If]q M !

;d"x (Sets)

!= kl/ro
k|frLg sfnb]lv g} xfd|f] b}lgs hLjgdf ;d"x l;b\wfGt -set theory_ k|of]u x“'b} cfPsf] 5 .
v]ns'b jf 3/fo;L sfddf klg o;sf] k|of]u klxnf] k6s ul0ft1 hh{ sfG6/n] u/]sf x'g\ .
To;}n] pgnfO{ ;d"xsf hGdbftfsf ¿kdf lnOG5 . hf]g e]g (John Venn) n] HofldtLo
cfs[ltaf6 ;d"xsf] ;DaGwnfO{ cem k|:6\ofPsf] kfOG5 . cfw'lgs ul0ftsf] cWoogdf ;d"x
l;b\wfGt (set theory) geO{ gx'g] ljifo j:t' ag]sf] 5 . ;d"x zAbnfO{ group, collection,
aggregate, class cflb zAbx¿af6 klg lrlgG5 .
dfWolds lzIff kf7\oj|md 2064 / ljlzi6Ls/0f tflnsf 2065 n] sIff 10 sf] clgjfo{ ul0ft
ljifodf ;d"x kf7af6 lgDg lnlvt ljifoj:t', k/LIffsf nflu ;+1fgfTds If]qcg';f/sf k|Zg
;ª\Vof / cª\s ef/sf] Joj:yf u/]sf] 5 .
j|m=;= ljifoj:t' 1fg l;k hDdf l;k ;d:of hDdf s'n If]qut
tyf ;+1fg / cª\s ;dfwfg cª\s hDdf s'n hDdf
If]q af]w cª\s cª\s
5f]6f pTt/ nfdf] pTt/
cfpg] cfpg] k|Zg
! ;d"x ! $ $ $
(Set)

@= cfwf/e"t tYo tyf ;"q

of] PsfO cWoogkZrft\ ljb\ofyL{x¿ ;d"x ;DaGwL zflAbs ;d:ofx¿df e]g lrqsf] k|of]uåf/f
;dfwfg ug{ ;Sg] 5g\ . o;sf nflu b'O{ / ltg ;d"xsf] zflAbs ;d:ofx¿ ;f]lwg] 5g\ . oL
pb\b]Zo k|fKt ug{ lgDg lnlvt ljifo j:t'sf af/]df ljb\ofyL{nfO{ hfgsf/L x'g cfjZos 5 .
-s_ ;d"xsf] ;+of]hg (
olb A / B b'O{ cf]6f ;d"x eP ( eGgfn] A
df dfq, B df dfq cyjf A tyf B b'j}df ePsf ;a} ;b:o
k5{g\ .

1
 -v_ ;d"xx¿sf] k|ltR5]bg ( )
olb A / B b'O{ cf]6f ;d"x eP ;d"xsf] k|ltR5]bg
( ) df A / B b'j}df ePsf ;femf ;b:ox¿ k5{g\ .

-u_ ;d"xx¿sf] km/s

-c_ ;d"x A km/s B, ( ) df A df dfq ePsf
;b:ox¿ k5{g\ .
-cf_ ;d"x B km/s A, ( ) df B df dfq ePsf

;b:ox¿ k5{g\ .
-3_ ;d"xsf] k'/s cyjf A' df A df gePsf t/ ;j{Jofks
;d"x U df ePsf ;b:ox¿ k5{g\ .

-ª_ ;d"xsf] u0ffTdstf

;d"xdf /x]sf ;b:ox¿sf] ;ª\VofnfO{ To; ;d"xsf] u0fgfTdstf elgG5 . o;nfO{ n(A),
n(B) cflbn] hgfOG5 .

o; kf7df k|of]u x'g] s]xL ;"qx¿

;"qx¿ e]glrqsf db\btaf6 agfpg] k|of; ul/g'k5{ . s]xL dxTTjk"0f{ ;'qx¿ tn k|:t't ul/Psf 5g\
M
-s_ olb / b'O{ cf]6f ;d"x / ;j{Jofks ;d"x xf] eg]
-c_ gvlK6Psf] ;d"xsf nflu
-cf_ vlK6Psf] ;d"xsf nflu
-O_
-O{_
-p_
-v_ ;d"xx¿ A, B / C 5g\ ;fy} U ;j{Jofks ;d"x xf] eg] M
-c_
-cf_
-O_
-O{_
-p_
2
#= k|Zgsf] pTt/ lb“bf Wofg lbg'kg]{ s'/fx¿
-s_ k|ZgnfO{ Wofg lbP/ k9\g'xf];\ .
-v_ k|Zgdf lbOPsf s'/fx¿nfO{ 5'6\ofP/ n]Vg'xf];\ .
-u_ e]glrq k]lG;nn] sf]g'{xf];\ .
-3_ e]glrqdf gfds/0f jf ;ª\s]t lbg'k/]df klg k]lG;nn] g} n]Vg'xf];\ .
-ª_ e]glrq klxn] agfpg] jf kl5 agfpg] :ki6 x'g'xf];\ .
$= gd'gf k|Zgf]Tt/ tyf cEof;
gd'gf k|Zg !
/ ;j{Jofks ;d"x sf pk;d"xx¿ x'g\ . olb
eP,
-s_ dflysf] tYonfO{ e]glrqdf k|:t't ug'{xf];\ .
-v_ sf] dfg lgsfNg'xf];\ .
pTt/ M oxfF, lbOPsf]

-s_

-v_ ;"qcg';f/,

ca ,

cEof;sf nflu k|Zg

dflysf pbfx/0f x]/L tnsf k|Zgx¿ xn ug{'xf];\ .
!= olb eP,
-s_ sf] dfg kTtf nufpg'xf];\ .
-v_ of] tYonfO{ e]g lrqdf b]vfpg'xf];\ .

3
@= olb
eP,
-s_ sf] dfg kTtf nufpg'xf];\ .
-v_ pSt tYonfO{ e]g lrqdf b]vfpg'xf];\ .
#= olb
/ eP,
-s_ x sf] dfg kTtf nufpg'xf];\ .
-v_ of] tYonfO{ e]g lrqdf b]vfpg'xf];\ .
pTt/x¿ M ! s_ 65 @ s_ 15 # s_ 80
gd'gf k|Zg @
120 hgfsf] Pp6f ;d"xdf hgfnfO{ ul0ft dg kbf]{ /x]5, hgfnfO{ lj1fg dg kbf]{ /x]5 /
hgfnfO{ b'j} dg gkg]{ /x]5 eg],
-s_ pSt tYofª\snfO{ e]g lrqdf k|:t't ug{'xf];\ .
-v_ b'j} ljifo dg k/fpg]sf] ;ª\Vof kTtf nufpg'xf];\ .
pTt/ M -s_ oxfF lbOPsf], hDdf ;ª\Vof hgf,
ul0ft dg k/fpg]sf] ;ª\Vof hgf,
lj1fg dg k/fpg]sf] ;ª\Vof hgf,
b'j} dg gk/fpg] hgf
-v_ b'j} ljifo dg k/fpg]sf] ;ª\Vof
;"qcg';f/,

hgf
ca,

hgf
b'j} ljifo dg k/fpg]sf] ;ª\Vof 52 hgf /x]5 .
cEof;sf nflu k|Zg
dflysf pbfx/0f x]/L tnsf k|Zgx¿ xn ug{'xf];\ .
!= 12 hgfsf] Pp6f ;d"xdf 9 hgfnfO{ ddM dg kbf]{ /x]5, 7 hgfnfO{ rfpldg dg kbf]{ /x]5 /
2 hgfnfO{ b'j} dg gkg]{ /x]5 eg],
-s_ pSt tYofª\snfO{ e]glrqdf k|:t't ug'{xf];\ .
-v_ b'j} dg k/fpg]sf] ;ª\Vof kTtf nufpg'xf];\ .
@= s'g} ufp“sf] hg;ª\Vof 4000 dWo] 2000 n] sflGtk'/, 1800 n] gful/s b}lgs klqsf
k9\bf /x]5g\ / 400 hgfn] s'g} klg klqsf k9\b}gg\ eg],
4
 -s_ pSt tYonfO{ e]glrqdf b]vfpg'xf];\ .
-v_ b'j} klqsf k9\g] dflg;sf] ;ª\Vof kTtf nufpg'xf];\ .
#= Pp6f ljb\ofnosf 60 hgf ljb\ofyL{dWo] 35 hgfn] ;ª\uLt / 45 hgfn] g[Tosf] sIff lnG5g\
eg] b'j} sIff lng] ljb\ofyL{sf] ;ª\Vof kTtf nufO{ e]glrqdf b]vfpg'xf];\ .
pTt/x¿ M
! v_ 6 hgf @ v_ 200 hgf # 20 hgf

gd'gf k|Zg #
200 hgfdf ul/Psf] ;j]{If0fdf 30 hgf g[To ug{ / ufpg b'j} ghfGg], 60 hgf ufpg dfq hfGg] /
50 hgf g[To dfq hfGg] kfOof] eg],
-s_ pSt tYofª\snfO{ e]glrqdf k|:t't ug{'xf];\ .
v_ g[To hfGg]sf] ;ª\Vof kTtf nufpg'xf];\ .
pTt/ M
oxfF lbOPsf] hDdf ;ª\Vof hgf,
g[To / ufpg b'j} ghfGg] hgf,
ufpg dfq hfGg] hgf,
g[To dfq hfGg] hgf /
lut ufpg hfGg] ;ª\Vof n (D) = ?
-s_ e]glrqdf b]vfpFbf,

-v_ ;"qcg';f/ g[To / ufpg hfGg] hDdf,

hgf
g[To / ufpg b'j} hfGg]

= hgf
ca, g[To hfGg]sf] ;ª\Vof

hgf

5
cEof;sf nflu k|Zg
dflysf pbfx/0f x]/L tnsf k|Zgx¿ xn ug{'xf];\ .
!= 50 hgfdf ul/Psf] ;j]{If0fdf 10 hgf g]kfnL / cª\u|]hL b'j} ghfGg], 25 hgf g]kfnL dfq hfGg]
/ 8 hgf cª\u|]hL dfq hfGg] kfOof] eg],
-s_ pSt tYofª\snfO{ e]glrqdf k|:t't ug'{xf];\ .
-v_ cª\u|]hL hfGg]sf] hDdf ;ª\Vof kTtf nufpg'xf];\ .
@= 400 hgfsf] Pp6f ;j]{If0fdf 100 hgf lj|ms]6 / km'6an b'j} gv]Ng], 250 hgf km'6an dfq
v]Ng] / 88 hgf lj|ms]6 dfq hfGg] kfOof] eg],
-s_ pSt tYofª\snfO{ e]glrqdf k|:t't ug'{xf];\ .
-v_ lj|ms]6 v]Ng]sf] hDdf ;ª\Vof kTtf nufpg'xf];\ .
#= Pp6f k/LIffdf 40 % ul0ftdf dfq pTtL0f{{, 30 % lj1fgdf dfq pTtL0f{{ / % % b'j}df cg'TtL0f{
eP eg],
-s_ b'j} ljifodf slt k|ltzt pTtL0f{{ eP <
-v_ ul0ftdf hDdf slt k|ltzt pTtL0f{{ eP <
! v_ 15 hgf @ v_ 126 hgf # v_ 65 % v_ 25 %

gd'gf k|Zg $
hgf ufp“n]df ul/Psf] ;j]{If0f0fdf hgfn] b'w, hgfn] bxL / hgfn] b'j} dg k/fpg]
atfP eg],
-s_ pSt tYofª\snfO{ e]glrqdf k|:t't ug'{xf];\ .
-v_ b'w / bxL b'j} glkpg]sf] ;ª\Vof kTtf nufpg'xf];\ .

pTt/ M
oxfF lbOPsf], hDdf ;ª\Vof hgf
b'w dg k/fpg] hgf
bxL dg k/fpg] hgf
b'j} dg k/fpg] hgf
b'j} dg gk/fpg]
-s_ e]glrqdf b]vfp“bf,

6
-v_ ;"qcg';f/,

hgf
ca, hgf
cEof;sf nflu k|Zg
dflysf pbfx/0f x]/L tnsf k|Zgx¿ xn ug{'xf];\ .
!= sIff !! sf hgf ljb\ofyL{nfO{ k|Zg ubf{ hgfn] kf]v/f dg k/fP, hgfn]
lrtjg dg k/fP / hgfn] b'j} 7fp“ dg k/fPsf] kfOof] eg],
-s_ pSt tYofª\snfO{ e]glrqdf k|:t't ug'{xf];\ .
-v_ b'j} 7fp“ dg gk/fpg]sf] ;ª\Vof kTtf nufpg'xf];\ .
@= hgf l;kfxLdf ul/Psf] ;j]{If0fdf hgfn] lrof, hgfn] skmL / hgfn] b'j} dg
k/fpg] atfP eg],
-s_ pSt tYofª\snfO{ e]glrqdf k|:t't ug'{xf];\ .
-v_ lrof / skmL b'j} dg gk/fpg]sf] ;ª\Vof kTtf nufpg'xf];\ .
#= Pp6f sn]hdf hgf ljb\ofyL{ /x]5g\ . To;dWo] hgf ufpg / g[To b'j} hfGg] /x]5g\,
hgf ufpg hfGg] / hgf g[To hfGg] /x]5g\ eg],
-s_ pSt tYofª\snfO{ e]glrqdf k|:t't ug'{xf];\ .
-v_ ufpg / g[To b'j} ghfGg]sf] ;ª\Vof kTtf nufpg'xf];\ .
pTt/x¿
! v_ 50 hgf @ v_ 125 hgf # v_ 30 hgf
gd'gf k|Zg %
hgfn] lbPsf] Pp6f k/LIffdf cª\u|]hLdf pTtL0f{, ul0ftdf pTtL0f{ / b'j}df
pTtL0f{ eP eg],
-s_ b'j} ljifodf cg'TtL0f{ x'g] slt hgf /x]5g\ <
-v_ Ps ljifodf dfq pTtL0f{ x'g] slt /x]5g\ <
oxfF, hDdf ;ª\Vof
cª\u|]hLdf pTtL0f{
ul0ftdf pTtL0f{
b'j}df pTtL0f{
-s_ ;"qcg';f/,

km]l/,

7
 ca b'j} ljifodf cg'TtL0f{ x'g] ;ª\Vof hgf
-v_ Ps ljifodf dfq pTtL0f{

ca, Ps ljifodf dfq pTtL0f{

hgf
cEof;sf nflu k|Zg
dflysf pbfx/0f x]/L tnsf k|Zgx¿ xn ug{'xf];\ .
!= Pp6f k/LIffdf 70 % cª\u|]hLdf pTtL0f{, 75 % ul0ftdf pTtL0f{ / 60 % b'j}df pTtL0f{ ePsf
/x]5g\ olb b'j}df 9 hgf cg'TtL0f{ ePsf /x]5g\ eg] hDdf slt hgfn] k/LIff lbPsf /x]5g\ kTtf
nufpg'xf];\ . pTt/ M
@= ljhok'/ ufp“sf dlxnfsf] ;j]{If0fdf 45 % dlxnfn] lth dgfp“bf /x]5g\, 65% dlxnfn] 57
dgfp“bf /x]5g\ / 28 % n] b'j} dgfpFbf /x]5g\ eg] e]glrq agfO{ sDtLdf Pp6f kj{ dg k/fpg]
dlxnfsf] k|ltzt kTtf nufpg'xf];\ . pTt/ M
#= ;~hLjgLsf ljb\ofyL{x¿ dWo] 50% n] lrof dg k/fpbf /x]5g\ , 70% n] slkm dg k/fpbf
/x]5g\ , 10 % nfO{ b'a} dg gkg]{ /x]5 . 120 hgfn] b'a} dg k/fpg] kfO{of] eg] e]g lrqsf]
k|of]u u/L hDdf ljb\ofyL{ ;ª\Vof kTtf nufpg'xf];\ .
pTt/x¿ M
!= 60 hgf @= 82 % #= 400 hgf

gd'gf k|Zg ^
Pp6f ;d"xdf ul/Psf] ;j]{If0fdf lrof dg k/fpg] 60 hgf, slk dg k/fpg] 45 hgf, b'w dg
k/fpg] 30 hgf, skmL / lrof b'j} dg k/fpg] 25 hgf, lrof / b'w b'j} dg k/fpg] 20 hgf, skmL
/ b'w b'j} dg k/fpg] 15 hgf / ltg}cf]6f dg k/fpg] 10 hgf e]l6P eg] hDdf slt hgfnfO{ ;f]
k|Zg ;f]lwPsf] lyof] < e]g lrq agfO{ xn ug'{xf];\ .
;dfwfg M T
C
;j}If0fdf ;xefuL hDdf JolStx¿nfO{ 'U' ;fy} lro,
15
slkm / b'w nfO{ T, C / M j|md;“u} dfGbf, 25
n(T) = 60 15
10
n(c) = 45 10 5
n(M) = 30
5

8
ca,

= 60 + 45 + 30 - 25 - 20 - 15 + 1-
= 85 hgf

ctM ;j]{If0fdf ;f]lwPsf hDdf dflg; 85 hgf /x]5g\ .

9
 If]q M @ cª\s ul0ft (Arithmetic)
If]q kl/ro
dfWolds lzIff kf7\oj|md 2064 / ljlzi6Ls/0f tflnsf 2065 n] sIff 10 sf] clgjfo{ ul0ft ljifodf
lgDgfg';f/sf ljifo j:t', k/LIffsf nflu ;+1fgfTds If]qcg';f/sf k|Zg ;ª\Vof / cª\s ef/sf]
Joj:yf u/]sf] 5 .
j|m; ljifo 1fg / l;k hDdf l;k ;d:of hDdf s'n If]qut
j:t'÷;+1fg af]w cª\s ;dfwfg cª\s hDdf s'n
If]q cª\s hDdf
cª\s
2 5f]6f pTt/ nfdf] pTt/
cfpg] cfpg] k|Zg
2.1 ;do / 1 4 4 16
sfo{÷sfd
2.2 Gffkmf / 1 2 1 4 4
gf]S;fg
2.3 rlj|mo Aofh, 1 2 1 4 6
hg;ª\Vof
j[b\lw / ld>
x|f;

kf7 2.1 M ;do / sfo{÷sfd (Time and Work)

!= kl/ro
o; kf7df P]sLs lgodsf cfwf/df ;do / sfo{÷sfd;“u ;DalGwt ;d:ofx¿ ;dfwfg ug]{ vfnsf
;d:ofx¿ ;dfj]z ul/Psf 5g\ . k/LIffdf o;sf ljifo j:t'af6 ;d:of ;dfwfg txsf] Pp6f nfdf]
pTt/ cfpg] k|Zg ;f]lwg] Joj:yf 5 .
!= cfwf/e"t tYo tyf ;"qx¿
-s_ ljr/0f (Variation)
xfd|f] b}lgs hLjgdf cfO/xg] Jofjxfl/s ;d:ofnfO{ j:t'sf] kl/0ffd, ltgsf] d"No, sfdbf/sf]
;ª\Vof, sfd ug{ nfUg] ;ª\Vof h:tf ljleGg r/x¿sf] cGt/ ;DaGwnfO{ ljr/0f elgG5 .
ljr/0f b'O{ lsl;dsf x'G5g\ .
-c_ k|ToIf ljr/0f (Direct Variation) M olb xfdLn] Pp6f kl/0ffdnfO{ 36fp“bf csf]{
kl/0ffd klg 36\5 eg] / Pp6f kl/0ffd a9fp“bf csf]{ kl/0ffd klg a9\5 eg] To:tf]

10
 ljr/0fsf] k|lj|mofnfO{ k|ToIf ljr/0f elgG5 . pbfx/0fsf nflu olb b'O{cf]6f sfkLsf] d"No ?=
$) k5{ eg] Pp6f sfkLsf] d"No ?=$) ÷ @ = ?= @) x'G5 . To;} u/L $ cf]6f sfkLsf] d"No
?= @) × $ = ?= *) x'G5 .
-cf_ ck|ToIf ljr/0f (Indirect Variation) M olb xfdLn] Pp6f kl/0ffdnfO{ 36fp“bf csf]{
kl/0ffd a9\5 eg] / Pp6f kl/0ffd a9fp“bf csf]{ kl/0ffd 36\5 eg] To:tf] ljr/0fsf]
k|lj|mofnfO{ ck|ToIf ljr/0f elgG5 . pbfx/0fsf nflu olb b'O{ hgf dflg;n] Pp6f sfd $
lbgdf ug{ ;S5g\ eg] ! hgf dflg;n] ;f] sfd ug{ $ × @ lbg = * lbg nufp“5g\ . To:t},
u/L ToxL sfd $ hgf dflg;n] * ÷ $ = @ lbg nufp“5g\ .
@= ljz]if Wofg lbg'kg]{ s'/fx¿
• lbOPsf] kl/df0fnfO{ Pp6} PsfOdf abNg'kb{5 . Ps PsfOdf kl/jlt{t kl/df0fnfO{
;f]lwPsf] kl/df0fdf n}hfg'kb{5 . kl/df0f w]/} x'g] eP u'0ff / yf]/} x'g] eP efu ug]{ s'/f
ofb ug'{kb{5 . t/ leGg dfg PseGbf sd x'g] ePdf of] lj|mof pN6f] x'G5 . abNg'kg]{
s'/fx¿ w]/} ePdf Ps k6sdf Pp6fnfO{ dfq abNg'kb{5 .
• h'g PsfOdf pTt/ lgsfNg'kg]{ xf] To;nfO{ bfof“tk{m n]Vg'kb{5 .
• lbOPsf] ;d:ofnfO{ tflnsLs/0f u/]/ ;dfwfg ubf{ ;lhnf] x'G5 .
• tflnsf ljlw jf ;do / sfo{÷sfd h'g ljlwaf6 ug]{ eP klg ;'?df af“sL /x]sf]
sfo{÷sfd / lbgx¿ kTtf nufOxfNg'kb{5 .
• tflnsf lgdf{0f ubf{ Pp6f k'/} jfSosf] gDa/x¿ Pp6} nx/df n]Vg'kb{5 .
• h'g s'/fnfO{ rnåf/f dflgPsf] 5 To;};“u x/]s s'/fsf] ;DaGw k|ToIf jf ck|ToIf s:tf]
5 < yfxf kfpg'kb{5, h:t} M dfG5]sf] ;DaGw lbg;“u ck|ToIf x'G5 eg] dfG5] /
sfo{÷sfdsf] ;DaGw k|ToIf x'G5 .
• k|ToIf ;DaGw x'g]sf] jf0f Ps}lt/ kmsf{pg ;lsG5 eg] ck|ToIf ;DaGwdf pN6f] lbzflt/
uPsf] b]vfOG5 . o;/L ;dLs/0f agfp“bf ;f]xLcg'¿k n]Vg Wofg k'¥ofpg'k5{ .
• cfotleq sf]8 u/]sf s'/fx¿nfO{ ljz]if Wofg lbg'k5{ .
• ;do / sfo{÷sfd ljlwdf dflg;sf ;ª\Vof rflxPdf ;lhnf]sf nflu o;nfO{ ;a}eGbf
bfof“k6\l6 n]Vg] ug'{k5{ .
• dflgPsf] rn x, y z cflbsf] dfg lg:s]kl5 To;sf] cy{ v'nfO{ pTt/ n]Vg'k5{ .

11
#= gd'gf k|Zgf]Tt/ tyf cEof;
#=! cfwf/e"t tyf k'g/fjnf]sg cEof;
-b'O{hgf;“u ;DalGwt ;do / sfdsf ;d:ofx¿_
gd'gf 1 :
A / B ldn]/ s'g} sfd j|mdzM 20 lbgdf ug{ ;S5g\\ . B n] 5 lbgdf ug]{ sfd A n] 4 lbgdf ug]{
eP, k|To]snfO{ PSn} PSn} sfd ;Sg slt slt ;do nfUnf < kTtf nufpg'xf];\ .
;dfwfg
oxfF, k|Zgcg';f/ B sf] 5 lbgsf] sfd = A sf] 4 lbgsf] sfd
∴ 5B = 4A
4 5
B = 5 A / A = 4 B.
ca, (A + B) = (A + B)
 4  5 
cyjf, A + 5 A= 4 B + B
   
9A 9B
cyjf, 5 = 4 E
A A

k|Zgfg';f/,
(A + B) n] 1 sfd 20 lbgdf ug{ ;S5g\\ .
9A
A

5 n] 1 sfd 20 lbgdf ug{ ;S5g\\ .

E
A

9
A n] 1 sfd 20 × 5 lbgdf ug{ ;S5g\\ . A

E
A

= 36 lbg .
90
km]l/, 4 n] 1 sfd 20 lbgdf ug{ ;S5g\\ .
A

E
A

9
B n] 1 sfd 20 4 lbgdf ug{ ;S5g\\ . = 45 lbg .
A A

ctM A nfO{ ;f] sfd ;Sg 36 lbg / B nfO{ ;f] sfd ;Sg 45 lbg nfUbf] /x]5 .
csf]{ tl/sf
dfgf}“, A n] 4 lbgdf x sfd l;Wofp“5g\\ .
x
∴ A n] 1 lbgdf 4 sfd l;Wofp“5g\\ . A A

∴ B n] 5 lbgdf x sfd l;Wofp“5g\\ .

x x
∴ A / B ldnL 1 lbgdf 4 + 5 Error! Bookmark not defined.sfd l;Wofp“5g\\ .
 
A E A A A

9x
= 20A

12
∴ A / B ldnL 20 lbgdf 9x sfd l;Wofp“5g\\ .
k|Zgcg';f/, A / B ldnL 20 lbgdf 1 sfd l;Wofp“5g\\ .
1
∴ 9x = 1 or, x = 9 A

1
To;}n] A n] 4 lbgdf 9 sfd l;Wofp“5g\\ .
A A

1
∴ To;}n] A n] 4 lbgdf 9 sfd l;Wofp“5g\\ . A A

∴ A n] 36 lbgdf 1 sfd l;Wofp“5g\\ .

1
B n] 5 lbgdf 9 sfd l;Wofp“5g\\ .
A A

B n] 45 lbgdf 1 sfd l;Wofp“5g\\ . cfjZos lbg = 45 lbg .

cEof;sf nflu k|Zg

ca, dflysf] pbfx/0fsf] ;xfotfaf6 tnsf k|Zgx¿ xn ug]{ k|of; ug'{xf];\ .
1. /fd / Zofdn] s'g} sfd 60 lbgdf ug{ ;S5g\\ . /fdn] 2 lbgdf ug]{ sfd Zofdn] 3 lbgdf ug]{ eP
k|To]snfO{ PSnfPSn} sfd ug{ slt slt ;do nfUnf < kTtf nufpg'xf];\ .
-pTt/ M 100 lbg, 150 lbg_
2. Zofd / s[i0fn] s'g} sfd 50 lbgdf ug{ ;S5g\\ . Zofdn] 5 lbgdf ug]{ sfd s[i0fn] 3 lbgdf ug]{
eP k|To]snfO{ PSnfPSn} sfd ;Sg slt slt ;do nfUnf < kTtf nufpg'xf];\ .
1
-pTt/ M 1333 , 80 lbg_
A E A

3. ph]nL / h'gls/Ln] s'g} sfd 70 lbgdf ug{ ;S5g\\ . olb ph]nLn] 4 lbgdf ug]{ sfd h'gls/Ln]
3 lbgdf ug]{ eP k|To]snfO{ PSnfPSn} sfd ;Sg slt ;do nfUnf < kTtf nufpg'xf];\ .
1 1
pTt/ M 1633 lbg, 1222 lbg_
A E A A E A

gd'gf 2 :
A / B n] s'g} Ps sfd j|mdzM 12 / 18 lbgdf ;S5g\\ . b'j}n] sfd ;'? u/sf] s]xL lbgkl5 A n] 5f]8\5
/ af“sL sfd B n] 6 lbgdf k'/f u/]5 eg] A n] slt lbg;Dd ;“u} sfd u/]/ 5f8]sf] lyof] < kTtf
nufpg'xf];\ .
;dfwfg M
oxfF dfgf}“, A n] x lbg;Dd ;“u} sfd u/]sf] lyof] .
1
A sf] 1 lbgsf] sfd = 12 A

1
B sf] 1 lbgsf] sfd = 18 A

13
 1 1
(A + B) sf] 1 lbgsf] sfd = 12 + 18 A

E
A A

1 1
(A + B) sf] x lbgsf] sfd = x 12 + 18
 
A E

1 1
B n] 6 lbgdf u/]sf] sfd = 6 × 18 = 3 A

E
A A

1 1 1
k|Zgcg';f/, x 12 + 18 + 3 = 1
 
A E A A A

E E

5x 1
cyjf, 36 A

E
A =1–3 A

2 36
cyjf, x = 3 × A A

E
A

5 E

24 4
cyjf, x= A

5 E
A ∴ x=45 A

4
ctM A n] 4 5 lbg ;“u} sfd u/]/ sfd 5f8]sf] lyof] .
A A

cEof;sf nflu k|Zg

ca, dflysf] pbfx/0fsf] ;xfotfaf6 tnsf k|Zgx¿ xn ug]{ k|of; ug'{xf];\ .
1. P / Q n] s'g} sfd 15 / 18 lbgdf ug{ ;S5g\\ . b'j}n] sfd ;'? u/]sf] s]xL lbgkl5 P n]
5f]8]5 / af“sL sfd Q n] 3 lbgdf k'/f ub{5 eg] P n] slt lbg ;“u} sfd u/]/ 5f8]sf] lyof] <
9
-pTt/ M 611 lbg_ A E A

2. /fd / xl/n] s'g} sfd 20 / 24 lbgdf ug{ ;S5g\\ . b'j}n] sfd ;'? u/]sf] s]xL lbgkl5 /fdn]
5f]8]5 / af“sL sfd xl/n] 4 lbg sfd u/]/ 5f8]sf lyP . b'j}hgf ldnL slt lbg;Dd sfd
u/]5g\ < kTtf nufpg'xf];\ .
1
-pTt/ M 911 lbg_ A E A

3. wgdfof / dgdfofn] s'g} sfd 25 / 30 lbgdf ug{ ;S5g\\ . b'j}n] sfd ;'? u/sf] s]xL
lbgkl5 wgdfofn] sfd 5f]l85g\\ / af“sL sfd dgdfofn] 5 lbgdf k'/f ul/g\ eg] wgdfofn]
slt lbg sfd u/]/ 5f8]sL lyOg\ <
4
-pTt/ M 1111 lbg_
A E A

gd'gf 3 :
A nfO{ Pp6f sfd k'/f ug{ 20 lbg nfU5 . PSn}n] 5 lbg;Dd sfd u/]/ af“sL sfd B ;“u ldnL u5{ . olb
;Dk"0f{ sfd k'/f x'g 15 lbg nfu]5 eg] B PSn}n] Tof] sfd k'/f ug{ slt lbg nfUnf < kTtf
nufpg'xf];\ .
;dfwfg M
oxfF, A n] 1 sfd 20 lbgdf ub{5 .
14
 1
A n] 20 sfd 1 lbgdf ub{5 .
A

E
A

5 1
A n] 20 = 4 sfd 5 lbgdf ub{5 .
A

E
A A

E
A

 1 3
(A + B) n] 1 – 4 = 4 sfd (15 – 5) = 10 lbgdf ;S5 .
 
A E A A A

3 1 3
(A + B) n] 4 × 10 = 40 sfd 1 lbgdf ;S5 . A A

E
A

E
A A

E
A

3 1
B PSn}]n] 1 lbgdf 40 – 20 sfd l;Wofp“5 .
 
A E A

3–2 1
B PSn}]n] 1 lbgdf = 40 = 40 sfd l;Wofp“5 . A

E
A A

E
A

B PSn}n] 1 sfd 40 lbgdf ;S5 . ctM B nfO{ ;f] sfd ;Sg nfUg] ;do 40 lbg .

cEof;sf nflu k|Zg

ca, dflysf] pbfx/0fsf] ;xfotfaf6 tnsf k|Zgx¿ xn ug]{ k|of; ug'{xf];\ .
1. P nfO{ Pp6f sfd k'/f ug{ 30 lbg nfU5 . pm PSn}n] 5 lbg;Dd sfd u/]/ af“sL sfd Q
;“u ldnL u5{ . olb ;Dk"0f{ sfd k'/f x'g 16 lbg nfu]5 eg] Q PSn}nfO{ Tof] sfd k'/f ug{
1
slt lbg nfUnf < -pTt/ M 533 lbg_ A E A

2. dLgfnfO{ Pp6f sfd k'/f ug{ 50 lbg nfU5 . pm PSn}n] 20 lbg;Dd sfd u/]/ af“sL sfd
aLgf;“u ldnL ub{l5g\ . olb ;Dk"0f{ sfd k'/f x'g 15 lbg nfu]5 eg] aLgf PSn}nfO{ Tof] sfd
k'/f ug{ slt lbg nfUnf < -pTt/ M 50 lbg_
3. k|ltefnfO{ Pp6f sfd k'/f ug{ 60 lbg nfU5 . pm PSn}n] 15 lbg;Dd sfd u/]/ af“sL sfd
/]lhgf;“u ldnL ub{l5g\\ . olb ;Dk"0f{ sfd k'/f x'g 20 lbg nfu]5 eg] /]lhgf PSn}nfO{ Tof]
sfd k'/f ug{ slt lbg nfUnf < -pTt/ M 48 lbg_
gd'gf 4 :
df]xg / ;f]xg ldn]/ s'g} Pp6f sfd 30 lbgdf l;Wofp“5g\\ . b'j} ldnL 20 lbg;Dd sfd u/]/ ;f]xgn]
5f8]5 . olb df]xg PSn}n] af“sL sfd 35 lbgdf l;Wofpg ;S5 eg] ;f]xg PSn}n] k'/} sfd slt
lbgdf l;Wofpnfg\ <
;dfwfg
oxfF, df]xg / ;f]xgsf] 30 lbgsf] sfd = 1
1
cyjf, df]xg / ;f]xgsf] 1 lbgsf] sfd = 30 A

1 2
cyjf, df]xg / ;f]xgsf] 20 lbgsf] sfd = 30×20 = 3 A

E
A A

15
 2 1
af“sL sfd = 1 – 3 = 3A A

E
A

1
df]xgn] 35 lbgdf af“sL = 3 sfd u5{ . A A

1
;f]xgn] 1 lbgdf af“sL = 3 × 35 sfd u5{ . A

E
A

1 1
;f]xgn] 1 lbgdf af“sL = 30 – 3 × 35 [∴ ;f]xg = -df]xg + ;f]xg_ – df]xg ]
A

E
A A

E
A

7-2
= 210 A E

5
= 210 A E

1
= 42 A

ctM ;f]xgnfO{ 1 sfd ug{ nfUg] ;do = 42 lbg .

cEof;sf nflu k|Zg

ca, dflysf] pbfx/0fsf] ;xfotfaf6 tnsf k|Zgx¿ xn ug]{ k|of; ug'{xf];\ M
1. k|df]b / /fdrGb|n] s'g} sfd j|mdzM 30 / 20 lbgdf ug{ ;S5g\\ . b'j} ldnL s]xL lbg;Dd sfd
u/]kl5 /fdrGb|n] 5f]8] . olb af“sL sfd k|df]bn] 15 lbgdf k'/f u/] eg] /fdrGb|n] slt
lbg;Dd ;“u} sfd u/]/ 5f8]sf lyP < -pTt/ M 6 lbg_
2. /fd / Zofdn] s'g} sfd j|mdzM 15 / 10 lbgdf ug{ ;S5g\\ . b'j} ldnL s]xL lbg;Dd u/]kl5
/fdn] 5f]8] / af“sL sfd Zofdn] 5 lbgdf k'/f u/]sf lyP eg] /fdn] slt lbg;Dd ;“u} sfd
u/]/ 5f8]sf lyP . -pTt/ M 3 lbg_
3. k|df]b / /fdrGb|n] s'g} sfd j|mdzM 30 / 20 lbgdf ug{ ;S5g\\ . b'j} ldnL s]xL lbg;Dd sfd
u/]kl5 /fdrGb|n] 5f]8] . olb af“sL sfd k|df]bn] 15 lbgdf k'/f u/] eg] /fdrGb|n] slt
lbg;Dd ;“u} sfd u/]/ 5f]8]sf lyP . -pTt/ M 6 lbg_

gd'gf 5:
2
;f}/enfO{ Ps lbg nfUg] sfd uf}/jnfO{ b'O{ lbg nfU5 . olb uf}/jn] Pp6f sfdsf] 5 efu ;Sg A A

24 lbg nufp“5 eg] b'j} ldn]/ ;f] k'/f sfd ;Sg slt lbg nufpnfg\ <

;dfwfg M
2
oxf“, uf}/jn] 5 efu sfd 24 lbgdf ;S5 .
A A

16
 2 1
uf}/jn] 5×24 = 60 sfd 1 lbgdf u5{ .
A

E
A A

E
A

1 1 1
ctM -;f}/e ± uf}/j_ n] 30 + 60 = 20 sfd 1 lbgdf ug{ ;S5 . A

E
A A

E
A A

E
A

,, ,, 1 sfd 20 lbgdf ug{ ;S5 .

cEof;sf nflu k|Zg

ca, dflysf] pbfx/0fsf] ;xfotfaf6 tnsf k|Zgx¿ xn ug]{ k|of; ug'{xf];\ .
1. uf]kL / s[i0f ldnL s'g} sfd 24 lbgdf ug{ ;S5g\\ . uf]kLn] 4 lbgdf u/]sf] sfd s[i0fn] 6 lbgdf ug{
;S5 eg] k|To]sn] PSnfPSn} slt lbgdf k'/f ug{ ;S5g\ < -pTt/ M 40 lbg, 60 lbg_
2
2. bofnfO{ 1 lbg nfUg] sfd dfofnfO{ 3 lbg nfU5 . olb bofn efu 5 sfd 30 lbgdf ug{ A A

1
;S5 eg] b'j} ldnL ;f] k'/f sfd ;Sg slt lbg nufpnfg\ < -pTt/ M 564 lbg_ A E A

2
3. s?0ffnfO{ 3 lbg nfUg] sfd ;fu/nfO{ 5 lbg nfU5 . olb s?0ffn] Pp6f sfdsf] 3 efu sfd ;Sg A

E
A

3
30 lbg nufp“5 eg] b'j} ldn]/ ;f] k'/f ug{ slt lbg nufpnfg\ < -pTt/ M 184 lbg_ A E A

gd'gf 6:
2
dbgn] Pp6f sfdsf] 5 efu 9 lbgdf l;Wofp“5 . af“sL sfd ug{ p;n] cd/nfO{ af]nfp“5 . b'j} A

E
A

ldnL ;f] sfd 6 lbgdf l;Wofp“5g\\ eg] cd/ PSn}n] k'/} sfd l;Wofpg slt lbg nfUnf <
;dfwfg M
2
oxfF, dbgn] 5 efu sfd 9 lbgdf l;Wofp“5 .
A A

2 2
dbgn] 5 × 9 = 45 efu sfd 1 lbgdf l;Wofp“5 .
A

E
A A

E
A

2 3
dbgn] / cd/n] ldn]/ u/sf] sfd = 1 – 5 = 5 A A

E
A

3
oxf“, dbg / cd/n] 5 efu sfd 6 lbgdf l;Wofp“5 . A A

3 1
dbg / cd/n] 5 × 6 = 10 efu sfd 1 lbgdf l;Wofp“5 .
A

E
A A E A

1 2
cd/ PSn}n] 10 – 45 efu sfd 1 lbgdf l;Wofp“5 .
 
A E A

cd/ PSn}n] k'/f (1) sfd 18 lbgdf l;Wofp“5 .

17
cEof;sf nflu k|Zg
ca, dflysf] pbfx/0fsf] ;xfotfaf6 tnsf k|Zgx¿ xn ug]{ k|of; ug'{xf];\ .
2
1. /fdn] Pp6f sfdsf] 3 efu 30 lbgdf l;Wofp“5 . af“sL sfd ug{ p;n] xl/nfO{ af]nfp“5 /
A

E
A

b'j} ldnL ;f] sfd 8 lbgdf l;Wofp“5g\\ eg] xl/ PSn}nfO{ k'/} sfd ;Sg slt lbg nfUnf <
4
-pTt/ M 537 lbg_
A E A

2
2. /dfn] Pp6f sfdsf] 5 efu 18 lbgdf l;Wofp“5 . af“sL sfd ug{ p;n] cl:dtfnfO{ af]nfp“5
A

E
A

/ b'j} ldnL ;f] sfd 9 lbgdf l;Wofp“5g\\ eg] cl:dtf PSn}nfO{ k'/} sfd l;Wofpg slt lbg
1
nfUnf < -pTt/ M 222 lbg_
A E A

3
3. A n] Pp6f sfdsf] 5 efu 25 lbgdf l;Wofp“5 . af“sL sfd ug{ p;n] B nfO{ af]nfp“5 / b'j}
A A

ldnL ;f] sfd 10 lbgdf l;Wofp“5g\\ eg] B PSn}nfO{ k'/} sfd l;Wofpg slt lbg nfUnf <
1
-pTt/ M 622 lbg_ A E A

gd'gf 7 :
ljsf;n] s'g} sfd 6 lbgdf k'/f ug{ ;S5 . lgdf{0fn] 9 lbgdf ;f]xL sfd k'/f ug{ ;S5 . ;f] sfd
3 lbg;Dd b'j} ldn]/ ul/;s]kl5 ljzfnn] 5f]8]5 eg] M
-s_ ljsf; PSn}n] af“sL sfd slt slt lbgdf k'/f unf{ <
-v_ k'/f sfd slt lbgdf ;lsPnf <

;dfwfg M
oxf“, ljsf;n] 6 lbgdf 1 sfd u5{ .
1
lgdf{0fn] 1 lbgdf 6 sfd u5{g\ .
A A

lgdf{0fn] 9 lbgdf 1 sfd u5{g\ .

1
lgdf{0fn] 1 lbgdf 9 sfd u5{g\ .
A A

1 1  5
-ljsf; ± lgdf{0f_ n] 1 lbgdf 6 + 9 = 18
 
A E A A

E E

5 5
ljsf; ± lgdf{0fn] 3 lbgdf 3 × 18 = 6 sfd l;Wofp“5g\ . A

E
A A A

18
 5 1
af“sL sfd = 1 – 6 = 6
A A

E
A

1
(i) ljsf;n] af“sL sfd 6 sfd 1 lbgdf k'/f ug{ ;S5 .
A A

jf af“sL sfd 6 lbgdf k'/f ug{ ;S5 .

(ii) k'/} sfd l;Wofpg nfu]sf] ;do
= 3 lbg + 1 lbg = 4 lbg

cEof;sf nflu k|Zg

ca, dflysf] pbfx/0fsf] ;xfotfaf6 tnsf k|Zgx¿ xn ug]{ k|of; ug'{xf];\ .
1. A n] s'g} sfd 18 lbgdf k'/f ug{ ;S5 / B n] 24 lbgdf ;f]xL sfd k'/f ug{ ;S5 . ;f] sfd
6 lbg;Dd b'j} ldn]/ ul/;s]kl5 B n] 5f]8]5 eg],
1
(i) A Psn}n] af“sL sfd slt lbgdf k'/f unf{ < -pTt/ M 72 lbg_ A E A

1
(ii) k'/f sfd slt lbgdf ;lsPnf < -pTt/ M 132 lbg_
A E A

2. P n] s'g} sfd 20 lbgdf k'/f ug{ ;S5 / n] 30 lbgdf ;f]xL sfd k'/f ug{ ;S5 . ;f] sfd
10 lbg;Dd b'j} ldn]/ ul/;s]kl5 Q n] 5f]8]5 eg],
1
(i) P PSn}n] af“sL sfd slt lbgdf k'/f unf{ < -pTt/ M 33 lbg_
A E A

1
(ii) k'/f sfd slt lbgdf ;lsPnf < -pTt/ M 33 lbg_
A E A

19
#=@ nfdf] pTt/ cfpg] k|Zgx¿ -ltg hgf;“u ;DalGwt ;do / sfdsf ;d:ofx¿_

gd'gf 1 :
A n] s'g} sfd 10 lbgdf / B n] 12 lbgdf ug{ ;S5g\\ . olb C sf] ;xof]u lnO{ ltgLx¿n] ;f] sfd
4 lbgdf ug{ ;S5g\\ eg] C PSn}n] ;f] sfd slt lbgdf ;S5 < kTtf nufpg'xf];\ .
;dfwfg
oxf“, A sf] 10 lbgsf] sfd = 1
1
A sf] 1 lbgsf] sfd = 10 A

B sf] 12 lbgsf] sfd = 1

1
B sf] 1 lbgsf] sfd = 12A

A + B + C) sf] 4 lbgsf] sfd = 1

1
(A + B + C) sf] 1 lbgsf] sfd = 4 A

1 1 1 1
∴ C sf] 1 lbgsf] sfd = 4 – 10 – 12 = 15
A A

E
A

E
A A

E
A A

1 1 1
= 4 – 10 – 12
A

E
A A

E
A A

E
A

= (15-6-5)/60
= 4/60
1
= 15 A

∴ C sf] 15 lbgsf] sfd = 1

ctM C PSn}n] ;f] sfd 15 lbgdf ;S5 .

cEof;sf nflu k|Zg

ca, dflysf] pbfx/0fsf] ;xfotfaf6 tnsf k|Zgx¿ xn ug]{ k|of; ug'{xf];\ .
1. /fdn] s'g} sfd 10 lbgdf / xl/n] 12 lbgdf ug{ ;S5g\, s[i0fsf] ;xof]u lnO{ ltgLx¿n] ;f]
sfd 5 lbgdf ug{ ;S5g\\ eg] s[i0f PSn}n] ;f] sfd slt lbgdf ;S5 <
-pTt/ M 60 lbg_
2. P n] s'g} sfd 16 lbgdf / Q n] 24 lbgdf ug{ ;S5g\\ . R sf] ;xof]u lnO{ ltgLx¿sf] ;f] sfd 12
lbgdf ug{ ;S5g\\ eg] R PSn}n] ;f] sfd slt lbgdf ;S5 < -pTt/ M 48 lbg_
3. dLgf / /Lgfn] s'g} Pp6f sfd j|mdzM 24 / 32 lbgdf ug{ ;S5g\\ . lbgfsf] ;xof]u lnO{
ltgLx¿n] ;f] sfd 12 lbgdf ug{ ;S5g\\ eg] aLgf PSn}n] ;f] sfd slt lbgdf ;lS5g\\ <
-pTt/ M 96 lbg_

20
gd'gf 2:
A, B / C n] s'g} sfd j|mdzM 10 lbg, 20 lbg / 30 lbgdf ;S5g\\ . ltg} hgf ldnL sfd ;'?
u/]sf] 4 lbgkl5 A n] 5f]8\5 . C n] sfd ;dfKt x'g'eGbf 5 lbg klxn] 5f]8]5 eg] sfd ;dfKt x'g
slt lbg nfUnf <
;dfwfg M
oxfF, dfgf}“ x lbgdf ;f] sfd ;dfKt x'G5 .
A sf] 10 lbgsf] sfd = 1
1
A sf] 1 lbgsf] sfd = 10 A

4
A sf] 4 lbgsf] sfd = 10 A

B sf] 20 lbgsf] sfd = 1

1
B sf] 1 lbgsf] sfd = 20 A

E
A

x
B sf] x lbgsf] sfd = 20 A

C sf] 30 lbgsf] sfd = 1

1
C sf] 1 lbgsf] sfd = 30 A

x–5
C sf] (x – 5) lbgsf] sfd = 30 A

4 x x–5
To;sf/0f, 10 + 20 + 30 = 1
A A

E
A

E
A A

E
A

24 + 3x + 2x – 10
cyjf, A

60 =1 E
A

cyjf, 5x + 14 = 60
cyjf, 5x = 46
1
x = 9 5 lbg A E A

1
ctM ;f] sfd ;dfKt x'g 9 5 lbg nfU5 . A E A

cEof;sf nflu k|Zg

ca, dflysf] pbfx/0fsf] ;xfotfaf6 tnsf k|Zgx¿ xn ug]{ k|of; ug'{xf];\ .
1. A n] s'g} sfd 10 lbgdf ug{ ;S5 . B n] 12 lbgdf / C n] 15 lbgdf ug{ ;S5g\ . ;a} ldnL
sfd ;'? u5{g\ t/ A n] 2 lbgkl5 sfd 5f8\5 . B n] sfd ;lsg'eGbf 3 lbg cl3 sfd
5f]8\5 eg] ;f] sfd slt lbgdf ;dfKt xf]nf < -pTt/ M 7 lbg_
2. s'g} sfd A n] 10 lbgdf B n] 20 lbgdf / C n] 30 lbgdf ;S5g\ . ltg} hgf ldnL sfd
;'? u/]sf] 4 lbgkl5 A n] sfd 5f]8\5 . C n] sfd ;dfKt x'g'eGbf 3 lbg klxn] 5f]8]5 eg]
2
sfd ;dfKt x'g slt lbg nfUnf < -pTt/ M 8 5 lbg_ A E A

21
3. s'g} sfd P n] 20 lbgdf, Q n] 30 lbgdf / R n] 40 lbgdf ;f]xL ;S5g\\ . ltg} hgf ldnL
sfd ;'? u/]sf] 8 lbgkl5 P n] sfd 5f]8\5 . R n] sfd ;dfKt x'g'eGbf 3 lbg klxnf] 5f]8]5
4
eg] sfd ;dfKt x'g slt lbg nfUnf < -pTt/ M 117 lbg_ A E A

yk cEof;sf nflu k|Zg

1. 20 hgf dflg;n] Pp6f sfd 30 lbgdf ug{ ;S5g\\ . olb ;f] sfd 6 lbg cufj} k'/f ug'{5 eg]
slt hgf dflg; yKg' knf{ < -pTt/ M 5 hgf_
2. 45 hgf dflg;n] k|ltlbg 12 306fsf] b/n] 30 lbgdf ug{ ;Sg] sfd 60 hgf dflg;n] 10
306fsf] b/n] slt lbgdf l;Wofpnfg\ < -pTt/ M 27 lbg_
3. Pp6f 5fqfjf;df 500 hgf ljb\ofyL{nfO{ 40 lbg k'Ug] /;b 5 . Tof] /;bn] yk 10 lbg
k'¥ofpg slt hgfnfO{ 5fqjf;af6 cGoq ;fg'{ k5{{ < -pTt/ M 100 hgf_
4
4. P n] Pp6f sfdsf] 9 efu 36 lbgdf l;Wofp“5 . af“sL sfd ug{ p;n] Q nfO{ af]nfp“5 / b'j}
A E A

ldnL ;f] sfd 12 lbgdf l;Wofp“5g\\ eg] Q PSn}nfO{ k'/} sfd l;Wofpg slt lbg nfUnf < -pTt/
5
M 2911 lbg_ A E A

5. Pp6f Aof/]sdf /x]sf s]xL l;kfxLnfO{ 30 lbg k'Ug] /;b lyof] . 10 lbgkl5 ;f] Aof/]saf6 60
hgf l;kfxLn] 5f]8] . ca ;f] /;b tf]lsPsf] lbgeGbf yk 10 lbgnfO{ k'Uof] eg] ;f] Aof/]sdf
;'?df slt l;kfxL lyP < -pTt/ M 180 hgf_
6. P / Q ldn]/ s'g} sfd 40 lbgdf ug{ ;S5g\\ . Q n] 5 lbgdf ug]{ sfd P n] 4 lbgdf ug]{ eP
2
k|To]snfO{ PSnfPSn} sfd ;Sg slt slt ;do nfUnf < -pTt/ M663 lbg, 100 lbg A E A

8. A / B n] s'g} sfd j|mdzM 32 / 24 lbgdf ;S5g\\ . b'j}n] sfd ;'? u/]sf] s]xL lbgkl5 A n]
5f]8\5 / af“sL sfd B n] 8 lbgdf k'/f u/]5 eg] A n] slt lbg ;“u} sfd u/]/ 5f8]sf] lyof] <
1
-pTt/ M 97 lbg_
A E A

9. A nfO{ Pp6f sfd k'/f ug{ 40 lbg nfU5 . pm PSn}n] 10 lbg;Dd sfd u/]/ af“sL sfd B
;“u ldnL u5{ . olb ;Dk"0f{ sfd k'/f x'g 20 lbg nfu]5 eg] B PSn}nfO{ Tof] sfd k'/f ug{
slt lbg nfUnf < -pTt/ M 22 lbg_
10. /Lt' / gLt' ldn]/ s'g} Pp6f sfd 40 lbgdf l;Wofpg ;S5g\\ . b'j} ldnL 25 lbg;Dd sfd
u/]/ gLt'n] 5fl85g\\ . olb /Lt' Psn}n] af“sL sfd 45 lbgdf l;Wofpg ;lS5g\\ eg] gLt' PSn}n]
k'/} sfd slt lbgdf l;Wofplng\ < -pTt/ M 60 lbg_
11. D n] s'g} sfd 40 lbgdf k'/f ug{ ;S5 / R n] 50 lbgdf ;f]xL sfd k'/f ug{ ;S5 . olb ;f]
sfd 20 lbg;Dd b'j} ldn]/ ul/;s]kl5 R 5f]8]5 eg],
(i) D PSn}n] af“sL sfd slt lbgdf k'/f unf{ < -pTt/ M 4 lbg_
(ii) k'/f sfd slt lbgdf ;lsPnf < -pTt/ M 24 lbg_
12. A, B / C n] s'g} Pp6f sfd j|mdzM 30, 40 / 60 lbgdf k'/f ug{ ;S5g\ . ltg} hgf ldn]/ sfd
;'? u/]sf] 10 lbgkl5 B n] 5f]8\of] . B n] 5f8]sf] 4 lbgkl5 A n] 5f]8\5 . af“sL sfd C n] k'/f
ub{5 eg] C n] hDdf slt lbg sfd u¥of] < -pTt/ M 3 lbg_

22
kf7 2.2 gfkmf / gf]S;fg, 5'6 tyf d"No clej[b\lw s/
(Profit and Loss, Discount and Value Added Tax)

o; kf7df gfkmf / gf]S;fg ;DaGwL ;d:of -cª\lst d"No, 5'6, d"No clej[b\lw s/ ;d]t_ ;dfwfg
ug]{ vfnsf ;d:ofx¿ ;dfj]z ul/Psf 5g\ . k/LIffdf o;sf ljifoj:t'af6 Pp6f b'O{ cª\s ef/sf]
5f]6f] l;k txsf] tyf Pp6f rf/ cª\s ef/sf] ;d:of ;dfwfg txsf] nfdf] pTt/ cfpg]] k|Zg u/L
hDdf 5 cª\s ef/sf k|Zgx¿ ;f]lwg] Joj:yf 5 .
2. cfwf/e"t tYo tyf ;"qx¿
-s_ cfwf/e"t wf/0ffx¿
gfkmf / gf]S;fg cl3Nnf sIffx¿af6 ;'?jft ul/Psf] 5 . lgDg lnlvt kl/efiffx¿sf] cWoog
u/f}+“ M
-c_ j|mo d"No (Cost Price)
;fdfg vl/b ubf{ ltl/Psf] d"NonfO{ ;f] ;fdfgsf] j|mo d"No elgG5 / o;nfO{ ;ª\lIfKt ¿kdf
j|m=d"= (C.P.) n]lvG5 .
-cf_ ljj|mo d"No (Selling Price)
s'g} klg ;fdfg laj|mL ubf{ k|fKt x'g] /sdnfO{ ;f] ;fdfgsf] ljj|mo d"No elgG5 . o;nfO{
;ª\lIfKt ¿kdf lj=d"=(S.P.) åf/f hgfOG5 .
-O_ gfkmf (Gain or Profit)
olb s'g} klg ;fdfgsf] vl/b d"No (C.P.) eGbf ljj|mo d"No (S.P) w]/} eP S.P. / C.P. larsf]
cGt/nfO{ gfkmf (profit or gain) elgG5 .
To;sf/0f, gfkmf = S.P. – C.P. x'G5 .
-O{_ 3f6f jf gf]S;fg (Loss)
s'g} klg j:t'sf] ljj|mo d"No (S.P.) eGbf j|mo d"No (C.P. ) w]/} eP C.P, / S.P larsf] cGt/nfO{
pSt ;fdfgdf x'g] 3f6f (Loss) elgG5 .
To;sf/0f 3f6f (Loss) = C.P. – S.P. x'G5 .
ctM olb S.P. > C.P. eP gfkmf = SP – CP /
olb S.P. < CP eP 3f6f = CP – SP x'G5 .

cGTodf,
SP × 100
(i) S.P. / gfkmf % (G%) lbOPdf, CP = (100 + G%)
A E

SP × 100
(ii) S.P. / gf]S;fg % (L%) lbOPdf, CP = (100 + L%)
A E

23
-p_ 5'6 (Discount)
s'g} klg j:t' ljqmL ubf{ To; j:t'sf] ljj|mo d"Noaf6 s]xL /sd jf s]xL k|ltzt /sd
36fP/ lbOG5 eg] o;/L 36fPsf] /sdnfO{ 5'6 elgG5 . a]Rgsf nflu tf]lsPsf] cª\lst d"No
(marked price) df s]xL /sd sd u/]/ jf 36fP/ a]lrG5 eg] To:tf] /sd 5'6 /sd xf] .
k|ltztdf n]lvPsf] 5'6nfO{ 5'6 k|ltzt elgG5 .
t;y{, jf:tljs 5'6 = cª\lst d"No – ljj|mo d"No
5'6 k|ltzt = jf:tljs 5'6 / cª\lst d"No × 100 %
• s/ nfUg] /sd = hDdf cfDbfgL – s/ gnfUg] /sd
Taxable Income = Total Imcome – Exempted Limit
• ljj|mo d"No -Eof6 /sd;lxt_ = -c=d= – 5'6_ ++ lj= d"= sf] Eof6 %
• SP1= (MP–D) + VAT
• Eof6 /sd = lj=d"=sf] Eof6 k|ltzt , (VAT = VAT% of SP)

@= ljz]if Wofg lbg'{kg]{ s'/fx¿

1. olb j|mo d"No / ljj|mo d"Nosf kl/df0fx¿km/s km/s ePdf ;aeGbf klxnf a/fa/
kl/df0fsf j|mo d"No / laj|mo d"No lgsfn]/ gfkmf jf gf]S;fg lgsfNg' k5{ .
2. ;fdfGotof gfkmf jf gf]S;fg k|ltzt j|mo d"Noaf6 lgsflnG5 .
3. 5'6 hlxn] klg d"Noaf6 lgsflnG5 .
4. ljleGg lzif{sx¿sf vr{x¿ -3/fo;L, oftfoft, dd{t ;+xf/, tyf cGo ljljw_
5. d"No clej[blw s/ hlxn] klg a:t'sf] clGtd 5'6 kl5sf] ljj|mo d"Nodf nfUg] u5{ . o;sf]
b/ pNn]v gePsf] cj:yfdf 13 % /xg] 5 .
6. d"= c= s= jf VAT h] n]v] klg d"No clej[b\lw s/ eGg] a'em\g'kb{5 .

#= gd'gf k|Zgf]Tt/ tyf cEof;

#=! 5f]6f] pTt/ cfpg] k|Zg ;DaGwL cEof;
gd'gf 1 :
k|ltefn] Pp6f 38L ?= 1800 df lsg]/ ?= 2250 df laj|mL ubf{ slt k|ltzt kmfObf ul/g\ <
;dfwfg
oxf“,
j|mo d"No (CP) = ?=1800
ljj|mo d"No (S.P.) = ?. 2250
S.P. > CP ePsfn], gfkmf (G) = SP – CP
= ?=2250 – ?=1800
= ?= 450
ca, gfkmf k|ltzt (%) = P/CP × 100
450
= 2250 × 100
A E A

24
 = 25%
ctM gfkmf k|ltzt = 20%
ca, dflysf] pbfx/0fsf] ;xfotfaf6 tnsf k|Zgx¿ xn ug]{ k|of; ug'{xf];\ M
[

1. gfkmf k|ltzt kTtf nufpg'xf];\ .

-s_ C.P. = ?= 750 / S.P. = ?= 825 -pTt/ M 10% )
-v_ C.P. = ?= 850 / S.P. = ?= 1120 (pTt/ M 31.76%)
2. 3f6f k|ltzt kTtf nufpg'xf];\ .
-s_ C.P. = ?= 650 / S.P. = ?= 585 (pTt/ M 10%)
-v_ C.P. = ?= 1875 / S.P. = ?= 1425 (pTt/ M 24%)
gd'gf 2 :
/Ghgn] Pp6f SofNs'n]6/ ?= 800 df vl/b u/L 25% 3f6f vfO{ laj|mL u/]5 eg] p;n] sltdf ;f]
SofNs'n]6/ laj|mL u/]5 <
;dfwfg M
oxf“, j|mo d"No (C.P.) = ?= 800, 3f6f % (L%) = 25%
ljj|mo d"No (S.P.) = ?
klxnf] tl/sf,
xfdLnfO{ yfxf 5, ljj|mo d"No (S.P.) = C.P. (100
100
– L%)
A

E
E A

800(100 – 25)
= A

100 E
E

= ?= 600
bf];|f] tl/sf,
25% gf]S;fgsf] cyf{g';f/,
C.P. ?= 100 eP S.P. = ?= 75 x'G5 .
75
cyjf, C.P. ?= 1 eP S.P. = ?= 100 A x'G5 . E A

∴ CP ?= 800 eP S.P. = ?= A
75
100 × 800E A x'G5 .
∴ laj|mo d"No (S.P.) = ?=600
t];|f] tl/sf,
25% 3f6fsf] cy{af6,
ljj|mo d"No (S.P.) = CP sf] (100% – L%)
= ?= 800 × (100% – 25%)
= ?= 800 × 75%
= ?= 800
= ?=600
rf}yf] tl/sf M
ljj|mo d"No = C.P. – gf]S;fg (l)

25
 = ?= 800 – ?= 800 sf] 25%
= ?= 800 – ?= 800 x
∴ ljj|mo d"No (S.P.) = ?= 600

cEof;sf nflu k|Zg

ca, dflysf] pbfx/0fsf] ;xfotfaf6 tnsf k|Zgx¿ xn ug]{ k|of; ug'{xf];\ .
1. ljj|mo d"No (S.P.) kTtf nufpg'xf];\ . (pTt/ M ?= 1767)
-s_ C.P. = ?= 1425, gfkmf (G) = 24%
-v_ C.P. = ?= 1875, gfkmf (G) = 20% (pTt/ M ?= 2250)
2. ljj|mo d"No (S.P.) kTtf nufpg'xf];\ .
-s_ C.P. = ?= 4235, gf]S;fg (L) = 20% (pTt/ M ?= 3388)
-v_ C.P. = ?= 5050, gf]S;fg (L) = 10% (pTt/ M ?= 4545)
gd'gf 3 :
s'g} ;fdfg ?= 1620 df laj|mL ubf{ 10% gf]S;fg x'G5 eg] ;f] ;fdfgsf] j|mo d"No kTtf nufpg'xf];\ .
;dfwfgM
oxf“, ;fdfgsf] ljj|mo d"No (S.P) = ?= 1620
k|ltzt gf]S;fg (L) = 10%
;fdfgsf] j|mo d"No (C.P.) = ?
ca, 10% gf]S;fgsf] cyf{g';f/,
S.P. = C.P. sf] 90%
90
cyjf, 1620 = C.P. × 100 A

100
cyjf, 1620 × 90 = C.P.
A

E
A

∴ C.P. = 1800
ctM ;f] ;fdfgsf] j|mo d"No (C.P.) = ?= 1800 k5{ .

cEof;sf nflu k|Zg

ca, dflysf] pbfx/0fsf] ;xfotfaf6 tnsf k|Zgx¿ xn ug]{ k|of; ug'{xf];\ M
1. j|mo d"No (C.P.) kTtf nufpg'xf];\ .
-s_ lj=d"= (S.P.) = ?= 4545, gfkmf (G) = 25% pTt/ M ?= 3636
-v_ lj=d"= (S.P) = ?= 3798, gfkm (G) = 10% pTt/ M ?= 3450
2. j|mo d"No (C.P.) kTtf nufpg'xf];\ .
-s_ 10% gf]S;fg ;x]/ Pp6f /]l8of] ?= 5454 df laj|mL ul/of] . pTt/ M ?= 6060
-v_ 20% gf]S;fg ;x]/ Pp6f sDKo'6/ ?= 56560 df laj|mL ul/of] . pTt/ M ?= 70700

26
gd'gf –4 :
s'g} ljj|m]tfn] Pp6f /]l8of] ?= 225 df vl/b u/L ?=15 dd{tdf vr{ ub{5 . olb p;n] ;f] /]l8of] ?=
300 df laj|mL u5{ eg] p;sf] gfkmf k|ltzt slt xf]nf <
;dfwfg M
oxf“, /]l8of]sf] j|mo d"No (CP) = ?= 225
yk dd{t vr{ = ?= 15
/]l8of]sf] ljj|mo d"No (S.P.) = ? =300
∴ /]l8of]sf] hDdf j|mo d"No (C.P.) = ?=225 + ?=15 = ?= 240

csf]{ tl/sf M
oxf“, /]l8of]sf] j|mo d"No (C.P)
= ?= 225 + ?= 15 = ?= 240
/]l8of]sf] ljj|mo d"No (S.P) = ?= 300
gfkmf k|ltzt (G%) = ?
xfdLnfO{ yfxf 5,
gfkmf k|ltzt =
=
= 25%
ca, gfkmf = SP - CP ?= 300 - ?= 240 = ?= 60
60
∴ gfkmf k|ltzt = = 240 × 100 = 25%
A A

ctM cfjZos gfkmf k|ltzt = 25%

cEof;sf nflu k|Zg

ca, dflysf] pbfx/0fsf] ;xfotfaf6 tnsf k|Zgx¿ xn ug]{ k|of; ug'{xf];\ .
1. v'b|f k;n]n] Pp6f SofNs'n]6/ ?= 1350 df vl/b u/L dd{t vr{ ?= 50 nufO{ dd{t u¥of] .
olb ;f] SofNs'n]6/sf] ljj|mo d"No ?= 1540 eP gfkmf k|ltzt kTtf nufpg'xf];\ .
pTt/ M 10%)
2. /fdn] Ps 3f/L s]/f ?= 225 df vl/b u/L l/S;f ef8f ?= 15 vr{ u/]/ cfk\mgf] k;ndf
Nofof] . olb p;n] pSt s]/fwf/L ?= 300 df laj|mL u¥of] eg] p;sf] gfkmf k|ltzt kTtf
nufpg'xf];\ . -pTt/ M 25%)
3. ;lagn] 1 bh{g sfkL ?= 240 df vl/b u/L k|lt uf]6f ?= 18 sf b/n] a]r]5 eg] hDdf
gf]S;fg k|ltzt kTtf nufpm . -pTt/ M 10%)
gd'gf 5 :
Pp6f sf]6sf] cª\lst d"No ?= 3500 k5{ . k;n]n] 15% 5'6 lbof] eg],
(i) 5'6 /sd slt xf]nf < (ii) ljj|mo d"No slt xf]nf <

27
;dfwfg M
oxf“, sf]6sf] cª\lst d"No (M.P.) = ?= 3500
5'6 (d) = 15%
(i) 5'6 /sd = M.P. sf] d%
15
= ?= 3500 sf] 15% = 3500 × 10 = ?= 525A E A

∴ 5'6 /sd = ?= 525

(ii) S.P. = ?

xfdLnfO{ yfxf 5,
SP = MP – discount = ?= 3500 – ?= 525
∴ ljj|mod"No (S.P.) = ?= 2975

cEof;sf nflu k|Zg

ca, dflysf] pbfx/0fsf] ;xfotfaf6 tnsf k|Zgx¿ xn ug]{ k|of; ug'{xf];\ .
1. Pp6f df]6/;fOsnsf] cª\lst d"No ?= 120000 k5{ . b;}+sf] pknIodf 15% 5'6 lbOPsf] eP
;f]sf] d"No slt xf]nf < -pTt/ M ?= 102000)
2. Pp6f j:t'sf] jf:tljs d"Nodf 15% a9fO cª\lst d"Nodf ?= 1380 sfod ul/of] ;f] j:t'sf]
jf:tljs d"No kTtf nufpg'xf];\ . -pTt/ M ?= 1200)
3. Pp6f SofNs'n]6/sf] cª\lst d"No ?. 260 5 . 5% 5'6 lb“bf ljj|mo d"No slt xf]nf <
-pTt/ M ?= 247)
4. /htn] Pp6f j:t' 25% 5'6 lbO{ ?= 2250 df a]Rof] eg] o;sf] cª\lst d"No slt xf]nf <
-pTt/ M 3000)
gd'gf 6 :
?= 7500 d"No n]lvPsf] Pp6f 6]lnlehg ?= 5700 df laj|mL ul/of] eg] 5'6 k|ltzt kTtf nufpg'xf];\ .
;dfwfg M
oxf“, /]l8of]sf] cª\lst d"No (M.P.) = ?= 7500
/]l8of]sf] ljj|mo d"No (S.P.) = ?= 5700
5'6 /sd (d) = MP – SP = ?= 7500 – ?= 5700 = ?= 1800
d 1800
∴ 5'6 k|ltzt (d %) = M.P.
A × 100 = 7500 × 100 = 24
E A A E A

∴ 5'6 k|ltzt ( d %) = 24%

cEof;sf nflu k|Zg

ca, dflysf] pbfx/0fsf] ;xfotfaf6 tnsf k|Zgx¿ xn ug]{ k|of; ug'{xf];\ .
!= ?= 15,000 d"No n]lvPsf] Pp6f Pn l; 8L nfO{ ?= 12,000 df laj|mL ul/of] eg] 5'6 k|ltzt kTtf
nufpg'xf];\ . -pTt/ M 20%_
@= slt k|ltzt 5'6 lbP/ lalqm ubf{ ?= 38,000 c+lst d"No ePsf] :df6{ kmf]gnfO{ ?= 32,300 df laj|mL
ug{ ;lsG5 <-pTt/ M 15%_
28
gd'gf 7 :
Pp6f b/fhdf df 15% 5'6 kfP/ ¿ 7650 df lslgof] eg] b/fhsf] cª\lst d"No kTtf nufpg'xf];\ .
;dfwfg M
oxF“,
b/fhdf 5'6 (D) = 15%
b/fhsf] ljj|mo d"No (SP) = ¿ 7650
xfdLnfO{ yfxf 5,
S.P = M=P – 5'6
cyjf, ?= 7650 = MP – M=P sf] 15%
cyjf, ?= 7650 = M=P – M=P
cyjf, ?= 7650 =
cyjf, M.P = ?= 9000
ca, dflysf] pbfx/0fsf] ;xfotfaf6 tnsf ;"qx¿ xn ug]{ k|of; ug'{xf];\ .
1= ;'hgn] Pp6f SofNs'n]6/ 8 % 5'6 kfO{ ?= 888 df lsGof] eg] pSt SofNs'n]6/sf] cª\lst
d"No slt lyof] < -pTt/ M ?= 965.21_
2= Pp6f ;fdfgsf] ljj|mo d"No ?= 9680 5 . olb cª\lst d"Nodf 12% 5'6 lbP/ laj|mL ul/G5
eg] To; ;fdfgsf] cª\lst d"No kTtf nufpg'xf];\ . -pTt/ M ?= 11,000_
3= s'g} j:t'sf] ljj|mo d"No ?= 320 5 . olb To;sf] cª\lst d"Nodf 29 % 5'6 lbP/ laj|mL
ul/Psf] eP cª\lst d"No slt lyof] xf]nf < -pTt/ M ?= 450.70_
4= s]zjn] Pp6f sf]6 245 % 5'6 kfP/ ?= 5625 df lsg] eg] sf]6sf] cª\lst d"No
lgsfNg'xf];\ . -pTt/ M ?= 7500_

gd'gf 8 :
Pp6f d"lt{sf] cª\lst d"No ?= 8,500 5 . 10 % VAT ;lxt ;f] d"lt{sf] ljj|mo d"No kTtf
nufpg'xf];\ .
;dfwfg M
oxf“
d"lt{sf] cª\lst d"No -M.P) ?= 8,500
VAT = 10%
ljj|mo d"No -S.P) = <
xfdLnfO{ yfxf 5,
SP = M=P + M=P sf] 10%
= ?= 8500 +?= 8500 x
= ?= 9350
ca, dflysf] pbfx/0fsf] ;xfotfaf6 tnsf ;"qx¿ xn ug]{ k|of; ug'{xf];\ .
29
1= Pp6f cfO/gsf] cª\lst d"No ? 4200 5 . olb 13% VAT hf]8]/ laj|mL ubf{ ;f] cfO/gsf]
d"No slt k5{ < -pTt/ M ?= 4746_
2. 15% VAT ;lxt cª\lst d"No ?= 17,500 ePsf] s'g} /]lk|mlh/]6/sf] ljj|mo d"No slt k5{ <
-pTt/ M ?= 20,125_
3= Pp6f lx6/ Uof;sf] cª\lst d""No ? 7,800 cª\lst ul/Psf] 5 / 10% VAT tf]lsPsf] 5
eg] j|m]tfn] pSt Uof;sf] d"No slt ltg'{k5{ < -pTt/ M ?= 8,580_

gd'gf 9 :
olb lbksn] Pp6f df]afOn kmf]g 13 % VAT lt/L ?= 4,800 df lsGof] eg] df]afOn kmf]gsf] cª\lst
d"No slt
lyof] <
;dfwfg M
oxfF,
df]afOn kmf]gsf] ljj|mo d"No (S=P) = ?= 4,800
VAT = 135
cª\lst d"No (M=P) = <
cª\lst d"No (M=P) = ?= x -dfgf+}_
k|Zgg';f/,
S.P = M=P + VAT
cyjf, ?= 4800 = x + x sf] 13%
cyjf, ?= 4800 = x + x x
cyjf, ?= 4800 =
cyjf, x = ?= 4247=79
ctM pSt df]afOn kmf]gsf] cª\lst d"No ?= 4247=78 /x]5 .

ca, dflysf] pbfx/0fsf] ;xfotfaf6 tnsf ;"qx¿ xn ug]{ k|of; ug'{xf];\ M

1= olb /ljgn] 6]lnlehg 10% VAT ;lxt ?= 9625 df lsg]/ NofP eg] 6]lnlehgsf] cª\lst
d"No slt xf]nf < -pTt/ M ?= 8,750_
2. s'g} j:t'sf] ljj|mo d"No ?= 15255 5 . olb ;f] j:t'df 13% VAT nfU5 eg] j:t'sf] cªlst
d"No lgsfNg'xf];\ . -pTt/ M ?= 13,500_
3. s'g} j:t'sf] d"Nodf 15 % VAT lt/L ?= 172.50 df laj|mL ul/of] eg] VAT afx]ssf] d"No
lgsfNg'xf];\ . -pTt/ M ?= 150_

30
@ nfdf] pTt/ cfpg] k|Zgx¿ ;DaGwL cEof;
gd'gf 1 :
s'g} ;fdfg ?= 1620 df laj|mL ubf{ 10% gf]S;fg x'G5 eg] 5% gfkmf ug{ pSt ;fdfg s'g d"Nodf
a]Rg'knf{ <
;dfwfg M
oxf“, klxnf] cj:yf S.P. = ?= 1620 , gf]S;fg = 10%
bf];f] cj:yf S.P. = ?, gfkmf = 5%
klxnf] cj:yfaf6
CP sf] (100% – 10%) = ?= 1620
cyjf, CP × 90% = ? 1620
90
cyjf, CP × 100
A = ?=1620
E A

cyjf, CP = ?= 1620 × 100 90 A E

∴ C.P. = ?= 1800
bf];|f] cj:yfaf6,
CP sf] (100% + 5%) = S.P.
cyjf, 1800 × 105% = SP
cyjf, 1800 × 105
A

100 = S.P.
E A

∴ SP = ?= 1890
ctM 10% gfkmf ug{ ;f] ;fdfg ?= 1890 df a]Rg'k5{ .
cEof;sf nflu k|Zg
ca, dflysf] pbfx/0fsf] ;xfotfaf6 tnsf k|Zgx¿ xn ug]{ k|of; ug'{xf];\ .
1. olb Pp6f ;fdfgnfO{ ?= 23 df laj|mL ul/G5 eg], 15% gfkmf x'G5 . 10% sf] gfkmf agfpg pSt
;fdfg sltdf laj|mL ug'{k5{ < kTtf nufpg'xf];\ . -pTt/ M ?= 22)
2. Pp6f 38L ?= 2700 df a]Rbf 10% gf]S;fg x'G5 . pSt 38Lsf]
-s_ j|mo d"No slt xf]nf < -pTt/ M ?= 3000)
-v_ 20% gfkmf ug{ sltdf a]Rg'knf{ < (pTt/ M ?= 3600)
3. Pp6f l;nfO d]l;g ?= 12690 df laj|mL ubf{ 6% gfkmf x'G5 . 4% gfkmf lng ;f] d]l;g sltdf
a]Rg'knf{ < -pTt/ M ?= 12450.57)
gd'gf 2 :
A n] Pp6f /]l8of] B nfO{ a]r]5 . B n] ;f] /]l8of] C nfO{ 10% gfkmf lnP/ a]r]5 . olb C n] B nfO{
?= 726 lt/]sf] eP B n] A nfO{ slt lt/]5 <
;dfwfg M
oxf“, B n] A ;“u lsg]sf] 38Lsf] d"No = x dfgf}“
k|Zgaf6,
x + x sf] 10% = ?= 726

31
 10
cyjf, x + x × 100 = ?= 726
A A

11x 10
cyjf, 10 = ?= 726 × 11
A

E
A A

cyjf, x = ?= 660
ctM B n] A nfO{ lt/]sf] d"No = ?= 660.

cEof;sf nflu k|Zg

ca, dflysf] pbfx/0fsf] ;xfotfaf6 tnsf k|Zgx¿ xn ug]{ k|of; ug'{xf];\ .
1. Pp6f sDkgLn] ?= 12000 df sfk]{6 tof/ u5{ . ;f] sDkgLn] l8n/nfO{ 15% gfkmfdf laj|mL
u5{ . l8n/n] v'b|f Jofkf/LnfO{ 10% kmfObf / v'b|f Jofkf/Ln] u|fxsnfO{ 5% 3f6fdf laj|mL
ub{5 eg] u|fxsn] ;f] sfk]{6nfO{ slt lt5{ xf]nf < kTtf nufpg'xf];\ .
(pTt/ M ?=14421)
2. A n] Pp6f 6]lnlehgdf B nfO{ 10% kmfObfdf laj|mL u¥of] . B n] ;f] 6]lnlehg C nfO{ 20%
kmfObfdf laj|mL u¥of] . olb C n] pS6 6]lnlehg ?= 1650 df vl/b u/]sf] lyof] eg] A n]
sltdf vl/b u/]sf] lyof] xf]nf < kTtf nufpg'xf];\ . (pTt/ M ?=1250)
3. /fhgn] Pp6f lstfa ?= 180 df lsg]/ 20% gfkmfdf ;fhgnfO{ a]Rof] . ;fhgn] ;f] lstfa 20
% gf]S;fgdf lg/fhgnfO{ a]Rof] eg] lg/fhgn] 5% gfkmf lng ;f] lstfa sltdf laj|mL ug'{knf{
< kTtf nufpg'xf];\ . (pTt/ M ?= 181.44)
gd'gf 3 :
Pp6f Jofkf/Ln] b'O{ cf]6f 38L ?= 400 df lsg]5 . Pp6fdf 5% gfkmf / csf]{df 5% 3f6f x'g] u/L a]r]5 .
olb b'j} 38Lsf] ljj|mo d"No a/fa/ eP 38Lsf] Jofkf/df p;nfO{ k|fKt ePsf] gfkmf jf gf]S;fg slt x'G5
< kTtf nufpg'xf];\ .
;dfwfg M
oxf“, b'O{ cf]6f 38Lsf] j|m=d"= (C.P.) = ?= 400
klxnf] 38Lsf] j|m=d"= (C.P) = x
bf];|f] 38Lsf] j|m=d"= (C. P) = ?= (400 – x)
klxnf] 38Lsf] lj=d"= (S.P) = x + x sf] 5%
= x sf] 105%
105
= x × 100 A

21x
= 20 A

bf];|f] 38Lsf] la=d"= (S.P) = (400 – x) sf] 95%

95
= (400 – x) × 100 A

32
 19
= 20 (400 – x)
A

E
A

k|Zgcg';f/, b'j} 38Lsf] S.P. a/fa/ ePsfn],

21x 19(400 – x)
20 = 20 A

E
A A

E E

cyjf, 21x = 7600 – 19x

cyjf, 40x = 7600
7600
cyjf, x = 40 = ?=190
A

E
A

21x
hDdf S.P. = 20 × 2
A

E
A

21 × 190 × 2
=A

20 E
A

= ?= 399
hDdf j|m=d"= (C.P) = ?= 400 ∴ C.P. > S.P. ePsfn]
gf]S;fg = CP – S.P. = ?= 400 – ?= 339 = ?= 1
gf]S;fg
gf]S;fg k|ltzt = C.P × 100 A

E
A

1
= 400 × 100
A

E
A

ctM s"n gf]S;fg k|ltzt = 0.25%

cEof;sf nflu k|Zg
ca, dflysf] pbfx/0fsf] ;xfotfaf6 tnsf k|Zgx¿ xn ug]{ k|of; ug'{xf];\ .
1. Pp6f Jofkf/Ln] b'O{ cf]6f 38Lx¿ ?= 1000 df lsg]5 . p;n] Pp6fdf 20% gfkmf / csf]{df
20% 3f6f x'g] u/L a]r]5 . olb b'j} 38Lsf] ljj|mo d"No Pp6} eP p;nfO{ k|fKt ePsf] gfkmf
jf gf]S;fg k|ltzt slt xf] < lgsfNg'xf];\ . -pTt/ M 4% gf]S;fg_
2. k|sfzn] b'O{ cf]6f ;fOsnx¿ ?= 3000 df lsg]5g\ . pgn] Pp6fdf 15% gf]S;fg / csf]{df
19% gfkmf x'g] u/L tL ;fOsnx¿ laj|mL u/]5g\ . kl5 pgn] k|To]s ;fOsnsf] laj|mo d"No
Pp6} ePsf] yfxf kfP5g\ . k|To]s ;fOsnsf] j|mo d"No kTtf nufpg'xf];\ .
(pTt/ M ?= 1750, ?= 1250)
3. k|tLsn] b'O{ cf]6f sDKo'6/x¿ hDdf ?= 130000 df lsg]5g\ . Pp6fnfO{ 20% gfkmf / csf]{nfO{
12% 3f6fdf laj|mL ubf{ of] yfxf eof] ls b'j} sDKo'6/sf] ljj|mo d"No Pp6} /x]5 . k|To]s
sDKo'6/sf] j|mo d"No kTtf nufpg'xf];\ . (pTt/ M ?= 55000, ?= 75000)
gd'gf 4 :
Ps hgf Jofkf/Ln] 2 cf]6f ;fOsn k|To]ssf] ?= 10000 df a]Rbf Pp6fdf 20% gfkmf / csf]{df
20% gf]S;fg eof] eg] p;nfO{ slt k|ltzt gfkmf jf gf]S;fg eof] xf]nf <

33
;dfwfg M
oxf“, 2 cf]6f ;fOsnsf] S. P. = 2 × 10000 = ?= 20000
100
klxnf] ;fOsnsf] C.P. = ?= 10000 sf] (100 + 20) A

25000
= ?= A

3 E

100
bf];|f] ;fOsnsf] C.P. = ?= 10000 × (100 – 20) A

= ?= 12500
25000
∴ hDdf C.P = ?= 12500 + ?= A

3 E

62500 1
= ?= A

3 E
A = ?= 20833 3 A

C.P. > S.P. ePsfn],

C.P. – S.P
gf]S;fg k|ltzt = C.P × 100%
A

E
A

1
?= 20833 3 – ?= 20000
= A

1 × 100% A

?= 208333 E E

= 4%
ctM gf]S;fg k|ltzt = 4%
csf]{ tl/sf
klxnf] ;fOsn
lj=d"\ (S.P) = j|m=d" (C.P) ± gfkmf
cyjf, 1000 = CP +
cyjf, 1000 =

cyjf,
cyjf, ?=

bf];|f] ;fOsn
la=d"\ = (S.P) = j|m=d"= (CP) – gf]S;fg
cyjf, 10000
cyjf, 10000 =

34
cyjf,
cyjf ?= 12500 - CP
hDdf (CP) = ?=
= ?=
oxf“ CP > SP ePsfn]
gf]S;fg k|ltzt
=
62500
3 -20000
A E A

= A

62500 ×100% EA

3
EA

= 4%

cEof;sf nflu k|Zg

ca, dflysf] pbfx/0fsf] ;xfotfaf6 tnsf k|Zgx¿ xn ug]{ k|of; ug'{xf];\ M
1. Pp6f dflg;n] b'O{ cf]6f l;nfO{ d]l;gx¿ k|To]ssf] ?= 2970 sf] b/df a]Rbf Pp6fdf 10%
gfkmf / csf]{df 10% gf]S;fg x'G5 eg] To; sf/f]jf/df p;nfO{ slt k|ltzt gfkmf jf
gf]S;fg eof] xf]nf < kTtf nufpg'xf];\ . -pTt/ M 1 % gf]S;fg_
2. Pp6f k;n]n] 2 cf]6f /]l8of]x¿ k|To]sssf] ?= 1500 df a]Rbf Pp6fdf 30% gfkmf /
csf]{df 30% gf]S;fg eof] eg] p;nfO{ slt k|ltzt gfkmf jf gf]S;fg eof] xf]nf <
-pTt/ M 9% gf]S;fg_
3. s]zjn] 2 cf]6f lstfax¿ k|To]ssf] ?= 800 df a]Rbf Pp6fdf 40% gfkmf / csf]{df 40%
gf]S;fg eof] eg] p;nfO{ slt k|ltzt gfkmf jf gf]S;fg eof] xf]nf < kTtf nufpg'xf];\ .
-pTt/ M 16% gf]S;fg_
gd'gf 5 :
Pp6f Sofd]/fsf] cª\lst d"No ?= 3200 5 . o;nfO{ k;n]n] 8% 5'6 lbPkl5sf] d"Nodf 10%
Eof6 nufO{ lbG5 eg] pSt Sofd]/fnfO{ u|fxsn] lsGg slt ltg'{knf{ <
;dfwfg M
oxf“, cª\lst d"No (M. P.) = ?= 3200
5'6 (d) = 8% / Eof6 = 10%
5'6kl5sf] d"No = ?= 3200 – ?= 3200 sf] 8%
= ?= 3200 sf] 92 %
92
= ?= 3200 × 100
A

= ?= 2944
Eof6;lxtsf] d"No = ?= 2944 + ?= 2944 sf] 10%

35
 = ?= 2944 sf] 110%
110
= ?= 2994 × 100 A

= ?= 3238. 40
ctM ;f] Sofd]/fsf] Eof6;lxtsf] d"No = ?= 3238.40

cEof;sf nflu k|Zg

ca, dflysf] pbfx/0fsf] ;xfotfaf6 tnsf k|Zgx¿ xn ug]{ k|of; ug'{xf];\ M
1. Pp6f ;fOsnsf] cª\lst d"No ?= 5550 /flvPsf] lyof] . ;f] ;fOsndf 10% 5'6 lbO{ 15%
d"No clej[b\lw s/ nufp“bf ;f]sf] d"No slt knf{ < kTtf nufpg'xf];\ .
-pTt/ M ?= 5744.25)
2. Ps ;]6 sDKo'6/sf] cª\lst d"No ?= 65500 /flvPsf] lyof] . ;f] sDKo'6/df 12% 5'6 lbO{
15% d"No clej[b\lw s/ -VAT_ nufp“bf ;f]sf] d"No slt knf{ < kTtf nufpg'xf];\ .
-pTt/ M ?= 66286)
gd'gf 6 :
Pp6f ;fOsnsf] cª\lst d"Nodf 25% 5'6 lbO{ 15% Eof6 (VAT) nufOPsf] lyof] . olb ;f]
;fOsn ?= 3450 df a]lrPsf] lyof] eg] ;f] ;fOsnsf] cª\lst d"No slt x'G5 < kTtf nufpg'xf];\ .
;dfwfg M
dfgf}“, ;fOsnsf] cª\lst d"No = ?= x
Eof6 = 15%
5'6 = 25%
Eof6;lxtsf] d"No = ?=3450
ca, 5'6kl5sf] d"No = ?= x – ?= x sf] 25%
= ?= x sf] 75%
75
= ?= x × 100 A

3x
= ?= 4 A

3x 3x
Eof6;lxtsf] d"No = ?= A

4 +E
A ?= A

4
E
A sf] 15%
= ?=
= ?=
= ?=

cyjf

36
3450 =

cyjf,
cyjf, x - 4000
ctM ;f] ;fOsnsf] cª\lst d"No = ?= 4000.

cEof;sf nflu k|Zg

ca, dflysf] pbfx/0fsf] ;xfotfaf6 tnsf k|Zgx¿ xn ug]{ k|of; ug'{xf];\ .
1. Pp6f 38Lsf] cª\lst d"Nodf 25% 5'6 lbO{ 15% Eof6 (VAT) nufp“bf ?= 1725 eof] eg]
;f] 38Lsf] cª\lst d"No kTtf nufpg'xf];\ . -pTt/ M ? 2000)
2. Pp6f b/fhsf] cª\lst d"Nodf 25% 5'6 lbO{ 15% Eof6 (VAT) nufp“bf ?= 5085 eof] eg]
;f] b/fhsf] cª\lst d"No kTtf nufpg'xf];\ . -pTt/ M ?= 5895.65)
3. Pp6f df]afOn ;]6sf] cª\lst d"Nodf 15% 5'6 lbO{ 13% Eof6 (VAT) nufp“bf ?= 7684
eof] eg] ;f] df]afOnsf] cª\lst d"No kTtf nufpg'xf];\ . -pTt/ M ?= 8000)

gd'gf 7 :
Pp6f 38Lsf] cª\lst d"No j|mo d"NoeGbf 30% a9L /flvPsf] lyof] . olb ;f] 38LnfO{ 20% 5'6
u/L a]Rbf ?= 150 gfkmf x'g] eP5 eg] 38Lsf] cª\lst d"No / j|mo d"No kTtf nufpg'xf];\ .
;dfwfg M
dfgf}“, 38Lsf] j|mo d"No = ?= 100x
38Lsf] cª\lst d"No = ?= (100x + = ?= 130x

5'6 = 20%
gfkmf = ?= 150
5'6kl5sf] d"No = ?= 130x sf] (100 – 20)%
80
= ?= 130x 100
A

∴ S.P. = ?= 104x
gfkmf = ?= 150
cyjf, SP - CP = ?= 150
cyjf, 104x - 100x = ?= 150
cyjf, 4x = ?= 150
cyjf x = ?=
= ?= 37.50
ctM 38Lsf] j|mo d"No = ?= 100 × 37.50
= ?= 3750

37
38Lsf] cª\lst d"No = ?= 130 × 37.50
= ? 4875
cEof;sf nflu k|Zg
ca, dflysf] pbfx/0fsf] ;xfotfaf6 tnsf k|Zgx¿ xn ug]{ k|of; ug'{xf];\ .
1. s'g} ;fdfgsf] j|mo d"No eGbf 25% n] a9L cª\lst d"No sfod ul/Psf] 5 . olb 15% 5'6
lbO{ a]Rbf ?= 125 gfkmf x'G5 eg] ;f] ;fdfgsf] ljj|mo d"No slt xf]nf < kTtf nufpg'xf];\ .
-pTt/ M ?= 2125)
2. Pp6f /]l8of]sf] cª\lst d"No j|mo d"No eGbf 40% a9L /flvPsf] lyof] . ;f] /]l8of]df 30%
5'6 u/L a]Rbf ?= 100 gf]S;fg eP5 eg] ;f] /]l8of]sf] cª\lst d"No slt sfod ul/Psf] lyof]
< kTtf nufpg'xf];\ . -pTt/ M ?= 7000)
3. Pp6f Sofd]/fsf] cª\lst d"No j|mo d"NoeGbf 20% a9L /flvPsf] lyof] . ;f] /]l8of]df 10%
5'6 u/L a]Rbf ?= 208 gfkmf eP5 eg] ;f] /]l8of]sf] cª\lst d"No slt sfod ul/Psf] lyof] <
kTtf nufpg'xf];\ . -pTt/ M ?= 3120)
gd'gf 8:
Pp6f Jofkf/Ln] s'g} j:t'sf] d"No ?= 550 cª\lst u/]5 / 10% u|fxsnfO{ 5'6 lbP5 . o;/L
Jofkf/Ln] ?= 75 gfkmf ub{5 eg] cª\lst d"No j|mo d"No eGbf slt k|ltzt a9L xf]nf < kTtf
nufpg'xf];\ .

;dfwfg M
oxf“, cª\lst d"No = ?= 550
5'6 = 10%
gfkmf = ?= 7.5
ljj|mo d"No (S.P.) = ?= 550 – ? = 550 sf] 10%
= ?= 550 - ?=
= ?= 550 - ?= 55
j|mo d"No (C.P.) = S.P. – gfkmf
= ?= 495 – ?= 75
= ?= 420
cª\lst d"No – j|mo d"No = ?= 550 – ?= 420
= ?= 130
130
ca, 420 × 100% = 30. 95%
A

E
A

ctM ;f] ;fdfgsf] cª\lst d"No j|mo d"No eGbf 30.95% w]/}n] a9L 5 .
cEof;sf nflu k|Zg
ca, dflysf] pbfx/0fsf] ;xfotfaf6 tnsf k|Zgx¿ xn ug]{ k|of; ug'x{ f];\ .

38
1. k;n]n] s'g} j:t'sf] d"No ?= 4000 cª\lst u/]5 / 10% u|fxsnfO{ 5'6 lbP5 . o;/L
Jofkf/Ln] ?= 600 gfkmf ub{5 eg] slt k|ltzt ;f] j:t'sf] cª\lst d"No j|mo d"NoeGbf a9L
1
xf]nf < kTtf nufpg'xf];\ . -pTt/ M 33 3%)
A A

2. Zofdn] s'g} j:t'sf] d"No ?= 5000 cª\lst u/]5 / 15% u|fxsnfO{ 5'6 lbP5 . o;/L
Jofkf/Ln] ?= 250 gfkmf ub{5 eg] slt k|ltzt ;f] j:t'sf] cª\lst d"No j|mo d"NoeGbf a9L
xf]nf < kTtf nufpg'xf];\ . -pTt/ M 25%)
3. gf/fo0fn] s'g} j:t'sf] d"No ?= 3000 cª\lst u/]5 / 12% u|fxsnfO{ 5'6 lbP5 . o;/L
Jofk/Ln] ?= 140 gfkmf ub{5 eg] slt k|ltzt ;f] j:t'sf] cª\lst d"No j|mo d"NoeGbf a9L
xf]nf < kTtf nufpg'xf];\ . -pTt/ M 20%)
gd'gf 9 :
olb Pp6f 38L cª\lst d"Nodf laj|mL ubf{ ljj|m]tfnfO{ 20% gfkmf x'G5 t/ 5% 5'6 lb“bf
?= 140 dfq gfkmf x'G5 eg] ljj|m]tfn] pSt 38L sltdf lsg]sf] x'g'k5{ < kTtf nufpg'xf];\ .
;dfwfgM
oxf“, 38Lsf] cª\lst d"No = ?= x /
38Lsf] j|mo d"No (C. P.) = ?= y -dfgf}“_
ca, ljj|mo d"No (S.P.) = C.P + gfkmf
= y + y sf] 20%
6y
= 5
A

cª\lst d"Nodf 38L laj|mL ul/Psf] 5 . To;}n] S.P. = x

6y
∴ x= 5 A

6y
∴ cyjf, 5 × 95% – y = 140A

E
A

cyjf,
cyjf,
cyjf, =
14y
cyjf, 100 = 140
A

E
A

cyjf, y = 1000
ctM 38Lsf] j|mo d"No (C.P.) = ?= 1000
cEof;sf nflu k|Zg
ca dflysf] pbfx/0fsf] ;xfotfaf6 tnsf k|Zgx¿ xn ug]{ k|of; ug'{xf];\ M

39
1. s'g} Jofkf/Ln] Pp6f j:t'sf] cª\lst d"Nodf laj|mL ubf{ 50 % gfkmf ub{5 . olb u|fxsnfO{ 12
% 5'6 lb“bf ;f] j:t'sf] laj|mLdf ? 16 kmfObf u5{ eg] j:t'sf] j|mo d"No lgsfNg'xf];\ .
-pTt/ M ?= 50)
2. Pp6f Jofkf/Ln] s'g} j:t'nfO{ cª\lst d"Nodf laj|mL ubf{ 40 % gfkmf u5{ . olb p;n]
u|fxsnfO{ 15% 5'6 lb“bf ?= 76 gfkmf ub{5 eg] ;f] j:t'sf] cª\lst d"No kTtf nufpg'xf];\ .
-pTt/ M ?= 560)
3. Pp6f Jofkf/Ln] Pp6f ;fOsnsf] cª\lst d"Nodf laj|mL ubf{ -5'6 glb“bf_ 20 % gfkmf x'G5 t/
5% 5'6 lb“bf ?= 180 gfkmf x'G5 eg] laj|m]tfn] pSt ;fOsn sltdf lsg]sf] x'g'kb{5 < (pTt/
M ?= 545.45)
gd'gf 10 :
Ps k;n]n] s'g} ;fdfgdf cfkm"n] lsg]sf] d"Nodf 20% yk u/L cª\lst d"No lgwf{/0f ub{5 . olb
u|fxsx¿nfO{ 10% 5'6 lb“bf p;nfO{ ?= 120 gfkmf x'G5 eg] p;n] ;f] ;fdfg p;n] sltdf lsg]sf]
lyof] <
;dfwfg M
dfgf}“, j|mo d"No (C.P.) = 100x
k|Zgfg';f/, cª\lst d"No (M.P.) = 120x
ljj|mo d"No (S.P.) = M. P – 5'6 /sd
= 120x – 120x sf] 10%
= 120x × 120x ×
= 120x -12x
= 108x
ca, SP – CP = gfkmf
cyjf, 108x – 100x = 120
cyjf, 8x = 120
120
cyjf, x = 8
A

∴ x = 15
ctM 38Lsf] j|mo d"No = 100x
= 100 × 15
= ?= 1500
cEof;sf nflu k|Zg
ca dflysf] pbfx/0fsf] ;xfotfaf6 tnsf k|Zgx¿ xn ug]{ k|of; ug'{xf];\ M
1. Ps hgf kmlg{r/ k;n]n] P6f 6]a'nsf] lsg]sf] d"Nodf 25% yk u/L cª\lst d"No lgwf{/0f ub{5 .
u|fxsnfO{ 10% 5'6 lbG5 o;f] ubf{ p;nfO{ ?= 500 gfkmf x'G5 eg] p;n] ;f] ;fdfg sltdf
lsg]sf] lyof] < kTtf nufpg'xf];\ . -pTt/ M ?=4000)

40
2. Ps hgf sDKo'6/ k;n]n] Pp6f sDKo'6/sf] lsg]sf] d"Nodf 30% yk u/L cª\lst d"No lgwf{/0f
ub{5 . u|fxsnfO{ 20 % 5'6 lbG5 . o;f] ubf{ p;nfO{ ?= 1800 gfkmf x'G5 eg] p;n] ;f] sDKo'6/
sltdf lsg]sf] lyof] < kTtf nufpg'xf];\ . -pTt/ M ?= 45000)
3. Pp6f :6];g/L k;n]n] Pp6f SofNs'n]6/sf] lsg]sf] d"Nodf 20% yk u/L cª\lst d"No
/fv]5 . olb 25 % 5'6 lbof] eg] p;nfO{ ?=100 gf]S;fg x'G5 . ;f] SofNs'n]6/sf] j|mod"No
slt lyof] xf]nf < kTtf nufpg'xf];\ . -pTt/ M ?= 1000)
gd'gf 11 :
Pp6f /]l8of]sf] cª\lst d"Nodf 20 % 5'6 lbO{ ;f] /sddf 10% d"No clej[b\lw s/ (VAT) hf]8]/
a]lrof] . olb lsGg] dflg;n] ?= 320 Eof6 lt/sf] lyof] eg] slt /sd p;n] 5'6 kfPsf] lyof] <
;dfwfg M
dfgf}“, cª\lst d"No = x
SP = x - x sf] 20%
=x-

Eof6 /sd = sf] 10%

=
=

t/, = 320
cyjf, 8x = 32000
cyjf, x = 4000
∴ 5'6 =
=
= ?= 800

cEof;sf nflu k|Zg

ca dflysf] pbfx/0fsf] ;xfotfaf6 tnsf k|Zgx¿ xn ug]{ k|of; ug'{xf];\ M
1. Pp6f ;'6s];sf] cª\lst d"Nodf 10 % 5'6 lbO{ 13 % Eof6 hf]8]/ a]lrof] . olb lsGg] dflg;n] ?=
468 Eof6 lt/]sf] lyof] eg] p;n] slt /sd 5'6 kfPsf] lyof] <
-pTt/ M ?= 400)

41
2. Pp6f knª\sf] cª\lst d"Nodf 15% 5'6 lbO{ 13% Eof6 hf]8]/ a]lrof] . olb lsGg] dflg;n] ?=
636 Eof6 lt/]sf] lyof] eg] slt /sd 5'6 kfPsf] lyof] <
-pTt/ M ?. 863.34)
3. Pp6f ;fOsnsf] cª\lst d"Nodf 25% 5'6 lbO{ ;f] /sddf 10% Eof6 hf]8]/ a]lrof] . olb
5'6 /sd ?= 750 lyof] eg] ;f] ;fOsndf slt /sd Eof6 nufOPsf] lyof] <
-pTt/ M ?= 225)
gd'gf 12 :
s'g} j:t'sf] cª\lst d"No o;sf] ljj|mo d"No eGbf 25% a9L 5 . ;fy} o;sf] j|mo d"No o;}sf]
cª\lst d"No eGbf 30% sd 5 eg] 5'6 k|ltzt / gfkmf k|ltzt lgsfNg'xf];\ .
;dfwfg M
dfgf}“, laj|mo d"No (S.P.) = ?= x
MP = x sf] 25%
=x+x+
=
=
CP = sf] 30%
=
=
=
=
5'6 k|ltzt
5'6
5'6 % = M.P. × 100
A

E
A

20x
= 100x × 100
A

E
A

∴ 5'6 = 20%
gfkmf /sd = S.P. C.P = 80x – 70 x = 10x
gfkmf
gfkmf k|ltzt = CP × 100
A

E
A

10x
= 70x × 100
A

E
A

∴ gfkmf = 14.28%

42
cEof;sf nflu k|Zg
ca dflysf] pbfx/0fsf] ;xfotfaf6 tnsf k|Zgx¿ xn ug]{ k|of; ug'{xf];\ M
1. s'g} j:t'sf] cª\lst d"No o;sf] ljj|mo d"NoeGbf 25% n] a9L 5 . ;fy} o;sf] j|mo d"No
o;}sf] cª\lst d"NoeGbf 30 % sd 5 eg] 5'6 k|ltzt / gfkmf k|ltzt kTtf
2
nufpg'xf];\ . -pTt/ M 20% / 147 %)
A E A

2. s'g} j:t'sf] cª\lst d"No o;sf] ljj|mo d"NoeGbf 25 % a9L 5 . ;fy} o;sf] j|mo d"No o;}sf]
1
cª\lst d"NoeGbf 33 3 % sd 5 eg] 5'6 k|ltzt / gfkmf k|ltzt kTtf nufpg'xf];\ . pTt/
A A

M 2%, 20 %
3. s'g} j:t'sf] cª\lst d"No o;sf] ljj|mo d"NoeGbf 60 % n] a9L 5 . ;fy} o;sf] j|mo d"No
o;}sf] cª\lst d"NoeGbf 50 % n] sd 5 eg] 5'6 k|ltzt / gfkmf k|ltzt kTtf nufpg'xf];\ .
-pTt/ M 37.5% / 25 %)
1
4. s'g} j:t'sf] cª\lst d"No o;sf] ljj|mo d"NoeGbf 11 9 % n] a9L 5 . ;fy} o;sf] j|mo d"No
A A

o;}sf] cª\lst d"NoeGbf 20% n] sd 5 eg] 5'6 k|ltzt / gfkmf k|ltzt kTtf nufpg'xf];\ .
-pTt/ M 10% / 12.5 %)
gd'gf 13 :
Pp6f 38L cª\lst d"Nodf ?= 255 5'6 lbO{ 15 % VAT hf]8]/ a]lrof] . olb VAT /sd
?= 450 lyof] eg] 38Lsf] cª\lst d"No / 5'6 /sd kTtf nufpg'xf];\ .
;dfwfg M
oxf“,
38Ldf 5'6 = 25%
38Ldf VAT = 15%
VAT /sd = ?= 450
cª\lst d"No (M.P) = <
5'6 /sd = <
cª\lst d"No (M.P) = x (dfgf“)}
5'k = MP sf] 5'6 %
= x sf] 25 %
=x
=
5'6kl5sf] lj=d" = M.P – 5'6
=x–
=

43
VAT /sd = sf] 15%
cyjf, ?= 450 =
cyjf, x = ?= 4000
5'6 /sd =
ctM ;f] 38Lsf] cª\lst d"No ?= 4000 / 5'6 /sd ?= 1000 /x]5 .
cEof;sf nflu k|Zg
1. Pp6f kª\vfsf] cª\lst d"nodf 20 % 5'6 lbO{ 10 % VAT hf]8]/ a]lrof] . olb VAT /sd
?= 200 lyof] eg] kª\vfsf] cª\lst d"No / 5'6 /sd kTtf nufpg'xf];\ .
-pTt/ M ?= 2500, ?= 500)
2. Pp6f lk|mhsf] cª\lst d"Nodf 25% 5'6 lbO{ 13% d"No clej[b\lw s/ hf]8]/ lslgof] . olb
d"No clej[b\lw s/ /sd ?= 780 eP cª\lst d"No / 5'6 /sd lgsfNg'xf];\ .
-pTt/ M ?= 8000, ?= 2000)
yk cEof;sf nflu k|Zg
1. k|jL0fn] ?= 800 df Pp6f SofNs'n]6/ vl/b u/L C.P sf] 98 df laj|mL u¥of] eg] p;n] slt
A E A

k|ltzt gfkmf k|fKt u¥of] < (pTt/ M 12.5%)

2. Pp6f Sofd/f ?= 8100 df a]Rbf 10% gf]S;fg x'G5 eg] 5% gfkmf lng s'g d"Nodf
a]Rg'knf{ < kTtf nufpg'xf];\ . kTtf nufpg'xf];\ . -pTt/ M ?= 9450)
3. A n] Pp6f 6]k/]s8{/ 20% gfkmf x'g]u/L B nfO{ a]r]5 . B n] ;f] 6]kf]l/s8{/ 10% gfkmf
lnP/ C nfO{ a]r]5 . olb C n] B nfO{ ?= 1320 lt/]sf] eP A sf] j|mo d"No kTtf nufpg'xf];\ .
-pTt/ M ?= 1000)
4. Pp6f Jofkf/Ln] b'O{ cf]6f 38Lx¿ ?= 1200 df lsg]5g\ . p;n] Pp6fdf 20% gfkmf /
csf]{df 20% 3f6f x'g] u/L a]r]5g\ . b'j} 38Lsf] ljj|mo d"No Pp6} eP p;nfO{ k|fKt ePsf]
gfkmf jf gf]S;fg k|ltzt slt xf] < lgsfNg'xf];\ . -pTt/ M gf]S;fg = 4% )
5. b'O{ cf]6f sDKo'6/x¿ k|To]ssf] ? 2000 df a]Rbf Pp6fdf 50% gfkmf / csf]{df 50%
gf]S;fg eof] eg] p;nfO{ slt k|ltzt gfkmf jf gf]S;fg eof] xf]nf < -pTt/ M 25% gf]S;fg_
6. Pp6f sDKo'6/sf] cª\lst d"No ?=4000 5 . o;nfO{ k;n]n] 10% 5'6 lbPkl5 13% d"No
clej[b\lw s/ (VAT) nufOlbG5 eg] pSt sDKo'6/nfO{ u|fxsn] lsGg slt
ltg'{knf{ < kTtf nufpg'xf];\ . -pTt/ M ?= 4068)
7. Pp6f ;fdfgdf ?=200 5'6 lnP/ ?= 1300 df lslgof] eg] 5'6sf] b/ lgsfNg'xf];\ .
-pTt/ M 1313 %) A E A

8. Pp6f 38Lsf] cª\lst d"No j|mo d"No eGbf 30% a9L /flvPsf] lyof] . ;f] /]l8of]df 15% 5'6
u/L a]Rbf ?= 210 gfkmf eP5 eg] ;f] /]l8of]sf] cª\lst d"No slt sfod ul/Psf] lyof] <
-pTt/ M ?= 2600)

44
9. j?0fn] s'g} j:t'sf] d"No ?= 5000 cª\lst u/]5 / 20% u|fxsnfO{ 5'6 lbP5 . o;/L
Jofkf/Ln] ?= 1000 gfkmf ub{5 eg] slt k|ltztn] ;f] j:t'sf] cª\lst d"No j|mo d"NoeGbf
2
a9L xf]nf < kTtf nufpg'xf];\ . -pTt/ M 66 3 %)
A A

10. olb 38L cª\lst d"Nodf laj|mL ubf{ laj|m]tfnfO{ 20 % gfkmf x'G5 t/ 5% 5'6 lb“bf
?= 280 dfq gfkmf x'G5 eg] laj|m]tfn] pSt 38L sltdf lsg]sf] x'g'kb{5 <
-pTt/ M ?= 848.48)
11. Pp6f :6];g/L k;n]n] Pp6f k':tssf] lsg]sf] d"Nodf 20% yk u/L cª\lst d"No /fv]5 . olb
30 % 5'6 lbof] eg] p;nfO{ ?= 64 gf]S;fg x'G5 . ;f] k':tssf]] j|mo d"No slt lyof] xf]nf <
-pTt/ M ?= 400)
12. Pp6f lk|mhsf] cª\lst d"Nodf 20 % 5'6 lbO{ ;f] /sddf 13 % Eof6 hf]8]/ a]lrof] . olb 5'6
/sd ?= 2400 eP ;f] lk|mhdf slt /sd Eof6 nufOPsf] lyof] < -pTt/ M ?= 1248)
13. Pp6f ;fdfgsf] cª\lst d"Nodf 15 % 5'6 lbO{ 10 % VAT hf]8]/ a]lrof] . olb 5'6 /sd
?= 2025 eP cª\lst d"No / VAT /sd lgsfNg'xf];\ . -pTt/ M ?= 13500, ?= 1147.50)

kf7 M 3. rj|mLo Aofh, hg;ª\Vof j[b\lw / ld> x|f; (Compound Interest,

Population Growth and Compound Depreciation)

o; kf7df kf7\oj|md / ljlzi6Ls/0f tflnsfn] Joj:yf u/] adf]lhd rj|mLo Aofh cw{ jflif{s k|0ffnLdf
a9Ldf b'O{ jif{;Dd / jflif{s k|0ffnLdf a9Ldf b'O{ jif{;Dddf ;d:ofx¿ hg;ª\Vof j[b\lw / ld>x|f;
;DjlGwt ;d:ofx¿ ;dfj]z ul/Psf 5g\ . To;} u/L hg;ª\Vof j[b\lw / ld> x|f;sf aflif{ssf dfq
;d:ofx¿ ;dfj]; ul/Psf 5g\ . P;= Pn= ;L= k/LIffdf o; kf7;“u ;DjlGwt ljifo j:t'x¿af6 Pp6f
b'O{ cª\s ef/sf] 5f]6f] k|Zg 1fg tyf af]w txsf] / Pp6f ;d:of ;dfwfg txsf] u/L hDdf b'O{ cf]6f
hDdf 6 cª\sef/sf k|Zgx¿ ;f]Wg] Joj:yf /x]sf] 5 .
3.1 rj|mLo Aofh (Compound Interest)
1. cfwf/e"t tYo tyf tyf ;"qx¿
-s_ cfwf/e"t wf/0ffx¿
rlj|mo Aofh
s'g} wg/flz lglZrt ;dofjlw h:t} M rf}dfl;s, cw{ jflif{s, dfl;s cfpg] ld> wgn] bf];|f]
;dofjlwsf] nflu d"n wgsf ¿kdf /xL km/s Aofh lbG5 eg] o;/L cfPsf] Aofh g} rj|mLo
Aofh (Compound Interest) xf] .
csf]{ cy{df
rj|mLo ld> wg / jf:tljs d"n wglarsf] leGgtf (difference) nfO{ rj|mLo Aofh
elgG5 . ;fwf/0f Aofhsf] Aofh nfUb}g . t/ rj|mLo Aofhsf] klg Aofh nfUg] k|rng 5 .
-v_ dxTTjk"0f{ ;"qx¿

45
 R
1. rj|mLo ld> wg (A) = P (1+ 100 )T A E A

rj|mLo Aofh (C.I) = P {(1+ 100) –1)}

R T
2. A E A

oxf“, P = ;fjf“ -d"n wg_ R = Aofhb/ k|lt jif{, T = ;do jif{df x'G5 .
rj|mLo AofhnfO{ jflif{s, cw{ jflif{s rf}dfl;s cyjf dfl;s b/n] lgsfNg ;lsG5 .
3. olb rj|mLo Aofh k|To]s cfwf cfwf jif{df jf cw{ jflif{s lnOG5 eg] xfdLn] ;donfO{ bf]Aa/
agfpg'k5{ / Aofh b/nfO{ cfwf agfpg'k5{ .
R T
oxf“, C.I= P [(1+ 100)
A E A –1]) jflif{s
R1 T1
C.I = [(1+ 100) –1]) cw{ jflif{s
A E A

R 2T R
∴C.I = P [(1+ 100) –1]) hxf“, R1= 2 / T1= 2T
A E A A E A

4. jif{ / dlxgf sf] rlj|mo ld>wg

/ rlj|mo Aofh (CI) = CA - P

2. pTt/ n]Vbf Wofglbg'kg]{ s'/fx¿

1. rj|mLo Aofh ;DaGwL ;"qx¿ s]xL sl7g 5g\ . xf]l;of/Lk"j{s ofb ug'{kb{5 .
2. PsfOx¿ ;fwf/0f Aofhdf h:t} u/L k|of]u ug'{k5{ .
3. jflif{snfO{ cw{ jflif{sdf n}hf“bf ;donfO{ 2 n] u'0fg / Aofh b/nfO{ cfwf ug'{kb{5 .
4. ;dfgfGt/ ;"q agfp“bf ljz]if Wofg lbg'k5{ .
5. pTt/ n]Vbf PsfO n]Vg la;{g' x'“b}g .

3. gd'gf k|Zg / cEof;

3.1 5f]6f] pTt/ cfpg] k|Zg ;DaGwL cEof;

gd'gf 1 :
?= 8000 sf] jflif{s 15% sf] b/n] 2 jif{sf] rj|mLo ld>wg kTtf nufpg'xf];\ .
;dfwfg M
oxf“, d"n wg (P) = ?= 8000
Aofh b/ (R) = 15%
;do (T) = 2 jif{
;"qcg';f/,

46
  R T
rj|mLo ld> wg (C.A.) = P 1 + 100
E

 
A

 15 2
= ?= 8000 1 + 100
E

 
A

= ?= 8000 × 13.225
= ?= 10580
ctM rj|mLo ld> wg (C.A.) = ?= 10580

cEof;sf nflu k|Zg

ca, dflysf] pbfx/0fsf] ;xfotfaf6 tnsf k|Zgx¿ xn ug]{ k|of; ug'{xf];\ M
1. d"n wg (P) = ?= 8000, ;do (T) = 2 jif{ / Aofh b/ (R) = 10% k|lt jif{
-pTt/ M ?= 9680)
2. d"n wg (P) = ? 10000, ;do (T) = 2 jif{ / Aofh b/ (R) = 12% k|lt jif{
-pTt/ M ?= 12544)
3. d"nwg (P) = ? 15000, ;do (T) = 3 jif{ / Aofh b/ (R) = 15% k|lt jif{ -pTt/ M ?= 22813.13)
4. ;fjf“ (P) = ?= 9000, ;do (T) = 3 jif{ / Aofh b/ (R) = 8% k|lt jif{
-pTt/ M ?= 11337.41)
gd'gf 2 :
?= 5000 sf] jflif{s 10% sf] b/n] 2 jif{sf] rj|mLo Aofh kTtf nufpg'xf];\ .
;dfwfg M
oxf“, d"n wg (P) = ?= 5000
;do (T) = 2 jif{
Aofh b/ (R) = 10%
;"qcg';f/,
rj|mLo Aofh (C.I.) =
= = 1050
ctM rj|mLo Aofh (C.I.) = ?=1050
cEof;sf nflu k|Zg
ca, dflysf] pbfx/0fsf] ;xfotfaf6 tnsf k|Zgx¿ xn ug]{ k|of; ug'{xf];\ M
1. ?= 8000 sf] jflif{s 10% k|lt jif{sf] Aofh b/n] 3 jif{sf] rj|mLo Aofh kTtf nufpg'xf];\ .
-pTt/ M ?= 2648)
2. ?= 10000 sf] jflif{s 8 % k|lt jif{sf] Aofh b/n] 2 jif{sf] rj|mLo Aofh kTtf nufpg'xf];\ .
-pTt/ M ?=1664)
3. ?= 6000 sf] jflif{s 15 % k|lt jif{sf] Aofh b/n] 2 jif{sf] rj|mLo Aofh kTtf
nufpg'xf];\ . -pTt/ M ?=1935)
47
4. s[i0fn] g]kfn a}ª\s lnld6]8 6f“8L ahf/af6 rj|mLo jflif{s Aofh 10% sf b/n] ?=9000
C0f lnof] . ca 2 jif{kl5 p;n] a}ª\snfO{ slt Aofh a'emfpg'knf{ < kTtf nufpg'xf];\ .
-pTt/ M ?= 1890)
gd'gf 3 :
?= 2000 sf] jflif{s 10% Aofh b/n] 2 jif{sf] cw{ jflif{s rj|mLo Aofh kTtf nufpg'xf];\ .
;dfwfg M
oxf“, d"n wg (P) = ?= 2000
Aofh b/ (R) = 10% k|lt jif{sf]
;do (T) = 2 jif{
;"qfg';f/,
cw{ jflif{s rj|mLo Aofh (C.I.) =
 10 2×2 
= 2000 1 + 200 – 1
  
A E

= 200 × 0.21550625
= 431.0125
ctM cw{ jflif{s rj|mLo Aofh (C.I.) = ?= 431.01

cEof;sf nflu k|Zg

ca, dflysf] pbfx/0fsf] ;xfotfaf6 tnsf k|Zgx¿ xn ug]{ k|of; ug'{xf];\ M
cw{ jflif{s rj|mLo Aofh kTtf nufpg'xf];\ .
1
1. d"n wg (P) = ?= 12000, jflif{s Aofh b/ (R) = 8% / ;do (T) = 1 2 jif{ A

E
A

-pTt/ M ?= 1498.37)
2. d"n wg (P) = ?= 2000, jflif{s Aofh b/ (R) = 10% / ;do (T) = 2 jif{
-pTt/ M ?=431.01)
1
3. d"nwg (P) = ?= 1000, jflif{s Aofh b/ (R) = 10% / ;do (T) = 1 2 jif{ A A

-pTt/ M ?= 157.63)
3.2 nfdf] pTt/ cfpg] k|Zgf]Tt/ / cEof;
gd'gf 1 :
?=15000 sf] 1 jif{ 6 dlxgfsf] jflif{s 10% sf] b/n] ;fwf/0f Aofh / rj|mLo Aofh kTtf
nufpg'xf];\ .
;dfwfgM
 6 3
oxf“ d"n wg (P) = ?= 15000 ;do (T) = 1 jif{ 6 dlxgf = 1 + 12 = 2 jif{
 
A E A A A

Aofh b/ (R) = 10% c k|lt jif{

48
 PTR 15000 × 3 × 10
(i) ;fwf/0f Aofh (S.I.) = 100 =
100 × 2
A A A

E
E

= ?= 2250
 R 1 
(ii) 1 jif{sf] rj|mLo Aofh (C.I.) = P 1 + 100 – 1
  
A E

 R 1 
= 15000 1 + 100 – 1
  
A E

1
= 1500 × 10 A

= ?= 1500
af“sL 6 dlxgfsf nflu
;fjf“ (P) = ?= 15000 + ?= 1500 = ?= 16500
6 1
Aofhb/ (R) = 10% ;do (T) = 6 dlxgf = 12 = 2 jif{ A

E
A A A

P×T× R 16500 × 10
∴ Aofh = 100 = = ?= 825
100 × 2
A A A A

E
E

ctM hDdf rj|mLo Aofh = ?= 1500 + ?= 825

= ?= 2325
ctM 1 jif{ 6 dlxgfsf] rj|mLo Aofh = ?= 2325

csf]{ tl/sf
CA =
n = jif{sf] ;ª\Vof, m = dlxgfsf] ;ª\Vof
=
= ?= 17325
∴ CI = CA - P
= ?= 17325 - ?= 15000 = ?= 2325

cEof;sf nflu k|Zg

ca, dflysf] pbfx/0fsf] ;xfotfaf6 tnsf k|Zgx¿ xn ug]{ k|of; ug'{xf];\ .
1. Pshgf dflg;n] ?= 5000 jflif{s 12 % sf b/n] 3 jif{nfO{ nufgL u/]5 eg],
(i) p;n] 3 jif{sf] cGtdf slt ;fwf/0f Aofh kfpnf < -pTt/ M ?= 1800)
(ii) 3 jif{sf] cGtdf slt rlj|mo Aofh kfpnf < -pTt/ M ?= 2024.64)
2. jflif{s 20 % Aofhb/df ;Ltfn] lgzf;“u ?= 5500 ;fk6L lnOl5g\ eg] 2 jif{sf] cGTodf
;fwf/0f Aofh / rj|mLo Aofh lgsfNg'xf];\ . -pTt/ M ?= 2200, ?= 2420)

49
3. jflif{s 21 k|ltzt Aofhb/df ;Gtf]ifn] ;'/]z;“u ?= 1,30,000 ;fk6L lnP5 eg] 3 jif{sf]
cGTodf x'g] ;fwf/0f Aofh / jflif{s rj|mLo Aofh lgsfNg'xf];\ .
-pTt/ M ?= 81900, ?= 100302.93)
4. jflif{s 21 k|ltzt Aofh b/df ;Gtf]ifn] ;'/]z;“u ?= 1,50,000 ;fk6L lnP5 eg] 2 jif{sf] cGTodf
x'g] ;fwf/0f Aofh / jflif{s rj|mLo Aofh lgsfNg'xf];\ . pTt/ M ?= 63000, ?= 69615

gd'gf 2 :
?= 4000 sf] jflif{s 10 % Aofh b/n] 1 jif{sf] cGTodf ;fwf/0f Aofh / cw{ jflif{s rj|mLo Aofh
slt slt xf]nf < kTtf nufpg'xf];\ .
;dfwfg M
oxf“, d"n wg (P) = ?= 40000
Aofh b/ (R) = 10% k|lt jif{
;do (T) = 1 jif{
P×T×R
(i) ;fwf/0f Aofh (S. I.) = 100 A

E
A

4000 × 1 × 10
= A

100 E

= ?= 400

(ii) ;"qaf6, cw{jflif{s rj|mLo Aofh (C.I.) =

=
41
= 4000 × 400 = ?= 410
A

E
A

ctM ;fwf/0f Aofh = ?= 400 / cw{jflif{s rj|mLo Aofh = ?= 410

cEof;sf nflu k|Zg

ca, dflysf] pbfx/0fsf] ;xfotfaf6 tnsf k|Zgx¿ xn ug]{ k|of; ug'{xf];\ .
1. ?= 5000 sf] jflif{s 10% Aofh b/n] 1 jif{sf] cGTodf ;fwf/0f Aofh / cw{ jflif{s rj|mLo
Aofh kTtf nufpg'xf];\ . -pTt/ M ?= 500, ?= 512.5)
1
2. ?= 8000 sf] jflif{s 8% Aofhb/n] 12 jif{sf] cGTodf ;fwf/0f Aofh / cw{jflif{s rj|mLo
A

E
A

Aofh kTtf nufpg'xf];\ . -pTt/ M ?= 960, ?= 998.91)

3. jflif{s 10 % Aofh b/df /fdn] ;Ltf;“u ?= 4800 ;fk6L lnP5 eg] 1 jif{sf] cGTodf
(i) pgn] ;fwf/0f Aofh slt ltg'{knf{ < -pTt/ M ?= 480)
(ii) cw{ jflif{s rj|mLo Aofh slt ltg'{knf < -pTt/ M ?= 492)

50
4. jflif{s 12 % Aofhsf b/n] /fdn] Zofd;“u ?=4250 ;fk6L lnP5 eg] Ps jif{sf] cGTodf
(i) /fdn] slt ;fwf/0f Aofh ltg'{knf{ < -pTt/ M ?= 510)
(ii) cw{ jflif{s rj|mLo Aofh slt ltg'{knf{ < -pTt/ M ?= 525.30)
gd'gf 3 :
?= 5000 sf] 3 jif{sf] 10% jflif{s Aofh b/n] ;fwf/0f Aofh / rj|mLo Aofhdf slt km/s knf{
< kTtf nufpg'xf];\ .
;dfwfg M
oxf“, d"n wg (P) = ?= 5000
;do (T) = 3 jif{
Aofhb/ (R) = 10% k|lt jif{
PTR
(i) ;fwf/0f Aofh (S.I.) = 100 A

E
A

5000 × 3 × 10
= 100
A

= ?= 1500
 R T 
(ii) rj|mLo Aofh (C.I.) = 1 + 100 – 1
  
A E

 10  3

= 1 + 100 – 1
  
A E

331
= 5000 × 1000 A

= ?= 1655
ctM ;fwf/0f Aofh / rj|mLo Aofhdf km/s ?= 1655 – ?= 1500 = ?= 155

cEof;sf nflu k|Zg

ca, dflysf] pbfx/0fsf] ;xfotfaf6 tnsf k|Zgx¿ xn ug]{ k|of; ug'{xf];\ .
1. ?= 7500 sf] jflif{s 10% Aofh b/n] 2 jif{df x'g] rlj|mo Aofh / ;fwf/0f Aofhsf] km/s kTtf
nufpg'xf];\ < -pTt/M– ?= 75)
2 .?= 5120 sf] 3 jif{sf] 12.5% jflif{s Aofh b/n] x'g] rlj|mo Aofh / ;fwf/0f Aofhdf slt
km/s knf{ < -pTt/ M ?= 250)
3. ?= 18000 sf] jflif{s 15 % Aofhb/df 2 jif{df x'g] ;fwf/0f Aofh / jflif{s rj|mLo Aofhdf
x'g] km/s kTtf nufpg'xf];\ < -pTt/ M ?= 405)
gd'gf 4 :
s'g} /sdsf] k|ltjif{ 5% Aofhsf] b/n] 2 jif{df ldl>t Aofh / ;fwf/0f Aofhsf] km/s 120 eP ;f]
/sd slt xf]nf < kTtf nufpg'xf];\ .
;dfwfg M
oxf“, Aofh b/ (R) = 5% k|lt jif{, ;do (T) = 2 jif{, ;fjf“ (P) = ?
51
 PTR P × 2 × 5 P
;"qcg';f/, ;fwf/0f Aofh = 100 = A

100 = 10
E
A A

E
A A

rj|mLo Aofh CI =
41 41P
= P × 400 = 400
A

E
A A

k|Zgfg';f/, C. I. – S. I. = ?= 120
41P P
cyjf, 400 – 10 = ?= 120
A

E
A A

E
A

41 P – 40P
cyjf, A

400 = ?= 120 E
A

P
cyjf, 400 = ?= 120
A

E
A

∴ P = ?= 48000.
ctM ;f] /sd ?= 48000 /x]5 .

cEof;sf nflu k|Zg

ca, dflysf] pbfx/0fsf] ;xfotfaf6 tnsf k|Zgx¿ xn ug]{ k|of; ug'{xf];\ .
1. jflif{s 20 % Aofh b/n] 2 jif{df x'g] s'g} wgsf] ldl>t Aofh / ;fwf/0f Aofhsf] km/s
Rs. 400 x'G5 eg] ;f] wg kTtf nufpg'xf];\ < -pTt/ M ?= 10,000)
2. s'g} ?lkof“sf] jflif{s 15%sf b/n] 3 jif{df rj|mLo Aofh b/ / ;fwf/0f Aofhdf km/s
?= 283.5 x'G5 eg] ;fjf“ kTtf nufpg'xf];\ . -pTt/ M ?= 4000)
3. s'g} ?lkof“sf] jflif{s 8% b/n] 2 jif{df rj|mLo Aofh / ;fwf/0f Aofhdf km/s ?= 32 x'G5
eg] ;fjf“ kTtf nufpg'xf];\ . -pTt/ M ?= 5000)
gd'gf 5 :
s'g} /sdsf] 2 jif{df x'g] ;fwf/0f Aofh rj|m:o AofheGbf ?= 90 n] sd 5 . olb jflif{s Aofhb/
15% sfod ul/Psf] lyof] eg] d"n wg slt x'G5 < kTtf nufpg'xf];\ .

;dfwfgM
oxf“,
;do (T) = 2 jif{
Aofhb/ (R) = 15%
d"nwg (P) = ?
k|Zgcg';f/,
CI – SI = 90

52
  R T 
cyjf, P 1 + 100 – 1 –
  
A E A

 15  2
 P × 2 × 15
cyjf, P 1 + 100 – 1 – 100 = 90
  
A E A A A

129 3
cyjf, P × 400 – P × 10 = 90
A

E
A A

E
A

129P – 120P
cyjf, A

400 = 90 E
A

9P
cyjf, 400 = 90
A

E
A

400
cyjf, P = 90 × 9 = 4000 A

E
A

ctM d"nwg = ?= 4000.

cEof;sf nflu k|Zg
ca, dflysf] pbfx/0fsf] ;xfotfaf6 tnsf k|Zgx¿ xn ug]{ k|of; ug'{xf];\ .
1. k|ltjif{ 10% Aofhsf] b/n] 2 jif{df s'g] /sdsf] ;/n;fwf/0f Aofh rj|mLo AofheGbf
?=360 sd x'g cfp“5 eg] d"n wg kTtf nufpg'xf];\ . -pTt/ M ?= 36000)
2. k|ltjif{ 15% Aofhsf b/n] 2 jif{df s'g} /sdsf] ;fwf/0f Aofh rj|mLo AofheGbf
?= 180 sd x'g cfp“5 eg] d"n wg kTtf nufpg'xf];\ . -pTt/ M ?= 8000)
3. k|ltjif{ 10% Aofhsf b/n] 2 jif{df s'g} /sdsf] ;fwf/0f Aofh rj|mLo AofheGbf
?= 420 sd x'g cfp“5 eg] d"n wg kTtf nufpg'xf];\ . -pTt/ M?= 42000)
gd'gf 6 :
s'g} wg/flzsf] 10% jflif{s Aofhsf] b/n] b'O{ jif{df rj|mLo Aofh ?= 420 x'G5 eg] TolTts}
;dosf nflu pxL g} jflif{s Aofhb/n] ;f]xL ;fjf“df nfUg] ;fwf/0f Aofh slt x'G5 < lgsfNg'xf];\ .
;dfwfgM
oxf“, Aofhb/ (R) = 10% k|lt jif{
;do (T) = 2 jif{
;fjf“ (P) = ?
;"qcg';f/,
 R T 
rj|mLo Aofh = P 1 + 100 – 1
  
A E

cyjf, 420 = P
cyjf, 420 = P
cyjf, 420 = P

53
 21
cyjf, 420 = P × 100 A

420 × 100
cyjf, A

21 E
=P A

∴ P = ?= 2000

km]l/, ;"qcg';f/
PTR
;fwf/0f Aofh (SI) = 100 A

?= 2000 × 2 × 20
100 = A

= ?= 200
ctM ;fwf/0f Aofh (C.P.) = ?= 200

cEof;sf nflu k|Zg

ca, dflysf] pbfx/0fsf] ;xfotfaf6 tnsf k|Zgx¿ xn ug]{ k|of; ug'{xf];\ M
1. ?= 5000 sf] 2 jif{df jflif{s 20% sf b/n] x'g] rlj|mo Aofh slt xf]nf < ;f]xL b/df ;f] /sdsf]
;fwf/0f Aofhcg'¿k TolTt g} Aofh kfpg slt ;do nfUnf < -pTt/ M ?= 2200, 2.2 jif{_
2. jflif{s 10% Aofhb/df 2000 sf] 2 jif{df ldl>t Aofh slt xf]nf . ;f]xL /sdsf] Tolt g} Aofh
;f]xL cjlwdf kfpg ;fwf/0f Aofh b/ slt sfod ug'{knf{ <-pTt/ M ?=420, 10.5% k|lt jif{_
3. s'g} wg/fzLsf] 10% jflif{s Aofhsf] b/n] 2 jif{df rj|mLo Aofh ?= 840 x'G5 eg] TolTts}
;dosf nflu pxL g} Aofh b/n] ;f]xL ;fjfdf nfUg] ;fwf/0f Aofh lgsfNg'xf];\ .
-pTt/ M ?= 800)
gd'gf 7 :
jflif{s rj|mLo Aofh k|0ffnLcg';f/ s'g} /sdsf] ld> wg 2 jif{df ?= 7260 / 3 jif{df ?= 7986
k'U5 eg] ;fjf“ / Aofhsf] b/ kTtf nufpg'xf];\ .
;dfwfgM
oxf“, 2 jif{sf] ld> wg (A1) = ?= 7260 / 3 jif{sf] ld> wg (A2) = ?= 7986
dfgf}“, ;"qcg';f/,
 R T
A = P 1 + 100 E

 
A A

 R 2
T = 2 jif{ x'“bf, A1 = P 1 + 100 E

 
A

 R 2
∴ 7260 = P 1 + 100 ……………….. (i) E

 
A A

 R 3
T = 3 jif{ x'“bf, A2 = P 1 + 100 E

 
A

54
  R 3
∴ 7986 = P 1 + 100 ………………… (ii) E

 
A A

cj ;dLs/0f (ii) nfO{ ;= (i) n] efu ubf{–

 R 3
P 1 + 100
7986  
7260 = 
A A

R 2
A A

P 1 + 100
E

 
E E

R 11
cyjf, 1 + 100 = 10 A A

E
A

R 11
cyjf, 100 = 10 – 1
A

E
A A

E
A

R 1
cyjf, 100 = 10
A

E
A A

∴ R = 10% k|lt jif{

R sf] dfg ;= (i) df /fVbf
 10 2
7260 = P 1 + 100 E

 
A

121
cyjf, 7260 = P × 100 A

100
cyjf, P = 7260 × 121 A

∴ P = ?= 6000
ctM ;fjf“ = ? 6000 / Aofhb/ 10% k|lt jif{

cEof;sf nflu k|Zg

ca, dflysf] pbfx/0fsf] ;xfotfaf6 tnsf k|Zgx¿ xn ug]{ k|of; ug'{xf];\ M
1= jflif{s rj|mLo Aofh b/df s'g} wg/fzLsf] 2 jif{ / 3 jif{sf] cGTodf j|mdzM ld> wgx¿ ?=
26460 / ?= 27783 x'G5 eg] rj|mLo Aofhb/ / pSt wg/fzL kTtf nufpg'xf];\ .
-pTt/ M 5%, ?= 240900)
2. k|ltjif{ 20% Aofhsf b/n] s'g} /sdsf] rlj|mo jflif{s Aofh 3 jif{df ;fwf/0f AofheGbf ?=
384 n] a9L x'g cfp“5 eg] d"nwg lgsfNg'xf];\ . -pTt/ M ?= 3000)
3 jflif{s ldl>t Aofhcg';f/ 2 jif{ / 3 jif{df s'g} /sdsf] ld>wgx¿ j|mdzM ?=12100 /
13310 k'Ub5 eg] d"nwg / Aofhb/ kTtf nufpg'xf];\ . -pTt/ M ?= 10,000, 10%)

gd'gf 8 :
?= 10000 sf] 2 jif{df cw{jflif{s rj|mLo Aofhb/df rj|mLo Aofh ?= 4641 x'G5 eg] Aofhsf] b/
kTtf nufpg'xf];\ .
55
;dfwfg M
oxf“, d"nwg (P) = ?=10000
;do (T) = 2 jif{
Aofhb/ (R) = ?
cw{jflif{s ld>Aofh (C. I.) = ?= 4641
 R 2T 
;"qcg';f/, C. I. = P 1 + 100 – 1
  
A E

14641  R 4
cyjf, 10000  200 – 1
=  1 +
E

A A A A

4641 R 4
cyjf, 1 + 10000 = 1 + 200
A A

E
A
E

4641 R 4
cyjf, 10000 = 1 + 200
A A

E
A
E

11 4 R 4
cyjf, 10 = 1 + 200
A E A A
E

R 11
cyjf, 1 + 200 = 10 – 1
A A

E
A A

R 1
cyjf, 200= 10
A A

E
A

∴ R = 20%

cEof;sf nflu k|Zg

ca, dflysf] pbfx/0fsf] ;xfotfaf6 tnsf k|Zgx¿ xn ug]{ k|of; ug'{xf];\ M
1
1. ?= 5000 sf] 1 2 jif{df cw{jflif{s rj|mLo Aofh ?= 624.32 x'G5 eg] Aofhsf] b/ kTtf A A

nufpg'xf];\ . -pTt/ M 8% k|lt jif{_

1
2. ?= 8000 sf] 1 2 jif{df cw{jflif{s rj|mLo Aofh ?= 998.912 x'G5 eg] Aofhsf] b/ kTtf A A

nufpg'xf];\ . -pTt/ M 8% k|lt jif{_

3. ?= 12000 sf] 2 jif{df cw{jflif{s rj|mLo Aofh b/df rj|mLo Aofh ?= 2586.075 eP Aofhsf]
b/ kTtf nufpg'xf];\ . -pTt/ M 10%)

4. ?= 15000 sf] 2 jif{df cw{jflif{s rj|mLo Aofh b/df rj|mLo Aofh ?= 3232.59375 eP
Aofhsf] b/ kTtf nufpg'xf];\ . -pTt/ M 10% k|lt jif{_

yk cEof;sf nflu k|Zg

56
ca, dflysf] pbfx/0fsf] ;xfotfaf6 tnsf k|Zgx¿ xn ug]{ k|of; ug'{xf];\ M
1. ?= 12000 sf] jflif{s 20% Aofh b/df 2 jif{df x'g] ;fwf/0f Aofh / jflif{s rj|mLo Aofhdf
x'g] km/s kTtf nufpg'xf];\ < -pTt/ M ?= 480)
2. jflif{s 10% Aofhsf b/n] 2 jif{df x'g] s'g} wg /fzLsf] rj|mLo Aofh / ;fwf/0f Aofhsf]
km/s ?. 20 x'G5 eg] ;f] wg kTtf nufpg'xf];\ . -pTt/ M ?= 2000)
3. k|ltjif{ 5% Aofhsf b/n] 2 jif{df s'g} /sdsf] ;fwf/0f Aofh rj|mLo AofheGbf
?= 120 sd 5 eg] d"n wg kTtf nufpg'xf];\ . (pTt/ M ?= 48000)
4. s'g} wg/flzsf] 15% jflif{s Aofhsf] b/n] 2 jif{df rj|mLo Aofh ?= 1935 x'G5 eg] TolTts}
;dosf nflu pxL g} Aofhb/n] ;f]xL ;fjf“df nfUg] ;fwf/0f Aofh lgsfNg'xf];\ .
-pTt/ M ?= 1800)
5. jflif{s rj|mLo Aofh k|0ffnLcg';f/ s'g} /sdsf] ld> wg 2 jif{df ?= 6050 k'Ub5 / 3 jif{df
?= 6655 k'Ub5 eg] ;f] /sd slt lyof] . -pTt/ M ?= 5000)
kf7 M 3.1.2 hg;ª\Vof j[b\lw / ld> x|f;
(Population Growth and Compound Depreciation)
1. cfwf/e"t tYo tyf ;"qx¿
-s_ cfwf/e"t wf/0ffx¿
-c_ hg;ª\Vof j[b\lw
Pp6f lglZrt ;dofjlwdf hg;ª\Vofdf ePsf] ;fk]lIft j[b\lwnfO{ hg;ª\Vof j[b\lw
elgG5 .
-cf_ ;fwf/0f x|f; (Simple Depreciation)
s'g} j:t'sf] d"Nodf k|To]s jif{ jf lglZrt ;dosf] cGt/fndf Pp6} b/ (same rate)
df x|f; cfp“5 eg] To:tf] x|f;nfO{ ;fwf/0f x|f; elgG5 .
-O_ ld> x|f; (Compound depreciation)
s'g} j:t'sf] d"Nodf k|To]s jif{df lglZrt ;dodf x'g] ;fk]lIfs x|f;nfO{ ld> x|f;
elgG5 .
-v_ dxTTjk"0f{ a'“bf tyf ;"qx¿
1. dfgf}+, R % n] k|ltjif{ hg;ª\Vofdf x'g] j[b\lwsf b/n] ;'?sf] hg;ª\VofnfO{ hgfp“5 eg] T
jif{kl5sf] hg;ª\Vof
P1 =P ( 1+ 100
A
R T
) E

hxf“ P / PT n] j|mdzM z'?sf] / T jif{ kl5sf] hg;ª\VofnfO{ hgfp“5g\ .

2. olb R % k|ltjif{df x'g] hg;ª\Vof x|f;sf] b/ eP T jif{kl5sf] hg;ª\Vof
PT =P ( 1+ 100
A
R T
) E

57
3. olb R1 % n] Pp6f lglZrt ;dosf] j[b\lw b/ / R% n] csf]{ lglZrt ;dosf] j[b\lw b/nfO{
hgfp“5 eg] T jif{sf] hg;ª\Vof
PT =P ( 1+ 100
A
R T
) 1 1
E A x P1 =P ( 1+ 100
A
R T
2
) 2
E x'G5 .
A

4. olb ‘P’ ;'?sf] d"No / ‘F’ clGtd d"No xf] eg] jflif{s ;fwf/0f x|f;nfO{ o;/L lgsflnG5
P–F
;fwf/0f x|f; (D) = n A E

hxf“, n = ;fdfgsf] cg'dflgt l6sfp jif{ ;ª\Vof

o;nfO{ x|f;sf] a/fa/ ls:tf (equal installment) tl/sf klg elgG5 .
P, PT, R / T n] j|mdM z'?sf] d"No, T jif{ kl5sf] d"No, x|f; b/ / ;do jif{df nfO{ hgfp“5 eg]
x'G5 .

-u_ pTt/ n]Vbf Wofg lbg'kg]{ s'/fx¿

1. ;dfgfGt/ ;"q agfp“bf ;Demg'kb{5 .
2. pTt/ n]Vbf PsfO n]Vg la;{g' x“'b}g .

3. gd'gf k|Zgf]Tt/ / cEof;

3.1. 5f]6f] pTt/ cfpg] k|Zg ;DaGwL cEof;
gd'gf 1 :
s'g} ufp“sf] hg;ª\Vof 2 jif{ klxn] 5400 lyof] . Toxf“sf] hg;ª\Vof j[b\lw b/ 5 % 5 eg] xfnsf]
hg;ª\Vof kTtf nufpg'xf];\ .
;dfwfg M
;'?sf] hg;ª\Vof (P) = 5400
hg;ª\Vof j[blwb/ (R) = 5%
xfnsf] hg;ª\Vof (PT) = ?
;do (T) = 2 jif{
 R T
;"qcg';f/, xfnsf] hg;ª\Vof (PT) = P 1 + 100 E

 
A

R 2

cyjf P2 = 540 1 + 100 E

 
A

2
21
= 5400 × 20
E

 
A

441
= 5400 × 400 A

∴ xfnsf] hg;ª\Vof (PT) = 5934 nueu

cEof;sf nflu k|Zg

58
ca, dflysf] pbfx/0fsf] ;xfotfaf6 tnsf k|Zgx¿ xn ug]{ k|of; ug'{xf];\ M
1. s'g} ufp“sf] hg;ª\Vof 2 jif{ klxn] 7000 lyof] . Toxf“sf] hg;ª\Vof j[b\lw b/ 3.1 % 5 eg]
xfnsf] hg;ª\Vof lgsfNg'xf];\ < -pTt/M– 7441 hgf_
2. 2 jif{ klxn] Pp6f ufp“sf] hg;ª\Vof 16000 lyof] . pSt ufp“sf] hg;ª\Vof j[b\lw b/ 5% 5
eg] clxn]sf] hg;ª\Vof slt xf]nf < -pTt/ M17640 hgf_
3. Pp6f ;x/sf] hg;ª\Vof k|To]s jif{ 3 k|ltztsf] b/n] j[b\lw x'G5 . olb clxn]sf] hg;ª\Vof
480000 eP b'O{ jif{kl5 pSt ;x/sf] hg;ª\Vof slt xf]nf < -pTt/ M 50923 hgf_
gd'gf 2 :
Pp6f ufp“sf] hg;ª\Vof Ps jif{ klxn] 10,000 lyof] . clxn]sf] hg;ª\Vof 10,210 5 eg]
hg;ª\Vof j[b\lw b/ slt x'G5, lgsfNg'xf];\ .
;dfwfg M
;'?sf] hg;ª\Vof = 10000
xfnsf] hg;ª\Vof (P) = (PT) = 10210
hg;ª\Vof j[b\lw b/ (R) = ?
;do (T) = 1 jif{
;"qcg';f/,
 R T
PT = P 1 + 100 E

 
A

 R 2
cyjf, 10210 = 10000 1 + 100 E

 
A

 R  2 1021
cyjf, 1 + 100 = 1000 E

 
A A A

R 21
cyjf, 100 = 1000
A

E
A A

21
cyjf, R = 10 A

∴ R = 2.1
ctM cfjZos hg;ª\Vof j[b\lw b/ = 2.1%
cEof;sf nflu k|Zg
ca, dflysf] pbfx/0fsf] ;xfotfaf6 tnsf k|Zgx¿ xn ug]{ k|of; ug'{xf];\ .
1. Pp6f ufp“sf] 1 jif{ klxn]sf] hg;ª\Vof 8000 lyof] . olb xfnsf] hg;ª\Vof 8400 eP 5 eg]
hg;ª\Vof j[b\lw b/ slt x'G5, kTtf nufpg'xf];\ . -pTt/ M 5%)
2. 2062 ;fndf wflbª a]+;Lsf] hg;ª\Vof 20,000 lyof] . olb 2063 ;fndf Toxf“sf] hg;ª\Vof
21000 eP hg;ª\Vof j[b\lw b/ kTtf nufpg'xf];\ . -pTt/ M 5)
3. /Tggu/sf] 2062 ;fnsf] hg;ª\Vof 40000 lyof] . olb 2063 ;fndf Toxf“sf] hg;ª\Vof
416000 eP hg;ª\Vof j[b\lw b/ kTtf nufpg'xf];\ . -pTt/ M 4% )
gd'gf 3 :
59
Pp6f d]l;gsf] d"Nodf k|To]s jif{ 10 % x|f; s6\6f x'“b} hfG5 eg] ?= 24000 kg]{ Pp6f d]l;gsf] 2
jif{kl5sf] d"No kTtf nufpg'xf];\ .
;dfwfg M
oxf“, x|f; b/ (R) = 10%
;'?sf] d"No (P) = ?=24000
2 jif{kl5sf] d"No (P2) = ?
 R T
;"qcg';f/, T jif{kl5sf] d"No (P1) = P 1 – 100 E

 
A

 10  2
∴ 2 jif{kl5sf] d"No (P2) = 2400 1 – 100 E

 
A

81
= 24000 × 100 A

= 19440
ctM ;f] d]l;gsf] 2 jif{kl5sf] d"No = ?=19440

cEof;sf nflu k|Zg

ca, dflysf] pbfx/0fsf] ;xfotfaf6 tnsf k|Zgx¿ xn ug]{ k|of; ug'{xf];\ .
1. ? 10,000 kg]{ d]l;gsf] d"Nodf 10% x|f; s6\6f x'G5 eg] ;f] d]l;gsf] b'O{ jif{kl5 x'g] d"No
kTtf nufpg'xf];\ . -pTt/ M ?= 8100)
2.. Pp6f ufp“sf] hg;ª\Vof 64,000 5 . 5% k|lt jif{sf] b/n] hg;ª\Vof j[b\lw x'G5 . b'O{
jif{kl5sf] hg;ª\Vof slt x'g]5 < -pTt/ M 70560)
3.. ?= 12000 kg]{ d]l;gsf] d"Nodf k|ltjif{ 10 % x|f; s6\6f x'G5 eg] ;f] d]l;gsf] b'O{ jif{ kl5
x'g] d"No kTtf nufpg'xf];\ . -pTt/ M ?= 9720
gd'gf 4 :
k|ltjif{ 15 k|ltzt x|f; s6\6f u/L Pp6f :s'6/ Ps jif{kl5 ?= 72250 df a]lrof] eg] pSt :s'6/
sltdf lslgPsf] lyof] < kTtf nufpg'xf];\ .
;dfwfg M
x|f; s6\6f b/ (R) = 15%
;do (T) = 1 jif{
x|f; kl5sf] d"No (PT) = ?= 72250
;'?sf] d"No (P) = ?
;"qcg';f/,
PT = P
 5 
cyjf, 72250 = 1 – 100
 
A E

60
 85
cyjf, 72250 = P × 100 A

100
cyjf, P = 72250 × 85 A

∴ P = 85000
ctM ;'?sf] d"No (P) = ?= 85000

cEof;sf nflu k|Zg

ca, dflysf] pbfx/0fsf] ;xfotfaf6 tnsf k|Zgx¿ xn ug]{ k|of; ug'{xf];\ M
1. k|ltjif{ 5 % x|f; s6\6f u/L Pp6f df]6/;fOsn Ps jif{kl5 ?= 57,000 df a]lrof] eg] pSt
df]6/;fOsn sltdf lslgPsf] lyof] < -pTt/ M ?= 60,000)
2. k|ltjif{ 12% x|f; s6\6f u/L Pp6f df]6/;fOsn Ps jif{kl5 ?= 30,976 df a]lrof] eg] pSt
df]6/;fOsn sltdf lslgPsf] lyof] < -pTt/ M ?= 35200)
3. k|ltjif{ 8% x|f; s6\6f u/L Pp6f df]6/;fOsn Ps jif{kl5 ?= 76176 df a]lrof] eg] pSt
df]6/;fOsn sltdf lslgPsf] lyof] < -pTt/ M ?= 82800)
gd'gf 5 :
?= 90000 kg]{ Pp6f df]6/;fOsnsf] a;]{lg 10% x|f; s6\6f ul/G5 eg] 2 jif{kl5sf] x|f; s6\6f
kTtf nufpg'xf];\ .
;dfwfg M
;'?sf] d"No (P) = ?= 90000
d"No x|f; b/ (R) = 10%
;do (T) = 2 jif{
x|f; s6\6f /sd = ?
;"qcg';f/,

 R T
x|f;s§f /sd = P 1 – 1 – 100 

 
A E

  10 2
= 90000 1 – 1 – 100 
  
A E

 81 
= 90000 1 – 100
 
A E

19
= 90000 × 100 A

= 17100
ctM x|f; s6\6f /sd = ?= 17100

cEof;sf nflu k|Zg

ca, dflysf] pbfx/0fsf] ;xfotfaf6 tnsf k|Zgx¿ xn ug]{ k|of; ug'{xf];\ M
61
1. ?= 5000 kg]{ Pp6f ;fOsnsf] a;]{lg 33% x|f; s§f ul/G5 eg] 2 jif{ kl5sf] x|f; s§f kTtf
nufpg'xf];\ . -pTt/ M ?= 2755.5)
2. ?= 75000 kg]{ Pp6f 6fOk/fO6/sf] a;]{lg 10% x|f; s§f ul/G5 eg] 2 jif{ kl5sf] x|f; s§f
kTtf nufpg'xf];\ . -pTt/ M ?= 14250)
3. ?= 80000 kg]{ Pp6f d]l;gsf] a;]{lg 20% x|f; s6\6f ul/G5 eg] 2 jif{kl5sf] x|f; s6\6f
kTtf nufpg'xf];\ . -pTt/ M ?= 28800)

gd'gf 6 :
Pp6f d]l;gsf] d"No 2 jif{df ?= 32000 af6 36]/ ?= 25920 sfod eP5 eg] jflif{s x|f; k|ltzt
kTtf nufpg'xf];\ .
;dfwfg M
;do (T) = 2 jif{
d]l;gsf] ;'?sf] d"No (P) = ?= 32000
2 jif{kl5sf] d"No (P2) = ?= 25920
x|f; b/ (R) = ?
;"qcg';f/,
 R T
PT = P 1 – 100 E

 
A

 R 2
cyjf, 25920 = 32000 1 – 100 E

 
A

25920  R 2
cyjf, 32000 = 1 – 100 E

 
A A A

 9 2  R 2
cyjf, 10 = 1 – 100
E
E

   
A A A

9 R
cyjf, 10 = 1 – 100
A

E
A A

R 9
cyjf, 100 = 1– 10
A

E
A A

R 1
cyjf, 100 = 10 A

E
A A

E
A

∴ R = 10%
ctM d"No x|f; b/ = 10% k|lt jif{
cEof;sf nflu k|Zg
ca, dflysf] pbfx/0fsf] ;xfotfaf6 tnsf k|Zgx¿ xn ug]{ k|of; ug'{xf];\ M
1. ?= 450000 df lsg]sf] Pp6f sf/ Ps jif{kl5 ?= 405000 df a]lrof] eg] slt k|ltzt x|f;
s6\6f ul/Psf] lyof] < -pTt/ M 10%)

62
2. ?= 90000 df lsg]sf] Pp6f df]6/;fOsn Psjif{ kl5 ?= 81000 df a]lrPsf] lyof] eg]
To;df slt k|ltzt x|f; s6\6f ul/Psf] lyof] < -pTt/ M 10%)
3. ?=. 200000 df lslgPsf] Pp6f 6]Dk' 1 jif{kl5 ?= 170000 df a]lrPsf] lyof] eg] slt
k|ltzt x|f; s6\6f ul/Psf] lyof] < -pTt/ M 10%)

3.2 nfdf] pTt/ cfpg] k|Zgf]Tt/ / cEof;

gd'gf 1 :
Pp6f ufp“sf] hg;ª\Vof k|To]s jif{ 5% sf] b/n] a9\b} hfG5 . olb 2 jif{sf] cGTodf 1025 hgf
a;fO“ ;/]/ cGoq hf“bf ;f] ufp“sf] hg;ª\Vof 10000 eof] eg] ;'?df ;f] ufp“sf] hg;ª\Vof slt
lyof] <
;dfwfg M
oxf“, hg;ª\Vof j[b\lw b/ (R) = 5%
;do (T) = 2 jif{
2 jif{sf] cGTodf ufp“sf] hg;ª\Vof = 1025 + 10000
PT = 11025
 R T
;"qcg';f/, PT = P 1 + 100 
 
A E

 R 2 
cyjf, 11025 = P 1 + 100 
 
A E

441
cyjf, 11025 = P × 400 A

11025 × 400
cyjf, A

441 E
=P A

∴ P = 10000
ctM ;'?sf] hg;ª\Vof = 10,000 hgf pTt/

cEof;sf nflu k|Zg

ca, dflysf] pbfx/0fsf] ;xfotfaf6 tnsf k|Zgx¿ xn ug]{ k|of; ug'{xf];\ M
1. Pp6f ;x/sf] hg;ª\Vof k|To]s jif{ 10 k|ltzt a9\b} hfG5 . olb b'O{ jif{sf] cGTodf ;f] ;x/sf]
hDdf hg;ª\Vof 30000 k'u]sf] lyof] . olb ;f] ;ª\Vofdf 5800 hgf clGtddf a;fO“ ;/fOaf6
ylkPsf lyP eg] ;'?sf] hg;ª\Vof slt lyof] < -pTt/ M 20000 hgf_
2. /Tggu/sf] hg;ª\Vof k|To]s jif{ 5 % a9\b} hfG5 . olb b'O{ jif{sf] cGTodf ;f] ;x/sf] hDdf
hg;ª\Vof 15000 k'u]sf] lyof] . olb ;f] ;ª\Vofdf 3975 hgf clGtddf a;fO“ ;/fOsf]
ylkPsf lyP eg] ;'?sf] hg;ª\Vof slt lyof] < -pTt/ M 10000 hgf_

63
3. sf7df8f}“sf] hg;ª\Vof k|To]s jif{ 10% a9\b} hfG5 . olb b'O{ jif{sf] cGTodf ;f] ;x/sf] hDdf
hg;ª\Vof 700000 k'u]sf] lyof] . olb ;f] ;ª\Vofdf 95000 hgf clGtddf a;fO“ ;/fOsf]
ylkPsf lyP eg] ;'?sf] hg;ª\Vof slt lyof] < -pTt/ M 20000 hgf_

gd'gf 2 :
2057 ;fnsf] ;'?df sf7df8f}“ pkTosfsf] hg;ª\Vof 10,00,000 / hg;ª\Vof j[b\lw b/ 4.5%
lyof] . 2058 ;fnsf] ;'?df 25000 dflg;x¿ cGoqaf6 oxf“ a;fO“ ;/fO u/L oxf“ :yfoL a;f]af;
ug{ cfP eg] 2060 ;fnsf] ;'?sf] hg;ª\Vof slt xf]nf <
;dfwfgM
2057 ;fnsf] hg;ª\vof = 1000000
hg;ª\Vof j[b\lw b/ = 4.5%
2058 ;fndf ylkPsf] dflg;sf] ;ª\Vof = 25000
2060 ;fnsf] hg;ª\Vof = ?
2058 ;fnsf] hg;ª\Vof = 1000000 + 1000000 sf] 4.5 % + 25000
= 10,00000 + 45000 + 25000
= 10, 70, 000
2060 ;fnsf] hg;ª\Vof lgsfNgsf nflu
;'?sf] hg;ª\Vof (P) = 10 70000
;do (T) = 2060 – 2058 = 2 jif{
j[b\lw b/ (R) = 4.5%
;"qcg';f/
 R T
PT = P 1 + 100
 
A E

 4.5 2
= 107000 1 + 100
 
A E

209 T
= 1070000 × 200
E

 
A

= 107000 × 1.092025
= 1168467 nueu
ctM 2060 ;fnsf] hg;ª\Vof = 1168467 hgf

cEof;sf nflu k|Zg

ca, dflysf] pbfx/0fsf] ;xfotfaf6 tnsf k|Zgx¿ xn ug]{ k|of; ug'{xf];\ M
1. Pp6f ufp“sf] 2060 ;fnsf] hg;ª\Vof 10000 lyof] / hg;ª\Vofdf j[b\lw b/ 5 % lyof] .
2061 ;fnsf] ;'?df Tof] ufp“df 500 dflg; cGoqaf6 a;fO“ ;/fO u/L :yfoL a;f]af; ug{
cfP eg] 2063 ;fnsf] hg;ª\Vof kTtf nufpg'xf];\ . -pTt/ M 12127)

64
2. Pp6f ufp“sf] 2059 ;fnsf] hg;ª\Vof 20000 lyof] / hg;ª\Vofdf j[b\lw b/ 5 %
lyof] . 2060 ;fnsf] ;'?df Tof] ufp“df 9192 dflg; cGoqaf6 a;fO“ ;/fO u/L :yfoL
a;f]af; ug{ cfP eg] 2062 ;fnsf] hg;ª\Vof kTtf nufpg'xf];\ . -pTt/ M 31212)
3. Pp6f ;x/sf] hg;ª\Vof k|To]s jif{ 4% n] a9\5 . olb xfnsf] hg;ª\Vof 50000 eP,
(a) ltg jif{ klxn] ;f] ;x/sf] hg;ª\Vof slt lyof] xf]nf < -pTt/ M 44450 hgf_
(b) 1 jif{kl5 slt x'G5 xf]nf < -pTt/ M 52000 hgf_
4. Pp6f ;x/sf] hg;ª\Vof 1622400 5 . 2 jif{ klxn] ;f] ;x/sf] hg;ª\Vof 1500000 lyof] .
;f] ;x/sf] 1 jif{kl5sf] hg;ª\Vof ;f]xL j[b\lw b/cg';f/ kTtf nufpg'xf];\ .
-pTt/ M 1687296 hgf_
gd'gf 3 :
Ps hgf dflg;n] ?= 44100 lt/]/ lsg]sf] Pp6f sDKo'6/ 2 jif{sf] k|of]ukl5 ?= 40000 df laj|mL
u¥of] eg] pSt sDKo'6/sf] ld> x|f; b/ kTtf nufpg'xf];\ .
;dfwfg M
sDKo'6/sf] ;'?sf] d"No (P) = ?= 44100
;do (T) = 2 jif{
2 jif{ kl5sf] d"No (P2) = 40000
ld> x|f; b/ (R) = ?
;"qcg';f/
 R T
PT = P 1 – 100
E

 
A

R 2 
cyjf, P2 = 44100 1 – 100
E

 
A

 R 2
cyjf, 40000 = 44100 1 – 100
E

 
A

40000  R  2
cyjf, 44100 = 1 – 100
E

A A A

400  R 2
cyjf 441  100
=  1 –
E

A A A

202  R 2
cyjf, 21 = 1 – 100
E E

   
A A A

20 R
cyjf, 21 = 1 – 100
A A

E
A

R 20
cyjf, 100
A = 1 – 21
A

E
A

R 1
cyjf, =
100 21
A A

E
A

65
 100
cyjf, R = 21
A

16
ctM ld> x|f; b/ (R) = 4 21%
A A

cEof;sf nflu k|Zg

ca, dflysf] pbfx/0fsf] ;xfotfaf6 tnsf k|Zgx¿ xn ug]{ k|of; ug'{xf];\ M
1. ?= 5000 df lslgPsf] Pp6f d]l;g ?=2560 df 3 jif{kl5 laj|mL ul/of] eg] ldl>t x|f; b/
kTtf nufpg'xf];\ . -pTt/ M 5%)
2. ?= 5000 df lslgPsf] Pp6f :s'6/ 3 jif{sf] k|of]ukl5 ?= 36450 df laj|mL ul/of] eg] ;f]
:s'6/sf] d"Nodf ePsf] ldl>t x|f; b/ kTtf nufpg'xf];\ . -pTt/ M 10%)
3. ?= 9000 df lslgPsf] Pp6f df]6/;fOsn 3 jif{sf] k|of]ukl5 ?= 65610 df laj|mL ul/of] eg]
;f] df]6/;fOsnsf] d"Nodf ePsf] ldl>t x|f; b/ kTtf nufpg'xf];\ . -pTt/ M 10%)

yk cEof;sf nflu k|Zg

tnsf k|Zgx¿ ;dfwfg ug'{xf];\ M
1. ljb'/ gu/kflnsfsf] hg;ª\Vof 50000 5 . olb jflif{s hg;ª\Vof j[b\lw b/ 4 % eP 2
jif{kl5sf] hg;ª\Vof kTtf nufpg'xf];\ . -pTt/ M 54080 hgf_
2. ;Nofg6f/sf] hg;ª\Vof j|mdzM 12000 / 12240 eP hg;ª\Vof j[b\lw b/ kTtf nufpg'xf];\ .
-pTt/ M 2%)
3. ?= 2500 kg]{ 38Lsf] d"Nodf Ps jif{df 15 % x|f; s6\6f x'G5 eg] b'O{ jif{kl5 ;f] 38Lsf]
d"No kTtf nufpg'xf];\ . -pTt/ M ?= 1806.25)
4. k|ltjif{ 6% x|f; s6\6f u/L Pp6f df]6/;fOsn Ps jif{kl5 ?= 70688 df a]lrof] eg] pSt
df]6/;fOsn sltdf lslgPsf] lyof] < -pTt/ M ?= 75200)
5. ?= 500000 kg]{ Pp6f sf/sf] a;]{lg 10% x|f; s6\6f ul/G5 eg] 2 jif{kl5sf] x|f; s6\6f
kTtf nufpg'xf];\ . -pTt/ M ?= 95000)
6. xl/n] ?= 5500 kg]{ Pp6f ;fOsn Ps jif{kl5 ? 5060 df a]Rof] eg] To;df slt k|ltzt x|f;
s6\6f ul/Psf] lyof] < -pTt/ M 8%)
7. Pp6f ;x/sf] hg;ª\Vof k|To]s jif{ 8% a9\b} hfG5 . olb b'O{ jif{sf] cGTodf ;f] ;x/sf] hDdf
hg;ª\Vof 40000 k'u]sf] lyof] . olb ;f] ;ª\Vofdf 50080 hgf clGtddf a;fO“ ;/fOsf]
ylkPsf lyP eg] ;'?sf] hg;ª\Vof slt lyof] < -pTt/ M 30000 hgf_

66
8. Pp6f sDKo'6/sf] 2060 ;fnsf] d"No ?= 80000 lyof] . olb ;f] sDKo'6/ 2063 ;fndf vl/b
ubf{ ?= 58320 df kfOG5 eg] pSt sDKo'6/df ePsf] ldl>t x|f; b/ kTtf nufpg'xf];\ .
-pTt/ M 10%)
gd'gf 4 :
Pp6f b/fhdf df 15% 5'6 kfP/ ¿ 7650 df lslgof] eg] b/fhsf] cª\lst d"No kTtf nufpg'xf];\ .
;dfwfg M
oxF“,
b/fhdf 5'6 (D) = 15%
b/fhsf] ljj|mo d"No (SP) = ¿ 7650
xfdLnfO{ yfxf 5,
S.P = M=P – 5'6
cyjf, ?= 7650 = MP – M=P sf] 15%
cyjf, ?= 7650 = M=P – M=P
cyjf, ?= 7650 =
cyjf, M.P = ?= 9000
ca, dflysf] pbfx/0fsf] ;xfotfaf6 tnsf ;"qx¿ xn ug]{ k|of; ug'{xf];\ .
1= ;'hgn] Pp6f SofNs'n]6/ 8 % 5'6 kfO{ ?= 888 df lsGof] eg] pSt SofNs'n]6/sf] cª\lst
d"No slt
lyof] <
2= Pp6f ;fdfgsf] ljj|mo d"No ?= 9680 5 . olb cª\lst d"Nodf 12% 5'6 lbP/ laj|mL ul/G5
eg] To; ;fdfgsf] cª\lst d"No kTtf nufpg'xf];\ .
3= s'g} j:t'sf] ljj|mo d"No ?= 320 5 . olb To;sf] cª\lst d"Nodf 29 % 5'6 lbP/ laj|mL
ul/Psf] eP cª\lst d"No slt lyof] xf]nf <
4= s]zjn] Pp6f sf]6 245 % 5'6 kfP/ ?= 5625 df lsg] eg] sf]6sf] cª\lst d"No
lgsfNg'xf];\ .
gd'gf 5 :
Pp6f d"lt{sf] cª\lst d"No ?= 8,500 5 . 10 % VAT ;lxt ;f] d"lt{sf] ljj|mo d"No kTtf
nufpg'xf];\ .
;dfwfg
oxf“
d"lt{sf] cª\lst d"No -M.P) ?= 8,500
VAT = 10%
ljj|mo d"No -S.P) = <
xfdLnfO{ yfxf 5,
SP = M=P + M=P sf] 10%

67
= ?= 8500 +?= 8500 x
= ?= 9350
ca,
1= Pp6f cfO/gsf] cª\lst d"No ? 4200 5 . olb 13% VAT hf]8]/ laj|mL ubf{ ;f] cfO/gsf]
d"No slt k5{ <
2. 15% VAT ;lxt cª\lst d"No ?= 17,500 ePsf] s'g} /]lk|mlh/]6/sf] ljj|mo d"No slt k5{
<
3= Pp6f lx6/ Uof;sf] cª\lst d""No ? 7,800 cª\lst ul/Psf] 5 / 10% VAT tf]lsPsf] 5
eg] j|m]tfn] pSt Uof;sf] d"No slt ltg'{k5{ <

gd'gf 6 :
olb lbksn] Pp6f df]afOn kmf]g 13 % VAT lt/L ?= 4,800 df lsGof] eg] df]afOn kmf]gsf] cª\lst
d"No slt
lyof] <
;dfwfg
oxf“,
df]afOn kmf]gsf] ljj|mo d"No (S=P) = ?= 4,800
VAT = 135
cª\lst d"No (M=P) = <
cª\lst d"No (M=P) = ?= x -dfgf+}_
k|Zgg';f/,
S.P = M=P + VAT
cyjf, ?= 4800 = x + x sf] 13%
cyjf, ?= 4800 = x + x x
cyjf, ?= 4800 =
cyjf, x = ?= 4247=79
ctM pSt df]afOn kmf]gsf] cª\lst d"No ?= 4247=78 /x]5 .

ca, dflysf] pbfx/0fsf] ;xfotfaf6 tnsf ;"qx¿ xn ug]{ k|of; ug'{xf];\ M

1= olb /ljgn] 6]lnlehg 10% VAT ;lxt ?= 9625 df lsg]/ NofP eg] 6]lnlehgsf] cª\lst
d"No slt xf]nf <
2. s'g} j:t'sf] ljj|mo d"No ?= 15255 5 . olb ;f] j:t'df 13% VAT nfU5 eg] j:t'sf] cªlst
d"No lgsfNg'xf];\ .
3. s'g} j:t'sf] d"Nodf 15 % VAT lt/L ?= 172.50 df laj|mL ul/of] eg] VAT afx]ssf] d"No
lgsfNg'xf];\ .
68
gd'gf 7 :
cd[tn] Pp6f SofNs'n]6/ 10 % gfkmf u/L ljsf;nfO{ a]Rof] ljsf;n] 10 % gf]S;fgdf k/L ;f]
SofNs'n]6/ rGb|nfO{ a]Rof] . olb rGb|n] ;f] SofNs'n]6/ 500 df lsg]sf] eP cd[tsf] j|mo d"No
lgsfNg'xf];\ .
;dfwfg
oxf“
cd[tsf] gfkmf = 1%
ljsf;sf] gf]S;fg = 10%
rGb|sf] j|m=d" (C=P) = ?= 500
cd[tsf] j|m=d" (C=P) = <
ca, cd[tsf] j|m=d" (C=P) = x dfgf+}
cd[tn] ljsf;nfO{ a]r]sf] x'“bf,
ljsf;sf] j|m=d" = cd[tsf] j|m=d" + gfkmf
= x + x sf] 10%
=x+xx

=
km]l/, ljsf;n] rGb|nfO{ a]r]sf] x'“bf,
rGb|sf]s j|m=d" = ljsf;sf] j|m=d" – gf]S;fg
cyjf, ?= 500 = – sf] 10%
cyjf, ?= 500 = –
cyjf, ?= 500
cyjf, 99x = ?= 50000
cyjf, x = ?= 505.05
ctM cd[tn] pSt SofNs'n]6/ / 505=05 df lsg]sf] /x]5 .
1. 'A' n] Pp6f ;fOsn 15% gfknf u/L 'B' nfO{ laj|mL u¥of] 'B' n] 10% gf]S;fgdf 'C' nfO{
laj|mL u¥of] . olb C n] ;f] ;fOsn ?= 7500 df lsg]sf] eP 'A' sf] j|mo d"No kTtf
nufpg'xf];\ .
2= chon] ljhonfO{ 20 % gfkmf u/L ljhon] uf]kfnnfO{ 20 % gf]S;fg u/L a]r]sf]
;fOsnnfO{ uf]kfnn] ?= 13,440 df lsg]sf] /x]5 eg] chon] ;f] ;fOsn sltdf lsg]sf]
/x]5 <
3= dLgfn] zfGtfnfO{ Pp6f ;f8L 10 % gf]S;fgdf a]lrg\ . zfGtfn] ;f]xL ;f8L /fwfnfO{ 210 %
gfkmf u/L a]lrg\ . olb /fwfn] ;f] ;f8L ?= 4556 df lslgg\ eg] ldgfn] sltdf lsg]sL
/lx5g\ < lgsfNg'xf];\ .
69
gd'gf 8 :
;'zLnfn] b'O{ cf]6f jf:s]6 an ?= 3200 df lslgg\ .
klxnf]nfO{ 10 % gfkmfdf / bf];|f]nfO{ 10 % gf]S;fgdf a]lrg\ . olb b'j}sf] ljj|mo d"No a/fa/ eP
k|To]ssf] j|mo d"No lgsfNg'xf];\ .
;dfwfg M
oxf“,
b'O{ cf]6f jf:s]6 ansf] j|m=d" (C=P) = ?= 3200
klxnf] af:s]6 ansf] gfkmf = 10%
bf];|f] af:s]6 ansf] gf]S;fg = 10%
klxnf] af:s]6 ansf] j|m=d" (C=P) = x dfgf}
bf];|f] af:s]6 ansf] j|m=d" =(C=P) = ¿ 3200 – x
k|Zgfg';f/,
klxnf] af:s]6 ansf] lj=d" = bf];|f] af:s]6 ansf] lj=d"
cyjf, x + x sf] 10% = (?= 3200 - x) – (?= 3200 – x) x
cyjf, =
cyjf, 20x = ?= 28800
cyjf, x = ?= 1440
ctM klxnf] af:s]6 ansf] j|m=d" = ?= 1440
/ bf];|f] af:s]6ansf] j|m=d" = ?= 3200 – ?= 1440
= ?= 1760
1= ;~hon] b'O{cf]6f ;fO{sn ¿ 14,000 df lsGof] . klxnf] ;fO{snnfO{ 15% gf]S;fgdf /
bf];|f] ;fO}snnfO{ 15% gfkmf u/L a]Rof] . olb b'j}sf] ljj|mo d"No a/fa/ eP k|To]s
;fO{snsf] j|mod"No lgsfNg'xf];\ .
2= lblnkn] b'Ocf]6f kfgL tfGg] kDk ?= 9000 df lsg]/ Pp6f 20% gfkmf / csf]{ 20%
gf]S;fgdf a]r]5 . olb b'j}sf] ljj|mo d"No Pp6} eP b'j}sf] j|mod"No lgsfNg'xf];\ . s'n
sf/f]af/df ePsf] gfkmf jf gf]S;fg k|ltzt klg kTtf nufpg'xf];\ .
3= e"jgn] b'O{cf]6f l6=le ¿ 30,000 df lsg]/ Pp6fnfO{ 10% gfkmf / csf{nfO{ 10%
gf]S;fgdf a]r]5 . olb b'j}sf] lj=d" a/fa/ eP s'n sf/f]af/df ePsf] gfkmf jf gf]S;fg
k|ltzt kTtf nufpg'xf];\ .
gd'gf 9 :
;ª\udn] b'O{ cf]6f HoldtLo afs; k|To]ssf] ?= 180 df a]Rof] . p;nfO{ klxnf] afs;df 20 % gfkmf
/ bf];|f]df 20 % gf]S;fg eP5 eg] p;nfO{ gfkmf jf gf]S;fg s] eP5 < k|ltztdf lgsfNg'xf];\ .
;dfwfg M
oxf“,
b'O{ cf]6f HofldtLo afs;sf] lj=d" (S=P) =?= 180 x 2 = ?= 360

70
klxnf] afs;sf] gfkmf = 20 %
bf];|f] afs;sf] gf]S;fg = 205
20 % gfkmf cg';f/,
klxnf] afs;sf] j|m=d" (C.P) =
=
= ?= 150
20% gf]S;fg cg';f/,
bf];|f] afs;sf] j|m=d" (C=P) =
–

= ?=
–
= ?= 225
b'O c{ f]6f afs;sf] hDdf j|m=d" (C=P) = ¿ 150 + ¿ 225
= ?= 375
ca, C.P>S=P
To;}n] gf]S;fg = S.P = ¿ 375 – ?= 360 = ?= 15
ctM
gf]S;fg % = jf:tljs gf]S;fg x 100%
j|m=d"
=
= 4%
1= xg'dfg ;fx'n] b'O{ cf]6f ;f8L k|To]ssf] ?= 7,500 df af]r]5 . o;af6 p;nfO{ klxnf]
;f8Ldf 15 % gfkmf / bf];|f]df 15 % gf]S;fg eP5 eg] o; sf/f]af/af6 p;nfO{ ePsf]
gfkmf jf gf]S;fg k|ltzt kTtf nufpg'xf];\ .
2= s'df/Lkf6L l6=eL= ;]G6/n] b'O{ cf]6f l6=eL= k|To]ssf] ?= 12,000 df a]r]5 . o;af6 p;nfO{
Pp6fdf 105 gfkmf / k|ltzt s] eP5 < lgsfNg'xf];\ .
3= jhfh ;f]¿dn] b'O{ cf]6f df]6/;fOsn k|To]ssf] ?= 1,25000 sf b/n] a]r]5 . o;/L laj|mL
ubf{ o;nfO{ klxnf]df ePsf] gfkmf jf gf]S;fg k|ltzt kTtf nufpg'xf];\ .
gd'gf 10 :
Pp6f ;'k/ dfs]{6df ePsf] k'm6andf / 5000 cª\lst ul/Psf] d"No slt x'G5 < kTtf nufpg'xf];\ .
;dfwfg M
oxfF,
k'm6ansf] cª\lst d"No (M=P) = ?= 5000
5'6 = 10 %
VAT = 135

71
lj=d" (S=P) = M=P – 5'6
= ?= 5000 – ?= 5000 sf] 10%
= ?= 5000 – ?= 5000 x
= ?= 4500
13% VAT cg';f/,
VAT ;lxtsf] lj=d" = 5'6kl5sf] lj=d" + VAT
= ?= 4500 + ?= 4500 sf] 13%
= ?= 4500 + ?= 4500 x
= ?= 5085
ctM pSt k"m6ansf] hDdf d"No ?= 5085 k5{ .
1. Ps hgf r:df k;n]n] ?= 1150 cª\lst u/]sf] Pp6f r:dfnfO{ 10 % 5'6 lbO{ 13 %
VAT hf]8]/ laj|mL ub{5 eg] ;f] r:dfsf] laj|mL d"No lgsfNg'xf];\ .
2= ?= 2100 cª\lst ul/Psf] :s'n emf]nfdf 20 % 5'6 lbO{ 15 % VAT hf]8]/ a]Rbf cfpg]
emf]nsf] laj|mL d"No kTtf nufpg'xf];\ .
3= ?= 6900 cª\lst ul/Psf] /]l8of]df 15 % 5'6 lbO{ 10 % d"No clej[b\lw s/ nufp“bf ;f]
/]l8of]sf] d"No slt knf{ <
gd'gf 11 :
c+=d" df 15 % 5'6 lbO{ 13 % d"No clej[b\lw s/ nup“bf s'g} ;fdfgsf] d"No ?= 2881.5 eof] eg]
;f] ;fdfgsf] d"No kTtf nufpg'xf];\ .
;dfwfg
oxfF,
;fdfgdf 5'6 = 15 %
;fdfgdf VAT = 13 %
;fdfgsf] lj=d" (S=P) =?=¿ 2881.5
;fdfgsf] c=d" (M=P) = <
15 % 5'6 cg';f/,
5'6kl5sf] ljd" (S=P) = M=P – 5'6
= x – x sf] 15 %
=x–

=
k]ml/,
13% VAT ;lxt lj=d" = 5'6kl5sf] lj=d" + VAT
cyjf, ?= 2881.5 = sf] 13%

72
cyjf, ?= 2882.5 =
cyjf, ?= 2881.5 =
cyjf, x = ?= 3000
ctM ;f] ;fdfgsf] cª\lst d"No ?= 3000 sfod ul/Psf] /x]5 .
1. Pp6f d"lt{sf] 20 % 5'l6;lxtsf] d"Nodf 10 % d"No clej[b\lw s/ hf]8]/ laj|mL ubf{ /
6336 k5{ eg] ;f] d"lt{sf] cª\lst d"No kTtf nufpg'xf];\ .
2. 15 % 5'6 u/L laj|mL u/]sf] Sofd/fdf 10 % VAT hf]8\bf Sofd/fsf] d"No kTtf
nufpg'xf];\ .
3= sdnfn] Pp6f knªnfO{ ?= 1,20,750 lt/L 3/df NofOg\ olb o;nfO{ 25 % 5'6kl5 15 %
VAT hf]8L a]lrPsf] eP knªsf] cª\lst d"No kTtf nufpg'xf];\ .
gd'gf 12 :
cª\lst d"Nodf 15 % 5'6 lbO{ 10 % VAT hf]8L laj|mL ubf{ Pp6f ;fOsnsf] d"No ?= 4207.5 x'g
cfp“5 eg]
(i) ;fOsnsf] 5'6 /sd kTtf nufpg'xf];\ .
(ii) ;fOsnsf] VAT /sd kTtf nufpg'xf]\ .

;dfwfg M
oxf,
;fOsnsf] lj=d" (S.P) = ?= 4207.5
5'6 = 15 %
VAT = 10 %
c+=d" (M.P) = x -dfgf+}_
15 % 5'6 cg';f/,
5'6kl5sf] lj=d" (S=P) = M=P – M=P sf] 15%
= x – x x sf]

VAT ;lxtsf] lj=d" = sf] 10%

cyjf, ?= 4207.5 =
cyjf, ?= 4207.5 =
cyjf, x = ?= 4500
ctM M=P = ?= 4500
(i) 5'6 /sd = M=P sf] 15%

73
= ?= 4500 sf] 15%
= ?= 4500 x
= ?= 675
-ii_ VAT /sd = sf] 10%
=
= -∴ x = ?= 4500_
= ?= 382.5
1= Pp6f /]l8of]df 20 % 5'6 lbO{ 15 % VAT nufp“bf pSt /]l8of]sf] d"No / 1428 x'G5 eg]
pSt /]l8of]sf] VAT /sd lgsfNg'xf];\ .
2= Pp6f 38Lsf] cª\lst d"Nodf 15 % 5'6 lbO{ 13 % d"=c=5'6 nufp“bf laj|m d"No / 9605
sd x'G5 eg] 38Lsf] 5'6 /sd kTtf nufpg'xf];\ .
3= Pp6f j:t'sf] cª\lst d"Nodf 18 % 5'6 lbO{ 10 % VAT nufp“bf ¿ 40, 590 x'g cfp“5 eg]
(i) 5'6 /sd
(ii) VAT /sd kTtf nufpg'xf];\ .

gd'gf 13 :
s'g} ;fdfgsf] cª\lst d"No j|mo d"NoeGbf 30 % a9fP/ /flvPsf] 5 . olb 205 5'6 lbg] xf] eg] ?=
500 gfkmf x'G5 . ;f] ;fdfgsf] cª\lst d"No / j|mo d"No kTtf nufpg'xf];\ .
;dfwfg
oxf“, j|mo d"No (C.P) = x -dfgf+“}_
ist Case,
M=P = C=P + C=P + sf] 30 %
=x+xx

=
S=P = M=p – 5'6
= – sf] 20%
= –
–
=

=
ca,

74
gfkmf = S=P – C=P
cyjf, ?= 500 = –
cyjf, ?= 12,500 = x
M=P = – = ?= 16250
ctM ;f] ;fdfgsf] cª\lst d"No ?= 16250 / j|mo d"No ?= 12,500 /x]5 .
1. Pp6f l6=eL=sf] cª\lst d"No j|mo d"NoeGbf 40 % a9L /flvPsf] 5 . olb 25 % 5'6 u/L
laj|mL ubf{ ?= 200 gfkmf x'G5 eg] ;f] l6=eL=sf] cª\lst d"No / j|mo d"No kTtf
nufpg'xf];\ .
2. Pp6f /]lk|mlh/6]/ cª\lst d"No j|mo d"NoeGbf 40 % a9L /flvPsf] 5 . olb 30 % 5'6 u/L
laj|mL ubf{ ?= 1000 3f6f x'G5 eg] ;f] /]lk|mlh/6]/sf] cª\lst d"No / j|mo d"No
lgsfNg'xf];\ .
3. Ps hgf k;n]n] s'g} j:t'sf] cª\lst d"No, j|mo d"NoeGbf 30 % a9fP/ /fv]5 . olb ;f]
j:t'df 25 % 5'6 lbg d"No / j|mo d"No lgsfNg'xf];\ .

gd'gf 14 :
Ps hgf Jokf/Ln] pp6f Nofk6ksf] cª\lst d"No ljj|mo d"NoeGbf 30 % a9fP/ /fv]5 . olb To;sf]
j|mo d"No ljj|mo d"NoeGbf 20 % sd eP 5'6 k|ltzt / gfkmf jf gf]S;fg k|ltzt kTtf
nufpg'xf];\ .
;dfwfg
oxfF,
Nofk6ksf] ljj|mo d"No (S.P) = x -dfgf}_
M=P = S.P + S.P sf] 30%
=x+xx

=
k]m/L, j|mod"No (C=P) = S=P – S=P sf] 20%
=x–xx

=
ca, 5'6 = M.P – S.P
= –x

=
75
5'6% = 5'6 x 100%
M.P
= 100%

= 23.08%
Gffkmf = S.P –C.P
=x–
=
Gfkmf% = gfkmf x 100%
C.P
= 100%

= 25%
ctM ;f] Nofk6kdf 23.08 % 5'6 lbOPsfn] 25 % gfkmf eP5 .
1. Pp6f 38Lsf] cª\lst d"No ljj|mo d"NoeGbf 40 % a9L / j|mo d"No ljj|mo d"NoeGbf 30 %
sd eP 5'6 k|ltzt / gfkmf jf gf]S;fg k|ltzt kTtf nufpg'xf];\ .
2= Pp6f emf]nfsf] cª\lst d"No ljj|mo d"NoeGbf 30 % a9fP/ / j|mo d"No ljj|mo d"NoeGbf
35 % sd eP 5'6 / gfkmf jf gf]S;fg k|ltzt lgsfNg'xf];\ .
3. Pp6f Sofd]/fssf] cª\lst d"No ljj|mo d"NoeGbf 20 % a9fP/ / j|mo d"No cª\lst
d"NoeGbf 20 % sd eP 5'6 / gfkmf k|ltzt lgsfNg'xf];\ .
gd'gf 15 :
Pp6f leTt] 38Lsf] cª\lst d"Nodf 20 % 5'6 lbPdf ?= 80 3f6f x'G5 . olb 10 % dfq 5'6 lbg] xf]
eg] ?= 10 gfkmf x'G5 eg] ;f] 38Lsf] cª\lst d"No / j|mo d"No kTtf nufpg'xf];\ .
;dfwfg
oxfF, cª\lst d"No (M.P) = x
/ j|mo d"No (C.P) = y (dfgf“)}
1st Case,
M=P – 5'6 = C.P – gf]S;fg
cyjf, x – x sf] 20% = y - 80
cyjf, x – x x = y – 80

cyjf, = y – 80
cyjf, 4x = 5y – 400
–
cyjf, x = ===== (i)
76
IInd case
M=P – 5'6 = C.P + gfkmf
cyjf, x – x x = y + 10

cyjf, = y + 10
cyjf, 9x = 10y + 100
cyjf, x = === (ii)
–
;dL g+ (i) / (ii) af6 =
cyjf, 45y – 3600 = 40y + 400
cyjf, 5y = 4000
cyjf, y = 800
;dL g+ (i) af6
–
x=
–
cyjf, x =
cyjf, x = 900
ctM ;f] 38Lsf] cª\lst d"No ?= 900 / j|mo d"No ?= 800 /x]5 .
1. Pp6f vf]nf}gfsf] cª\lst d"Nodf 25 % 5'6 lbg] xf] eg] ?= 150 gf]S;fg x'G5 t/ 105 dfq
5'6 lbPdf ?= 120 gfkmf x'G5 . ca pSt v]nf}gfsf] cª\lst d"No / j|mo d"No lgsfNg'xf];\ .
2. Pp6f Aof6«Lsf] cª\lst d"Nodf 25 % 5'6 lb“bf 10 % gf]S;fg x'G5 . t/ 10 % 5'6 lb“bf ?=
500 gfkmf x'G5 eg] Aof6«Lsf] cª\lst d"No / j|mo d"No kTtf nufpg'xf];\ .
3. Pp6f ;fOsnsf] cª\lst d"Nodf 15 % 5'6 lbg] xf] eg] 5 % gf]S;fg x'G5 . t/ 5% 5'6
lb“bf ?= 420 gfkmf x'G5 eg] ;fOsnsf] cª\lst d"No / j|mo d"No kTtf nufpg'xf];\ .
4. Pp6f l6=eL=sf] cª\lst d"Nodf 20 % 5'6 lb“bf ?= 400 gfkmf x'G5 . 10 % dfq x'g] lbg] xf]
eg] ?= 1400 gfkmf x'G5 . l6=eL=sf] cª\lst d"No / j|mo d"No lgsfNg'xf];\ .
gd'gf 16 :
Pp6f 38L cª\lst d"Nodf 255 5'6 lbO{ 15 % VAT hf]8]/ a]lrof] . olb VAT /sd ?= 450 lyof]
eg] 38Lsf] cª\lst d"No / 5'6 /sd kTtf nufpg'xf];\ .
;dfwfg
oxfF,
38Ldf 5'6 = 25%
38Ldf VAT = 15%
VAT /sd = ?= 450
cª\lst d"No (M.P) = <

77
5'6 /sd = <
cª\lst d"No (M.P) = x (dfgf“)}
5'6 /sd = M.P sf] 5'6 %
= x sf] 25 %
=x

=
5'6kl5sf] lj=d" = M.P – 5'6
=x–

VAT /sd = sf] 15%

cyjf, ?= 450 =
cyjf, x =?= 4000
5'6 /sd =
ctM ;f] 38Lsf] cª\lst d"No ?= 4000 / 5'6 /sd ?= 1000 /x]5 .
1. Pp6f kª\vfsf] cª\lst d"Nodf 20 % 5'6 lbO{ 10 % VAT hf]8]/ a]lrof] . olb VAT /sd
?= 200 lyof] eg] kª\vfsf] cª\lst d"No / 5'6 /sd kTtf nufpg'xf];\ .
2. Pp6f lk|mhsf] cª\lst d"Nodf 255 5'6 lbO{ 135 d"No clej[b\lw s/ hf]8]/ lslgof] . olb
d"No clej[b\lw s/ /sd ?= 780 eP cª\lst d"No / 5'6 /sd lgsfNg'xf];\ .
3. Pp6f ;fdfgsf] cª\lst d"Nodf 15 % 5'6 lbO{ 10 % VAT hf]8]/ a]lrof] . olb 5'6 /sd
?= 2025 eP cª\lst d"No / VAT /sd lgsfNg'xf];\ .

78
 +If]q M# If]qldlt (Mensuration)
If]q kl/ro
If]qldltcGtu{t lqe'hfsf/ lk|Hd, uf]nf, cw{uf]nf a]ngf, ;f]nL, ;fy} a]ngf / cw{uf]nf, ;f]nL /
cw{uf]nf, a]ngf / ;f]nL, juf{sf/ lk/fld8, juf{sf/ lk/fld8 / if8d'vf s'g} b'O{ cf]6f 7f]; j:t'x¿sf]
;+o'St jj|m ;txsf] If]qkmn, k'/f ;txsf] If]qkmn / cfotg h:tf 7f]; j:t' ;DaGwL ;d:ofx¿ o;
If]qleq ;dfj]z ul/Psf] kfOG5 . o; If]qcGtu{t P;Pn;L k/LIffdf # cf]6f 5f]6f] pTt/ cfpg]
juf{sf/ / @ cf]6f nfdf] pTt/ cfpg] k|Zgx¿ clxn]sf] kf7\oj|mdn] lgwf{/0f u/]sf] kfOG5 .

kf7 #=! lqe'hfsf/ lk|Hd

!= kl/ro
o; kf7df lqe'hfsf/ lk|Hdsf] jj|m ;txsf] If]qkmn / cfotg h:tf ;d:ofx¿dWo] s'g} Pp6f ;d:of
5f]6f] k|Zgdf P;Pn;L k/LIffdf ;f]Wg] ul/G5 . h;sf] k"0ff{ª\s @ x'G5 . lqe'h cfwf/ ePsf] lk|Hd g}
lqe'hfsf/ lk|Hd xf] . h:t} M lrqdf ABC / A'B'C' p:t} / pq} lqe'hx¿ x'g\ . of] lrqsf] jqm ;tx -
j|m; ;]S;g_ sf] If]qkmn klg cfwf/sf] If]qkmn;Fu a/fa/ x'g] ePsfn] of] lk|Hd xf] .
A A'

B'
B

P C'

@= cfwf/e"t tYo tyf ;"qx¿

-s_ lqe'hfsf/ lk|Hdsf] cfotsf/ ;txsf] If]qkmn (L.S.A.)
= cfwf/sf] kl/ldlt × lk|Hdsf] prfO -nDafO_
lqe'hfsf/ lk|Hdsf] cfoftsf/ ;txsf] If]qkmn (L.S.A.) = ( a + b + c) × h
-v_ k'/f ;txsf] If]qkmn (T.S.A.) = 2 × cfwf/sf] If]qkmn + cfotsf/ ;txsf] If]qkmn
= 2 × lqe'hsf] If]qkmn + (a + b + c) h
lqe'hfsf/ lk|Hdsf] cfotg (Volume) (V) = cfwf/sf] If]qkmn × prfO
=A×h
C C'

b
a
A
A'
c
B B'
#= ljz]if Wofg lbg'kg]{ s'/fx¿
– cfwf/ - lqe'h _ s:tf] 5 < o;sf] kl/ldlt / If]qkmn lgsfNg] ;"q yfxf kfpg'k5{ .
– kl/ldlt ;a} e'hfx¿sf] of]ukmn (a+b+c) x'G5 .

79
 – lqe'h ;dsf]0fL eP If]qkmn = × cfwf/ × prfO x'G5 .
– jj|m ;txsf] If]qkmn eP cfotsf/ efusf] dfq If]qkmn xf] eGg] yfxf kfpg'k5{ .
– k'/f ;txsf] If]qkmn eP cfwf/;lxtsf] If]qkmn eGg] yfxf kfpg'k5{ .
– cfotgsf k|Zg ;f]w]sfdf ;f]xLcg';f/sf] ;"q nufpg''kg]{ x'G5 .
$= gd'gf k|Zgf]Tt/ tyf cEof;
$=! 5f]6f] pTt/ cfpg] k|Zgf]Tt/
gd'gf ! M
lbOPsf] lrqaf6 jj|m ;txsf] If]qkmn k'/f ;txsf] If]qkmn / cfotg lgsfNg'xf];\ .
B 20 cm E

A
D

C
of] lqe'h ;dsf]0fL lqe'h xf] . ljGb' A sf] sf]0f 90o 5 .
oxf“, AB = 8 cm, AC = 6 cm
BC = - kfOyfuf]/; ;fWocg';f/ _
=
=
=
= 10 cm
lqe'h M cfwf/sf] kl/ldlt (P) = AB + AC + BC
= 8 + 6 + 10
= 24 cm
lk|Hdsf] prfO ( h ) = 20 cm
jj|m ;txsf] If]qkmn (L.S.A ) = ?
xfdLnfO{ yfxf 5, jj|m ;txsf] If]qkmn (L.S.A ) = cfwf/sf] kl/ldlt (P) × prfO (h)
= 27 × 20
= 480 cm2
km]/L, lk|Hdsf] k'/f ;txsf] If]qkmn (T.S.A.) x'gsf nflu,
cfwf/sf] If]qkmn (A) = × cfwf/ × prfO
= ×8×6
= 24 cm2 Ans
k'/f ;txsf] If]qkmn (T.S.A.) = 2A + P×h
= 2 × 24 + 24 × 20
= 48 + 480
= 528 cm2
lk|Hdsf] cfotgsf nflu (V) = cfwf/sf] If]qkmn × prfO
= 24 cm2 × 20 cm
= 480 cm3

80
cEof;sf nflu k|Zg
1= lbOPsf lk|Hdaf6 jj|m ;txsf] If]qkmn lgsfNg'xf];\ M
-s_ -v_ -u_
P M
5
C
Q M
N
R O

pTt/ M -s_ 120cm2 -v_ 900 cm2 -u_ 240 cm2

@= lbOPsf lrqx¿sf] k'/f ;txsf] If]qkmn lgsfNg'xf];\ M

A E’
5
D cm
5
cm
B
C 30cm 10cm 20c m

-s_ -v_ -u_

pTt/ M -s_1848 cm2 -v_132 cm2 -u_ 252 cm2

#= lbOPsf lk|Hdx¿sf] cfotg lgsfNg'xf];\ M

-s_ -v_ -u_

8cm 13
C
M

pTt/ M -s_ 480 cm3 -v_ 450cm3 -u_ 2880cm3

gd'gf k|Zg g+= @ M

lbOPsf] lrqaf6 cfotg lgsfNg'xf];\ M

4 3 cm
of] lrq ;dafx' lqe'h cfwf/ ePsf] lqe'hfsf/ lk|Hd xf] .
hxf“, e'hfsf] nDafO (a) = 6cm

81
;dafx' lqe'hsf] cfwf/sf] If]qkmn (A) = a2
= ×62 = 9 cm2
lk|Hdsf] cfotg (V) = A × h 3g PsfO
=9 cm3
= 36 × 3 cm3 = 108 cm3
cEof;sf nflu k|Zg
!= lbOPsf lrqaf6 cfotg lgsfNg'xf];\ .
-s_ -v_

10 3 cm

pTt/ M -s_ 270cm3 -v_ 180 cm3

yk cEof;
58\s] ;tx / k'/f ;txsf] M

!= tn lbOPsf lk|Hdsf] If]qkmn lgsfNg'xf];\ M

-s_ -v_

20 cm

pTt/ M -s_ 360cm2 / 391.176cm2, -v_1200cm2 / 1416cm2

82
@= lbOPsf] lk|Hdsf] cfotg lgsfNg'xf];\ M

pTt/ M -s_ 39.80cm3

#= Pp6f lqe'hsf e'hfx¿ 7 : 24 : 25 sf] cg'kftdf 5g\ . olb ;f] lqe'hsf] If]qkmn 84cm2 eP pSt
lqe'hsf] kl/ldlt lgsfNg'xf];\ .
pTt/ M 56 cm
$= olb Pp6f ;dafx' lqe'hsf] If]qkmn 25 ju{ ;]=ld= 5 eg] To;sf] kl/ldlt lgsfNg'xf];\ .
pTt/ M 30 cm
%= 30 lkm6, 36 lkm6 / 40 lkm6 e'hfx¿ ePsf] Pp6f lqe'hfsf/ d}bfgsf] If]qkmn lgsfNg'xf];\ .
pTt/ M 519 .04 ft2
^= 30 ld=, 30 ld=/ 32 ld= e'hfx¿ ePsf] lqe'hfsf/ d}bfgsf] If]qkmn lgsfNg'xf];\ .
pTt/M 406.03 m2
&= lbOPsf] lqe'hfsf/ lk|Hdsf] 58\s] ;txsf] If]qkmn kTtf nufpg'xf];\ .

12.5 cm

pTt/ M 840 cm2

*= lbOPsf] lqe'hfsf/ lk|Hdsf] 58\s] ;txsf] If]qkmn kTtf nufpg'xf];\ .

8 cm

pTt/ M 480cm2

83
(= lbOPsf] lqe'hfsf/ lk|Hdsf] 58\s] ;txsf] If]qkmn kTtf nufpg'xf];\ .

pTt/ M 378cm
!)= lrqdf lbOPsf] lk|Hdsf] cfotsf/ kf6fx¿sf] If]qkmn 144cm2 eP
lk|Hdsf] prfO kTtf nufpg'xf];\ .

pTt/ M 12 cm
!!= olb Pp6f lqe'hfsf/ lk|Hdsf] cfotsf/ ;txsf] If]qkmn / prfO j|mdz M 324 ju{ ;]=ld= / 18
;]=ld= eP o;sf] cfwf/sf] kl/ldlt kTtf nufpg'xf];\ .
pTt/ M 18 cm
!@= olb Pp6f lqe'hfsf/ lk|Hdsf] cfotsf/ ;txx¿sf] If]qkmn / prfO j|mdzM 1056 ju{ ;]=ld= /
24 ;]=ld= eP o;sf] cfsf/sf] kl/ldlt kTtf nufpg'xf];\ . pTt/ M 44 cm

84
kf7 #=@ a]ngf, uf]nf, cw{uf]nf / ;f]nL
(Cylinder, Sphere, Hemisphere and Cone)
!= kl/ro
o; kf7cGt/ut a]ngf, uf]nf, cw{uf]nf / ;f]nL h:tf 7f]; j:t'x¿sf] jj|m ;txsf] If]qkmn, k'/f
;txsf] If]qkmn / cfotg ;DaGwL ;d:ofx¿ ;dfj]z ul/Psf 5g\ . o; kf7af6 @ cf]6f 5f]6f]
pTt/ cfpg] k|Zg / csf]{ Pp6f nfdf] pTt/ cfpg] k|Zg @ cf]6f 7f]; j:t'sf] ;+o'St lrqsf] jj|m
;txsf] If]qkmn, k'/f ;txsf] If]qkmn / cfotg dWo] s'g} Ps P;Pn;L k/LIffdf ;f]Wg] ul/G5 .
@= cfwf/e"t tYo jf ;"qx¿
@=! a]ngf
of] a]ngfsf] lrq xf] . o;df cw{Jof; / prfO jf nDafO x'G5 . lrqdf r n] cw{Jof; / h («l) prfO
jf nDafOnfO{ hgfp“5 .
a]ngfsf] jj|m ;txsf] If]qkmn (C.S.A.) = 2 π r h,
a]ngfsf] k'/f ;txsf] If]qkmn (T.S.A.) = 2 π r (r + h)
/ a]ngfsf] cfotg (V) = π r2 h

@=@ uf]nf (Sphere)

uf]nfsf] k'/f ;txsf] If]qkmn (T.S.A.) = 4 πr2 = πd2
4
uf]nfsf] cfotg (V)= 3 π r3 = 16 A E A A E A π d3

@=# cw{uf]nf ( Hemisphere)

cw{uf]nfsf] jj|m ;txsf] If]qkmn (C.S.A.)= 2πr2 = πd2
cw{uf]nfsf] k'/f ;txsf] If]qkmn (T.S.A.)= 3 πr2 = πd2
2 1
cw{uf]nfsf] cfotg (V) E
3
3 π r =12 π d
A A
3
A E A

@=# ;f]nL (Cone)

;f]nLsf] jj|m ;txsf] If]qkmn (C.S.A.) = πrl
;f]nLsf] k'/f ;txsf] If]qkmn (T.S.A.) = πr( r + l)
;f]nLsf] cfotg (V)= πr2h
yk l= h2 + r2 A E

2 2
h= l - r
A E

2 2
r= l - h
A E

85
@=$ ldl>t 7f]; j:t' ;DaGwL ;"qx¿

a]ngf / ;f]nL ;+o'St 7f]; j:t'sf] jj|m ;txsf] If]qkmn(C.S.A.)= πrl + 2πrh
= πr (l + 2h) ju{ PsfO
k'/f ;txsf] If]qkmn (T.S.A.) = πrl + 2πrh + πr2
= πr (l + 2h + r) ju{ PsfO
1 2
cfotg (V) =3 πr h1+ πr2h
A A

1
= πr2 (3 h1 + h ) #g PsfO
A

E
A

a]ngf / cw{uf]nfsf] ;+o'St 7f]; a:t'sf] jj|m ;txsf] If]qkmn (T.S.A) = 2πr2 + 2πrh
= 2πr(r + h) ju{ PsfO
k'/f ;txsf] If]qkmn (T.S.A.) = 2πr2 +2πrh + πr2
= 3πr2 +2πrh
= πr(3r + 2h ) ju{ PsfO
cfotg (V) = πr3 + πr2h
= πr2( ) #g PsfO
@=% cw{uf]nf / ;f]nL

jj|m ;txsf] If]qkmn +(C.S A ) = 2πr2 +2πrl

= πr (2r + l ) ju{ PsfO

k'/f ;txsf] If]qkmn (T.S.A.) = 2πr2 + πrl

= πr (2r+l ) ju{ PsfO
2 2
cfotg (V) = 3 π r3 +3 πr2h
A E A A E A

86
 1
= 3 πr2 (2r + h) #g PsfO
A E A

#= ljz]if Wofg lbg'kg]{ s'/fx¿

— 7f]; j:t' lrGg] / To;sf] ;"q klxrfg u/L k|of]u ug'{k5{ .
—;+o'St 7f]; j:t' s] s] ldn]/ ag]sf] 5 . To;nfO{ cnu cnu lrqsf] ¿kdf lrGg'k5{ .
—;"q n]vL To;nfO{ ;/nsf] lgod cg';f/ SofNs'n]6/ k|of]u u/L ;d:of ;dfwfg ug]{ .

$= gd"gf k|Zgf]Tt/ tyf cEof;

$=!= 5f]6f] pTt/ cfpg] k|Zg / ;dfwfg
$=!=! a]ngf
lbOPsf] 7f]; j:t'sf] jj|m ;txsf] If]qkmn lgsfNg'xf];\ .

;dfwfg
of] lrq a]ngf xf] .
oxf“ cw{Jof; (r) = 7cm
a]ngfsf] prfO (h) = 10 cm
jj|m ;txsf] If]qkmn (C.S.A ) = ?
xfdLnfO{ yfxf 5,
jj|m ;txsf] If]qkmn (C.S.A ) = 2πrh ju{ PsfO
2
=2× × 7 × 10 cm
= 44 × 10 cm2
= 440 cm2
o;} u/L k'/f ;txsf] If]qkmn jf cfotg h] ;f]w]sf] 5 eg] To;s} ;"q k|of]u u/L ;d:of
;dfwfg ug'{k5{ .

cEof;sf nflu k|Zg

-s_ lrqdf lbOPsf] a]ngfsf] jj|m ;txsf] If]qkmn
lgsfNg'xf];\ .

pTt/ M 3520 cm2

-v_ olb prfO / cw{Jof; a/fa/ ePsf] a]ngfsf/ sf7sf] jj|m ;txsf] If]qkmn 308 ju{ ;]=ld=
5 eg] pSt sf7sf] prfO lgsfNg'xf];\ . pTt/ M 7cm
-u_ a]ngfsf] jj|m ;txsf] If]qkmn / prfO j|mdzM 6\6 ju{ ;]=ld= / 14 ;]=ld= eP pSt a]ngfsf]
cw{Jof; lgsfNg'xf];\ . pTt/ M 7cm
87
-3_ lbOPsf] 7f]; j:t'sf] cfotg lgsfNg'xf];\ .

pTt/ M 57.75cm3
$=!=@ gd"gf @ uf]nf ;DaGwL
lbOPsf] uf]nfsf] cfotg lgsfNg'xf];\ .

;dfwfg
oxfF lbOPcg';f/,
Jof; (d) = 18cm
d 18
cw{Jof; (r) = 2
A E A = 2 = 9 cm
A E A

cfotg(V) = ?
xfdLnfO{ yfxf 5, V = r3 #g PsfO
= × 93
=

= 3054.86 cm 3
cEof;sf nflu k|Zg
-s_ 36π cm3 cfotg ePsf] uf]nfsf] k'/f ;txsf] If]qkmn kTtf nufpg'xf];\ . pTt/ M 36π cm2
-v_ 7 ;]=ld= cw{Jof; ePsf] uf]nfsf] cfotg lgsfNg'xf];\ . pTt/ M 1437.33 cm3
-u_ 14 ;]=ld= Jof; ePsf] Pp6f uf]nfsf] k'/f ;txsf] If]qkmn lgsfNg'xf];\ . pTt/ M 616 cm2
-3_ 288π cm cfotg ePsf] uf]nfsf] k'/f ;txsf] If]qkmn kTtf nufpg'xf];\ .
3
pTt/ M 144π cm2
-ª_ Pp6f uf]nfsf] k'/f ;txsf] If]qkmn 22176 ju{ ;]=ld= 5 eg] pSt uf]nfsf] Jof; kTtf
nufpg'xf];\ . pTt/ M 84cm
$=!=# gd"gf # cw{uf]nf ;DaGwL
k|Zgf]Tt/ / cEof;
lbOPsf] cw{uf]nfsf] k'/f ;txsf] If]qkmn lgsfNg'xf];\ .
35 cm

88
;dfwfg
oxfF lbPcg';f/,
cw{uf]nfsf] cw{Jof; (r) = 3.5 cm
cw{uf]nfsf] k'/f ;txsf] If]qkmn (T.S.A.) = ?
xfdLnfO{ yfxf 5 M
cw{uf]nfsf] k'/f ;txsf] If]qkmn (T.S.A.) = 3πr2
= 3× 22/7 × (3.5)2
= 66 × 12.25
= 808.5 cm2
cEof;sf nflu k|Zg
-s_ tn lbOPsf] cw{uf]nfsf] jj|m ;txsf] If]qkmn, k'/f ;txsf] If]qkmn / cfotg lgsfNg'xf];\ .
-c_ cf_ -O_
21 cm

17.5 cm 3.5 cm

pTt/ M 693 cm2 ,1039.5 cm2 , 11229.16 cm3;

pTt/ 77cm2 ,115.5cm2 , 89.83cm3
pTt/ M M 693 cm2, 1039.5 cm2 / 2425.5 cm3
-v_ Pp6f cw{uf]nfsf] k'/f ;txsf] If]qkmn 108π ju{ ;]=ld= 5 eg] pSt cw{uf]nfsf] cw{Jof;
lgsfNg'xf];\ . pTt/ M 12 cm
-u_ k'/f ;txsf] If]qkmn 27π cm2 ePsf] cw{uf]nfsf] cfotg kTtf nufpg'xf];\ . pTt/ M 56.57cm3
-3_ Pp6f cw{uf]nfsf] k'/f ;txsf] If]qkmn 108π ju{ ;]=ld= eP pSt cw{uf]nfsf] 7'nf] j[Ttsf]
If]qkmn lgsfNg'xf];\ . pTt/ M 113.14 cm2
-ª_ 14 ;]=ld= Jof; ePsf] Pp6f cw{uf]nfsf] k'/f ;txsf] If]qkmn lgsfNg'xf];\ . pTt/ M 462cm2

$=!=$ ;f]nL (Cone)

lbOPsf] ;f]nLsf] k'/f ;txsf] If]qkmn lgsfNg'xf];\ .

;f]nLsf] l;wf prfO (h ) = 24 cm

;f]nLsf] cw{Jof; (r) = 7 cm
;f]nLsf] k'/f ;txsf] If]qkmn (T.S.A.) = ?

89
 xfdLnfO{ yfxf 5,
l = h2 + r 2 = 242 + 7 2
A E A A E A = A 576 +49 = 625 = 25
E A A E A

;f]nLsf] k'/f ;txsf] If]qkmn (T.S.A.) = πr (r + l)

= 22/7 × 7(7 + 25)
= 22 × 32
= 704 cm2
cEof;sf nflu k|Zgx¿
-s_ tn lbOPsf] ;f]nLsf] cfotg lgsfNg'xf];\ .

-c_ -cf_ -O_

pTt/ M -c_ 1232 cm -cf_ 314.28 cm3
3
-O_ 37.71 cm3
-v_ cfwf/sf] cw{Jof; 4 ;]=ld= / 58\s] prfO 5 ;]=ld= ePsf] ;f]nLsf] cfotg lgsfNg'xf];\ .
pTt/ M 50.28cm3
-u_ Pp6f ;f]nLsf] prfO 15 ;]=ld= / 58\s] prfO 25 ;]=ld= eP ;f] ;f]nLsf] cfotg lgsfNg'xf];\ .
pTt/ M 6285.71cm3
-3_ ;uF } lbOPsf] ;f]nLsf] jj|m ;txsf] If]qk,mn kTtf nufpg'xf];\ .

pTt/ M 550cm2
-ª_ cfwf/sf] cw{Jof; 6 ;]=ld= / 58\s] prfO 8 ;]=ld= ePsf] ;f]nLsf] k'/f ;txsf] If]qkmn kTtf
nufpg'xf];\ . pTt/ M 264 cm2
$=!=% yk cEof;
!= Pp6f a]ngfsf/ j:t'sf] cfwf/sf] If]qkmn cfwf/sf] kl/lw / prfO j|mdz M 38.5 cm2, 22cm /
15cm 5 . cj kTtf nufpg'xf];\ M
-s_ jj|m ;txsf] If]qkmn
-v_ k'/f ;txsf] If]qkmn
-u_ cfotg
pTt/ M 330cm2, 407cm2, 5775cm3
@= Pp6f a]ngfsf/ j:t'sf] cfwf/sf] If]qkmn, cfwf/sf] kl/lw / prfO j|mdz M 1386cm2, 132cm
/ 18cm 5 eg] cj kTtf nufpg'xf];\ M
-s_ jj|m ;txsf] If]qkmn,
-v_ k'/f ;txsf] If]qkmn,

90
 -u_ cfotg
pTt/ M 2376 cm2, 5148 cm2, 24948 cm3
#= Pp6f a]ngfsf/ j:t'sf] cfwf/sf] Jof; / prfO j|mdz M 14cm / 12cm eP kTtf
nufpg'xf];\ .
-s_ jj|m ;txsf] If]qkmn
-v_ k'/f ;txsf] If]qkmn
-u_ cfotg
pTt/ M 528cm2, 836cm2, 1848cm3
$= Pp6f a]ngfsf/ j:t'sf] cfwf/sf] Jof; / prfO j|mdzM7cm / 30cm eP kTtf nufpg'xf];\ .
-s_ jj|m ;txsf] If]qkmn
-v_ k'/f ;txsf] If]qkmn
-u_ cfotg
pTt/ M 660cm2,737cm2, 1155cm3
%= Pp6f a]ngfsf/ j:t'sf] cfwf/sf] Jof; / prfO j|mdz M 56cm / 8cm eP kTtf
nufpg'xf];\ .
-s_ jj|m ;txsf] If]qkmn
-v_ k'/f ;txsf] If]qkmn
-u_ cfotg
pTt/ M 1408 cm2, 6336 cm2, 197122 cm3)
^= Pp6f a]ngfsf/ j6\6fsf] cfotg 1.10 ln6/ 5 . olb cfwf/sf] If]qkmn 550cm2 5 eg] o;sf]
prfO lgsfNg'xf];\ . (Ans 2cm )
&= Pp6f a]ngfsf/ j6\6fsf] cfotg 2.60 ln6/ 5 . olb cfwf/sf] If]qkmn 130 cm2 5 o;sf]
prfO lgsfNg'xf];\ .
pTt/ M 20cm
*= prfO / cfotg j|mdz M 9cm / 1848 3g ;]=ld= ePsf] ;f]nL cfsf/sf] 7f];sf] k'/f ;txsf]
If]qkmn kTtf nufpg'xf];\ .
pTt/ M 1348.305 cm2
(= lbPsf] ;f]nLsf] jj|m ;txsf] If]qkmn kTtf nufpg'xf];\ .

pTt/ M 305.63cm2

91
!)= lbPsf] ;f]nLsf] jj|m ;txsf] If]qkmn kTtf nufpg'xf];\ .

pTt/ M 487cm2
!!= olb lbPsf] ;f]nLsf] cfwf/sf] cw{Jof; / prfO j|mdzM 3.5 ;]=ld= / 12 ;]=ld= 5g\ eg] ;f]nLsf]
cfotg kTtf nufpg'xf];\ .

pTt/ M 154 cm3

!@= olb lbPsf] ;f]nLsf] cfwf/sf] cw{Jof; prfO j|mdzM 1.4 ;]=ld= / 9 ;]=ld= 5g\ eg] ;f]nLsf]
cfotg kTtf nufpg'xf];\ .

pTt/ M 18.48cm3
!#= olb lbOPsf] ;f]nLsf] cfwf/sf] cw{Jof; / prfO j|mdz M 2.1 cm / 9
;]=ld= 5g\ eg] ;f]nLsf] cfotg kTtf nufpg'xf];\ .

pTt/ M 41.58cm3
!$= olb lbOPsf] ;f]nLsf] cfotg 2200 3g ;]=ld= / o;sf] cw{Jof; 10cm eP o;sf] prfO slt
xf]nf <

pTt/ M 21cm
92
!%= olb lbOPsf] ;f]nLsf] cfotg 1584 3g ;]=ld= / o;sf] cw{Jof; 6 ;]=ld= eP o;sf] prfO slt
xf]nf <

pTt/ M 14cm
!^= k'/f ;txsf] If]qkmn 48π cm2 ePsf] cw{uf]nfsf] cfotg kTtf nufpg'xf];\ . ( π = 3.14)
pTt/ M 133.97cm3
!&= k'/f ;txsf] If]qkmn 108π cm2 ePsf] cw{uf]nfsf] cfotg kTtf nufpg'xf];\ . (π = 3.14)
pTt/ M 452.16 cm3
!*= Pp6f uf]nfsf] ;txsf] If]qkmn 2456 ;]=ld= eP o;sf] cw{Jof; kTtf nufpg'xf];\ .
pTt/ M 14cm
!(= cfwf/sf] cw{Jof; 7cm ePsf] Pp6f ;f]nLsf] k'/f ;txsf] If]qkmn 704 ju{ ;]=ld= eP ;f]nLsf]
prfO kTtf nufpg'xf];\ . pTt/ M 2.4 cm
@)= 7 cm Jof; ePsf] Pp6f cw{uf]nfsf] k'/f ;txsf] If]qkmn lgsfNg'xf];\ . pTt/ M 1155 cm2
@!= 21 cm Jof; ePsf] cw{uf]nfsf] k'/f ;txsf] If]qkmn lgsfNg'xf];\ . pTt/ M 1039.5 cm2
@@= 972πcm3 cfotg ePsf] uf]nfsf] k'/f ;txsf] If]qkmn kTtf nufpg'xf];\ . pTt/ M 324 π cm2
@#= Pp6f uf]nfsf/ j:t'sf] cfotg 1047816 cm3 5 eg] o;sf] cw{Jof; slt x'G5 . kTtf
nufpg'xf];\ .
pTt/ M 63 cm
@$= uf]nfsf] 7'nf] j[Ttsf] kl/lw 8.8 cm eP o;sf] ;txsf] If]qkmn kTtf nufpg'xf];\ .
pTt/ M 55.44 cm2
@%= uf]nfsf] 7'nf] j[Ttsf] kl/lw 39.6cm eP o;sf] ;txsf] If]qkmn kTtf nufpg'xf];\ .
pTt/ M 498.96cm2
@^= tLg cf]6f uf]nfx¿ h;sf] cfwf/sf] cw{Jof; j|mdz M 12 cm 16 cm / 20 cm 5g\ . pSt
uf]nfx¿nfO{ kufn]/ Pp6} uf]nf agfpFbf ;f] uf]nfsf] cw{Jof; slt x'G5 < kTtf nufpg'xf];\ .
pTt/ M 24cm
@&= lbOPsf] ;f]nLdf OA=7cm / OB =24 cm 5 . ;f] ;f]nLsf] jj|m ;txsf] If]qkmn kTtf nufpg'xf];\ .

pTt/ M 550cm2

93
@*= cfotg 6600 3g ;]=ld= / cw{Jof; 10 cm ePsf] ;f]nLsf] prfO kTtf nufpg'xf];\ . pTt/ M 63cm
@(= Pp6f ;f]nL jj|m ;txsf] If]qkmn 44 π ju{ ;]=ld= 5 . olb o;sf] cfwf/sf] cw{Jof; 4cm eP
;f]nLsf] prfO kTtf nufpg'xf];\ . pTt/ M 11cm
#)= Pp6f ;f]nLsf] jj|m ;txsf] If]qkmn 20 π ju{ ;]=ld= 5 . olb o;sf] cw{Jof; 4 cm eP ;f]nLsf]
prfO kTtf nufpg'xf];\ . pTt/ M 3cm
#!= Pp6f ;f]nLsf] 58\s] prfO 1.5 ld= / cw{Jof; 0.9 ld= eP pSt ;f]nLsf] cfotg lgsfNg'xf];\ .
pTt/ M 1.01m3
#@= Pp6f ;f]nLsf] cfwf/sf] kl/lw 66 ;]=ld= / jf:tljs prfO 5 ;]= ld= eP pSt ;f]nLsf] cfotg
lgsfNg'xf];\ . pTt/ M 5775cm3
##= Pp6f ;f]nLsf] cfwf/sf] Jof; 24 ;]=ld= / 58\s] prfO 15 ;]=ld= eP pSt ;f]nLsf] cfotg
lgsfNg'xf];\ . pTt/ M 1357.71cm3
#$= Pp6f ;f]nLsf] prfO 7 ;]=ld= / cw{Jof; 24 ;]= ld= eP pSt ;f]nLsf] jj|m ;txsf] If]qkmn
cfotg lgsfNg'xf];\ . pTt/ M 1885.71cm2 , 4224cm3
#%= Pp6f ;f]nLsf] cfwf/sf] kl/ldlt 44 cm / 58\s] prfO{ 10cm ;f]nLsf] cfotg / k'/f ;txsf]
If]qkmn lgsfNg'xf];\ . pTt/ M 36652cm3, 374cm2
#^= Pp6f ;f]nLsf] l;wf prfO / 58\s] prfOsf] cg'kft 3:5 / cfotg 2816/7 3g ;]=ld= eP
pSt ;f]nLsf] 58\s] prfO kTtf nufpg'xf];\ . pTt/ M 10 cm
#&= Pp6f ;f]nLsf] cw{Jof; / 58\s] prfO 7: 25 sf] cg'kftdf 5g\ . olb o;sf] jj|m ;txsf]
If]qkmn 2200 ju{ ;]= ld= eP cw{Jof; kTtf nufpg'xf];\ . pTt/ M 14 cm

#=@ nfdf] pTt/fTds k|Zgf]Tt/

gd'gf ! M
lbOPsf] ;+o'St 7f]; j:t'sf] cfotg lgsfNg'xf];\ .

;dfwfg M
;f]nLsf] 58\s] prfO (l)=13 mm
cw{Jof; (r) = 10/2 mm = 5 mm
;f]nLsf] vf; prfO (h1) =
=
=
=
= 12 mm
a]ngfsf] prfO (h) = 140 mm - 12 mm
=128 mm

94
 ;+o'St 7f]; j:t'sf] cfotg (V) = a]ngfsf] cfotg (V1) + ;f]nLsf] cfotg (V2)
= πr2h + 1/3 πr2h1
= πr2(h + h1/3 )
= )
=
=
= 10371.43 mm3
cEof;sf nflu k|Zg

!= tn lbOPsf] ;+o'St 7f]; j:t'sf] k'/f ;txsf] If]qkmn lgsfNg'xf];\ .

-s_ -v_ u_

pTt/ M -s_ 1606cm2 -v_ 1339.48cm2 -u_ 4697.47cm2

gd'gf @ M
lbOPsf] ;+o'St 7f]; j:t'sf] k'/f ;txsf] If]qkmn lgsfNg'xf];\ .
;dfwfg M
;f]]nLsf] 58\s] prfO (l) =13cm+
;f]nLsf] vf; prfO (h) = 12cm
;+o'St 7f]; j:t'sf] k'/f ;txsf] If]qkmn (T.S.A.) = ?
xfdLnfO{ yfxf 5, ;f]nLsf] cw{Jof; (r) =

=
=
=
= 5cm
;+o'St 7f]; j:t'sf] k'/f ;txsf] If]qkmn (T.S.A.) = πrl + 2 πr2 = πr(l +2 r) ju{ PsfO
22
= 7 × 5( 13+ 5×2) cm2
A E A

110×23
= 7
A cm2 E A

= 2530 cm2
cEof;sf nflu k|Zg
lbOPsf] ;+o'St 7f]; j:t'sf] k'/f ;txsf] If]qkmn lgsfNg'xf];\ .

95
-s_ -v_ -u_ -3_ -ª_
14 cm 7cm
14 cm
31 cm 31 cm 15.5 cm
3.5 cm

pTt/ M -s_ 858 cm2, -v_ 831.72 cm2, -u_ 858 cm2, -3_ 858 cm2,-ª_ 214.50 cm2

gd"gf # M
;+o'St 7f]; j:t'sf] k'/f ;txsf] If]qkmn lgsfNg'xf];\ .
25 cm

39 cm
;dfwfg M
a]ngfsf] prfO (h) = 25cm
cw{Jof; (r ) =(39cm- 25cm)
= 14cm
;+o'St 7f]; j:t'sf] k'/f ;txsf] If]qkmn (T.S.A.) = ?
xfdLnfO{ yfxf 5,
;+o'St 7f]; j:t'sf] k'/f ;txsf] If]qkmn (T.S.A.) = πr(2h + 3r) ju{ PsfO
= 22/7 × 14 (2×25+3×14 )
= 44 (50+42)
= 44×92
= 4048cm2
#= tn lbOPsf ;+o'St 7f]; j:t'sf] k'/f ;txsf] If]qkmn lgsfNg'xf];\ .

-s_ -v_ -u_

10 cm 49 cm

17 cm 140 cm 140 cm
-s_ 902 cm 2
-v_ 2618 cm
2
) -u_ 19866 cm2

cEof;sf nflu k|Zg

!= lbOPsf] lrq cw{uf]nf / ;f]nLåf/f ag]sf] 7f]; j:t' xf] . hxfF cw{uf]nfsf] Jof; 14cm 5.
;f]nLsf] 58\s] prfO 25cm 5 . eg] pSt 7f]; j:t'sf] cfotg kTtf nufpg'xf];\ .
96
 pTt/ M 1950.66cm3

@= lbOPsf] lrq cw{uf]nf / ;f]nLåfjf/f ag]sf] 7f]; j:t' xf] . hxfF cw{uf]nfsf] Jof; 18 cm 5 . /
;f]nLsf] 58\s] 15 cm 5 eg] pSt j:t'sf] cfotg kTtf nufpg'xf];\ .

pTt/ M 2545.71cm3

#= lbOPsf] lrqsf] cw{uf]nf / ;f]nLåf/f ag]sf] 7f]; j:t' xf] hxfF cw{uf]nf Jof; 42cm 5 /
;f]nLsf] 58\s] prfO 75cm 5 eg] pSt j:t'sf] cfotg kTtf nufpg'xf];\ .

pTt/ M 52668c

$= lbOPsf] ;+o'St 7f]; j:t'sf] cfotg kTtf nufpg'xf];\ . hxfF ;f]nLsf] 58\s] prfO 10 cm /
prfO 8cm 5 .

8 cm

pTt/ M 754.28cm3
%= lbOPsf] ;+o'St 7f]; j:t'sf] cfotg kTtf nufpg'xf];\ hxfF ;f]nLsf] 58\s] prfO 13cm / prfO
12cm 5 .
pTt/ M 576.19cm3

97
 ^= Pp6f 7f]; j:t' a]ngf / cw{uf]nf ldnL ag]sf] 5 . k'/f 7f];sf] prfO cfwf/sf] Jof;;Fu a/fa/ 5 .
k'/f 7f];sf] cfotg 360πcm3 5 eg] 7f];sf] prfO kTtf nufpg'xf];\ . pTt/ M 12cm
&= Pp6f 7f]; j:t' a]ngf / cw{uf]nf ldnL ag]sf] 5 . k'/f 7f];sf] prfO cfwf/sf] Jof;;Fu
a/fa/ 5 . k'/f 7f];sf] cfotg 360π cm3 5 eg] 7f];sf] prfO kTtf nufpg'xf];\ .
pTt/ M 21 cm
*= Pp6f 7f]; j:t' a]ngf / cw{uf]nf ldnL ag]sf] 5 . k'/f 7f];sf] prfO cfwf/sf] Jof;;uF
a/fa/ 5 / k"/f 7f]; j:t'sf] cfotg 4608π cm3 5 eg] 7f];sf] prfO kTtf nufpg'xf];\ .
pTt/ M 28.07cm
(= lbOPsf] lrq Pp6f a]ngf / pxL cw{Jof; ePsf] Pp6f cw{uf]nf ldnL 30cm

ag]sf] Pp6f ;+o'St j:t' xf] . olb 7f]; j:t'sf] k'/f nDafO 35cm /
a]ngfsf] nDafO 28cm eP ;f] j:t'sf] k'/f ;txsf] If]qkmn kTtf 44cm
nufpg'xf];\ .
pTt/ M 1694cm2
!)= lbOPsf] lrq Pp6f a]ngf / pxL cw{Jof; ePsf] Pp6f cw{uf]nf ldnL ag]sf] Pp6f ;+o'St j:t'
xf] . olb 7f]; j:t'sf] k'/f nDafO 44cm / a]ngfsf] nDafO 30cm eP ;f] 7f]; j:t'sf] k'/f
;txsf] If]qkmn kTtf nufpg'xf];\ .
30 cm

44 cm
pTt/ M 4488cm 2

!!= lbOPsf] lrqdf Pp6f v]nf}gf cw{uf]nf / ;f]nL ldnL ag]sf] 5 . cw{uf]nfsf] Jof; 28cm /
v]nf}gfsf] k'/f prfO 62cm eP v]nf}gf]sf] cfotg / k'/f ;txsf] If]qkmn kTtf nufpg'xf];\ .

pTt/ M 10266.66 cm3 ,3432cm2

!@= lbOPsf] lrqdf Pp6f v]nf}gf cw{uf]nf / ;f]nL ldnL ag]sf] 5 . cw{uf]nf Jof; 6cm /
v]nf}gfsf] k'/f prfO 7cm eP v]nf}gfsf] cfotg / k'/f ;txsf] If]qkmn kTtf nufpg'xf];\ .

- pTt/ M 94.28 cm3 103.71cm2 _

98
!#= ;Fu} lbOPsf] 7f]; a]ngf / ;f]nL ldnL ag]sf] 5 . pSt 7f];sf] jj|m ;txsf] If]qkmn kTtf
nufpg'xf];\ . (π = 3.14)
24 cm

48 cm
pTt/ M 1604.54cm2
!$= ;“u} lbOPsf] 7f]; a]ngf / ;f]nL ldnL ag]sf] 5 . pSt 7f];sf] jj|m ;txsf] If]qkmn kTtf
nufpg'xf];\ . (π = 3.14)
24 cm

36 cm
pTt/ M 957.70cm 2

!%= ;Fu}sf] lrqdf 7f];sf] k'/f nDafO 45cm / 7f];sf] cfotg 5390 3g ;]=ld= eP 7f];sf]
cw{Jof; lgsfNg'xf];\ .
15 cm

(pTt/ M 7cm )
!^= Pp6f 7f];sf] k'/f ;txsf] If]qkmn 175 ju{ ;]= ld= / a]ngfsf] nDafO 10cm eP pSt a]ngfsf]
cw{Jof; lgsfNg'xf];\ . (pTt/ M 5cm )
!&= h cm nDafO / 3cm cw{Jof; ePsf] Pp6f a]ngf / pxL cw{Jof; ePsf] Pp6f cw{uf]nf ldnL
ag]sf] 7f]; j:t'sf] cfotg 792 3g ;]=ld= eP h sf] dfg lgsfNg'xf];\ . pTt/ M 26 cm
!*= a]ngfsf] cfwf/sf] If]qkmn 100 ju{ ;]=ld= / a]ngfsf] prfO 3cm 5 . olb lbPsf] k'/f 7f];
j:t'sf] cfotg 500 3g ;]=ld= eP 7f]; j:t'sf] prfO kTtf nufpg'xf];\ . (pTt/ M 9cm )
!(= lbOPsf] lrqdf dfly cw{j[Ttfsf/ x'g] u/L Pp6f ;f]nLdf a/km 25 cm

el/Psf] 5 . olb ;f]nLsf] 58\s] prfO 25cm / cw{Jof; 7cm O

P
eP ;f] ;f]nLdf ePsf] a/kmsf] cfotg lgsfNg'xf];\ .
21 cm A
pTt/ M 1950.67cm3
@)= lbOPsf] 7f]; j:t'sf] k'/f ;txsf] If]qkmn kTtf nufpg'xf];\ .
38 cm

O
P

27 cm A
(pTt/ M 471.42m2 )
99
@!= Pp6f a]ngfsf] jj|m ;txsf] If]qkmn 1232 ju{ ;]=ld= 5 / o;sf] cfwf/sf] cw{Jof; / prfO
a/fa/ 5g\ eg] a]ngfsf] cw{Jof; / a]ngfsf] k'/f ;txsf] If]qkmn lgsfNg'xf];\ .
(pTt/ M 1956cm3, 28.01cm2)
@@= lrqdf lbOPsf] l;;fsndsf] cfotg / jj|m ;txsf] If]qkmn lgsfNg'xf];\ .

24 cm
(pTt/ M 1037.13cm2 )

@#= ;Fu}sf] lrqdf lbOPsf] 7f]; j:t'sf] k'/f ;txsf] If]qkmn lgsfNg'xf];\ .
28 cm

40 cm
(pTt/ M 753.50cm2
)

@$= b'O{cf]6f cfO;lj|mdsf] ;f]nLx¿ lrqdf b]vfpg'xf];\ . h:t} Ps cfk;df vlK6Psf 5g\ . olb
cfwf/sf] Jof;fw{ 3cm eP hDdf cfotg kTtf nufpg'xf];\ . olb 100 3g ;]=ld= cfO;lj|mdsf]
tf}n 20 u|fd eP cfO;lj|mdsf] tf}n kTtf nufpg'xf];\ .

(pTt/
M 94.28cm3 )
@%= cfotg / Jof;fw{ a/fa/ ePsf Pp6f uf]nf] / Pp6f a]ngf 5g\ . a]ngfsf] jj|m ;txsf]
If]qkmn / uf]nf]sf] ;txsf] If]qkmnsf] cg'kftdf 2:3 x'G5 egL l;b\w ug'{xf];\ .
(pTt/ M 18.85 gram )
@^= Pp6f a]ngfsf] prfO To;sf] cfwf/sf] Jof;sf] b'OltxfO{ 5 pSt a]ngfsf] cfotg 4cm
cw{Jof; ePsf] uf]nLsf] cfotg ;Fu} a/fa/ eP a]ngfsf] cw{Jof; kTtf nufpg'xf];\ .
(pTt/ M 4cm)

100
kf7 #=# lk/fld8 (Pyramid)

!= kl/ro
o; kf7cGtu{t ju{ cfwf/ ePsf] juf{sf/ lk/fld8 / ;dafx' lqe'h cfwf/ ePsf] ;dafx',
lqe'hfsf/ lk/fld8sf] jqm ;txsf] If]qkmn, k'/f ;txsf] If]qkmn / cfotg ;DaGwL ;d:of ;fy}
if9d'vf / lk/fld8sf] ;+o'Qm 7f]; j:t'sf] jqm;txsf] If]qkmn, k'/f ;txsf] If]qkmn / cfotg ;DaGwL
;d:of P;Pn;L k/LIffdf Pp6f nfdf] pTt/ cfpg] k|Zg cfp“5 .

@= cfwf/e"t tYo tyf ;"qx¿

58\s] prfO
prfO s
h
b

juf{sf/ cfwf/ ePsf] lk/fld8sf] jqm ;txsf] If]qkmn (LSA) =2bs

k'/f ;txsf] If]qkmn (TSL) = 2bs + b2 = b(2s+b)
cfotg (V) = 13 b2 × h = 13 b2h A E A A E A

b2 b2
s= A h2+ 4 , h =
A EA A s2 - 4 , b =
A EA A s 2 - h2
E

lqe'hfsf/ cfwf/ ePsf]

k'/f ;txsf] If]qkmn (T.S.A.) = rf/cf]6f lqe'hsf] If]qkmnsf] of]u
cfotg = 13 -cfwf/sf] If]qkmn_ × prfO
A E A

#= ljz]if Wofg lbg'kg]{ s'/fx¿

— s:tf] vfnsf] lk/fld8 xf] klxrfg u/L ;f]xL cg';f/ ;"q k|of]u ug]{ .
— ;xL ;"qsf] k}of]u u/L SofNs'n]6/sf] k|of]u u/L ;d:of ;dfwfg ug]{ .
$= gd"gf k|Zgf]Tt/ / cEof;
$=!= lbOPsf] juf{sf/ lk/fld8sf] cfotg lgsfNg'xf];\ .
10 cm

16 cm

101
 juf{sf/ cfwf/ ePsf] lk/fld8sf] cfef/e"hf (b) = 16 cm
58\s] prfO (s) = 10 cm
juf{sf/ lk/fld8sf] vf; prfO (h) = ?
juf{sf/ lk/fld8sf] cfotg (V) = ?
b2 162
xfdLnfO{ yfxf 5, prfO (h) = A l2 - 4 =
A AE A 102 - 4 =
A AE A 100 -64 =E A A 36 = 6 cm
E A

1 1 1
cfotg (V) = 3 b2 h = 3 × 162 × 6 = 3 × 256 × 6 = 512 cm3
A E A A E A A E A

$=!=!= cEof;sf nflu k|Zg

-s_ tnlbOPsf] juf{sf/ lk/fld8sf] cfotg lgsfNg'xf]];\ .
s_ v_ u_

17 cm

8 cm
12 cm

10 cm
12 cm 10 cm 16 cm

( Ans 384cm3 ) (Ans 400cm3) (Ans 1280cm3 )

3_ ª_

5 cm

7 cm

(Ans 1568cm3 ) (Ans 960cm3 )

$=!=!= lbOPsf] juf{sf/ lk/fld8sf] k'/f ;txsf] If]qkmn lgsfNg'xf];\ . . lrq ldnfpg]]

3 cm

8 cm

juf{sf/ cfwf/ ePsf] lk/fld8sf] cfwf/sf] e"hfsf] nDafO (a) = 8 cm

102
 juf{sf/ cfwf/ ePsf] lk/fld8sf] l;wf prfO (h) = 3 cm
juf{sf/ cfwf/ ePsf] lk/fld8sf] 58\s] prfO (s) = ?
juf{sf/ cfwf/ ePsf] k'/f ;txsf] If]qkmn (TSA) = ?
a2 82
xfdLnfO yfxf 5 , s = A h2 + 4 =
A EA 32 + 4 =
A A EA A 9 + 16 = 5 cm.
E A

TSA = 2as + a2 = 2. 8. 5 + 82 = 80 + 64 = 144 cm2

cEof;sf nflu k|Zg

!=tnsf lk/fld8x¿sf] cfotg kTtf nufpg'xf];\ .
s_ v_ u_

5 cm 8 cm

9 cm

24 cm 12 cm

3_ ª_
10 cm 12 cm

12 cm 18 cm

pTt/ M s_ 1200 cm3, v_ 384 cm3, u_ 1296cm3, 3_ 384 cm3,ª_ 756 cm3

%= ldl>t cEof;
!= Pp6f lrq juf{sf/ cfwf/ ePsf] 7f]; lk/fld8 xf] pSt lk/fld8sf] k'/f ;txsf] If]qkmn 189
ju{ ;]=ld= 5 . olb juf{sf/ cfwf/sf] e"hf 7 ;]=dL+ eP pSt lk/fld8sf] 58\s] prfO{ kTtf
nufpg'xf];\ . (Ans: 10. 5 cm )
@= Pp6f lrq juf{sf/ cfwf/ ePsf] 7f]; lk/fld8 xf] pSt lk/fld8sf] k'/f ;txsf] If]qkmn 675
ju{ ;]=ld= 5 . olb juf{sf/ cfwf/sf] e'hf 15 ;]=ld= eP pSt lk/fld8sf] 58\s] prfO kTtf
nufpg'xf];\ .(Ans 15 cm)
#= lbOPsf] lrq Pp6f juf{sf/ lk/fld8sf] xf] . h;sf] 7f8f] prfO 12 ;]=ld= / cfwf/sf] e'hfsf]
nDafO 10 ;]=ld=5g\ . eg] o;sf] k'/f ;txsf] If]qkmn kTtf nufpg'xf];\ . (Ans 360 cm2)

103
 A

12 cm
O

Q
10 cm E

$= lbOPsf] lrq Pp6f juf{sf/ lk/ld8 xf] h;sf] 7f8f] prfO 24 ;]=ld=/ cfwf/sf] e'hfsf] nDafO
14 ;]=ld= 5g\ eg] o;sf] k'/f ;txsf] If]qkmn kTtf nufpg'xf];\ . (Ans 896 cm2)
A

24 cm
O
Q
14 cm E

%= lbOPsf] lrq Pp6f juf{sf/ lk/fld8 xf] . h;sf] 7f8f] prfO 7cm / cfwf/sf] e'hfsf] nDafO 5
3
cm 5g\ eg] o;sf] k'/f ;txsf] If]qkmn kTtf nufpg'xf];\ . (99.33 cm )
A

7 cm
O

Q
5 cm E

^= ;“u}sf] lrqdf juf{sf/ cfwf/ ePsf] lk/fld8sf] k"/f ;txsf] If]qkmn 576 cm2 / juf{sf/
lsgf/fsf] nDafO 16 cm eP pSt lk/fld8sf] 58\s] prfO l;wf prfO / cfotg kTtf
nufpg'xf];\ . (10cm, 6 cm, 512 cm3)

16 cm

&= ;“u}sf] lrqdf juf{sf/ cfwf/ ePsf] lk/fld8sf] k'/f ;txsf] If]qkmn 800 cm2 / juf{sf/
lsgf/fsf] nDafO 16cm eP pSt lk/fld8sf] 58\s] prfO, l;wf prfO / cfotg kTtf
nufpg'xf];\ .
(17cm, 15 cm, 1280 cm3)
104
*= cfwf/ ju{ ePsf] Pp6f lk/ld8 lqe'hfsf/ ;txsf] lsgf/fsf] nDafO 17cm / cfwf/sf] e'hfsf]
nDafO 16cm eP pSt lk/fld8sf] k'/f ;txsf] If]qkmn lgsfNg'xf];\ . (736 cm2)
A

17 cm

16 cm

(= cfwf/ ju{ ePsf] Pp6f lk/fld8sf] lqe'hfsf/ ;txsf] lsgf/fsf] nDafO 20.8 cm / cfwf/sf]
e'hfsf] nDafO 16cm eP ;f] lk/fld8sf] k'/f ;txsf] If]qkmn kTtf nufpg'xf];\ . (870.4 cm2)
20.8 cm

16 cm

!)= cfwf/ ju{ ePsf] Pp6f lk/fld8sf] lqe'hfsf/ ;txsf] lsgf/fsf] nDafO 20.4cm / cfwf/sf]
e'hfsf] nDafO 19.2cm eP ;f] lk/fld8sf] k'/f ;txsf] If]qkmn lgsfNg'xf];\ . (1059.84 cm2)
20.4 cm

19.2 cm

!!= lbOPsf] lrq cfwf/ ju{ ePsf] 7f]; lk/fld8 xf] olb OB =13cm / OP= 12cm eP pSt lk/fld8sf]
k'/f ;txsf] If]qkmn lgsfNg'xf];\ . (360 cm2)
O

D C

A B
P

!@= lbOPsf] lk/fld8sf] k'/f ;txsf] If]qkmn kTtf nufpg'xf];] h;df AD prfO, AE 58\s] prfO /
2
24cm e'hf ePsf] juf{sf/ cfwf/ NMPQ 5 . (1390.56 cm )

105
 A

M N
12 cm

O E
Q
24 cm P

!#= lbOPsf] lk/fld8sf] k'/f ;txsf] If]qkmn kTtf nufpg'xf];\ h;df AO prfO AE 58\s] prfO /
2
25cm e'hf ePsf] juf{sf/ cfwf/ NMPQ 5 . (2250 cm )
A

M N
30 cm

O E
Q
25 cm P

!$= lbOPsf] juf{sf/ cfwf/ ePsf] lk/fld8sf] cfotg 750 cm3 / cfwf/sf] e'hfsf] cfwf/ (OM)
2
=7.5cm eP lqe'hfsf/ ;txsf] If]qkmn kTtf nufpg'xf];\ . (375 cm )
P

D C

7.5 cm
M
O

Q
B

!%= lbOPsf] juf{sf/ cfwf/ ePsf] lk/fld8sf] cfotg 1053.696 cm3 / cfwf/sf] e'hfsf] cfwf/
2
OM=8.4cm eP lqe'hfsf/ ;txsf] If]qkmn kTtf nufpg'xf];\ . (752.64 cm )
P

D
C

O 8.4 cm M

A
9.6 cm B

106
 If]q M 4 aLh ul0ft (Algebra)
If]q kl/ro
dfWolds lzIff kf7\oj|md 2064 / ljlzi6Ls/0f tflnsf 2065 n] sIff 10 sf] clgjfo{ ul0ft ljifodf
aLh ul0ftsf] lgDgfg';f/sf ljifo j:t', k/LIffsf nflu ;+1fgfTds If]qcg';f/sf k|Zg ;ª\Vof /
cª\sef/sf] Joj:yf u/]sf] 5 .
j|m; ljifoj:t'÷;+1fg If]q 1fg l;k hDdf l;k ;d:of hDdf s'n If]qut
/ cª\s ;dfwfg cª\s hDdf s'n
af]w cª\s hDdf
cª\s
4
5f]6f pTt/ nfdf] pTt/
cfpg] cfpg] k|Zg
4.1
n=;= (L.C.M.) / d=;= 1 4 4
(H.C.F.)
lahLo leGgx¿sf]
4.2 ;/nLs/0f 1 4 4
(implification of algebric
fraction)
4.3 3ftfª\s (indices) 1 2
1 4 10
4.4
d"n / ;8{ 1 1 4 24
(roots and surds)
o'ukt/]vLo ;dLs/0f
/ ju{ ;dLs/0f
;DaGwL zflAbs
4.5 1 2 1 4 6
;d:ofx¿ (verbal
problems of simultaneous
equation and quadratic
equation)
Total 1 3 8 1 3 16 24 24

kf7 : 4.1 dxTtd ;dfjt{s / n3'Ttd ;dfjt{s -d=;= / n=;=_

(Highest Common Factor and Lowest Common Multiple)

1= kl/ro
lbOPsf b'O{ jf b'O{eGbf a9L aLh ul0ftLo cleJo~hsx¿sf] ;femf u'0fg v08 jf ;femf u'0fg
v08x¿sf] u'0fgkmnnfO{ d=;= (H.C.F) elgG5 eg] b'O{ jf b'O{eGbf a9L aLh ul0ftLo
cleJo~hsx¿n] lgz]if efu hfg] ;a}eGbf ;fgf] cleJo~hsnfO{ n=;= (L.C.M.) elgG5 . o;
107
PsfOdf aLh ul0ftLo cleJo~hsx¿sf] u'0fg v08 ljlwaf6 d=;= / n=;= lgsfNg] tl/sfx¿sf
af/]df rrf{ ul/g] 5 .
o; kf7af6 nfdf] pTt/ cfpg] Pp6f ;d:of ;dfwfg txsf] k|Zg ;f]lwg] 5 . h;sf] cª\s ef/ 4
/xg] 5 .
2. o; kf7df k|of]u x'g] cfwf/e"t ;"qx¿
-s_ d=;= (H.C.F) = ;femf u'0fg v08x¿ (common factors)
-v_ n=;= (L.C.M) = ;femf u'0fg v08 × af“sL u'0fg v08x¿
(common factors × remaining factors)
-u_ n=;= × d=;= = klxnf] cleJo~hs × bf];|f] cleJo~hs
L.C.M × H.C.F = First Expression × Second Expression
-3_ aLh ul0ftLo ;"qx¿
-c_ (a + b)2 = a2 + 2ab + b2 or, (a + b)2 = (a – b)2 + 4ab
-cf_ (a – b)2 = a2 – 2ab + b2
or (a + b)2 – 4ab = (a – b)2
-O_ a2 – b2 = (a + b) (a – b)
-O{_ a2 + b2 = (a + b)2 – 2ab = (a – b)2 + 2ab
-p_ a3 + b3 = (a + b) (a2 – ab + b2) = (a + b)3 – 3ab (a+ b)
-pm_ a3 – b3 = (a – b) (a2 + ab + b2) = (a – b)3 + 3ab (a–b)
-P_ (a + b)3 = a3 + 3a2b + 3ab2 + b3
-P]_ (a – b)3 = a3 – 3a2b + 3ab2 – b3
-cf]_ (a + b + c)2 = a2 + b2 + c2+ 2ab + 2bc + 2ca
-cf}_ a3 + b3 + c3– 3abc =(a + b + c) (a2 + b2 + c2 – ab – bc – ca)

3. ljz]if Wofg lbg'kg{] s'/fx¿

o; PsfOdf
1. aLh ul0ftLo cleJo~hsx¿sf] u'0fg v08 ljlwaf6 d=;= lgsfNg'kg{] x'G5 .
2. aLh ul0ftLo cleJo~hsx¿sf] u'0fg v08 ljlwaf6 n=;= kTtf nufpg'kg{] x'G5 .
3. lbOPsf b'O{ jf ltg cf]6f dfq aLh ul0ftLo cleJo~hsx¿sf] d=;=/ n=;= lgsfNg'kg{]
x'G5 .
4. lbOPsf cleJo~hsx¿sf] d=;= eGgfn] ;a} cleJo~hsx¿sf] ;femf u'0fg v08 jf u'0fg
v08x¿sf] u'0fg kmn xf] .

5. d=;= lgsfNbf,
– ;a} cleJo~hsx¿sf] v08Ls/0f ug]{,
– tL cleJo~hsx¿sf] ;femf u'0fg v08x¿ lng]
– ;femf u'0fg v08x¿nfO{ u'0fgkmnsf] ¿k (product form) df n]Vg]

108
6. n=;= eGgfn] b'O{ jf b'O{eGbf a9L aLh ul0ftLo cleJo~hsx¿sf] ;femf u'0fg v08 × af“sL
u'0fg v08nfO{ a'emfp“5 .

4.gd'gf k|Zgf]Tt/ tyf cEof;

gd'gf 1 :
d=;= lgsfNg'xf];\ M (Find the H.C.F)
x 3 - 64y3, x 2 - 6x y + 8y2 and x 2 - 16y2
;dfwfg
oxfF, klxnf] cleJo~hs : x 3 - 64y3
= x 3 – (4y)3
= (x -4y){x2+ xx 4y+(4y)2}
= (x -4y)(x2+ 4xy+16y2)
bf];|f] cleJo~hs : x 2 - 6xy + 8y2
= x 2 – (4+2) xy + 8y2
= x 2 – 4xy - 2xy + 8y2
= x(x - 4y) – 2y(x – 4y)
= (x - 4y) (x – 2y)
t];|f] cleJo~hs : x 2 - 16y2
= x 2 – (4y)2
= (x – 4y)(x + 4y)
d=;= ( H.C.F) = ;femf u'0fg v08x¿ (common factors)
= (x - 4y)
1. cEof;sf nflu k|Zgx?
(a) 4x 2 - 9, 2x 2 + x - 3 and 8x 3 + 27
(b) 3p2 - 9pq - 12q2, 4p2 - 18pq + 8q2 & 3p2 - 18pq + 24q2
(c) x 3 - 9x y2, 2x 2 + 11x y + 15y2 &x 3 + 27y3
(d) 4x 4 + 16x 3 - 20x 2, 3x 3 + 14x 2 - 5x & x 4 +125x
(e) 16x3y - 4 x 2y2 - 30x y3, 32x 4 - 50x 2y2 & 4x 2y2 + 9x y3 + 5y4
(f) (a + 2b)2, (a -2b)2 + 8ab & a4 - 16b4
(g) 3t4 - 8t3 + 4t2, t5 - 8t2 & 4t3 - 10t2 + 4t
(h) 36 x 4 - 81x 2, 24x 3 - 72x 2 + 54x & 24x 6 - 81x 3
pTt/x¿
(a) (2x +3), (b) p - 4q, (c) x + 3y, (d) x (x + 5), (e) 4 x + 5y, (f) a + 2b
(g) t (t - 2), (h) 3 x (2 x - 3)

gd'gf 2 :
d=; lgsfNg'xf];\ M (Find the H.C.F)
x + x y + y , x 4 + x 3y + x2y2 and x4 - xy3
4 2 2 4

109
;dfwfg M
oxfF, klxnf] cleJo~hs M x 4 + x2y2 + y4
= (x 2)2 + (y2)2 + x 2y2
= (x 2 + y2)2 – 2x2y2+ x 2y2
= (x 2 + y2)2 – x2y2
= (x 2 + y2)2 – (xy)2
= (x 2 + y2 + xy) (x 2 + y2 - x y)
= (x 2 +xy+ y2 ) (x 2 - x y + y2 )
bf];|f] cleJo~hs M x 4 + x 3y + x 2y2
= x 2 (x 2 + x y + y2)

t];|f] cleJo~hs M x4 - xy3

= x (x3 – y3)
= x (x – y) (x 2+ xy+y2)
d=;= ( H.C.F) = ;femf u'0fg v08x¿ (common factors)
= (x 2 + x y + y2)

cEof;sf nflu k|Zgx?

(a) 8a3 + b3, 16a4 + 4a2b2 + b4
(b) x3 - 8y3, x4 + 4x 2y2 + 16y4
(c) a4 + 4a2 + 16, a3 + 8
(d) a4 + a2 + 1, a3 - 1
(e) x 4 + 4x2 + 16, x 3 - 8
(f) m4 + m2n2 + n4, m2n3 + m5
(g) 4x4 + 19x 2y2 + 49y4, 4x3 + 14x y2 - 6 x 2y
(h) x 3 - 1, x4 + x 2 + 1, x3 + 1 + 2x2 + 2x
(i) 4x 4 +16x 3 - 20x 2, 3x3 + 14x 2 - 5x and x 4 + 125x
pTt/x¿
(a) 4a2 - 2ab + b2, (b) x 2 + 2 x y + 4y2, (c) a2 - 2a + 4, (d) a2 + a + 1, (e) x 2 + 2 x + 4,
(f) m + mn +n2, (g) 2x 2 - 3xy +7y2, (h) x 2 + x +1, (i) x (x+5)
2

gd'gf 3 :
d=;= lgsfNg'xf];\ (Find the H.C.F)
x - 2x - x + 2, x 3 - 3x 2 + 2x and x 2 - x - 2
3 2

;dfwfg M
oxfF, klxnf] cleJo~hs M x3 - 2x 2 - x + 2
= x 2(x – 2) - 1(x – 2)
= (x – 2) (x 2 – 1)
= (x – 2) (x 2 – 12)
= (x – 2) (x + 1) (x – 1)

110
bf];|f] cleJo~hs M x 3 - 3 x 2 + 2x
2
= x (x - 3x + 2)
= x {x 2 - (2+1) x+2}
= x {x 2 – 2x – x +2}
= x { x (x – 2) – 1(x – 2)}
= x (x – 2) (x – 1)
t];|f] cleJo~hs M x2 - x - 2
= x 2 – (2-1) x - 2
= x2 – 2 x + x - 2
= x (x – 2) + 1(x – 2)
= (x – 2) (x + 1)
d=;= ( H.C.F) = ;femf u'0fg v08x¿ (common factors)
= (x – 2)
yk cEof;sf nflu k|Zgx?
d=;= lgsfNg'xf];\ M (Find the H.C.F)
(a) a3 - a2 + a - 1 / 2a3 - a2 + a - 2
(b) x 3 - x 2 + x - 1 / 2x 3 - x 2 + x - 2
(c) p3+1, p4+p2+1 / p3+1+2p2+2p
(d) 8a3 + 1 / 16a4 - 4a2 + 4a - 1
(e) 8x 3 - 1 / 16x 4 - 4x 2 - 4 x - 1
(f) 2x 3 - 16, x 2 - 4x + 4 / x 2 - 3x + 2

pTt/x?
(a) a - 1, (b) x – 1, (c) 1, (d) 4a2 - 2a + 1, (e) 4x 2 + 2x + 1, (f) x - 2

gd'gf 4 :
d=;= lgsfNg'xf];\ M(Find the H.C.F)
x - y + z + 2x z, x 2 + y2 - z2 + 2xy, z2 + y2 - x 2 + 2yz
2 2 2

;dfwfg M
oxfF, klxnf] cleJo~hs : x 2 - y2 + z2 + 2xz
= x 2 + 2x z+ z2 - y2
= (x + z)2 – y2
= (x + z + y) (x + z - y)
= (x + y + z) (x + z - y)
bf];|f] cleJo~hs : x 2 + y2 - z2 + 2 x y
= x 2 + 2xy + y2 - z2
= (x + y)2 – z2
= (x + y + z) (x + y - z)
t];|f] cleJo~hs : z2 + y2 - x2 + 2yz

111
 = y2+ 2yz + z2 - x2
= (y + z)2 – x 2
= (y + z + x) (y + z - x)
= (x + y + z) (y + z - x)
d=;= ( H.C.F) = ;femf u'0fg v08x¿ (common factors)
= (x + y + z)
cEof;sf nflu k|Zgx?
d=; lgsfNg'xf];\ M (Find the H.C.F)
(a) a2 + 2ab + b2 - c2, b2 + c2 - a2 + 2bc, c2 + a2 + 2ca - b2
(b) e2 + f2 - g2 + 2ef, e2 - f2 + g2 + 2ge, g2 + f2 + 2fg - e2
(c) 9m2 - 4n2 - 4nr - r2, r2 - 4n2 - 9m2 - 12mn and 9m2 + 6mr + r2 - 4n2
pTt/x¿
(a) a + b+ c, (b) e + f + g, (c) 3m + 2n + r

gd'gf 5 :
n=;= lgsfNg'xf];\ (Find the L.C.M.)
a5 + a3b2 + ab4 and a4b + ab4

;dfwfg M
oxfF, klxnf] cleJo~hs : a5 + a3b2 + ab4
= a{(a2)2 + (b2)2+ a2b2}
=a{ (a2)2 + (b2)2+ a2b2}
= a{(a2 + b2)2 - 2a2b2 + a2b2}
= a{(a2 + b2)2 – (ab)2}
= a (a2 + b2 + ab) (a2 + b2 – ab)
bf];|f] cleJo~hs M a4b + ab4
= ab(a3 – b3)
= ab(a – b) (a2 + ab + b2)
n=;= (L.C.M.) = ;femf u'0fg v08 × af“sL u'0fg v08x¿
(common factors × remaining factors)
= ab (a – b) (a2 + ab + b2) (a2 + b2 – ab)

cEof;sf nflu k|Zgx?M

n=;= lgsfNg'xf];\ (Find the L.C.M.)
(a) x 4 + x 2y2 + y4 & x 4 - xy3
(b) 6x 2 - x - 1 & 54 x 4 + 2x
(c) (2x 2 - x - 1) & 24x4 + 3x
(d) x 3 - 3x 2 - x + 3 & x 3 - x 2 - 9x + 9
(e) x 3 + 2x 2 - x - 2 & x3 + x 2 - 4x - 4
(f) x 3 - 9x, x 4 - 2x3 - 3x 2 & x3 - 27
(g) a3 - 4a, a4 + a3 - 2a2 and a3 - 8

112
pTt/x¿
(a) x (x -y) (x 2 + x y + y2) (x 2 - x y + y2)
(b) 2x (3x +1) (2x -1) (9x 2-3x +1)
(c) 3x (x -1), (2x +1)(4x 2 - 2x + 1)
(d) (x + 3) (x + 1) (x - 1) (x - 3)
(e) (x - 1) (x - 2) (x +1) (x + 2)
(f) x 2 (x + 3) (x 3 - 27) (x + 1)
(g) a2(a2 - 4) ( a2 + 2a + 4)(a - 1)

gd'gfM 6 :
d=;= / n=;= lgsfNg'xf];\ (Find the H.C.F. and L.C.M.)
2(x 2 - y2), 4(x 3 - y3) / 6(x 4 - y4)
;dfwfg M
oxfF, klxnf] cleJo~hs : 2(x 2 - y2),
= 2(x + y) (x - y)
bf];|f] cleJo~hs M 4(x 3 - y3)
= 2×2(x - y) (x 2 + xy + y2)
t];|f] cleJo~hs M 6(x 4 - y4)
= 6{(x 2)2 – (y2)2}
= 6(x 2 + y2) (x 2 – y2)
= 2×3(x 2 + y2) (x + y)( x – y)
d=;. ( H.C.F) = ;femf u'0fg v08x¿ (common factors)
= 2(x – y)
n=;= (L.C.M.) = ;femf u'0fg v08 × af“sL u'0fg v08x¿
(common factors × remaining factors )
= 2(x – y)(x + y)2×3(x 2 + y2) (x 2 + x y + y2)
= 12(x – y)(x + y)(x 2 + y2) (x2 + xy + y2)
= 12(x 2– y2)(x 2 + y2) (x 2 + x y + y2)

cEof;sf nflu k|Zgx?

d=;= / n=;= lgsfNg'xf];\ (Find the H.C.F. and L.C.M.)
3 3 4 2
(a) x - 1, (x - 1) , x + x + 1
(b) x 3y + y4, x 4 + x 2y2 + y4, 2ax 3 - 2ax 2y + 2axy2
(c) 2 x 3 - x 2 - x, 4x 4 - 10x 3 - 6x 2, 8x 2 + 4x
(d) 8a6x - 8a3 x 4, 4a6 x 2 - 4a2x 6, 6x 3a3 + 24a2x 4 - 30ax 5
(e) 3a3b + a2b2 - 10ab3, 6a4b - a3b2 - 15a2b3, 6a3b - 19a2b2+15ab3
(f) x2 - 2xy + y2, 2x2 + xy - 3y2, x3 - 3x2y + 3xy2 - y3

113
pTt/x?
(a) H.C.F = 1, L.C.M. = (x - 1)3 ( x2 + x + 1) (x2 - x +1)
(b) H.C.F. = x2 - xy +y2, L.C.M = 2axy ( x+ y) (x2 - xy + y2) (x2 + xy + y2)
(c) H.C.F.= x(2x+1), L.C.M.= (4x2 (2x +1) ( x- 1) (x - 3)
(d) H.C.F. = 2ax ( a - x), L.C.M.= 24a3x3 ( a - x) (a + x) (a2 + x2) (a + 5x) (a2 + ax + x2)
(e) H.C.F. = ab(3a - 5b), L.C.M.= a2b(3a - 5b) (a + 2b) (2a + 3b) (2a - 3b)
(f) H.C.F.= x - y, L.C.M.= (x- y)3 ( 2x + 3y)

yk cEof;sf nflu k|Zgx? (Challenging Problems)

d=;= lgsfNg'xf];\ M (Find the H.C.F.)
(a) x4 + 2x2 + 9 & x3 - x + 6
(b) x3 - 3x - 2 & x3 - x2 - 4
(c) a4 - 6a2 - 7 + 8x - x2 & a3 - ax + a
(d) x4 - 10x2 + 24 + 6y - 9y2 & x3 + 3xy - 6x
Hints (d):
;dfwfg M
oxfF, klxnf] cleJo~hs : x 4 - 10 x 2 + 24 + 6y - 9y2
= (x 2)2 - 2. x 2.5 + 52 - 1 + 6y - 9y2
= (x 2 - 5)2 - [12-2.1.3y + (3y)2
= (x 2 - 5)2 - ( 1- 3y)2
= (x 2 + 3y-6) (x 2 - 3y - 4)
pTt/x¿
(a) x 2 - 2x + 3 (b) x - 2 (c) (a2 - x + 1)
2
(d) x + 3y - 6

yk cEof;sf nflu k|Zgx?

d=;= / n=;= lgsfNg'xf];\ (Find the H.C.F. and L.C.M.)
(a) x 2 + 2x - 3, x 3 + 3x 2 - x - 3
(b) x 2 - 4x + 3, x 3 - x 2 - 14x + 24
(c) 3x 2 + 16x - 35, x3 + 343, 2x 2 - 9x - 161
(d) x 3 - 3x + 2, x 3 + 4x 2 - 8x + 3
(e) 3x 2 - 22 x + 19, 2x 3 + 3x - 5
pTt/x?
(a) H.C.F. = (x - 1) (x + 3), L.C.M.= (x - 1) (x + 3) (x + 1)
(b) H.C.F.= x - 3, L.C.M.= (x - 1) (x - 2) (x - 3) (x + 4)
(c) H.C.F. = (x + 7), L.C.M = (x + 7) (3x - 5) (2x - 23) (x 2 - 7x + 49)
(d) H.C.F. = (x - 1), L.C.M.= (x - 1)2 (x + 2) (x2 + 5x - 3)
(e) H.C.F. = (x - 1), L.C.M.= (x - 1) (3x - 19) (2x 2 + 2x + 5)

114
kf7 : 4.2 aLhLo leGgx¿sf] ;/nLs/0f
(Simplification of Algebraic Fractions)

1. kl/ro
o; PsfOdf rf/ cf]6f;Dd lahLo leGgx¿ ;dfj]z ePsf ;d:ofx¿sf] ;/nLs/0f ug{] tl/sfx¿sf
af/]df rrf{ ul/Psf] 5 . o; PsfOaf6 nfdf] pTt/ cfpg] Ps cf]6f ;d:of ;dfwfg d"ns k|Zg
;f]lwg] ul/G5 . h;sf] cª\s ef/ 4 x'G5 .

2= Wofg lbg'kg{] s'/fx¿ M

a) lahLo leGg ;DaGwL o;eGbf cufl8 cWoog u/]sf s'/fx¿sf] k'g/fjnf]sg ug]{] .
b) aLh ul0ftLo ;"qx¿af/] k'g/fjnf]sg ug]{ .
c) ul0ft ;DalGwt cfwf/e"t lj|mofsf af/]df k'g/fjnf]sg ug]{] .
d) aLh ul0ftLo n=;= af/] k'g/fjnf]sg ug{] .
e) rf/ cf]6f;Dd aLhLo leGgx¿ ;dfj]z ePsf ;d:ofx¿sf] ;/nLs/0f ug]{] .
f) lahLo leGgx¿sf] ;/nLs/0f ubf{ cem ljz]if lgDg lnlvt s'/fx¿df Wofg lbg'kg]{M
• aLhLo leGgx¿sf x/nfO{ v08Ls/0f ug]{ .
• b'O{ b'O{ kbx¿sf] of]u÷cGt/ lgsfNg] .
• n3'Ttd kbdf n}hfg] .

3= gd'gf k|Zgf]Tt/ tyf cEof;

3.1 nfdf] pTt/ cfpg] k|Zgx¿

gd'gf 1 :
;/n ug'{xf];\ (Simplify)
1 1 2 4
A

x-1 - x+1 - x2 + 1 - x4 + 1
E A A E A A E A A E

;dfwfg M
1 1 2 4
oxfF, x-1 - x+1 - x2 + 1 - x4 + 1
A E A A E A A E A A E A

1(x+1) - 1(x-1) 2 4
= (x-1)(x+1)
A

E
- x2 + 1 - x4 + 1 E A A E A A E

x+1 - x+1 2 4
= x2-1
A - x2 + 1 - x4 + 1 E A A E A A E

2 2 4
= x2-1 - x2 + 1 - x4 + 1
A E A A E A A E

2(x2+1) -2(x2-1) 4
= (x2-1)(x2+1)
A

E
- x4 + 1 E A A E

2x2+2 -2x2+2 4
= A

x4-1 - x4 + 1 E A A E

4 4
= x4-1 - x4 + 1
A E A A E

115
 4(x4+1) - 4(x4-1)
= (x4 + 1)(x4-1)
A

E
E

4x4+4 - 4x4+4
= x8-1 A E

8
= x8-1 A E

1.1 cEof;sf nflu k|Zgx¿ (Questions for Practice)

;/n ug'{xf];\ (Simplify)
1 1 2 4
1. a - 1 - a+1 - a2 + 1 - a4 + 1
A

E
E A

b b 2a 4a3b
2. a - b + a+b + a2 + b2 + a4 + b4
A

E
E

a a 2a 16a3
3. 1 - 2a + 1 + 2a + 1+4a2 + 16a4 - 1
A

E
E A

a a 6a2 8a4
4. a-b + a + b - a2 - b2 + a4 - b4
A

E
E A

a a 2a 16a3
5. 1-2a + 1+2a + 1+4a2 +16a4-1
A

E
E A A E

pTt/x¿
8 8a7b 4a
1.1 1. a8-1 A E A 2. a8-b8 A E A 3. 1+4a2A E A

4a2 4a
4. a2 + b2
A E A 5. 1+4a2 A E A

gd'gf 2 :
;/n ug'{xf];\ (Simplify)
1 - a 1 +a 4a 8a
A

1 + a - 1 - a - 1 + a2 - 1+ a4 E
E A

;dfwfg M
oxfF, 11 +- aa - 11 -+aa - 1 4a 8a
+ a2 -
1+ a4 A

E
E

(1 - a)2 - (1+a)2 4a 8a
= (1 + a)(1 - a) - 1 + a2 - 1+ a4
A

E
E

1 - 2a +a2 - 1-2a - a2 4a 8a
= A

1 - a2 - 1 + a2 - 1+ a4 E
E

- 4a 4a 8a
= 1 - a2 - 1 + a2 - 1+ a4
A

E
E

- 4a(1+ a2) - 4a(1 - a2) 8a

= A

(1 - a2)(1+ a2) - 1+ a4 E
E

- 4a - 4a3 - 4a + 4a3 8a
= A

1 - a4 - 1+ a4 E
E

116
 - 8a 8a
= 1 - a4 - 1+ a4
A

E
E

- 8a(1+a4) - 8a(1 - a4 )
= A

(1 - a4 ) (1+ a4)
E
E

- 8a - 8a5 - 8a + 8a5
= 1 - a8
A
E

- 16a
= 1 - a8 A E

- 16a
= - ( a8 - 1)
A E

16a
= a8 - 1 A E

1.2cEof;sf nflu k|Zgx¿ (Questions for Practice)

;/n ug'{xf];\ (Simplify)
1-x 1+x 4x 8x
1. 1+x - 1-x - 1+x2 - 1+ x4
A E A A E A A E A

1+y 1-y 4y 8y3

2. - -
1-y 1+y 1+y
A 2+
1+y4 E
E A

b 2ab a 4a3b
3. 1 + a-b + a2 + b2 - a+b + a4 + b4
A

E
E A

1 2x x 4x3
4. 1 + x-1 + x2 + 1 - x + 1 + x4 + 1
A

E
E

1 1 1 1
5. y + x y - y2 - x2 y2
A E A

1+x y - x 1+x2 y2 - x2 E

1 1 1 1
6. b + a b - b2 - a2 b2
A E

1+a b - a 1+a2 b2 - a2 E

pTt/x¿
16x 16y3 8a7b
1.2 1. x8-1 A E A 2. 1-y8 A E A 3. a8 - b8 A E A

8x7 x2y2 a2b2

4. x8 - 1 A E A 5. x4 - y4
A E A 6. a4 - b4 A E A

gd'gf 2 :
;/n ug'{xf];\ (Simplify)
1 2x 4x3 8x7
A + 2 2+ 4 4- 8
x+y x + y x + y x - y8 E
E A

;dfwfg M
1 2x 4x3 8x7
oxfF, x+y + x2 + y2 + x4 + y4 - (x4)2 - (y4)2
A

E
E

1 2x 4x3 8x7
= x+y + x2 + y2 + x4 + y4 - (x4 - y4)(x4+y4)
A

E
E

117
 1 2x 4x3(x4 - y4) - 8x7
= x+y + x2 + y2 + (x4 - y4)(x4+y4)
A

E
E

7 3 4 7
1 2x 4x - 4x y - 8x
= x+y + x2 + y2 + (x4 - y4)(x4+y4)
A

E
E

1 2x - 4x7- 4x3 y4
= x+y + x2 + y2 + (x4 - y4)(x4+y4)
A

E
E

1 2x - 4x3 (x4 + y4)

=x+y + x2 + y2 + (x4 - y4)(x4+y4)
A

E
E

1 2x - 4x3 (x4 + y4)

= x+y + x2 + y2 + (x4 - y4)(x4+y4)
A

E
E

1 2x 4x3
= x+y + x2 + y2 - x4 - y4
A

E
E

1 2x 4x3
= x+y + x2 + y2 - (x2)2- (y2)2
A

E
E

1 2x 4x3
=x+y + x2 + y2 - (x2- y2)(x2+y2)
A

E
E

1 2x(x2 - y2) - 4x3×1

= x+y + (x2 + y2)(x2- y2)
A

E
E

1 2x3 -2xy2 - 4x3

= x+y + (x2 + y2)(x2- y2)
A

E
E

1 -2x3 -2xy2
= x+y + (x2 + y2)(x2- y2)
A

E
E

1 -2x(x2 + y2 )
= x+y + (x2 + y2)(x2- y2)
A

E
E

1 2x
= x+y - x2- y2
A

E
E

1 2x
= x+y - (x- y)(x+y)
A

E
E

1(x-y) - 2x ×1
= (x- y)(x+y)
A

E
E

x-y - 2x
=(x- y)(x+y)
A E

-x-y
= (x- y)(x+y)
A E

-(x+y)
= (x- y)(x+y)
A

E
E

-1
= x- y
A E

-1
= -(-x+ y )
A E

1
= y-xA E

118
cEof;sf nflu k|Zgx¿ (Questions for Practice)
;/n ug'{xf];\ (Simplify)
1 2 4 8
1. x+1 + x2 + 1 + x4+1 + x8 - 1
A

E
E A

1 2a 4a3 8a7
2. a+b + a2+b2 + a4+b4 - a8-b8
A

E
E A

1 2b 4b3 8b7
3. a+b + a2 + b 2 + a4 + b4 - b8 - a8
A

E
E A

1 2a 4a3 8a7
4. a+x - a2+x2 - a4+x4 + a8-x8
A

E
E A

1 2y 8y7 4y3
5. y+p - y2+p2 + y8-p8 - y4+p4
A

E
E

pTt/x¿
1 1 1
1. x-1 A E A 2. b-a
A E A 3. a-b A E A

3a-x 3y-p
4. a2-x2 A E A 5. y2-p2 A E

gd'gf 3.1 :
;/n ug'{xf];\ (Simplify)
1 1 2x
1+ x + x2 - 1- x + x2 + 1 + x2 + x4
A

E
E A

;dfwfg M
oxfF, 1+ x1+ x2 - 1- x1+ x2 + 1 + x2x2 + x4
A

E
E A

2 2
1(1- x + x ) - 1(1+ x + x ) 2x
= ( 1+ x + x2)(1- x + x2) + 1 + x2 + x4
A

E
E

1- x + x2 - 1 - x - x2 2x
= 1+ x + x2 - x-x2-x3 + x2+x3+x4 + 1 + x2 + x4
A

E
E

- 2x 2x
= 1+ x2+x4 + 1 + x2 + x4
A

E
E

- 2x +2x
= 1+ x2+x4
A E

0
= 1+ x2+x4
A E

=0

3.1 cEof;sf nflu k|Zgx¿ (Questions for Practice)

;/n ug'{xf];\ (Simplify)
1 1 2a
1. 1+a+a2 - 1-a+a2 + 1+a2+a4
A

E
E A

1 1 2x
2. 1 - x + x2 - 1 + x + x2 + 1 + x2 + x4
A

E
E A

a+2 a-2 2a2

3. 1 + a+ a2 - 1 - a + a2 - 1 + a2 + a4
A

E
E

119
pTt/x¿
4x 4
3.1. 1. 0 2. 1 + x2 + x4
A E A 3. a4 +a2 +1
A E

gd'gf 3.2 M
;/n ug'{xf];\ (Simplify)
a-5 a+5 250
A

a2 - 5a + 25 + a2 + 5a + 25 - a4 + 25a2 + 625 E A A E A A E A

;dfwfg M
oxfF, a2 - a5a- 5+ 25 A E A
a+5
A
250
+ a2 + 5a + 25 - a4 + 25a2 + 625 E A A E

2 2
(a - 5) (a + 5a + 25)+(a+5)( a - 5a + 25) 250
= A

(a2 - 5a + 25)( a2 + 5a + 25) - a4 + 25a2 + 625

E
E A A E

a3 - 53+a3+53 250
= a4+5a3+25a2 - 5a3- 25a2 - 125a+25a2+125a+625 - a4 + 25a2 + 625
A E A A E

2a3 250
= a4+25a2+625 - a4 + 25a2 + 625
A E A A E

2a3 -250
= a4+25a2+625
A E

2(a3 -125)
= a4+25a2+625
A

E
E

2(a3 -53)
= a4+25a2+625
A

E
E

2(a -5)(a2+5a+25)
=( a2 - 5a + 25)( a2 + 5a + 25)
A

E
E

2(a -5)
= a2 - 5a + 25
A

E
E

3.2 cEof;sf nflu k|Zgx¿ (Questions for Practice)

;/n ug'{xf];\ (Simplify)
a-4 a+4 128
1. a2-4a+16 + a2+4a+16 - a4-16a2+256
A E A A E A A E A

a-2 a+2 16
2. a2-2a+4 + a2+2a+4 - a4+16+4a2
A E A A E A A E A

p +3 p-3 54
3. p2+3p+9 + p2- 3p+9 + p4+9p2+81
A E A A E A A E A

2x - y 2x + y 2y3
4. A 2
4x - 2xy + y 2 + 2 -
4x + 2xy + y 16x + 4x2y2 + y4
2 4 E A A E A A E A

3x-1 3x + 1 54x3
5. A - 2 +
9x - 3x + 1 9x +3x+1 81x4+9x2+1
2
E
E A

2a+b 2a-b 2b3

6. A 2+ 2 2-
4a +2ab+b 4a -2ab+b 16a +4a2b2+b4
2 4
E
E A

pTt/x¿
2(a - 4) 2(y-2) 2p-3
3:2. 1. (a2 - 4a + 16) 2. y2 - 2y+4 E
A E A A

E
E A 3. p4+9p2+81
A E

120
 2(2x - y) 2(3x-1) 2(2a-b)
4. 4x2 - 2xy + y2
A

E
E A 5. 9x2-3x+1
A

E
E A 6. 4a2-2ab+b2
E
A E A

gd'gf 4 :
;/n ug'{xf];\ (Simplify)
1 2 1 2
A

x-a - 2x+a + x+a -2x-a E

E

;dfwfg M
1 1 2 2
oxfF, x-a + x+a- 2x+a -2x-a A

E
E

1 1 2 2
= [x-a + x+a]- [2x+a +2x-a ]
A

E
E A

x+a+x - a 2(2x - a)+2(2x+a)

=(x-a)(x+a) - (2x+a)(2x -a)
A

E
E

4x - 2a+4x+2a
2x
=x2-a2 -
A

4x2 - a2 E E

2x 8x
= x2-a2 - 4x2 - a2
A

E E

2x(4x2 - a2) - 8x(x2-a2)

= (x2-a2 ) (4x2 - a2)
A

E E

8x3 -2xa2 - 8x3+ 8xa2

= (x2-a2 ) (4x2 - a2)
A

6xa2
= (x2-a2 ) (4x2 - a2)
A

cEof;sf nflu k|Zgx? (Questions for Practice)

;/n ug'{xf];\ (Simplify)
1 2 1 2
1. x-1 - 2x+1 + x+1 - 2x-1
A

E
E

1 1 1 1
2. x-5 - x-3 + x+5 - x+3
A

E
E A

a3 a3 1 1
3 . a-1 + a+1 - a-1 + a+1
A

E
E

x3 x2 x 1
4. x-1 - x+1 - x-1 + x+1
A

E
E

2 m 1 3
5. m-1 - m2+1 - m+1 - m2-1
A

E
E A

4 2m 2 6
6. m-1 - m2+1 -m+1 + 1-m2
A

E
E

pTt/x¿
6x 32x
1. A

(x2 - 1) (4x2 - 1) E A 2. (x2 - 25) (x2 - 9) A E A 3. 2(a2 + 1)

2m 4m
4. x2 + 1 5. m4-1 A E A 6. m4-1
A E A

121
gd'gf 5 :
;/n ug'{xf];\ (Simplify)
(n-r)2 - m2 (m - n)2 - r2 (r-m)2 - n2
n2 - (r+ m)2 + m2 -(n+r)2 + r2 - (m+n)2
A

E
E A A

E E
A A

E
E A

;dfwfg M
2
- m2 (m - n)2 - r2 (r-m)2 - n2
oxfF, n(n-r)
2
- (r+ m)2 + m2 -(n+r)2 + r2 - (m+n)2
A

E
E A A

E E
A A

E
E A

(n-r+ m)(n-r-m) (m - n+ r)(m - n -r ) (r-m+ n)(r -m-n)

= (n + r+ m)(n-r-m) + (m +n+r)(m-n-r) + (r+m+n)(r - m-n.)
A

E
E A A

E E
A A

E
E A

n-r+ m m - n+ r r-m+ n
= n + r+ m + m +n+r + r+m+n
A E A A

E
A A E A

n-r+ m m - n+ r r-m+ n
= m + n+ r + m +n+r + m+n+r
A E A A

E
A A E A

n-r+ m+m-n+r+r-m+n
= m + n+ r
A E

m+n+r
= m + n+ r
A E

=1

cEof;sf nflu k|Zgx¿ (Questions for Practice)

;/n ug'{xf];\ (Simplify):
(a-b)2 -c2 (b-c)2 - a2 (c - a)2 - b2
1. a - (b+ c) b - (c +a)2 + c2 - (a+b)2
A

E
2 2+ 2 E

(x - y)2 - z2 (z - x)2 - y2 (y-z)2 - x2

2. x2 - (y+z)2 + z2 - (x + y)2 + y2 - (z +x)2
A

E
E

(g-e)2 - f2 (e-f)2 - g2 (f-g)2 - e2

3. E
2
g - (e+f)
A 2+ 2
e - (f+g) 2+ 2
f - (g+e)2 E A A

E
E A A

E
E A

9x2 -(y-z)2 y2 -(z-3x)2 z2 -(3x-y)2

4. (3x+z)2-y2 + (3x+y)2-z2 + (y+z)2-9x2
A

E
E A A

E
E A A

E
E

x2 -(2y-3m)2 4y2 -(3m-x)2 9m2-(x-2y)2

5. (3m+x)2-4y2 +(x+2y)2-9m2 +(2y+3m)2 - x2
A

E
E A A

E
E A A

E
E

pTt/x¿
1. 1 2. 1 3. 1 4. 1 5. 1

gd'gf 6 :
;/n ug'{xf];\ (Simplify)
a- 1 a-2 a-5
A

a2 - 3a + 2 + a2 - 5a+6 + a2 - 8a + 15 E
E A

;dfwfg M
oxfF, a2 -a-3a1+ 2 + a2 a- 5a+6
A
-2 a-5
+ a2 - 8a + 15 E
E

a- 1 a-2 a-5
= a2 - (2+1)a + 2 + a2 - (2+3)a+6 + a2 - (5+3)a + 15
A

E
E

a- 1 a-2 a-5
= a2 - 2a-1a + 2 + a2 - 2a-3a+6 + a2 - 5a-3a + 15
A

E
E

122
 a- 1 a-2 a-5
= a(a - 2)-1(a - 2)+ a(a - 2) -3(a-2) + a(a - 5)-3(a-5)
A

E
E

a- 1 a-2 a-5
= (a - 2)(a-1)+ (a - 2)(a -3) + (a - 5)(a-3)
A

E
E

1 1 1
= a - 2+ a -3 + a-3
A

E
E

a-3+a-2+a-2
= (a - 2)(a -3)
A E

3a - 7
= (a - 2) (a - 3)
A E

cEof;sf nflu k|Zgx¿ (Questions for Practice)

;/n ug'{xf];\ (Simplify)
1 2 1
1. x2 - 5x + 6 - x2 - 4x + 3 + x2 - 3x + 2
A

E
E

1 2 1
2. a2-5a+6 - a2-4a+3 - a2-3a+2
A

E
E

2a-6 a-1 a-2

3. a2-9a+20 - a2-7a+12 - a2-8a+15
A E A A E A A E

2(a-3) a-1 a-2

4. (a-4) (a-5) +(3-a) (a-4) + (5-a) (a-3)
A

E
E

x-1 3 1
5. (2x-1)(x+2) + (x+2) (x-1) - (1-x) (1 - 2x)
A

E
E

pTt/x¿
2 5 5 x+4
1. 0 2. (1-a)(a-2) 3. (a-3)(a-4)(a-5) 4. (a-3) (a - 4) (a - 5) 5. (2x - 1) ( x+2)
A E A A E A A E A A E

gd'gf 7 :
;/n ug'{xf];\ (Simplify):
1 1 2 2
A

(a+1)2 (a+2)2 - (a+1)2 + a + 1 - a+2 E

E A

;dfwfg M
A
1
oxfF, (a+1)21(a+2)2 - (a+1) 2+
2 2
a + 1 - a+2 E
E A

2
1 - (a+2) 2(a+2) - 2(a+1)
= (a+1)2 (a+2)2 + (a + 1)(a+2)
A

E E

1 - a2 - 4a-4 2a+4 - 2a-2

= (a+1)2 (a+2)2 + (a + 1)(a+2)
A

E E

- a2 - 4a-3 2
= (a+1)2 (a+2)2 + (a + 1)(a+2)
A

E E

- a2- 4a-3+2(a+1)(a+2)
= A

(a+1)2 (a+2)2 E E

123
 - a2 - 4a-3+2(a2+2a+a+2)
= A

(a+1)2 (a+2)2 E E

- a - 4a-3+2a2+4a+2a+4
2
= A

(a+1)2 (a+2)2 E

a2+2a+1
= (a+1)2 (a+2)2
A

(a+1)2
= (a+1)2 (a+2)2
A

E E

1
= (a+2)2
A

cEof;sf nflu k|Zgx¿ (Questions for Practice)

;/n ug'{xf];\ (Simplify)
1 1 2 2
1. (y+1)2 (y+2)2 - (y+1)2 + y + 1 - y+2
A

E
E

x(x+3) 1 2 2
2. (x+1)2 (x+2)2 + (x+1)2 + x + 2 - x+1
A

E
E

pTt/x¿
1 x
1. (y+2)2A E A 2. (x+1)2(x+2)2 A E

pRr bIftfsf nflu k|Zgx¿ (Higher Ability Questions)

cEof;sf nflu k|Zgx¿ (Questions for Practice)
;/n ug'{xf];\ (Simplify)
5x - 3x2 5x + 7 2
1. A

1 - x3 - x2 + x + 1 + x- 1 E
E A

2 2
2. A

x3 + x2 + x+1 - x3 + x2 - x -1 E
E

1 1 1 ax bx cx
3. x+a + x+b + x+c + x3 + ax2 + x3 + bx2 + x3 + cx2
A

E
E A

1 ax 1 bx 1 cx
[Hints: x+a +x2(x+a) +x+b +x2(x+b)+x+c +x2(x+c) ]
A E A A E A A E A A A

E
A E A A E A

1 1 1 a b c
4. m - a + m -b + m -c - m(m-a) - m(m-b) - m(m-c)
A

E
E

1 a 1 b 1 c
[Hints m-a -m(m-a) +m-b -m(m-b) +m-c -m(m-c)
A E A A E A A E A A E A A E A A E A

m-a m-b m-c 1 1 1 3

= m(m-a) +m(m-b) +m(m-c) = m + m + m = m ]
A E A A E A A E A A E A A E A A E A A E A

(a+b) (a + b -c) (b+c) (b + c -a) (c+a) (c + a - b)

5. A

2ab + E 2bc + 2ca E

6. 1 + 1 ( a+ b -c) + 1 + 1 (b + c -a) + 1 + 1 (c + a -b)

a b
A

b c
E c a E

x+ 2 x- 2
7. -
2 2 (x2 + 2x + 2) 2 2(x2 - 2x + 2)
A E

124
 4 1 1 2(x-1)
8. x2 - x+ 1 + x+ x + 1 + x - x + 1 - x2 + x +1
A E

x 2  x a2
9. 
a+x x2
- - - +
 A

x a+x  a x a(x+a) E A A E A A E

2 2
10.  +  -2 2 - 2 +  - 
x y 4 x y 2 x y 4
y x y x  y x
A E A A E A A E

[Hints y+x -y - x  ]

x y 2 x y 2 2
A E A

pTt/x¿ M
9-5x 4 3 3
1. x3 - 1
A A

E
2. (1+x)(1-x4) 3. x
A 4. m E A A E A A E A 5. 3
2 8x2 + 8
6. 6 7. x4 + 2x2 + 4 8. x4 + x2 + 1 9. 1
A E A A E A 10. 16

125
4.3 3ftfª\s (Indices)

1= kl/ro
a df a nfO{ cfwf/ / n nfO{ 3ftfª\s elgG5 . o; kf7df xfdL oxL 3ftfª\ssf lgodx¿ k|of]u u/L
n

aLhLo cleJo~hsx¿sf] ;/nLs/0f ug{ tyf 3ftfª\s o''St ;dLs/0fx¿sf] xn ug]{ tl/sfx¿sf
af/]df rrf{ ul/Psf] 5 .
o; kf7af6 k/LIffdf 5f]6f] pTt/ cfpg] Pp6f 1fg, af]w tyf l;k ;DaGwL b'O{ 2 cª\s ef/sf]] k|Zg
/ nfdf] pTt/ cfpg] Pp6f l;k ;DaGwL 2 cª\sf] k|Zg ;f]lwg] 5 .
2= 3ftfª\ssf lgodx¿ (Laws of Indices)
jf:tljs ;ª\Vofx¿ a, b, m / n sf nflu (For any real numbers a, b, m and n)
1. am × an = am + n
xa
2. am ÷ an = am-n or xb = xa-b
A E A

3. (am)n = amn
4. (ab)n = anbn
5. (xy)ab = xab yab
6.
n
a an
  = n
b b
0 0
7. x =y =1
8. olb xm = xn eP m = n x'G5, hxfF x ≠ 0
9. olb xm = ym eP x = y x'G5 .

3= 3ftfª\s ;lDdlnt ;dLs/0fx? (Equation involving indices)

1. olb a = ay eP x = y x'G5 (If a = ay then x = y)
x x

2. olb x
a =b
x
eP a = b x'G5 (If a = b then a = b)
x x

4= ljz]if Wofg lbg'kg{] s'/fx¿

o; kf7df,
• 3ftfª\s o''St ;dLs/0fsf] wf/0ff a'em\g] .
• 3ftfª\s o''St ;dLs/0f xn ug]{ tl/sfsf] lgodx¿sf] hfgsf/L lng] .
• k|sfzsf] ult 3×108 ld6/ k|lt ;]s]G8 x'G5 . o;nfO{ u'0fg u/]/ cEof; ug]{ . oL b'O{
tl/sfnfO{ bfh]/ x]bf{ 3ftfª\ssf] lgodåf/f 7'nf 7'nf cª\snfO{ ;/n ¿kdf JoSt ug{
;lsG5 . of] Pp6f j}1flgs ljlw xf] .
3ftfª\ssf] lgodaf6 cfg'kflts ;ª\Vofx¿ (irrational number) sf lj|mofx¿ klg ;lhn}l;t ug{
;lsG5 .

126
5. gd'gf k|Zgf]Tt/ tyf cEof;
5.1 5f]6f] pTt/ cfpg] k|Zg ;DaGwL cEof;
gd'gf 1 :
xn ug{'xf];\ M 3x+1 + 3x = 108
;dfwfg M
oxfF, 3x+1 + 3x = 108
x x
or, 3 . 31 + 3 = 108
x
or, 3 (3 + 1) = 108
x
or, 3 .4 = 108
x
or, 3 = 108/4
x
or, 3 = 27
x
or, 3 = 33
∴ x=3

cEof;sf nflu k|Zgx¿

x+3 x x+2 x+1 1
(a) 2 + 2 = 36 (b) 3 +3 = 13A E

x x-2
(c) 2 - 2 =6 (d) 2y - 2y-2 = 3
x+1 x+2 x+1
(e) 2 - 2x-1 = 12 (f) 4 = 22 + 14
2x+1 x+2 2x+3 x+1
(g) 3 =9 - 26 (h) 3 - 9 = 2.9 -6
pTt/x¿
1 1
(a) 2 b) -2 (c) 3 (d) 2 (e) 3 (f) 0 (g) - 2
A E A (h) - 2
A E

5.2 nfdf] pTt/ cfpg] k|Zg ;DaGwL cEof;

gd'gf 2 :
1 1
xn ug{'xf];\ (Solve) : 5a + 5 a = 25 25
A A

E
A E A

;dfwfg M
oxfF, dfgf“}
5a = x
1 626
∴x + x = 25 A A

E
A E A

x2+1 626
Or, x = 25
A

E
A A E

Or, 25x2 + 25 = 626x

Or, 25x2 - 626x + 25 = 0
Or, 25x2 - (625 + 1)x + 25 = 0
Or, 25x2 - 625x - x + 25 = 0

127
Or, 25x(x - 25) -1(x – 25) = 0
Or, (x - 25) (25x -1) = 0
x sf] dfg /fVbf,
(5a - 25) (25×5a -1) = 0
Either,
(5a - 25) = 0
Or, 5a = 25
Or, 5a = 52
∴a = 2
or, (25×5a -1) = 0
or, 25×5a = 1
1
or, 5a = 25 A E

1
or, 5a = 52
A E

or, 5a = 5 -2
∴ a= - 2
∴ a =2 or - 2

cEof;sf nflu k|Zgx¿

x 1
(a) 2 + = 22 A E A

x 16
(b) 2 + 2x = 10 A E A

x 9
(c) 3 + 3x = 10 A E A

x 64
(d) 4 + 4x = 20 A E A

1 1
(e) 4x + 4x = 1616
A E A A E A

x 1 1
(f) 3 + 3x = 99 A E A A E

x 1
(g) 5 -1 + 5-x = 15 A E

x x 26
(h) 51- + 5 -1 = 5 A E

(i) 5a+1 + 52-a = 126

(j) 2a-2 + 23-a = 3

pTt/x¿
(a) 1, -1 (b) 1,3 (c) 2,0 (d) 1, 2
(e) 2, -2 (f) 2, -2 (g) 1, 0 (h) 1,2
(i) -1, 2 (j) 3, 2

128
gd'gf 3 :
xn ug{'xf];\ (Solve) : 5.4x+1 – 16x = 64
;dfwfg M
oxfF, 5.4x+1 – 16x = 64
x x
or, 5. 4 4 – (42) = 64
x x
or, 20. 4 - (4 )2 = 64
dfgf}“ 4x = a
∴ 20a – a2 = 64
or, a2 – 20a + 64 = 0
or, a2 – 16a – 4a + 64 = 0
or, a(a – 16) – 4(a – 16) = 0
or, (a – 16)(a – 4) = 0
a sf] dfg /fVbf,
x x
(4 – 16)(4 – 4) = 0
Either,
x
(4 – 16) = 0
x
Or, 4 = 16
x
Or, 4 = 42
∴ x=2
Or,
x
4 –4=0
x
Or, 4 = 4
x
Or, 4 = 41
∴ x=1
∴ x =1 or 2
cEof;sf nflu k|Zgx¿
x x
(a) 4 - 6.2 +1 + 32 = 0
(b) 9a – 10×3a+9=0
1
(c) 3a+3 + 3 a = 28 A A

x+2
(d) 10.3 = 9 + 9x+2
x x
(e) 4 +1 - 9.2 +2 + 32 = 0
(f) 25x - 30.5x + 125 = 0
(g) 16v - 5.4v+1 + 64 = 0
(h) 4.3x+1 - 9x = 27
(i) 4x + 128 = 6.2x+2
(j) 4x + 128 = 3.2x+3

129
pTt/x¿
a) 2, 3 b) 0,2 c) 0, –3 d) 0, -2 e) 3, 0
f) 1, 2 g) 1, 2 h) 1, 2 i) 3, 4 j) 3, 4

gd'gf 4 :
;/n ug{'xf];\ (Simplify)
1 1 1
A

1 + ax-y + az- y +1 + ay-z + ax - z + 1 + az-x + ay - x

E A A

E
E

;dfwfg M
oxfF, 1 + ax-y1 + az- y +1 + ay-z1 + ax - z +
A E A A

E
1
1 + az-x + ay - x E

1 1 1
= ax az +
A

ay ax + az ayA A E

1 + ay + a y 1 + az + a z 1 + ax + a x
E
E

1 1 1
= ay + ax + az +az + ay+ ax + ax + az + ay
A A A E

ay az ax E E

ay az ax
= ay + ax + az + az + ay + az + ax + az + ay
A A

E
A

E
A A

ay az ax
= ay + az + ax + ay + az + ax + ay + az + ax
A A

E
A A

E
A

ay + az + ax
= ay + az + ax
A

E
A

=1

cEof;sf nflu k|Zgx¿

1 1 1
(a) A

1 + xb-a + xc -a + 1 + xc-b + xa-b + 1+xa-c + xb-c E A A

E
E

1 1 1
(b) 1+xl-m + xl-n + 1+xm-n + xm-l + 1+xn-l + xn-m
A

E
E

1 1 1
(c) 1+ax-y + ax-z + 1+ay-x + ay-z + 1+az-x + xz-y
A

E
E

1 1 1
(d) 1+ma-b + ma-c + 1+mb-c + mb-a + 1+mc-a + mc-b
A

E
E

1 1 1
(e) 1+mp-q + mr-q + 1+mq-r + mp-r + 1+mr-p + mq-p
A

E
E

1 1 1
(f) 1+xa-b+xc-b + 1+xb-c+xa-c + 1+xc-a+xb-a
A E A A E A A E

1 1 1
(g) 1+xp-q+xr-q + 1+xq-r+xp-r + 1+xr-p+xq-p
A E A A E A A E

1 1 1
(h) 1+xa-b+xa-c + 1+xb-c+xb-a + 1+xc-a+xc-b
A E A A E A A E

pTt/x¿
a) 1 b) 1 c) 1 d) 1 e)1 f) 1 g) 1 h) 1

130
gd'gf 5 :
a2 2a 1
(x-a)n + (x - a)n-1 + (x - a)n-2
A E A A E A A E A

a2 2a 1
= (x-a)n + (x - a)n + (x - a)n
A E A A E A A E A

(x-a)1 (x-a)2 E E

2
a 2a(x - a) (x-a)2
= (x-a)n + (x - a)n + (x - a)n
A E A A

E
E A A

E
E A

a2 + 2a(x - a) + (x-a)2
= A

(x - a)n E E

(a +x - a)2
= (x - a)n
A

E E

x2
= (x - a)n
A

4. cEof;sf nflu k|Zgx¿

p2 2p 1
a) A

(p-y)y - (p-y)y-1 +(p-y)y-2 E A A E A A E A

p2 2p 1
b) A

(y-p)y - (y-p)y-1 + (y - p)y-2 E

E

m2 2m 1
c) A

(n-m)n - (n-m)n-1 + (n - m)n-2 E

E

pTt/x¿
y2 (2p-y)2 (2m-n)2
a) A

(p-y)y E A b) (y-p)y A

E E
A c) (n-m)n
A

E E
A

gd'gf 6:
olb abc = 1 eP, k|dfl0ft ug{'xf];\ M
1 1 1
If abc = 1, prove that: 1+a + b-1 + 1+b+c-1 1 + c + a-1 = 1
A

E
E

;dfwfg M
oxfF, abc = 1
1 1
Or, bc = a and a = bc A

E
E A

1 1 1
L.H.S.= A

1 1+ 1 + 1 + c+ bc E

1+ bc + b 1+b + c E

bc c 1
= bc + 1 + c + c + bc + 1 + 1+c + bc
A

E
E A

bc+c+1
= bc+c+1 A E A

=1
= R.H.S.

131
 ∴ L.H.S. = R.H.S.

5. cEof;sf nflu k|Zgx¿

(a) olb pqr = 1 eP k|dfl0ft ug{'xf];\ .
1 1 1
If pqr = 1 show that 1+p+q-1 + 1+q+r -1 +1+r+p-1 = 1 A E A A E A A E A

(b) olb a + b + c = 0 eP k|dfl0ft ug{'xf];\ .

1 1 1
If a + b + c = 0 then prove that, 1+xa +x-b + 1+xb + x-c + 1+xc +x-a = 1 A

E
E A

(c) olb m + n + r = 0 eP k|dfl0ft ug{'xf];\ .

1 1 1
If m + n + r = 0 then prove that, 1+xm +x-n + 1+xn + x-r + 1+xr +x-m = 1 A

E
E A

(d) olb p+q+r= 0 eP k|dfl0ft ug{'xf];\ M If p+q+r= 0, then prove that:

1 1 1
A

1+xp+x-q +1+xq+x-r +1+xr+x-p = 1 E A A E A A E A

gd'gf 7:
a+b a-b b-c c-a
x  xb+c xc+a
;/n ug{'xf];\ (Simplify) : × x  × x 
 x 
A E A

  A E A

 A E

;dfwfg M
a+b a-b c-a b-c
xc+a xb+c
oxfF, xx 
A × x 
 E × x 
A

  A E A A E

(a+b)(a-b) (b+c)(b-c)
x(c+a)(c-a)
=  x(a-b) ×  x(b-c)  ×  x(c-a) 
x x

A

  E    E A A

E
E A A

E
E

2 2 2 2 2 2
xa -b  xb -c  xc -a 
=  xa - b  ×  xb - c  × xc - a 

A

  E    E A A

E
E A A

E
E

a2-b2+ b2-c2+ c2-a2

x
= A

xa - b + b - c +c - a
E
E

0
x
= x0
A E

=1

6. cEof;sf nflu k|Zgx¿

n-l l-m m-n
l+m
x l-m  xm+n xn+l
(a) ×  xm-n  ×  xn-l 
x 
A E A

  A E A

  A E

a +b b+c c +a
2 2 2 2 2 2
xa +b  xb +c  xc +a 
(b)  ab  ×  xbc  ×  xca 
 x 
A

E
E A

  A

E
E A

  A

E
E

2 2 a -b 2 2 b-c 2 2 c-a
xa +b  xb +c  xc +a 
(c)  -ab  ×  x-bc  ×  x-ca 
 x 
A

E
E A

  A

E
E A

  A

E
E

132
 a+b b-c a-b c-a
x c  xb+c xc+a
(d) ×  xa  ×  xb 
x 
A

    E A A E A A E

2 2
a a +ab+b b +bc+c2
2
c2 +ca+a2
xb  xb  xc
(e) × xc × xa
 
A

x E A

 
A E A

 
A E

a+b c-a a-b b-c

x a-b  xb+c xc+a
(f) ×  xb-c  ×  xc-a 
x 
A E A


A

 E A


A

 E

b b+c-a c+a-b a+b-c

x c  xc xa
(g) × xa × xb
x 
A E A

 
A E A

 
A E A

a a-b b-c c-a

 x-b xb xc
(h) × x-c × x-a
x 
A E A

 
A E A

 
A E

x x+y y+z
ay ay
× az
x-z
(i) ÷ (ax × az)
a 
A E A

 
A E A A E A

2 2 2
(xa+b) .(xb+c) .(xc+a)
(j) A

4 E A

(xaxbxc) E

1 1 1

xb bc
 c ×  a ×  b xc ca xa ab
( k)
 x  x  x 
A E A A E A A E A

(l) (xa+byc)a-b (xb+cya)b-c (xc+ayb)c-a

A E

a-b b-c c-a

(m) (xa . xb ÷ xc)
A E .(xb ÷ x-c . xa)
A A E .(xa . x-b . xc)
A A E

pTt/x?
x 2 ( a +b + c )
3 3 3

a) 1 b) c) 1 d) 1 e) 1 f) 1 g) 1
h)1 i) 1 j) 1 k) 1 l) 1 m) 1

6:cEof;sf nflu yk k|Zgx?

pRr bIftfsf k|Zgx? (Challenging Problems)
;/n u/ M (Simplify)
x y−x a +b a +b
 2 1   1  1  1
 x − 2   x −   a +  b − 
 y   y  b  a
y x− y b a
 2 1   1   2 1   2 1 
y − 2  y+  b − 2   a − 2 
(a)  x   x (b)  a   b 
x+ y x+ y b a
 1  1  b  a −b  a  a −b
y+   x − y  1 −  1 + 
 x    a  b
x b a
 2 1  2 1
y

 x − 2   y − 2  a  a −b  b  a −b
 − 1  + 1
(c)  y   x  (d)  b  a 

133
 p q
x y
 p  p−q  q  p−q
 x
1 + 
x−y
 y x−y 1 +  1 − 
1 −  q  p
 y  x 
y
p q
x
 y x−y  x x−y  q  p −q  p  p −q
 + 1  − 1  + 1  − 1
(e)  x  y (f)   q 
 p

pTt/x¿
x x+y a a-b y x-y
(a) y (b) b (c) x
 
A E A

 
A E A

 
A E A

a x p
(d) b A E A (e) y
A E A (f) q A E A

4.4 d"n / ;8{x? (Root and Surds)

1= kl/ro
;8{x¿ ljleGg k|sf/sf x'G5g\ . cGo cg'kflts ;ª\Vofx¿ u'0fg v08sf ¿kdf gePsf] ;8{nfO{
z'4 ;8{ (pure surd) elgG5 . o:t} rational ;ª\Vof / surd sf] ;+o'St ¿kdf ePsf] ;ª\VofnfO{ ldl>t
;8{ elgG5 .
o; kf7df cg'kflts ;ª\Vofsf ;fy;fy} cfg'kflts ;ª\Vof (irrational number) ;“u ;DalGwt ;d:of
;dfwfg ug]{ ljlw k|of]u ul/G5 . o; kf7leq cg'kflts ;ª\Vofsf] hf]8, u'0fgsf ;fy} ;fwf/0f d"n
;dLs/0fsf ;d:ofx¿ /flvPsf 5g\ . ;fwf/0f jf rational ;ª\Vofsf] ;d:of ;dfwfg ug]{ tl/sf
irrational sf] lx;fa xn ug]{ tl/sfaLr tfnd]n Nofpg vf]lhPsf] 5 .
o; kf7af6 k/LIffdf 5f]6f] pTt/ cfpg] 1fg, af]w / l;kd"ns Ps Ps cf]6f 2 – 2 cª\sf] k|Zg
;f]Wg] ul/G5 .
2= ;8{x? (Surds) sf ;"qx?
1. x sf] ju{d"n + x x'G5 . (The square root of x is +
A E A A x .)
E A

2. olb x = 0 eP x = 0 x'G5 (If x = o then x = 0).

A E A A E A

n
3. A a = a1/n
E A

ju{ ;dLs/0fsf] ;fwf/0f ?k (General form of Quadratic Equation) ax2 + bx + c = 0

-b+ b2 - 4ac
ju{ ;dLs/0fsf] d"nx¿ (Roots of Quadratic Equations) x= A

2a E
E

3. ljz]if Wofg lbg'kg{] s'/fx?

3.1 cg'kflts tyf cfg'kflts ;ª\Voflarsf] cGt/ ;DaGw :ki6 ug{] .
3.2 cfg'kflts ;ª\Vof (irrational number) sf] juL{s/0f ug{] .
3.3 ;8{;“u ;DalGwt ;ª\Vofsf ;fwf/0f d"n ;dLs/0f xn ug{] .
3.4 lbPsf] cleJo~hsnfO{ x/ tyf c+zdf x/sf] cg'ab\w (conjugate) n] u'0fg u/]/
x/nfO{ cg'kflts ;ª\Vof (rational) agfpg ;lsG5 .

134
 3.5 olb lbPsf] leGg ;8{ xf] eg] ToxL leGgx¿sf] x/df ePsf] ;8{ jf To;sf]
cg'ab\wn] To; leGgsf] x/ / c+znfO{ u'0fg u/]/ x/sf] ;8{ x6fpg] k|lj|mofnfO{
cg'kflts ug]{ elgG5 .
3.6 ;dLs/0f n]lvPsf rf6{x¿ k|bz{g u/L rn /flzsf] dfg kTtf nufpg ;lsG5 .
3.7 cg'kflts / cgfg'kflts ;ª\Vofx¿larsf] cGt/ ;DaGw kTtf nufO{ cg'kflts /
cgfg'kflts ;ª\Vofx¿sf] wf/0ff :ki6 agfpg]{ .
gf]6 M cgfg'kflts ;ª\VofnfO{ ;8{ (surd) klg elgG5 . an df n nfO{ ;8{sf] l8u|L (order)
klg elgG5 eg] a nfO{ d'ns (radicand) elgG5 hxf“ a wgfTds cg'kflts ;ª\Vof xf] .

4= gd'gf k|Zgf]Tt/ tyf cEof;

4.1 5f]6f] pTt/ cfpg] k|Zg ;DaGwL cEof;
gd'gf 1 :
;/n ug'{xf];\ (Simplify)
3 2 + 4 2500 + 4 64 + 6 8

;dfwfg M
oxfF,
3 2 + 4 2500 + 4 64 + 6 8
= 3 2 + 4 2× 2×5×5×5×5 + 4 2× 2× 2× 2× 2× 2 + 6 2× 2× 2
= 3 2 + 5× 2× 2 + 2 2× 2 + 6× 2
4 4
2
= (3 + 12) 2 + (5 + 2) × 4 2 × 2

= 15 2 + 7 4 2 × 2
1
2×
= 15 2 + 7 × (2) 4

= 15 2 + 7 × (2) 2

= 15 2 + 7 2
= 22 2
1. cEof;sf nflu k|Zgx?
;/n ug'{xf];\ (Simplify)
1. 27 + 75 − 8 3
2. 50 + 18 − 8 2
3. 200 + 3 32 + 2 72
135
4.
3
128 + 23 54 − 23 250
5. 16a + 154a − 250a
3 3 3 3 3 3

6. 16 + 3 54 + 3 192 − 3 375 − 3 128 + 3 3

3

pTt/x¿
3
1. 0 2. 0 3. 34 2 4. 0 5. 0 6. 2
gd'gf 2 :
;/n ug'{xf];\ (Simplify) : 4 12 xy 4 × 4 2 x 3 y 9 z 7
= 4
12 xy 4 × 2 x 3 y 9 z 7
= 4
2 × 2 × 2 × 3x 1+3 y 4+9 z 7
= 4
24 x 4 y 13 z 7
= 4
24 x 4 y 4 y 4 y 4 yz 4 z 3
3
= xyyyz 4 24 yz
3 3
= xy z 4 24 yz

2. cEof;sf nflu k|Zgx¿M

;/n ug'{xf]; (Simplify) :
1. 5 × 10 × 2
2. 5 2 x × 8 x 3 × 2 3x 5
3. 30ab 5 × 3a 2 b
4. 2 6 x 5 × 3x × 5 20 x 3
5. 3
4 xy 2 × 3 4 xy 4
pTt/x¿
1. 10 2. 40 x 4 3x 2. 3ab 3 10a 4. 60 x 4 10 x 5. 2 y 2 3 2x 2
gd'gf 3:
5 54 − 24 + 7 216
;/n ug'{xf];\ (Simplify) :
53 24
;dfwfg M

136
 5 54 − 24 + 7 216
oxfF,
53 24
5 2 × 3× 3× 3 − 2 × 2 × 2 × 3 + 7 2 × 2 × 2 × 2 × 3× 3× 3
=
53 2 × 2 × 2 × 3
5×3 2×3 − 2 2×3 + 7× 2× 2×3 2×3
=
53 × 2 2 × 3
15 6 − 2 6 + 84 6
=
106 6
(15 − 2 + 84) 6
=
106 6
97
=
106

3. cEof;sf nflu k|Zgx?

;/n ug'{xf];\ (Simplify)
3
128 − 3 16
1.
23 2
5 18 − 32 + 7 50
2.
23 8
5 27 − 12 + 7 75
3.
8 108
1
4. 4 2a − + 32a 2
2

8a 2
45 x 4 − 80 x 4 + 6 x 2 5
5.
5 5x 4
x 8 x 3 a + 18 x 5 a − x 2 32 xa
6.
x 2 2ax
23 192 − 43 81 + 33 24
7. 3
648 − 3 375
pTt/x¿
2 (32a 2 − 1)
1. 1 2. 2 3. 3 4. 5. 1 6. 1 7. 2
4a
137
gd'gf 4:
3 5+ 3
;/n ug{'xf];\ (Simplify) :
5− 3
;dfwfg M
3 5+ 3
oxfF,
5− 3
3 5+ 3 5+ 3
= ×
5− 3 5+ 3
3 × 5 + 3 15 + 15 + 3
=
( 5) 2 − ( 3) 2
15 + 4 15 + 3
=
5−3
18 + 4 15
=
2
2(9 + 2 15 )
=
2
= Error! Bookmark not defined. 9 + 2
A A
15

4. cEof;sf nflu k|Zgx?

2 2+ 3
1.
2− 3
2 7 +3 5
2.
7 −2 5
pTt/x¿
16 + 35
1. 6 −1 2. 3

138
4.2= nfdf] pTt/ cfpg] k|Zg ;DaGwL cEof;
gd'gf 1:
1/ab x1/a 1/bc x1/b 1/ca x1/c
;/n ug{'xf];\ (Simplify) : A

x 1/b ×
x1/c × x1/a
E A A E A A E

;dfwfg M
1/ab x1/a 1/bc x1/b 1/ca x1/c
oxfF, x 1/b ×
A E A A

x1/c × x1/a
E A A E

1/a ab
(x ) (x1/b)bc 1/c ca
(x )
= E
1/b ab ×
(x )
A

(x1/c)bc × E A A

E
E A

(x1/a)ca
A

E
E

(xb) (xc) (xa)

= a ×
(x )
A

E (x )b × (xc)
E A

E
A E A A

E
E

b c a
x x x
= xa × xb × xc
A E A A E A A E A

xb+c+a
= xa+b+c A E

= 1
1. cEof;sf nflu k|Zgx?
a b c
b c a
x x x
(a) ab
b
× bc
c
× ca
a
a b c
x x x
x y
z
y
xy
a a z
ax
(b) × yz × zx
y z x
x y
a a az

xy
a yz a zx a z
x y
(c) . .
ay az ax
2
/ y2
ay
2
/ z2
az
2
/ x2 2 2 ax
(d) y2z2 ×z x 2 2 ×x y 2
/ x2
az
2
/ y2
ax
2
/ z2
ay
2
/ c2 2
/ a2 2
/ b2
mb mc ma
(e) b 2 c 2 2
× c2a2 × a 2b 2
/ b2 2
/ c2 2
/ a2
mc ma mb

1
a a + b a 1− b a b + c b 1− c a c + a
(f) c − a × ×
a a −b ab −c ac−a
2 2 2
xb xc xa
(g) b + c 2
× c+a 2
× a +b 2
xc xa xb
139
 1 1
1 y 1

(h)
1
a × x
yz
a × 1 az
xy zx
1 1 1

ay az ax
1
ab
xa/b 1
bc
xb / c 1
ca
xc / a
(i) × ×
xb / a xc /b xa/c

mn
xm/ n np
xn/ p pm
x p/m
(j) × ×
xn/ m x p/n xm/ p
pTt/x¿
a) 1 b) 1 c) 1 d) 1 e) 1 f) 1 g) 1 h) 1 i) 1 j) 1

gd'gf 2:
5x - 4 5x - 3
xn ug{'xf];\ (Solve) : A

5x +2
=4- 2E A A

E
E A

;dfwfg M
5x - 4 5x - 3
oxfF, A

5x +2
=4- 2 E A A

E
E

(5x)2 - 22 8 -( 5x - 3)
or, A = 2 E A A E

E 5x +2 E

( 5x + 2) ( 5x - 2) 8 - 5x + 3
or, A = 2 E A A E

5x +2 E
E

( 5x - 2) 5 - 5x
or, A

1 = 2 E
E A A

E
E

or, 2 5x - 4 = 5 - 5x
A E A A E

or, 2 5x + 5x = 5 + 4
A E A

or, 3 5x = 9 A E A

or, 5x = 3 A E A

or, ( 5x)2= (3)2 A E A

or, 5x = 3
3
∴ x=5 A E

2. cEof;sf nflu k|Zgx?

5x - 4 5x + 2 7x - 36 5 7x-11
(a) A =2- 2 E A A E A (b) A =9- 3 E A A E A

5x + 2 E
6+ 7x E

6x-49 4 6x-3 5y-4 5y-3

(c) A =6- 3 E A A E A (d) A =2- 2 E A A E A

7+ 6x E
5y -2 E

5x - 9 5x -3 3x - 4 3x -2
(e) A =1+ 2 E A A E A ( f) A =2+ 2 E A A E A

3 + 5x E
3x +2 E

140
 5y - 4 5y - 3 x-4 x-2
(g) A =2+ 2
E A A E A (h) A E=2+ 2
A A E A

5y + 2 E
2+ x E

pTt/x¿
4 1
(a) 5A E A (b) 7 (c) 6 (d) 5 A E A

(e) 5 (f) 12 (g) 5 (h) 36

kf7 : 4.5 o'ukt /]vLo ;dLs/0f / ju{ ;dLs/0f ;DaGwL zflAbs ;d:ofx¿
(Verbal Problems of Simultaneous Equation and Quadratic Equation)

1. kl/ro
xfd|f] b}lgs hLjgdf Pp6f kl/df0fnfO{ csf]{ kl/df0f;“u bfFh]/ x]g]{ ul/G5, h:t} M b'O{ hgf JolStsf]
pd]/ leGgtf, sfdbf/x¿sf] sfo{Ifdtfcg'¿k Hofnfb/ sfod ug]{ cflb 5g\ . o:tf vfnsf ;d:of
;dfwfg ug{ ;d:ofnfO{ ;dLs/0fdf JoSt ul/G5 . o:tf b'O{ rno'St o'ukt /]vLo ;dLs/0fnfO{
ljleGg tl/sf -x6fpg] jf k|lt:yfkg ljlw_ af6 ;dfwfg u/fpg] cEof; ul/G5 .
o; kf7af6 l;k ;DaGwL Pp6f 5f]6f] pTt/ cfpg] cª\s 2 sf] k|Zg / nfdf] pTt/ cfpg] 4 cª\ssf] Pp6f
;d:of ;dfwfg d'ns k|Zg ;f]lwg] ul/G5 .

2. ljz]if Wofg lbg'kg{] s'/fx?

– ;a}eGbf klxn] lbOPsf] k|ZgnfO{ /fd|f];Fu k9L gePsf] /flznfO{ x, y, z cflbn] hgfpg] .
– ;d:ofdf lbOPsf] egfOx¿nfO{ ul0ftLo jfSodf n]Vg] .
– k|fKt ;dLs/0fnfO{ x6fpg] ljlw, k|lt:yfkg ljlw / n]vflrq ljlwdWo] s'g} Ps ljlw k|of]u
u/L ;dfwfg ug]{ .
– b'O{ rno'St o'ukt /]vLo ;dLs/0fsf] ljleGg tl/sfaf6 xn u/L ;s]sf 5f“} . oxfF, b'O{
rno'St ;dLs/0fsf] k|of]u u/]/ Jofjfxfl/s ;d:ofsf] ;dfwfg ug{ ;lsG5 .

3= cfwf/e't hfgsf/L / ;'qx?

3.1 ;ª\Vofx?sf af/]df hfgsf/L
1. cg'atL{ ;ª\Vofx¿ -consecutive numbers_ M x, x+1, x+2 ………………..
2. cg'atL{ hf]/ ;ª\Vofx¿ -Consetiver even numbers_
3. cg'atL{ lahf]/ ;ª\Vofx¿ -Consetive odd numbers_
4. x sf] Jo'Tj|mdfg'kftL 1/x x'G5 -Reciporcal of x is 1/x_

4= b'O{ cª\ssf] ;ª\Vofx¿df

y / x j|mdzM Ps :yfg / b; :yfgdf ePsf] b'O{ cª\ssf] ;ª\VofnfO{ 10x+y n] hgfOG5 / To;sf]
ljk/Lt ;ª\VofnfO{ 10y+x n] hgfOG5 .
3.2 b'/L / j]u;Fu ;DalGwt ;d:ofx¿ ePdf

141
 b''/L
– j]u jf ult (speed)= A

;do E

– b'/L Ö k|j]u jf ult × ;do

b''/L
– ;do Ö j]u A E

3.3 pd]/;Fu ;DalGwt k|Zgx? ePdf M

b'O{ hgf JolSt ePdf ltgLx¿sf] clxn]sf] pd]/ x jif{ / y jif{ lng] .
‘a’ jif{kl5sf] pd]/ -x+a_ jif{ / -y+a_ jif{ x'G5 . To;} u/L ‘a’jif{cl3sf] pd]/ -x-a_ jif{
/ -y-a_ jif{ lng'k5{ .
3.4 ju{ ;dLs/0f (Quadratic Equation) M
− b ± b 2 − 4ac
ju{ ;dLs/0f -ax +bx+c=0_
2
eP M x sf] dfg { x =
2a
4= gd'gf k|Zgf]Tt/ tyf cEof;
4.1 5f]6f] pTt/ cfpg] k|Zg ;DaGwL cEof;
gd'gf 1 :
olb Pp6f ;ª\Vofsf] ju{af6 3 36fp“bf 6 afFsL /xG5 eg] pSt ;ª\Vof kTtf nufpg'xf];\ .
;dfwfg M
oxfF, pSt ;ª\Vof x dfgf}F .
ta, k|Zgaf6,
x2-3 = 6
or,x2 = 6+3
or, x2= 9
2
or, x2= (±3)
x=±3
pSt ;ª\Vof ± 3

1. cEof;sf nflu k|Zgx¿

(a) olb Pp6f ;ª\Vofsf] ju{af6 5 36fpFbf 44 afFsL /xG5 eg] pSt ;ª\Vof kTtf nufpg'xf];\ .
(b) Pp6f ;ª\Vofsf] ju{af6 ToxL ;ª\Vofsf] 4 bf]Aa/ 36fpFbf 4 afFsL /xG5 eg] pSt ;ª\Vof kTtf
nufpg'xf];\ .
(c) Pp6f ;ª\Vofsf] 6 bf]Aa/af6 ToxL ;ª\Vofsf] ju{ 36fpFbf 9 x'G5 eg] pSt ;ª\Vof kTTff
nufpg'xf];\ .
(d) olb Pp6f k|fs[lts ;ª\Vofsf] ju{sf] t]Aa/af6 12 36fp“bf kl/0ffd 36L x'G5 eg] ;f] ;ª\Vof
kTtf nufpg'xf];\ .
(e) Pp6f ;ª\Vofsf] ju{ / 23 sf] km/s 2 5 eg] pSt ;ª\Vof kTtf nufpg'xf];\ .
(f) olb Pp6f wgfTds ;ª\Vofsf] ju{sf] bf]Aa/af6 17 36fpFbf kl/0ffd 111 x'G5 eg] ;f] ;ª\Vof
kTtf nufpg'xf];\ .
142
(g) Pp6f ;ª\Vofsf] ju{ / 81 sf] km/s z"Go 5 eg] pSt ;ª\Vof kTtf nufpg'xf];\ .

pTt/x¿
(a) ±7 (b) 2 (c) 3 (d) ±4 (e) ± 5 (f) 8 (g) ±9
gd'gf 2:
olb Pp6f ;ª\Vofsf] ju{;Fu 11 hf]8\bf 47 x'G5 eg] pSt ;ª\Vof kTtf nufpg'xf];\ .
;dfwfg M
oxfF, pSt ;ª\Vof x dfgf}F .
ta, k|Zgaf6,
x2+11 = 47
or, x2 = 47-11
or, x2= 36
2
or, x2 = (±6)
x=±6
pSt ;ª\Vof ± 6 /x]5 .
2. cEof;sf nflu k|Zgx¿
(a) olb Pp6f wgfTds ;ª\Vofsf] ju{df 7 hf]8\bf of]ukmn 71 x'G5 eg] ;f] ;ª\Vof
kTtf nufpg'xf];\ .
(b) olb Pp6f ;ª\Vofsf] ju{df 6 hf]8\bf 31 x'G5 eg] ;f] ;ª\Vof kTtf nufpg'xf];\ .
(c) Pp6f ;ª\Vofsf] ju{sf] b'O{ u'0ffdf 20 hf]8\bf 52 x'G5 eg] pSt ;ª\Vof kTtf nufpg'xf];\ .
(d) olb Pp6f ;ª\Vofsf] ju{df 3 hf]8\bf of]ukmn 28 x'G5 eg] pSt ;ª\Vof kTtf nufpg'x]f;\ .
(e) Pp6f To:tf] ;ª\Vof kTtf nufpg'xf];\ h;nfO{ To;}sf] ju{ ;ª\Vof;“u hf]8\bf 90 x'G5 <
(f) olb Pp6f wgfTds ;ª\Vofsf] ju{sf] 7 u'0ffdf 10 hf]8\bf of]ukmn 353 x'G5 eg] pSt ;ª\Vof
lgsfNg'xf];\ .
(g) olb Pp6f k|fs[lts ;ª\Vofsf] ju{sf] bf]Aa/df 5 hf]8\bf of]ukmn 23 x'G5 eg] ;f] ;ª\Vof kTtf
nufpg'xf];\ .
pTt/x?
(a) 8 (b) +5 (c) 4 (d) +5 (e) 9, -10 (f) 7 (g) 3
gd'gf 3 :
b'O{ ;ª\Vofx¿sf] of]u 17 / cGt/ 3 eP tL ;ª\Vofx¿ kTtf nufpg'xf];\ .
;dfwfg M
oxfF, b'O{ ;ª\Vofx¿ x / y dfgf}F .
ta, k|Zgaf6,
x+y = 17 ..................(i)
x-y = 3 ....................(ii)
;dLs/0f (i) / (ii) hf]8\bf,
x+y = 17
x-y = 3

143
2x = 20
x = 10
x sf] dfg ;dLs/0f (ii) df /fVbf,
10-y = 3
or, 10-3 = y
y=7
pSt ;ª\Vofx¿ 10 / 7 /x]5g\ .

3. cEof;sf nflu k|Zgx¿

(a) b'O{ cf]6f wgfTds ;ª\Vofx¿sf] of]ukmn / cGt/ j|mdzM 15 / 5 eP tL ;ª\Vofx¿ kTtf
nufpg'xf];\ .
(b) Pp6f lqe'hsf b'O{ cf]6f sf]0fx¿sf] of]ukmn 900 / km/s 600 5 eg] tL sf]0fx¿ kTtf
nufpg'xf];\ .
(c) Pp6f rt'e{'hsf ljk/Lt sf]0fx¿sf] of]ukmn 1800 / km/s 320 5 eg] tL sf]0fx¿ kTtf
nufpg'xf];\ .
(d) b'O{ cf]6f ;ª\Vofsf] of]ukmn 10 5 . ltgLx¿dWo] klxnf]sf] 2 u'0ff;Fu bf];|f]sf] 3 u'0ff a/fa/
x'G5 . tL ;ª\Vofx¿ kTtf nufpg'xf];\ .
(e) s'g} Pp6f ;ª\Vof csf{] ;ª\Vofsf] bf]Aa/ 5 . olb ltgLx¿sf] of]u 30 eP tL ;ª\Vof¿ kTtf
nufpg'xf];\ .
(f) A / B sf] pd]/sf] cGt/ 15 jif{ 5 . olb A sf] pd]/ B sf] pd]/sf] 4 u'0ff 5 eg] ltgLx¿sf]
pd]/ slt xf]nf kTtf nufpg'x]f;\ .
(g) cfdfsf] pd]/sf] 16 efu 5f]/Lsf] pd]/ 5 . olb ltgLx¿sf] pd]/sf] cGt/ 35 jif{ eP cfdfsf]
A E A

pd]/ kTtf nufpg'xf];\ .

pTt/x¿
a. 10, 5 b. 750, 150 c. 1060, 740 d. 6, 4 e. 20, 10 f. 20, 5 g. 42yrs

gd'gf 4:
of]ukmn 45 x'g] ltg cf]6f nuftf/ cfpg] wgfTds k"0ffª\sx¿ (integers) kTtf nufpg'xf];\ .
;dfwfg M
oxfF, nuftf/ cfpg] 3gfTds k"0ff{ª\sx¿ x, x+1 / x+2 dfgf}F .
ta, k|Zgaf6,
x +(x+1) + (x+2) = 45
or, 3x+3 = 45
or, 3x = 45-3
42
or, x = 3 A

x = 14
x+1 = 14+1 = 15
x+2 = 14+2 = 16
nuftf/ cfpg] 3gfTds ltg cf]6f k"0ff{ª\sx¿ 14, 15 / 16 /x]5g\ .
144
4. cEof;sf nflu k|Zgx¿
(a) s'g} b'O{ cf]6f j|mdfut ;ª\Vofx¿sf] u'0fgkmn 156 5 eg] tL ;ª\Vofx¿ kTtf nufpg'xf];\ .
(b) s'g} b'O{ cf]6f j|mdfut lahf]/ ;ª\Vofx¿sf] u'0fg kmn 255 5 eg] tL ;ª\Vofx¿ kTtf
nufpg'xf];\ .
(c) s'g} b'O{ cf]6f j|mdfut hf]/ ;ª\Vof¿sf] u'0fgkmn 288 5 eg] tL ;ª\Vofx¿ kTtf nufpg'xf];\ .
pTt/x¿
a. 12, 13 b.15, 17 c. 16, 18

gd'gf 5:
Pp6f wgfTds ;ª\Vof / To;sf] Jo'Tj|mdsf] 16 u'0ffsf] of]ukmn 8 x'G5 eg] pSt ;ª\Vof kTtf
nufpg'xf];\ .
;dfwfg M
oxfF, pSt ;ª\Vof x dfgf}F .
ta, k|Zgaf6,
1
x+ x×16 = 8 A A

2
x +16
or, x A =8 E
A

or, x2+16 = 8x
or, x2-8x+16 = 0
or, x2 -4x-4x+16 = 0
or, x(x-4) -4(x-4) = 0
or, (x-4) (x-4) = 0
or, (x-4)2 = 0
or, (x-4) = 0
x=4
pSt ;ª\Vof 4 /x]5 .
5 . cEof;sf nflu k|Zgx¿
(a) Pp6f ;ª\Vofsf] 25 u'0ff / ToxL ;ª\Vofsf] Jo'Tj|mdsf] cGt/ z"Go 5 eg] pSt ;ª\Vof kTtf
nufpg'xf];\ .
(b) 2 : 3 sf] cg'kftdf ePsf b'O{ cf]6f wgfTds ;ª\Vofx¿sf] u'0fgkmn 96 5 eg] tL ;ª\Vofx¿
kTtf nufpg'xf];\ .
(c) Pp6f ;ª\Vof To;}sf] Jo'Tj|md;Fu a/fa/ x'G5 eg] pSt ;ª\Vof kTtf nufpg'xf];\ .
(d) Pp6f ;ª\Vofsf] Jo'Tj|mdsf] 4 u'0ff ToxL ;ª\Vofsf] 9 u'0ff;Fu a/fa/ x'G5 eg] pSt ;ª\Vof kTtf
nufpg'xf];\ .
pTt/x¿
1 2
a. ±5
A E A b. 8, 12 c. ±1 d. ± 3
A E

145
nfdf] pTt/ cfpg] k|Zg ;DaGwL cEof;
4.6 o'ukt /]vLo ;dLs/0f (Simultaneous equation)
gd'gf 1 :
b'O{ cª\ssf] ;ª\Vofdf cª\sx¿sf] of]u 11 5 . pSt ;ª\Vofsf] cª\snfO{ :yfg dfg kl/jt{g
ubf{ aGg] ;ª\Vof ;'?sf] ;ª\VofeGbf 45 n] a9L x'G5 eg] ;'?sf] ;ª\Vof kTtf nufpg'xf];\ .
;dfwfg M
oxfF, b'O{ cª\s j|mdzM x / y dfgf}F .
b'O{ cª\sn] ag]sf] ;ª\Vof = 10x±y.
cª\sx¿sf] :yfgdfg kl/jt{g ubf{ aGg] ;ª\Vof = 10y±x.
ta, k|Zgsf] klxnf] ;t{cg';f/,
x+y = 11
or, y = 11- x ________(i)
k|Zgsf] bf];|f] ;t{cg';f/,
10y+x = (10x+y)+45
or, 10y+x-10x+y = 45
or, 9y - 9x = 45
or, 9(y-x) = 45
or, y-x =
or, y-x = 5
or, (11-x)-x = 5 [;dLs/0f (1) af6]
or, 11-x-x = 5
or, 11-5 = 2x
6
or, 2 = x
A E A

∴ x=3
x sf] dfg ;dLs/0f (1) df /fVbf,
y = 11- 3
∴y=8
pSt ;ª\Vof= 10x +y
= 10×3+8 = 38
pSt ;ª\Vof 38 /x]5 .

1. cEof;sf nflu k|Zgx¿

(a) b'O{ cª\sx¿ ldnL ag]sf] Pp6f ;ª\Vof 5 . tL cª\sx¿sf] of]u kmn 16 x'G5 . To; ;ª\Vofdf
18 hf]8\bf cª\sx¿sf] :yfg ablnG5 eg] ;f] ;ª\Vof kTtf nufpg'xf];\ .
(b) b'O{ cª\ssf] s'g} ;ª\Vof To;sf cª\sx¿sf] of]u kmnsf] ltg u'0ff 5 . olb ;f] ;ª\Vofdf 45
hf]8\g] xf] eg] cª\sx¿sf] :yfg ablnG5 eg] ;'?sf] ;ª\Vof kTtf nufpg'xf];\ .
(c) b'O{ cª\ssf] ;ª\Vof To;sf cª\sx¿sf] of]usf] 6 u'0ff 5 . Tof] ;ª\Vofaf6 9 36fof] eg] To;sf
cª\sx¿ plN6G5g\ . Tof] ;ª\Vof slt xf]nf <

146
(d) b'O{ cª\ssf] Pp6f ;ª\Vof To;sf cª\sx¿sf] of]usf] rf/ u'0ff 5 . olb Tof] ;ª\Vofdf 18
hf]8\of] eg] Tof] ;ª\Vofsf] ljk/Lt ;ª\Vof aG5, Tof] ;ª\Vof kTtf nufpg'xf];\ .
(e) b'O{ cª\ssf] s'g} ;ª\Vof o;sf cª\sx¿sf] of]u kmnsf] rf/ bf]Aa/eGbf 3 n] a9L 5 . Tof]
;ª\Vofdf 36 hf]8\of] eg] cª\sx¿sf] :yfg ablnG5 eg] Tof] ;ª\Vof kTtf nufpg'xf];\ .
(f) b'O{ cª\ssf] Pp6f ;ª\Vof tL cª\sx¿sf] of]usf] ltg bf]Aa/ 5 . tL cª\sx¿nfO{ pN6fP/
cfPsf] ;ª\Vofdf 9 hf]8\bf aGg] ;ª\Vof klxn]sf] ;ª\Vofsf] 3 bf]Aa/ x'G5 . Tof] ;ª\Vof kTtf
nufpg'xf];\ .
(g) Pp6f ;ª\Vof h'g 10 / 100 sf lardf k5{ . ;f] ;ª\Vof To;sf cª\sx¿sf] of]usf] 8 u'0ff 5 /
olb Tof] ;ª\Vofaf6 45 36fof] eg] Tof] ;ª\Vofsf] ljk/Lt ;ª\Vof aG5 eg] ;f] ;ª\Vof kTtf
nufpg'xf];\ .
pTt/x¿
a. 79 b. 27 c. 54 d. 24 e. 59 f. 27 g. 72
gd'gf 2:
s'g} ;ª\Vofsf] Pssf] :yfgdf /x]sf] cª\s b;sf] :yfgdf /x]sf] cª\ssf] eGbf 2 n] a9L 5 / tL
cª\sx¿sf] of]u kmnsf] t]Aa/ Tof] ;ª\Vof;Fu hf]l8of] eg] To; ;ª\Vofx¿sf] :yfg ablnG5 . ca
Tof] ;ª\Vof kTtf nufpg'xf];\ .
;dfwfg M
oxfF, Pssf] :yfgdf /x]sf] cª\s y / b;sf] :yfgdf /x]sf] cª\s x dfgf}F,
pSt cª\sn] ag]sf] ;ª\Vof=10x+y.
cª\sx¿sf] :yfg abNbf aGg] ;ª\Vof = 10y+x.
ta, k|Zgsf] klxnf] ;t{cg';f/,
y= x+2 ………………(i)
k|Zgsf] bf];|f] ;t{cg';f/,
3(x+y) + (10x+y) = 10y+x
or, 3x+3y+10x+y-10y-x = 0
or, 12x - 6y = 0
or, 6(2x-y) = 0
or, 2x-y = 0
or, y = 2x
or, x+2 = 2x [;dLs/0f (1) af6]
x=2
x sf] dfg ;dLs/0f (1) df /fVbf,
y = 2+2
y=4
pSt ;ª\Vof= 10x+y
= 10×2+4
= 24

147
2. cEof;sf nflu k|Zgx¿
(a) b'O{ cª\ssf] Pp6f ;ª\Vofdf klxnf] cª\s cf];|f] cª\seGbf 5 n] a9L 5 . olb cª\sx¿sf] :yfg
cbn abn ul/of] eg] gofF ;ª\Vof klxn]sf] ;ª\Vofsf] 83 efu;Fu a/fa/ x'G5 eg] ;'?sf] ;ª\Vof
A E A

kTtf nufpg'xf];\ .
(b) b'O{ cª\ssf] ;ª\Vofdf cª\sx¿sf] cGt/ 6 5 . olb o;sf] cª\sx¿sf :yfg abNof] eg] gofF
;ª\Vof klxn]sf] ;ª\Vofsf] 13
31 ;Fu a/fa/ x'G5 eg] ;'?sf] ;ª\Vof kTtf nufpg'xf];\ .
A E A

(c) b'O{ cf]6f ;ª\VofdWo] klxnf]nfO{ 36 ;Fu hf]8\of] eg] bf];|f] ;ª\Vofsf] 5 bf]Aa/ x'g cfpF5 . bf];|f]
;ª\VofnfO{ 36 af6 36fof] eg] klxnf] ;ª\Vofsf] 41 x'g cfpF5 . tL ;ª\Vofx¿ kTtf nufpg'xf];\ .
A E A

pTt/x¿
(a) 27 (b) 93 (c) 64, 20
gd'gf 3:
olb 5f]/fsf] pd]/sf] b'O{ u'0ff afa'sf] pd]/df hDdf ul/lbP pgLx¿sf] pd]/sf] hf]8 70 jif{ x'G5 . olb
afa'sf] pd]/sf] b'O{ u'0ff 5f]/fsf] pd]/df hf]l8lbP pgLx¿sf] pd]/sf] of]u 95 jif{ x'G5 eg] pgLx¿sf]
clxn]sf] pd]/ kTtf nufpg'xf];\ .
;dfwfg M
oxfF, afa' / 5f]/fsf] xfnsf] pd]/ x / y jif{ dfgf}F .
ta, k|Zgsf] klxnf] ;t{cg';f/,
2y+x = 70
∴ x = 70-2y .............................(i)
k|Zgsf] bf];|f] ;t{cg';f/,
2x+y = 95
or, 2 (70-2y) +y = 95 [;dLs/0f (i) af6]
or, 140-4y+y = 95
or, 140-95 = 3y
45
or, y = 3 A E A

∴ y = 15 jif{
y sf] dfg ;dLs/0f (i) df /fVbf,
2 ×15+x = 70
or, x = 70-30
∴ x = 40 jif{
∴afa' / 5f]/fsf] xfnsf] pd]/ 40 jif{ / 15 jif{ /x]5 .
3. cEof;sf nflu k|Zgx¿
(a) ltg jif{cl3 afa' / p;sf] 5f]/fsf] pd]/sf] of]u kmn 48 jif{ lyof], ltg jif{kl5 afa'sf] pd]/
5f]/fsf] pd]/sf] ltg u'0ff k'Ug] 5 eg] xfnsf] afa' / 5f]/fsf] pd]/ kTtf nufpg'xf];\ .
(b) 2 jif{cl3 afa'sf] pd]/ 5f]/fsf] pd]/sf] gf} u'0ff lyof] . t/ 3 jif{kl5 5 u'0ff dfq x'g] 5 eg] afa'
/ 5f]/fsf] xfnsf] pd]/ kTtf nufpg'xf]];\ .

148
(c) 6 jif{cl3 Pp6f dflg;sf] pd]/ 5f]/fsf] pd]/sf] 6 u'0ff a9L lyof] . ca 4 jif{kl5 p;sf] pd]/sf]
ltg u'0ff 5f]/fsf] pd]/sf] 8 u'0ff;FUf a/fa/ x'G5 . ca pgLx¿sf] xfnsf] pd]/ kTtf
nufpg'xf];\ .
(d) Ps jif{kl5 afa'sf] pd]/ 5f]/fsf] pd]/sf] 5 u'0ff x'g] 5 . b'O{ jif{cl3 afa'sf] pd]/ 5f]/fsf] ca 4
jif{kl5 x'g] pd]/sf] 3 bf]Aa/ lyof] . pgLx¿sf] clxn]sf] pd]/ kTtf nufpg'xf];\ .
(e) cfhsf] b'O{ jif{kl5 afa'sf] pd]/ 5f]/fsf] pd]/eGbf 6 u'0ff a9L 5 . ltg jif{ klxn] afa'sf] pd]/
5f]/fsf] b'O{ jif{kl5sf] pd]/eGbf kfFr u'0ff a9L lyof] eg] afa'sf] / 5f]/fsf] pd]/ lgsfNg'xf];\ .
(f) 14 jif{cl3 cfdfsf] pd]/ 5f]/Lsf] pd]/sf] 4 u'0ff lyof] . xfnsf] cfdfsf] pd]/ 5f]/Lsf] 4 jif{ kl5
x'g] pd]/sf] b'O{ u'0ff 5 eg] cfdf / 5f]/Lsf] slt slt pd]/ xf]nf < kTtf nufpg'xf];\ .

pTt/x¿
(a) 42 yrs, 12yrs (b) 47 yrs, 7 yrs (c) 36 yrs, 11 yrs,
(d) 29 yrs, 5 yrs (e) 28 yrs., 3 yrs. (f) 58yrs., 25 yrs.
gd'gf 4:
clxn] afa'sf] pd]/ 5f]/fsf] pd]/sf] b'O{ u'0ff 5 . olb 10 jif{kl5sf] 5f]/fsf] pd]/ / 15 jif{ cl3sf]
afa'sf] pd]/ a/fa/ eP afa' / 5f]/fsf] clxn]sf] pd]/ kTtf nufpg'xf];\ .
;dfwfg M
oxfF, afa' / 5f]/fsf] clxn]sf] pd]/ x jif{ / y jif{ dfgf}F,
ta, k|Zgsf] bf];|f] ;t{cg';f/,
x = 2y _______________(1)
k|Zgsf] bf];|f] ;t{cg';f/,
y+10 = x -15
or, y + 10 = 2y-15 [;dLs/0f (1) af6]
or, 10+15 = 2y-y
∴ y = 25 jif{
y sf] dfg ;dLs/0f (1) df /fVbf,
x =2×25
∴ x= 50 jif{
∴clxn]sf] afa' / 5f]/fsf] pd]/ j|mdzM 50 jif{ / 25 jif{ /x]5 .

4. cEof;sf nflu k|Zgx¿

(a) 8 jif{cl3 5f]/Lsf] pd]/ 5f]/fsf] t]Aa/ lyof] . clxn] 5f]/Lsf] pd]/ 5f]/fsf] eGbf 4 jif{n] a9L 5 .
eg] ltgLx¿sf] xfnsf] pd]/ lgsfNg'xf];\ .
(b) kfFr jif{cl3 afa'sf] pd]/ 5f]/fsf] pd]/sf] 4 u'0ff lyof] . clxn] afa' / 5f]/fsf] pd]/ hf]8\bf 45
jif{ 5 eg] afa' / 5f]/fsf] xfnsf] pd]/ kTtf nufpg'xf];\ .
(c) 5 jif{cl3 5f]/Lsf] pd]/ 5f]/fsf] eGbf 8 jif{n] a9L lyof] . clxn] 5f]/Lsf] pd]/ 5f]/fsf] pd]/sf] 2
u'0ff dfq 5 eg] ltgLx¿sf] xfnsf] pd]/ kTtf nufpg'xf];\ .
(d) 5 jif{cl3 Pp6f dflg;sf] pd]/ 5f]/Lsf] pd]/sf] 5 u'0ff lyof] . 3 jif{kl5 p;sf] pd]/sf] 2 u'0ff
5f]/Lsf] pd]/sf] 6 u'0ff;Fu a/fa/ x'G5 . pgLx¿sf] xfnsf] pd]/ slt slt xf]nf <
149
pTt/x¿
(a) 14 yrs, 10 yrs (b) 33 yrs, 12 yrs (c) 16 yrs, 8 yrs (d) 45 yrs, 13yrs
gd'gf 5:
s'g} leGgsf] c+z x/eGbf 1 n] sd 5 . olb pSt leGgsf] c+zdf 1 / x/df 5 hf]8\of] eg] gofF leGg
1
2 a5 eg] jf:tljs leGg kTtf nufpg'xf];\ .
A E A

;dfwfg M
oxfF, pSt jf:tljs leGg xy dfgf}F, hxfF c+z x / x/ y 5 . A E A

ta, k|Zgsf] klxnf] ;t{cg';f/,

x = y+1 ……………. (1)
k|Zgsf] bf];|f] ;t{cg';f/,
x+1 1
y+5 = 2 A E A A E

or, 2x+2 = y+5

or, 2(y+1)+2 = y+5 [;dLs/0f (1) af6]
or, 2y+2+2-y = 5
y = 5-4
∴y = 1
y sf] dfg ;dLs/0f (1) df /fVbf,
x = 1+1
∴x = 2
x 2
∴y =1 A E A A E

∴pSt jf:tljs leGg 12 /x]5 . A E A

5. cEof;sf nflu k|Zgx¿

(a) Pp6f leGgsf] c+znfO{ 4 n] u'0fg u/]/ x/af6 2 36fOof] eg] glthf 2 x'G5 . olb ;f] leGgsf]
c+zdf 15 hf]8L x/nfO{ bf]Aa/ u/L 2 36fp“bf 97 x'G5 eg] pSt leGg kTtf nufpg'x]f;\ .
A E A

(b) Pp6f leGgsf] c+znfO{ 4 n] u'0fg u/]/ x/af6 2 36fOof] eg] glthf 4 x'G5 . olb ;f] leGgsf]
c+zdf 10 hf]8L x/nfO{ bf]Aa/ u/L 2 36fpFbf 54 x'G5 eg] pSt leGg kTtf nufpg'xf];\ .
A E A

(c) b'O{ hgf s]6Lsf] pd]/sf] cg'kft 5.7 5 . cf7 jif{cl3 pgLx¿sf] pd]/sf] cg'kft 7:13 lyof] eg]
pgLx¿sf] xfnsf] pd]/ kTtf nufpg'xf];\ .
(d) A sf] pd]/sf] Ps ltxfO / B sf] pd]/sf] Ps rf}yfO hf]8\bf A sf] pd]/sf] b'O{ ltxfO x'G5 . olb
ltgLx¿sf] pd]/sf] of]u A sf] pd]/sf] b'O{ u'0ffeGbf klg 7 jif{n] a9L x'G5 eg] ltgLx¿sf] pd]/
slt xf]nf <
(e) ltg jif{ klxn] A / B sf] pd]/sf] cg'kft 4:3 lyof] . ltg jif{kl5 ltgLx¿sf] pd]/sf] cg'kft 11:
9 eP A / B sf] xfnsf] pd]/ kTtf nufpg'xf];\ .

150
pTt/x¿
3 5
(a) 8 A E A (b) 7
A E A (c) 15 yrs, 21 yrs (d) 21, 28 (e) 19yrs., 15yrs.

gd'gf: 6
Pp6f cfotsf/ rf}/sf] kl/ldlt 54 ld= 5 . o;sf] cfsf/ 36fpFbf nDafO / rf}8fO klxn]sf] nDafO
/ rf}8fOsf] j|mdzM 35 / 34 ;Fu a/fa/ x'G5 . 36]sf] cfotsf/ rf}/sf] kl/ldlt 36 ld= 5 . rp/sf]
A E A A E A

k|f/lDes nDafO / rf}8fO slt xf]nf <

;dfwfg M
cfotfsf/ rf}/sf] nDafO l / rf}8fO b dfgf}“,
k|Zgsf] klxnf] ;t{cg';f/,
2(l+b) = 54
or, l+b = 27
or, b = 27-l _________ (1)
gofF nDafO (L) = l×35 = 53 l A E A A E A

gof rf}8fO (B) = b × 43 = 34 b A E A A E A

k|Zgsf] bf];|f] ;t{cg';f/,

2(L+B) = 36
3 3 36
or, (5 l +4 b ) = 2
A E A A E A A E

3l×4+3b×5
or, A

20 = 18 E A

or, 3(4l+5b) = 360

or, 4l+5b= 120
or, 4l+5(27-l) = 120
or, 4l+135-5l = 120
or, 135-120 = l
∴ l = 15 m.
l sf] dfg ;dLs/0f (1) df /fVbf,
b = 27-l
= 27-15
= 12 m.
∴pSt rf}/sf] k|f/lDes nDafO / rf}8fO j|mdzM 15 m. / 12 m. /x]5 .

6. cEof;sf nflu k|Zgx¿

(a) Pp6f sf]7fsf] rf}8fOsf] rf/ u'0ff o;sf] nDafOsf] ltg u'0ff;“u a/fa/ 5 . olb rf}8fO 1 ld=n]
a9fO{ nDafO 1 ld=n] 36fof] eg] sf]7f juf{sf/ aGg] lyof] eg] sf]7fsf] nDafO / rf}8fO kTtf
nufpg'xf];\ .
(b) Pp6f cfotfsf/ sf]7fsf] If]qkmn 45 ju{ld6/ 5 . olb sf]7fsf] nDafO 3 ld6/ 36L / rf}8fO 1
ld6/ a9L eP sf]7f juf{sf/ x'g] lyof] eg] sf]7fsf] nDafO / rf}8fO kTtf nufpg'xf];\ .
151
pTt/x¿
(a) 8m, 6m (b) 9 m, 5m

gd'gf 7:
/fd;“u ePsf u'Rrfx¿dWo] Pp6f u'Rrf ;LtfnfO{ lb“bf b'j} hgf;“u a/fa/ ;ª\Vofdf u'Rrfx¿
x'G5 . olb ;Ltf;“u ePsf u'Rrfx¿dWo] Pp6f u'Rrf /fdnfO{ lb“bf, /fd;“u ;Ltfsf] eGbf bf]Aa/
u'Rrfx¿ x'G5 eg] ;'?df pgLx¿;“u ePsf] u'Rrfsf] ;ª\Vof kTtf nufpg'xf];\ .
;dfwfg M
;'¿df /fd;“u ePsf] u'R5f x / ;Ltf;“u ePsf] u'R5f y dfgf}“ .
ta, k|Zgsf] klxnf] ;t{cg';f/,
x-1 = Y+1
or, x= y+2 _________ (1)
k|Zgsf] bf];|f] ;t{cg';f/,
2(y-1) = (x+1)
or, x+1 = 2y-2
or, y+2+1 = 2y-2
or, 3+2 = 2y-y
∴y=5
y sf] dfg ;dLs/0f (1) df /fVbf,
x = 5+2
∴x=7
∴/fd / ;Ltf;“u ePsf u'R5fx¿ j|mdzM 7 cf]6f / 5 cf]6f /x]5g\ .

7. cEof;sf nflu k|Zgx¿

(a) Ps hgf sfdbf/n] sfd u/]sf] lbgdf ?= 30 Hofnf kfp“5 / cg'kl:yt ePsf] lbgdf ?= 6
hl/jfgf ltg{'k5{ . olb p;n] Ps dlxgfsf] cGt/df ?= 756 kfp“5 eg] ;f] dlxgf p;n] slt lbg
sfd u¥of] xf]nf <
(b) Ps hgf ;fx';“u b'O{ lsl;dsf] lrof 5 . klxnf] y/Lsf] lrof k|lt ls=u|f= ?= 100 / csf{] y/Lsf]
lrof k|lt ls=u|f= ?= 90 k5{ . b'j} y/Lsf] lrof slt slt ld;fp“bf 30 ls=u|f= ldl>t lrofsf]
d"No ?= 93 k|lt ls=u|f= knf{ <
(c) k/LIffdf /fdn] Zofdn] eGbf 12 gDa/ sd kfof] . olb p;n] hlt kfPsf] lyof] To;}sf] cfwf
c¿ a9L kfPsf] eP ZofdnfO{ 11 gDa/n] lhTg] lyof] . k|To]sn] slt slt gDa/ kfP5g\, kTtf
nufpg'xf];\ .
(d) /fdn] ZofdnfO{ eGof], ltdLn] dnfO{ ltdLl;t ePsf] bfdsf] cfwf lbof} eg] dl;t ?= 100 x'g]
5 . Zofdn] eGof], ltdLn] dnfO{ cfkm"l;t ePsf] bfdsf] Ps ltxfO dfTt lbof} eg] dl;t ?= 100
x'g] 5 . k|To]sl;t slt slt /x]5 <

pTt/x¿
(a) 26 (b) 9kg, 21kg (c) 46, 58 (d) Rs. 60, Rs. 80

152
s]xL pRr bIftf ljsf;sf nflu k|Zgx¿
(a) s]xL /sd s]xL s]6fnfO{ a/fa/ u/L af“l8of] . olb b'O{ hgf s]6f a9L ePsf] eP k|To]sn] ?= 10
sd kfp“y] / olb b'O{ s]6f sd ePsf] eP k|To]sn] ?= 15 a9L kfp“y] eg] s]6fx¿sf] ;ª\Vof /
k|To]sn] kfPsf] /sd kTtf nufpg'xf];\ .
[Hints: Let the sum of money be x & no. of boys be y.
x x
y+2 = y - 10
A E A A E 10y2 + 20y - 2x = 0 ..................... (i)
A

x x
y-2 = y + 15
A E A A 15y2 - 2x - 30y .............................(ii)
E A

x
Subtracted (i) & (ii) y = 10 & x = 600 ∴Each get = y = Rs. 60] A E A

(b) afa' / 5f]/fsf] clxn]sf] of]ukmn 82 jif{ 5 . ha afa' clxn]sf] 5f]/fsf] pd]/ hltsf lyP
ltgLx¿sf] pd]/sf] of]ukmn 46 jif{sf] lyof] eg] clxn]sf] afa' / 5f]/fsf] pd]/ kTtf
nufpg'xf];\ .
(c) afa' / 5f]/fsf] clxn]sf] pd]/sf] of]ukmn 73 jif{ 5 . b'j} hgf af“lr/x“bf 5f]/fsf] pd]/ clxn]
afa'sf] pd]/ hlTts} x'g] a]nf ltgLx¿sf] pd]/sf] of]ukmn 107 jif{ x'g] 5 eg] clxn]sf]
ltgLx¿sf] pd]/ kTtf nufpg'xf];\ .
[Hints Let the present age of father & son be x & y respectively.
x + y = 73 ......... (i) {x + (x - y)} + {y + (x - y)} = 107
∴ 3x - y = 107 ..................(ii) Solving (i) & (ii) x = 45 & y = 28]
pTt/x¿
(a) 10, Rs. 60 (b) 50 yrs, 32 yrs (c) 45 yrs, 28 yrs

153
4.7 ju{ ;dLs/0f (Quadratic Equations):

gd'gf 1 :
b'O{ cª\sx¿sf] ;ª\Vof df cª\sx¿sf] u'0fg kmn 18 / of]ukmn 9 5 eg] Tof] ;ª\Vof kTtf
nufpg'xf];\ . (If two digits number, the product of the digits is 18 and their sum is 9. Find the
number.)
;dfwfg
b'O{ cª\sx¿ x / y dfgf}F
b'O{ cª\sn] ag]sf] ;ª\Vof= 10x+y
ta, k|Zgsf] klxnf] ;t{cg';f/,
x ×y = 18
y= _________ (1)
k|Zgsf] bf];|f] ;t{cg';f/,
x+y = 9
18
or, x+ x = 9
A E [;dLs/0f (1) af6]
A

2
x +18
or, x = 9
A E A

or, x2+18 = 9x
or, x2-9x+18 = 0
or, x2-6x-3x+18 =0
or, x(x-6) -3 (x-6) =0
or, (x-6)-3(x-6) = 0
or, (x-6) (x-3) = 0
Either,
x-6 =0
∴ x= 6
or, x-3 = 0
∴x=3
x sf] dfg ;dLs/0f (1) df /fVbf,
18
x=6 x'“bf, y= 6 = 3 A E A

∴x=6/y=3
∴pSt ;ª\Vof= 10x+y = 10×6+3 = 63
k'g, x= 3 x'“bf, y =18
3 =6 A E A

∴x=3/ y=6
∴pSt ;ª\Vof=10x +y = 10×3+6 = 36
∴pSt ;ª\Vof= 63 jf 36 /x]5 .

154
1. cEof;sf nflu k|Zgx¿
(a) b'O{ ;ª\Vofsf] of]u kmn 7 / u'0fg kmn 12 5 eg] tL ;ª\Vofx¿ kTtf nufpg'xf];\ .
(b) b'O{ cª\ssf] Pp6f ;ª\Vof To;sf cª\sx¿sf] of]usf] rf/ u'0ff 5 . olb
cª\sx¿sf] u'0fg kmn 8 eP Tof] ;ª\Vof kTtf nufpg'xf];\ .
(c) b'O{ cf]6f wgfTds ;ª\Vofx¿dWo] 7'nf] ;ª\Vof ;fgf] ;ª\Vofsf] bf]Aa/eGbf 2 n] a9L 5 /
ltgLx¿sf] u'0fg kmn 12 5 eg] tL ;ª\Vofx¿ kTtf nufpg'xf];\ .
pTt/x¿
(a) 3, 4 (b) 24 (c) 2, 6

gd'gf 2:
b'O{ cf]6f ;ª\Vofx¿sf] of]u kmn 21 5 / tL ;ª\Vofx¿sf] ju{sf] of]u 261 5 eg] tL ;ª\Vofx¿
lgsfNg'x]f;\ .
;dfwfg
b'O{ cf]6f ;ª\Vof¿ j|mdzM x / y dfgf}F,
ta, k|Zgsf] klxnf] ;t{cg';f/,
x +y = 21
or, y = 21- x _________(1)
k|Zgsf] bf];|f] ;t{cg';f/,
+ = 261,
or, +(21- x)2 = 261
or, +(21)2-2×21 × x + 1
or, +441-42x + -261 = 0
or, 2 -42x +180 = 0
or, 2( -21x +90) = 0
or, -21x +90 = 0
or, -15x -6x +90 = 0
or, x (x -15)-6(x -15) = 0
or, (x -15_) (x -6) = 0
Either,
x -15 = 0
∴ x = 15
x = 15 /fVbf,
y = 21-15 = 6
∴ x = 15 / y = 6
or, x -6 = 0
x=6
x sf] dfg ;dLs/0f (1) df /fVbf,
x = 6 /fVbf, y = 21-6 = 15
∴ x = 6 / y = 15

155
∴pSt b'O{ ;ª\Vof¿ j|mdzM 15 / 6 jf 6 / 15 /x]5g\ .

2. cEof;sf nflu k|Zgx¿

(a) b'O{ ;ª\Vofx¿sf] of]u 16 5 / ltgLx¿sf] ju{sf] hf]8 130 5 eg] tL ;ªVofx¿ kTtf
nufpg'xf];\ .
The sum of two numbers is 16 and the sum of their sTtuares is 130. Find the numbers. [059A2]
(b) b'O{ cf]6f j|mdfut hf]/ ;ª\Vofx¿ kTtf nufpg'xf];\ h;sf ju{x¿sf] of]u kmn 340 5 .
Find two consecutive even numbers of which the sTtuares have the sum 340.
(c) b'O{ cf]6f nuftf/ cfpg] wgfTds k"0ff{ª\sx¿sf] u'0fg kmn 156 x'G5 eg] tL k"0ff{{ª\sx¿ s'g
s'g x'g\ <
The product of two consecutive positive integers is 156. Find the two numbers.
(d) nuftf/ cfpg] b'O{ cf]6f wgfTds lahf]/ ;ª\Vofx¿sf] u'0fg kmn 255 5 eg] tL ;ª\Vofx¿ kTtf
nufpg'xf];\ .
(The product of two consecutive positive odd numbers is 255. Find the two odd numbers.)
(e) Pskl5 csf{] cfpg] b'O{ cf]6f wgfTds hf]/ ;ª\Vofx¿sf] u'0fgkmn 288 5 eg] tL ;ª\Vofx¿
kTtf nufpg'xf];\ .
The product of two consecutive positive even numbers is 288. Find the number.
(f) b'O{ cf]6f wgfTds ;ª\Vofsf] cGt/ 3 5 . olb tL ;ª\Vofsf ju{x¿sf] of]u kmn 89 eP tL
;ª\Vofx¿ kTtf nufpg'xf];\ .
The difference between two positive numbers is 3. Find the numbers if the sum of their sttuares is
89.
pTt/x¿
(a) 7, 9 (b) 12, 14 (c)12, 13
(d) 15, 17 (e) 16, 18 (f) 5, 8
gd'gf 3:
b'O{ cª\ssf] ;ª\Vofdf cª\sx¿sf] u'0fg kmn 18 5 . pSt ;ª\Vofsf] cª\sx¿sf] :yfgdfg kl/jt{g
ubf{ aGg] ;ª\Vof ;'?sf] ;ª\VofeGbf 27 n] a9L x'G5 eg] ;'?sf] ;ª\Vof kTtf nufpg'xf];\ .
;dfwfg
b'O{ cª\sx¿ j|mdzM X / Y dfgf}+
b'O{ cª\sx¿n] ag]sf ;ª\Vof= 10x+y
:yfgdf kl/jt{g ubf{ aGg] ;ª\Vof= 10y+X
ta, k|Zgsf] klxnf] ;t{cg';f/,
x × y=18
or y = ………………(1)
ta, k|Zgsf] bf];|f] ;t{cg';f/
or, 10y+ x =(10 x +y)+27
or, 10y+ x -10-y=27
or, 9y-9 x =27

156
or, 9(y- x)=27
or, y- x =3
or,

or,
or, 18 - x 2= 3x
or,0 = x 2+3x -18
or, x 2+6x-2x -18=0
or, x (x +6)-3(x + 6)=0
or, (x +6) (x -3) =0
Either, x +6 =0 or x -3=0
x = -6( x'“b}g) x =3
x sf] dfg ;=s= (1) df /fVbf
y= =6

x = 3 / y=6
pSt ;ª\Vof =10x-y =10 × 3 +6 =36

3. cEof;sf nflu k|Zgx¿

(a) b'O{ cª\ssf] Pp6f ;ª\Vofdf cª\sx¿sf] u'0fg kmn 18 5 . pSt ;ª\Vofaf6 63 36fp“bf ;f]
;ª\Vofsf cª\sx¿ ablnG5g\ eg] ;ª\Vof kTtf nufpg'xf];\ .
(b) b'O{ cª\sx¿sf] ;ª\Vof df cª\sx¿sf] u'0fg kmn 24 5 . olb Tof] ;ª\Vofdf 45 hf]l8of] eg]
;ª\Vof¿sf] cª\sx¿sf] :yfg kl/jt{g x'G5 eg] Tof] ;ª\Vof kTTff nufpg'xf];\ .
(c) b'O{ cª\sx¿sf] Pp6f ;ª\Vofdf cª\sx¿sf] u'0fg kmn 8 5 . To; ;ª\Vofdf 18 yKbf ;f]
;ª\Vofsf cª\sxsf] :yfg ablnG5 eg] pSt ;ª\Vof kTtf nufpg'xf];\ .
(d) b'O{ cª\sx¿ ldnL ag]sf] Pp6f ;ª\Vof 5 . tL cª\sx¿sf] of]u kmn 16 x'G5 To; ;ª\Vofdf 18
36fp“bf cª\sx¿sf] :yfg ablnG5 eg] ;f] ;ª\Vof kTtf nufpg'xf];\ .
(e) Pp6f ;ª\Vof b'O{ cª\sn] ag]sf] 5 h;sf] of]u kmn 9 x'G5 olb ;f] ;ª\Vofsf] 3 u'0ff pSt
;ª\Vofsf] :yfg abNbf aGg] ;ª\Vofsf] 8 u'0ff;“u a/fa/ x'G5 eg] ;f] ;ª\Vof kTtf nufpg'xf];\ .

pTt/x¿
(a) 92 (b) 38 (c) 24 (d) 97 (e) 72

gd'gf 4:
(a) xfn afa' / 5f]/fsf] pd]/ j|mdzM 35 / 12 jif{ 5 . slt jif{cl3 ltgLx¿sf] pd]/sf] u'0fg kmn
210 lyof] kTtf nufpg'xf];\ .
;dfwfg M
pTt/, dfgf“}= x jif{cl3 afa' / 5f]/fsf] pd]/sf] u'0fg kmn 210 lyof] elg dfgf}+ .
ta, k|Zgsf] cg';f/,
157
 (35- x) (12- x) =210
or, 420-35x -12x + x 2-210=0
or, x 2-47x +210=0
or, x 2 -(42+5)x +210=0
or, x 2 -42x -5x +210=0
or, x (x -42)-5(x -42)=0
or, (x -42) (x -5)=0
Either, x -42=0 or, x - 5=0
x =42 x =5
x=42 jif{ x'“b}g lsgsL 42 jif{cl3 afa' g} hGd]sf lyPgg\ .
5 jif{cl3 plgx¿sf] pd]/sf] u'0fg kmn 210 x'G5 .
4. cEof;sf nflu k|Zgx¿
(a) b'O{ lbbL alxgLsf] xfnsf] pd]/sf] u'0fg kmn 150 5 . 5 jif{cl3 lbbLsf] pd]/ alxgLsf] pd]/sf]
bf]Aa/ eP ltgLx¿sf] xfnsf] pd]/ kTtf nufpg'xf];\ .
(b) bfh' / efOsf] pd]/sf] cGt/ 4 jif{ 5 / ltgLx¿sf] pd]/sf] u'0fg kmn 221 x'G5 eg] tL b'O{
efOsf] pd/] kTtf nfupg'xf];\ .
(c) ;Ltfsf] 4 jif{ cufl8 / 8 jif{kl5sf pd]/x¿sf] u'0fg kmn 28 eP pgsf] clxn]sf] pd]/ kTtf
nufpg'xf];\ .
(d) b'O{ bfh' efOsf] xfnsf] pd]/sf] u'0fg kmn 160 5 . 4 jif{cl3 bfh'sf] pd]/ efOsf] pd]/sf] bf]Aa/
eP ltgLx¿sf] xfnsf] pd]/ kTtf nufpg'xf];\ .
(e) Ps jif{cl3 Pp6f dflg; cfkm\gf] 5f]/feGbf 8 u'0ffn] h]7f] lyof] . clxn] p;sf] pd]/ 5f]/fsf]
pd]/sf] ju{;“u a/fa/ 5 . ltgLx¿sf] xfnsf] pd]/ kTtf nufpg'xf];\ .
(f) afa' / 5f]/fsf] clxn]sf] pd]/ j|mdzM 37 / 8 slt jif{ klxn] ltgLx¿sf] pd]/sf] u'0fg kmn 96
lyof] xf]nf <
pTt/x¿
(a) 15 yrs., 10 yrs. (b) 16 yrs, 10 yrs. (c) 6 yrs,
(d) 16 yrs, 10 yrs. (e) 7 yrs, 49 yrs. (f) 5 yrs.
gd'gf 5:
Pp6f cfotfsf/ hUufsf] If]qkmn 660 ju{ ld6/ / o;sf] kl/ldlt 104 eP hUufsf] nDafO / rf}8fO
lgsfNg'xf];\ .
;dfwfg M
cfotsf/ hUufsf] nDafO l / rf}8fO b dfgf}“ .
ta, k|Zgsf] klxnf] ;t{cg';f/,
l × b =660
or, b = (1)
ta, k|Zgsf] bf];|f] ;t{cg';f/,

158
 or

2 sf] dfg ;=s= (1) df /fVbf

/fVbf
or
k'gM /fVbf,
/
x'g ;Sb}g lsgls nDafOeGbf rf}8fO nfdf] x'g ;Sb}g .
pSt cfotsf/ hUufsf] nDafO / rf}8fO j|mdzM / /x]5 .

5. cEof;sf nflu k|Zgx¿

(a) olb Pp6f cfotsf/ rf}/sf] kl/ldlt 36 ld6/ / If]qkmn 77 ju{ ld6/ eP ;f] rf}/sf] nDafO /
rf}8fO kTtf nufpg'xf];\ .
(b) Pp6f sf]7fsf] nDafO To; sf]7fsf] rf}8fOeGbf 2 ld= nfdf] 5 . olb sf]7fsf] If]qkmn 63 ju{ ld=
5 eg] sf]7fsf] nDafO / rf}8fO kTtf nufpg'xf];\ .
(c) Pp6f cfotfsf/ hUufsf] rf}8fO nDafOeGbf 3 ld=eGbf sd 5 . olb pSt hUufsf] If]qkmn 88
ju{ ld= eP kl/ldlt kTtf nufpg'xf];\ .
(d) Pp6f sf]7fsf] If]qkmn 45 ju{ ld= 5 . olb nDafOdf 3 ld6/ sd / rf}8fOdf 1 ld6/ a9fp“bf
ju{ aG5 eg] pSt sf]7fsf] nDafO / rf}8fO kTtf nufpg'xf];\ .
(e) Pp6f sf]7fsf] If]qkmn 28 ju{ ld6/ 5 . olb nDafOdf 1 ld6/ sd / rf}8fOdf 2 ld6/ a9fp“bf
ju{ aG5 eg] pSt sf]7fsf] nDafO / rf}8fO kTtf nufpg'xf];\ .
(f) olb Pp6f cfotsf/ hUufsf] kl/ldlt 104 ld6/ / If]qkmn 640 ju{ ld6/ 5 eg] o;sf] nDafO
/ rf}8fO kTtf nufpg'xf];\ .
pTt/x¿
(a) 11m, 7m (b) 9m, 7 m (c) 38 m
(d) 9m, 5m (e) 7m, 4m (f) 32m, 20m

159
gd'gf 6:
Pp6f ;dsf]0f lqe'hsf] s0f{ e'hf 29 ;]=ld 5 / c¿ b'O{ e'hfx¿sf] km/s 1 ;]=ld= 5 eg] tL e'hfx¿
kTtf nufpg'xf];\ .
;dfwfg M
oxfF, ;dsf]0f lqe'hsf] s0f{ h / cGo b'O{ e'hfx¿ j|mdzM p / b 5g\ egL dfgf}F .
ta, h = 29 cm. / P-b = 1 cm.
or, p = (l+b) cm. ______________(1)
xfdLnfO{ yfxf 5,
+ = [kfOyfuf]/; ;fWocg';f/]
or, (l+b)2+b2=(29)2-840=0
or, 2b2 +2b - 840=0
or, 2(b2+b-420)=0
or, b2+b-420=0
or, b2 +21b-20b-420=0
or, b (b+21)-20(b+21)=0
or, (b+21) (b-20) =0
Either, or,
b+21=0 b-20=0
b = - 21cm x“'b}g b=20cm
dfg ;=s=(1) df /fVbf
p = 1+20 = 21cm
pSt b'O{ e'hfx¿ 21cm / 20cm /x]5g\ .
6. cEof;sf nflu k|Zgx?
(a) Pp6f ;dsf]0f lqe'hsf] s0f{ e'hf 5f]6f] e'hfsf] bf]Aa/eGbf klg 6 ld= nfdf] 5 . olb t];|f] e'hf
s0f{eGbf 2 ld6/ 5f]6f] 5 eg] lqe'hsf e'hfx¿ kTtf nufpg'xf];\ .
(b) Pp6f ;dsf]0fL lqe'hsf] s0f{ To;sf] cfwf/eGbf 2 cm a9L / prfOsf] bf]Aa/eGbf 1 cm a9L
5 eg] lqe'hsf k|To]s e'hf kTtf nufpg'xf];\ .
(c) Pp6f ;dsf]0fL lqe'hsf ;dsf]0f agfpg] e'hfx¿ To;sf] s0f{eGbf 5 cm /10 cm j|mdzM sd
5g\ eg] lqe'hsf e'hfx¿sf] nDafO kTtf nufpg'xf];\ .
pTt/x¿
(a) 10 cm, 24 cm, 26 cm (b) 8cm, 15cm, 17cm (c) 25cm, 20cm, 15cm

160
 If]q M 5 Hofldlt (Geometry)
If]q kl/ro
dfWolds lzIff kf7\oj|md 2064 / ljlzi6Ls/0f tflnsf 2065 n] sIff 10 sf] clgjfo{ ul0ft
ljifosf] Hofldlt If]qdf lgDgfg';f/sf ljifo j:t', k/LIffsf nflu ;+1fgfTds If]qcg';f/sf k|Zg
;ª\Vof / cª\s ef/sf] Joj:yf u/]sf] 5 .
j|m ljifoj:t'÷;+1fg If]q 1fg l;k hDd l;k ;d:o hDd s'n If]qut
; / f f f hDdf s'n
af]w cª\ ;dfw cª\ cª\s hDdf
s fg s cª\s
5 5f]6f pTt/ nfdf] pTt/ 24
cfpg] cfpg] k|Zg
5. lqe'h / rt'e'{hsf] 1 2 1 4 6
1 If]qkmn
5. j[Tt / ;f];DaGwL 1 1 4 1 4 8
2 ;fWo / ltgsf] k|of]u
5. :kz{ /]vf 1 2
3
5. rt'e'{hsf] If]qkmn;Fu 1 4 4
4 a/fa/ x'g] lqe'hsf]
/rgf
5. k|of]ufTds k/LIf0f u/L 1 4 4
5 l;b\w ug]{

5.1 M lqe'h / rt'e'{hsf If]qkmn

!= kl/ro
o; kf7df Pp6} cfwf/df / pxL ;dfgfGt/ /]vfx¿ lar aGg] lqe'h tyf ;dfgfGt/ rt'e{'hx¿sf]
If]qkmnsf ;DaGwsf af/]df rrf{ ul/Psf] 5 . P;Pn;L k/LIffdf o; kf7af6 Pp6f 5f]6f] pTt/
cfpg] / Pp6f nfdf] pTt/ cfpg] u/L hDdf b'O{ cf]6f k|Zgx¿ ;f]lwg] s'/f ljlzi6Ls/0f tflnsfdf
pNn]v ul/Psf] 5 .
@= cfwf/e"t tYo tyf ;"qx?
!= lqe'hsf] If]qkmn = ½ × cfwf/ × prfO
@= ;dfgfGt/ rt'e'{hsf] If]qkmn = cfwf/ × prfO
#= ;dafx' rt'{e'hsf] If]qkmn = ½ ljs0f{x¿sf] u'0fg kmn
$= ju{sf] If]qkmn = -e'hf_2 = ½ -ljs0f{_2
161
 %= cfotsf] If]qkmn = nDafO × rf}8fO
^= Pp6} jf a/fa/ cfwf/df / pxL ;dfgfGt/ /]vfx¿ lar ag]sf lqe'hx¿sf If]qkmn a/fa/
x'G5g\ .
&= Pp6} jf a/fa/ cfwf/df / pxL ;dfgfGt/ /]vfx¿ larsf ;dfgfGt/ rt'e{'hx¿sf
If]qkmnx¿ a/fa/ x'G5g\\ .
*= Pp6} jf a/fa/ cfwf/df / pxL ;dfgfGt/ /]vfx¿ lar ag]sf lqe'hsf] If]qkmn ;dfgfGt/
rt'e{'hsf] If]qkmnsf] cfwf x'G5 .
(= ;dfgfGt/ rt'e{'hsf] ljs0f{n] ;f] ;dfgfGt/ rt'e{'hnfO{ cfwf u5{ .
!)= lqe'hsf] dWo /]vfn] ;f] lqe'hnfO{ cfwf ub{5 .
!!= Pp6} cfwf/ / o;sf] Ps}lt/sf lqe'hx¿sf If]qkmn a/fa/ 5g\ eg] tL lqe'hx¿ pxL
;dfgfGt/ /]vfx¿ lar kb{5g\ .
!@= ;dnDa rt'e{'hsf] dWo /]vf cfwf/x¿;Fu ;dfgfGt/ x'G5g\ .
!#= ;dnDa rt'e'{hsf] dWo /]vfn] ljs0f{x¿nfO{ ;dlåefhg ub{5g\ .
!$= lqe'hsf] s'g} b'O{ e'hfx¿sf dWo ljGb'x¿ hf]8\g] /]vf v08 t];|f] e'hf;Fu ;dfgfGt/ eOsg
o;sf] cfwf klg x'G5 .
#= ljz]if Wofg lbg'kg{] s'/fx?
!= lqe'hsf] prfO eGgfn] s'g} zLif{ljGb'af6 To;sf] ;Dd'v e'hfdf lvlrPsf] nDa / ;Dd'v
e'hfnfO{ cfwf/ eg]/ a'‰g'k5{ .
h:t} M
A
A
A
b
F
Eb h

h
B D C
cfwf/ B C B
B
A A

h h

B b C D B b C
-oxfF lqe'h ABC sf] cfwf/ BC / prfO AD x'G5 ._
@= ;dfgfGt/ rt'e'{hsf] s'g} zLif{ ljGb'af6 ;Dd'v e'hfdf lvlrPsf] nDa -prfO_ / h'g e'hfdf
nDa lvlrPsf] xf] To;
e'hfnfO{ cfwf/ eg]/ a'‰g'k5{ .

162
 A D
A D
oxfF,
h F
h
b
AE = prfO = h
h:t}M B b E C B C M
E
A D
h

A D
h b

B C E
b B C

prfO = DE
cfwf/ = AB
oxfF, ;dfgfgt/ rt'e'{h ABCD sf] prfO DE / cfwf/ BC xf] .

#= lqe'hsf] s'g} zLif{ ljGb' / To;sf] ;Dd'v e'hfsf] dWo ljGb' hf]8\g] /]vf v08nfO{ dWo /]vf
elgG5 . h:t} M
A A A

M
M

B M C B CB C

To;}n], ∆ABM = ∆ACM = ½ ∆ABC / ∆ABM = ∆BCM ;fy} CM dWo /]vf xf] .
∴ ∆ACM = ∆BCM x'G5 .

163
$= ju{, ;dafx' rt'e{'h / cfoft ;dfgfGt/ rt'e{'hsf] k|sf/x¿ x'g\ .
h:t} M

;dfgfgG
t/

cfoft ju{ ;dafx', rt[e{'h

%= ;dfgfGt/ /]vfx¿ ;dfg b'/Ldf x'G5g\ .

h:t} M AB..CD 5g\ eg] EF = GH = IJ x'G5g\ .
E G I
A B

C D
F H J
^= Pshf]8f ljkl/t e'hfx¿ ;dfgfGt/ ePsf] rt'e{'h ;dnDa rt'e{'h xf] . ;dfgfGt/ e'hfx¿
cfwf/x¿ / ;dfgfGt/ gePsf e'hfx¿nfO{ kfb elgG5 . To:t} kfbx¿sf hf]8\g] /]vf dWo/]vf
xf] . h:t} M kfbx¿sf dWo ljGb'x¿ hf]8\g] AB / CD cfwf/x¿ tyf AC / BD kfbx¿ x'g\ . To:t}
XY dWo/]vf xf] .
A cfwf/ B
kb kb
X Y
dWo /]vf
C D
cfwf/
$= 5f]6f] gd'gf k|Zgf]Tt/ / cEof;
A
gd'gf ! M F D
lbOPsf] lrqdf ABCD Pp6f ;dfgfGt/ E
rt'e{'h / AE ⊥ CD 5g\ . olb ∆FBC sf]
If]qkmn 24 ju{ ;]=ld= / CD = 8 ;]=ld= 5g\ B C
eg] ;dfgfGt/ rt'e'{hsf] prfO AE sf] dfg kTtf nufpg'xf];\ .
;dfwfg M
oxfF, ABCD = 2 × ∆FBC x'G5 . -Pp6} cfwf/ BC / pxL ;dfgfGt/ /]vfx¿ BC / FD
= 2× 24 lar /x]sf] lqe'h / ;dfgfGt/ rt'e'{hsf] If]qkmnsf] ;DaGw_
= 48 cm2
164
 ca, ABCD = CD × AE ;dfgfGt/ rt'e'{h sf] If]qkmn = cfwf/ × prfO
or, 48 = 8 × AE
∴ AE = 6 cm.
cEof;sf nflu k|Zgx¿ M C
!= lbOPsf] lrqdf ABCD sf] e'hf D df E nDa 5 . olb AD = 5cm / D F

CE = 8cm 5 eg] ∆ABF sf] If]qkmn lgsfNg'xf];\ .

E

A B

A
@= lbOPsf] lrqdf TR // PQ, PS//QR / SU⊥QR F D
5g\ . olb ∆TPQ sf] If]qkmn 20cm2 / SU = 8cm eP QR sf] E
gfk kTtf nufpg'xf];\ .
B C
A E
D
#= lbOPsf] lrqdf ABCD Pp6f ;dfgfGt/ rt'e'{h xf] . olb BE =
F
12cm. / nDa CF = 6cm. eP ABCD sf] If]qkmn slt xf]nf < kTtf
nufpg'xf];\ . B C

gd'gf k|Zg @ M P S
lbOPsf] lrqdf PQRS Pp6f ;dafx' rt'e{'h xf] . olb PR=9cm. / QS=6cm.
5g\ eg] ∆PTS sf] If]qkmn lgsfNg'xf];\ .
T Q R
;dfwfg M
oxfF, ;dafx' rt'e{'h PQRS = ½ PR × QS
-;dafx' rt'e{'hsf] If]qkmn = ½ ljs0f{x¿sf] u'0fgkmn_
= 27cm2 -Pp6} cfwf/ PS / pxL ;dfgfGt/
ca, ∆PTS = ½× PQRS, /]vfx¿ PS / TR larsf] lqe'h /
= 13.5cm2 ;=r=sf] ;DaGwaf6 _

cEof;sf nflu k|Zgx¿ M

A D
E
!= lbOPsf] lrqdf ABCD Pp6f ju{ xf] . olb ljs0f{ AC = 10 2 cm
5 eg] ∆BCE sf] If]qkmn lgsfNg'xf];\ .

B C

165
 A D
@= lbOPsf] lrqdf AD//BE / AB=BC=CD=DA 5g\ . AC = 6cm. / BD =
8cm. eP ∆ADE sf] If]qkmn kTtf nufpg'xf];\ .
B C E

U S
#= lrqdf PQRS Pp6f ju{ xf] / QRTU Pp6f ;dfgfgt/ rt'e'{h xf] . P T
olb ljs0f{ QS = 10cm. eP ;dfgfGt/ rt'e'{h QRTU sf] If]qkmn slt
xf]nf < kTtf nufpg'xf];\ .

Q R
gd'gf k|Zg # M
A
lbOPsf] lrqdf AB sf] dWo ljGb' D / DE⊥BC 5g\ . olb ∆ABC sf] If]qkmn
120cm2 / BC = 12cm. eP DE sf] nDafO lgsfNg'xf];\ .
;dfwfg M D
oxfF, DC hf]l8Psf] 5 .
ca, ∆BCD = ½ ∆ABC -dWo /]vf DC n] ∆ABC nfO{ cfwf ug{] ePsfn]_
= ½×120 B E C

= 60cm2
km]l/, ∆BCD = ½ BC × DE -lqe'hsf] If]qkmn = ½ cfwf/ ×× prfO]_
or, 60 = ½×12×DE
∴ DE = 10cm
.
P

cEof;sf nflu k|Zgx¿ M S

!= lbOPsf] lrqdf PS = SR / ST ⊥ QR 5g\ . olb QR = 15cm / ST = 5cm.eP

Q R
∆PQR sf] If]qkmn kTtf nufpg'xf];\ . T

@= lbOPsf] lrqdf BC sf] dWo ljGb' D af6 AB df nDa DE lvlrPsf] 5 . E

olb AB = 10cm. / ∆ABC sf] If]qkmn 40cm2 eP DE sf] gfk lgsfNg'xf];\ .
B D C
#= lbOPsf] lrqdf MNOP Pp6f ;dfgfGt/ rt'e'{h xf] / MN sf] dWo ljGb'
A xf] . olb ∆AON sf] If]qkmn 15cm2 5 eg] MNOP sf] If]qkmn
M P
lgsfNg'xf];\ .
A

N O

166
gd'gf k|Zg $ M
lbOPsf] lrqdf AC//DE / BC=CE 5g\ . olb ∆ACD sf] If]qkmn 20cm2 eP A D

rt'e{'h ABCD sf] If]qkmn lgsfNg'xf];\ .

;dfwfg M
oxfF, ∆ACE = ∆ACD (Pp6} cfwf/ AC / pxL ;dfgfGt/ /]vfx¿ AC / DE B C
E
larsf lqe'hx¿_]
∴ ∆ACE = 20cm2
km]l/, ∆ABC = ∆ACE -dWo/]vf AC n] ∆ABE nfO{ cfwf ug{] ePsf]n] ._
∴ ∆ABC = 20cm 2

To;}n], rt'e{'h ABCD sf] If]qkmn = ∆ABC + ∆ACD

= 20 + 20
= 40cm2

cEof;sf nflu k|Zgx¿ A D

!= lbOPsf] lrqdf AC//DE / BC=CE 5g\ .
olb ∆ABC = 25cm2 eP rt'e{'h ABCD sf] If]qkmn lgsfNg'xf];\ .
E
B C
P S
@= lbOPsf] lrqdf PT//SQ / TQ=QR 5g\ . olb ∆PQS sf] If]qkmn 40cm2
eP ∆RST sf] If]qkmn kTtf nufpg'xf];\ .
T Q R
E
#= lbOPsf] lrqdf ABCD Pp6f ;dfgfGt/ rt'e{'h xf] . EC sf] dWo ljGb' F
xf] . olb ∆BEF sf] If]qkmn 15cm2 eP ABCD sf] If]qkmn slt xf]nf A F D
< kTtf nufpg'xf];\ .

B C
yk cEof;sf nflu k|Zgx¿ M C E D
!= lbOPsf] lrqdf AB//CD / AE//BD 5g\ . olb ∆BDE sf] If]qkmn 16cm2
5 eg] ∆ABC sf] If]qkmn slt xf]nf < kTtf nufpg'xf];\ .
A B
@= lbOPsf] lrqdf, PQ//TR / PS//QR 5g\ . olb ∆PST sf] If]qkmn 25cm2
/ ;=n=r= PQRT sf] If]qkmn 105cm2 5 eg] ∆PQT sf] If]qkmn T S R
lgsfNg'xf];\ .

P Q

167
 E D C

#= lbOPsf] lrqdf EC//AB, AD //BC / AB sf] dWo ljGb' M 5g\ . olb

ABCD sf] If]qkmn 80cm2 5 eg] ∆AME sf] If]qkmn kTtf nufpg'xf];\ .
A B
M
$= lrqdf ABCD Pp6f ju{ xf] . AD nfO{ E ;Dd nDJofOPsf] 5 . olb ∆BCE
sf] If]qkmn 25cm2 5eg] ljs0f{ AC sf] nDafO kTtf nufpg'xf];\ . A D E

%= lbOPsf] lrqdf ABCD sf] e'hf AD sf] dWo ljGb' M af6 CD df nDa
MN lvlrPsf] 5 . olb CD = 8cm. / N=4cm 5 eg] ∆BAM sf] If]qkmn B C
/ ABCD sf] If]qkmn kTtf nufpg'xf];\ .
A M D

N
nfdf] gd'gf k|Zgf]Tt/ / cEof;
C
gd'gf ! T
B
U S R
lbOPsf] lrqdf Pp6} cfwf/ PQ / pxL ;dfgfGt/ /]vfx¿ PQ / TR
lar ;dfgfGt/ rt'e{'hx¿ PQRS / PQUT 5g\ .
k|dfl0ft ug{'xf];\ M
i) ∆PST ≅ ∆QRU
P Q
ii) PQRS sf] If]qkmn = P QUT sf] If]qkmn
k|df0fM
tYox¿ sf/0fx¿
1. ∆PST / ∆QRU df 1.
i) PS = QR i) PQRS sf ;Ddv' e'hfx¿ ePsfn] .
ii) ∠PST = ∠QRU ii) ;dfgfGt/ /]vfx¿ PS / QR df ag]sf
;ª\ut sf]0fx¿ ePsfn] .
iii) ∠PTS = ∠QUR iii) ;ª\ut sf]0fx¿ M PT//QU ePsfn] .
2. ∆PST ≅ ∆QUR 2. e'=sf]=sf]= tYocg';f/ .
3. ∆PST = ∆QRU 3. cg'¿k lqe'hx¿sf If]qkmn a/fa/ x'g]
ePsfn] .
4. ;=n=r= PQRT-∆PST= 4. a/fa/ If]qkmn ePsf lqe'hx¿nfO{ pxL lrq
;=n=r= PQRT-∆QRU PQRT af6 36fpFbf
5. ∴ PQRS = PQUT 5. z]if tYo .
k|dfl0ft eof] . F E D C
cEof;sf nflu k|Zgx¿
!= lbOPsf] lrqdf Pp6} cfwf/ AB / pxL ;dfgfGt/ /]vfx¿ AB / FC lar
ag]sf ;dfgfGt/ rt'e{'hx¿ ABCD / ABEF 5g\ .
k|dfl0ft ug{'xf];\ M A B

168
 i) ∆AFD ≅ ∆BEC
ii) ABCD sf] If]qkmn = ABEF sf] If]qkmn .
A P B S
@= lbOPsf] lrqdf cfot AQRB / ;dfgfGt/ rt'e'{h PQRS Pp6} cfwf/
QR / pxL ;dfgfGt/ /]vfx¿ QR / AS lar 5g\ . k|dfl0ft ug{'xf];\ M
cfot AQRB sf] If]qkmn = ;dfgfGt/ rt'e'{h PQRS sf] If]qkmn
Q R
#= Pp6} cfwf/ WX / pxL ;dfgfGt/ /]vfx¿ WX / VY lar /x]sf
;dfgfGt/ rt'e{'hx¿ UVWX / WXYZ sf If]qkmnx¿ a/fa/ x'G5 egL
k|dfl0ft ug{'xf];\ .
gd'gf @ M
Pp6} cfwf/ XY / pxL ;dfgfGt/ /]vfx¿ XY / AZ lar ag]sf lqe'h AXY
sf] If]qkmn ;dfgfGt/ rt'e{'h WXYZ sf] If]qkmnsf] cfwf x'G5 egL A B W Z
k|dfl0ft ug{'xf];\ .

;dfwfg M
oxfF, yfxf lbOPsf] M Pp6} cfwf/ XY / pxL ;dfgfGt/ /]vfx¿ XY / X Y
AZ lar ∆AXY / WXYZ 5g\ .
k|dfl0ft ug{'kg{] M ∆AXY = ½ WXYZ
h'lSt -/rgf_ M XA//YB lvrf}+ .
k|df0f M
tYox¿ sf/0fx¿
1. AXYB Pp6f ;dfgfGt/ rt'e'{h 1. AX //BY / AB//XY ePsfn] .
xf] .
2. AXYB = WXYZ 2. Pp6} cfwf/ / pxL ;dfgfGt/ /]vfx¿ lar
ag]sf ;dfgfGt/ rt'e'{hx¿ ePsfn] .
3. ∆AXY = ½ AXYB 3. ljs0f{ AY n] AXYB nfO{ cfwf ug{] ePsfn] .
4. ∴ ∆AXY = ½ WXYZ 4. tYo 2 / 3 cg';f/ .
k|dfl0ft eof] .
a}slNks tl/sf M A W Z
h'lSt M XY df nDa WB lvrf}+ .

k|df0f M X B Y

tYox¿ sf/0fx¿
1. WXYZ = XY.WB 1. ;dfgfGt/ rt'e'{hsf] If]qkmn = cfwf/ × prfO
ePsfn] .
2. ∆AXY = ½ XY.WB 2. lqe'hsf] If]qkmn = ½ cfwf/ × prfO ePsfn] .

169
 3. ∴ ∆AXY = ½ WXYZ 3. tYo 1 / 2 cg';f/ .
k|dfl0ft eof] .
cEof;sf nflu k|Zgx¿ M A D E
!= lbOPsf] lrqdf ∆BCE / ABCD Pp6} cfwf/ BC / pxL ;dfgfGt/
/]vfx? BC / AE lar 5g\ eg] ∆BCE sf] If]qkmn ABCD sf] cfwf
x'G5, egL k|dfl0ft ug{'xf];\ . B C

@= Pp6} cfwf/ QR / pxL ;dfgfGt/ /]vfx¿ QR / PS lar ag]sf]

;dfgfGt/ rt'e'{h QRST sf] If]qkmn ∆PQR sf] If]qkmnsf] b'O{ u'0ff x'G5
egL l;b\w ug{'xf];\ .
M Q P
#= lbOPsf] lrqdf MNOP Pp6f ;dfgfGt/ rt'e{'h xf] . k|dfl0ft ug{'xf];\ M
∆QNO sf] If]qkmn = ½ MNOP sf] If]qkmnx¿ a/fa/ x'G5 egL
k|dfl0ft ug'{xf];\ .
N O

gd'gf k|Zg # M
Pp6} cfwf/ XY / pxL ;dfgfGt/ /]vfx¿ XY / WZ lar ag]sf lqe'hx¿
W
WXY / ZXY sf If]qkmnx¿ a/fa/ x'G5g\ egL l;b\w ug{'xf];\ . A Z

;dfwfg M
oxfF, yfxf lbOPsf] M Pp6} cfwf/ XY / pxL ;dfgfGt/ /]vfx¿ XY / WZ X Y
lar ∆WXY / ∆ZXY 5g\ .
l;b\w ug{'kg{] M ∆WXY = ∆ZXY
h'lSt M YZ//XA lvrf}+ .

k|df0f M
tYox¿ sf/0fx¿
1. AXYZ Pp6f ;dfgfGt/ rt'e'{h 1. AX ..ZY -h'lSt_ / XY..AZ -lbOPsf]_ ePsfn] .
xf] .
2. ∆ZXY= ½ AXYZ 2. ljs0f{ XZ n] AXYZ nfO{ cfwf ug{] ePsfn] .
3. ∆WXY = ½ AXYZ 3. Pp6} cfwf/ / pxL ;dfgfGt/ /]vfx¿ larsf
lqe'h / ;dfgfGt/ rt'e'{hsf] ;DaGw .
4. ∴ ∆ZXY = ∆WXY 4. tYo 2 / 3 cg';f/ . -a/fa/L tYo_
k|dfl0ft eof] . W Z
j}slNks tl/sf M
170
X Y
A
 h'lSt M WA⊥XY lvrf}+ .
k|df0f M
tYox¿ sf/0fx¿
1. ∆WXY=½ XY.WA 1. lqe'hsf] If]qkmn =½ cfwf/ × prfO ePsfn]
2. ∆ZXY= ½ XY.WA 2. sf/0f 1 cg';f/
3. ∴ ∆WXY = ∆ZXY 3. tYo 1 / 2 cg';f/
k|dfl0ft eof] .
cEof;sf nflu k|Zgx¿
!= lbOPsf] lrqdf ∆ABC / ∆ABD Pp6} cfwf/ AB / pxL ;dfgfGt/ C D
/]vfx¿ AB / CD lar 5g\ .
k|dfl0ft ug{'xf];\ M
∆ABC sf] If]qkmn = ∆ABD sf] If]qkmn A B

@= Pp6} cfwf/ QR / pxL ;dfgfGt/ /]vfx¿ QR / PS lar ag]sf

P Q
lqe'hx¿ PQR / SQR sf If]qkmn a/fa/ x'G5g\ egL l;b\w ug{'xf];\ .

#= lbOPsf] lrqdf PQ//MN 5g\ eg] k|dfl0ft ug{'xf];\ M ∆PMN=∆QMN

N
gd'gf k|Zg $ M M
lbOPsf] lrqdf AD//BC//EF, AB//DE / BE//AF A
D
5g\eg] ABCD / BEFH sf] If]qkmn a/fa/
x'G5g\ egL k|dfl0ft ug{'xf];\ . B
H
C

;dfwfg M E F
oxfF, k|dfl0ft ug{'kg{] M ABCD = BEFH
k|df0f M
tYox¿ sf/0fx¿
1. ABCD= ABEG 1. Pp6} cfwf/ AB / pxL ;dfgfGt/ /]vfx¿ AB / DE
larsf ;dfgfGt/ rt'e'{hx¿ ePsfn] .
2. BEFH = ABEG 2. Pp6} cfwf/ BE / pxL ;dfgfGt/ /]vfx¿ BE / AF
larsf ;dfgfGt/ rt'e'{hx¿ ePsfn] .
3. ∴ ABCD= BEFH 3. tYo 1 / 2 cg';f/ .
k|dfl0ft eof] .
cEof;sf nflu k|Zgx¿ M U

!= lbOPsf] lrqdf PTT////VR//UT, QU//RT / PT//QR eP PQRS / V

RTUV sf If]qkmn a/fa/ x'G5g\ egL k|dfl0ft ug{'xf];\ . P
W S
T

Q R

A B
171
E C
D
F G
@= lbOPsf] lrqdf AB//EC//FG, AG//BC / BF//CG 5g\ eg] k|dfl0ft
ug{'xf];\ M ABCD = CEFG.

#_ lbOPsf] lrqdf AD//BC//FE, AB//DC, BF//CE / A D

AF//DE 5g\ eg] k|dfl0ft ug{'xf];\M
H G
B
ADEF= ABCD + BCEF.
C
F E

gd'gf k|Zg % M P
lbOPsf] lrqdfPA//FB, PB//DA / DF//AB 5g\eg] k|dfl0ft ug{'xf];\ M F
D
PAD = ∆PBF E C

A B
;dfwfg M
k|df0f M
tYox¿ sf/0fx¿
1. ∆PAD=½ ABCD 1. Pp6} cfwf/ AD / pxL ;dfgfGt/ /]vfx¿ AD / BP
larsf lqe'h / ;dfgfGt/ rt'e'{hsf] ;DaGw .
2. ∆PBF=½ ABFE 2. Pp6} cfwf BF / pxL ;dfgfGt/ /]vfx¿ BF / AP
larsf lqe'h / ;dfgfGt/ rt'e'{hsf] ;DaGw .
3. ABCD= ABFE 3. Pp6} cfwf/ AB / pxL ;dfgfGt/ /]vfx¿ AB / DF
lardf ;dfgfGt/ rt'e'{hx¿sf] ;DaGw .
4. ∴ ∆PAD=∆PBF 4. tYo 1 , 2 / 3 cg';f/ .
k|dfl0ft eof] .

cEof;sf nflu k|Zgx¿ P A S

!= lrqdf PQRS Pp6f ;dfgfGt/ rt'e{'h xf] eg] k|dfl0ft ug{'xf];\ M
∆AQR=∆BPQ. B
Q R

C
P S
@= lrqdf PQRS / AQBC b'O{ ;dfgfGt/ rt'e{'hx¿ x'g\ eg] ltgLx¿sf] A B
If]qkmn a/fa/ x'G5 egL l;b\w ug{'xf];\ .
R
Q
P Q

T S U 172
R
V
#_ lrqdf PTT//TR, PT//QV / PV//QR 5g\ eg] ∆PVT / ∆QVR sf
If]qkmn a/fa/ x'G5g\ egL k|dfl0ft ug{'xf];\ .

gd'gf k|Zg ^ M
E
lrqdf ABCD Pp6f ;dfgfGt/ rt'e'{h xf] eg] k|dfl0ft ug{'xf];\ M ∆AOE =
∆COD A O
D
B
;dfwfg M C
h'lSt M B / D hf]l8Psf] 5 .
k|df0f M
tYox¿ k|df0fx¿
1. ∆ABE=∆ABD 1 Pp6} cfwf/ AB / pxL ;dfgfGt/ /]vfx¿ AB /
EC larsf lqe'hx¿ ePsfn] .
2. ∆ABE-∆BAO=∆ABD-∆BAO 2 tYo 1 sf] b'j}tkm{af6 pxL ∆BAO 36fp“bf .
3. ∴ ∆AOE=∆BOD 3 z]if tYo
4. ∆COD=∆BOD 4 Pp6} cfwf/ OD / pxL ;dfgfGt/ /]vfx¿ OD /
BC larsf lqe'hx¿ ePsfn] .
5. ∴ ∆AOE=∆COD 5 tYo 3 / 4 cg';f/ .
E k|dfl0ft eof] .
A O
D
B
C

a}slNks tl/sf M

k|df0f M
tYox¿ k|df0fx¿
1. ∆BOC=½ ABCD 1. Pp6} cfwf/ BC / pxL ;dfgfGt/ /]vfx¿ BC /
AD larsf lqe'h / ;dfgfGt/ rt'e'{hsf]
;DaGw .
2. ∴ ∆AOB+∆COD=½ 2. ∆AOB+∆COD+∆BOC= ABCD / tYo 1
ABCD cg';f/ .
3. ∆ABE=½ ABCD 3. Pp6} cfwf/ AB / pxL ;dfgfGt/ /]vfx¿ AB /
EC larsf lqe'h / ;dfgfGt/ rt'e'{hsf]
;DaGw .

173
 4. ∴ ∆AOB+∆AOE = ½ 4. l;ª\uf] 6'j|m] tYo .
ABCD
5. ∴ ∆AOB +∆COD=∆AOB 5. tYo 2 / 4 cg';f/ .
+∆AOE
6. ∴ ∆COD=∆AOE 6. tYo 5 sf] b'j}tkm{af6 ;femf ∆AOB x6fpFbf .
k|dfl0ft eof] . A D
cEof;sf nflu k|Zgx¿
!= lrqdf AD//BC 5g\ . O
B C
k|dfl0ft ug{'xf];\ M ∆AOB = ∆COD
P S
@= lrqdf PS//QR, QS//RM / SM//PR eP, k|dfl0ft ug{'xf];\ . O M
∆POQ=½ SORM Q
R

A X D
ABCD Pp6f
#= lbOPsf] lrqdf ;dfgfGt/ rt'e'{h xf] eg] k|dfl0ft ug{'xf];\ M
∆YAB=∆XAB+∆XDC Y
B C
$= lbOPsf] lrqdf PQRS Pp6f ;dfgfGt/ rt'e{'h xf] . P S
PQ nfO{ T ;Dd nDJofOPsf] 5 . olb TOS
Pp6f l;wf /]vf xf] eg] k|dfl0ft ug{'xf];\ M ∆POQ=∆ROT Q
O R

gd'gf k|Zg & M T

lrqdf ABCD Pp6f ;dnDa rt'e{'h xf] . A D
h:df AD‖BC, AC sf] dWo ljGb' Y / BD sf]
X Y
dWo ljGb' X 5g\ eg] k|dfl0ft ug{'xf];\ M
∆AXC=∆DYB, h'lSt M X / Y hf]8f}+ . C
k|df0f M
tYox¿ sf/0fx¿
1. AD//XY//BC 1 ;dnDa rt'e'{hsf] ljs0f{x¿sf dWo ljGb'x¿ dWo
/]vfdf kb{5g\ / dWo/]vf cfwf/x¿;Fu ;dfgfGt/ x'g]
ePsfn] .
2. ∆AXY=∆DXY 2 Pp6} cfwf/ XY / pxL ;dfgfGt/ /]vfx¿ XY / AD
larsf lqe'hx¿ .
3. ∆CXY=∆BXY 3 Pp6} cfwf/ XY / pxL ;dfgfGt/ /]vfx¿ XY / BC
larsf lqe'hx¿ .
4. ∆AXY+∆CXY = ∆DXY+∆BXY 4 tYo 2 / 3 hf]8\bf .
5. ∴ ∆AXC=∆DYB 5. l;ª\uf] 6'j|m] tYo .
k|dfl0ft eof] .

174
cEof;sf nflu k|Zgx¿ P S
!= lrqdf PS // QR / PR / QS sf dWo ljGb'x¿ j|mdzM A / B x'g\ .
∆PBR / ∆QAS sf A B
If]qkmnx¿ a/fa/ x'G5g\ egL k|dfl0ft ug{'xf];\ .

@= lbPsf] lrqdf ABCD Pp6f ;dnDa rt'e'{h xf] .h;df AD // BC 5g\ . A D

olb AB / DC sf dWo ljGb'x¿ j|mdz M M / N x'g\ eg] ∆ABN /
M N
∆CDM sf If]qkmnx¿ a/fa/ x'G5g\ egL k|dfl0ft ug{'xf];\ .

B C
gd'gf k|Zg * M A D
lrqdf AD // BG // EF / AB // DE // GF 5g\ olb ABCD /
CEFG sf If]qkmn a/fa/ 5g\ eg] k|dfl0ft ug{'xf];\ M BE // DG B C
G

;dfwfg M
E F
h'lSt M BD / EG hf]l8Psf 5g\ .
k|df0f M
tYox¿ k|df0fx¿
1. ∆BCD = ½ ABCD 1. ljs0f{ BD n] ABCD nfO{ cfwf ug{]
ePsfn] .
2. ∆CEG=½ CEFG 2. ljs0f{ EG n] CEFG nfO{ cfwf ug{] ePsfn]
3. ABCD= CEFG 3. lbOPsf]
4. ∴ ∆BCD=∆CEG 4. tYox¿ 1, 2 / 3 cg';f/ .
5. ∆BCD+∆BCE=∆CEG+∆BCE 5. of]u tYo .
6. ∆BDE=∆BGE 6. l;ª\uf] 6'j|m] tYo .
7. ∴ BE//DG 7. tYo 6 cg';f/ Pp6} cfwf/ BE / o;sf]
Ps}lt/sf lqe'hx¿sf] If]qkmn a/fa/
ePsfn] .
k|dfl0ft eof] .
A D
cEof;sf nflu k|Zgx¿ M
!= lbOPsf] lrqdf olb ∆AOB=∆COD 5g\ eg]
AD//BC x'G5 egL l;b\w ug{'xf];\ .
O
B C

175
 A S
@= lbOPsf] lrqdf olb PQRS / PABC sf If]qkmnx¿ a/fa/ 5g\ P

eg] QA//BR x'[G5 egL k|dfl0ft ug{'xf];\ . Q

R

C B
#= lbOPsf] lrqdf PS//QA//CB , PQ//SC//AB / QC//SA 5g\ eg]
P S
PQRS / RABC sf If]qkmn a/fa/ x'G5g\ egL k|dfl0ft ug{'xf];\ .
Q A
R
gd'gf k|Zg ( M
lbOPsf] lrqdf ABCD Pp6f ;dfgfGt/ rt'e'{h xf] . olb ∆ABX=∆ADY C B

eP k|dfl0ft ug{'xf];\ : BD//XY

A D
;dfwfg M
/rgf M BY / XD hf]8f}+ . Y

B C
k|df0f M
tYox¿ sf/0fx¿
1. ∆ABX = ∆DBX 1. Pp6} cfwf/ BX / pxL ;dfgfGt/ /]vfx¿ BX / AD
larsf lqe'hx¿ ePsfn] .
2. ∆ADY=∆BDY 2. Pp6} cfwf/ DY / pxL ;dfgfGt/ /]vfx¿ DY / AB
larsf lqe'hx¿ ePsfn] .
3. ∆ABX=∆ADY 3. lbOPsf]
4. ∴ ∆DBX=∆DBY 4. tYox¿ 1, 2 / 3 cg';f/ .
5. ∴ BD//XY 5. tYo 4 cg';f/ Pp6} cfwf/ BD / o;sf] Ps}lt/ k/]sf
lqe'hx¿sf] If]qkmn a/fa/ ePsfn] .
k|dfl0ft eof] .
cEof;sf nflu k|Zgx¿ M
!= lbOPsf] lrqdf PS//QR 5g\ . olb ∆PQR=∆QST eP k|dfl0ft ug{'xf];\ M P S

QS //RT T
Q
R

A
@= lbOPsf] lrqdf AD//BC 5g\ . olb ∆ABX / D

∆BDY sf] If]qkmn a/fa/ 5g\ eg] BD//XY

x'G5 egL l;b\w ug{'xf];\ M Y
B C

#= lbOPsf] lrqdf ABCD Pp6f ;dfgfGt/ rt'e'{h xf] / BD//EF A D

5 eg] k|dfl0ft ug{'xf];\ . ∆ABE = ∆ADF
F

B E C

176
 A
gd'gf k|Zg !) M
lbOPsf] lrqdf dWo/]vfx¿ BE / CD ljGb' O df k|ltR5]bg ePsf 5g\ D E
eg] k|dfl0ft ug{'xf];\ M rt'e'{h ADOE sf] If]qkmn = ∆BOC sf] If]qkmn
;dfwfg M O

k|df0f M B C

tYox¿ sf/0fx¿
1. ∆ABE =½ ∆ABC 1. dWo /]vf BE n] ∆ABC nfO{ cfwf u5{ .
2. ∆BCD=½∆ABC 2. dWo /]vf CD n] ∆ABC nfO{ cfwf u5{ .
3. ∴ ∆ABE=∆BCD 3. tYox¿ 1 / 2 cg';f/ .
4. ∆ABE-∆BOD=∆BCD-∆BOD 4. tYo 3 sf] b'j}tkm{af6 ∆BOC 36fpFbf .
5. ∴ rt'e{'h ADOE=∆BOC 5. z]if tYo .
k|dfl0ft eof] .
A
cEof;sf nflu k|Zgx¿ M
!= lbOPsf] ∆ABC df AB / AC sf dWo ljGb'x¿ j|mdzM X / Y 5g\ eg]
X
k|dfl0ft ug{'xf];\ M Y

i) ∆AXC sf] If]qkmn = ∆BYC sf] If]qkmn O

ii) rt'e{'h AXOY sf] If]qkmn = ∆BOC sf] If]qkmn
B C
A
@= lbOPsf] ∆ABC df BC sf] s'g} ljGb' D 5/ AD sf] dWo ljGb' 5 eg],
M
k|dfl0ft ug{'xf];\.
∆BMC = ½ ∆ABC306+52 B D C

#= lbOPsf] lrqdf ABCD Pp6f ;dfgfGt/ rt'e'{h xf] . olb ljs0f{x¿ AC / A D

BD ljGb' O df k|ltR5]bg ePsf 5g\ eg]k|dfl0ft ug{'xf];\ M ∆AOB =
∆BOC = ∆COD =∆ DOA O
B C
A

$= lbOPsf]∆ABC df olb BD = DE = CE 5g\ eg] k|dfl0ft ug{'xf];\ M ∆ADE =

⅓ ∆ABC

B D E C

gd'gf k|Zg !! M
lbOPsf] lrqdf PQRS Pp6f ;dfgfGt/ rt'e'{h xf] / ljs0f{ PR sf] s'g} P D
ljGb' T xf] . T
k|dfl0ft ug{'xf];\ M ∆PQT=∆PST
O
Q R
177
;dfwfg M
h'lSt M QS hf]8f}+ . h;n] PR nfO{ ljGb' O df k|ltR5]bg ul/Psf] 5 .
k|df0f M
tYox¿ sf/0fx¿
1. QS sf] dWo ljGb' O xf] . ;dfgfGt/ rt'e'{hsf ljs0f{x¿ k/:k/ ;dlåefhg
1.
x'G5g\ .
2. ∆POQ=∆POS 2. dWo/]vf PO n] ∆PQS nfO{ cfwf u5{ .
3. ∆TOQ=∆TOS 3. dWo/]vf TO n] ∆TQS nfO{ cfwf u5{ .
4. ∆POQ-∆TOQ=∆POS-∆TOS 4. tYo 2 af6 3 36fp“bf .
5. ∴ ∆PQT=∆PST 5. z]if tYo .
k|dfl0ft eof] . A
cEof;sf nflu k|Zgx¿
!= lbOPsf] lrqdf ∆ABC sf] dWo /]vf AD df s'g} ljGb' X 5 eg] ∆ABX /
∆ACX sf] If]qkmn a/fa/ x'G5g\ egL l;b\w ug{'xf];\ . X

B C
D
A D
@= lrqdf ABCD Pp6f ;dfgfGt/ rt'e'{h xf] . olb AC sf] s'g} ljGb' M
M
eP, k|dfl0ft ug{'xf];\ . ∆BMC = ∆DMC
B C

#= lrqdf PQRS Pp6f ;dfgfGt/ rt'e'{h xf] . QS sf] s'g} ljGb' A 5 . P S

k|dfl0ft ug{'xf];\ M ∆PAQ = ∆RAQ A

Q R
$= lbOPsf] lrqdf MNOP Pp6f ;dfgfGt/ rt'e'{h xf] . ljs0f{ OM nfO{ A
ljGb' A ;Dd nDAofOPsf] 5 . P
k|dfl0ft ug{'xf];\ M ∆AMN=∆AMP M
N O

A
gd'gf k|Zg !@=
lbOPsf] lrqdf∆ABC sf] e'hfx¿ AB / AC sf dWo ljGb'x¿ j|mdz M D
D E
/ E x'g\ . olb DG//EF 5g\ eg] k|dfl0ft ug{'xf];\ .
rt'e{'h DEFG sf] If]qkmn = ½∆ABC sf] If]qkmn C
B G F

178
 ;dfwfg M
h'lSt M BE hf]l8Psf] 5 .

k|df0f M
tYox¿ sf/0fx¿
1. ∆ADM / ∆CNM df 1.
i) DM = CM i) yfxf lbOPsf]af6 .
ii) ∠AMD =∠NMC ii) zLiff{led'v sf]0fx¿ .
iii)∠DAM =∠CNM iii) PsfGt/ sf]0fx¿ . AD//BN ePsfn] .
2. ∴ ∆ADM ≅ ∆CNM 2. e'=sf]=sf]=tYocg';f/ .
3. ∆ADM = ∆CNM 3. cg'¿k lqe'hx¿sf If]qkmnx¿ .
4. AM = NM 4. cg'¿k lqe'hx¿sf ;+ult e'hfx¿ .
5. ∆ADM + rt'e{'h ABCM = ∆CNM + 5. of]u tYo .
rt'e{'h ABCM
6. ;=n=r= ABCD = ∆ABN 6. l;ª\uf] 6'j|m] tYo .
7. ∆ABM = ½ ∆ABN 7. dWo/]vf BM n] ∆ABN nfO{ cfwf ug{]
ePsfn] .
8. ∴ ∆ABM = ½ ;dnDa rt'e'{h ABCD 8. tYo 6 / 7 cg';f/ .
k|dfl0ft eof] .
cEof;sf nflu k|Zgx¿ M P S
1= lbOPsf] lrqdf PQRS Pp6f ;dnDa rt'e'{h xf] . h;df PS//QR /
PA = QA 5g\ . olb SA / RQ nfO{ nSAofpFbf ljGb' B df e]6\5 eg]
A
k|dfl0ft ug'{xf];\ M
(i) ∆PAS ≅ ∆QAB
(ii) ∆BSR = ;dnDa rt'e'{h PQRS B Q R
1
(iii) ∆SAR = ;dnDa rt'e'{h PQRS
2

A D

2. lbOPsf] lrqdf
AD // BC / DX = CX 5g\ eg] k|dfl0ft ug'{xf];\ M X
∆ABX = ∆ADX + ∆BCX
B C

3. lbOPsf] lrqdf ABCD Pp6f ;dnDa rt'e'{h xf] . h;df AD // BC / DE = CE 5g\ . ljGb' E
af6 AB ;Fu ;dfgfGt/ x'g] u/L lvlrPsf] /]vfn] BC nfO{ F df / A D G
AD sf] nDAofOPsf] efunfO{ G df e]6]sf] 5 eg] k|dfl0ft ug{'xf];\ M
i) ∆DEG ≅ ∆CEF
B F C
179
 ii) ABFG = ;dnDa rt'e'{h ABCD
iii) ∆ABE = ½ ;dnDa rt'e'{h ABCD

yk cEof;sf nflu k|Zgx¿ M A D

1. lbOPsf] lrqdf AC // DE 5g\ eg] k|dfl0ft ug{'xf];\ . rt'e{'hABCD sf] If]qkmn

= ∆ABE sf] If]qkmn B C E

A E D

2. lrqdf, AD // BC // HG, AH // EG // DC / BE // GC 5g\ eg] B

C
ABCD / AEGH sf If]qkmnx¿ a/fa/ x'G5g\ egL k|dfl0ft ug'{xf];\ .
H G

3. lrqdfPS //QR / QS // RT 5g\ . k|dfl0ft ug'{xf];\ . P S

T
∆PQR = ∆QST
Q R

P A S
4. lrqdf PS // QR / AB// PR 5g\ eg] lqe'hx¿ APQ / BPR sf If]qkmn
B
a/fa/ x'G5g\ egL l;b\w ug'{xf];\ .
Q R

A
5. lbOPsf] ∆ABC df BC sf] dWo ljGb' M xf] / DA //MN N
5g\ eg] k|dfl0ft ug'{xf];\ M∆ABC = 2∆DNC
B D M C

A D
6. lrqdf ABCD Pp6f ;dfgfGt/ rt'e'{h xf] . olb BC = CE eP
ABCD / ∆ABE sf] If]qkmn a/fa/ x'G5g\ egL k|dfl0ft ug'{xf];\ .
B C E

Z
7. lbOPsf] lqe'h ABC df AB / AC sf dWo ljGb'x¿ j|mdzM D / E x'g\ . olb
BX = XY = YC eP k|dfl0ft ug'{xf];\ M ∆ABC = 3∆XYZ B
X Y C

D E

180
8. lbOPsf] lrqdf AO, BO / CO sf dWo ljGb'x¿ j|mdz M D, E / F x'g\ eg], k|dfl0ft
ug'{xf];\ . A

∆ABC sf] If]qkmn = 4∆DEF sf] If]qkmn D

E O F

9. lbOPsf] lqe'h PQR sf] dWo /]vf QS xf] . olb QS, B C

RA / PB sf dWo ljGb'x¿ j|mdzM A, B / C x'g\ eg],

k|dfl0ft ug'{xf];\ M ∆ABC = 1 ∆PQR P

8
C C
A
10. lrqdf ABCD Pp6f ;dfgfGt/ rt'e'{h xf] . DC nfO{ E ;Dd B
R
nDAofOPsf] 5 eg] ΔADE / ;dnDa rt'e'{h ABED sf A D

If]qkmnx¿ a/fa/ x'G5g\ egL l;b\w ug'{xf];\ .

C
B
E

181
 PsfO M 5
kf7 14. j[Tt (Circle)
1. kl/ro
o; kf7df j[Ttsf ljleGg efux¿ h:t} M s]Gb|Lo sf]0f, kl/lw sf]0f / rj|mLo rt'e'{h ;DaGwL
;fWox¿ / ltgsf] k|of]usf af/]df rrf{ ul/Psf] 5 . S.L.C . df o; kf7af6 b'O{ cf]6f 5f]6f]
pTt/ cfpg] / Pp6f nfdf] pTt/ cfpg] k|Zg ;f]lwg] 5 .
2. cfwf/e"t tYo tyf ;"qx¿
1. j[Ttsf] Pp6} rfkdf cfwfl/t s]Gb|Lo sf]0f kl/lw sf]0fsf] bf]Aa/ x'G5 .
2. j[Ttsf] Pp6} rfkdf cfwfl/t kl/lw sf]0fx¿ a/fa/ x'G5g\ . -Pp6} j[Tt v08sf
sf]0fx¿ a/fa/ x'G5g\ ._
3. cw{j[Ttsf] sf]0f Ps ;dsf]0f x'G5 .
4. rj|mLo rt'e'{tsf] ;Dd'v sf]0fx¿ kl/k'/s x'G5g\ .
5. rj|mLo rt'e'{hsf] afx\osf]0f To;sf] leqL ljkl/t sf]0f;Fu a/fa/ x'G5g\ .
6. j[Ttsf] s]Gb|Lo sf]0f To;sf] ;Dd'v rfk;Fu ;dk|efjL x'G5 .
7. j[Ttsf] kl/lwsf] sf]0f To;sf] ;Dd'v rfksf] cfwf;Fu ;dk|efjL x'G5 .
8. a/fa/ rfkx¿n] agfpg] s]Gb|Lo tyf kl/lw sf]0fx¿ a/fa/ x'G5g\ .
9. a/fa/ s]Gb|Lo sf]0fx¿ tyf kl/lw sf]0fx¿sf ;Dd'v rfkx¿ a/fa/ x'G5g\ .
10. a/fa/ hLjfx¿sf] ;ª\ut rfkx¿ a/fa/ x'G5g\ .
11. ;ª\ut rfkx¿ a/fa/ ePsf] hLjfx¿ a/fa/ x'G5g\ .
12. s'g} rt'e'{hsf] ;Dd'v sf]0fx¿ kl/k'/s 5g\ eg] Tof] rj|mLo rt'e'{h x'G5 .
13. s'g} rt'e{'hsf] afx\osf]0f leqL ljkl/t sf]0f;Fu a/fa/ 5g\ eg] Tof] rt'e{'h rj|mLo
x'G5 .
14. s'g} b'O{ ljGb'x¿ hf]8\g] /]vf v08n] Ps}lt/sf b'O{ ljGb'x¿df agfPsf sf]0fx¿
a/fa/ 5g\ eg] tL rf/ cf]6f ljGb'x¿ rj|mLo x'G5g\ .
#= Wofg lbg'kg]{ s'/fx¿
• Pp6} j[Ttsf cw{ Jof;x¿ a/fa/ x'G5g\ . h:t} M j[Tt ABC df
AO = BO = CO x'G5g\ .
B A
O

C
• s]Gb|Lo sf]0f M s]Gb|Lo sf]0fsf] zLif{ljGb' j[Ttsf] s]Gb| x'G5 / b'O{ cf]6f e'hfx¿ cw{ Jof;x¿
x'G5g\ . h:t} M ∠BOC / a[xt\ sf]0f ∠BOC s]Gb|Lo sf]0fx¿ x'g\ . kl/lw ABC sf c+zx¿
BC / BAC nfO{ rfkx¿ elgG5 . o;nfO{ BC / BAC n]lvG5 . ;fy} cfwf/e"t tYo 6 cg';f/
182
 ∴ ∠BOC ≡ BC / a[xt\ ∠BOC ≡ BAC
A

B C

• kl/lwsf]0f M j[Ttsf] s'g} b'O{ cf]6f hLjfx¿ kl/lwsf] s'g} Pp6f ljGb'df ldn]/ kl/lw sf]0f
aG5 . h:t} M ∠BAC Pp6f kl/lw sf]0f xf] . cfwf/e"t tYo 7 cg';f/∴ ∠BAC ≡ 1 BC.
2
• Jof;df plePsf] kl/lwsf] sf]0f cw{ j[Ttsf] sf]0f x'G5, h:t} M ∠ACB cw{ j[Tt -j[Ttf{w_ sf]
sf]0f xf] . To;}n] ∠ACB = 900 x'G5 .

O
O
A B A B

C
• rf/} cf]6f zLif{ ljGb'x¿ Pp6f j[Ttsf] kl/lwdf k/]sf] rt'e'{h
rj|mLo rt'e'{h xf] / zLif{ ljGb'x¿ rj|mLo ljGb'x¿ x'G5 .
h:t} M ABCD Pp6f rj|mLo rt'e'{h xf] .
A, B, C / D rj|mLo ljGb'x¿ x'g\ . ;fy} tYo 4 cg';f/
∴ ∠A + ∠C = 180 0 / ∴ ∠B + ∠D = 180 0
A
D

• rj|mLo rt'e'{hsf] afx\osf]0f eGgfn] o;sf] s'g} Pp6f e'hfnfO{ nDAofpFbf aGg] aflx/L sf]0f
xf] . h:t} M ∠BAF, ∠CBG, ∠DCE / ∠ADH afx\o sf]0fx¿ x'g\ .
cfwf/e"t tYo 5 cg';f/ ∠DEC = ∠BAD, ∠ADH =∠ABC, ∠BAF=∠BCD / ∠GBC =
∠ADC

183
 H
A
F D

B
E
C

• ;dfgfGt/ hLjfx¿n] sf6]sf rfkx¿ a/fa/ x'G5g\ . ;fy} a/fa/ rfkx¿ sf6\g] hLjfx¿
;dfgfGt/ x'G5g\ .
h:t} M olb AB // CD eP AC = BD x'G5 / olb AC = BD eP AB // CD x'G5 .

A B

C D

• cfwf/e"t tYo 8 / 9 cg';f/ lrq 1 df olb AB = BC eP, ∠AOB = ∠BOC x'G5 / ∠AOB =
∠BOC eP AB = BC x'G5 .
B A

P
A C

B Q
lrq
(i) lrq (ii)
lrq (ii) df olb PQ = QR eP ∠A = ∠B x'G5 / ∠A = ∠B eP QR = PQ x'G5 .

• cfwf/e"t tYo 10 / 11 cg';f/ olb PQ = RS eP, PQ = RS

x'G5 . / PQ = RS eP PQ = RS x'G5 .
Q

P S

184
 X

lqe'hsf] ltg cf]6f sf]0fx¿sf] of]u 1800 x'G5 . h:t} M ∠X +

∠Y + ∠Z =1800
Y
Z
• lqe'hsf] Pp6f e'hfnfO{ nDAofpFbf aGg] afx\osf]0f To;sf] leqL E

ljkl/t b'O{ sf]0fx¿sf] of]u;Fu a/fa/ x'G5 . h:t} M ∠ACD = A

∠ABC + ∠BAC,
∠BAE = ∠ABC + ∠ACB / B D

∠CBF = ∠BAC + ∠ACB F

C

• rt'e'{hsf] rf/ cf]6f sf]0fx¿sf] of]ukmn

A

3600 x'G5 .
∴ ∠A + ∠B +∠C +∠D = 3600 B
D

• Ps kl/lw = 3600 x'G5 . cyf{t\ s'g} ljGb'sf] jl/kl/sf] Ps rSs/sf] C

sf]0f 3600 x'G5 . h:t} M kl/lw ABC = 3600 / ∠AOB + a[xt\ ∠ AOB = 3600
A B

5f]6f] k|Zgf]Tt/ / cEof; M A

gd'gf k|Zg !M
lbOPsf] j[Ttdf O s]Gb| xf] . olb ∠ABO = 200 / ∠ACO = 300 eP
x0, y0 / z0 sf dfgx¿ lgsfNg'xf];\ . o
y 30
z
;dfwfg M B C
h'lSt M AO hf]8f}+ .
1. AO = BO = CO -Pp6} j[Ttsf cw{ Jof;x¿_
2. ∠BAO = ∠ABO = 200 -;dlåafx' ∆AOB sf cfwf/sf sf]0fx¿_
3. ∠CAO = ∠ACO = 300 -;dlåafx' ∆AOC sf cfwf/sf sf]0fx¿_
4. ∴ ∠BAC ( X ) = ∠BAO + ∠CAO
0

= 200 + 300
= 500
5. y 0 = 2 × x0 -Pp6} rfkdf cfwfl/t s]Gb|Lo sf]0f / kl/lwsf] sf]0flarsf] ;DaGw_
= 2 × 500
= 1000
185
6. km]l/,∠OBC = ∠OCB = z0 -;dlåafx' ∆BOC sf cfwf/sf sf]0fx¿_
0
7. ∠OBC + ∠OCB + ∠ BOC = 180 -∆BOC sf sf]0fx¿sf] of]u_
or, Z + Z + 100 = 1800
or, 2Z = 800
80 0
∴Z = = 40 0
2
∴ x0 = 500, y0 = 1000 / z0 = 400

cEof;sf nflu k|Zgx¿

lgDg lrqx¿af6 x0 / y0 sf] dfgx¿ lgsfNg'xf];\ .
-s_ -v_ B
A

0 O 0
60 A
x x
O y
0
D
C
40
B C

-u_
D

O 0
A 80
0
x
C
B

pTt/x¿ M -s_ 500 -v_1200, 300 -u_ 500

gd'gf k|Zg 2 : D

lbOPsf] j[Ttdf O j[Ttsf] s]Gb| xf] . olb AO // BC /

AB // OC eP ∠OCB (= x0) sf] dfg lgsfNg'xf];\ .
O

;dfwfg M A

(i) ∠AOC = ∠ABC = ao -dfgf}+_ -;dfgfGt/ rt'e'{h ABCO sf

C
;Dd'v sf]0fx¿ ._ B
(ii) a[xt ∠AOC = 2×∠ABC -Pp6} rfk ADC df cfwfl/t
186
 s]Gb|Lo sf]0f / kl/lw sf]0f larsf] ;DaGw_
=2a0
(iii) ∠AOC + a[xt ∠AOC = 3600 - ljGb' O sf] jl/kl/sf] Ps rSs/sf] sf]0f ePsfn]_
or, a0 + 2a0 = 3600
∴ a0 = 1200
(iv) ∠OCB + ∠ABC = 1800 -j|mdfut leqL sf]0fx¿sf] of]ukmn M AB ‖ OC_
or, x0 + a0 = 1800
∴ x0 = 600
cEof;sf nflu k|Zgx¿ M
!= tn lbOPsf lrqx¿af6 x0 sf] dfg lgsfNg'xf];\ .
-s_ -v_ -u_

P
140 0
Q R O
x0 O
0
M A 110

O x
0
0
x
P C
N B

pTt/x¿ M -s_ 1200 -v_ 1250 -u_ 500

D
gd'gf k|Zg 3 : A
lbOPsf j[Ttdf O j[Ttsf] s]Gb| xf] . olb 120 0
∠BAD = 1200 eP, ∠COD (x0) sf] gfk x0
B C
kTtf nufpg'xf];\ . O

;dfwfg M
1. a[xt ∠BOD = 2∠BAD – -Pp6} rfk BCD df cfwfl/t s]Gb|Lo = 2 × 1200 – sf]0f / kl/lw sf]0f
larsf] ;DaGw ._
= 2400
2. ∠BOC = 1800 – ;/n sf]0f ePsfn] .
3. ∴ ∠COD (x ) = 2400 - 1800 = 600
0

187
cEof;sf nflu k|Zgx¿ M
lbOPsf] lrqx¿af6 x0 / y0 sf] dfg lgsfNg'xf];\ .
-s_ -v_ -u_
A

D P
A 0
N
110 x0
x
0 O
CM
600 Q O
D
B 0
O 140
y0
450 C
B

pTt/x¿ M -s_ 400 -v_1200 -u_ 250, 1100

gd'gf k|Zg 4 :
lbOPsf] j[Ttdf s]Gb|ljGb' O 5 . olb QO = QR eP,
∠PTS sf] dfg kTtf nufpg'xf];\ .

B
P S

Q R
600

T
;dfwfg M
h'lSt M QS / RO hf]8f}+ .
oxfF,
(i) QO = QR – lbOPsf] .
(ii) QO = RO – cw{ Jof;x¿ .
(iii) ∴ QO = QR = RO – (i) / (ii) cg';f/ .
(iv) ∴ ∠QOR = 60 0
– ∆QOR ;dafx' lqe'h ePsfn] .
0
(v) ∠PQS = 90 – cw{ j[Ttsf] sf]0f ePsfn] .
1
(vi) ∠QSR = ∠QOR – Pp6} rfk QR df cfwfl/t kl/lw sf]0f / s]Gb|Lo sf]0fsf] ;DaGw .
2
1
= × 60 0
2
= 300
(vii) ∠PQS =∠PTS+∠QST – ∆QST sf] afx\o sf]0f / To;sf] leqL ljkl/t b'O{ sf]0fx¿sf] ;DaGw
188
 or, 900 = ∠PTS + 300
∴ ∠PTS = 600
cEof;sf nflu k|Zgx? M
lbOPsf] lrqx¿af6 x0 / y0 sf] dfgx¿ kTtf nufpg'xf];\ .
-s_ -v_ -u_

M
O
A D O B
P Q
500 0
x
O x0 P
B C S R
x0 A
600

E A N

pTt/x¿ M -s_ 650 -v_ 600 -u_ 600

gd'gf k|Zg 5 :
lbOPsf] lrqdf O j[Ttsf] s]Gb| xf] . olb AB//CD ∠BEC = 600 / ∠ABE = 500 5g\ eg] x0 /
sf y0 dfgx¿ lgsfNg'xf];\ .
E

600
O 500
A B
0
x
y0
C D

;dfwfg M h'lSt M AE hf]8f}+ .

ca, (i) ∠AEB = 900 – cw{ j[Ttsf] sf]0f ePsfn]
(ii) ∴ ∠AEC = 90 -60 0 0
= 300 – ∠AEC = ∠AEB - ∠BEC ePsfn] .
(iii) ∴ ∠ABC = ∠AEC = 300 – Pp6} rfk AC df plePsf kl/lw
sf]0fx¿
(iv) ∴ ∠BCD = ∠ABC – PsfGt/ sf]0fx¿ M AB//CD ePsfn]
∴ y0 = 300
km]l/, (v) ∠BAE = 900 - 500 –∆ABE df, ∠AEB = 900 / ∠ABE = 500 ePsfn]
= 400
(vi) ∴ ∠BCE = ∠ BAE – Pp6} rfk BE df plePsf kl/lw sf]0fx¿
∴ x0 = 400
To;}n], ∴ x0 = 400 / y0 = 300

189
cEof;sf nflu k|Zgx¿ M
lbOPsf] lrqx¿af6 xo / y0 sf dfgx¿ kTtf nufpg'xf];\ .
-s_ -v_ -u_
A
R S A
300

O O x0
B 500 P Q
C
5y0 4y0
B C
x0 0
x O

D T

-3_ -ª_
A
P
y0
700
D
O x0 O
Q R
600
x0 B C
T U

lrqdf AB=AC 5g\ .

pTt/x¿ M -s_ 400 -v_ 600 -u_ 500, 100 -3_ 200 -ª_ 600, 600

gd'gf k|Zg 6 :
lbOPsf] lrqdf PQ = PT / PT//SR 5g\ . olb
∠PQT = 800 5 eg] ∠QPS sf] dfg lgsfNg'xf];\ .
P S

800 T
Q R

;dfwfg M
oxfF,
(i) ∠PTQ = ∠PQT = 800 -PQ = PT ePsfn] ;dlåafx' lqe'hsf cfwf/sf sf]0fx¿_
(ii) ∠QRS = ∠PTQ =800 -;ª\ut sf]0fx¿ M PT ‖ SR ePsfn]_
(iii) ∠QPS + ∠QRS = 1800 -rj|mLo rt'e'{h PQRS sf ;Dd'v sf]0fx¿ ePsfn]_
or, ∠QPS + 800 = 1800

190
 ∴ ∠QPS = 1000
cEof;sf nflu k|Zgx¿ M
!= lbOPsf] lrqx¿af6 x0 sf] y0 dfg kTtf nufpg'xf];\ .
-s_ -v_ -u_ -3_ -ª_
A D R P
P S
x0
800 x0
E H O
300
M N A x0
B
B D x0 O
y0 y0
Q R
G D
F C

pTt/x¿ M -s_ 400 -v_ 1200, 600 -u_ 900 -3_ 1200 -ª_ 450, 1350

gd'gf k|Zg 7 M
lbOPsf] lrqdf ABCD Pp6f ;dfgfGt/ rt'e'{h xf] .
olb ∠CDE = 750 eP ∠CED sf] dfg kTtf nufpg'xf];\ .
A E D

750

B C
;dfwfg M
oxfF,
(i) ∠ABC = ∠ADC=750 –
;dfgfGt/ rt'e'{h sf ;Dd'v sf]0fx¿
(ii) ∴ ∠CED = ∠ABC – rlqmo rt'e'\{hsf] afx\o sf]0f Tof] sf]0fsf] leqL ljkl/t
∴ ∠CED = 750 sf]0f;Fu a/fa/ x'g] ePsfn] .
cEof;sf nflu k|Zgx¿ M
lbOPsf] lrqx¿df xo / yo sf] gfkx¿ lgsfNg'xf];\ .
-s_ -v_ -u_
T S
P
x 0 M P
800 A E D
0 0
y x0 x

O
640
Q
650
1400
B F y0
R
Q U R N P

191
pTt/x¿ M -s_ 800 -v_ 650, 660 -u_ 700 , 700
gd'gf k|Zg 8 :
lbOPsf] lrqdf ∠CDE sf] cw{s DF xf] / DF//BC 5g\ .
olb ∠ ABC = 860 eP x0 sf] dfg kTtf nufpg'xf];\ .
D
A E
x0
F
860
B
C

;dfwfg M
(i) ∠CDE = ∠ABC – rj|mLo rt'e'{h ABCD sf] afx\osf]0f / To;sf] leqL ljkl/t sf]0fsf] ;DaGw .
= 860
1
(ii) ∠CDF = ∠EDF = ∠CDE – ∠ CDE sf] cw{s DF ePsfn]
2
1
= × 860
2
= 430
(iii) ∠ BCD = ∠CDF –PsfGt/ sf]0fx¿ M BC//DF 5 .
= 430
(iv) ∴ ∠BAD + ∠BCD = 1800 –rj|mLo rt'e'{hsf] ;Dd'v sf]0fx¿ ePsfn] .
or, x0 + 430 = 1800
∴ x0 = 1370
cEof;sf nflu k|Zgx¿ M
!= tn lbOPsf lrqx¿df x0 / y0 sf] dfg lgsfNg'xf];\ .

-s_ -v_ -u_

E
R
P Q
960 200
A S
600
M S
D x0 y0
y0 x0

x0 960
Q
920
B R N P
0 0 0 0 0
pTt/x¿ M -s_ 48 -v_ 32 , 32 -u_104 , 76

192
yk cEof;sf nflu k|Zgx¿ M
tn lbOPsf lrqx¿df x0 / y0 sf dfgx¿ kTtf nufpg'xf];\ .
-s_ -v_ -u_
C D
300
D C S
x0 O
A 200 x0
O B x0
0
y0 R
y0 A 30 B
0 400
Q T
P
E

-3_ -ª_ -r_

A

B P Q
x0
0
x0 y

O B
1200 C
y 0
C
x0 400
D
300 900

A P Q S R
-5_ -h_ -em_
A A

x0
800
O B D
B x0
O C A
0
700 50
250 y0 x0
D E C
B C

pTt/x¿ M -s_800, 400 -v_ 300 -u_ 400, 800 600, 1200
-3_
-ª_ 550 -r_ 800, 1000 -5_ 400 -h_ 400, 300 -em_ 400

nfdf] pTt/ cfpg] k|Zgx¿

gd'gf k|Zg 1 :
Pp6f j[Tt PQR sf] Pp6} rfk QR df ag]sf s]Gb|Lo sf]0f QOR kl/lw sf]0f QPR sf] b'O{ u'0ff x'G5
egL l;b\w ug'{xf];\ .
yfxf lbOPsf] M O PQR sf] s]Gb| O xf] . pxL QR df s]Gb|Lo ∠QOR / kl/lw ∠QPR plePsf 5g\ .
l;b\w ug'{kg]{ M ∠QOR = 2∠QPR
193
h'lSt M PO hf]8f}+ / o;nfO{ ljGb' S ;Dd nDAofcf+} .
P

Q R
S

k|df0f M
tYox¿ sf/0fx¿
1. ∆POQ df PO = QO 1. Pp6} j[Ttsf cw{ Jof;x¿ ePsfn]
2. ∠PQO = ∠QPO 2. ;dlåafx' ∆POQ sf cfwf/sf sf]0fx¿
ePsfn] .
3. ∠QOS = ∠PQO + ∠QPO 3. lqe'hsf] afx\osf]0f / To;sf] leqL
= ∠QPO + ∠QPO ljkl/t b'O{ sf]0fx¿sf] ;DaGw .
= 2∠QPO
4. To:t}, ∠ROS =2∠RPO 4. dflysf] h:t} -∆POR af6_
5. ∠QOS + ∠ROS=2(∠QPO + 5. tYo 3 / 4 hf]8\bf
∠RPO)
6. ∴ ∠QOR = 2∠QPR 6. l;ª\uf]6'j|m] tYo
k|dfl0ft eof] .
csf]{ tl/sf M
P

Q S R

k|df0f M
tYox¿ sf/0fx¿
1. ∠QOR ≡ QR 1. j[Ttsf] s]Gb|Lo sf]0f / ;Dd'v rfksf]
;DaGwaf6 .
2. 2∠QPR ≡ QR 2. j[Ttsf] kl/lw sf]0f / ;Dd'v rfksf]
;DaGwaf6 .
3. ∴ ∠QOR = 2 ∠QPR 3. tYo 1 / 2 cg';f/ -a/fa/L tYo_
k|dfl0ft eof] .
gd'gf k|Zg 2:

194
s]Gb|ljGb' X ePsf] j[Ttsf] rfk BC df plePsf kl/lw sf]0fx¿ ∠BAC / ∠BDC a/fa/ x'G5g\ egL
k|dfl0ft ug'{xf];\ .
A D

B C

;dfwfg M
yfxf lbOPsf] M j[Ttsf] s]Gb|ljGb' X xf] . kl/lw sf]0fx¿ ∠BAC / ∠BDC pxL rfk BC df
plePsf 5g\ .
k|dfl0ft ug'{kg]{ M ∠BAC = ∠ BDC
h'lSt M BX / CX hf]l8Psf] 5 .

k|df0f M
tYox¿ sf/0fx¿
1. ∠BAC = 1 ∠ BXC 1. Pp6} rfk BC df cfwfl/t s]Gb|Lo sf]0f / kl/lw
2
sf]0fsf] ;DaGw .
2.
∠BDC=
1
∠BXC
2. sf/0f 1 cg';f/
2
3. ∴ ∠BAC= ∠BDC 3. tYo 1 / 2 cg';f/ .
k|dfl0ft eof] .
k|df0f M
tYox¿ sf/0fx¿
1. ∆POQ df PO = QO 1. Pp6} j[Ttsf cw{ Jof;x¿ ePsfn]
2. ∠PQO = ∠QPO 2. ;dlåafx' ∆POQ sf cfwf/sf sf]0fx¿
ePsfn] .
3. ∠QOS = ∠PQO + ∠QPO 3. lqe'hsf] afx\osf]0f / To;sf] leqL
= ∠QPO + ∠QPO ljkl/t b'O{ sf]0fx¿sf] ;DaGw .
= 2∠QPO
4. To:t}, ∠ROS =2∠RPO 4. dflysf] h:t} -∆POR af6_
5. ∠QOS + ∠ROS=2(∠QPO + 5. tYo 3 / 4 hf]8\bf
∠RPO)
6. ∴ ∠QOR = 2∠QPR 6. l;ª\uf]6'j|m] tYo
k|dfl0ft eof] .

195
csf]{ tl/sf M
P

Q R

k|df0f M
tYox¿ sf/0fx¿
1. ∠QOR ≡ QR 1. j[Ttsf] s]Gb|Lo sf]0f / ;Dd'v rfksf]
;DaGwaf6 .
2. 2∠QPR ≡ QR 2. j[Ttsf] kl/lw sf]0f / ;Dd'v rfksf]
;DaGwaf6 .
3. ∴ ∠QOR = 2 ∠QPR 3. tYo 1 / 2 cg';f/ -a/fa/L tYo_
k|dfl0ft eof] .
gd'gf k|Zg 3 :
s]Gb|ljGb' X ePsf] j[Ttsf] rfk BC df plePsf kl/lw sf]0fx¿ ∠BAC / ∠BDC a/fa/ x'G5g\ egL
k|dfl0ft ug'{xf];\ .
A D

B C

;dfwfg M
yfxf lbOPsf] M j[Ttsf] s]Gb|ljGb' X xf] . kl/lw sf]0fx¿ ∠BAC / ∠BDC pxL rfk BC df
plePsf 5g\ .
k|dfl0ft ug'{kg]{ M ∠BAC = ∠ BDC
h'lSt M BX / CX hf]l8Psf] 5 .
k|df0f M
tYox¿ sf/0fx¿
1.
∠BAC =
1
∠ BXC
1. Pp6} rfk BC df cfwfl/t s]Gb|Lo sf]0f / kl/lw sf]0fsf]
2 ;DaGw .
2.
∠BDC=
1
∠BXC
2. sf/0f 1 cg';f/
2
3. ∴ ∠BAC= ∠BDC 3. tYo 1 / 2 cg';f/ .
196
 k|dfl0ft eof] .

j}slNks tl/sf M
A D

B C

tYox¿ sf/0fx¿
1.
∠BAC
1
BC
1. j[Ttsf] kl/lw sf]0f / ;Dd'v rfksf]
2 ;DaGw
2.
∠BDC
1
BC
2. sf/0f 1 cg';f/
2
3. ∠BAC= ∠BDC 3. tYo 1 / 2 cg';f/
k|dfl0ft eof]

gd'gf k|Zg 4 :
rj|mLo rt'e'{h PQRS sf ;Dd'v sf]0fx¿ kl/k"/s x'G5g\ egL k|dfl0ft ug'{xf];\ .
;dfwfg M
yfxf lbOPsf] M O j[Ttsf] s]Gb| xf] / PQRS Pp6f rj|mLo rt'e'{h xf] .
k|dfl0ft ug'{kg]{ M (i) ∠P + ∠R = 1800 /
(ii) ∠Q + ∠S = 1800
h'lSt M (i) QO / SO hf]8f}+ .
(ii) ∠ QOS = x0 / a[xt ∠ QOS = y0 dfgf}+ .
P

x S
Q O
y

R
csf]{ t/Lsf M
tYox¿ sf/0fx¿
1.
∠P
1
≡ QRS
1. j[Ttsf] kl/lwsf]0f / ;Dd'v rfksf] ;DaGw
2

197
 2. 1
∠R ≡ QPS
2. sf/0f (1) cg';f/
2
3.
∠P +∠R =
1
(QRS + QPS)
3. tYo 1 / 2 hf]8\bf
2
4. 4. l;ª\uf]6'j|m] tYo / Ps kl/lw = 3600 x'g]
ce 0
QPS +QRS = O PQRS=360 ePsfn] .
5. ∴ ∠ P + ∠ R =1800 5. tYo 3 / 4 cg';f/
To;/L g} ∠Q + ∠ S = 1800
k|dfl0ft eof] .
cEof;sf nflu k|Zgx¿ M
1. s]Gb| X ePsf] Pp6f j[Ttsf] rfk BC df cfwfl/t s]Gb|Lo ∠BXC kl/lw ∠BAC sf] bf]Aa/ x'G5
egL k|dfl0ft ug'{xf];\ .
2. s]Gb| A ePsf] Pp6f j[Ttsf] kl/lwdf ljGb'x¿ X, Y / Z 5g\ eg] k|dfl0ft ug'{xf];\ .
1
∠YXZ = ∠YAZ
2
3. j[Ttsf] Pp6} j[Ttv08df ag]sf sf]0fx¿ a/fa/ x'G5g\ egL l;b\w ug'{xf];\ .
4. s]Gb| O ePsf] Pp6f j[Ttsf] kl/lwdf ljGb'x¿ P, Q, R / S kb{5g\ eg] ∠PRQ = ∠PSQ x'G5
egL k|dfl0ft ug'{xf];\ .
5. rj|mLo rt'e'{h ABCD sf] ;Dd'v sf]0fx¿ kl/k"/s x'G5g\ egL l;b\w ug'{xf];\ .
6. rj|mLo rt'{e'h PQRS df k|dfl0ft ug'{xf];\ M ∠Q + ∠S = 1800
k|df0f M
tYox¿ sf/0fx¿
1. 1 0
∠P = y
1. Pp6} rfk QRS df plePsf kl/lw sf]0f /
2 s]Gb|Lo sf]0fsf] ;DaGw .
2.
∠R =
1 0
x
2. Pp6} rfk QPS df plePsf kl/lw sf]0f /
2 s]Gb|Lo sf]0fsf] ;DaGw
∠P +∠R =
1 0 0
(x + y )
3. tYo 1 / 2 hf]8\bf
3. 2
4. x0 + y0 = 3600 4. ljGb' O sf] jl/kl/sf] Ps rSs/sf] sf]0f
ePsfn] .
5. ∴ ∠ P + ∠ R =1800 5. tYo 3 / 4 cg';f/
To;/L g} ∠Q + ∠ S = 1800
k|dfl0ft eof] . C D
gd'gf k|Zg 5 :
lbOPsf] j[Ttdf s]Gb|ljGb' O xf] / BC = DC 5g\ eg]
k|dfl0ft ug'{xf];\ . AD//OC B A
O

198
;dfwfg M
tYox¿ sf/0fx¿
(i) s]Gb|Lo sf]0f / ;Dd'v rfksf] ;DaGw .
(i) ∠BOC ≡ BC
(ii)
∠BAD ≡
1
BCD
(ii) kl/lw sf]0f / ;Dd'v rfksf] ;DaGw .
2
(iii) 1 (iii)
∠BAD ≡ 2 BC ≡
2 BC = CD ePsfn] BCD = 2 BC x'G5 .
BC
(iv) ∴ ∠ BOC = ∠BAD (iv) tYo (i) / (iii) cg';f/
(v) ∴ AD//OC (v) tYo (iv) cg';f/, ;ª\ut sf]0fx¿ a/fa/
ePsfn] .
k|dfl0ft eof] .
R S
cEof;sf nflu k|Zgx¿ M
1. lbOPsf] j[Ttdf O s]Gb| / rfk PS sf] dWo ljGb' R xf] eg], k|dfl0ft
Q
ug'x{ f]\;\ . RO//SQ P
O

2. lbOPsf] j[Ttdf O s]Gb| xf] . olb AD//OC 5 eg] BC = DC x'G5 egL C

D
k|dfl0ft ug'{xf];\ .
A B
3. ;dfgfGt/ hLjfx¿n] a/fa/ rfk k|ltR5]bg ub{5g\ egL l;b\w ug'{xf];\ . O

A
B
X
gd'gf k|Zg 6 : D
C
lbOPsf] lrqdf j[Ttsf] b'O{ hLjfx¿ AB / CD afx\oljGb' X df e]l6Psf
5g\ . k|dfl0ft ug{'xf];\ M
1
∠AXC ≡ (AC - BD)
2

;dfwfg M h'lSt M A / D hf]8f}+ .

tYox¿ sf/0fx¿
(i)
∠BAD ≡
1
BD / ∠ADC ≡
1
AC
(i) j[Ttsf] kl/lw sf]0f / ;Dd'v
2 2 rfksf] ;DaGw
(ii) ∠ADC = ∠BAD + ∠AXC (ii) ∆ADX sf] afx\o sf]0f To;sf]
leqL b'O{ ljkl/t sf]0fx¿sf]
of]u;Fu a/fa/ x'G5 .

199
 (iii) ∠AXC = ∠ADC - ∠BAD (iii) tYo 2 cg';f/ -kIffGt/ ubf{_

(iv)
∠ AXC ≡
1
AC -
1
BD
(iv) tYo (i) / (iii) cg';f/
2 2
cyf{t\
1
∠AXC ≡ (AC - BD )
2
k|dfl0ft eof] .

cEof; nflu k|Zgx¿ M

1. Pp6f j[Ttsf] b'O{ cf]6f hLjfx¿ PQ / RS afx\o ljGb' X df k|ltR5]bg x'G5g\ . k|dfl0ft
ug'{xf];\ M ∠PXR= 1 (PR - QS)
2
2. lrqdf j[Ttsf b'O{ hLjfx¿ PQ / RS leqL ljGb' X df k|ltR5]bg ePsf 5g\ . k|dfl0ft
ug'{xf];\ M
1
(i) ∠PXR = 1 (PR + QS) (ii) ∠PXS = (PS + QR)
2 2

P S
X

R Q

3. lbOPsf] j[Ttdf hLjfx¿ AB / CD cfk;df ljGb' X df nDa 5g\ . k|dfl0ft ug'{xf];\ .

(i) BC + AD = 1800

(ii) BC + AD = AC + BD
A

C
X D

B
gd'gf k|Zg 7 :
lbOPsf] j[Ttdf ∠AXD = ∠BYC eP k|dfl0ft ug'{xf];\ M AB//CD

200
 X Y

A B

C D
;dfwfg M
k|df0f M
tYox¿ sf/0fx¿
(i) ∠X = ∠Y (i) lbOPsf]

(ii) (ii) a/fa/ kl/lw sf]0fx¿sf ;Dd'v rfkx¿

ACD = BDC ePsfn]
(iii) (iii)
ACD - CD =BDC - CD tYo (ii) sf] b'j} tkm{af6 pxL CD 36fpFbf
(iv) (iv) z]if tYo
AC = BD
(v) AB//CD tYo 4 cg';f/ a/fa/ rfk k|ltR5]bg ug]{
hLjfx¿ ;dfgfGt/ x'g] ePsfn]
k|dfl0ft eof] .

cEof;sf nflu k|Zgx¿ M

1. lbOPsf] j[Ttdf ∠X = ∠Y eP AB//CD x'G5
egL k|dfl0ft ug'{xf];\ .
X Y

A B

C D
2. lbOPsf] j[Ttdf olb AB//CD eP ∠X =∠Y x'G5 egL l;b\w ug'{xf];\ .
X Y

A B

C D

201
gd'gf k|Zg M 8:
lbOPsf]j[Ttdf BC = AD / AE = BF 5g\ .
k|dfl0ft ug'{xf];\ M ∠AED = ∠BFC
D C

A E F B

;dfwfg M
h'lSt M AD / BC hf]8f}+ .
k|df0f M
tYox¿ sf/0fx¿
1. 1. lbOPsf]
AD = BC
2. 2. of]utYo
AD + DC = BC + DC
3. 3. l;ª\uf]6'j|m] tYo
ADC = BCD
4. ∠ ABC = ∠BAD 4. tYo 3 cg';f/ a/fa/ rfkx¿n] agfPsf] kl/lw
sf]0fx¿
5. cj ∆ADE / ∆BCF df, 5.
(i) AD = BC tYo 1 cg';f/ a/fa/ rfkx¿n] agfPsf]
(i)
hLjfx¿
(ii) ∠A = ∠B (ii) tYo 4 cg';f/
(iii) AE = BF (iii) yfxf lbOPsf]af6
6. ∴ ∆ADE ≅ ∆ BCF 6. e'=sf]=e'= tYo
7. ∴ ∠AED = ∠BFC 7. cg'¿k lqe'hx¿sf ;+utL sf]0fx¿ ePsfn]
k|dfl0ft eof] .
S R
cEof;sf nflu k|Zgx¿ M
1. lbOPsf] j[Ttdf PS = QR / ∠PAS = ∠QBR 5g\ eg], k|dfl0ft
ug'{xf];\ M PA = QB

P A B Q

P S
2. lbOPsf] lrqdf PQ = RT / ∠PQR sf] QS cw{s xf] .
k|dfl0ft ug'{xf];\ . SQ = ST
Q T
R
202
 A D

3. lbOPsf] lrqdf ABCD Pp6f ju{ xf] . olb AE = AF eP, BD//EF F

x'G5 egL k|dfl0ft ug'{xf];\ .
B C
E
yk cEof;sf nflu k|Zgx¿ M
1. lbOPsf] lrqdf hLjfx¿ AB / CD ljGb' X df k|ltR5]bg ePsf 5g\ . A D
olb CX = BX eP, X
k|dfl0ft ug'{xf];\ M
(i) ACB = CBD (ii) AB = CD C B

A
2. lbOPsf] j[Ttdf O s]Gb| xf] . olb AB = BC eP
k|dfl0ft ug'{xf];\ M O
(i) AD = CD (ii) ∠ABC sf] cw{s BD xf] . B D

3. lbOPsf] j[Ttdf AC = BD eP k|dfl0ft ug'{xf];\ M A D

(i) AB = CD (ii) AD//BC

B C

S T
4. lbOPsf] j[Ttdf rfk QP / rfk RP sf A B
dWo ljGb'x¿ j|mdzM S / T eP , k|dfl0ft ug'{xf];\ . PA = PB.
Q R

A
5. lbOPsf] j[Ttdf O s]Gb|ljGb' / BQ = BR 5g\ .
k|dfl0ft ug'{xf];\ M ∠B = 1 (∠Q - ∠R) P
2 O

Q R
gd'gf k|Zg 9:
B
203
 lbOPsf] j[Ttsf] s]Gb|
O xf] . k|dfl0ft ug'{xf];\ .
∠AOC -∠BOD = 2∠AEC
A

;dfwfg M B

h'lSt M A / D hf]8f}+ . O E
k|df0f M D
tYox¿ sf/0fx¿ C
1. ∠AOC = 2∠ADC 1. Pp6} rfk AC df cfwfl/t
s]Gb|Lo sf]0f / kl/lw
sf]0fsf] ;DaGw
2. ∠BOD = 2∠BAD 2. Pp6} rfk BD df cfwfl/t
s]Gb|Lo sf]0f / kl/lw
sf]0fsf] ;DaGw
3. ∠AOC- ∠BOD = 2 (∠ADC - 3. tYo 1 af6 2 36fpFbf
∠BAD)

4. ∠ ADC = ∠BAD + ∠AEC 4. ∆AED sf] afx\osf]0f /

To;sf] leqL b'O{ ljkl/t
sf]0fx¿sf] ;DaGw
5. ∴ ∠ADC - ∠BAD = ∠AEC 5. tYo 4 af6 kIffGt/ ubf{
6. ∴ ∠AOC - ∠BOD = 2 6. tYo 3 / 5 cg';f/
∠AEC
k|dfl0ft eof] .
cEof;sf nflu k|Zgx¿ M A

1. lbOPsf] lrqdf O j[Ttsf] s]Gb| xf] . k|dfl0ft ug'{xf];\ . B

X O
1
∠AXD = (∠AOD - ∠BOC)
2 C
D

A
2. lbOPsf] lrqdf
O j[Ttsf] s]Gb| xf] . k|dfl0ft ug'{xf];\ . B
∠AOC + ∠BOD = 2∠AXC O
D
A

X
C B

204
O

R S
X
3. lbOPsf] lrqdf O j[Ttsf] s]Gb| xf] . olb PQ ⊥ RS eP, k|dfl0ft ug'{xf];\ M ∠ROQ + ∠SOP =
1800

gd'gf k|Zg 10 :
lbOPsf] j[Ttdf MN//AB eP P Q

k|dfl0ft ug'{xf];\ M ∠P = ∠Q

;dfwfg M M N
h'lSt M AN hf]8f}+ . B
A
tYox¿ sf/0fx¿
1. ∠Q = ∠N 1. Pp6} rfk AM df plePsf kl/lw
sf]0fx¿
2. ∠P = ∠A 2. Pp6} rfk BN df plePsf kl/lw
sf]0fx¿
3. ∠N = ∠A 3. PsfGt/ sf]0fx¿ ; MN//AB ePsfn]
4. ∴ ∠ P = ∠Q 4. tYox¿ 1, 2 / 3 cg';f/
k|dfl0ft eof] . X Y

cEof;sf nflu k|Zgx¿ M

1. lbOPsf] j[Ttdf ∠X = ∠Y 5 eg] AB//CD x'G5 egL
k|dfl0ft ug'{xf];\ . A B

D
C

2. lbOPsf] lrqdf XAY / PAQ ;/n /]vfx¿ x'g\ . X A

Q
k|dfl0ft ug'{xf];\ M∠PBX = ∠QBY
P Y

B
P

3. lbOPsf] j[Ttdf PQ = PX eP k|dfl0ft ug'{xf];\ M Q

∆XRS Pp6f ;dlåafx' lqe'h xf] .

S
R

B 205
C
E F
D
4. lbOPsf] lrqdf ∠EDF sf] cw{s AD xf] eg] AB = AC x'G5 egL k|dfl0ft ug'{xf];\ .

O
gd'gf k|Zg 11 : C

lbPsf] j[Ttsf] s]Gb| O xf] / DO ⊥ AB 5g\ .

k|dfl0ft ug'{xf];\ . ∠ADO = ∠AEC A
O
B

;dfwfg M h'StL BC hf]8f}+ . E

k|df0f M
tYo sf/0f
i) ∠ACB = 90 0
i) cw{ j[Ttsf] sf]0f ePsfn] .
ii) ∴ ∠ABC + ∠BAC = 90 0
ii) ;dsf]0f ∆ABC sf Go"g sf]0fx¿
iii) ∠AOD = 90 0
iii) lbPsf]af6 DO ⊥ AB ePsfn]
iv) ∴ ∠ADO + ∠OAD = 90 0
iv) ;dsf]0f ∆ADO sf Go"g sf]0fx¿
v) ∴ ∠ADO = ∠ABC v) tYo (i) / (ii) cg';f/ Pp6} sf]0f
∠BAC.(∠OAD) sf ;dk'/sx¿
ePsfn]
iv) ∠ABC = ∠AEC iv) Pp6} rfk AC df plePsf kl/lw
sf]0fx¿
vii) ∴ ∠ADO = ∠AEC vii) tYo (v) / (vi) cg';f/ .
k|dfl0ft eof] .
B

cEof;sf nflu k|Zgx¿ M

1. lbOPsf] lrqdf O j[Ttsf] s]Gb| / EO ⊥ AC 5g\ . A
O
C
k|dfl0ft ug'{xf];\ . ∠B = ∠E
D
E

P
2. lbOPsf] lrqdf O j[Ttsf] s]Gb|
xf] / SO ⊥ QR 5g\ .
k|dfl0ft ug'{xf];\ M ∠OPR = ∠OSQ Q
O
R

206
3. lbOPsf] lrqdf O j[Ttsf] s]Gb| 5g\ . olb ∠OCB = ∠ADO
5g\ eg] DO ⊥ AB x'G5, egL k|dfl0ft ug'x{ f];\ . O
A B

gd'gf k|Zg 12 :
lbOPsf] lrqdf b'O{ cf]6f j[Ttx¿ A / B ljGb'df C
D
k|ltR5]bg ePsf 5g\ . O ;fgf] j[Ttsf] s]Gb| xf] . C
OA / OB nfO{ hf]8]/ 7"nf] j[Ttsf] kl/lwsf]
A
ljGb'x¿ j|mdz M C / D ;Dd k'Ug] u/L nDAofOPsf
5g\ . O
k|dfl0ft ug'{xf];\ M (i) AB // CD (ii) AC = BD
B
;dfwfg M D
k|df0f
tYox¿ sf/0fx¿
1. OA = OB 1. Pp6} j[Ttsf cw{ Jof;x¿
2. ∠OAB = ∠OBA 2. ;dlåafx' ∆OAB sf cfwf/sf sf]0fx¿
3. ∠OAB = ∠ODC 3. rj|mLo rt'e'{h ABDC sf] afx\o sf]0f /
To;sf] leqL ljkl/t sf]0fsf] ;DaGw
4. ∴ ∠OBA = ∠ODC 4. tYo 2 / 3 cg';f/
5. ∴ AB // CD 5. tYo 4 df ;ª\ut sf]0fx¿ a/fa/ ePsfn]
6. km]l/, ∠OBA = ∠OCD 6. sf/0f (3) cg';f/
7. ∴ ∠ODC = ∠OCD 7. tYo (4) / (6) cg';f/
8. OC = OD 8. ∆OCD df a/fa/ sf]0fx¿ ∠C / ∠D
sf ;Dd'v e'hfx¿ ePsfn]
9. ∴ OC − OA = OD − OB 9. tYo (8) af6 (1) 36fpFbf
10. ∴ AC = BD 10. z]if tYo
11. To;}n] AB // CD / AC = BD 11. tYo 5 / 10 af6
k|dfl0ft eof] .
B
A
cEof;sf nflu k|Zgx¿ M
1. lrqdf O ;fgf] j[Ttsf] s]Gb| xf] . olb OAB / OCD ;/n /]vfx¿ O
x'g\ eg] AB = CD x'G5g\ egL k|dfl0ft ug'{xf];\ .
C
D

2. lrqdf rj|mLo rt'{e'h ABCD sf] A

D

207
E

C
B
 e'hfx¿ AD / BC nfO{ E ;Dd nDAofOPsf] 5 . olb AE = BE 5 eg],
k|dfl0ft ug'{xf];\ M (i) AB // DC (ii) DE = CE

A X
C
3. lbOPsf] lrqdf b'O{ cf]6f j[Ttx¿ ljGb' X / Y df k|ltR5]bg
ePsf 5g\ . olb AXC / BYD ;/n /]vfx¿ x'g\ eg] AB // CD D
x'G5 egL k|dfl0ft ug'{xf];\ . B Y

4. lbOPsf] lrqdf olb ∠ EAC sf] cw{s AB xf] eg] k|dfl0ft E A

ug'{xf];\ M BC = BD D

B C

S
gd'gf k|Zg 13 :
lbPsf] lrqdf NPS, MAN / RMS ;/n /]vfx¿ x'g\ eg]
PQRS Pp6f rj|mLo rt'e{'h xf] egL k|dfl0ft ug'{xf];\ . P M
A

M R
;dfwfg M h'lSt M AQ hf]8f}+ . Q
tYox¿ sf/0fx¿
1. ∠NPQ = ∠NAQ 1. Pp6} rfk NQ df cfwfl/t kl/lw
sf]0fx¿ .
2. ∠NAQ = ∠MRQ 2. rj|mLo rt'{e'h AQRM sf] jfx|o sf]0f /
To;sf] leqL ljkl/t sf]0fsf] ;DaGw .
3. ∴ ∠NPQ = ∠MRQ 3. tYo 1 / 2 af6 .
4. ∴ PQRS Pp6f rj|mLo rt'e{'h xf] . 4. tYo 3 cg';f/ rt'e{'h PQRS sf] afx\o sf]0f
To;sf] leqL ljkl/t sf]0f;Fu a/fa/
ePsfn] .
k|dfl0ft eof] .
cEof;sf nflu k|Zgx¿ M A

1. lbOPsf] lrqdf k|dfl0ft ug'{xf];\ M ABCD Pp6f rj|mLo rt'{e'h G

F
D

xf] . E
B
C

2. lbOPsf] lrqdf AC=AB / XY // BC

X Y
208

B C
 eP XBCY Pp6f rj|mLo rt'{e'h xf] egL l;b\w ug'{xf];\ .

3. lbOPsf] lrqdf AB / EF cfk;df ;dfgfGt/ 5g\ eg] CDEF Pp6f

A
rj|mLo rt'e{'h x'G5 egL k|dfl0ft ug'{xf];\ . D E

F
C
B

4. lbOPsf] lrqdf AF // ED 5g\ eg] BCDE Pp6f rj|mLo

B E
rt'e{'h xf] egL k|dfl0ft ug'{xf];\ . A

D
C F
5. lbOPsf] lrqdf AD // BC / AB // DE 5g\ . olb DE=DC
eP ABCD Pp6f rj|mLo rt'e{'h xf] egL k|dfl0ft ug'{xf];\ . A D

gd'gf k|Zg 14 :
lbOPsf] lrqdf ∠BAC sf] cw{s AG xf] eg] k|dfl0ft ug'{xf];\ M B C
E
(i) A, E, F / D rj|mLo ljGb'x¿ x'g\ .
(ii) EF// BC A
D

;dfwfg E F

h'lSt M A / D hf]l8Psf 5g\ . B

k|df0f M G
C

tYox¿ sf/0fx¿
1. ∠BAG = ∠CAG 1. ∠BAC sf] cw{s AG ePsfn]
2. ∠BAG = ∠BDG 2. Pp6} rfk BG df plePsf kl/lw sf]0fx¿
3. ∴ ∠CAG = ∠BDG cyf{t 3. tYo 1 / 2 cg';f/
∠EAF = ∠EDF
4. ∴ A, E, F / D rj|mLo ljGb'x¿ x'g\ 4. tYo 3 cg';f/ EF n] A / D df a/fa/
sf]0fx¿ agfPsf]n]
5. ∠CAD = ∠DEF 5. Pp6} rfk DF df cfwfl/t kl/lw sf]0fx¿
6. ∠CAD = ∠DBC 6. Pp6} rfk DC df cfwfl/t kl/lw sf]0fx¿
7. ∴ ∠DEF = ∠DBC 7. tYo 5 / 6 cg';f/
8. ∴ EF//BC 8. tYo 7 df ;ª\ut sf]0fx¿ a/fa/ ePsfn]
k|dfl0ft eof] .

cEof;sf nflu k|Zgx¿ M A G

E 209

C F D
B
1. lbOPsf] j[Ttdf AB=AC 5g\ eg] k|dfl0ft ug'{xf];\ .
(i) D, E, G / F rflqms ljGb'x¿ x'g\ .
(ii) BC // FG

A
D

2. lbOPsf] j[Ttsf] s]Gb| O xf] . olb AE=CE eP k|dfl0ft ug{'xf];\ M EO

// CB
E O

C
B
P

S T
3. lbOPsf] lrqdf, olb ST // AB 5g\
eg] k|dfl0ft ug'{xf];\ PS = PT A B

Q R
yk cEof;sf nflu k|Zgx¿ M
1. lbOPsf] j[Ttsf] s]Gb| O xf] . olb ST = UT 5 eg] k|dfl0ft R
ug'{xf];\ . ∠R = ∠S S

P Q
O T
U

O
2. lbOPsf] j[Ttsf] s]Gb| O xf] .
k|dfl0ft ug'{xf];\ M ∠AOC = 2(∠ACB + ∠BAC ) A C

A
Y

D
210
C
B X
3. lbOPsf] j[Ttdf ∠BAD / ∠BCD sf cw{sx¿ j|mdzM AX / CY x'g\ eg] XY j[Ttsf] Jof;
x'G5 egL k|dfl0ft ug'{xf];\ .

E
4. lbOPsf] lrqdf b'O{ cf]6f j[Ttx¿ ljGb' E / F df k|ltR5]bg
ePsf 5g\ . olb ABCD Pp6f ;Lwf/]vf xf] eg] k|dfl0ft
ug'{xf];\ M ∠AFC + ∠BED = 1800
F
A B C D

5. lbOPsf] j[Ttdf ABCD Pp6f ;dfgfGt/ rt'e'{h xf] . A D

k|dfl0ft ug'{xf];\ . CD = ED

B C E

6. lbOPsf ∆ABE df AC ⊥ BE / BD ⊥ AE 5g\ eg] A

k|dfl0ft ug'{xf];\ M ∠DCE = ∠BAD
D

E
B C
7. lbOPsf] ∆ABC df AB, BC / CA sf dWo ljGb'x¿ j|mdz M M, O / N x'g\ . olb AP ⊥
BC 5 eg] MNOP Pp6f rj|mLo rt'e'{h xf] egL k|dfl0ft ug'{xf];\ .
A

M N

B C
P O

M 211
O
R S
X

Q
8. lbOPsf] j[Ttdf s]Gb|ljGb'
O5 . olb PQ ⊥ RS 5g\ eg] k|dfl0ft ug'{xf];\ M
(i) ∠POR = ∠PMQ
(ii) MNOP Pp6f rj|mLo rt'e'{h xf] .

212
kf7 12 :kz{ /]vf (TANGENT)

!= kl/ro B

o; kf7af6 5f]6f] pTt/ cfpg] Pp6f k|Zg P;Pn;L k/LIffdf D C

;f]lwg] Joj:yf 5 .
@= cfwf/e"t tYox¿ T N
1. j[Ttsf] s]Gb|ljGb' / :kz{ ljGb' hf]8\g] cw{ Jof; :kz{ /]vfdf A

nDa x'G5 .
2. j[Ttsf] afx\o ljGb'af6 lvlrPsf b'O{ :kz{ /]vfx¿sf] :kz{ ljGb'x¿;Ddsf] b'/L a/fa/
x'G5g\ .
3. j[Ttsf] :kz{ ljGb'af6 lvlrPsf] hLjfn] :kz{ /]vfl;t agfPsf sf]0fx¿ j|mdzM PsfGt/
j[Tt v08sf sf]0fx¿;Fu a/fa/ x'G5g\ .
2 2 2
4. ;dsf]0f lqe'hdf, -s0f{_ = -nDa_ + -cfwf/_ x'G5 .
Y P
Wofg lbg'kg]{ s'/fx¿
(i) j[Ttsf] :kz{ /]vf eGgfn] j[Ttsf] kl/lwsf] Pp6f ljGb'nfO{ 5f]P/ C B

hfg] /]vf xf] . :kz{ ljGb' eGgfn] Tof] 5f]Psf] ljGb'nfO{ a'‰g'kb{5 .
h:t} M TN, XY / PQ :kz{ /]vfx¿ x'g\ eg] A, B / C ljGb'x¿ X Q
:kz{ljGb'x¿ x'g\ . T A N

Q
A
(ii) cfwf/e"t tYo 2 cg';f/ PA = PB, QA = QC /
RB = RC x'G5g\ . C

P
B
R

B
(iii) lrqdf hLjf AB n] j[TtnfO{ b'O{ cf]6f v08dfljefhg ul/Psf]
D
5 . To;}n] ADB / ACB j[Tt v08x¿ x'g\ . To:t} AB n] :kz{ /]vf TN C
;Fu agfPsf sf]0fx¿ ∠BAN / ∠BAT agfPsf 5g\ . ca, T N
A
∠BAN sf] PsfGt/ j[Tt v08 ADB xf] eg] ∠BAT sf]
PsfGt/j[Ttv08 eGgfn]] j[Ttv08 ACB nfO{ a'‰g'kb{5 . To;}n] tYo 3 cg';f/ ∠BAN = ∠BDA
/ ∠BAT = ∠BCA x'G5g\ .

213
(iv) ;dsf]0f lqe'hsf] s0f{ eGgfn] ;dsf]0fsf] ;Dd'v e'hf / B
c? b'O{ e'hfnfO{ nDa / cfwf/ a'‰g'kb{5 . h:t}
∆ABC df ∠ABC = 90 0 To;}n] AC s0f{ eof] . D
∴ AC 2 = AB 2 + BC 2 -o;nfO{ kfOyfuf]/; ;fWo elgG5 _ C

A N
5f]6f] pTt/ cfpg] k|Zgx¿sf] pTt/ / cEof;sf nflu T
k|Zgx¿ M
1. lbOPsf] lrqdf TAN :kz{/]vf / A :kz{ ljGb' 5g\ . olb AN = 24cm / BN=18cm 5g\ eg]
j[Ttsf] cw{ Jof; kTtf nufpg'xf];\ .

T A 24 cm N

;dfwfg M
(i) OA ⊥ TN -j[Ttsf] cw{Jof; :kz{ /]vfsf] :kz{ ljGb'df nDa x'g] ePsf]n]_
(ii) OA = OB = x -dfgf}+_ -Pp6} j[Ttsf cw{ Jof;x¿_
(iii) ∴ ON 2 = OA 2 + AN 2 -kfOyfuf]/; ;fWo_
or, (18+ x)2 = x2 + 242
or 324 + 36x + x2 = x2 + 576
or 36x = 252
∴ x=7 To;}n] cw{Jof; = 7cm

cEof;sf nflu k|Zgx¿ M lbOPsf] lrqx¿df X sf] dfg lgsfNg'xf];\ .

-s_ -v_ -u_
A A A
T N

O O

B B
O

T A 8 cm N T A x cm N

pTt/x¿ M -s_ 4cm -v_ 4cm -u_ 5cm

214
 T
gd'gf k|Zg 1 : B C

lbOPsf] lrqdf O j[Ttsf] s]Gb|, TAN :kz{/]vf x0

/ A :kz{ ljGb' xf] . olb ∠OTA = 20 eP
0
D A
O
∠DCA(= x) sf] dfg lgsfNg'xf];\ M
T

;dfwfg M (i) ∠OAT = 90 0 - N

-cw{Jof; OA :kz{/]vf TN df nDa x'g] ePsfn]_

(ii) ∠AOT = 90 0
− 20 =70 - ∠AOT + ∠OTA = 90 0 x'G5 ._
0

1
(iii) ∠ADB = ∠AOB = 350 -Pp6} rfk AB df cfwfl/t kl/lw sf]0f / s]lGb|o
2
sf]0fsf] ;DjGw_
(iv) ∴ ∠ADC + ∠ACD = 90 0 - ∆ADC df ∠DAC = 90 o ePsfn]_
∴ ∠ACD = 90 0 − ∠ADC
= 900 - 350
∴ x = 550
0
cEof;sf nflu k|Zg M lbOPsf] lrqx¿df x sf] dfg lgsfNg'xf];\ .
-s_ -v_ -u_
T

B
B
O
x° B A
O
O
60° B
C M
x°
40° x°
N T
A N A 24 cm N

-3_ -ª_
A
O

x° B O x° 40° P
35°
T A N B

pTt/x¿ M -s_ 250 -v_ 300 -u_ 500 -3_ 700 -ª_ 1400

gd'gf k|Zg 2:

215
lbOPsf] lrqdf PA / PB b'O{ :kz{ /]vfx¿ x'g\ . olb ∴ ∠APB = 80 0 eP, A

∠ACB sf] dfg lgsfNg'xf];\ .

P 80° ?° C

;dfwfg M
(i) PA = PB-afx\o ljGb' P af6 j[Ttdf lvlrPsf :kz{ /]vfx¿_ B
(ii) ∠PAB = ∠PBA = x 0 -dfgf}_ -;dlåafx' ∆PAB sf
cfwf/sf sf]0fx¿_
(iii) 800 + x0 + x0 = 1800 - ∆PAB sf] ltg cf]6f sf]0fx¿sf] of]u_
∴ x = 50 0
(iv) To;}n], ∠PAB = ∠ACB -PsfGt/ j[Tt v08sf sf]0fx¿ ePsfn]_
∴ ∠ACB = 50 0
cEof;sf nflu k|Zgx¿ M
tn lbOPsf lrqx¿df x0 sf] dfg lgsfNg'xf];\ .
-s_ -v_
S
- u_
S
A
Q R
Q R

60° P

B
P P

pTt/x¿ M -s_ 10cm -v_ 800 -u_ 650

216
yk cEof;sf nflu k|Zgx¿ M
tn lbOPsf] lrqx¿df x0 / y0 sf] dfg lgsfNg'xf];\ .

-s_ -v_ -u_

P

B
N
C
C B x°
x° O x°
130°
Q
A
50° y°

T A N T A N T

-3_ -ª_ -r_

B T

50°
x°
A A
B O
y° x°
C
O y° P
x°
T 25° N D
A N B

(TA=AB / ∠ ATC = 250)

pTt/x¿ M
-s_ 400 -v_ 600, 600 -u_ 500
-3_ 400, 400 -ª_ 250 -r_ 560, 680

217
kf7 13 /rgf (Construction)
1. kl/ro
o; kf7df lqe'h / rt'e{'hsf] If]qkmn ;DaGwL ;fWox¿sf cfwf/df, a/fa/ If]qkmn x'g]
lqe'h / rt'e'{hsf] /rgf ug]{ tl/sfsf af/]df rrf{ ul/Psf] 5 . o; kf7af6 P;Pn;L
k/LIffdf nfdf] pTt/ cfpg] Pp6f k|Zg ;f]lwg] 5 eGg] s'/f ljlzi6Ls/0f tflnsfdf pNn]v
ul/Psf] 5 .

2. cfwf/e"t tYox¿
-s_ Pp6} cfwf/ / pxL ;dfgfGt/ /]vfx¿ lar /x]sf
(i) lqe'hx¿sf] If]qkmn a/fa/ x'G5g\\ .
(ii) ;dfgfGt/ rt'e'{hx¿sf] If]qkmn a/fa/ x'G5g\ .
(iii) lqe'hsf] If]qkmn ;dfgfGt/ rt'e'{hsf] If]qkmnsf] cfwf x'G5 .
-v_ ;dfgfGt/ rt'e'{hsf]
(i) ;Dd'v e'hfx¿ a/fa/ x'G5g\ .
(ii) ;Dd'v sf]0fx¿ a/fa/ x'G5g\ .
(ii) ljs0f{x¿ k/:k/ ;dlåefhg x'G5g\ .

3. Wofg lbg'kg]{ s'/fx¿

-s_ ;j{k|yd /rgf ug'{kg]{ gfds/0f;lxt lrqsf] v];|f -/km_ sf]/]df lrq agfpgdf ;xhtf
cfp“5 .
-v_ :ki6 / dl;gf] /]vf lvRg'kb{5 . /]vfsf] df]6fO{ sxL“ dl;gf] sxL“ df]6f] geO{ Psgf;sf] x'g'
/fd|f] x'G5 .
-u_ lbOPsf gfkx¿ lrqdf b]vfpg / gfds/0f ug{ lj;{g' x'“b}g .
-3_ /rgf u/]sf] lrqnfO{ ;}b\wflGts ¿kdf hf“Rg'kb{5 .
-ª_ lrq tof/ u/]kl5 cGTodf cfjZos lrq of] xf] egL n]Vg'kb{5 .
-r_ /rgfsf r/0fx¿ pNn]v ug'{kb}{g .

gd'gf k|Zg 1:
AB =BC = 5.5cm, CD=DA=4.5cm / ∠BAD = 75 sf] ePsf] rt'e'{h
0
ABCD sf]
If]qkmn;Fu a/fa/ x'g] Pp6f lqe'h ADE sf] /rgf ug'{xf];\ .

218
 oxfF rt'e{h' / lqe'hsf] ;femf e'hf AD
nfO{ lrqdf h:t} af“of kIfdf /flvPsf] .
∴ cfjZos ADE sf] /rgf ul/of] .

r/0fx¿
1. lbOPsf] rt'e'{h ABCD sf] /rgf ug]{ . To;sf nflu M
(i) cfwf/ AB=5.5cm lvrf}+ .
(ii) A df ∠BAD = 750 lvrL ;f] /]vfdf AD = 4.5cm sf] rfk sf6f}+ .
(iii) D af6 DC = 4.5cm sf] rfk / B af6 BC = 5.5cm sf] rfk Ps cfk;df sf6]/ DC / BC
hf]8f}+ . o;/L rt'e'{h ABCD k'/f tof/ eof] .
2. ljs0f{ BD hf]8]/ of];Fu C af6 ;dfgfGt/ /]vf lvRg] . h;n] AB nfO{ nDAofOPsf] efunfO{
ljGb' E df e]6\5 . ;dfgfGt/ /]vf CE lvRgsf nflu M
(i) lrqdf b]lvP h:t} B af6 BD / BC nfO{ sf6\g] u/L Pp6f rfk PQ lvRg] / ;f]xL rfk a/fa/
x'g]u/L C af6 CB nfO{ sf6\g] u/L csf]{ rfk RS lvRg] . rfk PQ gfk]/ R af6 RT rfk sf6\g] .
o;/L PsfGt/ sf]0fx¿ ∠DBC = ∠BCE eO{ DB//CE x'G5 . -k/LIffdf r/0fx¿ n]lv/xg'
kb}{g . rfk PQ / RT eg]/ gfds/0f klg ug'{kb}{g ._
3. A / E hf]8\g] .
cEof;sf nflu k|Zgx¿
1. PQ = QR = 5cm, RS = Sp = 6cm / ∠P = 60 0 x'g] rt'e'{h PQRS /rgf u/L ;f] rt'e'{hsf]
If]qkmn;Fu a/fa/ x'g]u/L ∆PST sf] /rgf ug'{xf];\ .
2. AB = 6.2cm, BC = 5.3cm / CD = 5cm, PS = 5.7cm / AC = 5.6cm ePsf] rt'e'{h ABCD

/rgf u/L pSt rt'e'{hsf] If]qkmn;Fu a/fa/ x'g] u/L ∆ BCE sf] klg /rgf ug'{xf];\ .

219
gd'gf k|Zg @ M
a = 7cm, b = 6.6 / c = 5.8cm ePsf] Pp6f ∆ ABC /rgf u/L To;sf] If]qkmn;Fu a/fa/
x'g] Pp6f ;dfgfGt/ rt'e'{h CDEF sf] /rgf ug'{xf];\ . h;sf],
(i) Pp6f sf]0f ∠CDE = 60 0 5 . (ii) Pp6f e'hf DE = 6.8cm 5 .
;dfwfg M
S
A E F
X y A
T
R
c=5.8cm b=6.6cm

b=6.6cm B C
c = 5.8cm a=7cm

D 60
B P a=7cm 0 C

:. cfjZos ;dfgfGt/ rt'e\{'h CDEF /rgf ul/of] .

r/0fx¿ M
1. ∆ ABC sf] /rgf M
(i) cfwf/ BC = a = 7cm lvrf}+ .
(ii) B af6 AB = c =5.8cm sf] rfk / C
af6 CA = b = `6.6cm sf] rfk Ps
cfk;df sf6]sf] ljGb' A ;Fu B / C
hf]8]/ ∆ ABC jgfcf}+ .

2. cfwf/ BC ;Fu ;dfgfGt/ x'g]u/L A af6

/]vf XY lvrf}+ . h;sf] lgldTt
(i) B af6 Pp6f rfk PQ lvrf}+ / ;f]xL gfk a/fj/sf] rfk RS ljGb' A af6 lvrf}+ .
(ii) ∠B agfpg] rfk PQ gfk]/ R af6 rfk RT sf6\g] .

220
 ;dfgfGt/ /]vf XY lvRg] . -PsfGt/ sf]0fx¿ ∠BAX = ∠ABC ePsfn]_
(iii) A ;Fu T hf]8]/
3. BC sf] nDjfw{s lvr]/ BC sf] dWo ljGb' D kTtf nufpg] . h;sf] lgldTt M
(i) BC sf] cfwfeGbf a9L gfksf] rfk lnP/ B / C af6 lrqdf b]vfP h:t} M o;sf] b'j} ;fO8df
rfkx¿ lvRg] . o;/L rfkx¿ sfl6Psf] ljGb'x¿ hf]8\g] . h;n] BC nfO{ D df sf6\5 .
4. D df ∠CDE = 60 0 sf] sf]0f lvrf}+ . DE n] XY nfO{ E df e]6\5 .
5. ca E af6 DC = EF x'g] u/L XY df rfk EF sf6\g] / FC hf]8\g] . o;/L ;dfgfGt/ rt'e'{h CDEF
sf] /rgf k'/f eof] .
(ii) ∆ ABC sf] If]qkmn a/fa/ x'g] u/L Pp6f e'hf DE = 5.5cm ePsf] ;dfgfGt/ rt'e'{h CDEF
sf] /rgf ug{ r/0fx¿ 1, 2 / 3 k"/f ug]{ .
6. 5.5cm sf] rfk gfk]/ D / C af6 XY sf] b'O{ ljGb'x¿ E / F df sf6\g] . DE / CF hf]8]/
CDEF sf] /rgf k'/f x'G5 .
∴ cfjZos ;dfgfGt/ rt'e'{h CDEF sf] /rgf ul/of] .

cEof;sf nflu k|Zgx¿

1. a = 7.8cm, b = 7.2cm / c = 6.3cm ePsf] ∆ ABC sf] If]qkmn;Fu a/fa/ x'g]u/L Pp6f sf]0f
0
45 ePsf] Pp6f ;dfgfGt/ rt'e'{hsf] /rgf ug'{xf];\ .
2. PQ = 7.5cm, QR = 6.8cm / RP = 6cm ePsf] Pp6f ∆ PQR /rgf u/L To;sf] If]qkmn;Fu
a/fa/ x'g] u/L Pp6f e'hf TQ = 6.4cm ePsf] ;dfgfGt/ rt'e'{h TQSU sf] /rgf ug'{xf];\ .
3. AB = 6cm, BC = 7cm / CA = 5cm ePsf] ∆ ABC sf] /rgf u/L To;sf] If]qkmn;Fu a/fa/ x'g]
Pp6f cfot CDEF sf] /rgf ug'{xf];\ .

gd'gf k|Zg 3. AB = 6.2cm, AC = 8cm / BD = 6cm ePsf] ;dfgfGt/ rt'e'{h ABCD sf] /rgf
u/L To;sf] If]qkmn;Fu a/fa/ x'g]u/L ∆ ADE sf] /rgf ug'{xf];\ .
∴ cfjZos ∆ ADE sf] /rgf eof] .

D C

6cm 8cm

A 6.2cm B
r/0fx¿ M1. lbOPsf] gfkcg';f/ ;dfgfGt/ rt'e'{h ABCD
lvRg] .
(i) cfwf/ AB = 6.2 lvrf}+ .
221
 (ii) A af6 4cm sf] rfk / B af6 3cm sf] rfk lvr]/ cfk;df ljGb' O df sf6f}+ .
(iii) AO / DO nfO{ hf]8]/ nDAofpg] .
(iv) AO = CO 4cm sf] rfk lnP/ O af6 rfk OC sf6\g] / BO = OD = 3cm x'g]u/L O
af6 rfk OD sf6\g] .
(v) AD, DC / BC hf]8]/ ;dfgfGt/ rt'e'{h ABCD k'/f ug]{ .
2. cfwf/ AB nfO{ nDAofP/ AB = BE = 6.2cm agfpg] .
3. D / E hf]8\g] .
∴ ∆ ADE tof/ eof] . h;sf] If]qkmn ABCD sf] If]qkmn;Fu a/fa/ x'G5 .
cEof;sf nflu k|Zgx¿ M
1. AB = 4cm, ∠ABC = 60 0 / BC = 6cm ePsf] Pp6f ;dfgfGt/ rt'e'{h ABCD sf]
0
/rgf u/L To;sf] If]qkmn;Fu a/fa/ x'g] / Pp6f sf]0f 30 ePsf] lqe'hsf] /rgf
ug'{xf];\ .
2. ;dfgfGt/ rt'e'{h PQRS sf] /rgf ug'{xf];\ . h;df PQ = 5cm ljs0f{ PR = 6cm /
ljs0f{ QS = 8cm 5g\ . pSt ;dfgfGt/ rt'e'{hsf If]qkmn;Fu a/fa/ x'g]u/L Pp6f
∆ PSA sf] /rgf klg ug'{xf];\ .

222
kf7 !$= k|of]ufTds k/LIf0f (EXPERIMENTAL VERIFICATION)

!= kl/ro
o; kf7df lqe'h / ;dfgfGt/ rt'e'{hsf] If]qkmn tyf j[Tt ;DaGwL ;fWox¿sf] k|of]uåf/f
k/LIf0f ug]{ tl/sfsf af/]df rrf{ ul/Psf] 5 . ljlzi6Ls/0f tflnsf cg';f/ P;Pn;L k/LIffdf
o; kf7af6 nfdf] pTt/ cfpg] Pp6f k|Zg ;f]lwg] 5 .
@= Wofg lbg'kg]{ s'/fx¿ M
1. b'O{ cf]6f km/s km/s gfksf / cfs[ltsf lrqx¿ agfpg'kb{5 .
2. lrqx¿ 7'nf] / :ki6 x'g'kb{5 / gfkx¿ k|Zgdf pNn]v u/] cg';f/ x'g'kb{5 .
3. cfjZos gfkx¿nfO{ tflnsfdf k|:t't u/L To;sf] cfwf/df lg:sif{ lgsfNg'kb{5
4. yfxf lbOPsf] n]Vbf k|of]udf cfpg] HofldtLo pks/0fx¿sf] -h:t} M k]lG;n,sDkf;,
k|f]6\ofS6/ cflbsf] gfd klg pNn]v ul/P /fd|f] x'G5 ._
5. S.L.C k/LIffdf yfxf lbOPsf] / k/LIf0f ug'{kg]{ pNn]v ul//xg' cfjZos 5}g .
#= gd'gf k|Zgf]Tt/ / cEof;
gd'gf k|Zg 1:
j[Ttsf] Pp6} rfkdf cfwfl/t s]Gb|Lo sf]0f kl/lw sf]0fsf] bf]Aa/ x'G5 egL k|of]uåf/f k/LIf0f ug'{xf];\ .
sDtLdf 3cm cw{Jof;x¿ ePsf km/s km/s gfksf b'O{ cf]6f j[Ttx¿ clgjfo{ 5 ._

1. yfxf lbPsf] M
(i) k]lG;n sDkf;sf] db\btn] 4cm / 4.5cm cw{Jof;x¿ ePsf b'O{ cf]6f j[Ttx¿ lvlrPsf
5g\ . s]Gb|nfO{ O n] hgfOPsf] 5 .
(ii) k|To]s j[Ttdf ?n/sf] ;xfotfn] rfk QR df cfwfl/t x'g] u/L km/s km/s gfksf s]Gb|Lo
sf]0f QOR / kl/lw sf]0f QPR agfOPsf 5g\ .
223
2. k/LIf0f ug'{kg]{ M ∠QOR = 2∠QPR
3. ljlw M b'j} lrqdf k|f]6\ofS6/n] ∠QOP / ∠QPR gflkPsf 5g\ . gfkx¿ lgDg tflnsfdf
k|:t''t ul/Psf 5g\ .

gfk tflnsf
lrq ∠QOR ∠QPR glthf
(i) 940 470 ∴ ∠QOR = 2∠QPR

(ii) 540 270 ∠QOR = 2∠QPR

4. lgisif{ M dflysf tflnsfaf6 j[Ttsf] Pp6} rfkdf cfwfl/t s]Gb|Lo sf]0f kl/lw sf]0fsf] bf]Aa/
x'G5 eGg] tYo k|dfl0ft x'G5 .

cEof;sf nflu k|Zgx¿ M

k|of]uåf/f l;b\w ug{'xf];\ M
(3cm.eGbf a9L cw{Jof;x¿ ePsf b'O{cf]6f j[Ttx¿ clgjfo{ 5)
1. s]Gb|ljGb' O ePsf] j[Ttsf] rfk BC df cfwfl/t s]Gb|Lo sf]0f ∠BOC kl/lw sf]0f ∠BAC
sf] bf]Aa/ x'G5 .
2. j[Ttsf] pxL rfk BC df ag]sf kl/lw sf]0fx¿ ∠BAC /∠BDC a/fa/ x'G5g\ .
3. cw{j[Ttdf aGg] sf]0f Ps ;dsf]0f x'G5 .
0
4. rj|mLo rt'e{'h PQRS df ∠P+∠R=180 x'G5 .
gd'gf k|Zg 2:
Pp6} cfwf/ / pxL ;dfgfGt/
/]vfx¿ lar ag]sf ;dfgfGt/ F D E C
M N
rt'e'{hx¿sf] If]qkmnx¿ a/fa/
x'G5g\ .
1. yfxf lbOPsf] M
(i) ;]6 :Sjfo/sf] d4tn] /]vfx¿
XY//MN lvlrPsf] 5 .
(ii) XY df km/s gfksf] cfwf/ X B Y
A G
AB lnO{ ;dfgfGt/ /]vfx¿
lardf km/s gfksf b'O{ cf]6f ;dfgfGt/ rt'e'{hx¿ ABCD / ABEF agfOPsf 5g\\ .
(iii) ;]6 :Sjfo/sf] ;xfotfn] AB df EG nDa lvlrPsf] 5 .

224
 M F E D C N

X B Y
A G

2. kl/If0f ug'{kg]{ M ABCD sf] If]qkmn = ABEF sf] If]qkmn

3. ljlw M l8efO8/ / ?n/sf] db\btn] k|To]s lrqdf cfwf/ AB / EG rfk gfk]/ ABCD
/ ABEF sf] If]qkmn u0fgf u/]/ tnsf] tflnsfdf k|:t't ul/Psf] 5 .
lrq AB EG ABCD sf] If]qkmn ABEF sf] If]qkmn = glthf
= (AB.EG) (AB.EG)
(i) 3.8cm 3.3cm 12.54cm2 12.54cm2 ABCD = ABEF
(ii) 5.5cm 4cm 22cm2 22cm2 ABCD = ABEF

4. lgisif{ M o;/L dflysf] tflnsfaf6 Pp6} cfwf/ / pxL ;dfgfGt/ /]vfx¿ larsf ;dfgfGt/
rt'e'{hx¿sf] If]qkmn a/fa/ x'G5g\ eGg] tYo k|dfl0ft x'G5 .

cEof;sf nflu k|Zgx¿ M

k|of]uåf/f k/LIf0f ug'{xf];\ .
(i) Pp6} cfwf/ / pxL ;dfgfGt/ /]vfx¿ larsf lqe'hx¿sf] If]qkmn a/fa/ x'G5g\ .
(ii) Pp6} cfwf/ / pxL ;dfgfGt/ /]vfx¿ lar /x]sf] lqe'hsf] If]qkmn ;dfgfGt/ rt'e'{hsf]
If]qkmnsf] cfwf x'G5 .
(iii) Pp6} cfwf/ PQ / pxL ;dfgfGt/ /]vfx¿ PQ / ST lar /x]sf ;dfgfGt/ rt'e'{hx¿ PQRS
/ PQTU sf] If]qkmn a/fa/ x'G5g\ .

225
 If]q M ^ lqsf]0fldlt (Trigonometry)
!= kl/ro -Introduction)
lqsf]0fldlt (Trigonometry) zAb lu|s
Word Triangle
Metron : To Measure cyjf the Measurement of the angle of a triangle cyf{t lqe'hsf]
ltg cf]6f sf]0fsf] gfknfO{ lqsf]0fldlt elgG5 . lqe'hsf] xn ug'{ g} o;sf] ljz]iftf xf] . lqsf]0fldlt
/ Hofldlt cfk;df ;DalGwt ljifox¿ x'g\ . lqsf]0fldltsf] ljsf;df lu|s Astronomer
Hipparchus sf] dxTTjk"0f{ e"ldsf b]lvG5 To;}n] pgnfO{ lqsf]0fldltsf hGdbftfsf ¿kdf lnG5 .
cfhfef]ln cfP/ lqsf]0fldltsf] k|of]u cGo If]qx¿ Engineerting, Geology, Astrological
Survey, Navigation / lj1fgsf cGo If]qx¿df ePsf] kfOG5 . lqsf]0fldltnfO{ ul0ftsf] ;a}eGbf
Jofjxfl/s cª\u dflgG5 . lj1fgsf] pGgltnfO{ lzv/df k'¥ofpgdf o;sf] 7'nf] e"ldsf /x]sf]
kfOG5 .
d'Vo ljifoj:t'x¿ (Main Subject matters)
-s_ lqsf]0fldlto cg'kftx¿ (trigonomeric ratios)
-v_ lqsf]0fldlto ;"qx¿ k|of]u u/L lqe'h tyf rt'e'{hsf If]qkmn
-u_ pGgtf+z sf]0f (angle of elevation) / cjgltsf]0f (angle of depression) sf]0f
-3_ lqsf]0fldlto cg'kftsf] k|of]u u/L prfO tyf b'/L;“u ;DalGwt ;d:ofx¿sf] ;dfwfg
S.L.C Exam Specification Grid – 2065
P;Pn;L k/LIff ljlzi6Ls/0f tflnsf M 2065
;+1fg 1fg (knowledge) l;k hDdf l;k ;d:of hDdf s"n hDdf
(cognitive af]w (skill) cª\s (skill) ;dfwfg cª\s (Grand
domain) (understanding) (total (problem (total total
If]q (Area) marks) solving) marks) marks)
ljifoj:t' 5f]6f] pTt/ cfpg] k|Zg nfdf] pTt/ cfpg] k|Zg
(content) (short answer) questions) (Long answer)
questions)
lqe'hsf] 11 - 2 - - - 2
If]qkn (Area
lqsf]0fldlt of Triangle)
(trigonometry) prfO / b"/L - - - - 1 4 4
(Height and
Distance)

226
 pTt/ s'l~hsf
Marking Scheme

5f]6f] pTt/ cfpg] k|Zgsf nflu

lqe'hsf] If]qkmn l7s ;"q k|of]u afkt 1 cª\s A
l7s pTt/ lgsn] afkt 1 cª\s
7cm
hDdf M 2 cª\s
pbfx/0f M lbOPsf] lrqdf AB = 7cm, BC = 16cm C
B
/ sf] If]qkmn eg] ∠ABC sf] dfg lgsfNg'xf];\ . 16cm
;dfwfg
ABC sf] If]qkmn (A) = 18m

or, Marks

or,
or,
Marks
nfdf] pTt/ cfpg] k|Zgsf nflu pTt/ s'l~hsf
l7s lrq tyf j0f{g afkt 1 cª\s
s'g} e'hf jf sf]0f kTtf nufP/ afkt 1 cª\s
k|Zgn] ;f]w]cg';f/sf] l7s dfg 1 cª\s
u0fgf u/] afkt hDdf 4 cª\s
pbfx/0f M Pp6f 3/sf] 5taf6 s]xL ld6/ 6f9f /x]sf] ?vsf] 6'Kkf]df x]bf{ 30) sf] pGgtf+z sf]0f
aG5 . olb pSt 3/ / ?vsf] prfO j|mdzM 6 ld6/ 18 ld6/ 5 eg] 3/ / ?v larsf] b'/L
kTtf nufpg'xf];\ .
;dfwfg E
dfgf“} AB Pp6f 3/ xf] h;sf] prfO 6m 5 . CE Pp6f ?v xf] h;sf] prfO
18 m 5 .
3/sf] 5t A af6 ?vsf] 6'Kkf] E df x]bf{ 300 sf] pGgtf+z sf]0f ag]sf] 5 . A 300 D
AB = CD = 6m
CE = 18m 6
EAD= 300
ED = EC – DC B C
= 18 – 6
=12 m …................................ 1 Marks
;dsf]0f ADE af6
227
Tan 300 = ...................... 1 Marks
AD
Or,
Or,
Or, AD = 12 x 1.732
Or, AD = 20. 78 m
3/ / ?v larsf] b'/L = 20.78 m ............................1 Marks
cfwf/e"t tYox¿ tyf ;"qx¿
(Basic Facts and Formula)
C
i). lqsf]0fldlt cg'kftx¿ (Trigonometric Ratios)
Sin A =

Cos B = B
A
Tan C =
A
(ii) ABC sf] If]qkmn =
of] ;"q k|of]u ug{ lqe'hdf c b

cj:yf ;Gt'i6 x'g'kb{5 hxf“ a, b, c lqe'hsf ltg B a

C

cf]6f e'hfx? x'g\ .

iii) ljleGg ljlzi6 sf]0fx¿sf] lqsf]0fldlt cg'kft
Angle 00 300 450 600 900
Ratio
Sin 0 1

cos 1 0

Tan 0 1
(undefiend)

228
iv) ABC sf] If]qkmn = cfwf/ prfO A

BC AD
D
A
B B
D
v) ;dfgfGt/ rt'e'{h ABCD sf] If]qkmn = 2ABC sf] If]qkmn

C
B
A D
vi) cfotg ABCD sf] If]qkmn = cfwf/ x prfO = BC x AB

B C B
vii) pGgtf+z sf]0f (angle of elevetion) b[li6/]vf
cjnf]sg stf{n] xf]rf] :yfgdf jl; cUnf] :yfgdf /x]sf] j:t'sf]
cjnf]sg ubf{ b[li6 /]vf n] hldg jf hldg ;+u ;dfgfGt/ /]vf ;+u agfPsf]
sf]0fnfO{ pGgtf+z sf]0f elgG5 . C A
lIflth /]vf
lrqdf ACB= pGgtf+z sf]0f xf] .
viii) cjglt sf]0f (angle of depression) B X
cjnf]sgstf{n] xf]rf] :yfgdf a;L cUnf] :yfgdf b[[li6 /]vf
(opticalize)
/x]sf] j:t'sf] cjnf]sg ubf{ b[li6 /]vfn] hldg jf hldg;“u
;dfgfGt/ /]vf;“u agfPsf] sf]0fnfO{ cjglt sf]0f elgG5 . C
A
lIflth /]vf (Horizontal line)
15.1.3 ljz]if Wofg lbg'kg]{ s'/fx¿ B X
cjgltsf]0f
i) cjglt sf]0f / pGgtf+z sf]0f hlxn] klg a/fa/ x'G5 lsgls

oL b'O{ sf]0fx¿ PsfGt/ sf]0f x'g\ . pGgtf+ztLsf]0f

C
cyf{t\, ∠ACB = ∠CBX x'G5 . cjgtLA sf]0f (Angle of depression)

A (Angle of elevation)

s0f{(h)
ii) ∠B 900 ;dsf]0f ∆ABC df ∠B = 900 ePsfn]
∠B ;fd'g]sf] e'hf AC nfO{ s0f{ (h) elgG5 .
To:t} nfO{ Go"gsf]0f elgG5 B C
/ o;nfO{ angle of reference elgG5 / of] angle of reference sf] ;fd'Gg]sf] e'hf AB nfO{
nDa (p) elgG5 / af“sL t];|f] e'hf BC nfO{ cfwf/ (b) elgG5 .
s0f{ ;w}“ 900 sf] ;Dd'v e'hf x'G5 t/ nDa / cfwf/ eg] Angle of reference sf cfwf/df
lgwf{/0f ul/g] ePsf]n] km/s kg{ ;S5 .

229
 A
h:t} M
Q
s0f{(h)
cfwf/ (b)

iii) dlWosfn] lqe'hnfO{ cfwf ub{5 . B C

nFDa (p)
lrqdf, dlWosf AD n] ABC nfO{ cfwf ub{5 To;}n]
A
ABD sf] If]qkmn = ADC sf] If]qkmn

vi) Pp6} cfwf/ AB / pxL ;dfgfGt/

/]vfx¿ AB / CD lar /x]sf B D C
lqe'h ABE / ;dfgfGt/ rt'e{'h ABCD df lqe'hsf]
If]qkmn ;dfgfGt/ rt'e{'hsf] If]qkmnsf] cfwf x'G5 . cyf{t\
D E
C
ABE sf] If]qkmn = sf] If]qkmn
cyf{t\ 2 ∆ ABE sf] If]qkmn = ABCD sf] If]qkmn
v) prfO / b'/L ;DaGwL ;d:ofnfO{ ;dsf]0f lqe'h (right A B
angled triangle) df abnL lrq agfpg' cjZos x'G5 .
vi) lqe'hsf] If]qkmn lgsfNbf k|of]u x'g] sf]0f yfxf lbOPsf] 2 cf]6f e'hfsf] lardf k/] gk/]sf] ljrf/
ug'{kb{5 / olb lardf gk/]sf] eP To;nfO{ lardf kfg]{ pkfox¿sf] vf]hL ug'{kb{5 .
vii) lqe'hsf] If]qkmnsf] kl/df0f n]lv;s]kl5 PsfO n]Vg la;{g' x“'b}g, h:t} M ∆ABC = 10 ju{ ;]=ld=
10 ju{ PsfO cflb .

gd'g k|Zg pTt/

5f]6f] pTt/ cfpg] k|Zgx¿
gd'gf k|Zg 1 M C
lbOPsf] ∆ABC df a = 12 cm, c = / ∠ABC = 45 5 eg]
0
12cm
∆ABC sf] If]qkmn lgsfNg'xf];\ .
450
;dfwfg A B
15 cm
∆ABC sf] If]qkmn =

=
= 90cm2
P
cEof;sf nflu k|Zgx¿
4cm
1. lbOPsf] ∆PQR df olb PQ = 4 cm QR = 7cm / ∠PQR = 300 eP
∆PQR sf] If]qkmn lgsfNg'xf];\ . Ans: 7cm2 Q 30
0
R
7cm

230
2. lbOPsf] ∆PQR df PQ = 4 P
0
PR = 8cm / ∠QPR = 450 5 eg] 45
8cm
4 cm
∆PQR sf] If]qkmn lgsfNg'xf];\ . Ans: 16cm2
Q R

3. lbOPsf] lqrdf KM = 17cm M

∠MKP = 600 / KP = 20cm 17cm
eP ∆MPK sf] If]qkmn lgsfNg'xf];\ . Ans: 147.22cm2 60
0

K 20cm P

4. lbOPsf] lrq ∆XYZ df XY = YZ = 12 cm

X
/ ∠XYZ = 300 eP ∆XYZ sf] If]qkmn lgsfNg'xf];\ .
Ans: 36cm2 0
30
Z Y

5. lbOPsf] lrqdf a = 4cm, b = 6cm / ∠C = 600 eP ∆ABC sf]

If]qkmn lgsfNg'xf];\ . A
6cm
2 60
0
Ans: 10.39cm B C
4cm

gd'gf k|Zg g+= 2 : X

lbOPsf] lrqdf ∠XYZ = 300, yz=12cm / ∆ xyz sf] If]qkmn= 27 cm2 5

eg] e'hf XY sf] gfk lgsfNg'xf];\ . Z 30
0
Y
12cm
;dfwfg
lbOPsf] ∆XYZ sf] If]qkmn (A) =
or, 27 =
or, 27 = 3xy
or, 3xy = 27
or, xy =
A
or, xy = 9cm
cEof;sf nflu k|Zgx¿ 6

1. lbOPsf] lrqdf AB = 6 , 45
0
C
B
∠ABC = ∆45 / ∆ABC sf] If]qkmn = 24cm2
0

5 eg] e'hf BC sf] gfk kTtf nufpg'xf];\ . Ans: 8cm

X
0
30
10cm 231

Y Z
2. lbOPsf] lrqdf xz = 10cm , ∠yxz = 300
/ ∆xyz sf] If]qkmn = 24cm2 5 eg]
xy sf] gfk kTtf nufpg'xf];\ .
Ans: 9.6cm
Y
0
3. lbOPsf] ∆xyz df olb xy = 14cm 60
14cm
∠xyz = 600 / ∆xyz sf] If]qkmn = 28 eP Z X
e'hf yz sf] nDafO kTtf nufpg'xf];\ . Ans: 8cm
A
4. lbOPsf] ∆ABC df BC = 12 cm, AB = AC,
∠B = 300 / ∆ABC sf] If]qkmn 27cm2 5 0
30
eg] e'hf AC sf] nDafO kTtf nufpg'xf];\ . B 12cm C

[Hint:(∆ABC ;dflåjfx' lqe'h ePsfn] o;sf cfwf/sf sf]0fx¿ a/fa/ x'G5 cyf{t\ ∠B =
∠C = 300 x'G5 To;kl5 ∆ABC sf] If]qkmnsf] ;"q k|of]u ug]{)] Ans: 12cm

5. ;“u} lbOPsf] ∆ABC df

∠A + ∠B = 1500, AC = 8cm / A
∆ABC sf] If]qkmn 24cm2 eP
8cm
e'hf BC sf] gfk kTtf nufpg'xf];\ . Ans: 12cm [Hint :
(;'?df ∠C sf] dfg kTtf nufpg] / To;kl5 ∆ABC sf] If]qkmnsf] ;"q B C
nufpg])]

A
gd'gf k|Zg g+= 3 : 6cm
;“u} lbOPsf] lrqdf AB = 6cm, BC = 8cm
B
/ ∆ABC sf] If]qkmn = 12 cm2 eP
8cm C
∠ABC sf] gfk kTtf nufpg'xf];\ .
;dfwfg
∆ABC sf] If]qkmn (A) =
Or, 12 =
Or, 12 = 24 sinABC

232
Or, 24 SinABC = 12

Or, SinABC =

Or, SinABC =
Or, Sin ABC = sin600
∴ ∠ABC = 600
P
cEof;sf nflu k|Zgx¿
1. lbOPsf] lqe'hdf PQ = 10cm, PR = 18cm 10cm
18cm

/ ∆PQR sf] If]qkmn = 45 eP

Q
∠QPR sf] dfg kTtf nufpg'xf];\ . Ans:600 R

D
2. lbOPsf] ∆DEF df DF = 7cm,
7cm
EF = 10cm, ∆DEF sf] If]qkmn =
0
50
/ ∠DEF = 500 eP ∠EDF sf] dfg kTtf nufpg'xf];\ . E 10cm
F

Hint: ;'?df ∆ sf] If]qkmnsf] ;"qaf6 Ans:700

∠DEF kTtf nufpg] To;kl5 clGtddf ∠EDF sf] dfg kTtf nufpg'xf];\ .
P
3. ;“u}sf] lrqdf PQ = 7cm, 7cm
QR = 16cm / ∆PQR sf] If]qkmn = 5
Q R
16cm
eg] ∠PQR sf] dfg kTtf nufpg'xf];\ . Ans:600

4. lbOPsf] ∆ LMN df LN= 7cm, MN = 10cm / ∆ L

7cm
LMN sf] If]qkmn = eP ∠MNL sf] dfg
lgsfNg'xf];\ . Ans: 600 M
? N
10cm

9 cm P
gd'gf k|Zg g+= 4 : S

lbOPsf] lrqdf PQRS Pp6f ;dfGgfGt/ rt'e{'h

xf] h;df QR nfO{ T laGb';Dd a9fPsf] 5 . 6 cm
?
olb ;dfgfGt/ rt'e{'hsf] If]qkmn = 27 T R Q

233
e'hf QP = 6cm / e'hdf PS = 9cm
eP ∠SRT sf] gfk kTtf nufpg'xf];\ .
;dfwfg
lbPsf], = ;=r= PQRS 5 . (A) = 27
PQ = 6cm = SR cyf{t PQ = SR =6cm
To:t}, PS = 9cm = RQ
cyf{t\ PS = RQ = 9cm
/rgf M ljs0f{ SQ nfO{ hf]8f“} .
ca,
;=r= PQRS sf] If]qkmn (A) = 2∆SRQ sf] If]= (;=r=sf] ljs{0fn] ;=r= nfO{ cfwf ug]{ ePsfn] ._
or, 27
or, 27 = 6 x 9 x sin SRQ
or, 27 = 54 sin SRQ
or, 54 sinSRQ = 27
or, 54 sinSRQ =27

or, sin SRQ =

or, sin SRQ =

or, sin SRQ = sin600
∴ ∠SRQ = 600
ca km]l/ ∠SRQ + ∠SRT = 1800 -;/n sf]0f 1800 x'g] ePsfn] ._
or, 600 +∠SRT = 1800
or, ∠SRT = 1800 - 600 = 1200

cEo;sf nflu k|Zgx¿ D C

1. lbOPsf] lrqdf ;dfgfGt/ rt'e{'h ABCD sf]
If]qkmn = 81 5 . olb BC = f
/ AB = eP ∠DAB sf] gfk kTtf nufpg'xf];\ . ? f
0 A B
Ans: 60 ff

2. lbOPsf] ;dfgfGt/ rt'e'{h ∆ABCD sf] If]qkmn D

600 C
kTtf nufpg'xf];\ h;df AD = 6cm, AB = 8cm 6cm
/ ∠ BCA = 60 5 .
0
Ans: 24
A 8cm B

234
3. ;“u}sf] lrqdf ;dfgfGt/
S R
rt'e'{h PQRS sf] If]qkmn = 5.
olb ∠P = 450 / QR = eP
e'hf SR sf] nDafO kTtf nufpg'xf];\ . 450
P Q
Ans: 7

gd'gf k|Zg 5 : P M
Q
lbOPsf] lrqdf 60 ju{ ;]=ld= If]qkmn ePsf]
;dfgfGt/ rt'e{'h PQRS / ∆MRS Pp6}
cfwf/ SR df /]xsf 5g\ . olb SM = S 8cm R

SR = 8cm eP ∠AMR sf] dfg lgsfNg'xf];\ .

;dfwfg
;=r= PQRS sf] If]qkmn (A) = 2∆SMR sf] If]qkmn -Pp6} cfwf/ / pxL ;dfgfGt/ /]vfx¿lar /x]sf
lqe'h / ;dfgfGt/ rt'e'{hdf ;+=r= sf] If]qkmn lqe'hsf] If]qkmnsf] 2 u'0ff;“u a/fa/ x'g] ePsfn] ._
or, 60 =
or, 60 = 5 x 8 x sin MSR
or, 60 = 40 sin MSR
or, 40 sin MSR = 60
or, sin MSR =

or,Sin MSR =

or, Sin MSR =

or, Sin MSR =

or, Sin MSR = sin6 00
∴∠MSR = 600

A P
cEof;sf nflu k|Zg D
1. lbOPsf] lrqdf 18 ju{ ;]=ld= If]qkmn ePsf]
;dfgfGt/ rt'e'{h ABCD / ∆BPC Pp6}
B 6cm C
cfwf/ BC df /x]sf 5g\ . olb BC = 6cm
/ BP = 6 cm eP ∠PBC lgsfNg'xf];\ . Ans: 300

235
 A D
2. lbOPsf] lrqdf 80 ju{ ;]=ld= If]qkmn ePsf]
;dfgfGt/ rt'e'{h ABCD / lqe'h PCD Pp6} P
8cm
cfwf/ DC df /x]sf 5g\ . olb PC = 8cm / PD = 10 cm B C
0
eP ∠CPD lgsfNg'xf];\ . Ans: 45
A E
1000 D
3. lbOPsf] lrqaf6 ;dfgfGt/ rt'e'{h ABCD sf]
If]qkmn kTtf nufpg'xf];\ . hxf“ BC = 10cm, 6cm
100

CE = 6cm, ∠DCE = 100 / ∠BAD = 1000 5g\ . B

2 10cm C
Ans: 60cm
[Hint:∠BAD =∠ BCD = 1000 (= ;=r=sf] ;Dd'vsf]0fx¿ a/fa/ x'g] ePsfn] . )
t/ ∠ECD = 100
To;}n] ∠ BCE = 100 - 100
= 900
ca dflysf] h:t} u/L ;=r= ABCD sf] If]qkmn lgsfNg] . ]
gd'gf k|Zg g+= 6 :
D C
lbOPsf] lrqdf ABCD Pp6f ;dafx' rt'e'{h xf] h;sf] If]qkmn 18 cm2 5
/ ∠ABC = 600 5 eg] e'hf AB sf] gfk kTtf nufpg'xf];\ . 600
;dfwfg A B
ca,
;dafx' rt'e'{h ABCD sf] If]qkmn = 2∆ABC sf] If]=
or, 18 =2x

or, 18 = AB x AB x [ AB = BC]

or, 18 = (AB)2

or, (AB)2 = 18

or, (AB)2 =

or, (AB)2 = 36
or, (AB)2 = (6)2
∴ AB = 6cm

236
cEof;sf nflu k|Zgx¿ S
R
1. lbOPsf] lrqdf PQRS Pp6f ;dafx' rt'e'{h xf] .
450 10 m
olb PQ = 10 cm / ∠PQR = 450 eP
pSt ;dafx' rt'{e'h PQRS sf] If]qkmn kTtf nufpg'xf];\ . P Q

Ans: 100 cm2

K N
2. lbOPsf] lrqdf KLMN Pp6f ;dafx' rt'e'{h xf] .
h;sf] If]qkmn 72 cm2 5 / ∠NML = 1800 eP 1200
L
e'hf KL sf] gfk lgsfNg'xf];\ . M
Ans: 12cm
gd'gf k|Zg g+= 7 :
lbOPsf] lrqdf ∆BPC Pp6f ;dafx' ∆ xf] . / ABCD Pp6f ;dfafx' rt'e'{h D C
2
xf] . olb APCD sf] If]qkmn 27 cm eP AB sf] gfk kTtf
nufpg'xf];\ .
A B P
;dfwfg M /rg M ljs0f{ BD hf]8f}“ .
∆BPC ;dafx' ∆ePsfn] ∠B = ∠C = ∠P = 600
km]/L, ∠B = ∠A = 600 ;+ut sf]0f ePsfn] .
;djfx' rt'e{'h ABCD sf] If]= (A1) = 2∆ABD sf] If]=
= 2x

= AB x AB x [ AD = AB]

= (AB)2
To:t}, ;dafx' ∆BPC sf] If]= (A2) =
=

=
k|Zgfg';f/,
APCD sf] If]= (A) = A1 + A2
or, 27 = (AB)2 + (AB)2

or, 27 =( + (AB)2

or, 27 =( (AB)2

or, (AB)2 = 27

237
or, (AB)2 =

or, (AB)2 =
or, (AB)2 = 36
or, (AB)2 = (6)2
or, AB = 6cm.
cyjf j}slNks tl/sf
APCD sf] If]qkmn (A) = 3 ∆ABCD sf] If]qkmn [∆APCD = ∆ABD + ∆BCD + ∆BPC
= ∆ABC+∆ABD+∆ABD
=3∆ABD]
Or, 27

Or, 27

Or, 27 (AB)2
Or, 3 (AB)2 = 108
Or, (AB)2 =
Or, (AB)2 = 36
Or, (AB)2 = (6)2
∴ AB = 6cm.

cEof;sf nflu k|Zgx¿ 8cm

P S
1. lbOPsf] lrqdf PQRS Ps ;dafx' rt'e'{h / ∆RTS Ps ;dfafx'
lqe'h eP ;dnDa rt'e'{h PQTS sf] If]qkmn kTtf nufpg'xf];\ .
Q R T
Ans: 48

T
2. ;“u}sf] lrqdf ∆QSR Pp6f ;dafx' lqe'h xf] / PQST Pp6f
;dafx' rt'e{'h xf] . olb ;dnDa rt'e{'h PQRT sf] If]qkmn P
48 ju{ ;]=ld= 5 eg] PT sf] nDafO kTtf nufpg'xf];\ . S

Ans: 8cm
Q
R

238
gd'gf k|Zg g+= 8 :
lbPsf] rt'e{'h ABCD sf] If]qkmn kTtf A
nufpg'xf];\ h;df AB = BC, AD = CD = 3cm
BD = 5cm / ∠BDC = 300 5g\ . B 300
D

;dfwfg
1. e'hf AB = BC – yfxf lbOPsf] . C

(i) e'hf AD = e'hf CD – yfxf lbPsf] .

(ii) e'hf BD = e'hf BD – ;femf e'hf ePsfn] .
(iii) 1. ABD ≅ BCD – e'= e'= e'= tYocg';f/ .

2. ∆ABD = ∆BCD – cg'¿k lqe'hsf] If]qkmn a/fa/ x'g] ePsfn] .

ca, ∆BCD sf] If]qkmn =

=
=
ca, k|Zgcg';f/
rt'e{'h ABCD sf] If]qkmn = 2 ∆BCD sf] If]qkmn
=2x
=
= 7.5 cm2

cEof;sf nflu k|Zgx¿ P

1. lbOPsf] rt'e'{h PQRS sf] If]qkmn
Q S
kTtf nufpg'xf];\ h;df PQ = PS, QR = ST = 8cm,
PR = 12 cm / ∠PRQ = 300 5 . Ans: 48cm2 300

R
2. lbOPsf] lrqdf rª\ufsf] If]qkmn
A
lgsfNg'xf];\ h;df AB = AD,
BC = CD = 27cm, AC = 30 cm B D
/ ∠ACD = 300 5 . Ans: 405cm 2
300

239
 gd'gf k|Zg g+=9:
lbOPsf] lrqdf BD dlWosf xf] . olb BD = 6cm
BC = 10cm / ∠DBC = 300 eP A
∆ABC sf] If]qkmn kTtf nufpg'xf];\ .
D
;dfwfg
∆ABD = ∆BDC – dlWosf BD n] ∆ABC nfO{ cfwf ug]{ ePsfn] . B 300
ca, 10cm C

∆BDC sf] If]qkmn =

=
=
ca k|Zgfg';f/, ∆ABC sf]=If]==2∆BDC sf
A
=2×15
0
= 30cm2 30
8cm
cEof;sf nflu k|Zgx¿ 6cm
fff
1. lbOPsf] lrqdf AD dlWosf xf] . olb ∠CAD = 300 AD = C
B D
6cm / AC = 8cm eP ∆ABC sf] If]qkmn kTtf
nufpg'xf];\ .
Ans: 24cm2 Q

2. lbOPsf] lrqdf PS dlWosf xf] . olb SP=4cm, 60

0
P
S 4cm
PQ= cm / ∠QPS=600 eP ∆PQR sf] If]qkmn
kTtf nufpg'xf];\ .
R
P
Ans: 6cm2 0
98
yk cEof;sf nflu ldl>t k|Zgx¿
1. lbOPsf] ∆PQR df ∠QPR = 980, ∠PQR = 370 0
37
Q R
PR = 6 cm / QR = 9 cm 5g\ eg] ∆PQR sf] If]qkmn 9cm
kTtf nufpg'xf];\ . Ans: 27cm2
A
2. ;“u} lbOPsf] lrqdf AB = 6cm, BC = 8cm / ∆ABC sf]
If]qkmn 12 j=;]=ld= eP ∠ABC sf] gfk kTtf
nufpg'xf];\ .
B C
Ans: 300 8 cm

240
 C
5x 16 cm
3. ∆ABC sf] If]qkmn 3 j=;]=ld= e'hf BC sf] nDafO 3cm /
0
∠ACB = 60 eP e'hf AC sf] nDafO kTtf nufpg'xf];\ . 3x 4x
B
A 20 cm
Ans: 4cm

Aa 12 cm
Ba
4. ;“u}sf] lrqdf ABCD Ps ;dafx' rt'e{'h xf] . olb AB = 12cm
/ rt'e'{h ABCD sf] If]qkmn 72 j=;]=ld= eP ∠BCD sf] gfk
kTtf nufpg'xf];\ . Ans: 300 Da Ca

5. lbOPsf] ∆ABC df BC = 16cm, AB = 20cm eP

W X
∆ABC sf] If]qkmn kTtf nufpg'xf];\ . 0
60
Ans:

T Z Y
6. lbOPsf] lrqdf wxyz Ps ;dafx' rt'e{'h / ∆WTZ Ps
;dafx' lqe'h eP ;dnDa rt'e'{h XYTW sf] If]qkmn kTtf nufpg'xf];\ .
Ans:

7. lbOPsf] lrqdf 64 j= ;]=ld= If]qkmn ePsf] ;dfgfGt/ rt'e{'h P

A D
ABCD / lqe'h BPCPp6} cfwf/ / BC df /x]sf 5g\ . olb BC =
8cm / BP = 8 cm eP ∠PBC lgsfNg'xf];\ . Ans: 450
B 8cm C

8. lbOPsf] ;dfgfGt/ rt'e'{h ABCD df A D

2
∆ADE sf] If]qkmn lgsfNg'xf];\ . Ans: 7.5cm
5cm
0
30
B 6cm C

9. ;“u}sf] lrqdf PQRS Pp6f ;dafx' rt'e{'h xf]

R
h;sf] If]qkmn 18 j=;]=dL= 5 . olb ∠PQR = 300 eP PS S
sf] nDafO kTtf nufpg'xf];\ .
Ans: 6cm 30
0
Q
P

241
 A D
10. lrqdf lbOPsf] ;dfgfGt/ rt'e{'h ABCD 70
0
10cm
sf] If]qkmn kTtf nufpg'xf];\ .
Ans: 65
0
B 50 13cm C

11. lbOPsf] lrqdf PQ = 12cm, P

0
QR = 16 cm, Rs 8cm, ∠PQR = 60 / 12cm
∠PSR = 300 5 . olb ∆PQR = 4∆PSR eP PS
0
30 S
8cm
R
sf] dfg kTtf nufpg'xf];\ . 0
60
Q
Ans: 18cm

12. lbOPsf] lrqdf AD//BC 5 . olb AD = 10cm, AB = 8c, D C

∠DAB = 300 / ∠DBC = 600 5 . olb ∆DBC sf]

If]qkmn = 2∆ABD sf] If]qkmn eP BC 10cm
sf] nDafO kTtf nufpg'xf];\ . 0
60
[Hint: ∠ADB = ∠DBC = 600 PsfGt/ sf]0f ePsfn] . 30
0
A
∴ ∠ABD = 900 x'G5 ca ;dsf]0f ∆ABD af6 BD e'hf kTtf nufO{ 8cm

;dfwfg ug]{ . ]
Ans: 15.39cm

nfdf] pTt/ cfpg] k|Zgx¿ X 0

A
30
gd'gf k|Zg g+= 1 :
91 m cUnf] Pp6f w/x/fsf] 6'Kkf]af6 ;f]xL ;dtndf /x]sf] 12m 0
C 30 E
cUnf] 3/sf] 5tdf x]bf{ 300 sf] cjglt sf]0f kfOof] eg] w/x/f /
91cm
3/ larsf] b'/L kTtf nufpg'xf];\ . 12cm
;dfwfg
D B
dfgf“} AB Pp6f w/x/f xf], h;sf] 6'Kkf] A / km]b B 5 . To;t}
CD Pp6f 3/ xf] pSt 3/sf] 5t C df w/x/fsf] 6'Kkf]af6 x]bf{
cjglt sf]0f ∠CAX = 300 agfPsf] 5 .
k|Zgg';f/,
AB = 91m
BE = CD = 12m
AE = AB – BE
= 91 – 12
= 79m
∠CAX = ∠ACE = 300
242
(PsfGt/ sf]0f ePsfn])w/x/f / 3/ larsf] b'/L (CE = BD = ?)
;dsf]0f ∆ACE af6
Tan 300 =

Or,

Or, CE = 79
= 79 x 1.732
= 136.83m
∴ w/x/f / 3/ larsf] b'/L = 136.83m=

cEof;sf nflu k|Zg

1. 2m cUnf] dflg;n] Pp6f vDafsf] 6'Kkf] x]bf{ 300 sf] pGgtf+z sf]0f kfP5 . olb vDafsf]
prfO 21m 5 eg] ;f] dflg; / vDaf larsf] b'/L kTtf nufpg'xf];\ . Ans: 32.91cm
2. b'O{ cf]6f 3/ xm sf] km/sdf 5g\ . bf];|f] 3/sf] 6'Kkf]af6 klxnf] 3/sf] 6'Kkf]sf] cjglt sf]0f
450 5 . olb klxnf] 3/sf] prfO 30m 5 / bf];|f] 3/sf] prfO 60m eP x sf] dfg kTtf
nufpg'xf];\ . Ans: 30m

gd'gf k|Zg g+= 2 :

1.7m cUnf] Pp6f s]6fn] p8fO/x]sf] rª\ufsf] wfuf]sf] nDafO 180m x'“bf lIflth;“u o;n] 300 sf]
sf]0f agfp“5 eg] hldgb]lv rª\uf ;Ddsf] prfO slt x'G5 <
;dfwfg E
dfgf} AB Pp6f dflg; xf] / rª\ufsf] wfufsf] 180m
nDafO AE xf] / rª\ufsf] wfufn] lIlth;“u 300
A 300 D
sf] pGGtf+z sf]0f agfPsf] 5 .
k|Zgg';f/, AB = CD = 1.7cm 1.7m
B C
AE = 180m
∠EAD = 300
hldgb]lv rª\uf ;Ddsf] prfO (CE) = ?
;dsf]0f ∆ADE af6
Sin 300 =

Or, =
Or, DE = 180
Or, 2DE =
Or, DE = 90m
243
∴ CE = CD + DE
= (1.7 + 90)m
= 91.7m
∴ hldgb]lv rª\uf;Ddsf] prfO (CE) = 91.7m

cEof;sf nflu k|Zg

1. 2m cUnf] dflg;n] Pp6f :tDesf] 6'Kkf]df x]bf{ pGgftf+z sf]0f 600 kfP5 . olb :tDe /
dflg; larsf] b'/L 45m eP :tDesf] prfO kTtf nufpg'xf];\ . (79.94m)
2. 1.7m cUnf] dflg;n] Pp6f :tDesf] 6'Kkf] cjnf]sg ubf{ pGgtf+z sf]0f 600 kfOof] . olb
:tDe / dflg; larsf] b'/L 25m eP :tDesf] prfO kTtf nufpg'xf];\ . Ans:45m

gd'gf k|Zg g+= 3 : D

Pp6f dflg;n] cfkm"eGbf cufl8sf] 52m cUnf] vDafsf] 6'Kkf]df x]bf{
pGGtf+z sf]0f 300 kfP5 . olb dflg; / vDa larsf] b'/L 86m 5 eg] A 300
52m
86m
To; dflg;sf] prfO kTtf nufpg'xf];\ .
B 86m C

;dfwfg
dfgf} AB Pp6f dflg; xf] h:t} CD Pp6f
vDaf xf] h;sf] 6'Kkf] D df x]bf{
pGgtf+z sf]0f 300 ag]sf] 5 .
k|Zgg';f/,
dflg; / vDaf larsf] b'/L (AE = BC = 86m)
vDafsf] prfO (CD) = 52m
∠DAE = 300
dflg;sf] prfO AB = EC = ?
;dsf]0f AED af6
Tan300 =

Or, =

Or, DE = 86
Or, DE =

Or, DE =
Or, DE = 49.65m
∴ EC = DC - DE

244
= (52 + 49.65)m
= 2.35m
∴ dflg;sf] prfO (AB = EC) = 2.35m

cEof;sf nflu k|Zg

1. 26m cUnf] Pp6f vDafsf] 6'Kkf]af6 To; vDafsf] l7s cufl8 15 b'/Ldf /x]sf] Pp6f
0
3/sf] 6'Kkf]nfO{ cjnf]sg ubf{ cjglt sf]0f 30 kfOof] eg] 3/sf] prfO slt xf]nf <
Ans:11m
2. b'O{ :tDe 60m sf] b'/Ldf 5g\ . cUnf] :tDesf] prfO 150m 5 . pSt :tDesf] 6'Kkf]af6
xf]rf] :tDesf] 6'Kkfsf] cjgltsf]0f 300 eP xf]rf] :tDesf] A
prfO kTtf nufpg'xf];\ .
Ans:115.35m

gd'gf k|Zg g+= 4 :

Ps hgf dflg;n] 80 cUnf] Pp6f w/x/fsf] 6'Kkf]nfO{ ;f] ?
C 240m B
w/x/fsf] km]bb]lv 240m k/ af6 x]bf{ aGg] pGgtf+z sf]0f kTtf
nufpg'xf];\ .
;dfwfg
dfgf} AB Pp6f w/x/f xf] h;sf] 6'Kkf] A nfO{ s'g} dflg;n] C laGb'af6 x]bf{ aGg] pGGtf+z
sf]0f∠ACB xf] .

k|Zgg';f/,
AB = 80
BC = 240m
∠ACB = ?
;dsf]0f ∆ABC af6
Tan ACB =

Or, Tan ACB =

Or, Tan ACB =

Or, Tan ACB =

Or, Tan ACB =

245
Or, Tan ACB = Tan 300
∴ ∠ACB = 300
∴ pGGt+fzsf]0f = 300

cEof;sf nflu k|Zgx¿

1. 10m cUnf] :yfgaf6 ;f]xL ;dtndf kg]{ l7s cufl8 /x]sf] 30m cUnf] :tDesf] 6'Kkfdf x]bf{
x0 pGgtf+z sf]0f kfOof] . olb pSt :yfg / :tDesf] b'/L 20 m eP x0 sf] dfg kTtf
nufpg'xf];\ . Ans:300
2. 16m cUnf] :yfgaf6 ;f]xL ;dtndf kg]{ l7s cufl8 /x]sf] 41m cUnf] :tDesf] 6'Kkfdf x]bf{
x0 sf] pGgtf+z sf]0f agfPsf] 5 . olb pSt :yfg / :tDesf] b'/L 25 m eP x0 sf] dfg
kTtf nufpg'xf];\ . Ans:300

gd'gf k|Zg g+= 5 :

9m cUnf] 3/sf] 5fgfdf 1.5m cUnf] dflg;n] rª\uf p8fO/x]sf] lyof] / A

rª\uf hldgb]lv 58m prfOdf lyof] olb To; rª\ufsf] wfuf]n] lIflth;Fu
300 sf] sf]0f agfPsf] /x]5 eg] ;f] rª\ufsf] wfuf]sf] nDafO kTtf
nufpg'xf];\ . C 300 B
58m

9m 1.5m
D E
;dfwfg
G F
dfgf} AF hldgb]lv rª\uf;Ddsf] prfO xf] .

CD dflg;sf] prfO xf], DG 3/sf] prfO xf] .

AC rª\ufsf] wfufsf] nDafO xf] .
k|Zgg';f/,
DG = EF = 9cm
CD = BF= 1.5m
AF = 58cm
AB = 58m
AB = AF – EF – BE
= 58 – 9 – 1.5
= 47. 5m
∠ACB = 300
;dsf]0f ABC af6
Sin 300 =

246
Or,
Or, AC = 95 m
rª\ufsf] wfufsf] nDafO (AC) = 95m

cEof;sf nflu k|Zgx¿

1= 5 m cUnf] 3/sf] 5fgfdf m cUnf] dflg;n] rª\uf p8fO/x]sf] lyof] / rª\uf
hldgb]lv 20 m prfOdf lyof] . olb To; rª\ufsf] wfufn] lIflth;“u 450 sf] sf]0f
agfPsf] /x]5 eg] ;f] rª\ufsf] wfufsf] nDafO kTtf nufpg'xf];\ . Ans:34.29m
2. 6m cUnf] 3/sf] 5fgfdf 1.6m cUnf] dflg;n] rª\uf p8fO/x]sf] lyof] / rª\uf hldgb]lv
60m prfOdf lyof] . olb To; rª\ufsf] wfuf]n] lIflth;Fu 300 sf] sf]0f agfPsf] /x]5 eg]
/f] rª\ufsf] wfufsf] nDafO{ kTtf nufpg'xf];\ . Ans:104.8m
gd'gf k|Zg g+= 6 :
Pp6f 18m cUnf] ?v xfjfn] ef“lrP/ 6'Kkfn] hldgdf x'“bf 300 sf] sf]0f ag]sf] 5 . pSt ?vsf]
ef“lrPsf] efusf] nDafO kTtf nufpg'xf];\ .
;dfwfg A
dfgf“} AC pp6f ?v xf] h'g xfjfn] C laGb'df ef“lrP/ x
ef“lrPsf] efu AC n] hldg sf] D laGb'df C
0
5'bf 30 sf]0f agfPsf] 5 . 18m
k|Zgg';f/, x

AB = 18m 300
D B
∠CDB = 300
ef“lrPsf] efusf] nDafO (AC = CD = ?)
dfgf“} AC = CD = xm df 5 .
BC = AB – AC
= (18 - x)
ca ;dsf]0f ∆BDC af6
Sin 300 =

Or,
Or, 36 – 2x = x
Or, 36 = x + 2x
Or, 36 = 3x
Or, x =
Or, x = 12

247
ef“lrPsf] efusf] nDafO (AC) = (CD) = 12m

cEof;sf nflu k|Zgx¿

1. Pp6f 14m cUnf] ?v xfjfn] ef“lrP/ 6'Kkf]n] hldgdf 5'bf hldg;“u 600 sf] sf]0f ag]sf]
5 . ?vsf] ef“lrPsf] efusf] nDafO kTtf nufpg'xf];\ . Ans:7.50 m
2. Pp6f ?v cf“wLn] ef“lrP/ hldg;“u ;Dsf]0f lqe'h agfp“5 . ?vsf] ef“lrPsf] efun] hldg;“u
600 sf] sf]0f agfp“5 . olb clxn] ?vsf] 6'Kkf] o;sf] km]baf6 15m 6f9f 5 eg] ?v slt
cUnf] lyof] < Ans:55.98m
3. x'/Ln] Pp6f l;wf ?v ef“lrbf o;sf] 6'Kkfn] km]baf6 4 m 6f9f hldgdf 5f]P/ 300 sf]
sf]0f agfp“5 eg] ef“lrg'eGbf klxnfsf] ?vsf] prfO kTtf nufpg'xf];\ . Ans:12m

gd'gf k|Zg g+= 7 : A

Pp6f j[Ttsf/ rf}/sf] dWo efudf Pp6f 60 m cUnf] vDaf

ufl8Psf] 5 . ;f] rf}/sf] kl/lwsf] s'g} Pp6f laGb'af6 ;f] vDafsf]
6'Kkfdf x]bf{ 300 sf] pGgtf+z sf]0f aG5 eg] ;f] rf}/sf] Jof;sf]
300
nDafO kTtf nufpg'xf];\ . C
0 B

;dfwfg
dfgf“} j[Ttsf/ rf}/sf] dWo efu 0 5 .
pSt dWo efudf ufl8Psf] vDaf -AO) 5 .
rf}/sf] kl/lwsf] B laGb'af6 vDafsf] 6'Kkf]
A df x]bf{ 300 sf] pGgtf+z sf]0f ag]sf] 5 .
k|Zgg';f/,
AO = 60 m
0
∠ABO = 30
Jof; (BC) = ?
;dsf]0f ∆AOB af6
Tan 300 =

Or,

Or, BO = 60 x 2
Or, BO = 60 x 3
Or, BO = 180m
ca k|Zgg';f/ Jof; (BC) = 2BO (d = 2r)
= 2 x 180
248
= 360m

cEof;sf nflu k|Zgx¿

1= Pp6f j[Ttsf/ kf]vf/Lsf] Jof; 60 m 5 . pSt kf]vf/Lsf] lardf ufl8Psf] vDafsf] 6'Kkfdf
;f] kf]vfl/sf] lsgf/faf6 x]bf{ 300 sf] pGgtf+z sf]0f aG5 eg] kfgLsf] ;txb]lv vDafsf]
6'Kkf] ;Ddsf] prfO kTtf nufpg'xf];\ . Ans:17.32m
2= Pp6f j[Ttsf/ kf]v/Lsf] s]Gb|df Pp6f vDafsf] 7f8f] pEofOPsf] 5 . vDafsf] 6'Kkf] kfgLsf]
;txaf6 30m prfOdf 5 . kf]vf/Lsf] kl/lwsf] Ps laGb'af6 o;sf] pGgtf+z sf]0f 600 eP
pSt kf]v/Lsf] cw{Jof;sf] nDafO kTtf nufpg'xf];\ . Ans:17.32m
3. Pp6f j[Ttsf/ kf]v/Lsf] s]Gb|df kflgsf] ;txb]lv dfly 11.62m cUnf] vDaf kf]v/Lsf]
lsgf/fsf] s'g} laGb'af6 1.62m cUnf] dflg;n] vDafsf]
A
6'Kkf] cjnf]sg ubf{ pGgtf+z sf]0f 300 kfof] eg] ;f]
kf]v/Lsf] Jof; slt xf]nf <
Ans:34.64m

gd'gf k|Zg g+= 8:

s'g} ?vsf] 5fofsf] nDafO ;"o{sf] prfO 450 ePsf] ;dodf ;"o{sf]
prfO 600 ePsf] ;dodf eGbf 20m nfdf] 5 eg] ;f] ?vsf] A
450
0
60
prfO kTtf nufpg'xf];\ . D 25m C B
xm
;dfwfg M x + 20m
dfgf} AB Pp6f ?v xf], BC / BD o;sf 5fofx¿ x'g\ .
k|Zgg';f/,
∠ACB = 600
∠ADB = 450 + BC = x -dfgf}“_
BD = (x + 20) m
?vsf] prfO (AB) = ?
;dsf]0f ∆ABC af6
Tan600 =

Or, =

Or, AB = ……… (i)

km]l/ ;ds0f ∆ABD af6
Tan 450 =

Or, 1 =

249
Or, 1 =
Or, = x + 20
Or, – x= 20
Or, = 20
Or,

Or,

Or,
Or,
∴ ?vsf] prfO (AB) =
= 27.32 x 1.732
= 47.32m

cEof;sf nflu k|Zgx¿

1. 30m cUnf] s'g} ?vsf] 5fofsf] nDafO ;"o{sf] prfO 450 ePsf] ;dodf ;"o{sf] prfO 600
ePsf] ;dodfeGbf x m nfdf] x'G5 eg] x sf] dfg kTtf nufpg'xf];\ . Ans:12.68m
2. xm cUnf] s'g} ?vsf] 5fofsf] nDafO ;"o{sf] prfO 300 ePsf] ;dodf ;"o{sf] prfO 450
ePsf] ;dodfeGbf 21.96m n] a9L x'G5 eg] x sf] dfg kTtf nufpg'xf];\ . Ans:30m
3. 30 cUnf] s'g} ?vsf] 5fofsf] nDAffO ;"o{sf] prfO 300 ePsf] ;dodf ;"o{sf] prfO
0
60 ePsf] ;dodfeGbf slt ld6/n] a9L x'G5 < Ans:6m

gd'gf k|Zg g+= 9 : A

1m cUnf] Pp6f s]6f aTtLsf] vDafaf6 6m 6f9f 5 . pSt s]6fsf] 5fof C

nfdf] 5 eg] aTtLsf] vDafsf] prfO kTtf nufpg'xf];\ . ?
1m
;dfwfg
dfgf“} AB Pp6f aTtLsf] vDaf xf] / CD Pp6f dflg; xf] . D 6m B
E
k|Zgg';f/, m
CD = 1m
BD = 6m
ED =
AB = ?
;dsf]0f ∆CDE af6

250
TanE =

Or,TanE=
Or, TanE = Tan 300
∴∠E = 300
km]l/ ;dsf]0f ∆ABE af6
TanE =

Or, Tan 300 =

Or,

Or, AB =

Or, AB =

Or, AB =

Or, AB =
Or, AB = 4.46m
alQsf] vDafsf] prfO = 4.46m

cEof;sf nflu k|Zgx¿

1. 1.5m cUnf] dflg;sf] 5fof 3m x“'bf 15m cUnf] ejgsf] 5fofsf] nDAffO kTtf
nufpg'xf];\ . Ans:30m
@ 12 lkm6 cUnf] j:t'sf] 5fof 24 lkm6 x'“bf 21 lkm6 cUnf] j:t'sf] 5fofsf] nDafO kTtf
nufpg'xf];\ . Ans:21 feet

yk cEof;sf nfuL ldl>t k|Zgx¿

1. b'O{ cf]6f 3/x¿ 45m sf] km/sdf 5g\ . bf];|f] 3/sf] 6'Kkfaf6 klxnf] 3/sf] 6'Kkfsf]
cjgltsf]0f 450 5 . olb klxnf] 3/sf] prfO 20m eP bf];|f] 3/sf] prfO kTtf
nufpg'xf];\ . Ans:65m
2. 1.5m prfO ePsf] Pp6f s]6fn] rª\uf p8fO/fv]sf] 5 . wfuf]sf] nDAffO 200m ePsf] a]nf
rª\ufsf] wfufn] lIflth;“u 300 sf] sf]0f agfp“5 eg] hldgaf6 rª\ufsf] prfO lgsfNg'xf];\ .
Ans:101.5m

251
3. s'g} cjnf]sgstf{ cfgf] 8m cUnf] 3/sf] dflyNnf] tnfaf6 l7s cufl8 /x]sf] dlGb/sf]
6'KkfnfO{ x]bf{ 600 sf] pGgtf+z sf]0f kfP5 . olb 3/ / cjnf]sgstf{ larsf] b'/L 15
eP dlGb/sf] prfO kTtf nufpg'xf];\ . Ans:53m
4. gbLsf] Ps lsg/df 60 lkm6 cUnf] 6fj/ 5 . gbLsf] csf]{ lsgf/af6 6fj/sf] 6'Kkf] x]bf{
pGgtf+z sf]0f x /x]5 . olb gbLsf] rf}8fO 120 lkm6 eP x0 sf] dfg kTtf nufpg'xf];\ .
0

Ans:600
5. 18 lkm6 cUnf] 3/sf] 5fgfdf 6 lkm6 cUnf] dflg;n] rª\uf p8fO/x]sf] lyof] / rª\uf
hldgb]lv x lkm6 prfOdf lyof] . olb To; rª\ufsf] wfuf]n] lIflth ;Fu 300 sf] sf]0f
agfPsf] /x]5 . olb wfufsf] nDafO 52 lkm6 eP x sf] dfg kTtf nufpg'xf];\ . (50 lkm6)
6. Pp6f ?v xfjfn] ef“lrP/ 6'Kkfn] hldg;“u 450 sf] sf]0f agfp“5 . pSt ef“lrP/ af“sL /x]sf]
efusf] prfO 18 lkm6 eP k'/f ?vsf] prfO kTtf nufpg'xf];\ . (36 lkm6)
7. j[Ttsf/ kf]v/Lsf] s]Gb|df kfgLsf] ;txb]lv dfly 11.62m cUnf] vDaf 5 . kf]vf/Lsf]
lsgf/sf] s'g} laGb'af6 1.62m cUnf] dflg;n] vDafsf] 6'Kkf] cjnf]sg ubf{ pGgtf+z sf]0f
600 kfof] eg] ;f] kf]v/Lsf] Jof; kTtf nufpg'xf];\ . (11.54 m)
8. Pp6f 3/ 10 lkm6 cUnf] 5 . pSt 3/sf] 5taf6 Pp6f 14 lkm6 cUnf] dlGb/sf] 6'Kkf]
cjnf]sg ul/Psf] 5 . olb 3/ / dlGb/larsf] b'/L 45 lkm6 eP pSt dlGb/sf] 6'Kkfsf]
0
pGgtf+z sf]0f kTtf nufpg'xf];\ . (60 )
9. 2 cUnf] kvf{nsf] 5fof 2m nfdf] ePsf a]nfdf ;"o{sf] prfO kTtf nufg'xf];\ . (600)
10. 6 lkm6 cUnf] dflg; pp6f j[Ttsf] vDafaf6 x lkm6 6f9f 5 . pSt s]6fsf] 5fof 2 lkm6
nfdf] 5 / j[Ttsf] vDafsf] prfO 28 lkm6 eP x sf] dfg kTtf nufpg'xf];\ . (4 lkm6)
11. j[Ttsf/ kf]vf/Lsf] s]Gb|df kfgLsf] ;txb]lv dfly 24 lkm6 cUnf] vDaf 5 . kf]v/Lsf]
lsgf/sf] s'g} ljGb'af6 x lkm6 cUnf] dflg;n] vDafsf] 6'Kkf] cjnf]sg ubf{ pGgtf+z
sf]0f 450 kfof] . olb ;f] kf]v/Lsf] Jof; 36 lkm6 eP x sf] dfg kTtf nufpg'xf];\ . (6lkm6)
12. Ps hgf dflg; vDafsf km]baf6 50, k/ plePsf] 5 . Toxf“af6 vDafsf] 6'Kkf] x]bf{ p;n]
300 sf] pGgtf+z sf]0f kfP5 . olb hldgaf6 p;sf] cf“vf 1.5cm dfly 5 eg] pSt
vDafsf] prfO kTtf nufpg'xf];\ . Ans:30.36 m

252
 If]q M & tYofª\s zf:q (Statistics)
1. If]q kl/ro
o; tYofª\s zf:q If]qdf dWos, dlWosf / rt'yf{+znfO{ ;dfj]z ul/Psf] 5 . o; If]qaf6 kl/Iffdf
5f]6f] k|Zg 2 cf]6f = 2 X 2 = 4 cª\ssf / nfdf] k|Zg Pp6f = 1 X 4 = 4 cª\ssf] cfpg] Joj:yf
5.
2. cfwf/e"t tYo / ;"qx¿
2.2 cfwf/e"t tYox¿
-s_ dWos (Mean) M lbPsf] tYofª\ssf] cf};t dfgfª\snfO{ dWos elgG5 . o;nfO{ Χ n]
hgfOG5 .
-v_ dlWosf (Median) M lbOPsf] tYofª\ssf] a9\bf] . 36\bf] j|mddf /fVbfsf] dWo dfgnfO{
dlWosf elgG5 . o;nfO{ md jf Q2 n] hgfOG5 .
-u_ rt'yf{+z (Quartice) M lbOPsf] tYofª\snfO{ a9\bf] jf 36\bf] j|mddf /fVbf klxnf] 25 % df
kg]{ / 75% df kg]{ dfgfª\s / 75 % df kg]{ dfgfª\snfO{ j|mdzM klxnf] / t];|f] rt'yf+{z
elgG5 . klxnf] rt'yf+{znfO{ Q1 / t];|f] rt'yf+{znfO{ Q3 n] hgfOG5 .
2.2 cfwf/e"t ;"qx¿
JolStut >]0fL vl08t >]0fL juL{s[t >]0fL

tYofª\s 21, 27,32,47 X 10 20 25 30 X 0–10 10–20 20–

f 2 7 4 5 30
f 2 7 4
dWos

dlWosf cf“} kb cf“} kb dlWosf kg]{ :yfg cf“} kb

md = L +
rt'yf{+z Q1= cf}“ Q1= cf} kb Q1= kg]{ :yfg cf}“ kb
kb Q3= cf} kb
Q1 = L
Q3= cf“}
kb

Q3= kg]{ :yfg cf}“ kb

253
 -

Q3 = L
gf]6 M N = -af/Daf/tfsf] of]ukmn_
L = juf{Gtfsf] tNnf] dfg
c.f. = dlWosf÷rt'yf{+z kg]{ juf{Gt/eGbf dflyNnf] juf{Gt/sf] c.f.
f = dlWosf÷rt'yf{+z kg]{ juf{Gt/sf] f.
I = juf{Gt/sf] km/s -h:t} M 0–10 sf] km/s 5_

3. ljz]if Wofg lbg'kg]{ s'/fx¿

-s_ dlWosf / rt'yf{+z lgsfNg] k|Zgdf tYofª\snfO{ a9\bf] 5 . 36\bf] j|mddf /fVg'k5{ . a9\bf]
j|mddf /fVg' pTtd dflgG5 .
-v_ dlWosf / rt'yf+{zdf JolStut >]0fLdf bzdnjdf kb cfP ljz]if Vofn ug'{k5{ . h:t} M
2.25 cf}“ kb cfPdf 2.25 cf}“ kb = 2 cf}“ kb + 0.25 -3 cf}“ kb_ x'G5 .
-u_ k|To]s k|Zgsf] cGTodf lgZsif{ n]Vg' clgjfo{ 5 .

4. gd'gf k|Zgf]Tt/ / cEof;

4.1 5f]6f] pTt/ cfpg] k|Zgx¿
gd'gf 1 :
s'g} ju{s[t tYofª\ssf] dWos ( ) = 50 / = 750 5 eg] N sf] dfg lgsfNg'xf];\ .
;fdfwfg
oxfF lbOPsf]
dWos ( ) = 50 / = 750
N=¿
xfdLnfO{ yfxf 5,
or, 50 =
or, 50 = × N = 750
or, N = = 15
∴ N = 15
To;}n] lbOPsf] tYofª\s N = 15 x'G5 .
cEof;sf nflu k|Zgx¿
-s_ olb = 320 / = 16 eP N sf] dfg slt xf]nf (N = 20)
-v_ olb N = 40,
-u_ olb = 150 eP n sf] dfg lgsfNg'xf];\ < (N= 40)
-3_ olb s'g} tYofª\sf] 40 / N = 20 eP

254
gd'gf 2 :
s'g} lg/Gt/ >]0fLdf kbx¿sf] ;ª\Vof (N) = 50 + a
dWos = 20 / kbx¿sf] of]ukmn = 1200 eP 'a' sf] dfg kTtf nufpg'xf];\ .
;dfwfg
oxfF lbOPsf],
dWos = 20
N= 50 + a
xfdLnfO{ yfxf 5,
=
Or, 20 =
Or, 20 × (50 + a) = 1200
Or, 1000 + 20a = 1200
Or, 20a = 1200 – 1000
Or, 20a = 200
Or, a =
∴ a = 10
To;}n] a = sf] dfg 10 x'G5 .
cEof;sf nflu k|Zgx¿
-s_ olb = 72 + 8k, = 6 / eP k sf] dfg slt xf]nf (k = 12)
-v_ olb Pp6f >]0fLdf = 10, = 700 + 5, / = N = 40 + 3m eP m sf] dfg kTtf
nufpg'xf];\ < (m = 12)
-u_ olb = 14.25, = 240 + 15p, / N = 17 + p eP p sf] dfg kTtf nufpg'xf];\ <
(p = 3)

gd'gf 3 :
lbPsf] n]vflrqaf6 dlWosf >]0fL kTtf nufpg'xf];\ .
;dfwfg
oxf“,
n]vflrqsf] dflyNnf] laGb' 55 5 .
To;}n] N= 55
dlWosf kg]{ >]0fL = cf}
=
= 27.5 cf}“ >]0fL
27.5 cf}“ >]0fL = 200 – 300

255
To;}n] lbPsf] n]vflrqsf] dlWosf >]0fL 200 - 300 x'G5 .
cEof;sf nflu k|Zgx¿
-s_ lbPsf] n]vflrqaf6 dlWosf >]0fL lgsfNg'xf];\ .

-v_ lbPsf] cf]hfOeaf6 klxnf] rt'yf{+z (Q1 = cf}“

>]0fL) / t];|f] rt'yf{z (Q3 = cf}“ >]0fL)
lgsfNg'xf];\ . (Q1 = 10-20, Q3 = 30-40)

-u_ ;“u} lbPsf] n]vflrqaf6 dlWosf >]0fL kTtf nufpg'xf];\ . (400-500)

256
4.2. nfdf] pTt/ cfpg] k|Zgx¿
gd'gf 1 :
lbOPsf] cf“sF8faf6 klxnf] rt'yf{+z kTtf nufpg'xf];\ .
k|fKtfª\s 0-20 20-40 40-60 60-80 80-100
ljb\ofyL{ ;ª\Vof 5 7 7 5 4

;dfwfgM
k|fKtfª\s ljb\ofyL{ ;ª\Vof c.f.
0-20 5 5
20-40 7 12
40-60 7 19
60-80 5 24
80-100 4 28
N =28
ca, Q1 kg]{ >]0fL = cf}“ kb
= cf“kb
oxfF, 7 -jf eGbf 7'nf]_ cf}“ kb = 20 – 40
Q1 kg]{ >]0fL = 20 – 40
To;}n], L = 20, =7, c.f = 5, f = 7, i = 20
-
Q1 = L

-
= 20

= 20

= = = 25.71
To;}n] lbPsf] tYofª\ssf] klxnf] rt'yf{+z (Q1) = 25.56
cEof;sf nflu k|Zgx¿
-s_ lbPsf] tYofª\saf6 dlWosf lgsfNg'xf];\ .
k|fKtfª\s 10-20 20-30 30-40 40-50 50-60 60-70
ljb\ofyL{ ;ª\Vof 2 5 7 6 3 2

257
-v_ lbPsf] tYofª\saf6 klxnf] rt'yf{+z lgsfNg'xf];\ .
pd]/ 0-10 10-20 20-30 30-40 40-50 50-60
Af/Daf/tf 4 8 12 20 18 6

-u_ lbOPsf] cf“s8faf6 t];|f] rt'yf{+z (Q3) lgsfNg'xf];\ .

Tf}n (kg) 40- 44-48 48-52 52-56 56-60 60-64
44
ljb\ofyL{ 8 10 14 16 3 1
;ª\Vof

gd'gf 2:
lbOPsf] tYofª\ssf] dlWosf 18 eP 'k' sf] dfg lgsfNg'xf];\ .
k|fKtfª\s 0-10 10-20 20-30 30-40 40-50
ljb\ofyL{ 5 K 20 4 2
;ª\Vof
;dfwfg
k|fKtfª\s ljb\ofyL{ ;ª\Vof c.f.
0-10 5 5
10-20 K 5+k
20-30 20 25+k
30-40 4 29+k
40-50 2 31+k
N = 31 + k
oxfF, dlWosf = 18 cf}“ kb
dlWosf kg]{ >]0fL = 10 - 20
ca, L = 10, c.f = 5, f = k, i = 10
To;}n],
dlWosf = L

or, 18 = 10 +

or, 18 – 10 =
or, 8 =
or, 8 x k = (21 = k) x 5
or, 8k – 5k = 105
258
or, 3k = 105
or, k =
∴k = 35
lbOPsf] tYofª\ssf] dlWosf 18 eP k = 35 x'G5 .

cEof;sf nflu k|Zgx¿

-s_ lbOPsf] tYofª\ssf] dlWosf 24 eP 'p' sf] dfg slt xf]nf <
juf{Gt/ 0-10 10-20 20-30 30-40 40-50
Af/Daf/tf 4 12 P 9 5

-v_ olb klxnf] rt'yf{+z 31 5 eg] 'm' sf] dfg kTtf nufpg'xf];\ .
juf{Gt/ 10-10 20-30 30-40 40-50 50-60 60-70
Aff/Daf/tf 4 5 M 8 7 6

-u_ olb Q3 = 60 eP 'x ' sf] dfg lgsfNg'xf];\ .

pd]/ -jif{df_ 10-20 20-30 30-40 40-50 50-60 60-70 70-80
;ª\Vof 3 5 4 5 4 X 3

5. cEof;sf nflu yk ldl>t k|Zgx¿

-s_ lbOPsf] tYofª\ssf] Q1 lgsfNg'xf];\ .
juf{Gt/ 0-20 0-40 0-60 0-80 0-100
Aff/Daf/tf 4 12 P 9 5

-v_ lbOPsf] tYofª\sf] dlWosf kTtf nufpg'xf];\ .

juf{Gt/ 0≤x<4 4≤ x <8 8≤ x <12 12≤ x <16 16≤ x <20
Aff/Daf/tf 5 7 8 7 5

-u_ lbOPsf] tYofª\ssf] dlWosf 32 eP a sf] dfg slt xf]nf <

k|fKtfª\s 5-15 15-25 25-35 35-45 45-55 55-65
ljb\ofyL{ 5 8 A 9 7 1
;ª\Vof

259
 If]q M * ;DefJotf (Probability)
!= If]q kl/ro
-s_ kf7\oj|mdn] sIff !) sf nflu tf]s]sf] ljifoj:t'
• kf/:kl/s lgif]ws 36gfx¿ (mutually exclusive events_ sf nflu ;DefJotfsf] hf]8
l;b\wfGt (addition law_ / u'0fg l;b\wfGt (multiplication law) ;DaGwL ;fwf/0f ;d:of
• ;DefJotf j[If lrq (tree diagram) ;DaGwL ;d:of -ltg 36gfdf b'O{ tx / b'O{ 36gfdf ltg
tx;Dd dfq}
• ;fwf/0f k/fl>t 36gfx¿sf ;DefJotf ;DaGwL ;d:of -y}nf]af6 an lgsfNg] ._
-v_ o; PsfOaf6 P;Pn;Ldf b'O{ cf]6f 5f]6f] pTt/ cfpg] k|Zgx¿ ;f]Wg] k|fjwfg /x]sf] 5 . b'j}
k|Zgx¿ @ cª\ssf x'g] x'gfn] hDdf $ cª\s o; PsfOaf6 k/LIffdf ;f]lwG5 .

@= cfwf/e"t cjwf/0ff / ;"qx¿

-s_ ;Defjgfsf] kl/ro
s'g} 36gf x'g ;S5 jf ;Sb}g / To;sf] lglZrttf slt 5 egL ul0ftLo ¿kdf JoSt ug]{
tl/sf g} ;DefJotf xf] . h:t} M ltdLn] kfFr cf]6f u'Rrf Ps} k6s kmfn]/ vf]kLdf k'¥ofpg
vf]Hof] . ltdLn] Pp6f vf]kLdf v;fof} eg] k'/l:s[t x'G5f} . s] Pp6f u'Rrf vf]kLdf k'U5 t < s]
of] lgZlrt 5 < kSs} klg lglZrt 5}g t/ Pp6f vf]kLdf v:g klg t ;S5 . xf] o;}nfO{
ul0ftLo efiffdf n]Vof] eg] Tof] ;DefJotf xf] .
;DefJotf xfdLn] k9\b} cfPsf] leGgsf] h:t} cjwf/0ff xf] . dflysf] pbfx/0fdf kfFr cf]6f
u'Rrfdf Pp6f vf]kLdf v;fNg' eg]sf] leGgdf o;/L b]vfpg ;lsG5 <
1
= 5A E

1
olb vf]lkdf v;fpg'kg]{ Pp6f eP o;nfO{ leGgdf 5
A E A n]lvG5 x}g t < xf] To;/L g} o;sf]
;DefJotf klg 15 g} x'G5 .
A E A

km]l/, # cf]6f sfnf] / Pp6f ;]tf] kTtLnfO{ ;Fu}} /fv]/ leGgdf n]vf}“ t .
leGgdf lbOPsf] lrqnfO{ 43 n]lvG5 lg .
A E A

o;}nfO{ ;DefJotfdf o;/L JofVof u/f}“ M

# cf]6f sfnf / Pp6f ;]tf] sfuhsf kTtLx¿ ePsf]af6 ltg cf]6f kTtL leSof] eg] Pp6f sfnf] kg]{
;DeJAotf slt x'G5 < of] eg]sf] klg l7s pxL leGg xf] 34 . A E A

ca eg, leGg / ;DefJotfdf s]xL km/s 5}g eg] lsg ;DefJotfnfO{ leGg elgPg t < xf]o;df
vf; km/s klg 5 . Tof] s] eg] Pp6f sfnf] kTtL lemSbf leGgdf 14 x'G5 eg] ;DefJotfdf 34 x'G5 .
A E A A E A

260
Tof] lsgeg] ;DefJotfdf sfnf] kTtL # cf]6fdWo] s'g} Pp6f lemSg'kg]{ ePsfn] c+zdf 3 n]Vg'k/]sf]
xf] .
;DefJotf lgsfNg] ;"q M ;Fu}sf] 6f]s/Ldf !! cf]6f an 5g\ h;df 6 cf]6f sfnf] / 5 cf]6f ;]tf] .
5
ca, Pp6f sfnf] an lemSof] eg] To;sf] ;DefJotf 11 x'G5 lsgls Pp6f sfnf] an 5
A E A

cf]6fdWo]af6 s'g} klg cfpg ;S5 h'g c+zdf n]lvof] / hDdf anx¿ 11
ePsfn] x/df 11 n]lvof] .
To:t}, Pp6f ;]tf] an cfpg] ;DefJotf slt xf]nf t < of] kTtf nufpg hDdf
;]tf anx¿ slt 5 egL hfGg'k¥of] . oxfF ^ cf]6f ;]tf anx¿ ePsfn] Pp6f
6
an lemSbf ;]tf] kg]{ ;DefJotf 11
A E x'G5 . oxfF sfnf] an leSbf lemSg ;lsg]
A

jf lemSg cg's"n x'g] anx¿ 5 cf]6f 5g\ . ;]tf] an lemSbf cg's"n jf ;Defljt anx¿sf] ;ª\Vof 6
eof] . oxfF ;]tf] an lemSg] jf sfnf] an lemSg] sfdnfO{ 36gf elgG5 . To;}n] ;DefJotf lgsfNg]
tl/sfnfO{ ;"qab\w u/f}+ t M

hDdf ;Defljt 36gf

s'g} 36gf x'g ;Sg] ;DefJotf = A

hDdf 36gfx? E

oxL ;"qsf cfwf/df,

hDdf ;Defljt 36gf 6
Pp6f ;]tf] an cfpg ;Sg] ;DefJotf = A

hDdf 36gfx? E = 11 = 0.54 pTt/

A A E A .
oxfF hDdf cg's"n 36gf 6 x'g'sf] sf/0f ;]tf anx¿sf] hDdf ;ª\Vof 6 5 . :jtGq ¿kdf lemSof]
eg] tLdWo] s'g} klg an cfpg ;S5 . To:t} u/L,
sfnf anx?sf] ;+Vof 5
Pp6f sfnf] an cfpg ;Sg] ;DefJotf = = A

hDdf anx?x? = E A

11 = 0.46 pTt/
A E A .

-v_ kf/:kl/s lgif]ws 36gf (Mutually Exclusive Events)

s'g} k/LIf0fdf olb b'O{ cf]6f 36gf Ps} k6s kg{ ;Sb}g eg] To:tf 36gfx¿ k/:k/ lgif]ws
36gfx¿ x'g\ .
h:t} M Pp6f 8fO;df ! b]lv ^ ;Ddsf ;txx¿ X'G5\ . pSt 8fO; u'8fof] eg] tL 5 cf]6fdWo] Pp6f
dfq ;tx dfly kb{5 lg . s] b'O{ cf]6f ;tx Ps} k6s dfly cfpg ;S5g\ t < ;Sb}g . xf] To;}n]
8fO;sf !, @, #, $, % / ^ Ps csf{sf k/:k/ lgif]ws 36gf cyjf Ps} k6s cfpg g;Sg]
36gfx¿ x'g\ .
To:t}, Pp6f l;SsfnfO{ 6; u¥of] eg] ls t ufO{ dfly kb{5 cyjf k'R5/ kb{5 . t/ b'j} ufO{ /
lqz"n t Ps} k6s dfly cfp“b}g . To;}n] ufO{ / lqz"n Pp6} l;Ssf Ps k6s kN6fp“bf k/:k/
lgif]ws 36gf x'g\ .
gf]]6 M Pp6} l;Ssf pkmfbf{ jf Pp6} 8fO; Ps k6s dfq u'8fpFbf cfpg] 36gf dfq k/:k/ lgif]ws
x'G5g\ .

261
pbfx/0f != Pp6f l;Ssf

Head Tail

• l;Ssf dfq pkmbf{ k/:k/ lgif]ws • 8fO;dfq u'8fpFbf k/:k/ lgif]ws 36gfx¿
36gfx¿ {H, T} 36gfx¿ {1, 2, 3, 4, 5, 6}
• H kg]{ ;DefJotf = ½ • 1 kg]{ ;DefJotf = 1/6
• T kg]{ ;DefJotf ½ • 2 kg]{ ;DefJotf = 1/6
• 3 kg]{ ;DefJotf = 1/6
• Ps k6s pkmfbf{ H / T b'j} kg{]
• 4 kg]{ ;DefJotf = 1/6
;DefJtf x'Fb}g . To;}n] H / T
• 5 kg]{ ;DefJotf = 1/6
k/:k/ lgif]ws 36gf x'g\ .
• 6 kg]{ ;DefJotf = 1/6
• Ps k6s u'8fpFbf s'g} klg b'O{ cf]6f cª\s Ps}
k6s dfly kg{] ;DefJtf x'Fb}g . To;}n] 1, 2, 3, 4, 5
/ 6 k/:k/ lgif]ws 36gfx¿ x'g\ .
-u_ k/:k/ :jtGq 36gfx¿ (Mutually Independent Events)
s'g} v]ndf 5gf]6 x'gsf nflu toss ul/of] . k'gM clGtd r/0fdf ;d"x 5gf]6 ug{sf nflu k'gM csf]{
k6s toss ul/of] eg] oL b'O{ k6s ul/Psf toss x¿ cfk;df c;DalGwt 5g\ . o:tf] cj:yfdf ul/g]
;DefJotfsf 36gfx¿nfO{ c;DalGwt 36gfx¿ elgG5 lsgls klxnf] 6;df cfpg] 36gf / kl5Nnf]
36gfdf x'g] 6;x¿ cfk;df c;DalGwt 5g\ tyf klxnf] 6;df x'g] kl/0ffd;Fu bf];|f] 6;df kg]{
36gfsf] s'g} c;/ x'Fb}g .

Head Tail Head Tail

klxnf] 6;df {H, T} bf];|f] 6; {H, T}

olb klxnf]df H / bf];|f]df T k¥of] eg] oL b'j} 36gf :jtGq 36gfx¿ x'g\ .
To:t}, b'O{ cf]6f 8fO; Ps} k6s u'8fp“bf jf 5'6\6f 5'6\6} u'8fp“bf kg{ ;Sg]
36gfx¿ b'j}df {1, 2, 3, 4, 5, 6} kb{5g\ .

oxf“ klxnf] 8fO;df cfpg] 36gf / bf];|f] 8fO;df cfpg] 36gf c;DalGwt 36gfx¿ x'g\ .

-3_ k/:k/ lgif]ws 36gfsf] hf]8 l;b\wfGt

(Additive Law of Mutually Exclusive Events)

Ps k6s dfq ul/Psf] k/LIf0faf6 k/:k/ lgif]ws 36gf k|fKt x'G5 . h:t} M Ps k6s 6; ug]{, Ps
k6s Pp6f 8fO; u'8fpg] k/LIf0faf6 cfpg] 36gfx¿ cflb o;sf pbfx/0fx¿ x'g\ .

262
k/:k/ lgif]ws 36gfsf] ;DefJotfsf] hf]8 eGgfn] s'g} 8fO; u'8fp“bf, {1, 2, 3, 4, 5, 6} dWo] s'g}
Pp6f 36gf kb{5 . t/ 1 jf 5 kg]{ ;DefJotf slt x'G5 egL ;f]Wof] eg] 1 jf 5 dWo] s'g} Pp6f dfq
36gf kg{ ;S5 . o; cj:yfdf,
1
1 kg]{ ;DefJotf, P (1) = 6 A E A

1
5 kg]{ ;DefJotf, P (5) = 6 A E A

1 1
1 jf 5 dWo] s'g} Pp6f dfq 36gf kg{] ;DefJotf = 6+6 A E A A E

1
= 12 A E

oxfF 1 kg]{ ;DefJotf hlt x'G5, 1 jf 5 dWo] s'g} Ps kg]{ ;DefJotf a9L x'G5 . To;}n] tL b'O{
;DefJotfnfO{ hf]l8G5 . ctM
sg} 36gf A sf] ;DefJotf P(A) / 36gf B sf] ;DefJotf P(B) eP,

A jf B s'g} Ps kg]{ ;DefJotf = P(A) + P(B)

#= ;dfwfg ul/Psf pbfx/0fx¿ M

gd'gf ! M
2 b]lv 20 ;Dd n]lvPsf ;ª\Vof kTtLx¿af6 gx]l/sg Pp6f ;ª\Vof kTtL y'Tbf ju{ ;ª\Vof
jf 3g ;ª\Vof g} kg]{ ;DefJotf kTtf nufpg'xf];\ .

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

oxfF, hDdf ;ª\Vof = 19 cf]6f

ju{ ;ª\Vofx¿ = 3 cf]6f
3g ;ª\Vof = 1 cf]6f

ju{;+Vof x?sf] ;+Vof 3

To;}n], ju{ ;ª\Vof kg]{ ;DefJotf hDdf ;+Vofx? = 19 = A E A A E

3g ;+Vof x?sf] ;+Vof 1

3g ;ª\Vofx¿ kg]{ ;DefJotf = hDdf ;+Vofx? = 19A E A A E

To;}n], ;a}vfn] ;ª\Vofx¿dWo]af6,

ju{ ;ª\Vof jf 3g;ª\Vof kg]{ ;DefJotf = ju{;ª\Vof kg]{ ;DefJotf + 3g;ª\Vof kg]{ ;DefJotf
3 1
= +
19 19
A E A A E

263
 2
=
19
A E

gd'gf @ M
1 b]lv 100 ;Dd cª\lst ;ª\Vof kTtL (number card) nfO{ /fd|/L lkm6]/ s'g} Pp6f sf8{
gx]/Lsg y'Tbf lgDg sf8{x¿ kg{] ;DefJotf slt x'G5 <
(i) 50 kg{] (ii) hf]/ ;ª\Vofx¿ kg{]
pTt/ M hDdf 36gfx¿sf] ;ª\Vof n(S) = 100
(i) ;Defljt 36gf , 50 kg]{ ;ª\Vof kTtL = 1 -Pp6f dfq_
hDdf ;Defljt 36gf 1
∴ P(50) = hDdf 36gfx? = 100 -lsgls 100 cf]6f kTtLx¿dWo] Pp6f dfq 50 sf]
A E A A E A

kTtL x'G5 .
(ii) hf]/ ;ª\Vof cfpg] ;ª\Vofsf ;Defljt 36gfx¿ = 50 cf]6f
hDdf ;Defljt 36gf 50 1
∴ P -hf]/ ;ª\Vof_ = hDdf 36gfx? = 100 = 2
A E A A

E
E

cEof;sf nflu k|Zgx¿

1. 2 b]lv 30 ;Dd n]lvPsf ;ª\Vof kTtLx¿af6 gx]l/sg Pp6f ;ª\Vof kTtL y'Tbf ju{ ;ª\Vof
4
g} kg]{ ;DefJotf kTtf nufpg'xf];\ . (Answer : 29 ) A E A

2. 2 b]lv 30 ;Ddsf ;ª\Vof kTtLx¿af6 gx]/Lsg Pp6f kTtL lgsfNbf ju{ ;ª\Vof jf 3g
6
;ª\Vof kTtL g} kg]{ ;DefJotf kTtf nufpg'xf];\ . (Answer : 29 )
A E A

3. 2 b]lv 30 ;Ddsf ;ªVof kTtLx¿af6 gx]l/sg Pp6f kTtL y'Tbf ¿9 ;ª\Vofs} kTtL kg]{
10
;DefJotf kTtf nufpg'xf];\ . (Answer : 29 ) A E A

4. Pp6f l:kg/nfO{ 8 cf]6f efudf 1 b]lv 8 ;Dd u/L af“l8Psf] 5 . pSt l:kg/nfO{ hf]8;“u
1
3'dfp“bf ;'O/fn] hf]/ ;ª\Vof g} b]vfpg] ;DefJotf kTtf nufpg'xf];\ . (Answer : 2 )
A E A

gd'gf # M
52 kTtL tf;sf] Ps Kofsaf6 s'g} Pp6f tf; y'Tg] / To;nfO{ lkmtf{ /fv]/ csf]{ k6s klg
Pp6f tf; y'Tbf lgDg tf;x¿ cfpg] ;DefJotf slt slt x'G5 <
(i) b'j} k6s O6f cfpg] (iii) klxnf] k6s kGhf / csf]{df lr8 cfpg]
(iv) b'j} k6s kfg cyjf ;'/y cfpg]
pTt/
(i) Ps Kofs]6 tf;sf] ;ª\Vof n(S) = 52, O6fsf] tf; ;ª\Vof = 13

264
 hDdf ;Defljt 36gf OF6fsf] tf; ;+Vof 13 1
P1 -O6f_ = A

hDdf 36gfx? = hDdf tf; = 52 = 4 E A A E A A

E
E

bf];|f] k6s kfg cfpg] ;DefJotf P2 -kfg_

hDdf ;Defljt 36gf kfg tf;sf] ;+Vof 13 1
P2 -O6f_ = hDdf 36gfx? = hDdf tf;sf] ;+Vof = 52 = 4
A E A A E A A

E
E

b'j} k6s O6f cfpg] ;DefJotfnfO{ P -O6f_ eGbf

1 1 1
P -O6f_ = P1 -O6f_ × P2 -O6f_ = 4 × 4 = 16 A

E
E

(iii) klxnf] k6s O6f cfpg] ;DefJotfnfO{ P1 -O6f_ eGbf

OF6f tf;sf] ;+Vof 13 1
P1 -O6f_ = hDdf tf;sf] ;+Vof = 52 = 4
A E A A

E
E

bf];|f] k6s kfg cfpg] ;DefJotfnfO{ P2 -kfg_ eGbf

kfg tf;sf] ;+Vof 13 1
P2 -kfg_ = hDdf tf;sf] ;+Vof = 52 = 4
A E A A

E
E

klxnf] k6s O6f / csf]{ k6s kfg cfpg] ;DefJotfnfO{ P -kfg / O6f_ eGbf
1 1 1
P -O6f_ / kfg_ = P1 -O6f_ × P2 -kfg_ = 4 × 4 = 16 A

E
E

cEof;sf nflu k|Zgx¿

1. /fd|/L lkm6]sf] 52 ktTLsf] Ps Kofs tf;af6 gx]l/ssg Pp6f tf; lemSbf afbzfx
1
;DefJotf tTtf nufpg'xf];\ . (Answer : 13 ) A E A

2. /fd|/L lkm6]sf] 52 kTtLsf] Ps Kofs tf;af6 gx]l/sg Pp6f tf; lemSbf ld:;L jf PSsf kg]{
2
;DefJotf kTtf nufpg'xf];\ . (Answer : 13 ) A E A

3. /fd|/L lkm6]sf] Ps u8\8L tf;af6 gx]l/sg Pp6f tf; lgsfNbf cg'xf/ ePsf] tf; jf k~hf
4
kg]{ ;DefJotf kTtf nufpg'xf];\ . . (Answer : 13 ) A E A

4. /fd|/L lkm6]sf] 52 kTtLsf] Ps Kofs tf;af6 gx]l/sg Pp6f tf; lgsfNbf O6f jf sfnf]
/ªsf] tf; kg]{ ;DefJotf kTtf nufpg'xf];\ . (Answer : 34 ) A E A

5. 52 kTtLsf] Ps Kofs tf;af6 Pp6f tf; y'Tbf PSsf jf ;'/t kg]{ ;DefJotf kTtf nufpg'xf];\ .
4
(Answer : 13 ) A E A

6. Pp6f l;Ssf / To;kl5 8fO;;“u} -Pskl5 csf]{_ pkmfbf{ l;Ssfdf ufO{ / 8fO;df 4 kg]{
1
;DefJotf kTtf nufpg'xf];\ . (Answer : 12) A

E
A

265
k/fl>t 36gf / ;DefJotf j[If lrq
olb b'O{ jf ;f]eGbf a9L 36gfx¿ Pp6f kg]{ ;DefJotf csf]{sf] ;ª\Vof jf ;DefJotfn] c;/ kfb{5
eg] tL 36gfx¿ k/fl>t x'g\ .

gd'gf $ M
Pp6f 6f]s/Ldf % cf]6f ;]tf / # cf]6f sfnf anx¿ 5g\ eg] @ cf]6f
anx¿ Pskl5 csf]{ u/L lemSg] / lemlsPsf] annfO{ k'gM y}nLdf g/fVg] xf]
eg] b'j} k6s sfnf] kg]{ ;DefJotf lgsfNg s] ug{ ;lsG5 <

oxf“ klxnf] an lemSbf,

sfnf anx?sf] ;+Vof 3
sfnf] kg]{ ;DefJotf P1(B) =
hDdf anx?sf] ;+Vof = 8
A E A A E

bf];|f] k6s an lemSbf, sfnf] an klxn] g} lemlsPsf] x'Fbf

sfnf anx¿sf] ;ª\Vof / hDdf anx¿sf] ;ª\Vof 36]sf]
5 . To;}n] o; k6s an lemSbf x'g] ;DefJotf klxn]sf]
an lemSbfsf] 36gfn] km/s kfb{5 .
bf];|f] k6s an lemSbf,
sfnf anx?sf] ;+Vof
sfnf] kg]{ ;DefJotf = hDdf anx?sf] ;+Vof = 72
A E A A E

b'j} an sfnf] kg]{ ;DefJotf

3 2 6 3
P1(B) / P2(B) = 8 × 7 = 56 = 28
A E A A E A A

E
E

o; k/LIf0fdf cfpg ;Sg] 36gfx¿nfO{ j[If lrqdf b]vfp“bf,

bf];|f] k6s an
klxnf] k6s an lemSbf, sfnf]
lemSbf, sfnf] cfPdf ;DefJotf
cfPdf ;DefJotf B 2 lsgls sfnf] an # cf]6f dfq afFsL
7 5g\ / hDdf an % cf]6f x'G5g\ .
3
8
5 klxnf] an sfnf] cfO;s]kl5 bf];|f]
B W 7
an ;]tf] cfpFbf
3B B 3 klxnf] an ;]tf] cfO;s]kl5 bf];|f] an
7
5W
W sfnf] cfPdf
5
hDdf an 8
8 W
4
klxnf] an ;]tf] cfO;s]kl5 bf];|f] an
7
klg ;]tf]] g} cfPdf
klxnf] k6s an lemSbf,
;f]tf] cfPdf ;DefJotf
266
 3 2 6 3
dflysf] j[If lrqaf6, b'j} k6s sfnf] cfpg] ;DefJotf 8 × 7 = 56 = 28
A E A A E A A

E
E A g} eof] .
gd'gf % M
Pp6f kl/jf/df 3 hgf aRRff hGdFbf x'g] 36gfx¿sf] j[If lrq agfO{ lgDg s'/fx¿ lgsfNg'xf];\ M
-s_ ltg} hgf s]6L x'g]
-v_ sDtLdf Pp6f s]6L x'g] Girl ½
pTt/ M oxfF, ltg aRrf hGdFbf,
klxnf] aRrf s]6L x'g] Boy ½
Girl ½
1
;Defjgf P1 -s]6L_ = 2 ,
Girl ½
A E A

Girl ½
bf];|f] aRrf s]6L x'g] ;Defjgf
1 Boy ½ Boy ½
P2 -s]6L_ = 2 A E
Girl ½
Girl ½
t];|f] aRrf s]6L x'g] ;Defjgf BOy ½ Girl ½
Boy ½
1
P3 -s]6L_ = 2 A E
Boy ½

ltg} k6s s]6L cfpg] Girl ½

Boy ½
;DefJotf
= P1 -s]6L_ × P2 -s]6L_ × Boy ½
1 1 1 1
P3 -s]6L_ = 2 × 2 × 2 = 8
A

E
E A A E A

-v_ sDtLdf Pp6f s]6L x'g] ;DefJotf = 1

− ;a} s]6f x'g] ;Defjgf
1 1 1 1 1
=1−( × × )=1−
2 2 2
A

E
E A

8 =7 A E A A E A

cEof;sf nflu k|Zgx¿

1. Pp6f emf]nfdf 4 cf]6f ;]tf / 5 cf]6f sfnf p:t} / pq} anx¿ /flvPsf 5g\ . kfn}kfnf] k'gM
g/flvsg b'O{ cf]6f an gx]/L y'Kbf cfpg] ;a} ;Defljt kl/0ffdx¿nfO{ j[If lrqdf
b]vfpg'xf];\ . (Ans..........)
2. Pp6f y}nLdf 7 cf]6f sfnf] / 9 cf]6f xl/of anx¿ 5g\ . gx]l/sg b'O{ cf]6f anx¿ Ps Ps
u/L -k'gM g/fvL_ lemSbf aGg] ;DefJotf j[If lrqdf k|:t't ug'{xf];\ . (Ans..........)
3. Ps bDktLaf6 ltg cf]6f aRrf hlGdPsf 5g\ . ;DefJo kl/0ffdx¿sf] ;DefJotfnfO{ j[If
lrqdf k|:t't ug'{xf];\ . (Ans..........)
267
4. Pp6f kl/jf/df 3 hgf aRrfx¿ hGd“bf b'O{ hgf 5f]/f x'g] ;DefJotfnfO{ j[Iflrq agfO{
33
b]vfpg'xf];\ . (Ans. 100 )
A E A

5. Pp6f afs;df 3 cf]6f /ftf, 5 cf]6f lgnf p:t} l;;fsndx¿ 5g\ . ltg cf]6f l;;fsnd
gx]l/sg k'gM afs;df g/fvL lgsfNbf aGg] gd'gfsf] j[If lrq n]Vg'xf];\ / ltg} cf]6f lgnf
l;;fsnd kg{] ;DefjJotf b]vfpg'xf];\ .
5
(Ans. 28 )
A E A

6. Pp6f afs;df 4 cf]6f /ftf, 7 cf]6f lgnf p:t} l;;fsndx¿ 5g\ . ltg cf]6f l;;fsnd
gx]l/sg k'gM afs;df g/fvL lgsfNbf aGg] gd'gf If]q n]Vg'xf];\ .
(Ans. S = {RRR, RRB, RBR, RBB, BRR, BRB, BBR, BBB})
7. Pp6f l;Ssf ltg k6s p5fNbf aGg] ;DefJo kl/0ffdx¿nfO{ j[If lrqdf k|:t't ug'{xf];\ / -i_
Pp6f dfq ufO{ kg]{ ;DefJotf kTtf nufpg'xf];\ . (Ans. 38 )
A E A

8. ;a} kf6fx¿ ;dfg ePsf] Pp6f 8fO; b'O{ k6s u'8fp“bf aGg] gd'gf If]q n]Vg'xf];\ / klN6Psf
5
;ª\Vofx¿sf] hf]8 8 cfpg] ;DefJotf lgsfNg'xf];\ . (Ans. 28 ) A E A

268
 (= P;= Pn= ;L= k/LIffsf] ljlzi6Ls/0f tflnsf, gd"gf k|Zgkq, gd"gf
pTt/ s'l~hsf / cEof;sf nflu gd"gf kZgkq

COMPULSORY MATHS
Model Set
Time: 3.00 hours Full Marks: 100
Pass Marks: 32
;d'x …sÚ (Group ‘A’) [9×(2+2)=36]
1. a. dfg lgsfNg'xf];\ (Evaluate):
45a 2 − 80a 2 + 6a 5
5 5a 2
b. ;/n ug{'xf];\ (Simplify):
a 2 p × a p −1 − a p
a 3 p − a p +1

2. a. xn ug{'xf];\ (Solve) : 3 2 x + 7 − 3 = 0
b. olb Pp6f wgfTds ;ª\Vofsf] ju{af6 7 36fpFbf kl/0ffd 9 x'G5 eg] pQm ;ª\Vof
kQf nufpg'xf];\ .
If 7 is subtracted from the square of a positive number, the result is 9. Find the
number.
3. a. olb ∑ fx = 400+20a, ∑ f =18+2a / lbOPsf] >]0fLsf] dWos @) eP a sf]
dfg
lgsfNg'xf]; .
If ∑ fx = 400+20a , ∑ f =18+2a, and the mean of the given series is 20, find the
value of a. 8

b. ;Fu} lbOPsf ;l~rt jf/Djf/tf js|jf6 dlWosf / 7

6
dlWsfsf] ju{ 5

kQf nufpg'xf];\ . 4
3
Find the median class and value of median from the 2
adjoining cumulative frequency curve. 1

4. a. /fd|/L lkml6Psf] 52 kQL ePsf] tf;sf] 0 10 20 30 40 50 60 70 80 90

u8\8Laf6 Pp6f kQL ->]0fL cGt/_ Class interval

lgsflnPsf] 5 eg] pQm klQ /fgL cyjf sfnf] /ªsf] PSsf kg{] ;DefJotf kQf
nufpg'xf];\ .
A card is drawn from a well-shuffled deck of 52 cards. Find the probability of
getting such card is queen or a black ace.
b. Pp6f l;SsfnfO{ b'O{k6s ;Dd pkmfbf{ aGg] ;DefJotfnfO{ j[Iflrqdf k|:t't
ug{'xf];\ .
A coin is tossed two times and represents the probabilities in a tree diagram.

269
5. a. lqe'h ABC df AB=8 ;]=ld= BC= 12 ;]=ld= / ∠ABC=60° eP ;f] lqe'hsf]
If]qkmn kQf nufpg'xf];\ .
In ∆ABC, AB=8cm, BC= 12 cm and ∠ABC=60°.
Find the area of the triangle.
b. lbOPsf] lrq lqe'hfsf/ cfwf/ ePsf] Pp6f 7f]; lk|Hd xf] . olb AB= 3;]=ld=
BC=5;]=ld= / CF= 12;]=ld= eP pQm lk|Hdsf] cfotg kQf A

nufpg'xf];\ . B C

The adjoining diagram is a triangular based solid prism. If

AB= 3cm, BC=5cm and CF= 12cm find the volume of the D

prism.
F
Pp6f a]ngfsf] cw{Jof; / prfOsf] of]ukmn ;]=ld / cfwf/sf]
E
6. a . 12
kl/lw 416 ;]=ld= eP pQm
a]ngfsf] k"/f ;txsf] If]qkmn lgsfNg'xf];\ .
If the sum of radius and height of cylinder is 12 cm and circumference of base is
416 cm find the total surface area of that cylinder.
b. ;Fu} lbOPsf] lrq Pp6f 7f]; uf]nfsf] xf] . olb cfwf/sf] Jof;
- AB)= 28 ;]=ld= eP pQm uf]nfsf] cfotg lgsfNg'xf];\ .
The diagram given alongside is of a solid sphere. If AB= 28 cm, A B

Find the volume of the sphere. 28cm

7. a. Pp6f j:t'sf] jf:tljs d"Nodf15% a9fO{ clª\st d"No ?=

2760 sfod ul/of] . ;f] j:t'sf] jf:tljs d"No kQf nufpg'xf];\ .
The marked price of an article was fixed to Rs. 2760 by increasing 15% in its actual
price. Find its actual price.
b. ?= 5000 sf] k|ltjif{ 10% jflif{s rqmLo Aofhb/n] 2 jif{df x'g] rlqmo ld>wg
kQf nufpg'xf];\ .
Find the compound amount on Rs. 5000 in 2 years at 10% per
Z T Y
annum.
8. a. lbOPsf] lrqdf, WXYZ Pp6f ;dfgfGt/ rt'e{'h 5 . olb
WX nfO laGb' V ;Dd a9fpFbf ag]sf] ∆ZYV sf] If]qkmn
15cm 2
eP W X V

∆WXT sf] If]qkmn kQf nufpg'xf];\ .

P
If the given figure, WXYZ is a parallelogram . WX is extended
up to U and ∆ZYU is formed. If Area of ∆ZYU is 15cm2,
find
the area of ∆WXT. 46°
S
Q
R

b.
lbOPsf] lrqdf PR⊥QS 5 . olb ∠PQS=46° eP ∠QSR D
C

kQf nufpg'xf];\ . In the given figure, PR⊥QS. If ∠PQS=46°,

find ∠QSR. 30 ° 70°
A O B

270
9. a. lbOPsf] lrqdf laGb'x? A, D, C / B cw{j[Qdf 5g\ . olb ∠DAC=30° / ∠ABC=
70° eP ∠ACD kQf nufpg'xf];\ .
In the given figure, points A, D, C and B are on semi-circle. If ∠DAC=30° and
∠ABC= 70°, find ∠ACD.
b. lbOPsf] lrqdf PT n] laGb' A df j[QnfO{ :kz{ u/]sf] 5 . B

olb∠BAT=60° / ∠BAC=50° eP ∠ABC kQf nufpg'xf];\ .

In the given figure, PT touches a circle at a point A. If ∠BAT=60° and
C
50° 60° T

∠BAC=50°, find ∠ABC. A

;d'x …vÚ (Group ‘B’) [16×4=36]

10 Pp6f ljBfyL{x¿sf] ;d"xdf ul/Psf] ;j{]If0fdf 70% ljBfyL{x¿n] j}1flgsx¿sf
;DaGwdf , 65% n] v]nf8Lx?sf ;DaGwdf / 430 hgfn] b'j}sf af/]df cWoog u/]sf]
kfOof] . olb 8% n] s'g} sf] af/]df klg cWoog u/]sf] kfOPg eg] M
In a survey of the group of students, it was found that 70% of students studied about
scientists, 65% about players and 430 studied about both scientists and players. If 8% did not
study about scientists and players, then,
i) dflysf] tYofªsnfO{ e]glrqdf k|:t't ug{'xf];\ .
Represent the above information in a Venn-diagram.
ii) ;j{]If0fdf efu lnPs]f hDdf ljBfyL{;ª\Vof kQf nufpg'xf];\ .
Find the total number of students who took part in the survey.
11. dxQd ;dfkjt{s lgsfNg'xf];\ . (Find the H.C.F of):
6 x 2 + 2 x,6 x 3 − 10 x 2 − 4 x 2(3x + 1)
2
and
1
12. xn ug{'xf];\ . (Solve) : 5x-1 + 5-x = 1
5
13. ;/n ug{'xf];\ . (Simplify):- 2 1 3y xy
− − 2 + 3
x+ y x− y y −x 2
x + y3
14. b'O{ cÍsf] s'g} Pp6f ;ª\Vof, Tof] ;ª\Vofsf] cª\sx¿sf] of]usf] rf/ u'0ff 5 . olb Tof]
;ª\Vofsf] cª\sx¿sf] :yfg abn]/ aGg] ;ª\Vof / 9 sf] of]ukmn] pQm ;ª\Vofsf] b'O{ u'0ff
x'G5 eg], ;f] ;ª\Vof kQf nufpg'xf];\ .
A two digit number is four times the sum of its digits. If the sum of the number formed by
reversing its digits and 9 is two times the original number, find the original number.

15. olb tn lbOPs]f cfFs8fsf] dWos 68 eP x sf] dfg kQf nufpg'xf];\ .

If the mean of the following data is 68, find the value of x.
Marks obtained 40-50 50-60 60-70 70-80 80-90 90-100
-k|fKtfÍ_
No.of students 17 22 28 26 x 12
-ljBfyL{sf] ;+Vof_

271
16. Pp6f gbLsf] lsgf/df ePsf] 40 ld6/ cUnf] ¿vsf] 6'Kkf]df pQm gbLsf] csf{] lsgf/af6
cjnf]sg ubf{ pGgtf+z sf]0f 30° kfOof] eg] ;f] gbLsf] rf}8fO kQf nufpg'xf];\ .
The angle of elevation of the top of a tree, 40m high situated at the bank of a river when
observed from the opposite bank of the river is found to be 30°, find the breadth of the river.

17. lbOPsf] 7f]; a:t' j]ngf / cw{uf]nf ldn]/ ag]sf] 5 . h;sf]

cfwf/sf] Jof; 14;]=ld= / k"/f nDafO 21;]=ld= 5 eg] ;f] 7f]; 14cm

a:t'sf] k"/f ;txsf] If]qkmn kQf nufpg'xf];\ .

21cm
The given solid object is made up of a cylinder and a
hemisphere whose diameter of the base is 14cm and total length is 21 cm. Find the total
surface area of the solid object.

O
18. tn lbOPsf ju{ cfwf/ ePsf] lk/fld8sf] k"/f ;txsf]
If]qkmn lgsfNg'xf];\ .
Find the total surface area of the following square based
C
pyramid. D

19. A / B n] Pp6f sfd s|dz M 12 / 16 lbgdf ug{ ;S5g\ .A E 7cm F

/ B b'j}hgf ldn]/ 4 lbg sfd u/]kl5 B n] ;f] sfd 5f]8\5 A

B

eg] afFsL sfd ug{ A nfO{ slt lbg nfUnf < kQf
nufpg'xf];\ .
A and B can do a piece of work in 12 and 16 days respectively. A and B work together for 4
days and B leaves the work, find how long will A take to complete the remaining work.
20. Pp6f Sofd/f 25% 5'6 lbP/ 10% d"No clej[l4 s/ nufO{ a]lrof] . olb 5'6 /sd ?=
750 eP pQm Sofd/fsf] d"Nodf d"No clej[l4 s/ /sd slt lyof] kQf nufpg'xf];\ .
A camera was sold after allowing 25% discount on the marked price and then levying 10%
value added tax (VAT). If the discounted amount was Rs.750, how much value added tax
(VAT) was levied on the price of the camera?
21. Pp6f ufpFsf] hg;ª\Vof k|To]s jif{ 5 k|ltztn] a9\b} hfG5 . olb b'O{ jif{sf] cGTodf
1025 hgf a;fOF ;/]/ cGoq hfFbf ;f] ufpFsf] hg;ª\Vof 10,000 eof] eg] ;'?df ;f]
ufpFsf] hg;ª\Vof slt lyof] <
The population of a village increases every year by 5%. If 1025 people leave the village at the
end of two years and the population of the village is 10,000, find the population of the village
in the beginning.
22. Pp6} cfwf/ QR / pxL ;dfgfGt/ /]vfx? QR / PS sf] jLrdf S R
ag]sf lqe'hx¿ PQR / SQR sf] If]qkmn a/fa/ x'G5g egL
k|dfl0ft ug'{xf];\ .
Prove that the triangles PQR and SQR standing on same base
P T
QR and between same parallel lines QR and PS are equal in area. O

23. ;Fu}sf] lrqdf PT j[Qsf] Jof; xf] / O s]Gb|laGb' xf] olb rfk
SR= rfk RT eP PS//OR.x'G5 egL l;4 ug'{xf];\ .

272
 In alongside diagram, PT is the diameter and O is centre of circle. If arc SR=arc RT, prove
that PS//OR.
24. Pp6f j[Qsf] pxL rfkdf cfwfl/t kl/lwsf]0fx¿ a/fa/ x'G5g\ egL k|of]u4f/f l;4
ug'{xf];\ . -slDtdf 3;]=ld= cw{Jof; ePsf b'O{ cf]6f j[Qx¿ cfjZos 5g\._ Verify
experimentally that the angles at the circumference standing on the same arc of a circle are
equal.(Two circles of radii at least 3 cm are necessary.)
25. rt'e'{h ABCD sf] /rgf ug'{xf];\ h;df AB=5.8 ;]=ld, BC=6.2;]=ld , CD=5.1;]=ld ,
DA=4.8;]=ld / ∠BAD=60° 5g\ .;fy} pQm rt'e'{h;Fu If]qkmn a/fa/ x'g] lqe'hsf] /rgf
ug'{xf];\ .
Construct a quadrilateral ABCD in which AB=5.8cm, BC=6.2cm , CD=5.1cm , DA=4.8cm
and ∠BAD=60°.also construct a triangle equal in area to the quadrilateral ABCD.

Marking Scheme
Model set 1
1. a.
9a 5 − 4a 5
i) ………………………………….. (1)
5a 5
ii) 1 ……………………………………… …… (1)
a2 p
p
a ( − 1)
b. i)
a ……………… …………………. (1)
a p (a 2 p − a)
1
ii) …………………………………………………. (1)
a

2. a. i)
(3
2x + 7 ) = (3)
3 3

x +7=27 ………………………………………(1)
⇒2
ii) 2 x =20⇒ x =10………………………………..........(1)
b. Let the positive number be x .
i) x 2 − 7 = 9 ……………………………………………(1)
ii) x = 4 …………………………………………………..(1)
3.
400 + 20a
a. (i) 20= ……………………………..……. (1)
18 + 2a
(ii) a=2………………………………………………. (1)

b . (i) Median class = 30-40 …………………………. (1)

Median = 4…………….. ……………………………..…(1)

273
4. a
4 2
i) P(Queen) = , P(B.ace) = ……….……………….1
52 52
3
ii) P(Queen or B.ace) = …………….…………...……….1
26
b.
(i) First outcome………...1
(ii) Second outcome …..1

5. a. (i) Area of triangle ABC =

1
× 8cm ×12cm × sin 60° ………….(1)
2
(ii) Area of triangle ABC=55.42cm2 …………………………(1)
b. (i) AC = 4 cm and area of triangle (A)=6 cm2……..…………....(1)
(ii) Volume of prism (V) = 72 cm3……………..………………..(1)
6. a.
i) T.S.A. of the cylinder= 2πr(r+h)………………….(1)
ii) T.S.A. of the cylinder=416×12=4992cm2……..….(1)

b.
3
4 22  28 
i) V= × ×   ……….(1)
3 7  2 
ii) V= 11498.67 cm3……….(1)

7. a.

i) x +15% of x = Rs. 2760………………….(1)

ii) Actual price( x ) = Rs. 2400 ………………(1)

b.

2
 10 
i) CA = Rs. 5000 1 +  ………………….(1)
 100 
ii) CA = Rs. 6050 …………….………………(1)

274
8. a.
i) Finding area of WXYZ = 30 cm2………… .. 1
ii) Finding area of ∆WXT=15cm2……………….1
iii)

b. (i) Finding ∠QPR=44°………………… ………....1

(ii) Finding ∠QSR= 44°………………………1

9 a. i) Getting ADC=110° with correct reason…………1

ii) Getting ACD=40° with correct reason …………….1

a. i) ∠BAC= 60, with correct reason……………………1

ii) ∠ABC= 70°, with correct reason……………………1

10. Let total number of students be x.

i) Representation of the information in Venn-diagram………1+1

70 x 65 x 8x
ii) − 430 + − 430 + 430 + = x …………..…1
100 100 100
iii) =1000 [i.e. Total number of students =1000]……….1

Alternatively

Let n(s∩p) be a% of total students.

i) Representation of information in Venn-
diagram…………....(1+1)

ii) 70-a+a+65-a=100-8 or
a=43…………………….…….(1)
iii) 43% of total students =430 or, total students =1000….(1)

11. i. 2x(3x+1) …………………………………………… ....(1)

ii. 2x(x-2) (3x+1) ………………………………………. (1)
iii. 2(3x+1) (3x+1) ……………………………………..… (1)
12.
iv. H.C.F = 2(3x+1) …………………………………...……(1)

275
 i. Let , 5x = a ⇒ a2-6a+5=0 …………………………….(1)
ii. (a-1) (a-5) = 0 …………………………..……………(1)
iii. a-1=0 ⇒ x=0 …………..……………….…………(1)
a-5=0 ⇒ x-1
iv. x=0, 1 ……………….………………………………..(1)

2x − 2 y − x − y 3y xy ………………….(1)
13. i. + 2 + 3
x −y
2 2
x −y 2
x + y3
x xy
+
( )
ii. ………..……….(1)
(x + y )(x − y ) (x + y ) x − xy + y 2
2

x − x y + xy + x y − xy
3 2 2 2 2
iii.
(
(x + y )(x − y ) x 2 − xy + y 2 ) …………..………………(1)
x3
iv.
(
(x − y ) x 3 + y 3 ) ……………….……………………..(1)
14.
Let a two digit number be 10x+y where x and y are the digits of ten's place and unit place
respectively.
1. 10 +y=4( +y)
⇒6 =3y
∴y=2 …………………………………………..(1)
2. 10y+ +9=2(10 +y)
⇒ 19 =8y+9 ………………………………..………(1)
3. =3 and y = 6……………….………..………………(1)
. The original number = 10×3+6=36……………………(1)

276
15

Mark obtain No. of students Mid value F(m)

40-50 17 45 765
50-60 22 55 1210
60-70 28 65 1820
70-80 26 75 1950
80-90 x 85 85x
90-100 12 95 1140

∑ f = 105 + x ∑ f .m = 6885 + 85x ………………..….1+1

6885 + 85 x
i. 68= …………………………….1
105 + x
ii. x=15 ………………………………………..1

16. (i) Appropriate figure with description ………….. (1)

40
(ii) In right angled triangle CAB, Tan30°= … (1)
AB
(iii) AB= 40 3 meter.….………..……………….. (1)
(iv) The breadth of the river (AB) = 69.28…….... (1)
….meter………….…..(1)

17.

i) Slant height of pyramid =25 ……………..………………..(1)

ii) Area of base of pyramid (A1)= 196 cm2 ………….………(1)
iii) Area of 4 triangles(A2) = 700 cm2 ………..………………(1)
iv) Total surface area of pyramid (A)=896 cm2………….…..(1)

277
18.
i) Radius of the cylinder (r)= Radius of the hemisphere
(r)=7cm………………………………………………..…...1
ii) T.S.A. of cylinder without one circular part(A1)=πr2+2πrh=
22 2 22
× 7 + 2 × × 7 × 21 = 1078 cm2…………………………1
7 7
22 2
iii) Curved surface area of the hemisphere (A2)= 2πr2= 2 × ×7
7
=308cm2………………………………………..1
iv) T.S.A. of the solid(A)= A1=A2
1078+308=1386cm2 ……………………………..……..…1

19. 7
i) A and B can do of work in 1day …………..………..(1)
48
5
ii) Remaining work after4 days= works ………….………(1)
12
5
iii) A do works in 5 days ………..………….……………(1)
12
iv) A can do remaining in 5 days………………………….…..(1)

20.

i) PT = 10000+1025= 11025 …………………………..….(1)

2
 5
ii) 11025 = P 1 +  ………...………….…………...…(1)
 10 
iii) 11025 = P × 1.1025 …………….………….…...…...…(1)
iv) P = Rs. 10,000 …………….………………...…..........…(1)

278
21.
i) Let the MP be Rs x
x
Discount amount = 25% of x = Rs ……………….(1)
4
x
ii) Rs. 750= , o r x=Rs3000 …………….…………...…(1)
4
 x
iii) VAT amount (VAT) = 10% of x−  =
3x
…...…...…(1)
 4 40
iv) VAT amount = Rs 225 ……………………...…..........…(1)

22.
i) Correct figure with given to prove and construction if
necessary…………………………1
ii) Ar(∆PQR) = ½ ar( PQRT)……..1
iii) Ar(∆SQR) = ½ ar( PQRT)……..1
iv) Ar(∆PQR) = Ar(∆SQR) …………1
Full marks will be given for correct and appropriate proof:
Note: If the first step is wrong zero score for this answer.
23

Following statements with correct reason.

i) SR=RT
……………………..………….1
ii) ∠SOR=∠ROT……………….…….1
iii) ∠SPT= ½ ∠SOT= ∠ROT…………1
iv) PS//OR……………………………...1
NB:Full marks will be given for alternative correct and appropriate proof.

24.
i) For the correct figures…………………………..….…..1
ii) For the correct measurements with table……… .. …1+1
iii) For the correct conclusion…………………………..…..1
25. (i) Construction of ∠BAD=60° on the line AB= 5.8 cm…...1
(ii) Construction of quadrilateral ABCD. …………………..1
(iii) Draw a line parallel to BD through the point C…….……1
(iv) Construction of required ∆ADE…………………………1

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