 + 24 ÷ 3 - 30 (2) 63 ÷ 9 x 7 + 4 - 52

(3) 6 x 64 ÷ 16 + 7 - 21 (4) 24 x 12 ÷ 12 - 24 + 17

(5) 55 ÷ 11 + 7 x 3 - 13 (6) 132 ÷ 12 x 12 - 124 ÷ 31

(7) 422 + 124 ÷ 4 x 2 - 355 (8) 144 ÷ 24 - 3 x 15 ÷ 5 + 16

(9) 625 ÷ 25 - 25 x 25 ÷ 5 + 100 (10) 576 ÷ 24 + 51 ÷ 17 - 20

(11) tnsf kT| os] zflAbs ;d:ofnfO{ ul0ftLo efiffdf JoSt u/L ;/n u/ M
-s_ 5 sf] 2 u'0ffdf 3 hf8] b\ f slt x'G5 <
-v_ 12 sf] 3 u'0ffdf 7 hf]8\bf slt x'G5 <
-u_ 36 sf] Ps ltxfOaf6 5 36fP/ 7 hf]8\bf slt xG' 5 <
-3_ 15 sf] 10 u'0ffdf 50 hf8] /] 200 36fp“bf slt xG' 5 <
-ª_ 15 af6 9 sf] Ps ltxfOsf] 2 u0' ff 36fpb“ f slt af“sL x'G5 <

d]/f] ul0ft M sIff % 35

3.2 ;/nLs/0fdf sfi] 7sf] k|of]u

k9, 5nkmn u/ / l;s M

?= 10 kg]{ sfkL / ?= 2 kg]{ O/h] / Ps Ps cf6] f ltg hgfnfO{ lsGg' kbf{ hDdf slt
vr{ xfn] f <

Ps hgfnfO{ nfUg] vr{ = 10 + 2
ltg hgfnfO{ nfUg] vr{ = 3 x (10+2)
= 3 x (10+2) nfO{ ;/n ubf{
= 3 x 12 -sfi] 7leqsf] ljm| of u/]sf]_
= ?= 36

pbfx/0f 1
;/n u/ M 12 - (20 - 12)
oxf“, 12 - (20 - 12)
= 12 - 8 -sf]i7 leqsf] lj|mof u/]sf]_

=4

pbfx/0f 2
;/n u/ M 5 + 8 x 2 - (5 - 2)
oxf,“ 5 + 8 x 2 - (5 - 2)
= 5 + 8 x 2 - 3 -sfi] 7 leqsf] lj|mof u/s] f_]
= 5 + 16 - 3 -u0' fg lj|mof klxn] u/s] f_]
= 21 - 3 -hf8] lj|mof u/s] f]_
= 18 -36fp lj|mof u/s] f]_

36 d/] f] ul0ft M sIff %

pbfx/0f 3

5 df 2 hf8] ]/ hf]8kmnsf] 3 u0' ffaf6 6 36fp“bf slt x'G5 < of] ;d:ofdf hf]8, 36fp
tyf u0' fg ljm| ofx¿ ;dfj]z ePsf 5g\ . oL lj|mofdWo] klxnf u0' fgsf] sfd ug'k{ 5{
t/ of] ;d:ofdf 5 / 2 sf] hf8] kmnsf] ltg u0' ff egs] fn] klxnf hf8] g\ ] sfd ugk'{ 5{ .

t;y,{ 5 / 2 sf] hf8] klxnf ugk{' g{] ePsfn] sfi] 7 ko| fu] u/L ul0ftLo efiffdf JoSt ubf{ –

(5 + 2) x 3 - 6 x'G5 .

= 7 x 3 - 6 [ klxnf sfi] 7leqsf] sfd u/s] f] . sf]i7leqsf]
sfd ul/;sk] l5 sfi] 7 x6fpg'k5{ . ]

= 21-6 = 15

t;y{ ;/nLs/0fdf sf]i7x¿ ko| f]u ePsf 5g\ eg] sfi] 7leq ;dfj]z ePsf ljm| ofnfO{
klxnf ul/;s]kl5 af“sL lj|mofx¿ ub{} hfgk' 5{ . ;/nLs/0fdf ko| fu] ePsf sfi] 7x¿
jm| d;} u“ - _ ;fgf] sfi] 7 / { } demf}nf sfi] 7sf] sfd ub{} hfg'k5{ .

pbfx/0f 4

;/n u/ M 8 + 14 x {(8 - 2) + 3} ÷ 18

oxf“, 8 + 14 x {(8 - 2) + 3} ÷ 18 cyjf

= 8 + 14 x {6 + 3} ÷ 18 8 + 14 x {(8 - 2) + 3} ÷ 18
= 8 + 14 x {6 + 3} ÷ 18
= 8 + 14 x 9 ÷ 18 = 8 + 14 x 9 ÷ 18
= 8 + 126 ÷ 18
= 8 + =8+7
= 15
= 8 + 7
37
= 15
d/] f] ul0ft M sIff %

cEof; 3.2

-s_ ;/n u/ M

(1) 6 - (5 - 2)
(2) (16 - 4) x (5 - 3)
(3) (16 + 4) ÷ 5 - 3
(4) 20 - {8 - (5 + 2)}
(5) 3 - {12 ÷ (2 x 3)}
(6) 19 - 7 + {4 - (5 - 2)} x 2
(7) 3 + 4 ÷ {2 + 4 ÷ (4 - 2)}
(8) 30 ÷ {60 - 3 (21 - 6)}
(9) 22 ÷ {20 ÷ (4 + 6)} x 2
(10) 80 - 5 {9 - (14 - 12)} ÷ 5

-v_ ul0ftLo jfSodf nv] L ;/n u/ M

(1) 12 / 5 sf] cGt/sf] 3 u0' ffnfO{ 7 n] efu ubf{ slt xG' 5 <
(2) 20 af6 9 / 5 sf km/ssf] 3 u'0ff 36fpb“ f slt x'G5 <
(3) 7 / 3 sf] ofu] kmnsf] 5 u'0ffaf6 70 / 30 sf] cGt/ 36fpb“ f slt xG' 5 <
(4) 16 / 20 sf] ofu] kmnsf] Ps rfy} fOaf6 16 / 7 sf] km/s 36fpb“ f slt xG' 5 <
(5) 50 af6 12 sf] b'O{ u'0ffsf] Ps ltxfO 36fP/ 6 efu nufpb“ f slt x'G5 <
(6) 7 af6 5 / 2 sf] cGt/df 4 hf8] /] cfpg] ;ªV\ of 36fO{ 9 hf8] b\ f slt xG' 5 <

38 d]/f] ul0ft M sIff %

$kf7 ;do
(Time)

dfly lbOPsf ;dosf PsfOx¿nfO{ Ps csf{] PsfO;u“ t'ngf ub}{ ;DaGwsf] vfh] L
u/ . 38Lsf] ;ae} Gbf dl;gf] -;s] G] 8_ ;O' {n] Ps kmGsf] dfbf{ 38Lsf dl;gf 60 cf6] f
w;fx{ ¿ kf/ u5{ . Toltan] f 38Lsf] nfdf] ;'O{ csf]{ w;fd]{ f dfq k'U5 . To;f] eP
38Lsf oL ;'Ox{ ¿larsf] ultsf] cfk;L ;DaGw ;DaGwL tnsf k|Zgx¿sf] pTt/ cfk"m
tyf ;fyLx¿lar 5nkmn u/L vfh] M

-s_ 38Lsf] ;s] G] 8 ;O' n{ ] 1 kmGsf] 3D' bf ldg6] ;O' n{ ] slt cf6] f ;fgf w;f{ kf/ u5{ <

-v_ ldg6] ;'On{ ] 1 kmGsf] 3'Dbf ;s] ]G8 ;O' n{ ] slt kmGsf] df5{ <

-u_ 38Lsf] 306f bv] fpg] ;O' {n] 1 kmGsf] 3D' g slt ;do nfU5 <

-3_ 306f hgfpg] ;O' n{ ] 1 lbgdf slt kmGsf] nufp5“ < ldg6] hgfpg] ;'On{ ] rflx“
slt kmGsf] nufp“5 lg <

d]/f] ul0ft M sIff % 39

pbfx/0f 1

3 jif{ 8 dlxgf 12 lbgnfO{ 3 n] u'0fg u/ M

jif,{ dlxgf / lbgnfO{ 3 n] 5'6\6f56' 6\ } u'0fg ubf{,

jif{ dlxgf lbg 30 lbg = 1 dlxgf, 36 lbg = 1 dlxgf 6 lbg
12 dlxgf = 1 jif,{ 25 dlxgf = 2 jif{ 1 dlxgf
3 8 12
x 3

9 24 36

9 25 6

11 1 6

pbfx/0f 2

15 306f 30 ldg6] 48 ;]sG] 8nfO{ 4 n] efu u/ .
oxf,“ 306f, ldg6] / ;s] G] 8nfO{ 56' \6f5'66\ } /fv/] 4 n] efu ubf{ –

3 3=, 52 ld=, 42 ;s] ]G8

4 15 30 48
–1 2

3 30 48 3 306f = 180 ldg6]

210 48

– 20

10
– 8

2 48

168 2 ldg]6 = 120 ;s] ]G8

– 16

8

– 8

0

pbfx/0f 3

10 jif{ 7 dlxgf 12 lbgnfO{ 6 n] efu u/ M d]/f] ul0ft M sIff %

jif{, dlxgf / lbgnfO{ 56' 6\ f56' \6} /fv]/ efu ubf{ –

40

1 jif{ 9 dlxgf 7 lbg

6 jif { dlxgf lbg
10 7 12

6

4 7 12

55 12 4 jif{ = 48 dlxgf
54

1 12 1 dlxgf = 30 lbg

42

42

x

pbfx/0f 4

Pp6f dfOj|mf]a; nuftf/ 6 k6s;Dd gf/fo0f u9af6 sf7df8f}“ cf]xf]/bf]xf]/
u¥of] . olb Tof] a;n] 6 k6ssf] ofqfdf 28 306f 34 ldg6] nufPsf] / Tof] a;n]
k|Tos] k6sdf plTt g} ;do nufpg] u/s] f] eP 1 k6s gf/fo0f u9af6 sf7df8f“}
cfOk'Ug] ;do slt xf] <

oxf“, 6 k6sdf nufPsf] hDdf ;do = 28 306f 34 ldg6]

Ps k6sdf nfUg] ;do lgsfNbf,

4 306f 45 ldg6] 40 ;]sG] 8

6 28 34
-24

4 x 60 240 + 34

274
- 24

34
– 30

4 x 60  240
– 240

0

d/] f] ul0ft M sIff % 41

cEof; 4

(1) u'0fg u/ M

-s_ 2 jif{ 7 dlxgf 16 lbgnfO{ 4 n]
-v_ 5 jif{ 6 dlxgf 12 lbgnfO{ 5 n]
-u_ 7 jif{ 4 dlxgf 18 lbgnfO{ 3 n]
-3_ 8 jif{ 9 dlxgf 6 lbgnfO{ 8 n]
-ª_ 6 306f 40 ldg]6 15 ;]s]G8nfO{ 5 n]

(2) efu u/

-s_ 10 jif{ 8 dlxgf 20 lbgnfO{ 2 n]
-v_ 15 jif{ 6 dlxgf 21 lbgnfO{ 3 n]
-u_ 13 jif{ 7 dlxgf 12 lbgnfO{ 6 n]
-3_ 5 306f 7 ldg6] 32 ;s] G] 8nfO{ 4 n]
-ª_ 8 306f 10 ldg]6 35 ;s] G] 8nfO{ 7 n]

(3) Pp6f sfdsf] efu ug{ 6 306f 30 ldg6] nfU5 eg] k'/} sfd slt
;dodf ;lsPnf <

(4) Pp6f juf{sf/ v]tdf sfpnLsf la?jf ;fg{ 2 306f 15 ldg]6
nfU5 . o:tf 6 cf]6f v]tdf la?jf ;fg{ slt ;do nfUnf <

(5) Pp6f gnn] Pp6f 6\ofª\sL 306fdf e5{ . ToxL gnn] 5 cf6] f 6\ofª\
sL slt ;dodf enf{ <

(6) zLnfsf] sIffdf 7 ljifosf] k9fO xG' 5 . Pp6f ljifosf] k9fO 45 ldg]6
x'g] eP zLnfn] hDdf slt ldg]6df 7 ljifo k9l\ 5g\ xfn] f <

(7) lzjn] 306fdf ;a} ljifosf] ux[ sfo{ ul/;S5 . p;n] kT| os] ljifonfO{
a/fa/ ;do lbG5 / 7 ljifosf] u[xsfo{ u5{ eg] 1 ljifonfO{ u[xsfo{
ug{ slt ;do nufp“5 <

42 d]/f] ul0ft M sIff %

(8) kf]v/faf6 ljleGg 7fp“df hfg] a;x¿sf] ;do tflnsf lbOPsf] 5 .
kfv] /faf6 sf7df8f“} hfg] A a; / B a; b'j} laxfg 5:30 df 56' ]5g\ /
lgDg lnlvt :yfgdf lgDg lnlvt ;dodf kU' 5g\ M

kfv] /f A a; B a;
v/} g] L6f/
bdf}nL 6:45 am 7:00 am
8d' ] | 7:50 am 8:45 am
cfa'v/} g] L 9:30 am 10:30 am
d'lUng 10:15 am 12:30 noon
dxfbj] a];“ L 11:00 am 1:00 pm
gfl} a;] 11:50 am 2:30 pm
sf7df8f“} 1:00 pm 4:00 pm
2:00 pm 5:30 pm

ca lgDg lnlvt k|Zgsf] hjfkm bp] m M

-s_ A a; bdfn} L slt ah] ku' 5] <

-v_ B a; A a;eGbf bdfn} L slt l9nf] kU' of] <

-u_ A a;df 8d' ]|df r9s] f ofqL slt ;dokl5 dl' Ung k'U5g\ <

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-5_ A a; sf7df8f“} k'us] f] slt ;dokl5 B a; sf7df8f“} k'U5 <

d/] f] ul0ft M sIff % 43

%kf7 d'bf|
(Money)

xfdLn] Pp6} wg/flznfO{ eS' tfgL ug{ xfdL;u“ ePsf l;Ssf / gf6] x¿ ko| fu] u/L
km/s km/s lsl;dsf] eS' tfgL ug{ ;S5f“} . pbfx/0fsf nflu 1 ?lkof“ 25 k;} f ltgs{ f nflu,
?= 1 sf] gf6] / 25 k};fsf] l;Ssf
b'O{ cf]6f 50 k};fsf l;Ssf / 25 k};fsf] Pp6f l;Ssf
5 cf]6f 25 k};fsf l;Ssfx¿ ====== cflb .
1 ?lkof“ 25 k};f ltg]{ oL dfq tl/sf geP/ c¿ klg 5g\ . sIffdf 5nkmn
u/L 1 ?lkof“ 25 k;} fnfO{ slt tl/sfn] e'StfgL ug{ ;lsG5, kTtf nufpm . 1 ?lkof“
25 k;} fnfO{ hgfpg xfdL ?= 1.25 nV] 5f“} . of] tl/sfdf ?lkof“ / k;} fsf lardf
bzdnj laGb' /flvG5 . lsg ?
?= 1 df 100 k};f x'G5 cyft{ \ 100 k;} f = 1 ?lkof“ x'G5 . To;}n] ?lkofn“ fO{ k;} fdf
abNg 100 n] u0' fg ugk'{ 5{ eg] 100 k};fn] 1 ?lkof“ agfpg] x'b“ f k};fnfO{ ?lkofd“ f
abNg 100 n] efu ug'k{ 5{ .
o; tYonfO{ tnsf] rj|maf6 a'‰g] ko| f; u/ M

dflysf] pbfx/0fdf 1 ?lkof“ 25 k;} fdf hDdf 125 k};f x'G5 / 125 k;} fnfO{ ?lkof“df
abNg 100 n] efu ubf{ ?= 1.25 cfp5“ .

44 d]/f] ul0ft M sIff %

pbfx/0f 1
?= 4 / k};f 35 nfO{ k;} fdf ¿kfGt/ u/ M
oxf,“ ?= 4 / k;} f 35 = (4 x 100 + 35) k}= = 435 k=}

pbfx/0f 2

525 k;} f nfO{ ?lkof“df kl/0ft u/ M
oxf,“
525 k};f = ?= (525 ÷ 100) = ?= 5.25

pbfx/0f 3

?= 10.75 ltg{ /dz] n] k;n]nfO{ ?= 20 sf] gf6] lbP eg] /dz] n] slt lkmtf{ kfp5“ g\ <
oxf,“ ?= 20 = 20 x 100 k}= = 2000 k}=
/ ?= 10.75 = 10.75 x 100 k}= = 1075 k=}
To;n} ] lkmtf{ kfpg] /sd = (2000 - 1075) k=} = 925 k}= = ?= 9.25
-k};fnfO{ ?lkof“df abNg bfof“lt/af6 2 cªs\ ug]/ bzdnj lrx\g /fvs] f_]
5f6] s/Ldf 36fp ubf{,
?= 20.00
– ?= 10.75
?= 9.25
csf]{ tl/sf M -?= nfO{ ?= af6 / k};fnfO{ k};faf6 36fpb“ f_
?= k=} ?= k}=

20 00 = 19 100

10 75 – 10 75

= ?= 9. 25

o;/L ?lkof“ / k};fsf] hf]8 cyjf 36fp ubf{ ?lkof;“ “u ?lkof“ / k};f;u“ k;} f hf]8g\ '
tyf 36fpgk' g]{ xG' 5 .

d]/f] ul0ft M sIff % 45

pbfx/0f 4
-s_ ?= 6.80 df ?= 7.25 hf]8\bf slt ?lkof“ xG' 5 <

?lkof“ / k};fnfO{ cnu cnu hf8] b\ f,

?= k=}
6 80

+ 7 25

13 105 105 k;} f eg]sf] ?= 1 / k;} f 5 xf] . To;}n] ?= 1 nfO{
14 0 5 ?= 1 3 df hf8] \bf ?= 14 / k};f 5 xG' 5 .
= ?= 14.05

5f]6s/Ldf ubf,{
?= 6.80
± ?= 7.25
= ?= 14.05

-v_ ?= 5 / k;} f 25 af6 ?= 3 / k};f 75 36fp“bf slt xG' 5 <
?lkof“ / k};fnfO{ cnu cnu 36fpb“ f,

?= k}= 125

4
5 25 25 k};faf6 75 k=} 36\bg} . To;n} ] ?= 1 = 100 k};f

- 3 75 ;fk6 ln“bf

1 50

= ?= 1.50

5f]6s/Ldf ubf{, of] bzdnjsf]
36fp g} xf] .
?= 5.25
— ?= 3.75
?= 1.50

46 d]/f] ul0ft M sIff %