MATHS GRADE 6

7.2 axe' h' sf 9fr“ fx¿

axe' h' x¿leq axe' h' x¿ lvr/] jf Pp6f axe' h' nfO{ c¿ w/] } axe' h' df ljefhg u//] ljleGg
/ª ko| fu] u/L cfsifs{ ¿k/v] f jf 9fr“ f agfpg ;lsG5 . sx] L o:tf 9fr“ fx¿ tn lbOPsf
5g\ . o:t} k|sf/sf 9f“rf agfP/ sIffsf]7f jf 3/df ;hfpg] k|of; u/ .

jus{ f] 9fr“ f 1111111111111112222222222222223333333333333331111111144444444444444422222222555555555555555333333336666666666666664444444477777777777777755555555888888888888888666666669999999999999997777777700000000000000088888888111111111111111222222222222222333333333333333444444444444444555555555555555
/ª gebf{
/ª ebf{

k|To]s gof“ ju{sf] e'hfsf] dWolaGb' hf]8\b} hf“bf c;ªV\ o ju{x¿ aGb} hfG5g\ . oxL
k|ljm| ofsf 9fr“ fx¿ ;dafx' lqe'h tyf cGo ax'e'hsf klg agfpg ;lsG5 . k|of; u/]/
x]/ .

k~reh' sf] 9fr“ f

k~reh' sf] eh' fnfO{ aflx/ nDAofpb“ f tf/f
aG5 . o:tf] tf/fleq c¿ tf/fx¿ agfpb“ } hfg
;lsG5 / /ª e//] cfsifs{ ¿k/v] f agfpg
;lsG5 .

tnsf] lrqnfO{ /ª\ufP/ b]vfOPsf] 5 . o:t} km/s km/s l8hfOg tof/ u/L x]/ .

11111111111111111111111111111111111111111111122222222222222222222222222222222222222222222233333333333333333333333333333333333333333333344444444444444444444444444444444444444444444455555555555555555555555555555555555555555555566666666666666666666666666666666666666666666677777777777777777777777777777777777777777777788888888888888888888888888888888888888888888899999999999999999999999999999999999999999999900000000000000000000000000000000000000000000011111111111111111111111111111111111111111111122222222222222222222222222222222222222222222233333333333333333333333333333333333333333333344444444444444444444444444444444444444444444411111111111111111115555555555555555555555555555555555555555555552222222222222222222666666666666666666666666666666666666666666666333333333333333333377777777777777777777777777777777777777777777744444444444444444448888888888888888888888888888888888888888888885555555555555555555999999999999999999999999999999999999999999999666666666666666666600000000000000000000000000000000000000000000077777777777777777771111111111111111111111111111111111111111111118888888888888888888222222222222222222222222222222222222222222222999999999999999999933333333333333333333333333333333333333333333300000000000000000004444444444444444444444444444444444444444444441111111111111111111555555555555555555555555555555555555555555555222222222222222222266666666666666666666666666666666666666666666633333333333333333337777777777777777777777777777777777777777777774444444444444444444888888888888888888888888888888888888888888888555555555555555555599999999999999999999999999999999999999999999966666666666666666660000000000000000000000000000000000000000000007777777777777777777111111111111111111111111111111111111111111111888888888888888888822222222222222222222222222222222222222222222299999999999999999991111111111111111111111111111111111111111111110000000000000000000222222222222222222222222222222222222222222222333333333333333333333333333333333333333333333444444444444444444444444444444444444444444444555555555555555555555555555555555555555555555666666666666666666666666666666666666666666666777777777777777777777777777777777777777777777888888888888888888888888888888888888888888888999999999999999999999999999999999999999999999000000000000000000000000000000000000000000000111111111111111111111111111111111111111111111222222222222222222222222222222222222222222222333333333333333333333333333333333333333333333444444444444444444444444444444444444444444444555555555555555555555555555555555555555555555666666666666666666666666666666666666666666666

46 ul0ft, sIff ^

if8e\ h' sf 9fr“ fx¿ 111111111111111111111111111111222222222222222222222222222222333333333333333333333333333333444444444444444444444444444444555555555555555555555555555555666666666666666666666666666666777777777777777777777777777777888888888888888888888888888888999999999999999999999999999999000000000000000000000000000000111111111111111111111111111111222222222222222222222222222222333333333333333333333333333333444444444444444444444444444444555555555555555555555555555555666666666666666666666666666666777777777777777777777777777777888888888888888888888888888888999999999999999999999999999999000000000000000000000000000000111111111111111111111111111111222222222222222222222222222222333333333333333333333333333333444444444444444444444444444444555555555555555555555555555555666666666666666666666666666666777777777777777777777777777777888888888888888888888888888888999999999999999999999999999999000000000000000000000000000000111111111111111111111111111111222222222222222222222222222222111111111111111111111111111111222222222222222222222222222222333333333333333333333333333333
11111111111111111111111111111111122222222222222222222222222222222233333333333333333333333333333333344444444444444444444444444444444455555555555555555555555555555555566666666666666666666666666666666677777777777777777777777777777777788888888888888888888888888888888899999999999999999999999999999999911111111111110000000000000000000000000000000002222222222222111111111111111111111111111111111333333333333322222222222222222222222222222222244444444444443333333333333333333333333333333335555555555555444444444444444444444444444444444666666666666655555555555555555555555555555555577777777777776666666666666666666666666666666668888888888888777777777777777777777777777777777999999999999988888888888888888888888888888888800000000000009999999999999999999999999999999991111111111111000000000000000000000000000000000222222222222211111111111111111111111111111111133333333333332222222222222222222222222222222224444444444444333333333333333333333333333333333555555555555544444444444444444444444444444444466666666666665555555555555555555555555555555557777777777777666666666666666666666666666666666777777777777777777777777777777777888888888888888888888888888888888999999999999999999999999999999999000000000000000000000000000000000111111111111111111111111111111111222222222222222222222222222222222111111111111111111111111111111111
if8e\ h' leq lqeh' x¿
if8e\ h' leq if8e\ h' x¿
hDdf slt 5g,\ ugL x/] .
jul{ eqsf /v] fx¿ hf8] L tnsf] h:t} lrq agfpg] ko| f; u/ .

o:t} ks| f/sf /v] Lo gdg' f jf 9fr“ f lqeh' , rte' h{' Pjd\ axe' h' leq lvrL x/] . s:tf s:tf
l8hfOgx¿ aGbf /x5] g\ <

cEof; 7.2

1. lbOPsf] 9fr“ fdf sltcf6] f lgoldt if8e\ h' x¿ 5g\ <
nv] .

2. lbOPsf] 9fr“ f h:t} cGo yk bO' { 9fr“ fx¿ agfpm .

3. ltdf| ] 3/df ko| fu] ul/g] afy?dsf 6fon, sfk6{] , b/L,
9fs] fdf agfOPsf lrqx¿, km' 6an, elnan cflbdf
agfOPsf lrqx¿df klg s] o; ks| f/sf
9fr“ fx¿ 5g,\ nv] .

ul0ft, sIff ^ 47

PsfO 8 ;dx" (Sets) Pp6f cf“k 5 .
Pp6f ;'Gtnf 5 .
8.1 ;dx" sf] kl/ro
tn lbOPsf pbfx/0fx¿ cWoog u/ M

-s_ l/sfkLdf s] s] 5g\ <

Pp6f s]/f 5 .
Pp6f :ofp 5 .

oL ;a} kmnk"mn x'g\ .
of] Pp6f l/sfkLdf ePsf kmnkm" nx¿sf] ;dx" (Set) xf] .

-v_ 3/] fleq s:tf cªs\ x¿ 5g\ < 2
2 Pp6f hf]/ ;ª\Vof xf] . 46
4 Pp6f hf]/ ;ª\Vof xf] .
6 klg Pp6f hf]/ ;ª\Vof xf] .
oL ;a} 7 eGbf ;fgf hf]/ ;ª\Vofx¿ x'g\ .
of] 7 eGbf ;fgf hf/] ;ªV\ ofx¿sf] ;d"x xf] .

-u_ 3/] fleqsf :yfgx¿nfO{ ss] f] ;dx" eGg ;lsG5 < sf7df8f}“

g]kfnsf] /fhwfgL sf7df8f}“ xf] . lbNnL lyDk'

ef/tsf] /fhwfgL lbNnL xf] . O:nfdfjfb 9fsf
To:t} lyDk,' 9fsf, O:nfdfjfb, dfn,]
sfn] Daf] / sfan' jm| dzM e6' fg, aªu\ nfbz] , dfn] sf]nDaf]
sfan'

kfls:tfg, dflNbE;, >Lnªs\ f /

ckmuflg:tfgsf /fhwfgLx¿ x'g\ .

of] ;fs{ (SAARC) /fi6«x¿sf /fhwfgLx¿sf] ;d"x xf] .

48 ul0ft, sIff ^

-3_ ;u“ s} f] lrq x/] / lgDg lnlvt kZ| gx¿sf pQ/ bp] m M
 lrqdf lbOPsf j:t'x¿ ss] f] ;ªs\ ng xf] <
 of] ss] f] ;dx" xf] <

lrqdf lbOPsf j:tx' ¿ oftfoftsf ;fwgx¿sf] ;dx" xf] .

-ª_ dfly lbOPsf] ;d"xaf6 l/S;f lgsfn M kml] / csf{]
;dx" aG5 <
s:tf] ;dx" aG5 <
of] OGwgaf6 rNg] oftfoftsf ;fwgx¿sf] ;dx"
xf] .
oL ;dx" af6 xjfOh{ xfh lgsfNbf s:tf] ;dx" aG5 <
5nkmn u/ .

-r_ pdz] , ljdnf, zflGt, zLnf / /fh' sIff 6 sf] klxnf] aG] rdf a:g] kfr“ hgf ljBfyLx{ ¿ xg' \ .
lzIfsn] eGge' of,] ltdLx¿dWo] pdz] / /fh' Psflt/ a; tyf ljdnf, zflGt / zLnf
csfl{] t/ a; . o;/L a:bf s] ss] f] ;dx" aG5 < 5nkmn u/ .

lzIfsn] ljBfyLx{ ¿nfO{ cUnfb] l] v xfr] fs] f] jm| ddf pleg nufpge' of] .

ca, lzIfsn] eGge' of,] ‘‘ltdLx¿n] cUnf ljBfyLx{ ¿sf] dfq} Pp6f ;dx" agfpm .’’ 49

ul0ft, sIff ^

cUnf ljBfyLx{ ¿sf] ;dx" df sf] sf] a:g] <
ljBfyL{x¿ cNdlnP .
ljdnf ;aeG} bf cUnL l5g\ . pm Tof] ;dx" df kl5{g\ . zLnfrflx“ zflGteGbf cUnL l5g\ t/ pgL
/fhe' Gbf xfr] L l5g\ . zLnf of] ;dx" df kl5g{ \ jf klbg{ g\ < To:t} c¿ lg < t;y,{ cUnf ljBfyLx{ ¿
sf] sf] x'g\ lglZrt ug{ ;lsb“ g} . To;n} ] o:tf ;ªs\ ngnfO{ ;d"x eGg ;lsb“ g} .

olb ;ªs\ ngdf sg' } j:t' k5{ ls kbg{} egL ls6fg ug{ ;lsG5 eg] To:tf ;ªs\ ngnfO{
kl/eflift (well-defined) ;ªs\ ng elgG5 . j:tx' ¿sf] kl/eflift ;ªs\ ngnfO{ ;dx" elgG5 .

dfly pbfx/0f -s_ df lbOPsf s]/f, :ofp, cf“k / ;'Gtnf kmnk"mnx¿sf] ;d"xsf ;b:ox¿
(members) xg' \ . To:t,} pbfx/0f -v_ df 2, 4 / 6, 7 eGbf ;fgf hf/] ;ªV\ ofx¿sf] ;dx" sf ;b:ox¿ xg' \ .

pbfx/0f 1

;dx" df gldNg] Pp6fnfO{ jm| ; (X) u/ . To;kl5 s:tf] -s]sf]_ ;d"x aG5, n]v .

1 10 3
57

pQ/ 1 10 3
57
oxf“ 1, 3, 5, 7 lahf]/ ;ª\Vofx¿ x'g\ .
10 Pp6f hf]/ ;ª\Vof xf] .
o;af6 10 lgsfnk] l5 8 eGbf ;fgf lahf/] ;ªV\ ofx¿sf] ;dx" aG5 .

pbfx/0f 2

tn lbOPsf dWo] kl/eflift ePsf ;ªs\ ngdf l7s (√) / gePsfdf al] 7s (X) lrx\g nufpm .
(i) sIff 6 sf la:tf/} cIf/ n]Vg] ljBfyL{x¿ .
(ii) cªu\ h]| L cIf/ J af6 ;?' xg' ] cªu\ h]| L dlxgfsf gfdx¿ .

pQ/
(i) (X) lsgeg,] oxf“ slt l56f] nV] g] ljBfyLn{ fO{ rf8“ f] / slt nV] g g;Sgn] fO{ l9nf] nV] g] ljBfyL{

eGg] :ki6 5}g . To;sf/0f, of] ;d"x kl/eflift 5}g .
(ii) (√) lsgeg,] J af6 ;?' xg' ] cªu\ h]| L dlxgfx¿ :ki6 5g\ . tL dlxgfx¿ January, June / July

x'g\ .

50 ul0ft, sIff ^

cEof; 8.1

tn kT| os] ;dx" (1-15) df lbOPsf dWo] gldNg] Pp6fnfO{ jm| ; (X) u/ . To;kl5 af“sL s]sf] ;d"x
aG5, n]v .

1. cfOtaf/, ;fd] af/, dªu\ naf/, aw' af/, laxLaf/, >fj0f, zj' m| af/, zlgaf/ .

2.

3. gk] fn, lrg, ef/t, e6' fg, aªu\ nfbz] , dflNbE;, kfls:tfg, >Lnªs\ f, ckmuflg:tfg .

4. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 17

5. 8 27 64 100

6.

7.

8.

9. 2, 3, 5, 7, 10, 11

10. 10, 15, 20, 25, 30, 32

11. 1 , 2 , 4
3 3 3

12.

13.

6:30 7:30 10:00 12:30

ul0ft, sIff ^ 51

14. x2+2x, 2x +3y, 2a +3b+4c, 5p-10q
15. a, e, i, o, p, u

16. lgDglnlvt ;ª\sng dWo] l7s kl/eflift ;ª\sngnfO{ (√) / gePsfnfO{ (x) lrx\g
nufpm M
-s_ nfdf] s]z x'g] s]6Lx¿sf] ;d"x .
-v_ Pp6f ljBfnodf sIff 6 df k9fpg] lzIfsx¿sf] ;d"x .
-u_ ltvf] :j/n] s/fP/ k9fpg] lzIfsx¿sf] ;d"x .
-3_ ltdf| ] ljBfnodf elnan vN] g] ljBfyLx{ ¿sf] ;dx" .

8.2 ;dx" sf] ;ªs\ t] tyf ;dx" nfO{ hgfpg] tl/sf

;dx" sf ;ªs\ t] (Notation of a set)

;fdfGotof ;dx" x¿nfO{ cªu\ h|] L j0fd{ fnfsf] 7n" f cIf/x¿ (capital letters) A, B, C, D,
……, X, Y, Z cflbn] hgfpg] ul/G5 . ;dx" sf kT| os] ;b:onfO{ cNklj/fd (,) af6 56' o\ fOG5
/ ;a} ;b:ox¿nfO{ demfn} f sfi] 7 { } leq} /flvG5, h:t} M
A = { s/] f, :ofp, cfk“ , ;G' tnf }

;dx" x¿nfO{ hgfpg] tl/sf (Methods of describing Sets)

;dx" nfO{ lgDg lnlvt ltg tl/sfn] hgfpg] ul/G5 M

JofVof ljlw ;"rLs/0f ljlw ;d"x lgdf{0f ljlw

(describing method) (listing method) (set builder method)

D = {xKtfsf ;ft D = {cfOtaf/, ;f]daf/, D = {x : x xKtfsf] Pp6f
af/x¿sf] ;dx" } dª\unaf/, aw' af/, laxLaf/, af/ xf] .}
zj' m| af/, zlgaf/}

V = {cªu\ ]|hL j0fd{ fnfsf V = { a, e, i, o, u } V = {x : x cªu\ ]|hL
:j/x¿ (Vowels) sf] j0f{dfnfsf] :j/ xf] .}
;dx" }

A = {5 eGbf 7n' f / 20 A = { 7,9,11,13,15,17,19 } A = { x : x, 5 eGbf 7n' f]
eGbf ;fgf lahf/] 20 eGbf ;fgf] Pp6f
lahf/] ;ª\Vof xf] .}
;ªV\ ofx¿sf] ;dx" }

-s_ JofVof ljlw M ;dx" df kg{] j:tx' ¿ jf ;b:ox¿sf] u0' fnfO{ ljrf/ u/L zAb jf jfSoåf/f
cleJoSt ul/G5 . o;df zAb jf jfSosf] g} k|of]u ul/G5 .

52 ul0ft, sIff ^

-v_ ;r" Ls/0f ljlw M ;dx" sf ;b:ox¿nfO{ demfn} f sfi] 7 { } leq} cNklj/fdn] 56' o\ fP/
/flvG5 .

-u_ ;dx" lgdf0{ f ljlw M o;df sg' } Pp6f ;dx" sf ;b:ox¿sf] ;femf u0' fsf cfwf/df pSt rnsf]
JofVof ul/G5 . dflysf] pbfx/0fdf x nfO{ jf/sf gfdsf] 7fpd“ f /flvPsf] 5 . of] Pp6f rn
xf] . o:tf] ( : ) lrxg\ n] egs] f] cyft{ \ such that ae' mfp5“ , h:t} M klxnf] 5 cf6] f uGtL
;ªV\ ofx¿ (Counting Numbers) sf] ;dx"

C = {1, 2, 3, 4, 5, 6}

klxnf] 5 ;Ddsf uGtL ;ªV\ ofx¿sf] ;dx" nfO{ tn lbOPcg;' f/ klg nV] g ;lsG5 M
{1, 2, 3, 4, 5}, {1, 3, 4, 2, 5} jf {2, 3, 1, 5, 4}

bi| 6Jo M demfn} f sfi] 7 { } leq ;dx" sf ;b:ox¿nfO{ hg' ;s' } jm| d (order) df klg n]Vg ;lsG5 .

csf{] pbfx/0f x/] f,“} bO' { nfv kt“} fln; xhf/ rf/ ;o klRr; (2,45,425) nV] bf ko| fu] ePsf
cªs\ x¿sf] ;dx" N nV] bf,

N = {2, 4, 5}

cyjf N = {4, 2, 5}
cyjf N = {5, 2, 4}

coffee zAb nV] bf ko| fu] ePsf cªu\ h]| L j0fd{ fnfsf cIf/x¿sf] ;dx" W agfpb“ f,

W = {c, o, f, e} xG' 5 t/ W = {c, o, f, f, e, e} n]Vg] rng 5}g .

bi| 6Jo M demfn} f sfi] 7 { } leq ;dx" sf sg' } klg ;b:onfO{ Ps k6seGbf a9L bfx] f¥] ofP/
nl] vb“ g} . sg' } klg ;dx" nfO{ cªu\ h]| L j0fd{ fnfsf 7n' f cIf/ (Capital Letter) n] hgfpg] ul/G5
eg] ;dx" sf ;b:ox¿nfO{ ;fgf] cIf/ (Small Letter) jf 7n' f] cIf/ bj' }n] hgfOG5 . h:t} M
cªu\ h]| L cIf/sf :j/x¿sf] ;dx" , V = {a, e, i, o, u} n]lvG5 .

pbfx/0f 1

olb P n] ;fo} { kl/jf/sf cf7 cf6] f ux| x¿sf] ;dx" hgfp5“ eg] ;r" Ls/0f ljlwaf6 ;dx" agfpm /
;d"x lgdf{0f ljlwaf6 klg n]v .

ul0ft, sIff ^ 53

pQ/
;r" Ls/0f ljlwaf6,
P = {aw' , zj' m| , kY[ jL, dªu\ n, ax[ :klt, zlg, c?0f, j?0f}
;dx" lgdf0{ f ljlwaf6,
P = {x : x ;f}o{ kl/jf/sf] Pp6f u|x xf] .}

pbfx/0f 2

lbOPsf] ;dx" nfO{ zAbdf JofVof u/L JoSt u/ M
T = {;dsf0] fL lqeh' , Gog" sf0] fL lqeh' , clwssf0] fL lqeh' }
pQ/
T = {sf0] fsf] cfwf/df juLs{ /0f ul/Psf lqeh' x¿sf] ;dx" }

pbfx/0f 3

J tLg cªs\ n] ags] f] ;ae} Gbf ;fgf] / ;ae} Gbf 7n' f] ;ªV\ ofx¿sf] ;dx" xf] eg] o;nfO{ ;r" Ls/0f
ljlwaf6 n]v .
pQ/
J = {100, 999} x'G5 .

cEof; 8.2

tn lbOPsf kT| os] ;dx" nfO{ ;r" Ls/0f tl/sfaf6 nv] M
1. gk] fnsf 7 k|b]zx¿sf] ;d"x .
2. 38Lsf] 8fondf ePsf ;ª\Vofx¿sf] ;d"x .
3. 12 dlxgfsf g]kfnL gfdx¿sf] ;d"x .
4. gk] fnsf] /fli6o« emG8fdf ko| fu] ul/Psf /ªx¿sf] gfdsf] ;dx" .
5. sIff 6 df k9\g'kg]{ ljifox¿sf] ;d"x .
6. 10 eGbf ;fgf lahf/] ;ªV\ ofx¿ (Odd numbers) sf] ;d"x .
7. 50 ;Ddsf 5 n] lgMz]if efu hfg] ;ªV\ ofx¿sf] ;d"x .

54 ul0ft, sIff ^

8. 20 ;Ddsf ¿9 ;ªV\ ofx¿ (Prime numbers) sf] ;d"x .
9. 20 ;Ddsf ;o+ S' t ;ªV\ ofx¿ (Composite numbers) sf] ;d"x .
10. 20 eGbf ;fgf 3 n] lgMz]if efu hfg] ;ªV\ ofx¿sf] ;d"x .
11. 1 bl] v 50 larsf Pssf] :yfgdf 4 cfpg] ;ª\Vofx¿sf] ;d"x .
12. 10 bl] v 50 larsf 5 n] efu ubf{ 2 z]if cfpg] ;ª\Vofx¿sf] ;d"x .
13. L = {3, 6, 9, 12, 15, 18} sf ;a} ;b:onfO{ 3 n] efu ubf{ cfpg efukmnx¿sf] ;d"x .
14. 15 / 25 bj' }nfO{ lgMz]if efu hfg] ;ªV\ ofsf] ;dx" .
15. olb A = {1, 2, 3, 4, 5} eP tnsf ;d"xx¿sf] ;"rL tof/ kf/L bv] fpm M

(i) B = ;dx" A sf kT| os] ;b:oaf6 1 36fp“bf aGg] ;d"x .
(ii) C = ;dx" A sf kT| os] ;b:onfO{ 3 n] u'0fg ubf{ aGg] ;d"x .
(iii) D = ;dx" A df ePsf hf]/ ;ª\Vofx¿sf] ;d"x .
(iv) E = ;dx" A df ePsf lahf]/ ;ª\Vofx¿sf] ;d"x .
(v) F = ;dx" A df ePsf] ;ae} Gbf ;fgf] / ;ae} Gbf 7n' f] ;ªV\ ofsf] ;dx" .
16. tn lbOPsf kT| os] ;dx" nfO{ zAbdf JofVof u//] nv] M

(i) R = {I, II, III, IV, V, VI, VII, VIII, IX, X}
(ii) C = {10, 12, 14, 16, 18, 20}
(iii) O = {21, 23, 25, 27, 29}
(iv) E = {a, b, c, d, e}

(v) U = {mm, cm, m, km}

17. kZ| g g=+ 1 bl] v 10 ;Ddsf ;dx" nfO{ ;dx" lgdf{0f ljlwaf6 n]v .

ul0ft, sIff ^ 55

8.3 ;dx" sf] ;b:otf (membership of a sets)

-s_ l/sfkLdf ePsf kmnkm" nx¿sf] ;dx" nfO{ F egf“} .
F = {cfk“ , s/] f, :ofp, cªu\ /' }
(i) of] ;d"xdf cf“k 5 .
To;n} ,] cfk“ ∈ {cfk“ , s/] f, :ofp, cªu\ /' }
cyft{ ,\ cfk“ ∈ F
To:t}, of] ;d"xdf s]/f klg 5 .
(ii) To;n} ,] s/] f ∈ {cfk“ , s/] f, :ofp, cªu\ /' }
cyft{ ,\ s/] f ∈ F
cfk“ , s/] f, :ofp / cgf/ F ;dx" sf ;b:ox¿ xg' \ . t/ of] ;dx" df ;G' tnf 5g} .
(iii) ;G' tnf ∉ {cfk“ , s/] f, :ofp, cªu\ /' }
cyft{ ,\ ;G' tnf ∉ F
cyft{ ,\ ;G' tnf ;dx" F df kb}{g .
;G' tnf ;dx" F sf] ;b:o xf]Og .

-v_ V = {a, e, i, o, u}
a ;dx" V sf] Pp6f ;b:o xf,] To;n} ] a ∈ V t/ b ∉ V, b ;dx" V sf] ;b:o xf]Og .
o ;dx" V sf] Pp6f ;b:o xf,] To;n} ] o ∈ V c ∉ V, c ;dx" V sf] ;b:o xf]Og .

lrx\g ∈ n] ;b:o xf] cyjf ;d"xdf kb5{ eGg] hgfp5“ .
lrx\g ∉ n] ;b:o xfO] g cyjf ;dx" df kb{g} eGg] hgfp5“ .

pbfx/0f 1

vfnL 7fpd“ f ∈ jf ∉ dWo] ldNg] lrx\g n]v M

(i) 3………. {1, 2, 3, 4} (ii) 5………… {1, 2, 3, 4}

56 ul0ft, sIff ^

(iii) H = { xKtfdf ;a} ljBfno aGb xg' ] lbgx¿sf] ;dx" }
dªu\ naf/ …………. H
zlgaf/ ……………. H

pQ/
(i) o; ;dx" df 3 k5,{ To;n} ] 3 of] ;d"xsf] ;b:o xf] .

3 ∈ {1, 2, 3, 4}
(ii) o; ;dx" df 5 kbg{} , To;n} ] 5 of] ;d"xsf] ;b:o xf]Og .

5 ∉ {1, 2, 3, 4}
(iii) dªu\ naf/ ljBfno aGb x“'b}g . To;n} ] dªu\ naf/ H ;d"xsf] ;b:o xf]Og . To;}n]

dªu\ naf/ ∉ H
zlgaf/ ljBfno aGb xG' 5 . To;n} ] zlgaf/ H ;dx" sf] ;b:o xf] . cyft{ \ zlgaf/ ∈ H

pbfx/0f 2

olb, P = {3/kfnj' f hgfj/x¿sf] ;dx" } /

W = {hªu\ nL hgfj/x¿sf] ;dx" } xg' \ eg] –

lgDglnlvt egfOdWo] l7s eP (T) / al] 7s eP (F) n]v M

(i) ss' /' ∈ W (ii) :ofn ∉ P (iii) ufO{ ∈ P (iv) af3 ∉ W

pQ/

(i) F, s's'/ hª\unL hgfj/ xf]Og .
(ii) T, :ofn 3/kfn'jf hgfj/ xf]Og .
(iii) T, ufO{ 3/kfn'jf hgfj/ xf] .
(iv) F, af3 hª\unL hgfj/ xf] .

cEof; 8.3

1. vfnL 7fpd“ f ∈ jf ∉ lrx\gdWo] ldNg] lrx\g n]v M

(i) 5 ………. {1, 2, 3, 4, 5}
(ii) 6 ………. {1, 2, 3, 4, 5}
(iii) 5 ……… {3, 5, 7, 11}

ul0ft, sIff ^ 57

(iv) 9 ……… {1, 3, 5, 7, 11} , }
(v) ........... { , ,

(vi) cm …….. {mm, cm, m, km}

2. W n] tfn} /] al] rg] j:tx' ¿sf] ;dx" hgfpg] eP vfnL 7fpd“ f ∈ jf ∉ dWo] ldNg] nv] M

(i) cfn' …………… W (ii) tn] ………… W

(iii) lrgL …………… W (iv) sk8f ……… W

(v) df;' …………… W (vi) kmn" ………== W

3. l7s eP (T) jf al] 7s eP (F) n]v M

(i) olb S n] ;fs{ /fi6x« ¿sf] ;dx" hgfp5“ eg,]

gk] fn ∉ S, ……… e6' fg ∈ S ……… rLg ∈ S ………

yfONofG8 ∉ S, ………… adf{ ∈ S ……… aªu\ nfbz] ∈ S ……

(ii) olb C n] ;a} bz] sf /fhwfgLx¿sf] ;dx" ae' mfp5“ eg,]

lbNnL ∈ C, …………, sf7df8f“} ∈ C, …… aO] lhª ∉ C ………

s/fr“ L ∈ C, ………, 9fsf ∈ C,……… lyDk" ∈ C …………

(iii) olb P = {df5f, df;,' km" n, bw' , u8] fu8' Lx¿} eP,

bw' ∈ P, ………, df;' ∉ P, ………, lrgL ∈ P ……… x'G5 .

4. olb A = {e, n, g, l, i, s, h} / B = {m, a, t, h, e, i, c, s} eP ;"rLs/0fåf/f ;d"x agfpm M

(i) A / B bj' } ;dx" df kg{] ;b:ox¿sf] ;dx" .

(ii) A df kg{] t/ B df gkg{] ;b:ox¿sf] ;dx" .

(iii) B df kg{] t/ A df gkg{] ;b:ox¿sf] ;dx" .

58 ul0ft, sIff ^

8.4 ;dx" sf] u0fgfTdstf (Cardinality of a set)

tn lbOPsf] pbfx/0f x]/ M

olb V n] cªu\ h]| L j0fd{ fnfsf :j/x¿sf] ;dx" hgfp5“ / E n] 11 eGbf ;fgf hf/] ;ªV\ ofx¿sf] ;dx"
hgfp5“ eg] V = {a, e, i, o, u} / E = {2, 4, 6, 8, 10} x'G5 .

oxf“ bj' } ;dx" x¿ V / E df ;b:ox¿sf] ;ªV\ of 5/5 5 . o;nfO{ ;ªs\ t] df lgDgfg;' f/ nl] vG5 M

n(V) = 5 / n(E) = 5

oxf“ n n] ;d"xsf] ;b:o ;ª\VofnfO{ hgfp“5 .

csf{] Pp6f pbfx/0f x/] f,“} Ps hgf lzIfsn] cfkm\ gf] b/fhdf 23 lsl;dsf lzIf0f ;fduL| x¿ /fvs] f
5g\ . b/fhdf ePsf ;fduL| x¿sf] ;dx" nfO{ F n] hgfpb“ f, n (F) = 23 x'G5 .

cyft{ \ ;dx" F sf ;b:ox¿sf] ;ªV\ of 23 x'G5 .

;dx" df ePsf ;b:ox¿sf] ;ªV\ ofnfO{ ;dx" sf] u0fgfTdstf (Cardinality or Cardinal number)
elgG5 .

;Lldt / c;Lldt ;dx" x¿ (Finite and Infinite Sets)

cªu\ h]| L j0fd{ fnfsf :j/x¿sf] ;dx" V = {a, e, i, o, u} xG' 5 . of] ;dx" df 5 cf]6f ;b:ox¿ 5g\ .
xKtfsf lbgx¿sf] ;dx" D = {cfOtaf/, ;fd] af/, dªu\ naf/, aw' af/, laxLaf/, zj' m| af/, zlgaf/}
o; ;dx" df 7 cf]6f ;b:ox¿ 5g\ .
gk] fnsf] /fhwfgLsf] ;dx" K = {sf7df8f}}“
o; ;d"xdf hDdf Pp6fdfq ;b:o 5 .
1 bl] v 100 ;Ddsf uGtLsf ;ªV\ ofx¿sf] ;dx" C = {1, 2, 3, …….. 98, 99, 100}
o; ;dx" sf] ;b:o ;ªV\ of 100 5g\ .
dflysf V, D, K / C rf/ cf]6} ;d"xx¿ ;Lldt ;d"x x'g\ .

;Lldt ;ªV\ ofdf ;b:ox¿ ePsf] cyjf ;b:o ;ªV\ of lglZrt ePsf] ;dx" nfO{ ;Lldt ;dx"
(Finite Set) elgG5 .

csf{] zAbdf, hg' ;dx" sf] ;b:o ;ªV\ of lglZrt ¿kdf olt g} xG' 5 eg/] eGg ;lsG5 To:tf] ;dx" nfO{
;Lldt ;d"x elgG5 .

ca, c¿ s]xL ;d"xx¿ x]/f}“ M

-s_ N = {1, 2, 3, 4, ……..}

ul0ft, sIff ^ 59

of] uGtL ;ªV\ ofx¿ (Counting Numbers) sf] ;d"x xf] .

cyft{ \ of] kf| sl[ ts ;ªV\ ofx¿ (Natural Numbers) sf] ;d"x xf] .

of] ;dx" df ;b:o ;ªV\ of slt 5 <

oxf,“ N sf] ;b:o ;ªV\ of slt xG' 5 <

lglZrt ¿kdf olt g} xG' 5 eg/] eGg ;lsG5 <

N Pp6f c;Lldt ;dx" (Infinite Set) xf] .

-v_ To:t,} hf/] ;ªV\ ofx¿sf] ;dx" E = {2, 4, 6, 8, 10, ……}

lahf/] ;ªV\ ofx¿sf] ;dx" O = {1, 3, 5, 7, 9, ……}

oxf“ klg E / O df olt g} ;b:o 5g\ eg/] lglZrt ¿kdf eGg ;lsb“ g} . oL ;dx" x¿df
c;Lldt ;b:ox¿ 5g\ .

To;n} ] E / O ;d"xx¿ c;Lldt ;d"x x'g\ .

;b:o ;ªV\ of lglZrt ¿kdf eGg g;lsg] jf cgGt ;b:ox¿ ePsf] ;dx" nfO{ c;Lldt ;dx" elgG5 .

o:tf] ;dx" nV] bf sx] L ;b:ox¿ nl] v;sk] l5 sDtLdf ltg cf6] f yfK] nfx¿ nV] g] rng 5 .

pbfx/0f 1

tn lbOPsf ;dx" x¿dWo] ;Lldt jf c;Lldt ;dx" 56' o\ fpm / ;Lldt ;dx" eP ;b:o ;ªV\ of klg
n]v M
(i) 10 eGbf ;fgf hf/] ;ªV\ ofx¿sf] ;dx"
(ii) 10 / 10 eGbf 7n' f hf/] ;ªV\ ofx¿sf] ;dx"
(iii) hf/] ;ªV\ ofx¿sf] ;dx"

pQ/
(i) 10 eGbf ;fgf hf/] ;ªV\ ofx¿sf] ;dx" nfO{ E1 dfgf“}

E1 = {2, 4, 6, 8}

E1 df hDdf 4 cf]6f ;b:o 5g\ .
To;n} ] E1 Pp6f ;Lldt ;d"x xf] .

(ii) 10 / 10 eGbf 7n' f hf/] ;ªV\ ofx¿sf] ;dx" nfO{ E2 dfgf“}

E2 = {10, 12, 14, 16 ……….}

E2 Pp6f c;Lldt ;d"x xf] .

60 ul0ft, sIff ^

(iii) hf/] ;ªV\ ofx¿sf] ;dx" nfO{ E3 dfgf“}

E3 = {2, 4, 6, 8, 10, 12, 14, 16 ……….}

E3 Pp6f c;Lldt ;d"x xf] .

pbfx/0f 2

Pssf] :yfgdf 5 cfpg] ;ªV\ ofx¿sf] ;dx" ;Lldt jf c;Lldt s:tf] ;dx" xf] <
pQ/

Pssf] :yfgdf 5 ePsf ;ªV\ ofx¿ M 5, 15, 25, 35, ……….
Pssf] :yfgdf 5 ePsf ;ªV\ ofx¿sf] ;dx" nfO{ F dfgf}“ .

F = {5, 15, 25, 35 ……..}

F Pp6f c;Lldt ;d"x xf] .

cEof; 8.4

1. tn lbOPsf ;dx" x¿dWo] sg' sg' ;Lldt ;dx" xg' \ < ;Lldt ;dx" sf] ;b:o ;ªV\ of klg nv] M

(i) A = {1, 3, 5, 7}
(ii) B = {1, 3, 5, 7, ……….,49}
(iii) C = {2, 4, 6, 8, ……., 100, 102, …..}

(iv) D = {100, 102, 104, 106, ……}

2. ;"rLs/0f ljlwaf6 ;d"x agfpm . To;df ;Lldt jf c;Lldt ;d"xx¿ 56' o\ fpm . ;Lldt eP
;b:o ;ª\Vof klg n]v M

(i) O1 = 20 eGbf ;fgf lahf/] ;ªV\ ofx¿sf] ;dx"
(ii) O2 = 20 bl] v 40 ;Ddsf lahf/] ;ªV\ ofx¿sf] ;dx"
(iii) O3 = 40 eGbf dflysf hf/] ;ªV\ ofx¿sf] ;dx"
(iv) T1 = Pssf] :yfgdf 3 cfpg] ;ªV\ ofx¿sf] ;dx"
(v) T2 = Pssf] :yfgdf 3 cfpg] 33 eGbf ;fgf ;ªV\ ofx¿sf] ;dx"
(vi) T3 = Pssf] :yfgdf 3 cfpg] 3 / 50 larsf ;ªV\ ofx¿sf] ;dx"
(vii) T4 = Pssf] :yfgdf 3 cfpg] 50 eGbf dflysf ;ªV\ ofx¿sf] ;dx"
(viii) 5 n] efu ubf{ 1 zi] f cfpg] ;ªV\ ofx¿sf] ;dx"

(ix) 5 n] efu ubf{ 1 zi] f cfpg] 1 bl] v 50 ;Ddsf ;ªV\ ofx¿sf] ;dx"
(x) D = b;sf] :yfgdf 4 cfpg] ;ªV\ ofx¿sf] ;dx"

ul0ft, sIff ^ 61

8.5 ;dtN' o / a/fa/ ;dx" x¿ (Equivalent and Equal Sets)

;dtN' o ;dx" x¿ (Equivalent sets)

olb X = {a,b} / Y = {1,2} 5g\ eg] Pp6f ;dx" X sf ;b:onfO{ csf{] ;dx" Y sf] ;b:o;u“
cnu cnu tl/sfaf6 hf8] f ldnfpg ;lsG5, h:t} M

XY XY

a1 a 1

b 2b 2

bj' } tl/sfdf X df ePsf kT| os] ;b:on] ;dx" Y df Ps Ps ;b:o hf8] f kfPsf] 5 . oxf“ ;d"xx¿
X / Y sf ;b:ox¿ lar Ps Ps ;ªu\ lttf (One to one correspondence) /x]sf] 5 . ;d"xx¿
X / Y sf ;b:ox¿sf] ;ªV\ of a/fa/ 5g\ .

dflysf lrqdf ;dx" x¿ X / Y cfk;df ;dt'No 5g\ .

olb bO' c{ f6] f ;dx" x¿ A / B df ePsf ;b:ox¿sf] ;ªV\ of a/fa/ 5 eg] n(A) = n(B) nV] g
;lsG5 . ;b:ox¿ ;ªV\ of a/fa/ ePsf ;dx" x¿nfO{ ;dtN' o ;dx" elgG5 / A~B n]lvG5 .

pbfx/0f 1

olb A = {10 eGbf ;fgf uGtLsf ;ªV\ ofx¿} / B = {18 eGbf ;fgf lahf/] ;ªV\ ofx¿}
eP n(A) / n(B) kQf nufpm .
s] n(A) = n(B) 5 <
s] ;dx" A / B ;dtN' o ;dx" x¿ xg' \ <

pQ/

oxf“, A = {1, 2, 3, 4, 5, 6, 7, 8, 9}
/ B = {1, 3, 5, 7, 9, 11, 13, 15, 17}
∴ n (A) = 9 / n (B) = 9 5 .

dfly n(A) / n(B) bj' s} f dfg 9 ePsfn] n(A) = n(B) xG' 5 . To;n} ] ;dx" A / B df ;dtN' o ;dx" x¿
x'g\ .

62 ul0ft, sIff ^

pbfx/0f 2

olb ;dx" P = {5 sf u0' fgv08x¿}
/ ;dx" Q = {6 sf u0' fgv08x¿} eP,
s] ;dx" P / ;dx" Q ;dtN' o ;dx" x¿ xg' \ <
pQ/ M
oxf,“ P = {1, 5} / n(P) = 2
kml] /, Q = {1, 2, 3, 6} / n(Q) = 4
∴ n (P) ≠ n(Q) 5 . To;}n] P / Q ;dt'No ;d"xx¿ xf]Ogg\ .

a/fa/ ;dx" x¿ (Equal sets)

;dx" A n] 1 eGbf 7n' f / 4 eGbf ;fgf uGtLsf ;ª\Vof hgfp“5 . ;d"x B n] 6 sf ¿9 u0' fg
v08x¿sf] ;dx" hgfp5“ eg] A = {2, 3} / B = {2, 3} xG' 5 .

oxf,“ bj' } ;dx" A / B sf ;b:ox¿sf af/d] f s] eGg ;S5f“} <

olb bO' c{ f6] f ;dx" sf ;b:ox¿ plTts} / pxL 5g\ eg] tL bO' { ;dx" x¿nfO{ a/fa/ ;dx" x¿ elgG5 .
dflysf] pbfx/0fdf ;dx" x¿ A / B a/fa/ ;d"xx¿ x'g\ . o;nfO{ A = B n]lvG5 .

tn lbOPsf] pbfx/0f x/] L sIff sf7] fdf 5nkmn u/ M

olb A = {2, 4, 6, 8, 10} / B = {a, b, c, d, e} 5 eg] n(A) = n(B) = 5 xG' 5
t/ A ≠ B, lsg < 5nkmn u/ .

vfnL ;dx" (Empty Sets)

tn lbOPsf pbfx/0fx¿df ;dx" sf] ;b:o ;ªV\ ofaf/] 5nkmn u/ M

M = { sIff 6 df k9fpg' xg' ] lzIfsx¿sf] ;dx" }

N = {2 / 7 larsf] k0" f{ 3g ;ªV\ ofx¿sf] ;dx" }

T = {200 jife{ Gbf a9L afr“ s] f dflg;sf] ;dx" }

dflysf ;dx" nfO{ ;r" Ls/0f ljlwåf/f sdnfnfO{ nV] g lbOP5 . pgn] ;dx" M sf ;b:ox¿ nV] g
;lsg\ t/ pgn] N / T ;dx" df kg{] Pp6f klg o:tf] ;b:o kfOgg\ lsgeg] 2 / 7 sf] lardf kg{] 3g
;ªV\ of g} 5g} . To:t} Pp6f klg dflg; hf] 200 jife{ Gbf a9L afr“ s] f] ;b:o kfOgg\ . To;n} ] pgn]
o;nfO{ lgDgfg;' f/ nl] vg\ M

N={ }

T={ }

ul0ft, sIff ^ 63

o;/L pgn] agfPsf] o; ;dx" nfO{ vfnL ;dx" (Empty Set) elgG5 . s'g} klg ;b:o gePsf]
;dx" nfO{ vfnL ;dx" elgG5 / o;nfO{ φ (Phi) jf { } n] hgfOG5 .

cEof; 8.5

1. tnsf ;dx" x¿af6 a/fa/ ;dx" 56' o\ fpm / (=) lrx\g k|of]u u/L n]v M

A = {2 ,4, 6}, B = {y, x}, C = {1, 3, 5, 7}, D = {x, y}, E = {1, 4, 9, 16}

F = {cªu\ h]| L j0fd{ fnfsf :j/x¿}, G = {2, 6}, H = {4, 2, 6}, I = {x, y, z},
J = {9, 4, 1, 16}, K = {klxnf 4 cf6] f lahf/] ;ªV\ ofx¿}, L = {a, e, i, o, u}

M = {n, i, l, e}, N = {r, e, a, d}, O = {d, e, a, r}, P = {l, i, n, e}

2. kZ| g g= 1 af6 ;dtN' o ;dx" x¿ 56' o\ fO{ (~) lrx\g k|of]u u/L n]v .
3. tnsf kT| os] ;dx" x¿dWo] sg' sg' a/fa/ ;dx" xg' ,\ a/fa/ eP '=' lrx\g k|of]u u/L n]v M

-s_ A = {2, 3, 5, 7}, B = {8 eGbf ;fgf ¿9 ;ªV\ ofx¿}
-v_ C = {p, q, r, s}, D = {r, q, p, s}
-u_ E = {A, B, C, D}, F = {a, b, c, d}
-3_ G = {G, O, L, F}, H = {F, L, O, G}
-ª_ I = {l, e, a, d}, J = {d, e, a, l}
-r_ K = {M, I, S, H, P}, L = {MISHISSIPPI df ko| fu] ePsf cªu\ h]| L cIf/x¿}
-5_ M = {2 n] efu hfg] ;ªV\ ofx¿}, N = {hf/] ;ªV\ ofx¿}
-h_ O = {d{ ªu\ n, ax[ :klt, c?0f}, P = {;fo} d{ 08nsf sg' } ltg ux| x¿}
-em_ Q = {aw' , zj' m| , kY[ jL}

R = {;fo} d{ 08nsf ;o" s{ f] glhsaf6 jm| dzM cfpg] klxnf tLg ux| x¿}
-`_ S = {1, 3, 5, 7, 9}, T = {10 eGbf ;fgf lahf/] ;ªV\ ofx¿}
-6_ U = {e, a, t}, V = {t, e, a}

4. kZ| g g= 3 sf sg' sg' ;dtN' o ;dx" x¿ a/fa/ 5g} g\ <

5. ;r" Ls/0f ljlwaf6 tnsf ;dx" x¿nfO{ nv] M
A = {1 bl] v 9 ;Ddsf k0" f{ ;ªV\ ofx¿}
B = {10 / 26 lardf /xs] f hf/] ;ªV\ ofx¿}
C = {1 bl] v 50 ;Ddsf 7 sf ckjTox{ ¿}

64 ul0ft, sIff ^

ca, lgDglnlvt k|Zgx¿sf] pQ/ bp] m M
-s_ n(A), n(B) / n(C) slt slt 5g\ <
-v_ sg' sg' bO' { ;dx" x¿ ;dtN' o 5g\ <

6. olb A = {1, 2, 3, 4} / B = {1, 2, 3, 4, 5} ePdf s] A = B xG' 5 < s] n(A) / n(B) Pp6} xg'
cfjZos 5 <

7. X n] 2310 sf ¿9 u0' fg v08x¿sf] ;dx" nfO{ hgfp5“ / Y n] 13 eGbf ;fgf ¿9
;ªV\ ofx¿sf] ;dx" nfO{ hgfp5“ eg,]

-s_ ;dx" X / Y sf ;b:ox¿sf larsf] Ps Ps ;ªu\ ltsf lrq agfP/ bv] fpm .

-v_ s] X = Y nV] g ;lsG5, lsg <

8. vfnL 7fp“ e/ M
-s_ olb A = {1, 2, 3} eP, n(A) = ……….
-v_ olb P = {w, a, y, b} eP, n(P) = ……..
-u_ olb R = {i, c, e, r, m} eP, n(R) = ……….
-3_ olb N = {2, 3, 4, 5, 6} eP, n(N) = …….....

9. olb A = {0, 2, 4, 6}, B = {2, 4, 6}, C = {0}, D = { }, E = {2, 4, 6} / F = {1} eP tnsf
k|Zgx¿sf] pQ/ n]v M
-s_ k|To]s ;d"xsf] ;b:o ;ª\Vof n]v .
-v_ sg' sg' ;dx" df ;b:o ;ªV\ of a/fa/ 5g\ <
-u_ ;b:o ;ªV\ of a/fa/ eP klg cfk;df a/fa/ gxg' ] ;dx" x¿ sg' sg' xg' \ <
-3_ sg' sg' ;dx" x¿ a/fa/ 5g\ <

10. tnsf ;dx" x¿dWo] sg' sg' vfnL ;dx" xg' ,\ vfnL ;dx" sf cufl8 φ -kmfO{_ /fVb} hfpm M
-s_ 3 / 4 sf] lardf /xs] f k0" f{ ;ªV\ ofx¿sf] ;dx"

-v_ 3 / 4 sf] lardf /xs] f ;ªV\ ofx¿sf] ;dx"
-u_ 5 jife{ Gbf ;fgf] pd/] sf sIff 6 df k9g\ ] ljBfyLx{ ¿sf] ;dx"
-3_ 2 n] efu hfg] lahf/] ;ªV\ ofsf] ;dx"
-ª_ hf/] ¿9 ;ªV\ ofsf] ;dx"
-r_ { 0 }
-5_ { }
-h_ 13 / 15 sf lardf /x]sf ¿9 ;ª\Vofx¿sf] ;d"x .

ul0ft, sIff ^ 65

PsfO 9 k0" f{ ;ªV\ ofx¿ (Whole Numbers)

9.1 k0" f{ ;ªV\ ofx¿sf] ljsf; (Development of whole numbers)
tn lrqdf lbOPsf kZ| gx¿sf] pQ/ nv] M

 dflysf kT| os] ;dx" df sltcf6] f km' nx¿ 5g\ <
 ltdf| ] sIffdf slt hgf ljBfyLx{ ¿ 5g\ <
 ltdf| ] kl/jf/df slt hgf ;b:o¿ 5g\ <
 gk] fndf sltcf6] f c~rnx¿ 5g\ <

o;/L cfpg] ;ªV\ ofx¿nfO{ 1 b]lv n]Vb} hf“bf cgGt;Dd hfG5 . ;ª\Vof 1, 2, 3, 4, …. 10, 11, 12,
……, 100, 101 ……. nfO{ k|fs[lts ;ª\Vof eGb5g\ .
To;f] eP kf| sl[ ts ;ªV\ of eGgfn] sxfb“ l] v ;?' xG' 5 / sxf“ cGTo xG' 5 <

kf| sl[ ts ;ªV\ ofx¿ (Natural Numbers) eGgfn] uGtL (Counting Numbers) sf ;ªV\ ofx¿ xg' \ . of]
1 bl] v ;?' xG' 5 / cgGt;Dd hfG5 . kf| sl[ ts ;ªV\ ofx¿sf] ;dx" nfO{ N n] hgfOG5 .
N df ;b:o ;ªV\ of c;Lldt (Infinite) x'G5g\ . N = {1, 2, 3, 4, 5, ………………….}

1 2 3 4 5 6 7 8 9 10

tnsf k|Zgx¿sf klg pQ/ vf]hf}“ M

 gk] fndf sltcf6] f ;db' | 5g\ <
5g} g\ cyft{ \ zG" o (0) cf]6f 5g\ .

 ltdf| ] sIffdf 20 jife{ Gbf a9L pd/] sf slt hgf ljBfyL{ 5g\ <
5g} g\ cyft{ \ zG" o (0) hgf 5g\ .

kf| sl[ ts ;ªV\ ofx¿n] dfq k0" f{ Jofjxfl/s ;d:of ;dfwfg xb“' g} . kf| sl[ ts ;ªV\ ofx¿sf] k0" f{ ;dx"
agfpg '0' yKbf aGg] ;dx" nfO{ k0" f{ ;ªV\ ofx¿ (Whole Numbers) elgG5 . o;nfO{ W n] hgfOG5 .

W = {0, 1, 2, 3, 4, 5, 6, ..............}

01234

66 ul0ft, sIff ^

lbOPsf] lrqdf lzIfsn] kfr“ cf6] f cfn“} fx¿ bv] fpb“ } xg' x' G' 5 .
lbOPsf] lrqsf] aG] rdf kfr“ hgf ljBfyL{ 5g\ .
lbOPsf] lrqsf] 6a] n' df kf“rcf6] f k:' tsx¿ 5g\ .

lrqdf,
ljBfyLs{ f] ;ªV\ of = k:' tssf] ;ªV\ of = Pp6f xftsf
cfn“} fx¿sf] ;ªV\ of = 5
of] ;ªV\ ofnfO{ tLg hgf ljBfyLn{ ] ltg ks| f/n] nv] /] JoSt u/] .
% – kfr“ ;ªV\ of hgfpg] of] ;ªs\ t] (Symbol) bj] gfu/L ;ªV\ ofªs\ (Devanagari Numeral) xf] .
5 - kfr“ ;ªV\ of hgfpg] of] ;ªs\ t] lxGb' c/l] as ;ªV\ ofªs\ (Hindu Arabic Numeral) xf] .
V - kf“r ;ªV\ of hgfpg] of] /fd] g ;ªV\ ofªs\ (Roman Numeral) xf] .
ltg cf]6} ;d"xsf] ;ª\Vof a/fa/ 5g\ .
t/ logLx¿nfO{ hgfpg] ;ª\s]tx¿ (Symbols) cyf{t\ ;ª\Vofª\s (Numeral) km/s km/s 5g\ .

;ªV\ ofnfO{ ljleGg ;ªs\ t] x¿n] hgfOG5 . o;/L ;ªV\ of (Number) nfO{ hgfpg] ljleGg
;ªs\ t] x¿nfO{ ;ªV\ ofªs\ (Numeral) elgG5 .

;ªV\ ofnfO{ nV] g 7fpc“ g;' f/ ;;+ f/sf ljleGg dflg;x¿n] ljleGg ks| f/sf cªs\ x¿ ko| fu] u/L
cfPsf 5g\ . tLdWo] sIff 4 bl] v ko| fu] ub{} cfPsf lxGb' c/l] as cªs\ x¿ 0, 1, 2, 3, 4, 5, 6, 7,
8, 9 x'g\ .

1 lbgdf 24 306f x'G5 . 365 lbgdf 365 × 24 = 8,760 306f x'G5 .

lxGb' c/l] as ;ªV\ ofªs\ g k0| ffnLdf 0 bl] v 9 ;Dd hDdf b; cf6] f cªs\ x¿ (Digits) dfq ko| fu]
ul/G5 . cª\sx¿ hltk6s klg bf]xf]¥ofpg ;lsG5 . 0 bl] v 9 ;Dd cªs\ x¿ ko| fu] u/L hlt;s' }
7'nf] ;ª\Vof klg n]Vg ;lsG5 .

Ps cªs\ b'O{ cª\s b'O{ cª\s ltg cªs\ ltg cª\s rf/ cªs\

8 9 10 98 99 100 998 999 1000

Ps cªs\ n] ags] f] ;ae} Gbf ;fgf] ;ªV\ of 1
Ps cªs\ n] ags] f] ;ae} Gbf 7n' f] ;ªV\ of 9
bO' { cªs\ n] ags] f] ;ae} Gbf ;fgf] ;ªV\ of 10

ul0ft, sIff ^ 67

bO' { cªs\ n] ags] f] ;ae} Gbf 7n' f] ;ªV\ of 99

tLg cªs\ n] ags] f] ;ae} Gbf ;fgf] ;ªV\ of 100

tLg cªs\ n] ags] f] ;ae} Gbf 7n' f] ;ªV\ of 999

oL ;ªV\ ofx¿ x/] f“} – 120, 201, 102, 210

oL ;a} ;ªV\ ofx¿df 0, 1, 2 u/L hDdf ltg cf6] f cªs\ x¿ ko| fu] ul/Psf 5g\ . s] oL ;a}
;ªV\ ofx¿sf] dfg (Value) Pp6} xf] <

;ªV\ ofªs\ 210 sf] :yfgdfgnfO{ :yflkt u/L uGbf ;osf] :yfgdf 2 5 / oxf“ 2 sf] :yfgcg;' f/sf]
dfg 200 x'G5 . To:t}, bzsf] :yfgdf 1 5, 1 n] 10 / To:t} Pssf] :yfgdf 0 5, 0 n] 0 hgfp“5 .
o;sf] sn' dfg (Total value) = 200 + 10 + 0 = 210 x'G5 .
lxGb' c/l] as ;ªV\ ofªs\ g k0| ffnLdf,
cªs\ x¿ Pp6} eP klg :yfgcg;' f/ To;sf] dfg km/s km/s xG' 5 . vfnL :yfgdf zG" o '0' /flvG5 .

O;fsf] 100 jifk{ Zrft\ cyft{ \ 100 A.D. lt/ lxGbx' ¿n] lxGb' c/l] as ;ªV\ ofªs\ g k0| ffnLcg;' f/ 0,
1, 2, 3, 4, 5, 6, 7, 8 / 9 u//] hDdf b; cf6] f cªs\ x¿ ko| fu] u//] ;ªV\ of nV] g] k4ltsf] ljsf;
u/] / c/aLx¿n] o; k0| ffnLnfO{ ;;+ f/el/ kr| f/k;| f/ u/] . To;sf/0f of] ;ªV\ ofªs\ g k4ltnfO{
lxGb' c/l] as ;ªV\ ofªs\ g k0| ffnL (Hindu Arabic Numeration System) eg/] ;;+ f/e/ kl| ;l4 kfPsf]
5.

lxGb' c/l] as ;ªV\ ofªs\ g k4ltsf lgDglnlvt ljzi] ftfx¿ xfdLn] 5nkmn u¥of“} M
1. 0 bl] v 9 ;Dd hDdf bzcf6] f cªs\ x¿ ko| fu] u/L hlt;s' } 7n' f] ;ªV\ of klg nV] g ;lsG5 .

2. cªs\ x¿sf] dfg ltgsf] :yfgcg;' f/ xg' ] xg' fn] 7n' f ;ªV\ ofx¿ nV] g ltgsf] :yfgdfq abn]
k'U5 .

3. o; k4ltdf zG" o '0' klg ePsf] xg' fn] vfnL :yfgdf klg '0' /fv]/ n]Vg ;lhnf] ePsf] 5 .

/fd] g ;ªV\ ofªs\ g k0| ffnL / cGo ;ªV\ ofªs\ g k0| ffnLdf zG" o '0' sf] cefj ePsfn] Tolt ljsf; /
k;| f/ xg' ;sg] t/ lxGb" c/l] as ;ªV\ ofªs\ g k0| ffnLdf '0´ sf] cfljisf/n] cªs\ x¿nfO{ :yfgcg;' f/sf]
dfg (Place Value) lbg ;Dej eof] . yf]/} cª\sn] klg 7'nf 7'nf ;ª\Vofx¿ n]Vg ;lhnf] x'g uof] .
To:t} hf8] , 36fp h:tf cfwf/et" ljm| ofx¿ klg ug{ ;lhnf] eof] . oxL sf/0fn] lxGb' c/l] as
;ªV\ ofªs\ g k0| ffnL nfdf] ;dobl] v ;kmntfkj" s{ ko| fu] eO{ cfPsf] xf] .

68 ul0ft, sIff ^

pbfx/0f 1

2, 5 / 7 af6 aGg] ;ae} Gbf 7n' f] ;ªV\ of / ;ae} Gbf ;fgf] ;ªV\ ofsf] km/s kQf nufpm .

pQ/

2, 5 / 7 df ;ae} Gbf 7n' f] ;ªV\ of 7 / ;ae} Gbf ;fgf] 2 xf] .

To;n} ,] 2, 5 / 7 n] aGg] ;ae} Gbf 7n' f] ;ªV\ of 752 / ;ae} Gbf ;fgf] ;ªV\ of 257 xg' \ . oL bO' {
;ªV\ ofx¿sf] km/s = 752 – 257 = 495

pbfx/0f 2

7, 0 / 8 af6 aGg] tLg cªs\ sf] ;ae} Gbf 7n' f] ;ªV\ of / ;ae} Gbf ;fgf] ;ªV\ ofsf] km/s kQf nufpm .

pQ/

oxf,“ 7, 0 / 8 df ;ae} Gbf 7n' f] ;ªV\ of 8 / ;ae} Gbf ;fgf] ;ªV\ of 0 xf] . To;}n] tLg cª\ssf]
;ae} Gbf 7n' f] ;ªV\ of 870 / ;ae} Gbf ;fgf] ;ªV\ of 708 xf] .

km/s 870 – 708 = 162

cEof; 9.1
1. tnsf ;ªV\ ofx¿df 5 sf] :yfgdfg n]v M

(i) 6503 (ii) 5761 (iii) 45678 (iv) 23456

2. (i) 5, 7, 9 af6 sg' sg' ;ªV\ ofx¿ aGg ;S5g\ <

(ii) sg' tLgcf6] f km/s km/s cªs\ x¿af6 ltg cªs\ sf] ;ae} Gbf ;fgf] ;ªV\ of aG5,
Tof] ;ªV\ of slt xG' 5 <

(iii) sg' } tLg cf6] f km/s km/s cªs\ x¿af6 tLg cªs\ sf] ;ae} Gbf 7n' f] ;ªV\ of aG5, Tof]
;ªV\ of slt xG' 5 <

3. 1, 7, 0, 2, 3 af6 aGg ;Sg] ;ae} Gbf 7n' f] / ;ae} Gbf ;fgf] ;ªV\ of nv] L ltgsf] ofu] kmn kQf
nufpm .

4. rf/ cªs\ x¿n] ags] f] ;ae} Gbf 7n' f] / ;ae} Gbf ;fgf] ;ªV\ of nv] L ltgsf]
(i) ofu] kmn / (ii) cGt/ kQf nufpm .

ul0ft, sIff ^ 69

9.2 sfi] 7x¿;lxtsf] ;/nLs/0f (Simplification with brackets)

8 / 12 sf] hf8] kmnnfO{ 4 n] efu ubf{ 5 xG' 5 .
8 / 12 sf] hf8] 20 xG' 5 / 20 nfO{ 4 n] efu ubf{ 5 x'G5 .
o;nfO{ ul0ftLo efiffdf JoSt u/L xn ubf,{
oxf,“ klxnf 8 / 12 sf] hf8] nfO{ Pp6} ;ªV\ of dfg/] ;fgf] sfi] 7 - _ leq /fVgk' b5{ .
clg dfq 4 n] efu ug'{kb{5 .
To;n} ] ul0ftLo jfSodf (8 + 12)÷ 4

= 20 ÷ 4
=5

To;sf/0f, ;/nLs/0f ubf{ ;w“} klxnf sfi] 7leqsf] lx;fa ugk{' b5{ .

pbfx/0f 1

;/n u/ M 8 ÷ (4 x 2)
pQ/ M

8 ÷ (4 x 2)
=8÷8
=1

pbfx/0f 2

;/n u/ M 55 ÷ 11[120 ÷ 2{4 + (10 + 5 – 7)}]

pQ/ M klxnf ;fgf] sfi] 7leq + lj|mof u/]sf] .
;fgf] sfi] 7leq — lj|mof u/]sf] .
55 ÷ 11[120 ÷ 2{4 + (10 + 5 – 7)}] demfn} f sfi] 7leqsf] + lj|mof u/]sf] .
= 55 ÷ 11[120 ÷ 2{4 + (15 – 7)}] demf}nf sf]i7 x6fPsf] .
= 55 ÷ 11[120 ÷ 2{4 + 8}] 7n' f] sfi] 7leqsf] ÷ lj|mof u/]sf] .
= 55 ÷ 11[120 ÷ 2{12}] 7'nf] sf]i7 x6fOPsf] .
= 55 ÷ 11[120 ÷ 24] ÷ lj|mof u/]sf] .
= 55 ÷ 11[5]
= 55 ÷ 55
=1

To;sf/0f,
 ;ae} Gbf klxnf ;fgf] sfi] 7 - _, clg demfn} f sfi] 7 { } / cGTodf 7'nf sfi] 7 [ ] leqsf ljm| ofx¿

ug'{kb{5 .

 sfi] 7leqsf ljm| ofx¿ ubf{ klxnf u0' fg / efu u/k] l5 hf8] / 36fpm ugk{' 5{ .

70 ul0ft, sIff ^

 u0' fg / efu tyf hf8] / 36fpdf klxnf cfPsf] ljm| of klxnf ugk{' 5{ .

 sfi] 7leqsf clGtd ljm| of u/L ;s/] Pp6fdfq ;ªV\ of afs“ L ePkl5 sfi] 7 x6fpgk' b5{ . o;/L
sfi] 7 x6fpb“ f sfi] 7;u“ sg' } ljm| ofsf] lrxg\ 5g} eg] sfi] 7sf] ;66\ f sfi] 7leq / aflx/sf] ;ªV\ of
u'0fg u/]/ /fVg'kb{5 .

pbfx/0f 3

ul0ftLo jfSodf n]vL xn u/ M

70 / 50 sf] ofu] kmnnfO{ 10 n] efu u/L efukmndf 13 hf8] L kml] / 5 n] efu ubf{ slt xG' 5 <

pQ/ M

ul0ftLo jfSodf nV] bf,

[{(70 + 50) ÷ 10} + 13] ÷ 5

xn ubf,{

cfjZos dfg = [{120 ÷ 10} + 13] ÷ 5

= [12 + 13] ÷ 5
= 25 ÷ 5

=5

pbfx/0f 4

;/n u/ M 80 ÷ 4(2 + 3) x 6
= 80 ÷ 4(5) x 6 -sfi] 7leq lx;fa u/s] f_]
= 80 ÷ 20 x 6 -sfi] 7 x6fpg 4 n] 5 nfO{ u0' fg u/s] f_]

=4x6
= 24

cEof; 9.2 2. 21 ÷ 3(10 - 3)
4. 3 + (6 × 12) ÷ 6
-s_ ;/n u/ M
6. 36 + (16 + 2 x 4 - 4)
1. 25 - (16 + 3)
8. 48 ÷ 3 (12 x 4 ÷ 2 – 20)
3. (39 + 16) ÷ 11 10. 3 {12 + 8 ÷ 2 (2 x 2)}
5. 27 ÷ (13 - 4) × 5 12. (22 + 16 x 2) ÷ (27 ÷ 9 x 3)
7. 39 ÷13(15 – 48 ÷ 4) 14. 26 – 3{24 ÷ (18 ÷ 6)}
9. 3 {12 + (8 ÷ 4 x 2)} 16. 39 – 4 {16 ÷ (7 - 3)} – 23
11. 4 {6 + 2(7 - 4)} ÷ 6 18. (20 -5 -10) ÷ {2(7 - 4)-1}
13. 16 – 8 {15 – (45 ÷ 3)}
15. 35 – 7{42 ÷ (56 ÷ 8)} + 7 71

17. {45 – (28 + 17)} x 4

ul0ft, sIff ^

-v_ ul0ftLo jfSodf n]vL xn u/ M
1. 49 nfO{ 7 n] efu u/L efukmnaf6 7 36fpm .
2. 52 nfO{ 13 n] efu u/L 4 hf]8 .
3. 3 nfO{ 5 n] u0' fg u//] 15 n] efu u/ .
4. 12 nfO{ 3 n] u0' fg u//] 9 n] efu u/ .
5. 12 df 3 / 5 sf] u'0fgkmn hf]8 .
6. 16 af6 3 / 4 sf] u0' fgkmn 36fpb“ f slt xG' 5 <
7. 3 / 4 sf] u0' fgkmnsf] 5 u0' ffaf6 25 36fpb“ f slt xG' 5 <
8. 10 / 7 sf] km/ssf] 6 u0' ffnfO{ 9 n] efu ubf{ efukmn slt xG' 5 <
9. 20 / 6 sf] km/snfO{ 7 / 2 sf] u0' fgkmnn] efu ubf{ efukmn slt xG' 5 <
10. 15 nfO{ 3 n] efu ubf{ cfpg] efukmn / 2 sf] u0' fgkmnaf6 10 36fpb“ f slt xG' 5 <

9.3 efHotf;DaGwL k/LIf0f (Divisibility test)

tnsf] pbfx/0f x]/f}“ M

347 nfO{ 8 n] / 234 nfO{ 3 n] efu u/L ;"qcg';f/ hf“r]/ x]/f}“ .

;q" , efukmn x efhs + zi] f = efHo
hf“r 43
43
→ ×8
8) 347

– 32 344

27 +3

- 24 347

3 zi] f zi] f cfof] . 78
78 hf“r

3) 234 → ×3

- 21 234

24 +0

- 24 234

0 lgMz]if efu nfUof] .

72 ul0ft, sIff ^

ca s:tf] ;ªV\ ofnfO{ s:tf] ;ªV\ ofn] lgMzi] f efu nfU5, sx] L pbfx/0f x/] f“} .

;ª\Vof efu hfg] ;ª\Vofsf] :j¿k pbfx/0f

2 s'g} ;ª\Vofsf] clGtd cª\s z"Go '0' jf hf]/ 30, 50, 100, 134, 758,
(Even) 5 eg] To:tf] ;ª\VofnfO{ 2 n] 1296 nfO{ 2 n] efu
hfG5 .
lgMz]if efu hfG5 .

3 olb s'g} ;ªV\ ofsf] cª\sx¿sf] 3252  3+2+5+2 = 12
ofu] kmnnfO{ 3 n] efu hfG5 eg] To:tf] 12 nfO{ 3 n] efu hfG5 .
;ª\VofnfO{ 3 n] lgMzi] f efu hfG5 .

5 s'g} ;ªV\ ofsf] clGtddf '0' jf 5 5 eg] 5 50,360 / 123,800 df
n] lgMz]if efu hfG5 . clGtd cª\s ‘)’ 5 .
175,435 / 193,895 df
clGtd cª\s ‘5’ 5 .

7 olb sg' } ;ªV\ ofsf] clGtdsf] ;ª\Vofsf] b'O{ 924 clGtd cª\s '4'
ug' f / af“sL cª\sn] ag]sf] ;ª\Vofsf] sf] bO' { ug' f 4 x 2 = 8
km/snfO{ 7 n] efu hfG5 eg] Tof] k'/}
;ªV\ ofnfO{ 7 n] lgMz]if efu afs“ L cª\sn] ags] f] ;ªV\ of
hfG5 .
92 – 8 = 84

84 nfO{ 7 n] efu hfG5 .

10 10 n] efu hfg] ;ªV\ ofsf] clGtddf (0) xG' 5 . 60, 100, 500, 15230

11. olb sg' } ;ªV\ ofsf] klxnf,] t;] f| ,] kfr“ f“}, ==== 9 5 9 6 4  (9+9+4) –
cª\ssf] of]u / bf];f| ,] rfy} f], 5}6f}“ =====
cªs\ sf] ofu] sf] km/s 11 sf] ckjTo{ jf (5+6) = 22-11=11
0 xG' 5 eg] Tof] ;ª\VofnfO{ 11 n] lgMzi] f
efu hfG5 . 1 9 8 4 4  (1+8+4) -

(9+4) = 13 – 13 = 0

bj' } ;ª\VofnfO{ 11 n]
lgMz]if efu hfG5 .

cEof; 9.3

1. tn lbOPsf sg' sg' ;ªV\ ofx¿nfO{ 2 n] lgMz]if efu hfG5 <

(i) 7111 (ii) 2376 (iii) 9230 (iv) 352

(v) 23702 (vi) 97812 (vii) 2371 (viii) 9233

2. tn lbOPsf sg' sg' ;ªV\ ofx¿nfO{ 3 n] lgMzi] f efu hfG5 <

(i) 2376 (ii) 9235 (iii) 352 (iv) 23702 (v) 97812

3. kZ| g g=+ 2 sf sg' sg' ;ªV\ ofx¿nfO{ 2 / 3 bj' n} ] lgMzi] f efu hfG5 < s] 2 / 3 bj' n} ] lgMzi] f
efu hfg] ;ªV\ ofx¿nfO{ 6 n] klg lgMzi] f efu hfG5 <

ul0ft, sIff ^ 73

4. tn lbOPsf s'g s'g ;ª\VofnfO{ 5 n] lgMz]if efu hfG5 < s] 5 n] efu hfg]
;ªV\ ofnfO{ 10 n] klg lgMzi] f efu hfG5 <

(i) 1250 (ii) 35765 (iii) 123530 (iv) 2345

5. tnsf dWo] sg' sg' ;ªV\ ofnfO{ 7 n] lgMzi] f efu hfG5 <

(i) 245 (ii) 735 (iii) 3976 (iv) 8855

6. tn lbOPsf sg' sg' ;ªV\ ofx¿nfO{ 11 n] lgMzi] f efu hfG5 <

(i) 407 (ii) 10813 (iii) 3453572 (iv) 10888878

9.4 ckjTox{ ¿ / u0' fg v08x¿ (Multiples and Factors)

ckjTox{ ¿ (Multiples)

2×1=2

2×2=4

2×3=6 {2, 4, 6, 8, 10, 12, 14, 16, 18, 20, …………}
2×4=8
2 × 5 = 10 of] 2 sf ckjTo{ (Multiple) x¿sf] ;dx" xf] . o;nfO{
2 ´× 6 = 12 M(2) n] hgfpg ;lsG5 .

2 × 7 = 14

2 × 8 = 16

2 × 9 = 18

2 × 10 = 20

M(2) ;dx" df 20 kl5 sg' ;ªV\ of cfp5“ <
M(2) ;dx" df 20 kl5 nuQ} cfpg] tLgcf6] f ;ªV\ ofx¿ klg nv] f“} .
c¿ ckjTo{x¿sf] ;d"x klg x]/f}“ .

3 sf ckjTox{ ¿sf] ;dx" M(3) = {3, 6, 9, 12, 15 ……..}
4 sf ckjTox{ ¿sf] ;dx" M(4) = {4, 8, 12, 16, 20 ……}
5 sf ckjTox{ ¿sf] ;dx" M(5) = {5, 10, 15, 20, 25 …..}
M(5) sf kT| os] ;b:onfO{ 5 n] lgMz]if efu hfG5 .

74 ul0ft, sIff ^

M(7) = {7, 14, 21, 28 ……}

sfM(7) kT| os] ;b:onfO{ 7 n] lgMz]if efu nfU5 .

pbfx/0f 1

tnsf kT| os] ;dx" nfO{ ;r" Ls/0f ljlwåf/f nv] M

-s_ 35 eGbf ;fgf 4 sf ckjTox{ ¿sf] ;dx" A

-v_ 10 / 50 larsf 6 sf ckjTox{ ¿sf] ;dx" B

-u_ ;dx" A / B sf ;femf ckjTox{ ¿sf] ;dx" C

pQ/
-s_ 4 sf ckjTox{ ¿sf] ;dx" , M(4) = {4, 8, 12, 16, 20, 24, 28, 32, 36, 40,……}

35, eGbf ;fgf 4 sf ckjTox{ ¿sf] ;dx" , A = {4, 8, 12, 16, 20, 24, 28, 32}
-v_ 6 sf ckjTox{ ¿sf] ;dx" , M(6) = {6, 12, 18, 24, 30, 36, 42, 48, 54, 60, ….}

10 / 50 larsf 6 sf ckjTox{ ¿sf] ;dx" , B = {12, 18, 24, 30, 36, 42, 48}
-u_ ;dx" A / B sf ;femf ckjTox{ ¿sf] ;dx" , C = {12, 24}

u0' fgv08x¿ (Factors) o; u0' fg tflnsfaf6

u0' fg tflnsf u'0fgkmn
× 1 2 3 4 56
(Product)
1 1 2 3 4 56
2 2 4 6 8 10 12 1×6=6
3 3 6 9 12 15 18 2×3=6
4 4 8 12 16 20 24 3×2=6
5 5 10 15 20 25 30 6×1=6
6 6 12 18 24 30 36

To:t,} 1, 2, 3 / 6, 6 sf u'0fgv08x¿ x'g\ .
24 = 1 × 24 1, 2, 3 / 6 n] 6 nfO{ lgMz]if efu nfU5 .
24 = 2 × 12
24 = 3 × 8 24 sf u0' fgv08x¿ 1, 2, 3, 6, 8, 12 /
24 = 4 × 6 24 n] 24 nfO{ lgMzi] f efu hfG5 .
24 = 6 × 4
24 = 8 × 3 75
24 = 12 × 2
24 = 24 × 1

ul0ft, sIff ^

sg' } klg ;ªV\ ofsf u0' fgv08x¿ egs] f] Tof] ;ªV\ ofnfO{ lgMzi] f efu hfg] ;ªV\ ofx¿ xg' \ .

1 n] ;a} ;ª\VofnfO{ efu nfU5 . To;}n] 1 h'g;'s} ;ª\Vofsf] klg u'0fgv08 xf] . 0 afxs] sg' } klg
;ªV\ ofn] cfkmn“} fO{ ;w“} efu nfU5 . To;n} ] lbOPsf] ;ªV\ of cfkm“} g} Pp6f u0' fgv08 xf] . oxf“ 2,
3 / 6 tLg kf| sl[ ts ;ªV\ ofx¿, cyft{ \ 2 × 3 = 6 eP, 2 / 3 bj' n} ] 6 nfO{ efu nfU5 .

2 / 3 bj' } 6 sf u'0fgv08x¿ 5g\ .

2 × 1 = 2 xG' 5 / 2 cfkmn“} ] 2 nfO{ lgMz]if efu nfU5 .

To;sf/0f 2 / 1 bj' } 2 sf u'0fgv08x¿ x'g\ .

bi| 6Jo M zG" o (0) n] sg' } klg ;ªV\ ofnfO{ efu kl/eflift gePsfn] k0" f;{ ªV\ ofsf] ;dx" df u0' fgv08
eGgl] alQs} '0' afx]ssf k"0f{;ª\Vof eGg] a'lemG5 .

pbfx/0f 2

F(12) n] 12 sf u0' fgv08x¿sf] ;dx" / F(20) n] 20 sf u0' fgv08x¿sf] ;dx" hgfp5“ eg] F(12) /
F(20) sf ;femf ;b:ox¿sf] ;dx" sf] ;r" L agfP/ nv] .

pQ/ M = 1 × 12 To;n} ] F(12) = {1, 2, 3, 4, 6, 12}
= 2×6
oxf“, 12 = 3×4

12
12

20 = 1 × 20 To:t} F(20) = {1, 2, 4, 5, 10, 20}
20 = 2 × 10
20 = 4×5

F(12) / F(20) sf ;femf ;b:ox¿sf] ;dx" = {1, 2, 4}

cEof; 9.4 ul0ft, sIff ^

1. tnsf kT| os] ;dx" nfO{ ;r" Ls/0f ljlwåf/f nv] M
(a) 25 eGbf ;fgf 2 sf ckjTox{ ¿
(b) 30 eGbf ;fgf 3 sf ckjTox{ ¿
(c) 28 ;Ddsf 4 sf ckjTox{ ¿
(d) 40 eGbf ;fgf 5 sf ckjTox{ ¿
(e) 20 eGbf 7n' f / 50 eGbf ;fgf 7 sf ckjTox{ ¿
(f) 60 / 100 larsf 8 sf ckjTox{ ¿

76

(g) 50 / 100 larsf 9 sf ckjTox{ ¿
(h) 6 sf klxnf 5 cf6] f ckjTox{ ¿
(i) 11 sf klxnf] 10 cf6] f ckjTox{ ¿
(j) 50 kl5sf 12 sf 4 cf6] f ckjTox{ ¿

2. kZ| g g=+ 1 (a) / (b) sf ;femf ckjTox{ ¿sf] ;dx" agfpm . s] of] ;dx" / (h) sf] ;dx"
Pp6} 5 <

3. (a) 100 eGbf ;fgf] 9 sf ckjTox{ ¿sf] ;r" L tof/ kf/L ;dx" agfpm .

(b) ;r" L (a) af6 b'O{ cª\sn] ag]sf ckjTo{ cª\sx¿sf] of]ukmn lgsfn . s] of]
ofu] kmnnfO{ 9 n] efu hfG5 <

4. (a) 20 eGbf ;fgf 2 sf ckjTox{ ¿sf] ;dx" A n]v .
(b) 20 eGbf ;fgf 3 sf ckjTox{ ¿sf] ;dx" B n]v .
(c) 20 eGbf ;fgf 6 sf ckjTox{ ¿sf] ;dx" C n]v .
(d) A, B / C sf ;femf ;b:osf] ;dx" D n]v . s] D / C km/s km/s ;dx" x¿ xg' \ <

5. (a) s] 4 sf ckjTox{ ¿ ;a} 2 sf klg ckjTox{ ¿ xg' \ <
(b) s] 2 sf ckjTox{ ¿ ;a} 4 sf klg ckjTox{ ¿ xg' \ <

6. lgDg lnlvt ;d"xx¿ n]v M

(a) 10 sf] u0' fgv08 ;dx" F(10) (b) 15 sf] u0' fgv08 ;dx" F(15)
(c) 11 sf] u0' fgv08 ;dx" F(11) (d) 17 sf u0' fgv08sf] ;dx" F(17)
(e) 25 sf u0' fgv08sf] ;dx" F(25) (f) 35 sf u0' fgv08sf] ;dx" F(35)
(g) 30 sf u0' fgv08sf] ;dx" F(30)
7. ;"rL agfP/ ;d"x n]v M

(a) 20 sf u0' fgv08x¿sf] ;dx" F(20)

(b) 21 eGbf ;fgf] 2 sf ckjTox{ ¿ (Multiples) sf] ;dx" M(2)

(c) F(20) / A(2) ;d"xsf ;femf ;b:ox¿sf] csf]{ ;d"x agfpm .

ul0ft, sIff ^ 77

8. 30 eGbf 7n' f 100 eGbf ;fgf 5 sf o:tf ckjTox{ ¿ kQf nufpm h;sf cªs\ x¿sf] ofu]
kmn 9 x'G5 .

9. Pp6f v/fof] sg' } :yfgaf6 Psk6sdf 2/2 lkm6 pkm| g] u5{ eg] csf{] v/fofn] ] pxL :yfgaf6
3/3 lkm6sf] b/' Ldf pkm| g ;S5 . olb tL bO' c{ f6] } v/fofx] ¿ Pp6f 16 lkm6 nfdf] l;wf af6fd] f
Psk} 6sdf pkm| /] hfg yfn] eg] tL bO' c{ f6] n} ] sg' sg' b/' Ldf Pp6} 7fpd“ f kfOnf 6S] 5g\ <

10. sf7df8fs“} f] yfgsf6] bl] v kfv] /f;Ddsf] nueu 200 ls=ld= nfdf] kY[ jL /fhdfud{ f ;?' df
25/25 ls=ld=sf] b/' Ldf :tDex¿ /flvPsf lyP . kl5 kml] / 10/10 ls=ld= sf] b/' Ldf csf{
:tDex¿ v8f ul/P eg] sf7df8fb“} l] v slt ls=ld=sf b/' Ldf klxns] f / kl5sf :tDex¿ Ps}
7fpd“ f k/] <

11. Pp6f sf7] fdf /flvPsf bO' { cf6] f 38LdWo] Pp6fnfO{ 3/3 306fdf aHg] u/L / csfn{] fO{ 4/4
306fdf 3G6L aHg] u/L l7s 12 ah] ldnfOof] eg] slt slt ah] tL bO' { cf6] } 38Lx¿ Ps;} fy
aHnfg\ <

9.5 ¿9 / ;o+ S' t ;ªV\ ofx¿ (Prime and Composite Numbers)

tn lbOPsf ;ªV\ of tflnsfnfO{ /fd/| L x/] M

¿9 ;ªV\ of ;ª\Vof u'0fgv08x¿ sx] L ¿9 ;ªV\ ofx¿ 2, 3, 5, 7,
¿9 ;ªV\ of
¿9 ;ªV\ of 1 1 11, 13, 17 ……..
2 1, 2
¿9 ;ªV\ of 3 1, 3
4 1, 2, 4
5 1, 5
6 1, 2, 3, 6
7 1, 7
8 1, 2, 4, 8
9 1, 3, 9
10 1, 2, 5, 10

dflysf] tflnsfdf ufn] f] 3/] f xflnPsf ;a} ¿9 ;ªV\ ofx¿ xg' \ . Psl} 5g ljrf/ u/f,“} o:tf ¿9
;ªV\ ofx¿sf u0' fgv08x¿ hDdf sltcf6] f xb“' f /x5] g\ < tL u0' fgv08x¿ s] s] xg' \ <

78 ul0ft, sIff ^

1 / cfkmd" fq u0' fgv08 xg' ] ;ªV\ ofnfO{ ¿9 ;ªV\ of elgG5 . bO' e{ Gbf a9L u0' fgv08x¿ ePsf
;ª\VofnfO{ ;+o'St ;ª\Vof elgG5 .

bi| 6Jo M 1 ¿9 klg xf]Og / ;+o'St klg xf]Og .

kf| sl[ ts ;ªV\ ofsf] ;dx" , N = {1, 2, 3, 4, 5 …….} kf| sl[ ts ;ªV\ ofsf] ;dx" af6 1 / ¿9
;ªV\ ofx¿ lemsk] l5 afs“ L /xs] f ;ªV\ ofx¿nfO{ ;o+ S' t ;ªV\ ofx¿ elgG5 .

2 × 12 2×8

24 Pp6f ;o+ S' t ;ªV\ of xf] . ;o+ S' t ;ªV\ ofnfO{ o;/L
cfotfsf/ (Rectangular) agfj6df /fVg ;lsg] xg' fn]
;o+ S' t ;ªV\ ofnfO{ cfotfsf/ ;ªV\ of (rectangular number)
klg elgG5 .

4×6

t/ ¿9 ;ªV\ of (Prime Number) nfO{ o:tf] cfotfsf/ agfj6df /fVg ;lsb“ g} .

pbfx/0f 1

P(15) n] 1 b]lv 15 ;Ddsf ¿9 ;ª\Vofx¿sf] ;d"x, A n] 15 eGbf ;fgf 3 sf ckjTo{x¿sf] ;d"x
/ F(21) n] 21 sf u'0fgv08x¿sf ;d"x hgfp“5g\ eg] kQf nufpm M

(i) P(15) / A sf ;femf ;b:ox¿ (ii) P15 / F(21) sf ;femf ;b:ox¿

pQ/ M

(i) P(15) = {2, 3, 5, 7, 11, 13}
A = {3, 6, 9, 12}

∴ P15 / A sf] ;femf ;b:o 3 xf] .

(ii) P(15) = {2, 3, 5, 7, 11, 13}
F(21) = {1, 3, 7, 21}

∴ P15 / F21 sf ;femf ;b:ox¿ 3 / 7 xg' \ .

ul0ft, sIff ^ 79

cEof; 9.5

1. 1-100 ;Ddsf ;ªV\ ofnfO{ 10/10 cf6] f nx/df tn tflnsfa4 ul/P h:t} u/L sfkLdf
n]v M

123456789 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50
51 52 53 54 55 56 57 58 59 60
61 62 63 64 65 66 67 68 69 70
71 72 73 74 75 76 77 78 79 80
81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100

ca j|mdzM lgDglnlvt sfdx¿ u/ M

 1 nfO{ ufn] f] 3/] f nufcf“} lsgeg] 1 ¿9 jf ;+o'St s'g} klg xf]Og .
 2 nfO{ 5f8] /] 2 n] efu nfUg] -cyft{ \ ;a} hf/] ;ªV\ ofx¿_ nfO{ sf6f“} .
 3 afxs] 3 n] lgMzi] f efu nfUg] ;a} ;ªV\ ofnfO{ sf6f“} .
 5 afxs] 5 n] lgMzi] f efu nfUg] ;a} ;ªV\ ofnfO{ sf6f“} .
 7 afxs] 7 n] lgMzi] f efu nfUg] ;a} ;ªV\ ofnfO{ sf6f“} .

(i) gsfl6Psf ;ªV\ ofx¿ s:tf ;ªV\ of xg' \ <
(ii) sfl6Psf ;ªV\ ofx¿ s:tf ;ªV\ of xg' \ <
(iii) 1 bl] v 20 ;Dddf hDdf sltcf6] f ¿9 ;ªV\ ofx¿ 5g\ <
(iv) 1 bl] v 50 ;Dddf hDdf sltcf6] f ¿9 ;ªV\ ofx¿ 5g\ <
(v) 1 bl] v 100 ;Dddf hDdf sltcf6] f ¿9 ;ªV\ ofx¿ 5g\ <
(vi) sg' bz lardf (1-10) jf (11-20) jf (21-30) df ;aeGbf a9L / ;aeGbf 36L -sd_

¿9 ;ªV\ ofx¿ 5g\ <

80 ul0ft, sIff ^

2. l7s (T) jf al] 7s (F) s] x'G5 < n]v M 81
(i) ;a} ¿9 ;ªV\ ofx¿ lahf/] (Odd) ;ª\Vof x'g\ .
(ii) ;a} lahf]/ ;ª\Vofx¿ ¿9 ;ª\Vof x'g\ .
(iii) hf]/ ;ª\Vofx¿ ¿9 slxNo} x'g ;Sb}gg\ .
(iv) ¿9 / hf/] b'j} ePsf] ;ªV\ of Pp6} dfq 5 .
(v) ;+o'St ;ª\Vof hlt ;a} hf]/ x'G5g\ .
(vi) hf]/ ;ª\Vof hlt ;a} ;+o'St x'G5g\ .
(vii) ¿9 ;ªV\ ofsf] u0' fgv08 2 cf]6f dfq} x'G5g\ .
(viii) ;+o'St ;ª\VofnfO{ cfotfsf/ ;ªV\ of elgG5 .

3. ;"rL agfP/ ;d"x n]v M
(i) 1 bl] v 20 ;Ddsf ¿9 ;ªV\ ofx¿sf] ;dx" P(20)
(ii) 1 bl] v 20 ;Ddsf ;o+ S' t ;ªV\ ofx¿sf] ;dx" C(20)
(iii) 1 bl] v 20 ;Ddsf hf/] ;ªV\ ofx¿sf] ;dx" E(20)
(iv) 1 bl] v 20 ;Ddsf lahf/] ;ªV\ ofx¿sf] ;dx" O(20)
(v) 20 sf] u0' fgv08 ;dx" F(20)
(vi) 20 eGbf ;fgf 7 sf ckjTox{ ¿sf] ;dx" A

4. kZ| g g=+ 3 sf cfwf/df kQf nufpm M
(i) s] ;dx" O(20) / E(20) sf ;femf ;b:o 5g\ <
(ii) s] ;dx" C(20) / E(20) sf ;femf ;b:o 5g\ <
(iii) ;dx" P(20) / E(20) sf ;femf ;b:ox¿sf] ;d"x agfpm .
(iv) ;dx" A / P(20) sf ;femf ;b:ox¿sf] ;d"x agfpm .
(v) ;dx" C(20) / F(20) sf ;femf ;b:ox¿sf] ;d"x agfpm .
(vi) C(20) / A sf ;femf ;b:ox¿sf] ;d"x agfpm .

ul0ft, sIff ^

9.6 ¿9 v08Ls/0f (Prime Factorization) 24
2 × 12
;o+ S' t ;ªV\ ofx¿ 18 / 24 nv] f,“} 2 × 2 ×6

18 2× 2 × 2 × 3

2 ¿9 ;ª\Vof xf] . 2× 9 ∴ 24 = 2 × 2 × 2 × 3

oL ;a} ¿9 ;ª\Vof x'g\ . 2 × 3×3

∴ 18 = 2 × 3 × 3

o;/L v08Ls/0f u//] bv] fOPsf] lrqnfO{ u0' fgv08 jI[ f (Factor Tree) elgG5 .

sg' } klg ;o+ S' t ;ªV\ ofnfO{ ¿9 ;ªV\ ofx¿sf] u0' fgkmnsf ¿kdf cleJoSt ug{ ;lsG5 . o;/L
;o+ S' t ;ªV\ ofnfO{ v08Ls/0f u/L ¿9 ;ªV\ ofx¿sf] u0' fgkmnsf ¿kdf nV] g] kl| jm| of (Process)
nfO{ ¿9 v08Ls/0f elgG5 .

24 sf] ¿9 v08Ls/0f kl| jm| ofnfO{ 5f6] s/Ldf ubf,{

24 hf/] ;ªV\ of xf] . 12 hf/] ;ªV\ of xf] . 6 klg hf/] ;ªV\ of xf] .
To;n} ] 2 n] efu ubf{ To;n} ] 2 n] efu ubf{ To;n} ] 2 n] efu ubf{

2 ) 24 ⇒ 2 ) 24 ⇒ 2) 24
12 2 ) 12 2) 12
2) 6
6
3
¿9 ;ªV\ of cfof] .

To;}n], 90 hf/] ;ªV\ of xf] . To;n} ] 2 n] efu nfU5 . 45
24 = 2 × 2 × 2 × 3 nfO{ 3 n] efu nfU5 . km]l/ 15 nfO{ 3 n] efu nfU5
ca 5 ¿9 ;ª\Vof xf] . To;}n] oxL /f]sf}“ .
To:t,} 90 nfO{ ¿9 v08Ls/0f ubf,{

2 90

3 45

3 15

5

¿9 v08Ls/0f ubf{ lbOPsf] ;o+ S' t ;ªV\ ofnfO{ 2, 3, 5, 7, 11 …… u/L ¿9 ;ªV\ ofx¿n] jm| dzM efu
u/L efukmn /fVg] / kml] / efukmnnfO{ g} efu ub{} hfgk' 5{ .

82 ul0ft, sIff ^

pbfx/0f 1

;ªV\ of 210 nfO{ ¿9 v08Ls/0f u/ .

(i) u0' fgv08 jI[ f (Factor Tree) agfP/ (ii) efu ljlwaf6, (∴ 1 + 0 + 5 = 6
pQ/ 210
nfO{ 3 n] lgMzi] f
(i) 2 × 105 (ii) 2 210 efu hfG5 ._
2 × 3 × 35 3 105
5 35

7

2×3×5× 7

∴ 210 = 2 × 3 × 5 × 7 ∴ 210 = 2 × 3 × 5 × 7

pbfx/0f 2

(i) 75 / 90 nfO{ ¿9 v08Ls/0f u/ .

(ii) ;femf ¿9 ;ª\Vofx¿sf u'0fgkmn klg n]v .

(iii) s] of] u0' fgkmnn] 75 / 90 bj' n} fO{ lgMzi] f efu nfU5 <
pQ/

(i) 3 75 2 90
5 25 3 45
5 3 15

75 = 3 × 5 × 5 5

90 = 2 × 3 × 3 × 5

(ii) 75 = 3×5 ×5

90 = 2 × 3 × 5 × 3

∴ ;femf ¿9 ;ªV\ ofx¿ 3 5

ltgLx¿sf] u0' fgkmn = 3 × 5 = 15

(iii) 5 6
15) 75 15) 90
- 75
- 90
0
0

To;n} ,] ;femf ¿9 ;ªV\ ofx¿sf] u0' fgkmn 15 n] 75 / 90 b'j}nfO{ lgMz]if efu nfU5 .

ul0ft, sIff ^ 83

cEof; 9.6 -s_

1. -s_ tnsf kT| os] ;ªV\ ofsf u0' fgv08nfO{ u0' fgv08 jI[ f (Factor Tree) agfP/ lgsfn M

(i) 18 (ii) 20 (iii) 46 (iv) 72

-v_ tnsf kT| os] ;ªV\ ofsf] efu ljlwaf6 ¿9 v08Ls/0f u/ M

(i) 21 (ii) 30 (iii) 56 (iv) 80

(v) 105 (vi) 144 (vii) 275 (viii) 625

2. tnsf ;ªV\ ofsf ;femf ¿9 ;ªV\ ofx¿sf u0' fgkmn lgsfn M

-s_ 18 / 20 -v_ 20 / 21 -u_ 72 / 144

-3_ 105 / 275 -ª_ 275 / 625

9.8 dxQd ;dfkjts{ / n3Q' d ;dfkjTo{ (H.C.F. and L.C.M.)

dxTtd ;dfjts{ (Highest Common Factor)

12 sf u0' fgv08x¿sf] ;dx"
F12 = {1, 2, 3, 4, 6, 12} 5 .
To:t,} 18 sf u'0fgv08x¿sf] ;d"x
F18 = {1, 2, 3, 6, 9, 18} 5 .

oL bO' { ;dx" F12 / F18 sf ;femf u0' fgv08x¿sf] ;dx" = {1, 2, 3, 6} x'G5 . cyf{t\ 12 / 18 sf ;femf
u0' fgv08x¿ 1, 2, 3 / 6 x'G5g\ . o;df ;a}eGbf 7'nf] ;femf u0' fgv08 6 5 .

ctM dxQd ;dfkjts{ 6 x'G5 . dxQd ;dfkjt{snfO{ 5f]6s/Ldf n]Vbf d=;= n]lvG5 . To:t}
d=;= sf] cªu\ h]| Ldf nV] bf Highest Common Factor n]lvG5 . o;sf] 5f]6s/L H.C.F. x'G5 .

lbOPsf kf| sl[ ts ;ªV\ ofx¿sf ;femf u0' fgv08dWo] ;ae} Gbf 7n' f] ;femf u0' fgv08nfO{ dxQd
;dfkjt{ s (Highest Common Factor) eGb5g\ . lbOPsf ;ªV\ ofnfO{ lgMzi] f efu hfg] ;ae} Gbf
7n' f] ;ªV\ of xf] .

pbfx/0f 1 ul0ft, sIff ^

15 / 20 sf] d=;= lgsfn .
oxf,“ 15 sf ?9 u0' fgv08x¿ lgsfNbf,
15 = 3 × 5 x'G5 .
To:t} 20 sf ?9 u0' fgv08x¿ lgsfNbf,
20 = 2 × 2 × 5 x'G5 .
o;df ;femf u0' fgv08 5 xf] .
t;y{ d=;= = 5 x'G5 .

84

pbfx/0f 2

18 cf6] f sfutL / 24 cf6] f :ofp a9Ldf slt hgfnfO{ a/fa/ xg' ] u/L af8“ g\ ;lsPnf <
kT| os] n] slt sltcf6] f kmnkm" n kfp5“ g\ xfn] f <
pQ/ M
oxf“ 18 = 2 × 3 × 3
/ 24 = 2 × 2 × 2 × 3
oxf,“ ;femf u0' fgv08x¿ 2 / 3 x'g\ .
t;y,{ d=;= = 2 × 3 = 6 x'G5 .
To;n} ] sfutL / :ofp 6 hgfnfO{ a/fa/ x'g] u/L af“8\g ;lsG5 .
ca 6 n] 18 nfO{ efu ubf,{

3
6 ) 18

18

×

kT| os] n] 3 cf]6f sfutL kfp“5g\ .
To:t} 6 n] 24 nfO{ efu ubf,{

4
6 ) 24

24

×

kT| os] n] 4 cf]6f :ofp kfp“5g\ .

cEof; 9.7 -s_

1. tn lbOPsf ;ªV\ ofx¿sf] u0' fgv08sf] ;dx" agfP/ d=;= lgsfn M

-s_ 4, 6 -v_ 6, 9 -u_ 8, 12

-3_ 9, 18 -ª_ 9, 12 -r_ 8, 16

2. tn lbOPsf ;ªV\ ofx¿sf] ¿9 v08Ls/0f ljlwaf6 d=;= lgsfn M

-s_ 12, 15 -v_ 12, 30 -u_ 16, 40

-3_ 18, 27 -ª_ 27, 36 -r_ 24, 60

3. 18 / 45 nfO{ lgMzi] f efu nfUg] ;ae} Gbf 7n' f] ;ªV\ of lgsfn .

4. 9 cf6] f ;G' tnf / 12 cf6] f :ofp a9Ldf slt hgfnfO{ a/fa/ xg' ] u/L af8“ g\ ;lsPnf < kT| os] n]
x/s] kmnkm" n slt sltcf6] f kfp5“ g\ xfn] f <

ul0ft, sIff ^ 85

5. 12 cf6] f sfutL / 18 cf6] f ;G' tnf a9Ldf slt hgfnfO{ a/fa/ xg' ] u/L af8“ g\ ;lsPnf /
kT| os] n] slt slt cf6] f kfp5“ g\ xfn] f <

6. Pp6f ef8“ fd] f 30 ln6/ / csfd{] f 50 ln6/ bw' /x5] . kT| os] ef8“ f] vfnL ug{] u/L gfKg ;lsg]
;ae} Gbf 7n' f] gfksf] csf{] 56' 6\ } ef8“ fd] f slt ln6/ c6fpnf <

7. Pp6f cfotfsf/ rfs] sf] nDafO 12 ld= / rf8} fO 9 ld= /x]5 . o;nfO{ Pp6} ;fOhsf
jufs{ f/ dfjn{ 5fKgk' bf{ ;ae} Gbf 7n' f] ;fOhsf] jufs{ f/ dfan{ sf] nDafO slt xfn] f <

8. Pp6f 6fs] /Ldf 25 cf6] f cDaf / csfd{] f 30 cf6] f gf;kftL 5g\ . kT| os] 6fs] /Laf6 Ps k6sdf
;a}eGbf w/] } cDaf / gf;kftL slt slt cf6] f lemSbf bj' } 6fs] /L Ps;fy vfnL xfn] fg\ <

n3Q' d ;dfkjTo{ (Lowest Common Multiple)

2 / 3 sf ckjTox{ ¿ (Multiples) nfO{ ;dx" agfpb“ f tn lbOPcg;' f/ agfpg ;lsG5 .
2 sf ckjTox{ ¿sf] ;dx"
M2 = {2, 4, 6, 8, 10, 12 ,14, 16, 18, 20 …} x'G5 .
3 sf ckjTox{ ¿sf] ;dx"
M3 = {3, 6, 9, 12, 15, 18, 21, 24, 27, 30 …} x'G5 .
ca ;ªV\ ofx¿ 2 / 3 sf ;femf ckjTox{ ¿ (Common Multiples) sf]
;dx" = {6, 12, 18 ……..} x'G5 .
of] ;dx" sf ;b:o 6 nfO{ ;ªV\ ofx¿ 2 / 3 sf] ;ae} Gbf ;fgf] ;dfkjTo{ cyjf n3Q' d ;dfkjTo{
(Lowest Common Multiple) elgG5 . o;nfO{ 5f6] s/Ldf nV] bf n=;= nl] vG5 / cªu\ h]| Ldf nV] bf
L.C.M. n]lvG5 .

bO' { jf bO' e{ Gbf a9L kf| sl[ ts ;ªV\ ofx¿sf] n3Q' d ;dfkjTo{ egs] f] tL ;ªV\ ofx¿n] lgMzi] f efu
nfUg] ;aeGbf ;fgf] k|fsl[ ts ;ª\Vof xf] .

pbfx/0f 3

4 / 6 sf] n3'Qd ;dfkjTo{ lgsfn .
pQ/ M
oxf“ 4 / 6 sf] ¿9 u0' fgv08 lgsfNbf,
4 = 2 × 2 / 6 = 2 × 3 x'G5 .
To;n} ] n=;= = 2 × 2 × 3 = 12 x'G5 .
lsgls ;femf u'0fgv08 eg]sf] d=;= xf] .
oxf“ d=;= 2 5 .

bO' { ;ªV\ ofsf nflu ;ªV\ ofx¿sf] n=;= = d=;= × afs“ L u0' fgv08x¿

86 ul0ft, sIff ^

pbfx/0f 4

bO' { cf6] f 3G6Lx¿ jm| dzM 20 ldg6] / 24 ldg]6sf] cGt/df aH5g\ . olb 10 ah] laxfg tL 3G6Lx¿
Psr} fl] 6 ah] eg] bf;] f| k] 6s slt ;dokl5 Ps;} fy aHnfg\ <

pQ/ M 60 ) 120 ( 2 306f
oxf“ 20 ldg6] = 2 × 2 × 5
/ 24 ldg6] = 2 × 2 × 2 × 3 - 120

o;df d=;= = 2 × 2 = 4 xG' 5 0

To;n} ,] n=;= = 4 × 5 × 2 × 3 = 120 ldg]6 x'G5 . 120 ldg6] = 2 3G6f xG' 5 .
∴ bf;] f| ] k6s 2 3G6fkl5 (12 ah]_ Ps};fy pSt 3G6Lx¿ aH5g\ .

cEof; 9.7 -v_

1. tnsf kT| os] ;ªV\ ofx¿sf] ckjTox{ ¿sf] ;dx" agfO{ n=;= lgsfn M

-s_ 3, 5 -v_ 4, 6 -u_ 6, 8 -3_ 8, 10

-ª_ 8, 12 -r_ 6, 7 -5_ 9, 12 -h_ 6, 9

2. tn lbOPsf kT| os] ;ªV\ ofx¿sf] ¿9 v08Ls/0f ljlwaf6 n=;= lgsfn M

-s_ 6, 9 -v_ 9, 12 -u_ 8, 12 -3_ 10, 14

-ª_ 14, 20 -r_ 20, 24 -5_ 24, 30 -h_ 24, 36

3. bO' { cf6] f 3G6Lx¿ jm| dzM 24 ldg6] / 30 ldg]6sf] cGt/df aH5g\ . olb 9 ah] laxfg tL
3G6Lx¿ Ps;} fy ah] eg] bf;] f| k] 6s slt ;dokl5 Ps} ;fy aHnfg\ <

4. bO' c{ f6] f ;:+ yfx¿dWo] klxnf] ;:+ yfsf] a7} s x/s] 4 xKtfdf / bf;] f| ] ;:+ yfsf] a7} s x/s] 6
xKtf a:bf] /x]5 . olb 2068 ;fn jz} fv 2 ut] bO' c{ f6] } ;:+ yfn] Ps} rfl] 6 a7} s u/] eg]
bf;] f| k] 6s slt xKtfkl5 kml] / Ps} lbg a7} s a:nf < 2068 ;fnsf] kfqf] x/] /] Tof] lbg sg'
dlxgfsf] slt ut] kbf{] /x]5, kQf nufpm .

5. Pp6f df6] /;fOsndf x/s] 80 ls=ld= u8' k] l5 k6] f« n] egk{' 5{ / 100 ls=ld= u8' k] l5 dfl] an
kmg] k{' 5{ . oL sfox{ ¿ Psk} 6s u/k] l5 ca slt b/' L kf/ u/k] l5 kg' M bj' } sfox{ ¿ Psk} N6
ugk{' nf{ <

ul0ft, sIff ^ 87

9.9 k0" f{ ju;{ ªV\ of / jud{ n" (Perfect square number and square root)

k0" f{ ju;{ ªV\ of (Perfect square number)
tnsf] lrq /fdf| ;] u“ x/] / tn ;fl] wPsf kZ| gsf] hjfkm bp] m M

1 23456 7 8 9

10 11 12 13 14 15 16

(a) lrqdf 3/] fleq /flvPsf] ;ªV\ of hgfpg] laGbx' ¿sf] 9fr“ f (Dot Pattern) c¿ ;ªV\ ofsf] laGbx' ¿sf]
9fr“ feGbf lsg km/s 5 <

(b) s] lrqdf ePsf c¿ ;ªV\ ofx¿nfO{ 3/] fleq /fvs] f] ;ªV\ ofsf] cyft{ \ laGbx' ¿sf] 9fr“ fdf
ldnfpg ;lsG5, s;/L <

(c) 3/] fleq /flvPsf laGbx' ¿sf] 9fr“ fn] ss] f] cfsl[ t agfPsf 5g\ <

(d) jul{ eq /flvPsf laGbx' ¿sf] 9fr“ fdf Psnx/sf] laGbs' f] ;ªV\ of yfxf eP k/' } 9fr“ fdf
sltcf6] f laGbx' ¿ 5g\ eg/] s;/L yfxf kfpg ;lsG5 <

(e) s] jul{ eq k/s] f laGbx' ¿sf] 9fr“ fn] hgfpg] ;ªV\ ofx¿nfO{ 1 × 1, 2 × 2, 3 ×3, 4 × 4 u//]
nV] g ;lsG5 <

(f) kZ| g (e) af6 cfpg] ;ªV\ ofsf] nx/df c¿ 3-3 cf6] f o:tf ;ªV\ ofx¿ yKg ;lsG5 <

sg' } ;ªV\ ofnfO{ laGbx' ¿sf] 9fr“ fdf JoSt ubf{ laGbx' ¿nfO{ Pp6f jufs{ f/ cfsl[ tdf ldnfpg
;lsG5 eg] To:tf ;ªV\ ofnfO{ k0" f{ ju;{ ªV\ of elgG5 cyjf,
sg' } ;ªV\ ofnfO{ bO' { cf6] f p:tfp:t} u0' fgv08x¿sf] u0' fgkmndf JoSt ug{ ;lsG5 eg] To:tf
;ªV\ ofnfO{ k0" f{ ju;{ ªV\ of elgG5 .

h:t} M 4=22 16 = 4  4 36 = 6  6
9=33 25 = 5  5 49 = 7  7

pbfx/0f 1

125 / 121 k"0f{ ju{;ª\Vofx¿ x'g\ jf x}gg\ 5'6\ofpm .
pQ/
oxf,“ 125 = 5 × 5 × 5 -¿9 v08Ls/0faf6_

/ 121 = 11 × 11
To;n} ,] 121 k0" f{ ju;{ ªV\ of xf] t/ 125 k"0f{ ju{;ª\Vof xf]Og .

88 ul0ft, sIff ^

pbfx/0f 2

125 nfO{ k0" f{ ju;{ ªV\ of agfpg sg' ;fgf] ;ªV\ ofn] u0' fg ugk{' nf{ <
pQ/
oxf“,

125 = 5 × 5 × 5
ca, bj' l} t/ 5 n] u0' ff ubf,{
125 × 5 = 5 × 5 × 5 × 5
cyjf, 625 = 25 × 25
To;n} ,] 125 nfO{ 5 n] u0' ff cfpg] ;ªV\ of 625 k"0f{ ju{;ª\Vof x'G5 .
pbfx/0f 3

13 sf] ju;{ ªV\ of slt xG' 5 <
pQ/
oxf,“ 13 sf] ju;{ ªV\ of lgsfNg' egs] f] 13 nfO{ 13 n] u'0fg u/]/ u'0fgkmndf JoSt ug'{ xf] .
To;n} ] 13 sf] ju;{ ªV\ of = 13 × 13 = 169 xf] .
bi| 6Jo M 169 = 13 ×13 cyjf 169 = 132 nl] vG5 . o;nfO{ k9b\ f Thirteen squared eg]/ kl9G5 .

k0" f{ ju;{ ªV\ ofsf] jud{ n" (square root of a perfect square number)

lrqdf Pp6f vt] df 64 cf6] f aGbfsf
la?jfx¿nfO{ juf{sf/ ¿kdf
ldnfP/ /fk] s] f] 5 . kT| os] lsgf/fdf
slt slt cf]6f la?jf /f]lkPsf
/x5] g\ <

lrqdf x/s] lsgf/fdf 8 cf6] f
la?jf kg]{ u/L /f]lkPsf] /x]5 .
To;n} ,] oxf“ 64 = 8 × 8 x'G5 .
ctM k0" f{ ju;{ ªV\ of 64 xf] / 64
sf] jud{ n" 8 xf] .

k0" f{ ju;{ ªV\ ofsf bO' { cf6] f p:tfp:t} u0' fgv08dWo] Pp6fnfO{ Tof] ;ªV\ ofsf] jud{ n" (square root)
elgG5, h:t} M a2 df k0" f{ ju;{ ªV\ of xf] . a2 = a × a xG' 5 / a jud{ n" xG' 5 .

ul0ft, sIff ^ 89

jud{ n" tflnsf (1 -100 ;Dd_ 10
100
123 4 5 6 7 8 9

11
24
39
4 16
5 25
6 36
7 49
8 64
9 81
10
dflysf] tflnsf cWoog u/ / lgDglnlvt kZ| gx¿sf] pQ/ bp] m M
-s_ olb 4 sf] ju,{ 4 × 4 = 42 = 16 x'G5 eg] 9 sf] ju{;ª\Vof slt x'G5 <
-v_ 5 / 8 sf ju;{ ªV\ ofx¿ nv] .
-u_ 36 / 100 sf jud{ n" slt slt xG' 5 <

cEof; 9.8

1. dfg lgsfn M

-s_ 12 -v_ 02 -u_ 42 -3_ 72
-ª_ 92 -r_ 32 -5_ 62 -h_ 102

2. tnsf kT| os] ;ªV\ ofsf] ju;{ ªV\ of slt xG' 5 M -3_ 4
-h_ 25
-s_ 1 -v_ 2 -u_ 3

-ª_ 9 -r_ 10 -5_ 15

90 ul0ft, sIff ^

3. jud{ n" lgsfn -v08Ls/0f ljlwaf6_ M

-s_ 25 -v_ 36 -u_ 64 -3_ 81

-ª_ 121 -r_ 144 -5_ 324 -h_ 625

4. tnsf kT| os] ;ªV\ ofnfO{ sg' ;fgf] ;ªV\ ofn] u0' fg ubf{ k0" f{ ju;{ ªV\ of xG' 5 <

-s_ 72 -v_ 108 -u_ 125 -3_ 192

5. sg' } ;g] fkltn] x/s] nx/df 64 hgf kg{] u/L l;kfxLx¿nfO{ jufs{ f/ ¿kdf ldnfP/ /fVbf
219 hgf l;kfxL a9L xg' cfP5g\ eg]

-s_ hDdf slt l;kfxLx¿ /x5] g\ <

-v_ ;an} fO{ jufs{ f/ ¿kdf ldnfpg sDtLdf slt hgf l;kfxL yKgk' nf{ <

6. nDafO / rf8} fO bj' l} t/ 49/49 cf6] f la?jf kg{] u/L jufs{ f/ ¿kdf /fK] bf slt la?jf rflxG5g\
xfn] f <

7. jI[ f/fk] 0f sfoj{ m| ddf hlthgf ;xefuL lyP Tolts} ;ªV\ ofdf kT| os] n] la?jf /fK] b} hfb“ f hDdf
1225 cf6] f la?jf /fl] kP5g\ eg] slt hgf ;xefuLx¿n] efu lnPsf /x5] g\ <

8. tn lbOPsf] 9fr“ f k/" f u/ M

1 = 12

1 + 3 = 22

1 + 3 + 5 = 32

1 + 3 + 5 + 7 = ... ...

1 + 3 + 5 + 7 + 9 = ... ...

1 + 3 + 5 + 7 + 9 + 11 = ... ...

1 + 3 + 5 + 7 + 9 + 11 + 13 = ... ...

ul0ft, sIff ^ 91

PsfO 10 k0" ffª{ s\ (Integers)

k0" ffª{ s\ sf] kl/ro (introduction of integers)

k0" f{ ;ªV\ ofx¿sf] ;dx" W = {0, 1, 2, 3, 4, 5, … } af6 sg' } bO' { cf6] f ;ªV\ ofx¿ np] m . dfgf,}“ tL
;ªV\ ofx¿ 3 / 4 x'g\ . ca 3 / 4 sf] ofu] kmn 7 xG' 5 / u0' fgkmn 12 x'G5 . oL b'O{ lj|mofnfO{ ;ª\Vof
/v] fdf bv] fpb“ f,

3+4=

3 x 4 = 12

oxf“ ofu] kmn hgfpg] ;ªV\ of 7 / u0' fgkmn hgfpg] ;ªV\ of 12 b'j} k"0f{ ;ª\Vof x'g\ . To;/L g} k"0f{
;ªV\ ofsf c¿ ;b:ox¿sf] hf8] jf u0' fg ljm| ofaf6 cfpg] ;ªV\ of klg k0" f{ ;ªV\ of g} xG' 5 t/ 36fp
ljm| of ubf{ s] xG' 5 xfn] f <
3 – 4 = slt xfn] f < dflysf] ;ªV\ of/v] fdf of] ljm| of bv] fpg ;lsb“ g} . ca tnsf] ;ªV\ of/v] fdf x/] f“} M

;ªV\ of/v] faf6 3 – 4 egs] f] 0 eGbf 1 PsfO sd xg' ] ;ªV\ of eGg] yfxf xG' 5 . o;nfO{ -1 n]vf}“ .
To;/L g} 3 – 5 = -2 (0 eGbf bO' { PsfO sd_; 3 – 6 = -3 OToflb . o;/L xb] f{ xfdf| ] ;ªV\ of/v] fsf] lj:tf/
0 b]lv afof“lt/ x'“b} hfG5 .

C0ffTds k0" ffª{ s\ zG" o wgfTds k0" ffª{ s\

;ªV\ ofx¿sf] of] ;dx" Z = {….., -2, -1, 0, 1, 2, …….} nfO{ k0" ffª{ s\ sf] ;dx" elgG5 / k0" ffª{ s\ sf]
;dx" df 36fp ljm| of kl/eflift xG' 5 . ;ªV\ of/v] fdf 0 nv] s] f] :yfgnfO{ pbu\ d laGb' (Point of
Reference) elgG5 . pbu\ d laGba' f6 bfofl“ t/sf ;ªV\ ofx¿ (+) wgfTds (Positive) 5g\ . oL

92 ul0ft, sIff ^

;ªV\ ofx¿sf] ;dx" Z+ = {+1, +2, +3, +4 …..} nfO{ wgfTds k0" ffª{ s\ x¿ (Positive Integers) sf]
;dx" elgG5 . pbu\ d ljGba' f6 afofl“ t/sf ;ªV\ ofx¿ C0ffTds (-) x'G5g\ . oL ;ª\Vofx¿sf] ;d"x
Z- = {-1, -2, -3, -4 …..} nfO{ C0ffTds k0" ffª{ s\ x¿ (Negative Integers) sf] ;d"x elgG5 .

bfofl“ t/ hfb“ f ;ªV\ of a9b\ } hfG5g\ / afofl“ t/ hfb“ f ;ªV\ ofsf] dfg 36b\ } hfG5g\ .

-3 -2 -1 0 1 2 3

k0" ffª{ s\ x¿sf] ;dx" eGgfn] wgfTds k0" ffª{ s\ x¿, C0ffTds k0" ffª{ s\ x¿ / 0 -zG" o_ ;dfjz] ePsf
;ª\Vofx¿sf] ;d"xnfO{ atfp“5 .

bi| 6Jo M– '0' C0ffTds jf wgfTds ;ª\Vof s'g} klg xf]Og .

cEof; 10

1. ;ªV\ of/v] fsf cfwf/df lgDg lnlvt kZ| gsf] hjfkm bp] m M
-s_ 0 eGbf ;fgf] ;ªV\ of pbu\ d laGba' f6 stfk6l\ 6 k5{ <
-v_ 0 eGbf 7n' f] ;ªV\ of pbu\ d laGba' f6 stfk6l\ 6 k5{ <
-u_ lbOPsf] sg' } ;ªV\ ofeGbf 1 PsfO ;fgf] ;ªV\ of lbOPsf] ;ªV\ ofaf6 stfk6l\ 6 k5{ <
-3_ lbOPsf] sg' } ;ªV\ ofeGbf 1 PsfO 7n' f] ;ªV\ of lbOPsf] ;ªV\ ofsf] stfk6l\ 6 /xs] f] xG' 5 <
-ª_ -6 / -5 df sg' 7n" f] <

-r_ -8 / -7 df sg' ;fgf] <
-5_ -5 / 3 sf] lardf slt cf6] f k0" ffª{ s\ x¿ 5g\ <

2. ;ªV\ of/v] fsf cfwf/af6 lgDglnlvt ;ªV\ ofeGbf 3 PsfO afofl“ t/ /xs] f ;ªV\ of nv] M

-s_ 5 -v_ 2 -u_ 0 -3_ -1 -ª_ -3

3. tnsf bO' { ;ªV\ ofx¿sf] lardf ldNgu] /L (>) jf (<) lrx\g /fv M

-s_ +7 -3 -v_ +3 +5 -u_ -3 -2

-3_ -5 -7 -ª_ -5 +2 -r_ +5 -5

4. -13 / +5 sf lardf sltcf6] f k0" ffª{ s\ x¿ xG' 5g\ <

5. xl/ Pp6f ;flnsaf6 l;wf 4 ls=ld= k"j{ kg]{ :yfgdf 5 . /fd 2 ls=ld= klZrddf kg{] :yfgdf
5 . of] hfgsf/LnfO{ ;ªV\ of/v] fdf k0" ffª{ s\ sf] ko| fu] u/L bv] fpm . ;fy} /fd / xl/larsf] b/' L
klg lgsfn .

ul0ft, sIff ^ 93

PsfO 11 cfgk' flts ;ªV\ ofx¿ (Rational Numbers)

kl/ro
tnsf lrqdf k0" ffª{ s\ x¿nfO{ ;ªV\ of/v] fdf bv] fOPsf] 5 .

-5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 ...

 ;ªV\ of/v] fdf /xs] f sg' } bO' { k0" ffª{ s\ x¿ hf8] .
h:t} M 5 + 3 = 8 xG' 5 . 8 k0" ffª{ s\ xf] .

 ;ªV\ of/v] fdf /xs] f sg' } bO' { k0" ffª{ s\ 36fpm .
h:t} M 5 - 8 = - 3 xG' 5 . -3 k0" ffª{ s\ xf] .

 ;ªV\ of/v] fdf /xs] f sg' } bO' { k0" ffª{ s\ x¿ u0' ff u/ .
h:t} M 5 x 3 = 15 xG' 5 . 15 k0" ffª{ s\ xf] .

 ;ªV\ of/v] fdf /xs] f sg' } bO' { k0" ffª{ s\ x¿sf] efu u/ .
h:t} M 3 ÷ 5 = 3/5 xG' 5 . s] 3/5 ;ªV\ of/v] fdf bl] vG5 <

oxf“, 8, -3 / 15 k0" ffª{ s\ x¿ xg' \ . t/ 3/5 k0" ffª{ s\ xfO] g .
3/5 nfO{ cfgk' flts ;ªV\ of (rational number) elgG5 .
-3/5, 1/2, 8, -1/5 cfgk' flts ;ªV\ ofx¿sf pbfx/0fx¿ xg' \ .

olb a / b bO' c{ f6] f k0" ffª{ s\ x¿ xg' \ / b  0 eP a/b sf ¿kdf JoSt ul/Psf ;ªV\ ofx¿nfO{
cfgk' flts ;ªV\ ofx¿ (Rational Numbers) elgG5 . o;nfO{ ‘Q’ cIf/n] hgfOG5 .
pbfx/0f

94 ul0ft, sIff ^

cEof; 11
1. tn lbOPsf ;ªV\ of /v] fx¿ (a) df bv] fOP h:t} k/" f u/ M

(a)
-1 0 1 2 3 4 5 6 7 8

(b)
3

(c)
-7

(d)
-4

(e)
10

(f)
12

2. tnsf sg' sg' ;ªV\ ofx¿ cfgk' flts ;ªV\ ofx¿ xg' \ <

3. ;ªV\ of/v] f lvrL bv] fpm .

4. tnsf tYodWo] l7s / al] 7s 56' o\ fpm M 95
(a) x/s] k0" ffª{ s\ ;ªV\ of cfgk' flts ;ªV\ of xf] .
(b) ;a} cgk' flts ;ªV\ ofx¿ k0" ffª{ s\ xg' \ .
(c) kf| sl[ ts ;ªV\ ofx¿ cgk' flts ;ªV\ of xfO] gg\ .
(d) hf/] / lahf/] bj' } ;ªV\ of cgk' flts ;ªV\ ofx¿ xfO] gg\ .
(e) ;a} ¿9 ;ªV\ ofnfO{ cgk' flts ;ªV\ of elgG5 .

ul0ft, sIff ^