गणित कक्षा ९

ul0ft

sIff (

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

kf7o\ jm| d ljsf; sG] b|
;fgfl] 7dL, eStk'/

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

kf7o\ j|md ljsf; s]Gb|

;fgf]l7dL, eStk'/

© ;jf{lwsf/ kf7\oj|md ljsf; sG] b|

978-9937-601- -

o; kf7o\ k':ts;DaGwL ;Dk"0f{ clwsf/ kf7\oj|md ljsf; sG] b| ;fgf]l7dL, eStk'/df lglxt
/x]sf] 5 . kf7o\ jm| d ljsf; sG] bs| f] lnlvt :jLs[ltlagf Jofkfl/s k|of]hgsf nflu o;sf] k'/}
jf cfl+ zs efu x'ax' k|sfzg ug{, kl/jt{g u/]/ k|sfzg ug,{ s'g} ljB'tLo ;fwg jf cGo
kl| jlwaf6 /s] 8{ ug{ / kl| tlnlk lgsfNg kfOg] 5}g .

k|yd ;:+ s/0f M lj=;+ @)&(

kf7o\ k:' ts;DaGwL kf7sx¿sf sg' } klg ks| f/sf ;´¬ fjx¿ ePdf kf7o\ jm| d ljsf;
sG] b|, ;dGjo tyf ks| fzg zfvfdf k7fOlbgx' 'g cg/' f]w 5 . kf7sx¿af6 cfpg]
;¬´fjx¿nfO{ s]Gb| xflbs{ :jfut ub5{ .

xfdf| ] egfO

lzIffnfO{ p2Z] od"ns, Jofjxfl/s, ;d;fdlos / /f]huf/dn" s agfpg ljleGg ;dodf kf7o\ j|md,
kf7\ok:' ts ljsf; tyf kl/dfhg{ sfon{ fO{ lg/Gt/tf lbObF } cfPsf] 5 . ljBfyL{df 1fgsf] vf]hL u/L
l;sfO / jf:tljs hLjglar ;DaGw :yflkt ug{], l;4fGt / Jojxf/sf] ;dGjo ug{], :jk/fjltt{ xF'b}
1fg, l;k / IfdtfnfO{ cBfjlws ug{] ;Ifdtfsf] ljsf; xg' ' cfjZos 5 . ljBfyLd{ f clwsf/, :jtGqtf
/ ;dfgtfsf] k|jwg{ ug,]{ :j:y hLjgsf] cEof; ug,{] tflss{ ljZni] f0f u/L lg0f{o ug],{ j1} flgs
ljZni] f0fsf cfwf/df JolSt, ;dfh / /fi6«sf] lbuf] ljsf;df ;l/s x'g] ;Ifdtfsf] ljsf; klg lzIffn]
ugk{' 5{ . ljBfyLd{ f gl} ts cfr/0f kb| zg{ ug,{] ;fdflhs ;be\ fjkl| t ;j+ b] gzLn xg' ,] kofj{ /0fLo ;Gtn' gkl| t
;j+ b] gzLn xg' ,] åGå Joj:yfkg ub}{ lbuf] zflGtsf nflu k|lta4 /xg] ;Ifdtfsf] ljsf; klg dfWolds
txsf] lzIffaf6 ck]lIft 5g\ . o; txsf] lzIffaf6 cfw'lgs 1fg, l;k, ;"rgf tyf ;~rf/ k|ljlwsf]
ko| f]u ug{], :jfjnDaL / Joj;fodv' L l;ksf] cEof; ug,{] /fi6,« /fli6«otf / /fli6«o cfbzs{ f] ;Ddfg ug]{,
;dfh :jLsfo{ cfr/0f / sfo{ ;:+ s[ltsf] cjnDag ug{], ;lxi0f' efj /fVg] ;Ifdtf ePsf] gful/s tof/
ug]{ ckI] ff /xs] f] 5 . To:t,} l;h{gzLn, sNkgfzLn, pBdzLn Pjd\ pRr ;f]r / cfbz{df cfwfl/t
Jojxf/ ug{], ;d;fdlos r'gf}tLx¿sf] ;kmn Joj:yfkg ugn]{ ufotsf ljz]iftfn] oS' t :jfjnDaL,
bz] eSt, kl/jt{gdv' L, lrGtgzLn Pjd\ ;dfjz] L ;dfh lgdf{0fdf ofu] bfg ug{ ;Sg] ;Ifdtf;lxtsf]
gful/s tof/ ug'{ dfWolds lzIffsf] nIo /xs] f] 5 . oxL nIo kl" ts{ f nflu dfWolds lzIffsf]
/fli6o« kf7o\ jm| d k|f¿k, @)&^ sf] dfu{bzg{ l;4fGtcg'¿k ljsf; ePsf] dfWolds lzIff -sIff (–!)_
kf7o\ j|mdcg;' f/ tof/ kfl/Psf] kf7\ok:' tsnfO{ b]zsf ljleGg ljBfnodf k/LIf0f u/L k|fKt ki[ 7kf]if0f
;d]6L of] gd'gf kf7o\ k':ts tof/ kfl/Psf] xf] .

o; kf7o\ k:' tssf] nv] g tyf kl/dfh{g sfo{ >L u0fz] ;fksf6] f, >L g/xl/ cfrfo,{ >L zlStk;| fb cfrfo,{
>L ljgfb] k;| fb kGt, >L ufd] f >i] 7, >L ljdnk;| fb e66\ /fO,{ >L /fdrGb| 9sfn / >L huGgfy clwsf/Laf6
ePsf] xf] . kf7o\ k:' tsnfO{ o; ¿kdf Nofpg] sfod{ f sG] bs| f dxflgbz{] s >L c0fk;| fb Gofk} fg,] 8f= /fdhLk;| fb
kl08t, >L sz] j/fh km' nf/f, >L kl| dnf avtL, >L /fd xf8f, >L lgdn{ f uft} d / ;l' :dtf zdfn{ ufotsf
dxfge' fjsf] ljzi] f ofu] bfg /xs] f] 5 . o;sf] efiff ;Dkfbg 8f= u0fz] k;| fb e66\ /fO,{ lrgfsd' f/L lg/fn} f / OGb'
vgfnaf6 ePsf] xf] . o; kf7o\ k:' tssf] nc] fp6 l8hfOg >L gj/fh k/' Laf6 ePsf] xf] . o; k:' tssf] ljsf;
sfod{ f ;n+ Ug ;ak} l| t kf7o\ jm| d ljsf; sG] b| wGojfb ks| 6 u5{ .

kf7o\ k':tsnfO{ lzIf0fl;sfOsf] dxŒjk"0f{ ;fwgsf ¿kdf lnOG5 . o;af6 ljBfyL{n] kf7\ojm| dåf/f

nlIft ;Ifdtf xfl;n ug{ dbt kU' g] ckI] ff ul/Psf] 5 . o; kf7o\ k':tsnfO{ ;s];Dd ljm| ofsnfkd'vL,

cge' js]lGb|t, p2Z] odn" s / ?lrs/ agfpg] k|oTg ul/Psf] 5 . l;sfO / ljBfyL{sf] hLjGt cge' jlar

tfbfTDo sfod ub]{ o;sf] ;xh k|ofu] ug{ lzIfsn] ;xhstf,{ pTk/|] s, kj| w{s / vf]hstf{sf ¿kdf

el" dsfsf] ck]Iff ul/Psf] 5 . kf7o\ k':tsnfO{ cem} kl/ist[ kfgs{ f nflu lzIfs, ljBfyL{, cleefjs,

al' 4hLjL Pjd\ ;Dk0" f{ kf7sx¿sf] ;d]t dxŒjk"0f{ el" dsf /xg] x'bF f ;Da4 ;as} f] /rgfTds ;e' mfjsf

nflu kf7\oj|md ljsf; s]Gb| xflb{s cg'/fw] u5{ .

g]kfn ;/sf/

lzIff, lj1fg tyf kl| jlw dGqfno

lj= ;=+ @)&( kf7\ojm| d ljsf; sG] b|

ljifo;"rL ki[ 7;ªV\ of

kf7 zLif{s ! – @$
@% – $(
! ;d"x (Sets) %) – %(
@ s/ (Tax) ^) – *&
# sld;g / nfefz+ (Commission and Dividend) ** – !#)
$ 3/fo;L cª\sul0ft (Household Arithmetic) !#! – !$^
% Ifq] kmn (Area) !$& – !^(
^ lk|Hd (Prism) !&) – !*#
& a]ngf / uf]nf (Cylinder and Sphere) !*$ – !(^
* cgj' |md / >]0fL (Sequence and Series) !(& – @!$
( v08Ls/0f (Factorization)
!) dxQd ;dfkjt{s / n3'Qd ;dfkjt{s (Highest Common @!% – @#^
@#& – @%!
Factor and Lowest Common Multiple) @%@ – @&#
@&$ – @*#
!! /v] Lo ;dLs/0f (Linear Equation) @*$ – @(%
!@ 3ftfªs\ (Indices) @(^ – #)%
!# lqeh' (Triangle) #)^ – #@%
!$ rt'eh'{ (Quadrilateral)
!% /rgf (Construction) #@^ – #$^
!^ jQ[ (Circle) #$& – #^#
!& tYofªs\ sf] juLs{ /0f / k:| tt' Ls/0f (Classification and #^$ – #*)

Presentation of Data)

!* sG] b|Lo kj| [lQsf] dfkg (Measures of Central Tendency)
!( ;DefJotf (Probability)
@) lqsf0] fldlt (Trigonometry)

kf7 1 ;dx" (Sets)

1.0 kg' /jnf]sg (Review)

sIff 9 sf rf/ hgf ljBfyL;{ Fu pgLx¿sf] vt] af/Ldf nufOPsf afnLgfnLx¿sf af/d] f 5nkmn
ubf{ o;ks| f/ hfgsf/L k|fKt eof] M

-s_ PGhnsf] v]taf/Ldf nufPsf afnL M wfg, ux'F, sfb] f], tf/] L, sfpnL, s/] fp, /fof]
-v_ ladnfsf] v]taf/Ldf nufPsf afnL M uxF,' kmfk/, tf]/L, uf]ne]F8f, eG6f, v;' f{gL, hf},

cfn,' l;dL
-u_ k]Dafsf] v]taf/Ldf nufPsf afnL M kmfk/, hf,} cfn', l;dL, ds}
-3_ /fdljnf;sf] vt] af/Ldf nufPsf afnL M ux',F tf]/L, wfg, ds}, s/] f, pv', aGbf
dflysf hfgsf/Lsf cfwf/df tnsf k|Zgsf] pQ/ vfH] g'xf];\ M
dflysf kT| os] tYox¿nfO{ ;r" Ls/0f ljlwaf6 n]Vgx' f];\ .
PGhnsf] v]taf/Ldf / kD] afsf] v]taf/Ldf nufOPsf afnLsf ;dx" x¿ s:Tff ;dx"

xfn] fg\ <
ladnfsf] vt] af/Ldf nufOPsf afnL / /fdljnf;sf] v]taf/Ldf nufPsf afnLsf

;d"xx¿ s:tf ;d"xx¿ xf]nfg\ <
ladnfsf] v]taf/Ldf / PGhnsf] v]taf/Ldf nufOPsf afnLsf ;dx" x¿ s:tf

;dx" x¿ xfn] fg\ <
/fdljnf;sf] v]taf/Lsf afNfLaf6 lgDgfg;' f/ pk;dx" x¿ lgdf0{ f ug{'xf];\ / s:tf

;d"x eP ;r" Ls/0f ljlwaf6 n]Vg'xf;] \ .
— cGgafnLsf] ;d"x
— kmnkm" nsf] ;dx"
— t/sf/Lsf] ;d"x

dflysf kZ| gx¿nfO{ ;d"xdf 5nkmn u/L lgisifn{ fO{ sIffdf k:| tt' ug'x{ f];\ .

1.1 ;dx" sf ljm| ofx¿ (Operation of sets)

ljm| ofsnfk 1

bO' { b'O{ hgfsf] ;dx" df kT| o]sn] ljleGg /ªsf km" nx¿ ;ªs\ ng ugx{' f];\ / hDdf u/]sf k"mnsf
/ªcg;' f/ ;d"xx¿ lgdf{0f ug'x{ f;] ,\ h:t} M

;'lk|ofn] hDdf u/s] f km" nsf /ªx¿sf] ;d"x (S) = {/ftf], kxn]F f,] lgnf,] u'nfkmL }

PlGhnfn] hDdf u/]sf km" nsf /ªx¿sf] ;d"x (A) = {kxFn] f,] ;]tf,] lgnf,] KofhL, ;G' tnf, /ftf}]

ul0ft, sIff ( 1

dflysf ;dx" x¿sf cfwf/df tnsf ;dx" x¿ lgdf{0f ug'x{ f;] \ M
-s_ ;'lk|of cyjf PlGhnf jf b'jn} ] ;ªs\ ng u/]sf km" nsf /ªx¿sf] ;dx"
-v_ ;l' k|of / PlGhnf bj' n} ] ;ª\sng u/s] f km" nsf ;femf /ªx¿sf] ;d"x
-u_ PlGhnfn] dfq ;ª\sng u/]sf k"mnsf /ªx¿sf] ;dx"
-3_ ;l' k|ofn] ;ªs\ ng u/]sf km" nsf /ª\x¿ afxs] sf /ªx¿sf] ;dx" .

o;/L lgdf{0f ul/Psf ;d"xx¿ s:tf ;dx" x¿ xfn] fg\ <

1.1.1 ;d"xx¿sf] ;+ofh] g (Union of sets)

dflysf] ;dx" x¿df, U
;'lko| fn] hDdf u/s] f km" nsf /ªx¿sf] ;dx" (S) SA

= {/ftf], kx]nF f,] lgnf,] un' fkmL} /ftf] ;t] f]
PGhLnfn] hDdf u/]sf k"mnsf /ªx¿sf] ;d"x (A) u'nfaL] kxF]nf] KofhL
= {kxn]F f], ;t] f,] lgnf], KofhL, ;G' tnf, /ftf}]
bj' }n] hDdf u/]sf km" nsf /ªx¿sf] ;d"x lgnf] ;'Gtnf

= {/ftf], kxF]nf], lgnf], un' fkmL, ;t] f], KofhL, ;'Gtnf}

o;/L bj' n} ] hDdf u/]sf km" nx¿sf /ªx¿sf] ;d"xnfO{ bj' }n]

cnu cnu ;ªsng u/s] f km" nsf /ªx¿sf] ;dx" sf] ;+ofh] g elgG5 .

;d"xx¿ A / B ;j{Jofks ;d"x U sf pk;d"x xg' \ U
eg] ;d"xx¿ A / B sf] ;o+ f]hg nfO{ A B n] A B

hgfOG5 . (A B) df ;dx" A df kg{] jf ;d"x B

df kg{] ;b:ox¿ k5{g\ . ;d"x lgdf0{ f ljlwcg;' f/,

A B = {x : x ∈ A jf x ∈ B} nl] vG5 . lbOPsf]

eg] lrqdf 5fof kfl/Psf] efun] ;dx" (A B) nfO{

hgfpF5 .

To;}u/L (A B C) = {x:x∈A jf x∈B jf x∈C} U
nl] vG5 . lbOPsf] eg] lrqdf 5fof kfl/Psf] efun] A B
(A B C) nfO{ hgfp5F .

;d"xsf] ;o+ fh] g ubf{ lbOPsf ;dx" sf ;femf
;b:ox¿nfO{ gbfx] f]¥ofOsg afsF L ;a} ;b:ox¿nfO{
lnP/ ;dx" sf] ¿kdf n]Vgk' 5{ .

C

2 ul0ft, sIff (

pbfx/0f 1

olb P = {20 eGbf ;fgf 3 sf ckjTo{x¿} / Q = {20 eGbf ;fgf 2 sf ckjTo{x¿} xg' \ eg] P Q
kQf nufO{ e]glrqdf k:| t't ug'x{ f;] \ .

;dfwfg P U
oxfF P = {20 eGbf ;fgf 3 sf ckjTox{ ¿} Q
= {3, 6, 9, 12, 15, 18}
Q = {20 eGbf ;fgf 2 sf ckjTo{x¿} 3 9 12 6 4 10
= {2, 4, 6, 8, 10, 12, 14, 16, 18} 15 18 16 2

14 8

ca P Q

= {3, 6, 9, 12, 15, 18} {2, 4, 6, 8, 10, 12, 14, 16, 18}
= {2, 3, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18}

eg] lrqdf 5fof kf/]sf] efun] P Q nfO{ hgfpF5 .

pbfx/0f 2

olb A = {ux'F, kmfk/, tf/] L, ufn] e8F] f, eG6f, v';f{gL, hf}, cfn,' l;dL} / B = {kmfk/, hf}, cfn',

l;dL} 5g\ eg], A B kQf nufO{ e]glrqdf k|:t't ug{x' f];\ .

;dfwfg

oxfF A = {ux'F, kmfk/, tf]/L, ufn] e8F] f, eG6f, v;' fg{ L, U

hf}, cfn', l;dL} v;' fg{ L A
B = {kmfk/, hf}, cfn', l;dL}
B

ca A B kmfk/
= {ux',F kmfk/, tf]/L, ufn] e8]F f, eG6f, v;' f{gL, uxF' hf}

cfn' eG6f

l;dL

hf}, cfn', l;dL} {kmfk/, hf}, cfn', tf/] L ufn] e]F8f
l;dL}

= {ux'F, kmfk/, tf]/L, uf]ne]F8f, eG6f, v';fg{ L, hf,}

cfn', l;dL} = A

;Fus} f] e]glrqdf 5fof kf/s] f] efun] A B nfO{ hgfp5F .

ul0ft, sIff ( 3

lj|mofsnfk 2

olb bO' {cf]6f ;dx" cfk;df cnlUuPsf 5g\ eg] Tof] cj:yfdf ltgLx¿sf] ;+ofh] g s] xf]nf <
cfk;df 5nkmn u/L eg] lrq;d]t agfP/ k|:t't ug'x{ f];\ .

gf6] M bO' { ;dx" dWo] Pp6f ;d"x csf]{ ;d"xsf] pk;dx" 5 eg] ltgLx¿sf] ;+of]hg 7n' f] ;dx"
g} xG' 5 . olb bO' c{ f6] f ;d"x cfk;df cnlUuPsf 5g\ eg] ltgLx¿sf] ;o+ f]hg tL ;dx" sf
;a} ;b:ox¿ ldn]/ ags] f] ;dx" x'G5 .

pbfx/0f 3

olb P = {10 eGbf ;fgf hf]/ ;ª\Vofx¿} / Q = {10 eGbf ;fgf lahf]/ ;ª\Vofx¿} eP, P Q
kQf nufO{ e]glrqdf k:| tt' ug{x' f];\ .

;dfwfg P U
oxfF P = {10 eGbf ;fgf hf/] ;ª\Vofx¿} Q

= {2, 4, 6, 8} 2 4 6 1 3
8 5 7
Q = {10 eGbf ;fgf lahf/] ;ª\Vofx¿} 9

= {1, 3, 5, 7, 9}

P Q = {2, 4, 6, 8} {1, 3, 5, 7, 9}
= {1, 2, 3, 4, 5, 6, 7, 8, 9}

gf]6 M eg] lrqdf 5fof kf/s] f] efun] P Q nfO{ hgfp5F .

pbfx/0f 4

olb A = {a, b, c, d, e}, B = {a, e, i, o, u}, C = {d, e, f, g} eP, A B C kQf nufO{

e]glrqdf k|:t't ug'x{ f];\ .

;dfwfg

oxfF A = {a, b, c, d, e}, B = {a, e, i, o, u}, U
A B
C = {d, e, f, g}
i
ca (A B C) b ao
c eu
= {a, b, c, d, e} {a, e, i, o, u} {d, e, f, g}
⸫ (A B C) = {a, b, c, d, e, f, g, i, o, u} d

;Fu}sf] e]glrqdf 5fof kf/s] f] efun] (A B C) nfO{

hgfpF5 . fg
C

4 ul0ft, sIff (

1.1.2 ;dx" sf] k|ltR5b] g (Intersection of sets)

sIff 9 sf ljBfyLd{ Wo] lj1fg dg k/fpg] / ul0ft dg k/fpg] ljBfyLx{ ¿ lgDgfg;' f/ /x]5g\ M

lj1fg dg k/fpg] ljBfyLx{ ¿sf] ;dx" (S) U

= {/fd, ;Ltf, kD] af, tl] Ghª, 8Lgf, ljgo} S M

ul0ft dg k/fpg] ljBfyL{x¿sf] ;d"x (M) /fd xl/
= {xl/, ;Ltf, d~h', 8f]Ndf, 8Lgf, clhdf} k]Daf l;tf d~h'
dflysf ;dx" nfO{ eg] lrqdf k:| t't ubf{, tl] Ghª l8gf 8fN] df
clhdf
;uF }sf] eg] lrqdf lj1fg / ul0ft b'j} dg k/fpg] ljBfyLs{ f] ljgo

;dx" {;Ltf, 8Lgf} xf] . of] g} lj1fg dg k/fpg] / ul0ft

dg k/fpg] ljBfyLs{ f ;d"xsf] kl| tR5b] g (Intersection of sets) xf] .

olb A / B vfnL ;d"xx¿ gePdf A / B b'j} ;dx" sf ;femf A U
;b:ox¿af6 ag]sf] ;d"xnfO{ ;dx" A / B sf] k|ltR5b] g
elgG5, o;nfO{ A B n] hgfOG5 . B

;d"x lgdf0{ f ljlwcg';f/ A B = {x : x ∈ A / x ∈ B}
nl] vG5 .

o;nfO{ eg] lrqsf] ko| fu] u/L lgDgfg;' f/ b]vfpg ;lsG5 . ;Fus} f]
e]glrqdf 5fof kf/s] f] efun] ;d"x A / B sf] k|ltR5b] gnfO{
hgfp5F .

pbfx/0f 5

olb P = {20 eGbf ;fgf 3 sf ckjTo{x¿} / Q = {20 eGbf ;fgf 2 sf ckjTo{x¿sf ;d"x} x'g\ eg]
;d"x P Q kQf nufO{ eg] lrqdf k|:tt' ug'{xf];\ .

;dfwfg U

oxfF P = {20eGbf ;fgf 3 sf ckjTox{ ¿} P Q

= {3, 6, 9, 12, 15, 18} 39 12 4
Q = {20eGbf ;fgf 2 sf ckjTo{x¿} 15 10
= {2, 4, 6, 8, 10, 12, 14, 16, 18} 6 2 16
18
8
14

ca P Q

= {3, 6, 9, 12, 15, 18} {2, 4, 6, 8, 10, 12, 14, 16, 18}

= {6, 12, 18}

;Fu}sf] e]glrqdf 5fof kf/s] f] efun] P Q nfO{ hgfpF5 .

ul0ft, sIff ( 5

pbfx/0f 6

olb A = {20 eGbf ;fgf 2 sf ckjTo{x¿} / B = {20 eGbf ;fgf 4 sf ckjTo{x¿} xg' \ eg] A B

kQf nufO{ eg] lrqdf k|:t't ug{'xf];\ .

;dfwfg U
oxfF A = {20 eGbf ;fgf 2 sf ckjTo{x¿} 6A
= {2, 4, 6, 8, 10, 12, 14, 16, 18}

B = {20 eGbf ;fgf 4 sf ckjTo{x¿} = {4, 8, 12, 16} 4 8B
ca A B = {2, 4, 6, 8, 10, 12, 14, 16, 18} {4, 8, 12, 16} 2 12 16 10

= {4, 8, 12, 16} 18 14

e]glrqdf 5fof kf/s] f] efun] A B nfO{ hgfpF5 .

bO' { ;dx" dWo] Pp6f ;d"x csf{] ;dx" sf] pk;d"x 5 eg] ltgLx¿sf] k|ltR5]bg
pk;d"x g} xG' 5 .

ljm| ofsnfk 3

olb bO' {cf6] f ;d"x cfk;df cnlUuPsf 5g\ eg] Tof] cj:yfdf ltgLx¿sf] k|ltR5b] g
s] xfn] f < cfk;df 5nkmn u/L k:| tt' ugx{' f;] \ .

olb b'O{cf6] f ;dx" cfk;df cnlUuPsf 5g\ eg] ltgLx¿sf kl| tR5b] g vfnL ;dx" x'G5 .

pbfx/0f 7

olb M = {a, e, i, o, u} / N = {p, q, r, s, t} 5g\ eg] M N kQf nufO{ eg] lrqdf b]vfpgx' f;] \ .

;dfwfg M U
oxfF M = {a, e, i, o, u}, N = {p, q, r, s, t} N
ae
M N = {a, e, i, o, u} {p, q, r, s, t} io pq
rs
⸫ M N = { }jf φ u
t

e]glrqdf ;femf ;b:o gePsfn] s'g} klg ;dx" df 5fof kfl/bF }g . ul0ft, sIff (

6

pbfx/0f 8

olb P= {1, 2, 3, 4, 5, 6}, Q = {3, 4, 5, 6, 7, 8} / R = {1, 3, 5, 7, 9} 5 eg] P Q R kQf nufO{
e]glrqdf b]vfpgx' f;] \ .

;dfwfg U
oxfF P = {1, 2, 3, 4, 5, 6} P Q

Q = {3, 4, 5, 6, 7, 8} 4
2 68
R = {1, 3, 5, 7, 9}
35
17
ca P Q R

= {1, 2, 3, 4, 5, 6} {3, 4, 5, 6, 7, 8}

{1, 3, 5, 7, 9} 9

= {3, 5} R
⸫ P Q R = {3, 5}

1.1.3 ;dx" x¿sf] km/s (Difference of sets)

ljm| ofsnfk 4 T U
P
lbOPsf] e]glrq cWoog ugx{' f];\ . 6]lnlehg xg] { ;Ltf
dg k/fpg] ljBfyLs{ f] ;d"x (T) = {;Ltf, l5l/ª, sfhL, ;f]gfd chLdf
;fg] fd, wd]G{ b}| ,
l5l/ª xl/
klqsf k9g\ dg k/fpg] ljBfyL{sf] ;dx" (P) = {clhdf, wdG]{ b|
xl/, lbks, ;f]gfd, wd]G{ b}| lbks
sfhL
6]lnlehg dfq} x]g{ dg k/fpg] ljBfyL{sf] ;dx" / klqsf
dfq k9g\ dg k/fpg] ljBfyL{sf] ;dx" n]Vg'xf;] \ .

oxfF klqsf dfq k9g\ dg k/fpg]n] 6]lnlehg xg] { dg k/fpFb}g . t;y{ o;nfO{ klqsf k9\g /

6l] nlehg x]g{ dg k/fpg] ;d"xsf] km/s elgG5 .

To:t}, 6]lnlehg xg] d{ fq dg k/fpg] klqsf k9g\ dgk/fpFb}gg\ t;y{ o;nfO{ 6l] nlehg xg] { /
klqsf k9\g dg k/fpg] ;d"xsf] km/s elgG5 .

;dx" A / ;d"x B ;jJ{ ofks ;dx" U sf pk;dx" x¿ xg' \ . ;dx" A df kg]{ t/ ;d"x B df
gkg{] ;d"x A sf afFsL ;b:ox¿sf] ;dx" nfO{ A / B sf] km/s (A difference B) elgG5,
o;nfO{ A – B n]lvG5 / k9b\ f A / B sf] km/s cyjf A difference B eg/] kl9G5 .
o;/L g} ;dx" B df dfq kg{] ;b:ox¿sf] ;d"xnfO{ B / A sf] km/s cyjf B difference
A elgG5 . o;nfO{ B – A nl] vG5 / k9b\ f B / A sf] km/s cyjf B difference A eg]/
kl9G5 . ;dx" lgdf0{ f ljlw (Set builder method) cg';f/,

A – B = {x : x ∈ A / x ∉ B} / B – A = {x : x ∈ B / x ∉ A} nl] vG5 .

ul0ft, sIff ( 7

eg] lrqdf A km/s B / B km/s A (B – A) U

-s_ olb ;d"x A / ;d"x B vlK6Psf ;d"x x'g\ eg,] A B

(A – B)
U

A B

-v_ olb ;dx" A / ;dx" B cnlUuPsf ;d"x xg' \ eg],

(A – B) U (B – A) U

A B A B

-u_ olb ;dx" B ;dx" A sf] Pp6f pkoS' t pk;dx" xf] eg,]

(A – B) U olb ;dx" A / ;d"x B a/fa/ ;d"x x'g\ eg,]
B A A–B=B–A=φ
olb ;d"x A af6 vfnL ;dx" sf] km/s lgsfNg] xf]

eg],
A–φ=A
olb vfnL ;dx" af6 ;dx" A sf] km/s lgsfNg] xf]

eg,]
φ–A=φ
olb ;dx" A ;d"x B sf] pk;d"x xf] eg],
A–B=φ

gf]6 M (A – B) / (B – A) sf] ;+of]hgnfO{ ;dldtLo km/s (Symmetric diference)
elgG5 . o;nfO{ A ∆ B n] hgfOG5 / A ∆ B = (A – B) (B – A) nl] vG5 .

8 ul0ft, sIff (

pbfx/0f 9

lbOPsf] eg] lrqaf6 lgDglnlvt ;d"xx¿nfO{ ;"rLs/0f ljlwaf6 nV] gx' f;] \ M

-s_ A -v_ B -u_ A – B -3_ B – A

;dfwfg U
oxfF,
A B
-s_ A = {1, 3, 5, 7, 9} 56
-v_ B = {6, 7, 8, 9, 10}
-u_ A — B = {1, 3, 5} 3 79 10
-3_ B — A = {6, 8, 10} 1 8

2 4

pbfx/0f 10

olb U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, A = {1, 2, 3, 4, 5} / B = {4, 5, 6, 7, 8, 9, 10} eP
A — B / B — A kQf nufO{ eg] lrqdf 5fof kf/]/ k|:t't ug{x' f];\ .

;dfwfg (A – B) U
oxf F A – B
A B
= {1, 2, 3, 4, 5}− {4, 5, 6, 7, 8, 9, 10}
6
= {1, 2, 3} /
1 7
B–A 8
= {4, 5, 6, 7, 8, 9, 10} – {1, 2, 3, 4, 5} 2 4 9
3 5
= {6, 7, 8, 9, 10}
10
;Fus} f] eg] lrqdf A − B / B − A nfO{ 5fof kf//]
(B – A) U
bv] fOPsf] 5 . A B

6

1 7
8
2 4 9
3 5

10

ul0ft, sIff ( 9

pbfx/0f 11

olb U = {x : x ≤ 30, x n] kf| s[lts ;ª\Vof hgfpF5},

A = {x : x n] 15 eGbf 7n' f] / 30 eGbf ;fgf] Pp6f kf| s[lts ;ªV\ of hgfp5F } /

B = {x : x n] 1 b]lv 15 ;Ddsf] Pp6f k|fs[lts ;ª\Vof hgfpF5} eg] A − B / B – A kQf
nufpgx' f];\ .

;dfwfg U
oxf F U = {1, 2, 3, 4, ..., 28, 29, 30}
A = {16, 17, 18, ..., 27, 28, 29} / A B
B = {1, 2, 3, 4, ..., 13, 14, 15}
16 17 1

18 19 20 21 2345

6 7 8 9 10
22 23 24 25

11 12 13

26 27 28 14 15
29 30

t;y{,

A − B = {16, 17, 18, ..., 27, 28, 29} – {1, 2, 3, 4, ..., 13, 14, 15}

= {16, 17, 18, ..., 27, 28, 29} = A /

B − A = {1, 2, 3, 4, ..., 13, 14, 15} – {16, 17, 18, ..., 27, 28, 29}
= {1, 2, 3, 4, ..., 13, 14, 15} = B

olb A / B cnlUuPsf (Disjoint) ;d"xx¿ ePdf A – B = A / B – A = B g} x'G5 .

1.1.4 ;dx" sf] k'/s (Complement of a set)

dfgf}F ;dx" A s'g} ;j{Jofks ;dx" U sf] pk;dx" xf] . U
hxfF ;dx" A dfeGbf ;d"x U df A sf ;a} ;b:ox¿;lxt A
sx] L yk ;b:ox¿ 5g\ . o:tf ;j{Jofks ;d"x U df
dfq xg' ] t/ ;dx" A df gx'g] ;b:ox¿sf] ;dx" nfO{ ;dx" U
A sf] k"/s ;dx" (Complement of set A) elgG5 .
o;nfO{ A' cyjf A cyjf Ac åf/f hgfOG5 . AA

csf]{ zAbdf s'g} ;jJ{ ofks ;dx" U / To;sf] pk;dx" A sf] km/s (Difference) nfO{ Tof]
pk;d"x A sf] k/" s elgG5 . o;nfO{ ;ª\s]tdf A = U − A nl] vG5 .

;dx" lgdf0{ f ljlwcg';f/,

10 ul0ft, sIff (

A = { x : x ∈ U t/ x ∉ A} nl] vG5 .
A nfO{ e]glrqdf ;Fu}sf] lrqdf h:t} 5fof kf/]/ bv] fpg ;lsG5 .

A / A sf] ;o+ fh] g U xG' 5 . ctM A / A nfO{ Pscsf{sf k/" s elgG5 .
A A = U / A = A xG' 5 .

A A s] xf]nf < ;d"xdf 5nkmn u/L n]Vg'xf];\ M

pbfx/0f 12

olb U = {1, 2, 3, ..., 18, 19, 20}, A = {1, 3, 5, 7, 9, 11, 13, 15, 17, 19} / B = {2, 4, 6,
8, 10, 12, 14, 16, 18, 20} eP lgDglnlvt ;d"xx¿sf ;b:ox¿sf] ;"rL tof/ kfg'x{ f;] \ .

-s_ A -v_ B -u_ A B -3_ A B

;dfwfg U
oxfF U = {1, 2, 3, ..., 18, 19, 20}
A B
A = {1, 3, 5, 7, 9, 11, 13, 15, 17, 19}
12
B = {2, 4, 6, 8, 10, 12, 14, 16, 18, 20}
3579 4 68
-s_ A = U – A 11 13 15 10 12 14

= {1, 2, 3, ..., 18, 19, 20} − {1, 3, 17 19 16 18
5, 7, 9, 11, 13, 15, 17, 19} 20

= {2, 4, 6, 8, 10, 12, 14, 16, 18,
20}

-v_ B = U – B

= {1, 2, 3, ... ,18, 19, 20} – {2, 4, 6, 8, 10, 12, 14, 16, 18, 20}

= {1, 3, 5, 7, 9, 11, 13, 15, 17, 19}

-u_ A ∪ B

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

= {1, 2, 3, 4, ..., 18, 19, 20}

-3_ A B

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

={}

ul0ft, sIff ( 11

pbfx/0f 13

olb U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} / A = {1, 3, 5, 7, 9} eP lgDglnlvt ;dx" x¿sf
;b:ox¿ ;"rLs/0f ljlwaf6 nV] g'xf];\ M

-s_ A -v_ A ∪ A -u_ A A -3_ A

;dfwfg U
oxfF U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} / 4

A = {1, 3, 5, 7, 9} 2 A 6
-s_ A = U − A
13
= {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} – {1, 3, 5, 7, 9} 57

9

= {2, 4, 6, 8, 10} 10 8

-v_ A ∪ A = {1, 3, 5, 7, 9} ∪ {2, 4, 6, 8, 10}

= {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} = U

-u_ A ∩ A = {1, 3, 5, 7, 9} ∩ {2, 4, 6, 8, 10} = φ

-3_ A = U − A

= {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} – {2, 4, 6, 8, 10}

= {1, 3, 5, 7, 9} = A

pbfx/0f 14

olb U = {a, b, c, d, e, f, g, h, i, j, k}, A = {a, b, c, d, i, j}, B = {c, d, e, f, g, h, i} / C
= {d, e, f, g, h, i, j, k} eP lgDglnlvt ;b:ox¿sf] ;r" L tof/ kfg'{xf;] \ M

-s_ (A ∪ B) C -v_ A∪B -u_ A B
-3_ (A C) ∪ A -ª_ (A C ) ∪B

;dfwfg

-s_ (A ∪ B) ∩ C

(A ∪ B) = {a, b, c, d, i, j} ∪ {c, d, e, f, g, h, i}

= {a, b, c, d, e, f, g, h, i, j}

(A ∪ B) ∩ C = {a, b, c, d, e, f, g, h, i, j} ∩{d, e, f, g, h, i, j, k}

= {d, e, f, g, h, i, j}

12 ul0ft, sIff (

km]l/ (A ∪ B) ∩ C 13

= U–{(A ∪ B) ∩ C}

= {a, b, c, d, e, f, g, h, i, j, k} – {d, e, f, g, h, i, j}
= {a, b, c, k}

-v_ A ∪ B

A= ∪ –A

= {a, b, c, d, e, f, g, h, i, j, k}– {a, b, c, d, i, j}
= {e, f, g, h, k}

B=∪–B

= {a, b, c, d, e, f, g, h, i, j, k}– {c, d, e, f, g, h, i}
= {a, b, j, k}

kml] / A ∪ B

= {e, f, g, h, k} ∪ {a, b, j, k}

= {a, b, e, f, g, h, j, k}

-u_ A ∩ B

= {a, b, c, d, i, j} ∩ {a, b, j, k}

= {a, b, j}

-3_ C = U – C

= {a, b, c, d, e, f, g, h, i, j, k} – {d, e, f, g, h, i, j, k}
= {a, b, c}

ca A C

= {a, b, c, d, i, j} {a, b, c}
= {a, b, c}

km]l/ (A C ) ∪ A

= {a, b, c} ∪ {e, f, g, h, k}

= {a, b, c, e, f, g, h, k}

-ª_ (A C)
= {e, f, g, h, k} ∩ {d, e, f, g, h, i, j, k}

= {e, f, g, h, k}

(A C) ∪ B
= {e, f, g, h, k} ∪ {a, b, j, k}

= {a, b, e, f, g, h, j, k}

ul0ft, sIff (

cEof; 1 U
FV
1. lbOPsf] eg] lrqdf F n] km'6an dg k/fpg] / V n] elnan
dg k/fpg] ljBfyL{sf] ;d"x hgfpF5 . eg] lrqsf] cWoog l5l/ª dfof af6n' L
u/L lgDgfg;' f/sf ;dx" x¿nfO{ ;"rLs/0f ljlwaf6 bf]h]{ cfzf 8f]Ndf
nV] gx' f];\ M xl/ u0f]z ;G' tnL
/fdaxfb'/ xs;{ G' b/
-s_ F -v_ V -u_ F ∪ V -3_ F ∩ V -ª_ U

2. olb U = {a, b, c, d, e, f, g, h, i, j, k}, A = {a, c,
e, f, g, i, k} / B = {b, d, i, j, k, h} eP lgDg ;dx" sf
;b:ox¿ kQf nufO{ eg] lrqdf k|:tt' ugx'{ f;] \ M

-s_ (A ∩ B) -v_ (B ∪ A) -u_ A — B -3_ B — A

3. olb U = {x : x n] 1 b]lv 30 ;Ddsf k"0f{ ;Vof hgfpF5}, A = {x : x n] 1 bl] v 30 ;Ddsf 3
sf] ckjTo{ hgfp5F } / B = {x : x n] 1 bl] v 30 ;Ddsf 4 sf] ckjTo{ hgfp5F } / C ={x : x
n] 1 bl] v 30 ;Ddsf 5 sf] ckjTo{ hgfpF5} eg] lgDglnlvt ;dx" x¿nfO{ ;r" Ls/0f ljlwaf6
nV] g'xf];\ / e]glrqdf b]vfpg'xf];\ M

-s_ (A — B) -v_ (B — A) -u_ (A — C) -3_ (B — C)

-ª_ (A ∪ B) -r_ A ∪ B ∪ C -5_ A ∩ B ∩ C -h_ (B ∪ C)

4. olb U = {a, b, c, d, e, f, g, h, i, j, k }, A = { a, c, d, f }, B = { g, h, i} eP,

-s_ lgDg ;dx" x¿ lgdf{0f ug{'xf];\ M

-c_ A -cf_ B -O_ A ∪ B

-O{_ A ∩ B -p_ A ∪ B -pm_ (A ∩ B)
-v_ kZ| g g= -s_ df agfOPsf ;dx" x¿ s'g s'g a/fa/ ;d"xx¿ xG' 5g,\ kQf nufpgx' f];\ .

5. olb U = {1 bl] v 12 ;Ddsf k0" ffª{ \sx¿sf] ;dx" }, E = {1 b]lv 12 ;Ddsf hf/] ;ªV\ ofx¿sf]
;dx" }, O = {1 bl] v 12 ;Ddsf lahf/] ;ªV\ ofx¿sf] ;d"x} / P = {1 bl] v 12 ;Ddsf ¿9
;ªV\ ofx¿sf] ;dx" } eP lgDglnlvt ;d"xx¿ kQf nufO{ eg] lrqdf k:| tt' ug'{xf];\ M

-s_ E -v_ O -u_ P -3_ (E ∪ P)
-h_ (E ∪ O ∪ R )
-ª_ P ∩ Q -r_ P – O -5_ P -6_ P ∪ (E ∩ O)

-em_ (E ∩ O ∩ R) -`_ (E ∪ P ) – (P ∩ O)

6. olb U = {m, n, o, p, q, r, s, t, u, v}, A = {q, r, s, t, u, v}, B = {n, o, p, q, r},
C = {m, u, s, t, q, r} eP lgDg ;DaGwx¿sf] ;d"x kQf nufO{ eg] lrqdf bv] fpgx' f];\ M

-s_ (A ∩ B) -v_ (A ∪ B) ∩ C -u_ (A ∪ B ∪ C) -3_ (A ∩ B ∩ C)

-ª_ (A – B) -r_ (A ∪ B ∪ C) -5_ A ∩ B -h_ A -em_ (A ∩ C) ∪ B

14 ul0ft, sIff (

7. tnsf eg] lrqx¿df 5fof kfl/Psf] efun] sg' ;d"x hgfpF5, ;ªs\ ]tdf pQ/ nV] g'xf;] \ M

-s_ U -v_ U -u_ U

A B A B A B

-3_ U -ª_ U -r_ U
A B
A B

A

C
C

8. olb P / Q ;j{Jofks ;dx" U sf k|ltR5l] bt pk;dx" x¿ eP lgDglnlvt ;dx" x¿nfO{
eg] lrqdf bv] fpg'xf];\ M

-s_ P – Q -v_ Q − P -u_ (P – Q) ∪ P -3_ P ∩ (Q – P)

9. s'g} Pp6f ;jJ{ ofks ;dx" U / To;sf bO' {cf]6f pk;dx" x¿ X / Y agfpg'xf;] \ . To;kl5
lgDglnlvt ;dx" x¿nfO{ ;r" Ls/0f ljlwaf6 nV] g'xf;] \ M

-s_ (X ∪ Y ) -v_ (X ∩ Y ) -u_ X -3_ X ∩ Y

10. tkfO{F k9\g] sIffdf ePsf ljBfyLa{ f6 lgDglnlvt ;d"xx¿ agfpgx' f;] \ M
-s_ ;a} ljBfyL{sf] ;dx"
-v_ 5fqx¿sf] ;dx"
-u_ 5fqfx¿sf] ;d"x

oL ;dx" x¿dWo] s'g ;d"x ;j{Jofks ;d"x / s'g sg' ;d"xx¿ o;sf pk;d"x x'g\ < pkoS' t
;ª\s]t;lxt nV] g'xf];\ . To;kl5 oL ;a} ;dx" sf k"/s ;dx" x¿ lgsfNgx' f];\ .

ul0ft, sIff ( 15

pQ/

1. -s_ {l5l/ª, bf]h],{ xl/, dfof, cfzf, u0f]z}
-v_ {af6n' L, 8f]Ndf, ;G' tnL, dfof, cfzf, u0f]z}
-u_ {l5l/ª, bf]h]{, xl/, af6'nL, 8fN] df, ;'GtnL, dfof, cfzf, u0fz] }
-3_ {dfof, cfzf, u0f]z }

-ª_ {l5l/ª, bfh] ]{, xl/, af6'nL, 8fN] df, ;'GtnL, dfof, cfzf, u0f]z, /fhaxfb'/,
xs{axfb/' }

2. -s_ {i, k} -v_ U u_ {a, c, e, f, g} -3_ {b, d, j, h}

-ª_ φ / eg] lrqx¿ lzIfsnfO{ b]vfpgx' f];\ .

3. -s_ {3, 6, 9, 15, 18, 21, 27, 30} -v_ {4, 8, 16, 20, 28}

-u_ {3, 6, 9, 12, 18, 21, 24, 27} -3_ {4, 8, 12, 16, 24, 28}

-ª_ {3, 4, 6, 8, 9, 12, 15, 16, 18, 20, 21, 24, 27, 28, 30}, eg] lrqx¿
lzIfsnfO{ bv] fpgx' f;] \ .

-r_ {3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 21, 24, 25, 27, 28, 30}
-5_ { }

-h_ {1, 2, 3, 4, 6, 7, 9, 11, 13, 14, 17, 18, 19, 21, 22, 23, 26, 27, 29}

4. -s_ -c_ { b, e, g, h, i, j, k } -cf_{a, b, c, d, e, f, j, k }

-O_ {a, b, c, d, e, f, g, h, i, j, k } -O{_ {b, e, j, k}

-p_ {b, e, j, k} -pm_ U

-v_ lzIfsnfO{ b]vfpgx' f];\ .

5. -s_ {1, 3, 5, 7, 9, 11} -v_ {2, 4, 6, 8, 10, 12}

-u_ {1, 4, 6, 8, 9, 10, 12} -3_ {1, 9} -ª_ U -r_ Q

-5_ {2, 3, 5, 7, 11} -h_ {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}

-em_ {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} -`_ {1, 9} -6_ {1, 4, 6, 8, 9, 10}

6. -s_ {q, r} -v_ {u, s, t, r} -u_ {m, n, o, p, q, r, s, t, u, v}

-3_ {q, r} -ª_ {s, t, u, v} -r_ { } -5_ {m, n, o, p, s, t, u, v }

-h_ {m, n, o, p} -em_ {q, r, s, t, u, n, o, p}

7. -s_ (A − B) -v_ (B − A) -u_ B jf B – A

-3_ A jf U – A 8 - 10 lzIfsnfO{ b]vfpg'xf];\ .

16 ul0ft, sIff (

1.2 ;dx" sf] u0fgfTdstf (Cardinality of sets)

sg' } klg ;d"xdf ePsf hDdf ;b:ox¿sf] ;ªV\ ofnfO{ g} To; ;d"xsf] u0fgfTdstf (Cardinality
of set) elgG5 . h:t} ;dx" A = {m, a, t, h} df hDdf ;b:o ;ªV\ of 4 5, To;}n] ;dx"
A sf] u0fgfTdstf 4 xf] . o;nfO{ ;ª\st] df n(A) = 4 n]lvG5 .
olb sg' } ;d"x vfnL ;dx" xf] eg] pSt ;d"xsf] u0fgfTdstf 0 -zG" o_ x'G5 .

h:t} M B = {9 sIffdf cWoog ug{] 5 jife{ Gbf sd pd]/sf ljBfyLx{ ¿sf] ;dx" } . o; ;d"xdf
Ps hgf klg ;b:o gxg' ] jf ;b:osf] ;ª\Vof z"Go xg' ] ePsfn] n(B) = 0 n]lvG5 .

ljm| ofsnfk 1 U
-s_ olb, A B

U = {;fs{ /fi6«x¿sf] ;d"x} gk] fn e'6fg
ef/t aªu\ nfb]z
A = {gk] fn, ef/t, kfls:tfg, ckmuflg:tfg} kfls:tfg >Lnª\sf
ckmuflg:tfg dflNbE;
B = {e6' fg, aªu\ nfbz] , >Lnªs\ f, dflNbE;} eP,
n(A B) slt xG' 5 <

oxfF ;d"x A / B df ;femf ;b:o gePsf]n] oL cnlUuPsf ;d"xdf n(A B) = 8 eof] .

km]l/ n(A) = 4, n(B) = 4, n(A B) = n(A) + n(B) = 4 + 4 = 8 x'G5 .

-v_ olb A = {a, b, c, d, e}, B = {d, e, f, g} 5g\ eg], U
A B
(A B) = {a, b, c, d, e, f, g} / (A ∩ B) = {d, e}
To;n} ] n(A B) = 7 / n(A ∩ B) = 2 xG' 5, a
df
⸫ olb sg' } b'Oc{ f]6f ;dx" k|ltR5l] bt ;d"x eP
n(A B) = n(A) + n(B) – n(A ∩ B) x'G5 . b eg
c

lj|mofsnfk 2

tnsf] e]g lrqsf] cfwf/df tnsf ;d"xsf u0fgfTdstf kQf nufpgx' f;] \ / sIffdf k|:tt' \

ug'{xf;] \ M U

-s_ n(A) -v_ n(B) -u_ n(C) A b a o B
cu
-3_ n(A B C) -ª_ n(A ∪ B ∪ C)

oxfF e]glrqaf6, ⸫ n(A) = 5 de i
⸫ n(B) = 5 f g
A = {a, b, c, d, e } h
B = {a, e, i, o, u}
C

ul0ft, sIff ( 17

C = {d, e, f, i} ⸫ n(C) = 4
(A B C) = {a, b, c, d, e, f, i, o, u} ⸫ n(A B C) = 9
⸫ n(A∪ B ∪ C) = 2
(A∪ B ∪ C) = {g, h}

lj|mofsnfk 3

sg' } b'O{ ;d"xx¿ M / N vlK6Psf (Overlapping) ;d"x x'g\ eg] ;d"x M sf] dfq ;b:o
;ª\VofnfO{ no(M) / ;d"x N sf] dfq ;b:o ;ªV\ ofnfO{ no(N) n] hgfOG5 .

lbOPsf] e]glrqdf, no(M) = 3 xG' 5 . U
To;}n] no(M) = no(M – N) xG' 5 . M N
o;nfO{ no(M) = n(M) – n(M ∩ N) klg nV] g ;lsG5 .
To;u} /L no(N) = 4 6
To;u} /L no(N) = n(N – M) = n(N) – n(M ∩ N) xG' 5 . 1 47
oxfF n(M) = 5, n(N) = 6, no(M) = 3, no(N) = 4, 2 58
n(M ∪ N) = 9 / n(M ∩ N) = 2 x'G5 . 39

10

sg' } bO' { ;d"xdWo] Pp6f ;dx" csf{] ;d"xsf] pko'St pk;dx" (Proper subset) eP,
cyft{ \ A ⸦ B eP n(A B) = n(B) / n(A ∩ B) = n(A) x'G5 .

pbfx/0f 1

olb A = {1, 2, 3} / B = {1, 2, 3, 4, 5} 5g\ eg] kd| fl0ft ug'{xf];\ M
-s_ n(A ∪ B) = n(B) -v_ n(A B) = n(A)

;dfwfg
olb A = {1, 2, 3} / B = {1, 2, 3, 4, 5} 5g\ .
-s_ n(A∪B) = {1, 2, 3} ∪ {1, 2, 3, 4, 5}

= {1, 2, 3, 4, 5}
⸫ n(A ∪ B) = 5
⸫ n(A ∪ B) = n(B)

18 ul0ft, sIff (

-v_ n(A B) = {1, 2, 3} {1, 2, 3, 4, 5}

= {1, 2, 3}
⸫ n(A B) = 3
n(A) = {1, 2, 3}
⸫ n(A B) = n(A)

;dx" sf] u0fgfTdstf kQf nufpg] tl/sf

tn lbOPsf eg] lrqx¿ cWoog u/L ;f]lwPsf kZ| gsf] pQ/ lbg'xf];\ M

UU

P Q P Q

ad af
be g
cf b d h
c e i
g
j kl

lrq g= 1 lrq g= 2

-s_ n(U) = ? -v_ n(P) = ? -u_ n(Q) = ? -3_ n(P ∪ Q) = ?

-ª_ n(P ∩ Q) = ? -r_ n(P ∪ Q) = ?

oxfF lrq g= 1 df ;dx" P / Q cnlUuPsf oxfF lrq g= 2 df ;dx" P / Q vlK6Psf
;dx" x'g\ . ;dx" xg' \ .

U = {a, b, c, d, e, f, g} U = {a, b, c, d, e, f, g, h, i, j, k, l}
n(U) = 7 n(U) = 12

P = {a, b, c} P = {a, b, c, d, e}
n(P) = 3 n(P) = 5

Q = {d, e, f, g} Q = {d, e, f, g, h, i}
n(Q) = 4 n(Q) = 6

(P ∪ Q) = {a, b, c, d, e, f, g} (P ∪ Q) = {a, b, c, d, e, f, g, h, i}
n(P ∪ Q) = 7 n(P ∪ Q) = 9

(P ∩ Q) = { } (P ∩ Q) = {d, e}
n(P ∩ Q) = 0 n(P ∩ Q) = 2
(P ∪ Q) = { } (P ∪ Q) = {j, k, l}
n(P ∪ Q) = 0
n(P ∪ Q) = 3
⸫ n(U) = n(P ∪ Q)
⸫ n(U) = n(P ∪ Q) + n(P ∪ Q)

ul0ft, sIff ( 19

;d"xsf] u0fgfTdstf (Cardinality) nfO{ lgDg ;"qdf nV] g ;lsG5 M

-s_ olb A / B bO' c{ f]6f cnlUuPsf ;d"x eP, n(A ∪ B) = n(A) + n(B) x'G5 .
-v_ olb A / B b'Oc{ f6] f vlK6Psf ;dx" eP, n(A ∪ B) = n(A) + n(B) – n(A ∩ B)

x'G5 . cyjf n(A ∪ B) = no(A) + no(B) + n(A ∩ B) x'G5 .
-u_ U leq A / B sf dfq ;b:o eP n(U) = n(A ∪ B) x'G5 .
-3_ U leq A / B sf ;b:ox¿ afxs] cGo ;b:ox¿ klg eP
n(A ∪ B) = n(U) – n(A ∪ B) x'G5 . To:t} n(U) = n(A ∪ B) + n(A ∪ B)

x'G5 .
-ª_ vlK6Psf ;d"xdf no(A) = n(A) – n(A ∩ B) / no(B) = n(B) – n(A ∩ B) x'G5 .

U

pbfx/0f 2 A a c j B
bk
lbOPsf] e]glrq cWoog u/L ;f]lwPsf k|Zgsf] hafkm lbg'xf];\ M
d
-s_ n(A) -v_ n(B) -u_ n(C) e g
h

-3_ n(A B) -ª_ n(A B C) l fi
mC
-r_ n(A∪ B ∪ C) -5_ n(A ∪ B ∪ C)

-h_ no(A) -em_ no(A B) -`_ n(A – B)

;dfwfg ⸫ n(A B C) = 1
lbOPsf] e]glrqnfO{ cWoog ubf{ -r_ (A ∪ B ∪ C) = {a, b, c, d, e, f, g,
-s_ A = {a, b, c, d, e}
⸫ n(A) = 5 h, i, j, k}
-v_ B = {c, d, g, h, j, k}
⸫ n(A ∪ B ∪ C) = 11
⸫ n(B) = 6 -5_ (A ∪ B ∪ C) = {l, m}
-u_ C = {e, d, f, g, h, i} ⸫ n(A ∪ B ∪ C) = 2
-h_ A = {a, b}
⸫ n(C) = 6 ⸫ no(A) = 2
-3_ A B = {c, d} -em_ A B = {c}

⸫ n(A B) = 2 ⸫ no(A B) = 1
-ª_ A B C = {d} -`_ A – B = {a, b, e}

⸫ n(A – B) = 3

20 ul0ft, sIff (

cEof; 1.2

1.(a) lbOPsf] eg] lrq cWoog u/L lgDg ;dx" sf] u0fgfTdstf kQf nufpgx' f];\ M

-s_ n(S) -v_ n(T) -u_ n(U) U
-3_ n(S T) -ª_ n(R ∪ T) -r_ n(S R T) S T
-5_ n(S ∪ R ∪ T) -h_ n(S ∪ R ∪ T)
-em_ no(S) -`_ no(S R) -6_ n(R) 7
56 2 8

19
34

11

10 12
R

(b) olb A / B cnlUuPsf ;d"x xg' \ / n(A) = 30, n(B) = 35 eP n(A ∪ B) sf] dfg
kQf nufpgx' f];\ .

2.(a) lbOPsf] eg] lrqsf] ko| fu] u/L tnsf ;d"xsf] u0fgfTdstf kQf nufpg'xf;] \ M

-s_ n(A) -v_ n(B) -u_ n(A ∪ B) U

-3_ n(A B) -ª_ no(A) -r_ no(B) A B
da h
(b) olb U = {a, b, c, d, e, f, g}, A = {c, d, e, f}, B e i
fb j
= {a, b, e, f} / C = {d, e, f, g} eP tnsf ;d"xsf] gc
u0fgfTdstf kQf nufpgx' f];\ . m
kl

-s_ n(A – B) -v_ n(B – C) -u_ n(A – C) -3_ n(A)

-ª_ n(A ∪ B) -r_ n{(A ∪ B) – (A B)} -5_ n{(A – B) ∪ (B – A)}

3.(a) olb U = {20 eGbf ;fgf k|fsl[ ts ;ªV\ ofx¿}, A = {20 eGbf ;fgf hf]/ ;ª\Vofx¿}, B = {20

eGbf ;fgf ?9 ;ªV\ ofx¿} / C = {20 eGbf ;fgf ju{ ;ª\Vofx¿} eP tnsf ;d"xsf] u0fgfTdstf

kQf nufpg'xf];\ M

-s_ n(U) -v_ n(C) -u_ n(A B) -3_ n(B – C)

-ª_ n(A) -r_ n(A ∪ C) -5_ n(A B)

(b) olb U = {x : x Pp6f 20 ;Ddsf] k|fs[lts ;ª\Vof xf}]

A = {y : y Pp6f ?9 ;ª\Vof xf}] , B = {z : z Pp6f 18 sf] u'0fgv08 xf}]

ul0ft, sIff ( 21

C = {p : p Pp6f 20 eGbf ;fgf] 3 sf] ckjTo{ xf]} eg] e]glrqdf k|:t't u/L lgDg
;d"xx¿sf] u0fgfTdstf kQf nufpgx' f;] \ .

-s_ n(A ∪ B) -v_ n(B ∪ C) -u_ n(A ∪ B ∪ C)

-3_ n(A B C) -ª_ no(A) -r_ no(A – B)

-5_ n(A B) -h_ n(C)

4. lbOPsf] e]glrqsf] k|ofu] u/L lgDglnlvt ;dx" x¿sf] u0fgfTdstf kQf nufpg'xf;] \ M
-s_ n(P ∪ Q) -v_ n(P Q) -u_ n(P ∪ Q) -3_ n(P Q)

U
PQ

ano ab
p qr cd
ef
st

5. lbOPsf] e]glrqsf] k|of]u u/L tnsf ;dx" x¿ kQf nufpg'xf];\ M

-s_ n(S) -v_ n(T) -u_ nn(oS(S) T) S T U
-3_ n(S ∪ T) -ª_ no(T) -r_

-5_ n(T) -h_ n(S ∪ T) 2 4 3 12 14
-em_ n(S T) -`_ n(U) 16 18
68 5
10
20

7

6. lbOPsf] eg] lrqsf] cWoog u/L tnsf ;d"xsf u0fgfTdstf kQf nufpg'xf];\ M

-s_ n(A) -v_ n(B) U

A B

-u_ n(C) -3_ n(A ∪ B) 5 26
-ª_ n(A∪B∪C) -r_ n(A B C) 1

34

-5_ n(A ∪ B ∪ C) -h_ no(A) 78
C
-em_ no(C) -`_ n(U)

7. ;Fus} f] lrqaf6 tnsf ;DaGwx¿ bv] fpg'xf;] \ M
-s_ n(A) = n(∪ – A)

22 ul0ft, sIff (

-v_ n(A ∪ B) = n(A) + n(B) – n(A B) U
-u_ no(A) = n(A – B) A B
-3_ n(A ∪ B) = no(A) + n(B)
-ª_ n(A ∪ C) = no(A) + n(C) 01 5 67
-r_ n(B ∪ C) = no(B) + n(C)
-5_ (A ∪ B ∪ C) = U–(A ∪ B ∪ C) 4
11 2 3 8

9 10
12 C

kl/ofh] gf sfo{

1. :ofp / ;G' tnfdWo] tkfOF{nfO{ sg' kmnkm" n dg k5{ egL sDtLdf 20 hgf dflg;nfO{
;fW] g'xf];\ . :ofp dg k/fpg] dflg;x¿sf] ;dx" nfO{ A / ;G' tnf dg k/fpg]
dflg;x¿sf] ;dx" nfO{ O n] hgfpg'xf];\ . kf| Kt k|ltljm| ofcg';f/ :ofp dg k/fpg],
;'Gtnf dg k/fpg,] bj' } kmn dg k/fpg,] :ofp dfq} dg k/fpg], ;G' tnf dfq} dg
k/fpg] tyf :ofp / ;G' tnf s'g} klg kmnkm" n dg gk/fpg] dflg;x¿sf] ;dx" nfO{
;"rLs/0f ljlwaf6 n]Vgx' f];\ . pSt hfgsf/Laf6 tnsf ;dx" x¿ lgdf0{ f u/L
eg] lrqdf k|:t't ug{x' f;] \ M

-s_ A -v_ O -u_ A ∪ O -3_ A
-ª_ A − O -r_ O − A
-5_ (O − A) ∪ ( A – O )

2. tkfOF{sf] sIffdf cWoog ug{] ;a} ljBfyL{nfO{ k'm6an, lj|ms6] / af:s]6an vn] dWo]
tkfOnF{ fO{ s'g vn] dg k5{ egL ;fW] gx' f;] \ . k'm6an dg kg{n] fO{ F, ljm| s6] dg kg{]nfO{
C / af:s]6an dg k/fpg]nfO{ B n] hgfpg'xf];\ . k|fKt k|ltljm| ofcg;' f/ km6' an
dg kg{,] ljm| s]6 dg kg]{ / af:s]6an dg kg,{] s'g} b'Oc{ f6] f v]n dg k/fpg,] tLgcf]6} v]n
dg k/fpg,] s'g} Ps vn] dfq dg k/fpg] / sg' } klg vn] dg gk/fpg] ;d"xnfO{
;r" Ls/0f ljlwaf6 nV] gx' f];\ . pSt hfgsf/Lsf cfwf/df tnsf ;dx" sf] u0fgfTdstf
kQf nufO{ eg] lrqdf k:| t't ug{'xf];\ M

-s_ n(F) -v_ n(C) -u_ n(B) -3_ n(F B)

-ª_ n(B C) -r_ n(F C B) -5_ n(F ∪ C ∪ B) -h_ n(F ∪ B)

-em_ n(B ∪ C) -`_ n(C) -6_ no(F) -7_ no(F C)
-8_ n(F – B)

ul0ft, sIff ( 23

pQ/

1. (a) -s_ 5 -v_ 6 -u_ 12 -3_ 2 -ª_ 8 -r_ 1

-5_ 10 -h_ 2 -em_ 2 -`_ 1 -6_ 6 (b) 55

2. (a) -s_ 6 -v_ 5 -u_ 9 -3_ 2 -ª_ 4 -r_ 3

(b) -s_ 2 -v_ 2 -u_ 1 -3_ 3 -ª_ 1 -r_ 4 -5_ 4

3. (a) -s_ 19 -v_ 4 -u_ 1 -3_ 8 -ª_ 10 -r_ 12 -5_ 18

(b) -s_ 12 -v_ 8 -u_ 14 -3_ 1 -ª_ 6 -r_ 6 -5_ 12 -h_ 8

4. -s_ 14 -v_ { } or φ -u_ { } or φ -3_ 14

5. -s_ 7 -v_ 7 -u_ 2 -3_ 12 -ª_ 12 -r_ 5

-5_ 5 -h_ 6 -em_ 1 -`_ 11 -6_ 13

6. -s_ 4 -v_ 4 -u_ 4 -3_ 6 -ª_ 7 -r_ 1

-5_ 1 -h_ 1 -em_ 1 -`_ 8

7. lzIfsnfO{ bv] fpg'xf];\ .

24 ul0ft, sIff (

kf7 2 s/ (Tax)

2.0 kg' /jnfs] g (Review)

ljBfyL{ pkoS' t ;ª\Vofsf ;d"xx¿df a;L tnsf cj:yfx¿sf af/d] f 5nkmn ugx{' f];\ M

-s_ ljzfnn] cfˆgf] dfl;s tnadf 1% 36fP/ kfP .
-v_ sdnfn] Pp6f dfa] fOn lsGbf df]afOnsf] d"Nodf 13% yk /sd ltl/g\ .
-u_ Pp6f pBfu] n] cfk"mn] jif{el/ sdfPsf] s]xL kl| tzt /sd ;/sf/nfO{ a'emfof] .
-3_ kw| fgfWofksn] Pp6f sfo{j|mdsf] kl| ta}7s eQfafkt ?= 600 dWo] hDdf ?= 510

dfq kf| Kt ug'{eof] .
-ª_ zlStn] oftfoft Joj:yf ;j] f sfofn{ o, afUdtL k|b]z cGtut{ btf{ ePsf] cfˆgf]

df6] /;fOsnsf] Ana' s' cf=j= 2078/2079 sf nflu gjLs/0f ubf{ ?= 300 lt/] .
dflysf cj:yfdf yk ul/Psf / s66\ L ul/Psf /sd s/afktsf /sd x'g\ . km/s km/s
zLifs{ df km/s km/s s/sf] b/ sfg'gdf pNn]v ul/Psf] x'G5 .

2.1 s/ (Tax)

ljm| ofsnfk 1

xfdf| ] bz] df kT| os] cflys{ jif{sf] ;'?df k|:tt' ul/g] ah6] jStJodf ;dfjz] ul/Psf] s/;DaGwL
Joj:yfnfO{ cfwf/ dfgL lgDglnlvt kZ| gx¿df ;fyLx¿lar 5nkmn ugx{' f];\ M

-s_ ;/sf/n] /fHosf] kz| f;lgs vr{ / ljsf; vr{ s;/L h'6fp5F xf]nf <
-v_ ;/sf/n] /fi6«;j] s sd{rf/LnfO{ lbg] tna eQf s;/L Joj:yfkg u5{ xfn] f <
-u_ ufpF 6f]ndf jif{}lkR5] vfg]kfgL 6o\ fª\sL lgdf{0f tyf dd{t, ;8s lgdf0{ f tyf

:t/f]Gglt / ljsf; lgdf{0fsf sfdx¿ h:t} M l;kdn" s sfoj{ |md cflbsf nflu ah6]
lgsf;f x'g] u/]sf] ;'Gg' ePs} xf]nf, o:tf vr{x¿ sxfFaf6 h6' fOG5 <

s/ egs] f] s'g} JolSt, kmd{ jf sDkgLn] sfg'gadfl] hd ;/sf/nfO{ ltg{k' g]{ clgjfo{ e'StfgL
xf] . /fHosf] cfDbfgLsf] dV' o ;f| ]t s/ xf] . ;/sf/n] b]zsf] lgoldt, cfsl:ds / ljsf;fTds
ultljlw oxL s/sf dfWodaf6 kf| Kt xg' ] cfDbfgLcg';f/ ;~rfng u/s] f] xG' 5 . To;}n] xfdL
;an} ] clgjfo{ ¿kdf s/ ltg'{k5{ . s/ ltg'{ c;n gful/ssf] stJ{ o k/" f ug{' xf] . g]kfndf
k|rngdf /xs] f s/x¿df ;jf/L s/, 3/axfn s/, eG;f/ s/, cfos/, dN" o clejl[ 4 s/,
;DklQ s/, ;fdflhs ;'/Iff s/ cflb xg' \ . s/nfO{ k|ltzt (%) df JoSt ul/G5 .
pbfx/0f M ljbz] cWoog ug{ hfg] ljBfyLn{ ] db' |f ;6xL ;'ljwf lnFbf ;6xL /sddf 1% s/ nfU5 .

ul0ft, sIff ( 25

2.1.1 cfos/ (Income tax)

ljm| ofsnfk 2

lbOPsf cfos/ ;Ldf / k|Zgx¿sf ;DaGwdf cfˆgf ;fyLx¿lardf ;d"xdf 5nkmn ug'x{ f];\ /
sIffsf]7fdf lgisif{ k|:t't ugx'{ f];\ M

cfo jif{ 2078/79 sf nflu
k|fs[lts JolStsf nflu nfu' xg' ] s/sf] b/

/fh] uf/Lsf] cfo dfq xg' s] f nflu

Psn JolStsf nflu bDktLsf nflu

zLifs{ s/sf] b/ zLif{s s/sf] b/

?= 4 nfv;Ddsf] cfodf 1% ?= 4 nfv 50 1%
xhf/;Ddsf] cfodf

4 nfveGbf a9L t/ 5 10% 4 nfv 50 xhf/eGbf 10%
nfv;Ddsf] cfodf a9L t/ 5 nfv 50
xhf/;Ddsf] cfodf

5 nfveGbf a9L t/ 7 20% 5 nfv 50 xhf/eGbf 20%
nfv;Ddsf] cfodf a9L t/ 7 nfv 50
xhf/;Ddsf] cfodf

7 nfveGbf a9L t/ 20 30% 7 nfv 50 xhf/eGbf 30%
nfv;Ddsf] cfodf a9L t/ 20 nfv;Ddsf]
cfodf

20 nfveGbf a9Lsf] cfodf 36% 20 nfveGbf a9Lsf] 36%
cfodf

Psnf}6L kmd{sf ¿kdf Joj;fo dfq x'g]sf nflu

Psn JolStsf nflu bDktLsf nflu

zLif{s s/ kl| tzt zLif{s s/ k|ltzt

?= 4 nfv;Ddsf] cfodf s/ gnfUg] ?= 4 nfv 50 xhf/;Ddsf] cfodf s/ gnfUg]

4 nfveGbf a9L t/ 5 10% 4 nfv 50 xhf/eGbf a9L t/ 5 10%
nfv;Ddsf] cfodf nfv 50 xhf/;Ddsf] cfodf

5 nfveGbf a9L t/ 7 20% 5 nfv 50 xhf/eGbf a9L t/ 7 20%
nfv;Ddsf] cfodf nfv 50 xhf/;Ddsf] cfodf

7 nfveGbf a9L t/ 20 30% 7 nfv 50 xhf/eGbf a9L t/ 20 30%
nfv;Ddsf] cfodf nfv;Ddsf] cfodf

20 nfveGbf a9Lsf] cfodf 36% 20 nfveGbf a9Lsf] cfodf 36%

26 ul0ft, sIff (

-s_ cfos/ egs] f] s] xf] <
-v_ sfgg' agfP/ xfdf| ] cfDbfgLdWo] lglZrt k|ltzt /sd ;/sf/n] p7fp5F , lsg xfn] f <
-u_ g]kfndf cfos/ Joj:yfkg ug]{ lgsfo s'g xf] <
-3_ cfGtl/s /fh:j ljefun] lgwf/{ 0f u/s] f] x/s] cflys{ jifs{ f nflu cfos/;DaGwL ;Ldfx¿,

kf| jwfgx¿ sxfaF f6 s;/L ;lhn} cWoog ug{ ;lsG5 < s:tf s:tf k|fjwfgx¿ pNn]v
ul/Psf] /x]5 <

JolStut jf ;:+ yfut -pBf]u, sDkgL cflb_ n] u/s] f] cfDbfgLdf nfUg] s/nfO{ cfos/
(Income tax) elgG5 . dn" tM cfo, kfl/>lds tyf gfkmfdf nfUg] s/ g} cfos/
xf] . ;/sf/sf] cfDbfgLsf] ;f| t] dWo] cfos/ klg Ps xf] . cfos/nfO{ kl| tztdf u0fgf
ul/G5 . JolStut cfos/b/ k];f, Joj;fo / jj} flxs l:yltcg';f/ km/s km/s xg' ] u/]sf]
kfOG5 . cfos/ Pg] 2058cg;' f/ cfosf rf/ zLifs{ -s_ /fh] uf/L -v_ Joj;fo -u_ nufgL
-3_ cfsl:ds nfe tf]lsPsf] 5 . g]kfndf cfos/sf] Joj:yfkg ug]{ lhDdj] f/L cfGtl/s
/fh:j ljefunfO{ tfl] sPsf] 5 . cfGtl/s /fh:j ljefun] x/]s jif{ cfos/ u0fgf sfo{ljlw
agfpg] ul/Psf] x'G5| , h;sf] ljj/0f ljefusf] ja] ;fO6 http://www.ird.gov.np af6
xg] { ;lsG5 .

cfos/df 56' xg' ] cj:yfx¿

zLif{s ;Ldf

-s_ sdr{ f/L ;~ro sfi] fdf hDdf ul/Psf] lgwf/{ 0f of]Uo cfosf] PsltxfO / tLg nfv

/sddf ?lkofdF f h'g sd xG' 5 ;f] a/fa/sf] /sddf

-v_ gful/s nufgL sf]ifdf hDdf ul/Psf]

/sddf

-u_ hLjg ladfafkt lt/s] f] lk|ldod vr{df clwstd ?= 25,000 ;Dd -bDktL;d]t_

-3_ wflds{ sfo{sf nflu u/]sf] vr{ tyf ;dfof]lht s/of]Uo cfosf] 5% /
rGbf lbOPsf] /sddf ?= 1,00,000 df h'g sd x'G5 ;f] a/fa/
sf] sddf

-ª_ bu' d{ eQfafkt k|fKt u/]sf] /sddf -s_ ?= 50,000 -v_ ?= 4,00,000
-bu' {d Ifq] sf cfwf/df_ -u_ ?= 20,000 -3_ ?= 10,000

-r_ jb} ]lzs eQfsf] 75% /sddf g]kfnsf] ljb]zl:yt s'6gLlts lgofu] df
sfo/{ t sd{rf/Lsf nflu

-5_ cf}ifwL pkrf/df nfus] f] vrd{ f s'n vrs{ f] 15% n] x'g cfpg] /sd 25%
df

ul0ft, sIff ( 27

-h_ ckfªu\ tf ePsf JolStn] kfpg] tf]lsPsf] ;Ldfdf yk krf; kl| tzt
5'6 ;'ljwf jflifs{ clwstd ?= 20,000 ;Dd
jflif{s clwstd ?= 5,00,000 ;Dd
-em_ :jf:Yo ladfafkt lt/s] f] lkl| dod cfos/ /sddf 10% 56'

-`_ ;fdflhs ;'/Iff sfi] fdf of]ubfg u/]sf] jflif{s clwstd ?= 5,000 ;Dd
/sddf tfl] sPsf] ;Ldfdf 25% yk /sddf
lgjl[ Qe/0fafktsf] cfo, lgj[lQe/0f sfi] f /
-6_ /fh] uf/Lsf] cfo dfq ePsf dlxnfsf of]ubfgdf cfwfl/t ;fdflhs ;'/Iff sfi] fdf
xsdf of]ubfg ug]{ k|fsl[ ts JolStsf] cfodf 1%
sf] s/ nfUg] 5}g .
-7_ ;fdflhs ;'/Iffsf] ¿kdf lbOg] ;a}
k|sf/sf eQf

-8_ bfOhf] 5fqjl[ Q, ckt' fnL, OR5fkqafkt
k|fKt /sddf

-9_ cfkmg\ f] :jfldŒjdf /x]sf] lghL ejgsf]
ladfafkt lt/s] f] lkl| dod /sddf

-0f_ lgj[lQe/0f cfodf

-t_ Psnf6} L kmd{ btf{ ePsf s/bftfsf]
xsdf

pbfx/0f 1

Ps hgf ljjflxt lzIfssf] dfl;s tna ?= 37,990 5 . cfo jif{ 2078/079 sf] cfos/
b/cg';f/ lzIfsn] k|fKt ug{] rf8kj{ vr;{ lxtsf] 13 dlxgfsf] cfDbfgL lx;fa ubf{ jflif{s slt cfos/
ltgk{' g]{ /x5] <

;dfwfg ?= 4,93,870

oxfF dfl;s tna = ?=37,990 ?= 4,50,000 ?= 43,870
jflif{s cfo = 13 × 37,990
= ?=49,3870 1% 10%

ca s/ofU] o /sd ?= 4,93,870 nfO{ cfos/b/sf cfwf/df lgDgfg';f/ nV] bf,

?= 4,93,870 = ?=4,50,000 + ?=43,870



1% 10%

jflifs{ cfos/ = ?= 4,50,000 sf] 1% + ?=43,870 sf] 10%

= ?= 1 43,870 × 10
4,50,000 × 100 + 100

= 4,500 + 4,387

= ?= 8,887

28 ul0ft, sIff (

pbfx/0f 2

Pp6f aª} \sdf sfd ug]{ cljjflxt dlxnf sdr{ f/Lsf] dfl;s tna ?= 30,000 5 . cfDbfgLdWo] jflifs{
?= 4,00,000 ;Dddf ;fdflhs ;'/Iff s/ 1% sf b/n] / ?= 4,00,000eGbf dflysf] cfodf 10%
sf b/n] cfos/ ltg{k' 5{ . pSt sd{rf/Ln] Ps jif{df 15 dlxgf a/fa/ tna kfpF5 eg] jflifs{ hDdf
slt cfos/ ltgk'{ 5{ < kQf nufpg'xf;] \ .

;dfwfg

oxfF sd{rf/Lsf] dfl;s cfDbfgL = ?= 30,000
jflif{s cfDbfgL = 15 × ?= 30,000
= ?= 4,50,000
oxfF s'n jflif{s cfDbfgL ?= 4,50,000 nfO{ lbOPsf] cfos/ ;Ldfcg;' f/ lgDgfg';f/ n]Vbf,
?= 4,50,000 = ?= 4,00,000 + ?= 50,000



1% 10%

⸫ hDdf jflifs{ cfos/ = ?= 4,00,000 sf] 1% + ?= 50,000 sf] 10%

= 4,00,000 × 1 + 50,000 × 10
100 100

= 4,000 + 5,000

= ?= 9,000

dlxnf ePsfn] cfos/df 10% n] 5'6 kfpg] xbF' f, = ?= 9,000 sf] 10%
56' /sd
= ?= 900
= ?= 9,000 – ?= 900

lgh sdr{ f/Ln] ltg'{kg]{ jflifs{ s/ = ?= 8,100



pbfx/0f 3

gk] fndf ?= 2,000 dxuF L eQf;lxt dfl;s ?= 40,500 sdfpg] Ps hgf ljjflxt k'?if sd{rf/Ln]
jflifs{ ?= 23,500 lkl| dod ltg]{ u/L hLjg ladf u/s] f 5g\ . pgn] k|fKt ug]{ rf8kj{ vr{;lxtsf] 13
dlxgf a/fa/ Ps jif{sf] cfDbfgLsf] u0fgfdf dxuF L eQf / rf8kj{ vra{ fxs] sf] cfDbfgLdWo] 10%
sdr{ f/L ;~ro sfi] fdf 56' \ofO;sk] l5 afFsL cfodf hDdf slt cfos/ ltg'k{ 5{ < kQf nufpgx' f;] \ .
-oxfF cufl8 ki[ 7df plNnlvt cfos/ ;Ldfcg';f/ u0fgf ul/Psf] 5 ._

ul0ft, sIff ( 29

;dfwfg
oxfF dfl;s tna = ?= 40,500 – ?= 2,000 = ?= 38,500

aflif{s tna = ?= 38,500 × 12 = ?= 4,62,000

dxFuL eQf = 2,000 × 12 = ?= 24,000

b;}F vr{ = ?= 38,500

sdr{ f/L ;~ro sf]if yk = 4,62,000 × 10 = ?= 46,200
100

ca lgwf{/0f of]Uo cfo = ?= 4,62,000 + ?= 24,000 + ?= 38,500 + ?= 46,200

= ?= 5,70,700

36fpg ]

(i) sd{rf/L ;~ro sfi] fdf hDdf xg' ] /sd

?= 46,200 + ?= 46,200 = ?= 92,400

(ii) ladf lk|ldod = ?= 23,500



hDdf ?== 1,15,900

ca lgwf/{ 0f ofU] o cfosf] Ps ltxfOn] xg' ] /sd

= = ?= 5,70,700 × 1
3

= ?= 1,90,233.33

lgwf/{ 0f of]Uo cfosf] Ps ltxfOeGbf sd{rf/L ;~ro sfi] f / ladf /sd hf]8b\ f cfpg] /sd
sd ePsfn,]

jf:tljs s/ 5'6 /sd = ?= 1,15,900

s/ of]Uo cfo = 5,70,700 – 1,15,900 = ?= 4,54,800

ljjflxt k'?if sd{rf/L ePsfn],

ltgk{' g{] cfos/ /sd = ?= 4,50,000 sf] 1% + ?= 4,800 sf] 10%

= ?= 4,50,000 × 1 + 4,800 × 10 ?= 4,54,800
= 4,500 + 480 100 100
?= 4,50,000 ?= 4,800

1% 10%

= ?= 4,980

t;y{ pSt sdr{ f/Ln] jflif{s ¿kdf hDdf ?= 4,980 cfos/ ltg'{k5{ .

30 ul0ft, sIff (

pbfx/0f 4

olb sg' } Joj;foLsf] jflif{s cfodWo] ?=4,50,000 ;Dd cfos/ 56' , ?= 4,50,001 b]lv ?= 5,50,000
;Dd 10% / ?= 5,50,001 b]lv ?=7,00,000 ;Dd 20% sf b/n] cfos/ nfU5 eg] jflifs{
?=6,75,000 cfDbfgL ug]{ Joj;foLn] hDdf slt cfos/ ltg{k' 5{ < kQf nufpg'xf;] \ .

;dfwfg

oxfF s'n jflif{s cfDbfgL ?= 6,75,000 nfO{ lbOPsf] cfos/ ;Ldfcg;' f/ lgDgfg;' f/ 6j' |mofP/
nV] bf,

?= 6,75,000 = ?= 4,50,000 + ?= 1,00,000 + ?= 1,25,000


cfos/ 5'6 10% 20%

⸫ hDdf jflifs{ cfos/ = ?= 1,00,000 sf] 10% + 1,25,000 sf] 20%

= 1,00,000 × 11000 + 1,25,000 × 20
100

= 10,000 + 25,000

= ?= 35,000

gdg' f lrq0f ljlwaf6,

?= 4,50,000 ?= 6,75,000
?= 1,00,000 ?= 1,25,000

56' 10% 20%

hDdf jflif{s cfos/ = ?= 1,00,000 sf] 10% + ?= 1,25,000 sf] 20%

= 1,00,000 × 11000 + 1,25,000 × 20
100

= 10,000 + 25,000

= ?= 35,000

⸫ pSt Joj;foLn] jflif{s ?= 35,000 cfos/ ltg'k{ 5{ .

ul0ft, sIff ( 31

pbfx/0f 5

wgaxfb'/n] ?= 35,000 sf] 4 jif{df 10% kl| t jif{sf b/n] ;fwf/0f Aofh kfpg] u/L Pp6f ;xsf/L
;:+ yfdf d'2tL artdf hDdf u/5] g\ . olb pgn] ;f] artdf kfpg] Aofhdf 5% s/ nfU5 eg] pgn] s/
s6\6Lkl5 slt Aofh k|fKt u5g{ \ < kQf nufpgx' f];\ .

;dfwfg

oxfF lbOPsf,]

;fjfF (P) = ?= 35,000

;do (T) = 4 jif{

;fwf/0f Aofhb/ (R) = 10%

s/b/ = 5%

;fwf/0f Aofh = <

s/ s66\ Lkl5 kf| Kt x'g] Aofh = <

Aofhafktsf] cfodf nfUg] s/ = <

;q" af6,

I= P×T×R
100

= 35,000 × 4 × 10
100

= ?= 14,000

⸫ ;fwf/0f Aofh = ?=14,000, oxfF o;/L kf| Kt Aofh /sd wgaxfb/' sf] nufgLaf6 kf| Kt
cfosf] ¿kdf u0fgf xG' 5 .

kml] / cfos/ = ?= 14,000 sf] 5%

= ?= 14,000 × 5
100

= ?= 700

⸫ Aofhafktsf] cfodf nfUg] s/ = ?= 700

⸫ s/ s6\6Lkl5sf] Aofh = ?= 14,000 − ?= 700

= ?= 13,300

;DalGwt ;xsf/L ;+:yfn] g} 5% n] xg' ] Aofhdf nfUg] s/ s66\ L u/L artstf{ wgaxfb/' nfO{
?= 13,300 dfq k|bfg u5g{ \ .

32 ul0ft, sIff (

cEof; 2.1

1. gk] fndf dfl;s ?= 38,000 sdfpg] / jflifs{ ?= 23,500 sf] hLjg ladf u/]sf Ps
hgf clwst[ txdf sfo/{ t ljjflxt sdr{ f/Ln] k|fKt ug]{ rf8kj{ vr;{ lxtsf] 13 dlxgf
a/fa/ Ps jifs{ f] cfDbfgL lx;fa ubf{ jflifs{ slt ?lkofF cfos/ lt5{g\, kQf nufpgx' f];\ .

-cufl8 k]hdf plNnlvt cfos/ ;Ldfcg';f/ lx;fa ug'{xf];\ ._

2. sg' } Pp6f ;:+ yfdf sfd ug{] Ps hgf sd{rf/Ln] p;sf] cfDbfgLdWo] ?= 4,50,000
;Dd 1%, ?= 4,50,000 eGbf dfly ?= 5,50,000 ;Dd 10%, ?= 5,50,000 eGbf dfly
?= 7,50,000 ;Dd 20%, ?= 7,50,000 eGbf dfly ?= 20,00,000 ;Dd 30% sf b/n]
cfos/ ltgk' 5{ eg] dfl;s ?= 65,000 sdfpg] pSt sd{rf/Ln] cfDbfgLafktsf] hDdf
slt s/afktsf] /sd ;/sf/nfO{ a'emfpgk' 5{ < lx;fa ugx{' f];\ .

3. tnsf] tflnsfdf Psnf6} L kmds{ f ¿kdf Joj;fo xg' ] Joj;foLsf nflu plNnlvt cfos/ ;Ldf
cWoog ug{x' f];\ M

jflifs{ cfDbfgL -?=_ s/b/
cfos/ 56'
1 - 4,50,000
10%
4,50,001 - 5,50,000

5,50,001 - 7,50,000 20%

7,50,001 - 20,00,000 30%
36%
20,00,000 eGbf dfly

ca cfos/ ;Ldfcg;' f/ lgDgadfl] hdsf] jflif{s cfDbfgL ug]{ Ps hgf Joj;foLn] slt cfos/
ltgk{' 5{, lx;fa ugx{' f];\ M

-s_ jflifs{ cfDbfgL = ?= 6,30,000

-v_ jflifs{ cfDbfgL = ?= 9,25,000

-u_ jflifs{ cfDbfgL = ?= 17,88,000

-3_ jflif{s cfDbfgL = ?= 22,25,000

4. a}ªs\ df hDdf u/]sf] ?=10 nfvsf] 4 jifd{ f 8.5% ;fwf/0f Aofhsf b/n] lx;fa xF'bf sn'
slt Aofh xG' 5 < olb pSt Aofhdf 5% cfos/ nufpbF f v'b ;fwf/0f Aofh slt xG' 5,
kQf nufpgx' f];\ . -o;/L Aofhdf ltl/Psf] cfos/n] nufgLaf6 ePsf] cfodf ltl/Psf]
s/nfO{ a'emfpF5 ._

ul0ft, sIff ( 33

kl/ofh] gf sfo{
-s_ cfkmg\ f] ljBfnodf sfo/{ t dfWolds lzIfssf] ;~ro sf]if, gful/s nufgL sfi] f,

ladf h:tf ljifox¿nfO{ ;d]6L tnasf] jf:tljs ljj/0f lngx' f];\ / k|Tos] lzIfsn]
jflifs{ cfodf slt slt ?lkofF s/ ltg{k' g{] /x]5 < lx;fa ug{x' f;] \ .
-v_ ljBfyLs{ f] cfcfˆgf] 3/ kl/jf/sf ;b:ox¿sf] gf]s/L ePdf lghx¿af6 cfDbfgL
u/s] f] lx;fa lstfa u/L jf:tljs ljj/0f lng'xf];\ / jflifs{ slt /sd s/afkt
ltgk'{ 5{, kQf nufpgx' f;] \ .

pQ/

1. ?= 6,550 2. ?=63,500 3. -s_ ?= 26,000 3. -v_ ?= 1,02,500
3.-u_ ?= 3,61,400 3. -3_ ?= 5,06,000 4. ?= 3,40,000 / ?= 3,23,000

34 ul0ft, sIff (

2.1.2 dN" o clej[l4 s/ (Value added tax)
lj|mofsnfk 1

lbOPsf] lan cWoog u/L ;fl] wPsf kZ| gsf ;DaGwdf 5nkmn ug{x' f;] \ M

ABC Electronics
Kathmandu

Invoice No. 883 TAX INVOICE Date: 2077-11-15

VAT No × × × × × × × × ×
M/s .......................................................................................................................................................
Address : ..............................................................................................................................................
Buyers VAT No. : × × × × × × × × × Mode of Payment: Cash/Cheque/Others

S.N. Particulars Quantity Rate Amount
1. Refrigerator Rs. Ps.
C,LD 201 ALLB
1 27,876.10 27,876 10

Amount in Words: Thirty One Total 27,876 10
Thousand Five hundred only.
....... Discount — 10
----------------------- Taxable Amount 27,876 90
Customer's Sign 3,623 00
ul0ft, sIff ( 13% VAT 31,500
Grand Total

------------------------
For. ABC Electronics

35

-s_ lbOPsf] landf /]lkm| h/] 6] /sf] lajm| L b/ slt /x5] <

-v_ pSt /l] km| h/] 6] / vl/bstf{n] 5'6 kfPsf] /x5] ls /xg] 5 <

-u_ vl/bstf{n] tf]lsPsf] laj|mLb/eGbf a9L /sd lt//] vl/b ul/Psf] bl] vG5, lsg o:tf]
ePsf] xfn] f <

-3_ pSt lancg';f/ lajm| L b/df ?= 3,623.90 yk /sd lt/s] f] b]lvG5 . o;/L lsg yk
/sd ltgk{' /s] f] xf]nf <

-ª_ s] xfdLn] h'g;s' } ;fduL| vl/b ubf{ klg tf]lsPsf] d"Nodf o;/L g} yk /sd ltg'k{ 5{ <

-r_ oxfF vl/bstfn{ ] lt/s] f] dN" o ?= 31,500 n] /l] k|mh]/]6/sf] sg' d"NonfO{ ae' mfpF5 <

d"No clej[l4 s/ (VAT)

dN" o clej[l4 s/ j:t' tyf ;]jfdf nfUg] Ps lsl;dsf] ck|ToIf s/ xf] . j:t' jf
;j] fsf] pTkfbgb]lv ljt/0f;Ddsf ljleGg tx÷r/0fdf j[l4 ePsf] dN" odf nfUg]
s/ g} d"No clejl[ 4 s/ xf] . xfn gk] fnsf] ;Gbed{ f dN" o clejl[ 4 s/sf] b/ 13%
/flvPsf] 5 .

ul0ft k|fljlws zAbsf]zaf6,

d"No clej[l4 s/M j:t' jf ;]jf laj|mL ubf{ 56' s6fP/ kT| o]s txdf j[l4 x'g] d"Nodf
nfUg] s/ dN" o clej[l4 s/ -d=" c=s=_ xf] . of] s/ cGTodf pkefS] tfn] g} rS' tf
ugk'{ 5{ t/ o;n] j:ts' f] dN" o cgfjZos ¿kdf a9g\ lbbF g} .

h:tM} Pp6f 6l] nlehg lgdf{0f ug{] sDkgLn] Pp6f 6]lnlehgsf] lgdf{0f vr{ / gfkmf hf8] L
d"No ?= 10,000 sfod u¥of] . p;n] lgdf{0f u/]sf] 6]lnlehg l8n/, xf]n;]n/, l/6]n/ x'Fb}
pkefS] tf;Dd cfOk'Ubf lgDgfg;' f/ kl| jm| of k/" f xG' 5 M

lgdf0{ fstf{ sDkgLn] l8n/nfO{ l8n/n] xf]n;n] /nfO{ laj|mL ubf{

lajm| L ubf{

Pp6f 6l] nlehgsf] dN" o = ?= 10,000 jm| o dN" o = pTkfbs sDkgLsf] lajm| L d"No

dN" o clejl[ 4 s/sf] b/ = 13% = ?= 11,300

laj|mL d"No yk vr{ / gfkmf ?= 1,200 hf8] L Pp6f 6l] nlehgsf]

= ?= 10,000 + 10,000 sf] 13% d"No = ?= 12,500
d"No clej[l4 s/sf] b/ = 13%
= 10,000 + 10,000 × 13 lajm| L d"No
100

= 10,000 + 1,300 = ?= 12,500 + ?= 12,500 sf] 13%
= ?= 12,500 + ?= 1,625
= ?= 11,300

= ?= 14,125

36 ul0ft, sIff (

xf]n;n] /n] l/6]n/nfO{ laj|mL ubf{ l/6]n/n] pkef]StfnfO{ laj|mL ubf{

jm| o d"No = l8n/sf] lajm| L d"No j|mo dN" o = xf]n;]nsf] lajm| L d"No

= ?= 14,125 = ?= 16,950

yk vr{ / gfkmf ?= 875 hf8] L Pp6f yk vr{ / gfkmf ?= 1,050 hf8] L Pp6f 6l] nlehgsf]
dN" o = ?= 18,000
6]lnlehgsf] d"No = ?= 15,000
dN" o clejl[ 4 s/sf] b/ = 13%
d"No clejl[ 4 s/sf] b/ = 13%
laj|mL dN" o
laj|mL d"No
= ?= 18,000 + ?= 18,000 sf] 13%
= ?= 15,000 + 15,000 sf] 13%
= ?= 18,000 + ?= 2340
= 15,000 + 1,950
= ?= 20,340
= ?= 16,950

oxfF pkefS] tfn] ?= 2,340 d"No clej[l4 s/ ltgk{' 5{ . pSt /sd ;/sf/L sfi] fdf hDdf xFb' f,

pTkfbsn] = ?= 1,300
l8n/n] = -1,625 – 1,300) = ?= 325

xf]n;n] /n] = (1,950 – 1,625) = ?= 325
l/6n] /n] = (2,340 – 1,950) = ?= 390

hDdf = 1,300 + 325 + 325 + 390) = ?= 2,340

ca k|Tos] txdf a9]sf] dN" osf cfwf/df

pTkfbsn] = ?= 10,000 sf] 13%

= 10,000 × 13 = 1,300
100

l8n/n] = ?= (12,500 – 10,000) sf] 13%

= 2500 × 13
100

= ?= 325

xf]n;]n/n] = ?= (15,000 – 12,500) sf] 13%

= 2500 × 13
100

= ?= 325

l/6n] /n] = ?= (18,000 – 15,000) sf] 13%

= 3,000 × 13
100

= ?= 390

hDdf d"No clejl[ 4 s/ = ?= 1,300 + ?= 325 + ?= 325 + ?= 390

= ?= 2,340

ul0ft, sIff ( 37

pbfx/0f 1

lbOPsf cj:yfdf dN" o clej[l4 s/ (VAT) /sd kQf nufpg'xf];\ M

-s_ dN" o clej[l4 s/afx]ssf] d"No = ?= 7,000 / d"No clejl[ 4 s/sf] b/ = 13%

-v_ d"No clej[l4 s/afx]ssf] cª\lst dN" o (MP) = ?=10,000, 5'6 = 15% / dN" o clejl[ 4
s/sf] b/ = 13%

;dfwfg

-s_ oxfF d"No clejl[ 4 s/afx]ssf] d"No = ?= 7,000

d"No clej[l4 s/sf] b/ = 13%

d"No clej[l4 s/ /sd = <

ca dN" o clej[l4 s/ /sd = ?=7,000 sf] 13%
13
= 7,000 × 100

= ?= 910

⸫ d"No clejl[ 4 s/ /sd = ?= 910
-v_ oxfF dN" o clej[l4 s/afxs] sf] cªl\ st d"No = ?=10,000
5'6 = 15%
dN" o clejl[ 4 s/sf] b/ = 13%

dN" o clej[l4 s/ /sd = <

ca 5'6 /sd lgsfNbf,

5'6 /sd = ?= 10,000 sf] 15%
15
= ?= 10,000 × 100

= ?= 1,500

dN" o clejl[ 4 s/ nfUg] /sd = ?= 10,000 − ?= 1500

= ?= 8,500

ca dN" o clej[l4 s/ /sd = ?= 8,500 sf] 13%

= 8,500 × 13
100

= ?= 1,105

⸫ d"No clej[l4 s/ (VAT) /sd = ?= 1,105

38 ul0ft, sIff (

pbfx/0f 2

Pp6f df]afOn ;6] sf] dN" o clej[l4 s/afx]ssf] cª\lst dN" o ?= 15,000 /flvPsf] lyof] . pSt
dfa] fOn ;6] df 15% 56' lbO{ 13% dN" o clej[l4 s/ nufpFbf ;fs] f] d"No slt kU' of] xfn] f < lx;fa
ug'{xf];\ .

;dfwfg

oxfF df]afOnsf] dN" o clej[l4s/afxs] sf] cª\lst d"No (MP) = ?= 15,000

5'6 = 15%

dN" o clejl[ 4 s/sf] b/ = 13%

dN" o clej[l4 s/;lxtsf] d"No = <

ca 56' /sd = ?= 15,000 sf] 15%
15
= ?= 15,000 × 100

= ?= 2,250

km]l/ df]afOnsf] d"No clej[l4 s/ nfUg] dN" o = ?= (15,000 – 2,250)

= ?= 12,750

d"No clej[l4 s/ /sd = ?= 12,750 sf] 13%

= 12,750 × 13
100

= ?= 1,657.50

⸫ d"No clej[l4 s/;lxtsf] d"No = 12,750 + 1,657.50

= ?= 14,407.50

js} lNks tl/sf
;dfwfg

df]afOnsf] d"No clej[l4 s/afxs] sf] cª\lst d"No (MP) = ?= 15000

15% 5'6 lbOPkZrft,

56' kl5sf] dN" o = 15,000 sf] 85%

= 15,000 × 85
100
= 12,750

13% d=" c=s=nufPkZrft,

d"=c=s=;lxtsf] d"No = ?= 12,750 sf] 113%

= ?= 12,750 × 113
100

= ?= 14,407.50

ul0ft, sIff ( 39

gdg' f lrq0f ljlwaf6, ?= 15x
?= 15,000 5'6

?= 85x



?= 100y ?= 13y

56' kl5sf] d"No Eof6

ca 100x = ?= 15,000

x = 15,000 = ?= 150
100

85x = ?= 150 × 85 = 12,750

kml] / 100y = 85x

= ?= 12,750

y = 12,750
100
113y
= 12,750 × 113
100

= ?= 14,407.50

⸫ dN" o clej[l4 s/;lxtsf] dN" o = ?= 14,407.50

pbfx/0f 3

Pp6f ;fdfgsf] d"No clej[l4 s/afxs] sf] dN" odf 15% 56' lbO{ 13% d"No clejl[ 4 s/ nufpFbf
vl/bstf{n] ?= 57630 ltg'{k5{ eg] pSt ;fdfgsf] cªl\ st dN" o kQf nufpg'xf];\ .

;dfwfg

oxfF dfgf}F dN" o clej[l4 s/afx]ssf] cªl\ st d"No (MP) = ?= x

lbOPsf,] 56' = 15%

dN" o clejl[ 4 s/sf] b/ = 13%

d"No clej[l4 s/;lxtsf] d"No = ?= 57,630

k|Zgcg';f/ 5'6 /sd = ?= x sf] 15%

= ?= x × 15
100

15x
= 100

40 ul0ft, sIff (

= ?= 3x gd'gf lrq0f ljlwaf6,
20 cª\lst d"No

d"No clejl[ 4 s/ nfUg] d"No ?= 85x

= ?= x– 3x ?= 15x
20 56'
?= 100y
= ?= 17x ?= 13y
20

dN" o clej[l4 s/ /sd = ?= 17x sf] 13% ?= 57,630
20
ca 113y = ?= 57,630
17x × 13
= 20 100 ?= 57630

221x y= 113
2,000
= ?= 100y = ?= 51,000

dN" o clej[l4 s/;lxtsf] dN" o km]l/ 85x = ?= 51,000
x = 51,000
= ?= 17x 221x
20 + 2000 85

= ?= 1700x + 221x 100x = ?= 60,000
2000
⸫ cª\lst d"No (MP) = ?= 60,000

= ?= 1921x
2000

k|Zgcg;' f/,

12902010x = 57630
cyjf 1921x = 2000 × 57630

cyjf x = 2,000 × 57,630

1921

cyjf x = 60,000

ctM cª\lst d"No (MP) = ?= 60,000

j}slNks tl/sf
dfgfF}, d"No clej[l4 s/afx]ssf] cªl\ st d"No (MP) = ?= x
oxfF lbOPsf] 5'6 = 15%
dN" o clejl[ 4 s/sf] b/ = 13%

ul0ft, sIff ( 41

d"No clej[l4 s/;lxtsf] d"No = ?= 57,630

d"No clej[l4 s/;lxtsf] d"No = ?= x sf] (100 − 15)% × (100 + 13)%

= x × 85 × 113
100 100

k|Zgcg;' f/,

?= 57,630 = x × 85 × 113
100 100

cyjf x = 57630 × 100 ×100

85 × 113

cyjf x = 60,000

ctM cªl\ st dN" o (MP) = ?= 60,000

pbfx/0f 4

Pp6f k;n]n] Pp6f ;fOsn d"No clej[l4 s/afx]s ?= 5,800 df lsg]/ NofPsf] /x]5 . p;n] jm| o
d"Nosf] 40% a9fP/ cªl\ st dN" o sfod u/]5 . cªl\ st dN" odf 10% 56' lbO{ 13% dN" o clejl[ 4
s/ nufpFbf pkef]Stfn] slt d"No ltg{'k5,{ kQf nufpgx' f;] \ .

;dfwfg

oxfF ;fOsnsf] dN" o clejl[ 4 s/afxs] sf] jm| o d"No (CP) = ?= 5,800

5'6 = 10%

dN" o clej[l4 s/sf] b/ = 13%

d"No clej[l4 s/;lxtsf] dN" o = <

gfkmf k|ltzt jf gf]S;fg k|ltzt = <

k|Zgaf6,

;fOsnsf] cª\lst dN" o (MP) = ?= 5,800 sf] (100 + 40)%

= ?= 5,800 × 140
?= 8,120 100
=

56' /sd = cªl\ st d"Nosf] 10%

= ?= 8120 × 10
?= 812 100
=

⸫ d"No clej[l4 s/ nfUg] d"No = ?= 8,120 − ?= 812 [cªl\ st dN" o — 5'6 /sd]
= ?= 7,308

42 ul0ft, sIff (

ca dN" o clej[l4 s/ /sd = ?=7,308 sf] 13%

= ?=7,308 × 13
100

= ?= 950.04

⸫ d"No clej[l4 s/;lxtsf] ;fOsnsf] dN" o = ?= 7,308 + ?=950.04

= ?= 8,258.04

t;y{ pkef]Stfn] ?= 8,258.04 ltg'k{ 5{ .

pbfx/0f 5

k;n]n] dN" o clejl[ 4 s/afx]s ?= 2,000 df lsgs] f] Pp6f 38L 25% gfkmf lnP/ a]R5 eg] 13%
dN" o clej[l4 s/ kl5 landf pkefS] tfn] ltg'k{ g]{ /sd slt x'G5 xfn] f, kQf nufpgx' f;] \ .

;dfwfg

oxfF 38Lsf] j|mo d"No (CP) = ?= 2,000
gfkmf = 25%
d"No clejl[ 4 s/sf] b/ = 13%

pkefS] tfn] ltg{k' g]{ /sd (VAT ;lxtsf] d"No_ = <

gfkmf = ?= 2,000 sf] 25%

= 2,000 × 25
100

= ?= 500

⸫ d"No clejl[ 4 s/ nfUg] d"No = jm| o d"No + gfkmf

= ?=(2,000 + 500)

= ?= 2500

dN" o clejl[ 4 s/ /sd = ?= 2500 sf] 13%

13
= 2,500 × 100

= ?= 325

⸫ pkef]Stfn] ltgk'{ g]{ /sd = ?= 2,500 + ?= 325

= ?= 2,825

ul0ft, sIff ( 43

pbfx/0f 6

vn] s'b ;fduL| sf] l8n/n] cfoftstf{af6 Pp6f 6]an 6]lg; af8] { d"No clejl[ 4 s/afxs] ?= 25,000 df
lsg/] NofP5 . p;n] 9'jfgL vr{ / gfkmf;dt] hf8] L pSt 6]an6l] g; af8] n{ fO{ d"No clejl[ 4 s/afx]s
?= 30,000 df v'b|f Jofkf/LnfO{ a]r5] . pSt vb' f| Jofkf/Ln] ;f] af8] n{ fO{ dN" o clejl[ 4 s/afxs]
?= 37,000 df ljSsLnfO{ a]r]5 . kT| os] txdf d"No clejl[ 4 s/sf] b/ 13% ePsf] cfwf/df tnsf
kZ| gx¿sf] pQ/ kQf nufpgx' f];\ M

-s_ l8n/sf] jm| o dN" o slt /x]5 <

-v_ v'bf| Jofkf/Lsf] jm| o d"No slt /x5] <

-u_ ljSsLn] slt ?lkofF lt/]/ lsg]sf /x]5g\ <

-3_ 6]an 6l] g;sf] lajm| Laf6 dN" o clej[l4 s/afkt slt /sd ;/sf/sf] sfi] fdf hDdf
x'G5 <

-ª_ l8n/ / v'b|f Jofkf/Ln] d"No clej[l4 s/afkt slt slt /sd ;/sf/L sfi] fdf hDdf
ug{k' 5{ <

;dfwfg
cfoftstfn{ ] sfod u/]sf] d"No clejl[ 4 s/afx]ssf] lajm| L dN" o = ? 25,000

d"No clej[l4 s/sf] b/ = 13%

-s_ l8n/sf] j|mo dN" o = 25,000 + 25,000 sf] 13%

= 25,000 + 25,000 × 13
100

= 25,000 + 3,250

= ?= 28,250

oxfF l8n/n] gfkmf / cGo vr;{ dt] hf8] L d"No clejl[ 4 s/afxs] sf,]

laj|mL dN" o = ?= 30,000

-v_ vb' |f Jofkf/Lsf] jm| o d"No = 30,000 + 30,000 sf] 13%

= 30,000 + 30,000 × 13
100

= 30,000 + 3,900

= ?= 33,900

oxfF v'b|f Jofkf/Ln] gfkmf / cGo vr{;dt] hf8] L d"No clej[l4 s/afx]ssf,]

laj|mL d"No = ?= 37,000

44 ul0ft, sIff (

-u_ ca ljSsLsf] jm| o dN" o = ?= 37,000 + ?= 37,000 sf] 13%

= ?= 37,000 + 37,000 × 13
100

= ?= 37,000 + 4,810

= ?= 41,810

-3_ d"No clej[l4 s/afkt ;/sf/sf] sf]ifdf hDdf xg' ] /sd = ?= 4,810

-ª_ l8n/n] dN" o clej[l4 s/afkt ;/sf/L sf]ifdf hDdf ug'{ kg{] /sd lgsfNbf,

klxnf] tl/sf bf;] |f] tl/sf
p;n] p7fPsf] d'cs = ?= 3,900 ;/sf/L sfi] fdf hDdf ug'{ kg{] /sd =

p;n] lt/]sf] dc' s = ?= 3,250 -?= 30,000 – ?= 25,000) sf] 13%
;/sf/L sfi] fdf hDdf ug'{ kg]{ /sd
= 5,000 × 13
= ?= 3,900 – ?= 3,250 100

= ?= 650

= ?= 650

vb' f| Jofkf/Ln] dN" o clej[l4 s/afkt ;/sf/L sfi] fdf hDdf ug'{ kg]{ /sd lgsfNbf,

klxnf] tl/sf bf];|f] tl/sf
p;n] p7fPsf] d'cs = ?= 4,810 ;/sf/L sfi] fdf hDdf ug'{ kg]{ /sd =
p;n] lt/]sf] dc' s = ?= 3,900
;/sf/L sfi] fdf hDdf ug'{ kg]{ /sd -?= 37,000 – ?= 30,000_ sf] 13%

= ?= 4,810 – ?= 3,900 = 7,000 × 13
= ?= 910 100

= ?= 910

ul0ft, sIff ( 45

cEof; 2.2

1. lbOPsf] tflnsfsf cfwf/df dN" o clejl[ 4 s/ /sd kQf nufpgx' f;] \ M

j|m=;= d"No clej[l4 s/afx]ssf] d"No clej[l4 dN" o clejl[ 4 s/ /sd
dN" o s/sf] b/ <

-s_ ?= 300 13%

-v_ ?= 750 13% <

-u_ ?= 6,000 13% <

-3_ ?= 3,75,000 13% <

-ª_ ?= 20,27,000 13% <

2. lbOPsf] tflnsfsf cfwf/df d"No clej[l4 s/ gnufpbF fsf] dN" o lx;fa ug'{xf;] \ M

j|m=;= d"No clej[l4 dN" o clej[l4 dN" o clejl[ 4 s/sf] b/
s/afxs] sf] d"No s/;lxtsf] d"No
?= 3,616 13%
-s_ < 13%
?= 30,510
-v_ <

-u_ < ?= 3,390 13%

-3_ < ?= 57,630 13%

-ª_ < ?= 1,19,328 13%

3. lbOPsf] ;fdfgsf] u|fxsn] ltgk'{ g]{ d"No lx;fa ug{'xf];\ M

-s_ VAT afxs] sf] cªl\ st dN" o -v_ VAT afx]ssf] cªl\ st d"No
= ?= 35,000 = ?= 6,500
5'6 = 10% 5'6 = 7.5%
d"=c=s = 13% d=" c=s = 13%

-u_ VAT afxs] sf] cªl\ st d"No -3_ VAT afxs] sf] cªl\ st dN" o
= ?= 25,700 = ?= 1,450

5'6 = 15% 5'6 = 22.75%

d=" c=s = 13% d=" c=s = 13%

46 ul0ft, sIff (