ແບບຮຽນຄະນິດສາດ ມ1

ສະຫງສ.ວວນ.ລິສຂະສິດ@D .97;k,

dD mt;u76o0v’9eo;ome,t-kfD
mt;u76o0v’9eo;ome,t-kf a c,jozqo76o0v’ a da[9eo;ome,t-kf !S @S #S DDD
8q;1jk’G d5j,mt;u76o0v’ # c,oj {#S *S _S !@S DDD} = 3 `

d5,j mt;u76o0v’ $ c,jo {$S )S !@S !*S DDD} = 4 `

0D mt;u76oIj;,|hvpl5f 0v’ a c]t b 0Posp=h mI|l(a,b) s^n LCM(a,b) D
gIqk,uG
d5j,mt;u76o0v’ # c,jo {#S *S _S !@S !&S !)S @!S @$S @(S DDD}= 3 `
d5j,mt;u76o0v’ $ c,oj {$S )S !@S !*S @WS @$S @)S #@S #*S DDD}= 4 `
d5j,mt;u76oIj;,0v’ # c]t $ c,oj {!@S @$S #*S $)S *WS DDD}= 12 `
mt;u76oI;j ,|hvpl5f0v’ # c]t $ c,jo !@ 0Posp=h LCM (3,4) =12 D

gIqk,uG
d5j,mt;u76o0v’ * c,oj {*S !@S !)S @$S #WS #*S $@S $)S DDD} = 6 `
d5,j mt;u76o0v’ ) c,oj {)S !*S @$S #@S $WS $)S &*S *$S DDD} = 8 `
d5,j mt;u76oIj;,0v’ * c]t ) c,jo {@$S $)S (@S _*S DDD}= 24 `
mt;u76oI;j ,|hvpl5f0v’ * c]t ) c,jo @$ 0Posp=h mI|lX*F )? = @$ s^n
LCM (6,8) = 24 D

7D v5xt76o0v’9eo;ome,t-kf
.shlv’9eo;ome,t-kf a c]t bF 4hk a sko0kf.sh b D gIqkg;Qk;kj b c,oj

v5xt76o0v’ aD

8q;1jk’G
#* sko0kf.sh $F gIqkwfh $ c,oj v5xt76o0v’ #*D
9qj’-vd9eo;ome,t-kfma’\qfmujgxaov5xt76o0v’ #*D
gIqk,uG #* = ! V #*S #* = @ V !)S #* = # V !@S #* = $ V _S #* = * V *
faj’oAo v5xt76o0v’ #* ,u G !S @S #S $S *S _S !@S !)S #*D
d5,j v5xt76o0v’ #* c,oj {!S @S #S $S *S _S !@S !)S #*}

47

ສະຫງສ.ວວນ.ລິສຂະສິດs^adgdoG
‚ m5dM9eo;omugj xaov5xt76o0v’9eo;o a c,oj v5xt76o0v’m5dM9eo;omujgxao

mt;u76o0v’ a.
‚ m5dMmt;u76o0v’ a sko0kf.shv5xt76om5dM8q;0v’ a.

8q;1kj ’G
{ mt;u76o0v’ * c,oj *S !@S !)S @$S #WS DDD c]t v5xt76o0v’ * c,oj !S @S # c]t *.
{ mt;u76o0v’ * c,jo *S !@S !)S @$S #WS DDD ]h;oc8jsko0kf.sh !S @S # c]t * g-yj’
c,oj v5xt76o0v’ * D

75o]adltot
‚ m5dM9eo;ome,t-kf nF gIqkwfh n sko0kf.sh ! c]t 8q;,aogv’ ltoAo
v5xt76o|hvpl5f0v’ n c,jo ! c]t v5xt76os^kpl5f0v’ n c,jo8q;,aogv’7n nD
‚ m5dM9eo;ome,t-kf nF 4hk b c,jov5xt76o0v’ n gIqkwfh 1≤b≤n D

ltoAo 9eo;ov5xt76o0v’9eo;ome,t-kf n oa[wfh XlYol5f?D

dko-vdskv5xt76o0v’9eo;ome,t-kfgIqkxt8y[affa’j oUG
8q;1kj ’G -vdskv5xt76o0v’ !@D

!@ = ! V !@ = @ V * = # V $D gIqk15fdko7yfw]j 4hkgsao;jkl;j o76oxUoda[7noD
gIqkwfh v5xt76o0v’ !@ c,oj !S @S #S $S * c]t !@
g-ajofP;dao le]a[ !)D
!) = ! V !) = @ V _ = # V *D
gIqkwfh v5xt76o0v’ !) c,jo !S @S #S *S _ c]t !)

’D v5xt76oIj;,s^kpl5f 0v’ a c]t b 0Posp=h vIs^l (a,b) s^n GCD(a,b) D
gIqkI6h d5,j v5xt76o0v’ !@ c,jo {!S @S #S $S *S !@ } c]t d5,j v5xt76o0v’ !)

c,jo {!S @S #S *S _S !) }
gIqkgsao;jkG

v5xt76oI;j ,0v’ !@ c]t !) ,uG !S @S # c]t *D le]a[ * c,oj v5xt76oI;j ,muj
s^kpd;kj \6j0v’!@ c]t !)D gryoj gvUo * ;jk v5xt76oIj;,s^kpl5f0v’!@ c]t !)
0Posp=h vIs^lX!@F !) ? = * s^n GCD(12,18) = 6 D

48

gIqkI6hG d5j,v5xt76o0v’ @$ c,oj {!S @S #S $S *S )S !@S @$}D
d5,j v5xt76o0v’ #* c,jo {!S @S #S $S *S _S !@S !)S #*} c]t d5,j v5xt76o0v’
$) c,jo {!S @S #S $S *S )S !@S !*S @$S $)}

‚ d5,j v5xt76oI;j ,0v’ @$S #* c]t $) c,oj {!S @S #S $S *S !@ }D
ltoAo GCD (24,36, 48) = 12 D

s^adgdo !
.sh9eo;ome,t-kf a, b c]t cF 4hk;kj GCD (a,b) = t c]h;

GCD (a,b,c) = GCD (t, c) .
s^adgdo@
.sh9eo;ome,t-kf a, bF b < a c]t r c,oj 8q;glf0v’dkosko gvu d]uf a .sh bD

4hk;kj r = W c]h; GCD (a,b) = b .

4hk;kj r ≠ W c]h; GCD (a,b) = GCD (b, r ) .
ສະຫງສ.ວວນ.ລິສຂະສິດ
9D dko-vdskv5xt76oIj;,s^kpl5f0v’lv’ s^n s^kp9eo;ome,t-kfD
s^addkoG v5xt76oI;j ,s^kpl5f0v’lv’9eo;ome,t-kf a, b c]t b < a, c,jo

8q;glf8jk’l6o 8q;l5fmhkp0v’dkoskogvud]yf8+gonvj ’dao0v’ a .sh bD

g-aoj ;jkG a sko.sh b c]h;glf r ≠ WS gIqksko b .sh r c]h;glf r′ ≠ WS

gIqksko r .sh r′ c]h;glf r″ ≠ WS gIqksko r′ .sh r″ c]h;glf r′′′ = W D

gIqkwfhG GCD (a,b) = GCD (b, r ) = GCD (r, r ') = r " D

8q;1kj ’G

!D -vdsk GCD (12,18) D

!) !@ !@ * gIqkwfh GCD (12,18) = 6.

6 1 0 2 

@D -vdsk GCD (24,32, 48) D

gIqk-vdGCD (24,32) D gIqkwfhG #@ @$ c]t @$ ) F fa’j oAoGCD (24,32) = 8.
8  1  0  3

gIqk-vd GCD (8, 48) gIqkwfhG $) ) faj’oAo GCD (8, 48) = 8D
06

ltoAo GCD (24,32, 48) = GCD (8, 48) = 8 D

49

lD 9eo;o,6o
‚ 9eo;ome,t-kfmu,j uv5xt76orP’c8lj v’8q; 7n ! c]t 8q;,aogv’gmqkj oAo gvUo;jk
9eo;o,6oD
8q;1kj ’G !# ,uv5xt76o ! c]t !# gmqkj oAoD ltoAo !# c,jo9eo;o,6oD

& ,uv5xt76o ! c]t & gmqkj oAoD ltoAo & c,oj 9eo;o,6oD
$ ,uv5xt76o !F @ c]t $. ltoAo $ [+c,oj 9eo;o,6oD
!@ ,uv5xt76o !F @F #F $F * c]t !@. ltoAo !@ [+c,jo9eo;o,6oD

‚ 9eo;o,6omu|j hvpd;kj !WWD

gryoj -vdsk9eo;o,6omuj|hvpd;jk !WW fa’j oUG

1  2  3  4  5  6  7  8  9  10  0Po9eo;ome,t-kfc8j !
11  12  13 14 15 16 17 18 19 20  sk !WW .l8j k8t]k’10×10F
21  22  23 24 25 26 27 28 29 30  0hk !F 0hkmt;u76o0v’ @F #F &F
31  32  33 34 35 36 37 38 39 40  ( c]t !!D
41  42  43 44 45 46 47 48 49 50  9eo;omugj s^nv.o8k8t]k’
51  52  53 54 55 56 57 58 59 60  c,oj 9eo;o,6o c8j ! sk !WW
61  62  63 64 65 66 67 68 69 70  g-yj’c,oj @F #F &F (F !!F !#F
71  72  73 74 75 76 77 78 79 80  !(F !_F @#F @_F #!F #(F $!F
81  82  83 84 85 86 87 88 89 90  ສະຫງສ.ວວນ.ລິສຂະສິດ
91  92  93 94 95 96 97 98 99 100

$#F $(F &#F &_F *!F *(F (!F(#F (_F )#F )_ c]t _(D

‚ 75o]adltot

Œ m5d9eo;ome,t-kfmuj[+c,jo9eo;o,6o,u1kj ’|hvpv5xt76ogxao9eo;o,6o|bj’8q;D

Œ grnvj d;fg[yj’9eo;ome,t-kf|b’j c,oj 9eo;o,6o s^n [+F gryjosko9eo;ooAo.sh

9eo;o,6o]Po8yfdaoD 4hkdkoskoskd[+0kf c]t gryoj 9t15fdkoskog,njvwfh
zqoskogmqjkda[ s^n |hvpd;kj 8q;sko c]t gIqkwfh;jk 9eo;ooAoc,jo9eo;o,6oD

g;Qkvud1jk’|bj’G 1kdI6h;kj 9eo;ome,t-kf A gxao9eo;o,6o s^n [+F gryoj sko
A .sh9eo;o,6o]Po4afdao 4hkskdsko[+0kfIvf9eo;o,6o P c]t P2 > A
c,jo A gxao9eo;o,6oD
8q;1kj ’G

_( sko.sh @S #S & [+0kfD gIqksko.sh (S !!S !#D gIqkwfhG _( = ( V !# + *
_( = !! V ) + _S _( = !# V ( + *D ltcf’;kj _( c,oj 9eo;o,6oD

50

-D dko0Po9eo;ome,t-kfgxaozqo76o0v’9eo;o,6oD

9eo;ome,t-kfm5dM8q;lk,kf0Pogxaozqo76o0v’9eo;o,6owfhD c8j]trqf

0v’zqo76ofaj’dkj ; gvUo;jk lj;o76o,6oD

8q;1kj ’G !& = # V &S

@! = # V (S
#!& = !& V @! = # V& V# V(= #@ V & V (S
#* = $ V _ = @@ V #@D

‚ ;ymu-vdl;j o76o,6o0v’9eo;ome,t-kfF gryjosko9eo;ooAo.sh9eo;o,6o3fp

g]uj,9kd9eo;o,6o|hvpl5fmujsko.sh9eo;ooAo0kf c]t skozqoskowfh,k8+wx

g]NvpMD

8q;1jk’G dD #!& # gIqkwfhG #!& = #@ V & V (D 0D #* @ gIqkwfhG #* = @@ V #@D
105  3  18  2
35 5  9 3 
7  7 
1  3  3 ສະຫງສ.ວວນ.ລິສຂະສິດ
1 

‚ dkooe.-hl;j o76o,6ogrnjv-vdsk v5xt76oIj;,s^kpl5f c]t mt;u76oI;j ,|hvpl5fD
8q;1kj ’G #!& = #@ V & V ( c]t !W) = @@ V ##

Œ mt;u76oI;j ,|hvpl5f c,jozqo76o0v’lj;o76o,6omu8j kj ’daoma’\qf.shlj;o76o,6o
muIj j;,daoD le]a[lj;o76o,6omuIj j;,dao .shgvqkl;j o76omuj,ude]a’.spjd;jk\6Dj

LCM (315,108) = 22 × 33 × 5× 7 = 3780.

Œ v5xt76oI;j ,s^kpl5f c,jozqo76o0v’lj;o76o,6omuIj ;j ,daoma’\qf0v’r;d,ao
Xc8jgvqkl;j o76o,6oI;j ,daomu,j ude]a’|hvpd;kj \6j?

GCD (315,108) = 32 = 9.

pD lv’9eo;o,6o8+daoD
lv’9eo;ome,t-kfmuj,uv5xt76oIj;,s^kpl5fc,oj ! gvUo;jk lv’9eo;o

,6o8+daoD
\kp7;k,;kj a c]t b gxao9eo;o,6o8+dao g,njvGCD (a,b) =1.

8q;1kj ’G GCD (8,13) = 1F ltoAo 8 c]t 13 gxao9eo;o,6o8+daoD

51

ສະຫງສ.ວວນ.ລິສຂະສິດ#D [qfg/ydsaf

!D 9qj’d;fg[yj’9eo;o]5,j oU;kj 9eo;o.fgxao9eo;o,6oD
$#S $(S $_S *#S *(S *_S (&S (_S )(S )_S _(S _)S !!#S !!(S @!(S #((S !#(_S $#(#D

@D dD 9qj’[vd9eo;o,6omu|j hvpd;jk #WD
0D 9qj’0Po9eo;o]5j,oUgxaozqo76o0v’9eo;o,6oD
)$S !@!S @&WS @_$S !)WS @$_S )*$S @&@WS (_@WS )WWWS &($WS
#* V $@S (@ V((S !W) V(&S )$ V@& V!@!S #* V @( V!$@S *& V $_ V@$D

#D dD 9qj’[vdmt;u76oIj;, !W 8q;mevyf0v’ *S ) c]t !WD
0D 9q’j [vdmt;u76oI;j ,!W 8q;mevyf0v’ * c]t ) rhv,ma’[vdmt;u76oIj;,|hvp
l5f0v’r;d,aoD
7D 9q’j [vdmt;u76oIj;,!W 8q;mevyf0v’ * c]t !W rhv,ma’[vdmt;u76oI;j ,|hvp
l5f0v’r;d,aoD
’D 9q’j [vdmt;u76oIj;, !W 8q;mevyf0v’ !W c]t ) rhv,ma’[vdmt;u76oIj;,
|hvpl5f0v’r;d,aoD
9D 9qj’[vdmt;u76oIj;, & 8q;mevyf0v’ *S ) c]t !W rhv,ma’[vdmt;u76oIj;,
|hvpl5f0v’r;d,aoD

$D dD 9q’j [vdmt;u76oIj;, !W 8q;mevyf0v’ &S ( c]t !&D
0D9q’j [vdmt;u76oI;j , !W 8q;mevyf0v’ & c]t !& rhv,ma’[vdmt;u76oI;j ,
|hvpl5f0v’r;d,aoD
7D 9q’j [vdmt;u76oI;j , !W 8q;mevyf0v’ & c]t ( rhv,ma’[vdmt;u76oI;j ,
|hvpl5f0v’r;d,aoD
’D 9qj’[vdmt;u76oIj;, !W 8q;mevyf0v’ !& c]t ( rhv,ma’[vdmt;u76oIj;,
|hvpl5f0v’r;d,aoD
9D 9q’j [vdmt;u76oIj;, & 8q;mevyf0v’ &S ( c]t !& rhv,ma’[vdmt;u76oIj;,
|hvpl5f0v’r;d,aoD

&D 9qj’-vdskG
LCM (36, 42) F LCM (120,96) F LCM (18, 22) F LCM (82,52) F
LCM (250,150) F LCM (254, 672) F LCM (720,900) F LCM (14, 26,32) F
LCM (15, 75,36) c]t LCM (84, 62,56, 42) D

52

*D 9q’j 0Pod5j,v5xt76o0v’9eo;o]5j,oUG
(@S $)S @*S )$S *_S *WS !@WS !#*S @!@S _*S &*S !W)D

(D dD 9q’j 0Pod5j,v5xt76o0v’ *$S #* c]t $@D
0D 9q’j [vdv5xt76oIj;,0v’ *$ c]t #* rhv,ma’[vdv5xt76oI;j ,s^kpl5f0v’
r;d,aoD
7D 9q’j [vdv5xt76oIj;,0v’ *$ c]t $@ rhv,ma’[vdv5xt76oIj;,s^kpl5f0v’
r;d,aoD
’D 9qj’[vdv5xt76oIj;,0v’ $@ c]t #* rhv,ma’[vdv5xt76oI;j ,s^kpl5f0v’
r;d,aoD
9D 9q’j [vdv5xt76oIj;,0v’ *$S $@ c]t #* rhv,ma’[vdv5xt76oI;j ,s^kpl5f0v’
r;d,aoD

)D dD 9q’j 0Pov5xt76o0v’ @$S &*S )@ c]t *WD ສະຫງສ.ວວນ.ລິສຂະສິດ
0D 9q’j [vdv5xt76oIj;,0v’ )@ c]t *W rhv,ma’[vdv5xt76oI;j ,s^kpl5f0v’
r;d,aoD
7D 9qj’[vdv5xt76oIj;,0v’ @$ c]t &* rhv,ma’[vdv5xt76oI;j ,s^kpl5f0v’
r;d,aoD
’D 9q’j [vdv5xt76oI;j ,0v’ @$S )@ c]t &* rhv,ma’[vdv5xt76oIj;,s^kpl5f0v’
r;d,aoD
9D 9qj’[vdv5xt76oI;j ,0v’ *WS )@ c]t &* rhv,ma’[vdv5xt76oIj;,s^kpl5f0v’
r;d,aoD
lD 9qj’[vdv5xt76oIj;,0v’ @$S *WS )@ c]t &* rhv,ma’[vdv5xt76oI;j ,s^kpl5f
0v’r;d,aoD

_D 9qj’[vdv5xt76oI;j ,s^kpl5f0v’9eo;o]5,j oUG

d !*) c]t #*WS 9D *_# c]t _$&S

0D ##* c]t $*@S lD *)S _@ c]t (@S

7D @&@ c]t *)$S -D #*S *W c]t )WS

’D !@W c]t @$@S pD #)WS $@& c]t &WW

53

!WD 9eo;ome,t-kfc8j]t76j]5j,oU ,u76.j fgxao9eo;o,6o8+daoL

dD !(! c]t _!S lD (& c]t @)S

0D (& c]t )WS -D $# c]t #*S

7D !* c]t )!S pD !& c]t #@S

’D @!) c]t !*@S fD #@( c]t !@(S

9D @! c]t !)S oD @_( c]t!!(D

!!D dD 9q’j -vdsk LCM (576,360) c]t GCD (576,360) D
0D 9qj’7yfw]zj qo76o 576×360 c]t LCM (576,360)×GCD (576,360)
c]h;xP[mP[D

!@D dD 9qj’-Uc9’;jk __ c]t !$W c,jolv’9eo;o,6o8+daoD
0D 9qj’-vdsk LCM (99,140) c]t GCD (99,140) D
7D 9q’j xP[mP[]ts;jk’ LCM (99,140) c]t 99×140 D
’D 9qj’xP[mP[]ts;jk’ LCM (a,b) c]t a ×b g,nvj a c]t b gxaolv’
9eo;o,6o8+daoD
ສະຫງສ.ວວນ.ລິສຂະສິດ
!#D dD 9q’j -Uc9’;kj #(& c,oj v5xt76o0v’ *#(&D
0D 9qj’-vdsk LCM (375,6375) c]t GCD (375,6375) D

7D gIqkg;Qkwfhco;.fdjP;da[ mI|l c]t vIs^l 0v’lv’9eo;omujsko0kf
.shdaoL

!$D gryjox6d8Qow,hvhv,l;o8vo|bj’mugj xaoI6[lk,c9muj,u0hk’cmdwfh !$$ mS !)W m
c]t @$W m. .og’njvow0c8]j t9v,8hv’,u8Qow,h c]t w]ptskj ’]ts;jk’8Qow,h
lv’8Qo]Po8yfdao gmqkj dao g-yj’skj ’ 5m, 6m, 7m, 8m, 9m s^n !W m.
9qj’7yfw]j9eo;o8Qow,hmuj8hv’x6d.sh4ndda[g’njvow0mu;j k’w;hD

!&D de,tdvoz6h|bj’Ia[g’yo7jkcI’’kolk,gfnvo]Podaofaj’oUG $*@ WWW du[S
&@) WWW du[ c]t &_$ WWW du[D
dD9q’j 7yfw]j7kj cI’’koxt9e;ao0v’de,tdvofa’j dkj ;D .shI6h;jkg’yo7kj cI’’ko
xt9e;ao0v’]k;16j]ts;jk’ @W WWW du[ sk #W WWW du[D
0D9qj’7yfw]j9eo;o,NgIaf;Pd0v’]k;c8]j tgfnvoD

54

ສະຫງສ.ວວນ.ລິສຂະສິດ!*D gryjox6rNoshv’ovogInvos^a’|b’j mujgxaoI6[lujc9lkd c]t ,u0t|kf#*Wcm×@@&cm
fh;pczjodk3iD .shI6hczjodk3igxaoI6[9t85]afmu,j u0hk’cmdwfh.o]ts;jk’ !W cm
sk @& cmD
dD 9qj’7yfw]]j ;’pk;0hk’0v’czoj dk3imuj.-hgrnjv[+.shwfh8afczoj dk3ig;]kx6D
0D 9q’j 7yfw]j9eo;oczoj dk3i|hvpl5fmu8j hv’-Ng,nvj x6c]h;8hv’.shr=fuda[rNomu0j v’
shv’fa’j dkj ;L

!(D dD 9q’j -vdskmt;u76oI;j ,0v’ *S ) c]t !W muj16]j ts;jk’ &WW sk !WWWD
0D 4k,;jk 9eo;omujwfh,kc,oj mt;u76o0v’9eo;o.fL c]t 9eo;ooAogxaospa’
0v’ *S ) c]t !WL

!)D mhk; fu wfhltlq,ltc8,lv’xtgrf7nG ltc8,]k7k #WW du[ ,u $& vao c]t
ltc8,]k 7k &WW du[,u #* vaoD ]k;wfh9af;k’.ljxB,sq;|bj’muj,u _ |hkF
.og’njvow0c8]j t|hk8hv’,ultc8,xtgrffP;dao c]t ,u9eo;ogmqjkdaoD
dD 9qj’-vdsk vIs^l 0v’ $& c]t #*D
0D 9qj’7yfw]jc8j]t|hk,ultc8,9advaoL
7D 4k,;kj ,u9ad|hkmu[j ao95ltc8,]k7k #WW du[L ,u9ad|hkmu[j ao95ltc8,]k7k
&WW du[L

!_D grnvj v;prvo76.o,N;ao76csj’-kfmu ( 85]kF z6hveo;pdko3I’IPocsj’|b’j wfh-N
fvdd5s^k[ lk,-toyf7nG fvdd5s^k[lucf’ !@W fvdF fvdd5s^k[lu0k; )W fvd
c]t fvdd5s^k[lugs^nv’ *W fvd ,v[.shshv’dko[=]ysko9afgxao-+D .og’nvj ow0
c8j]t-+8hv’ ,u9eo;ofvdd5s^k[gmqkj dao c]t c8]j t-toyfd=.sh,u7ndaoD
dD 4k,;kj gryoj 9t9af.shs^kpl5f9ad-+ grnvj .shfvdd5s^k[\qfr=fuL
0D c8]j t-+,ufvdd5s^k[c8j]t-toyf9adfvdL

@WD gryoj 9afl=lucf’9eo;o @!* glAo c]t l=lugs^nv’9eo;o !&@ glAo.ljda[.sh\qfr=fu
.og’nvj ow0F c8j]tda[.sh,ul=lufP;dao c]t ,u9eo;ogmqjkdaoD
dD 4k,;jk9eo;ol=16.j oc8j]tda[,ugmqjk.fL
0D 4k,;jk9eo;oda[l=ma’\qf,u9adda[ c]t c8]j t-toyf,u9adda[L

55

[qfmu ) 9eo;o.orNo4ko8kj ’M

!D dyf9tde

dyf9tdemu !
dD .og,nvj djvo-k;3ic,’wfh.-hg7nvj ’\kp9advao grnvj 0Po9eo;oL 9q’j 0Poc8j

]tg7njv’\kpD
0D c8]j tg7nvj ’\kp,u7jkgmqjk.f.o9eo;ome,t-kfL
7D gryoj wfhdeoqfs^addko0Po c]t vkj o9eo;omu0j Pofh;pg]d3ic,’7nco;.fL
’D 9q’j 0Po9eo;o8+wxoUgxao9eo;ome,t-kffh;p8q;g]dvkia[G

a. MDXXXVII; c. MMCMVI; e. MDLXXVIII;
b. CDLXXIII; d. CCCXL; g. MCII;
f. MMMCMXCIX.

9D 9qj’0Po9eo;o8+wxoUfh;pg]d3ic,’G ສະຫງສ.ວວນ.ລິສຂະສິດ
a. @WW(S b. !W(&S c. _)$S d. !$*#S e. #_)@

dyf9tdemu @
grnvj 0Po9eo;ome,t-kfF gryoj .-hg7nvj ’\kpgvUo;kj g]dvkia[D
dD 9qj’0Po8q;g]dvkia[,k.sh\qfD
0D 15 = 10 + 5,

203 = 2×100 + 0×10 + 3 = 2×102 + 0×10 + 3,

4530 = 4×1000 + 5×100 + 3×10 + 0 = 4×103 + 5×102 + 3×10 + 0

dko0Po0hk’gmy’oUc,oj dko0Po9eo;ome,t-kfc[[dt9kp.orNo4koly[D
9qj’0Po9eo;o]5,j oUc[[dt9kp.orNo4koly[ c]h;0Po9eo;ooAofh;p8q;|a’lnD

a. 18; d. 158; g. 37 500; j. 82 004 612;
b. 27; e. 368; h. 257 901; k. 3 400 783 256;
c. 94; f. 2 586; i. 7 403 179;
. 132 785 912.

56

dyf9tdemu #
.op5dltw\gvg]ad38ioydF gryoj wfhoe.-hg7nvj ’7yfw]jgvg]ad38ioydX7v,ru;g8u?

g0Qk.odko7eo;o8kj ’MD c8jg7njv’gvg]ad38ioydlk,kfvjko8q;g]dwfhlv’8q;
gmqjkoAo7nG W c]t ! Xdtclw2[+zkj oc,jo WF dtclw2zkj oc,jo !?D

9eo;omuj0Poc[[dt9kp3fp.-h W c]t ! gvUo9eo;ooU;jk 9eo;o.orNo4ko
lv’ g-ajoG

9eo;o.orNo4kolv’ 9eo;o.orNo4koly[

11 = 1× 2 +1 = 3
101 = 1× 22 + 0× 2 +1 = 5
1101 = 1× 23 +1× 22 + 0× 2 +1 = 13

9q’j 8nj,8k8t]k’]5,j oU.sh4nd8hv’c]t7q[4h;o

9eo;o.orNo4kolv’ 9eo;o.orNo4koly[
ສະຫງສ.ວວນ.ລິສຂະສິດ
111 …
1111 …
110111 …

0=h7;ogvqk.9.ljG

.orNo4kolv’G

♦ !W [+vjko;kj ly[ c]t vjko;jk < |bj’l6o < grkt,ao\kpg4y’ @ .o9eo;o

me,t-kfD

♦ !! [+vkj o;jk ly[gvaf c]t vjko;jk < |bj’|bj’ < grkt,ao\kpg4y’ # .o9eo;o

me,t-kfD

dD 8k8t]k’]5,j oU c,jo8k8t]k’-Ao0v’dko0Po c]t dkovkj o9eo;o

.orNo4kolv’D
DDD @*= *$ @& = #@ @$= !* @# = ) @@= $ @! = @ @W= !

!W

!!

!! W !!

57

8q;1kj ’G .orNo4koly[

2 = 1× 2 + 0 mP[gmqkj da[ 10 .orNo4kolv’

3 = 1× 2 +1 mP[gmqjkda[ 11 .orNo4kolv’

27 = 16 + 8 + 0 + 2 +1 = 1× 24 +1× 23 + 0× 22 +1× 2 +1 mP[gmqjkda[ 11011

.orNo4kolv’

9q’j 0Po9eo;o.orNo4koly[]5j,oUgxao9eo;o.orNo4kolv’D

a. )S cD !#S eD @!S gD !W!W

bD _S dD @WS f D !WWS hD !W!!W

0D 9q’j la’gdf[aofk8q;glf.odkoskogvu d]uf9eo;o8jk’M .sh @ 8+gonvj ’dao9qo

.shwfhzqoskogmqjk ! fa’j ]5,j oUG

!D @ @ gIqkwfh @ = ! V @ + WD
0 1 faj’oAo 9eo;ome,t-kf @ 0Pogxao9eo;orNo4kolv’c,jo !WD

@D # @ gIqkwfh # = ! V @ + !D

1 1 fa’j oAo 9eo;ome,t-kf # 0Po.orNo4kolv’c,oj !!D
gIqkwfh !@ = ! V @#+ ! V @@+W V @!+ W V !ສະຫງສ.ວວນ.ລິສຂະສິດ
#D !@ @

 0    6 2 faj’oAo !@ 0Po.orNo4kolv’c,oj !!WWD
0    3 2

1   1

3fp.-hdkoskogvu d]ufF 9qj’0Po9eo;ome,t-kf8+wxoUgxao9eo;o.orNo4kolv’D

*S _S )S !#S @$S !!S *WS #&S $@S !)S @&S @!)S !&WS @WWS )(S *$D

dyf9tdemu $
gryjo0Po9eo;o.orNo4ko a,( a ∈ ,a ≥ 2) mu.j -h8q;g]d WS !S @S # c]t $ gmqkj oAoD
dD 4k,;jk 9eo;omuj0Po0Bo.\joUgxao9eo;o.ornho4ko.fL
0D 9q’j 0Po9eo;ome,t-kf]5,j oU gxao9eo;o.orNo4ko a faj’dkj ;D

@S #S $S *S (S )S !WS !@S !(S @)S (*S !W@S @!(S &#@S *WWS $&#D
7D 9qj’0Po9eo;o.orNo4ko a ]5,j oUgxao9eo;ome,t-kfD

@S !S #S !WS !@S @$S #@!S $$W@S !#$W!S ##@S #WWS $W#D
’D 9q’j 0Po9eo;o.orNo4ko & ]5j,oUgxao9eo;ome,t-kf c]h;0Pogxao9eo;o.o

rNo4kolv’D
@S #S $S !#S $@S @#S !@S #$S @#$S #W@S !!!S $WWS !W#!S @!$@D

58

@D .97;k,

dD 9eo;o.orNo4ko8jk’MD
Œ gryjogvUo;jk 9eo;o.orNo4ko a, (a ∈ c]t a ≥ 2) c,oj 9eo;omu0j Pofh;p

8q;g]d WF !F @F DDD F a −1D
Œ 9eo;o.orNo4kolv’F gryjo.-hrP’c8lj v’8q;g]d7n W c]t ! grnjv0Po c]t oa[

faj’oUG
W S ! S !W S !! S !WW S !W! S !!! S DDD

7evjkoG l6oS |bj’S |bj’l6oS |bj’|b’j S |b’j l6ol6oS |bj’l6o|bj’S |b’j |b’j |bj’S DDD

8k8t]k’-Aodko0Po c]t dkovjko9eo;o.orNo4kolv’D
DDD @*= *$ @& = #@ @$= !* @# = ) @@= $ @! = @ @W= !

Œ 9eo;o.orNo4koshkF gryjo.-hrP’shk8q;g]dvkia[mevyf7nG WF !F @F # c]t $

grnjv0Po c]t vkj oF g-ajoG ສະຫງສ.ວວນ.ລິສຂະສິດ

!WS !!S !@S @@S @#S @$S #WS DDD

7evjkoG |bj’l6oS |bj’|b’j S |bj’lv’S lv’lv’S lv’lk,S lv’lujS lk,l6oS DDD

8k8t]k’-Aodko0Po c]t vjko9eo;o.orNoshkD
DDD &* && &$= *@& &#= !@& &@= @& &! = & &W= !

0D dkoxPj orNo4ko0v’9eo;oD
Œ xjPo9eo;o.orNo4kolv’gxao9eo;ome,t-kfD

8q;1kj ’G 9eo;orNo4kolv’ A = 111011; Β =1001111 c]t C = 10111111.
xjPogxao9eo;ome,t-kfG
A = ! V @&+ ! V @$+ ! V @#+ W V @@+ ! V @ + !
= #@ + !* + ) + W + @ + ! = &_
B = ! V @*+ W V @&+ W V @$+ ! V @#+ ! V @@+ ! V @ + !
= *$+ W + W + ) + $ + @ + ! = (_
C = ! V @(+W V @*+ ! V @&+ ! V @$+ ! V @#+ ! V @@+ ! V @ + !
= !@) + W +#@ + !* + ) + $ + @ + ! = !_!

59

ΠxPj o9eo;o.orNo4koshkgxao9eo;ome,t-kfD
8q;1kj ’G .sh9eo;o.orNo4koshk A = @!#S B = !!W#$ c]t C = @$!W# 9q’j xPj o
gxao9eo;ome,t-kfD

A = @ V &@+!V & + # = &W + & + # = &)
B = ! V &$+ ! V &#+ W V &@+ # V & + $ = *@& + !@& + W + !& + $ = (*_
C = @ V &$+ $ V &#+ ! V &@+ W V &+# = !(()

Œ xPj o9eo;ome,t-kfgxao9eo;o.orNo4kolv’D
;ymumu !G .-h8q;glf.odkoskogvu d]uf 9eo;omu.j sh sko.shlv’8yf8+dao9qozqosko
wfh,kgmqkj !D
8q;1jk’G
xjPo9eo;ome,t-kf A = !#S B = (S C = @) gxao9eo;o.orNo4kolv’D

!# @ (@ @) @ ສະຫງສ.ວວນ.ລິສຂະສິດ2 

1  6  2 1   3      2 0   14 2 
0  3 2 1    1  0   7  2 
1   1  1   3 
B = !!!
A = !!W! 1   1 

C = !!!WW

;ymumu @G 0Po9eo;omuj.shgxaoI6[dt9kp0v’g]dde]a’rNolv’D
8q;1kj ’G

A = !# = )+$ +! = ! V @#+ ! V @@+ W V @ + !F faj’oAo A = !!W! .orNo4kolv’D
B = ( = $ + @ + ! = ! V @@+ ! V @ + !F fa’j oAo A = !!! .orNo4kolv’D
C = @) = !* + ) + $ = ! V @$+ ! V @#+ ! V @@+W V @+ WF faj’oAo C = !!!WW .o
rNo4kolv’D

60

g-ajofP;dao le]a[9eo;o.orNo4koshkD

8q;1jk’G xjPo9eo;ome,t-kf A = !#@ S B = !( S C = @)W gxao9eo;o.o

rNo4koshkD

!#@ & !( & @)W &

2   26  5  2   3  0   56  5 

1     5  5 1    11  5 

0    1  1     2 

A = !W!@ B = #@ C = @!!W

s^n
A = !#@ = !@& + & + @ = !V&#+ WV&@+!V&+ @F fa’j oAo A = !W!@ .orNo4koshkD

B = !( = !& + @ = # V & + @F fa’j oAo A = #@ .orNo4koshkD
C = @)W = @&W + @& + & = @ V &# + ! V &@ + ! V & + WF faj’oAo C = @!!W .orNo

4koshkD

Œ xPj o9eo;o.orNo4ko a gxao9eo;o.orNo4ko b Xa∈ ,a ≥ 2; b∈ ,b ≥ 2) D

grnvj xPj o9eo;o.orNo4ko a gxao9eo;o.orNo4ko bF gIqkxPj o9eo;o.orNo
4ko a gxao9eo;ome,t-kfglpdjvo c]h;9b’j xPj ome,t-kfoUgxao9eo;o.o

rNo4ko b.
ສະຫງສ.ວວນ.ລິສຂະສິດ

8q;1kj ’G
9q’j xjPo9eo;o.orNo4koshk8+wxoU A= @!$S B = !W$ gxao9eo;o.orNo4kolv’D
{ xjPo9eo;o A c]t B gxao9eo;ome,t-kf fa’j oUG
A = @ V &@+ ! V & + $ = &W + & + $= &_S
B = ! V &@+ W V & + $ = @& + $= @_S
{ xPj o9eo;ome,t-kf &_ c]t @_ gxao9eo;orNolv’D
&_ = #@ + !* + ) + @ + ! = ! V @& +! V @$ +! V @#+ ! V @ + !F fa’j oAo
A=!!!W!! .orNo4kolv’D

@_ = !* + ) + $ + ! = ! V @$ +! V @# + ! V @@+ W V @ + !F fa’j oAo
B = !!!W! .orNo4kolv’D

61

#D [qfg/ydsaf

!D 9q’j xPj o9eo;ome,t-kf8+wxoUG #S $S &S (S _S !@S !&S !!WS !&*S @!_S _)D

dD gxao9eo;o.orNo4kolv’ 7D gxao9eo;o.orNo4kolju

0D gxao9eo;o.orNo4kolk, ’D gxao9eo;o.orNo4koshk

@D 9q’j xPj o9eo;o.orNo4ko8kj ’M8+wxoU gxao9eo;ome,t-kfD

dD 9eo;o.ornho4kolv’G

A = !W!W!WS B = !W!!!WS C = !WWWW!S D = !W!!!!!S E = !!!!WW!D

0D 9eo;o.ornho4kolk,G

A = !@S B = !W@S C = @@!WS D = @W!!!S E = !W@!@D

7D 9eo;o.ornho4koluGj

A = #!WS B = !@W#S C = @WW!#S D = #WWWS E = !WW!@D

’D 9eo;o.ornho4koshkG

A = !!WS B = !W!@S C = #$@!$S D = $$$##S E = #$@W!D

#D dD 9q’j xPj o9eo;o.orNo4kolv’8+wxoUgxao9eo;o.orNo4kolk,F gxao9eo;o.o
ສະຫງສ.ວວນ.ລິສຂະສິດ
rNo4koluj c]t gxao9eo;o.orNo4koshkD

A = !!!!!!W!S B =!!!S C = !WS D =!W!W!!!S E = !!!WW!!D

0D 9q’j xPj o9eo;o.orNo4kolk,8+wxoUgxao9eo;o.orNo4kolv’F gxao9eo;o.o

rNo4kolju c]t gxao9eo;o.orNo4koshkD

A = !W@S B = @!!!S C = @@!WS D = !@!W!S E = !!@W@D

7D 9qj’xPj o9eo;o.orNo4koshk8+wxoUgxao9eo;o.orNo4kolk,F gxao9eo;o.o

rNo4koluj c]t gxao9eo;o.orNo4kosqdD

A = #$!S B = $W!@S C = ##!WS D = @$W#S E =!!@W#D

$D 9eo;o.orNo4kolv’ wfh9af]Po8k,]efa[0Bofaj’oU W<1<10<11<100<101...

dD 9q’j 9af]Po9eo;o.orNo4kolv’]5,j oU8k,]efa[0BoD

!!!!S !W!!S !WS !!WS !W!WS !!!S !!W!!S !WWS !WW!!S !!!!!D

0D 9q’j 9af]Po9eo;o.orNo4kolv’]5,j oU8k,]efa[cI,D

!!S !WW!S !!!!!S !WWW!S !!!W!S !W!S !S !WWWWS !!W!!S !W!!!D

62

&D 9eo;o.orNo4koshkwfh9af]Po8k,]efa[0Bofaj’oU W<1< 2< 3< 4 < 10< 11<12...

dD 9q’j 9af]Po9eo;o.orNo4koshk]5,j oU8k,]efa[0BoD

###S @$$S @$S #!WS @WWS !$!S @$#S $S #W!S @W!S @#!D

0D 9qj’9af]Po9eo;o.orNo4koshk]5j,oU8k,]efa[cI,D

!W!S $W!S #@S @@!S #$S !!@S @#!S !!$S #W!S #$!D

*D gryjo.shrkdlj;o|b’j 0v’8k8t]k’dko[;d c]t 8k8t]k’dko76o9eo;o.orNo

4kolv’D

+ W ! !W !! × W ! !W !!

W W ! !W !! W WWW W

! ! !W !! !WW ! W ! !W !!

!W !W !! !WW !W! !W W !W !WW !!W

!! !! !WW !W! !!W !! W !! !!W !WW!

dD9qj’7yfw]jzqo[;d]5j,oU rhv,ma’d;f7nozqowfhIa[3fpdkoxPj ogxao9eo;ome,t-kfD
ສະຫງສ.ວວນ.ລິສຂະສິດ
a.!!W!! + !!W!S b.!!!WW! + !!!WS c. !WW!+ !!!S

d. !!!!+ !!!S e.!W!!!+ !W!!

0D9qj’7yfw]jzqo]q[]5,j oU rhv,ma’d;f7nozqowfhIa[3fpdkoxPj ogxao9eo;ome,t-kfD

a. !!W!! – !!W!S b. !W!W! – !WW!S c. !WW! – !!!S

d. !WW!W – !WW!S e. !!WW!!– !WW!W!

7D 9q’j 7yfw]zj qo76o]5,j oU rhv,ma’d;f7nozqowfhIa[3fpdkoxPj ogxao9eo;ome,t-kfD

a. !!W!!×!WS b. !!WW!×!!S c. !W!W×!W!S d. !W!!×!!WS e.!!!!!× !!!D

(D dD9q’j lhk’8k8t]k’dko[;d c]t 8k8t]k’dko76o 0v’9eo;o.orNo4kolk,

]5,j oUD

+ W ! @ !W × W ! @ !W

WW

!!

@@

!W !W

63

0D 9q’j 7yfw]jzqo[;d]5j,oU.orNo4kolk, rhv,ma’d;f7nozqowfhIa[3fpdkoxjPogxao

9eo;ome,t-kfD

a. !@@W + !!@S b. @!W@! + !@@!@S c. !WWW! + @@@@@S

d. @!!@W + !@@!S e.!!@@W!+ !@!!@@

7D 9qj’7yfw]zj qo]q[]5j,oU.orNo4kolk, rhv,ma’d;f7nozqowfhIa[3fpdkoxPj ogxao

9eo;ome,t-kfD

a. @@W – @!S b. @!W@! – !@@!@S c. @@@@ – !W!@S

d. @W!WW! – @@!@@S e. @!@W!W – !!@@@

’D 9qj’7yfw]zj qo76o]5,j oU.orNo4kolk, rhv,ma’d;f7nozqowfhIa[3fpdkoxPj ogxao

9eo;ome,t-kfD

a. @@W×@!S b. !W@×!@S c. !W@@!×@!WS d. @@!W!@×@W@S e. @@@@×@@D
)D 9qj’d;fg[y’j ;jkzqo[;d]5j,oU7yfw].j orNo4ko|hvpl5frNo4ko.fD

a. @@!W@ + @@@@ = #W##WS d. #$& + @@$ = !W!#S ສະຫງສ.ວວນ.ລິສຂະສິດ
b. @@!W@ + @@@@ = @$#@$ e. &#$ + @*& = !W@!S
c. @@!W@ + @@@@ = #@!W!S f. @!@ + !W! = #!#D

_D 9q’j 7yfw]7j jk\;f7eo;o]5j,oU X r4 = .orNo4ko ?D

a. X!!!?r4@ + X!!?r4@S i. X$W*?r4( − X@!&?r4(

b. X!W@?r4# + X@@W?r4#S j. X$#@?r4) − X!*&?r4)S

c. X!#@?r4$ + X!#?r4$ k. X$!$?r4& + X@!!?r4& + X#@@?r4&SS

d. X!$@?r4* + X@$#?r4*S . X$!$?r4* + X@!!?r4* + X#@@?r4*S

e. X#!&?r4) + X$(#?r4)S m. X$!$?r4( + X@!!?r4( + X#@@?r4(

f. X!@@?r4# − X!@?r4# n. X@!#?r4& + X!WW?r4& + X@@!?r4&S

g. X#@#?r4$ − X@@!?r4$S OD X@!#?r4* + X!WW?r4* + X@@!?r4*S

h. X@!@?r4* − X!W&?r4*S p. X@!#?r4$ + X!WW?r4$ + X@@!?r4$S

!WD dD 9qj’lhk’8k8t]k’0v’dko76o9eo;o.orNo4koluj c]h; 7yfw]jzqo76o]5j,oUD

a. X#!#?r4$ × X@@?r4$S b. X@#!?r4$ × X!#@?r4$ S c. X@W!?r4$× X#@?r4$
0D 9q’j lhk’8k8t]k’0v’dko76o9eo;orNo4koshk c]h; 7yfw]zj qo76o]5,j oUD

a. X#$?r4&× X@#?r4&S b. X@#$!?r4&× X#$?r4& S c. X@$!?r4&× X$@?r4&D

 

64

ສະຫງສ.ວວນ.ລິສຂະສິດ [qfmu _ 7jkle[6o

I6[rk[oadIPovvdde]a’dkp

w]ptskj ’]ts;jk’lv’9eo;o
!D dyf9tde

dyf9tdemu !
dD .sh7yfw]j − (+3) = DDDDDD ; − (−5)=…..;
0D 9qj’la’gdf9eo;o]5j,oU c]h;-vdsk

w]ptskj ’c8j W sk $S −@ sk #S
−$ sk !S −# sk @S −@ sk #D

$ sq;|;j p

−4 −3 −2 −1 0 1 2 3 4 5

& sq;|;j p

65

dyf9tdemu @
9q’j 8n,j 8k8t]k’]5,j oUD

ab a−b b−a w]ptskj ’c8j a sk b

15 10 15 − 10 = 5 10 − 15 = − 5 5
− 12 −7
− 10 0

7 −12
−3 2

9q’j .sh0=hla’gdfzqomuwj fhIa[

DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD

@D .97;k, ສະຫງສ.ວວນ.ລິສຂະສິດ

w]ptsjk’]tsj;k’9eo;o a c]t b gmqjkda[ a − b 4hk;jk a ≥ b
b − a 4hk;jk b ≥ a

8q;1jk’ !D 9qj’-vdskw]ptskj ’]ts;jk’ 7 c]t 35; −8 c]t – 2.
;ymucdhG

35 > 7, ltoAo w]ptsjk’]tsj;k’ 7 c]t 35 c,jo 35 − 7 = 22 sq;|j;pD
−2 > −8, ltoAo w]ptsjk’]tsjk’ − 2 c]t − 8 c,jo − 2 − (−8) = 6

8q;1jk’ @D vksko9ko|bj’164j hkooEdhvo,uv5ostr6, −5DC . g,jnvgvqk,k;k’16j
ovdw]ptg;]k.f|bj’ v5ostr6,wfhgruj,0Bogxao 20DC . 4k,;jk

20° v5ostr6,gruj,0Bo9kd]tfa[g]uj,8QoIvf]tfa[l5fmhkpgmqkj .fL

0°

−5° ;ymucdhG
v5ostr6,wfhgru,j 0BoG 20DC − (−5DC) = 25DC

66

#D [qfg/ydsaf
!D .sh-vdskw]ptmk’ c8j[hkofvo|6o XKm 10) skg0fmkj ’vj o XKm 28)F glAo
mk’wx mkj ]kfD

@D Ihko7hkcs’j |bj’lts^5[]kpwfhxt9egfnvoly’sk gsao;kj 0kfmbo $ ]hkodu[Dສະຫງສ.ວວນ.ລິສຂະສິດ
,kIvfgfnvodaopk.oxufP;dao lts^5[wfhdew] !W ]hkodu[D c8gj fnvo ly’sk
,kIvf gfnvodaopk wfhdew]gmqjk.fL

7kj le[6o

!D dyf9tde
9q’j -vdskD
dD w]ptskj ’c8j W sk −$ c]t c8j W sk $D
0D w]ptc8j W sk −# c]t c8j W sk # c]h;.sh0=hla’gdfzqomujwfhIa[D

@D .97;k,

oypk,G 7kj le[6o0v’ a g-yj’gryjo.shlaopt]adfh;p a c,oj w]ptskj ’c8j a sk WD

mk’fhkorbf-t7toyfgryjo0PoG a = a 4hk;jk a≥0
a = −a 4hk;jk a≤0

\kpgsfG le]a[m5dM9eo;o a, a ≥ 0

67

8q;1jk’ !D 9qj’-vd7jkle[6o0v’9eo;o8+wxoU: −3; 5; −5 ; 2.
;ymucdhG −3 = − (−3)= 3; 5 = 5; −5 = −(−5) = 5 ; 2 =2

8q;1jk’ @D 9qj’7yfw]jleo;o]5,j oUG

a. −5 ; b. 4 ; c. −7 ; d. 5 − −8 ;

e. − 7 +2 f. 8 − − 6 ; g. 4 − 3 +5

;ymucdhG

a. −5 = − (−5) =5 ; b. 4 = 4 ; c. −7 = −(−7) = 7 ;

d. 5 − −8 = 5 − 8 = −3; e. − 7 +2 = 7 + 2 = 9

f. 8 − − 6 = ) − X*? = 2; g. 4 − 3 +5 = 4×3 +5 = 17

8q;1kj ’ #D 9qj’[vd;jk ,u9eo;o.fcfmj uj,u7kj le[6ogmqkj #S $ c]t ( 8k,]efa[D
c]h;ltcf’9eo;ofaj’dkj ; c]t 7jkle[6o.lcj do9eo;oD

;ymucdhG
9eo;omu,j u7kj le[6ogmqkj # ,uG − 3 c]t 3.
9eo;omuj,u7jkle[6ogmqkj $ ,uG − 4 c]t 4.
9eo;omuj,u7kj le[6ogmqjk 7 ,uG −7 c]t 7 .
ສະຫງສ.ວວນ.ລິສຂະສິດ
# sq;|j;p # sq;|j;p
-
−7 −4 -−3- −1 0 1 34 7

$ sq;|j;p $ sq;|j;p

( sq;|;j p ( sq;|j;p

8q;1kj ’ $D 9qj’[vd7jk0v’ x 8k,g’njvow0]5j,oUG
a. x = 5, 9eo;omuj,u7jkle[6ogmqjk & ,uG −5 c]t 5.
b. x − 5 = 1, gIqklk,kf0Po x = 6, x ,u7jkgmqjk −6 c]t 6.
c. x − 2 = 3, mk’fhkog]0k7toyf leo;ofaj’dkj ;\kpg4y’w]ptc8j x
sk 2 gmqkj 3.

68

mk’fhkorbf-t7toyf leo;ofa’j djk;\kpg4y’ x – 2 gmqjk −3 s^n 3 (8k,oypk,?

gIqklk,kf0PoG

x −2=−3 s^n x − 2 = 3

x = −3 + 2 s^n x =3+2

x = −1 s^n x = 5

# sq;|;j p # sq;|;j p

−1 0 2 5

8q;1kj ’ &D -vd7jk0v’ x muj8v[lt|v’g’njvow0]5j,oUD

a. 5 − x = 3 b. 2 x = 8 c. 3 − x = −1 d. x = x

e. x + 1 = 0 f. 2 x = 0 g. x + x = 0
ສະຫງສ.ວວນ.ລິສຂະສິດ
;ymucdhG

a. grkt;jk 5 − 2 = 3 wfh x = 2 D 9kdoU 7jk0v’ x c,oj −@ c]t @D

b. grkt;jk 2× 4 = 8 wfh x = 4 D 9kdoU 7kj 0v’ x c,jo −$ c]t $D

c. grkt;jk 3− 4 = −1 wfh x = 4 D 9kdoU 7kj 0v’ x c,oj −$ c]t $D

d. grkt;jk x = x, x ≥ 0 9kdoU 7jk0v’ x c,oj [+c,jo9eo;o]q[D

e. grkt;jk x ≥ 0 9kdoU x + 1 ≥1 ≠ 0 fa’j oAo 9b’j [+,u7kj x .fmu8j v[lt|v’

x + 1 = 0D

f. grkt;kj 2×0 = 0 faj’oAo x = 0 9kdoU 7jk0v’ x = 0 D

g. grkt;jkzqo[;d0v’lv’9eo;omuj[+gxao9eo;o]q[ gmqjkda[ 0 d=8+g,njvlv’

9eo;ooAo]h;oc8jc,jo 0 D fa’j oAo x = 0 9kdoU 7kj 0v’ x = 0 D

8q;1jk’ *G dkooe.-h7kj le[6oD

lq,,5f;kj 76.shoadIPolhk’I6[c[[9e]v’0v’0q;mu,j u];’pk; 100cm 3fp

vto5pkf.sh,u 7jkzyffjP’[+gduo 1cm . mk’fhko7toyflkf gIqklk,kf0Po7jkzyf

fjP’gmqkj da[ 1 = 101−100 = 99 −100 . 9qj’[vd];’pk;0v’0q;mu7j 6Ia[wfhD

69

;ymucdhG 4hk7kj zyffjP’[+gduo 1 ];’pk;0v’0q;9t16j]ts;jk’ 99 cm sk101 cm.

75o]adltot
!D le]a[m5d9eo;o x, x = 0 mP[gmqkj da[;kj x = 0
2. le]a[m5dM9eo;o x c]t y gIqk,uG x.y = x . y
3. le]a[m5d9eo;o x c]t y gIqk,uG x + y ≤ x + y
ສະຫງສ.ວວນ.ລິສຂະສິດ
8q;1kj ’G b. x = −3; y = −4

a. x = −2; y = 5 xy = (−3)(−4) = 12 = 12,

xy = (−2)(5) = −10 = 10, x y = −3 −4 = 3× 4 = 12.

x y = −2 5 = 2×5 = 10.

c. x = −2; y = 5 d. x = −3; y = −4

x + y = (−2) + 5 = 3 = 3, x + y = (−3) + (−4) = −7 = 7,

x + y = −2 + 5 = 2 + 5 = 7, x + y = −3 + −4 = 3 + 4 = 7,
x+y < x + y. x+y = x + y.

#D [qfg/ydsaf

!D 9q’j 8nj,8k8t]k’]5j,oU c]h;d;f7nofh;pdkooe.-hcdo9eo;oD

x −3 −2 −1 0 12 3
x

70

2. 9qj’[vd7jk0v’ x c]t y grnjv.sh x + y = x + y c]t grnjv.sh

x+ y ≤ x + y.

#D 9qj’;k’ $ 9eo;o X-bj’gxao7jk0v’ x ? mu8j v[lt|v’g’nvj ow0 x − 2 ≤ 1
.ljcdo9eo;o c]h;mklug0fmuj9eo;o8v[lt|v’g’njvow0fa’j djk;D

−2 −1 0 1 2 3

dkooe.-h7kj le[6og0Qk.odko-vdskw]ptsjk’]ts;jk’lv’9eo;o

!D dyf9tde

dD 9qj’8n,j 8k8t]k’]5j,oUG

ab a−b a − b w]ptsjk’]ts;kj ’ a c]t b
15 10 15 − 10 = 5 ສະຫງສ.ວວນ.ລິສຂະສິດ
−12 − 7 15 −10 = 5 5

−10 0

0D 9qj’xP[mP[zqomujwfhIa[G
DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD

@D .97;k,

w]ptskj ’]ts;kj ’lv’9eo;o a c]t b gmqkj da[7jkle[6o a −b

8q;1kj ’ 1. w]ptsjk’]ts;kj ’lv’9eo;o −172 c]t 38 c,oj :

(−172) − 38 = −210 = 210 .

8q;1jk’ @D w]ptmk’c8sj ^ad $W (km 40) 4t|qog]dmu !# .8h sk3ro3I’ s^ad 70
(km 70) 4t|qog]dmu !# g|nv c,oj 70 − (−40) = 110 = 110 km.

71

#D [qfg/ydsaf

!D 9qj’7yfw]jG

dD 2 + 3 lD 7− − 3

0D − 2 + 3 -D − 5 × 2

7D − ( − 2 + 3 ) pD −5( − 5 − − 2 )

’D − ( − 2 + 3 ) fD − 3 − 2

9D − 7 − (5 − 7) 8D − 2 + − 3

@D 9qj’-vdlv’9eo;omuj,uw]ptsjk’gmqkj G

dD !@S 0D !&S 7D 4 S ’D 32D

#D 9qj’-vd7jk0v’ x muj8v[lt|v’g’njvow0]5j,oUG

dD x = 2; 0D x +1 = 4;

7D x = 3; ’D 6 − x = 2.

$D 16jgmy’cdo9eo;o 9q’j mklug0fmuj8v[lt|v’g’nvj ow0]j5,oUGສະຫງສ.ວວນ.ລິສຂະສິດ

dD x < 2

−4 −3 −2 −1 0 1 2 3 4 5

0D x > −2 c]t x < 2

−1 0 1

7D x > 2

−1 0 1

’D x < −2 s^n x > −2 1

−1 0

&D .sh[vd7kj 0v’ x + 2 c]t x +2 g,njv x = −# c]h;xP[mP[zqomuwj fhIa[D
g,njv x = −# 9tgxaoco;.fL

72

[qfmu !W leo;og]d7toyf ສະຫງສ.ວວນ.ລິສຂະສິດ

!D dyf9tde
dyf9tdemu !D

9q’j 8nj,9eo;omuj4nd8hv’.l8j k8t]k’8+wxoUG
dD

( 9x1)+2=
( 9x12)+3=
( 9x123)+4=
( 9x1234)+5=
( 9x12345)+6=
( 9x123456)+7=
( 9x1234567)+8=
( 9x12345678 ) +9=

0D

( 8x1)+1=
( 8x12)+2=
( 8x123)+3=
( 8x1234)+4=
( 8x12345)+5=
( 8x123456)+6=
( 8x1234567)+7=
( 8x12345678)+8=
( 8x123456789 ) +9=

dyf9tdemu @D

9q’j djkp7no c]t 8n,j ;q’g]a[.llj eo;o]5,j oU.sh4nd8hv’

dD 8 + 6 × 2 = 28 0D 7 + 8 ÷ 8 = 8 

  7D 20 + 12 ÷ 3 = 24     ’D 13 − 1 ÷ 4 = 3  

  9D 10 + 2 × 6 − 4 = 18     lD  6 − 3 + 9 × 0 = 0  

    -D 12 + 4 ÷ 24 ÷ 3 = 2    pD 2 × 6 − 4 + 4 = 4

73

dyf9tdemu #D

dD 9qj’0Poleo;og]d7toyf g-y’j 7jk0v’,aogmqjkda[ ! 3fp.-hdt8ydk8+wxoUG
D .shg]nvd.-hg7nvj ’\kp +F –F ×F ÷ c]t ;q’g]a[F
D .shoe.-h8q;g]dm5d8q; !F @F # c]t $ 3fp;kj c8]j t38g]d.-hwfhrP’7A’fP;

g-aoj G (3× 2) − (4 +1) = 1

0D .shoe.-hdt8ydk0hk’gmy’F 0Poleo;og]d7toyfmu7j jk0v’,aoc,jo @D
7D ln[8+F -vdskleo;o g]d7toyfmuj7kj 0v’,aoc,oj #F $F DDD F!@D

dyf9tdemu $D

dko7yfw]8j +wxoUF g7njv’\kp0v’dko7eo;o c]t ;q’g]a[4ndskpwxD

9q’j 7Qo7yfc]h;8nj,.lj grnvj Iadlkxt3spdlaopk]ad]5j,oU.sh4nd8hv’D

dD ! @ # $ & = !) 0D ! @ # $ & = @*

7D ! @ # $ & = @& ’D ! @ # $ & = &$

9D ! @ # $ & = @_ lD ! @ # $ & = !@W
ສະຫງສ.ວວນ.ລິສຂະສິດ
dyf9tdemu &D rNoohvp b

I6[7k’\6G

];’l6’ h

dD l6f7yfw]gj oNvmujD rNo.spj aF
9q’j la’gdfI6[8+wxoUD h
Rb

Ta bC

!D 9qj’-vd0t|kf0v’I6[lujlkd RECT 8k, b, a c]t h c]h;7yfw]jgoNvmuj0v’,aoD

74

@D 9q’j xP[mP[goNvmuj0v’I6[lujc9lkd RECT c]t goNvmuj0v’I6[7k’\6 RNPQ
#D 9qj’4vogvqkl6f7yfw]jgoNvmuj0v’I6[7k’\6D
0D .shI6[7k’\6 mu,j u7kj ;afcmd8+wxoUG ];’l6’ $ cmS rNo.spj ) cmS rNoohvp # cmD
!D 9q’j 7yfw]jgoNvmu0j v’I6[7k’\6oUD
@D 4hk;jk0t|kf0v’I6[7k’\64ndxjPocx’8k,c8]j td=]tou]5,j oU goNvmuj0v’,ao

9txjPocx’co;.fL
{ rNoohvpgruj,0Bogxaolv’gmnvj
{ rNoohvpgru,j 0Bo @ cm
{ rNo.spjs]5f]q’ ! cm
{ ];’l6’gruj,0Bogxaolk,gmnjv
{ ];’l6’gru,j 0Bo # cm

dyf9tdemu *D 3 1500 2000

dD mhk; lu;ao -Nl=fe $ glAoM]t !WWWdu[S 4 1000 x
xB,0Po # sq;M ]t !&WW du[ c]t ສະຫງສ.ວວນ.ລິສຂະສິດ
[aomaf ! vao]k7k @WWW du[D grnjv7yfw]j x +
]kp9jkpma’\qf mhk; lu;ao wfhc8h,I6[
dko7yfw]jc[[’kj fa’j I6[0hk’oUD
9qj’8nj,9eo;o.lj .oI6[faj’dkj ;
c]t 0Poleo;o7yfw]joUgxaoc4;D

0D .shI6[lujc9lkd CHOU ]5j,oUD +
!D 9q’j 0Poleo;o7yfw]jgoNvmuj0v’I6[lujc9lkd CHOU D
@D 9qj’8nj,9eo;o.lj .oI6[dko7yfw]jc[[’jk8+wxoUD

C 40
H

–
40

x

15 11
U O

75

dyf9tdemu (D
9q’j 8nj,g7njv’\kp + ; − ; × .l.j o;q’,qo c]t 8n,j 9eo;o.l.j oI6[lucj 9lkd.oI6[
dko7yfw]cj [[’kj 8+wxoUD
dD 4

50 x 0
50

9 −

0D

9

6

3 + 84 ສະຫງສ.ວວນ.ລິສຂະສິດ
75

@D .97;k,

dD dko7yfw]\j ;f7eo;omuj,u;q’g]a[.sh7yfw]j.o;q’g]a[djvoD

8q;1kj ’D

1. 2 × ( 3 − 5 ) = 2× (−2) = −4

2. ( 5 × 3 − 4 ) × 2 = (15 − 4)× 2 = 11× 2 = 22

3. ( 4 × 3 ) − ( 2 × 5 ) + (2× 3) = 12 −10 + 6 = 8

0D dko7yfw]j\;f7eo;omu,j u;q’g]a[-hvoF .sh7yfw]\j ;f7eo;omuj16j;q’g]a[.ol5f

djvo c]h;9b’j 7yfw]vj vd,kmk’ovd8k,]efa[D

8q;1jk’G

a = [ 25 − ( 3 × 2 )] − ( 5 − 1 ) b = ( 3 × 9 ) − [( 4 × 5) + 8]

= ( 25 − 6 ) − 4 = 27 − ( 20 + 8 )

= 19 −4 = 27 − 28

= 15 = −1.

76

#D [qfg/ydsafD -D 5 + [(3× 4) × 2]
pD 2 + [ 3 × ( 5 + ( 4 × 9))];
!D 9q’j 7yfw]7j jk0v’leo;o8+wxoUG fD 5 + [(3× 7)× 5]
dD (3 + 2)×5
0D ( 2 ×15 ) − ( 3 ×9 ); 8D (5 + 6)× (2× 4)
7D (3 + 2)×(−2) 4D (9× 2) + (9× 3)
’D (4× 7) − (4× 3) mD (21+1999) × (2555 − 2011)
9D 7 × (3 − 9)
lD (7 ×1505)× 2010

∗ le]a[[qfg/ydsaf0=h @ sk 0=h $F 9j’q .lj;q’g]a[grnvj .shwfhdko7yfw]4j nd8hv’8k,zqoIa[

muj.shD

@D dD 5 + 3× 4 = 32 7D 19 − 3× 5 = 4

0D 6 × 5 − 3 = 27 ’D 7 + 1 ÷ 3 + 5 = 1

#D dD 4 + 4 × 0 = 0 7D 9×8 + 4× 7 = 100ສະຫງສ.ວວນ.ລິສຂະສິດ
0D 36 −16 ÷ 4 + 8 = 40 ’D 1+ 2× 3× 4 + 5× 6 + 7 = 73

$D dD 18 + 3 − 7 ÷ 7 = 2 7D 13 −1÷ 4 = 3
0D 15 − 3 + 4 = 8 ’D 9 − 2 × 0 = 9
9D 6 + 2 − 6 + 2 = 0

&D 9qj’0Po9eo;o8nj,.lj grnjv.sh4nd8hv’8k,zqoIa[mu.j shD

dD ( + 20) × 2 = 60 7D 25+(4× ) =37

0D ( − 3) × 25 = 100 ’D (3× ) −8 = 7

*D de,tdvozh6|b’j gIaf;Pd !& -qj;3,’F wfh7jkcI’’ko $&WW du[8+ ! -qj;3,’ c]t

[=]ylafwfhsad )W du[8+ ! -q;j 3,’ grnjvg0Qk7a’lts;affudkoD 4k,;jk]k;wfhIa[g’yo

lqfgmqkj .fL

(D ok’ pvfgraf 4ng’yo *&WWW du[ wx8ts^kfF ]k;-NoExk ! d;f (WWW du[F -Nxkoyo

@ dy3]dik,M ]t !)WWW du[ c]t |kdok; ! dy3]dik, g,njvda[g,nvg4y’gInvo

g’yo0v’]k;pa’gs^nv 1j6 !$WWW du[D 4k,;jk ]k;9jkpg’yoma’\qfgmqjk.fL 9kj pgrnjv

-N\kdok;gmqjk.fLD

77

ສະຫງສ.ວວນ.ລິສຂະສິດ)D ok’ l5fmyfk wfh-N\kdoaf @W dy3]dik, grnjvgIaf\kdoafd;oS g,njvxkfgxnvd
vvd,;olko0v’\kdoafglpwxma’\qf @ )WW dik,D fa’j oAoF ]k;8hv’8nj,oE
8ko.lgj -yj’,uxt]y,kogmqkj da[,;olko0v’\kdoafmujglpwxD g,nvj 0Q;0hP;c]h;
,;olkoma’\qfwfhglpwxvud ! *WW dik,D 4k,;jk ok’ l5fmyfk 9twfh\kdoaf
d;o9addy3]dik,L

_D mhk; l5]ypk -N]qf9ad|bj’7ao wfh9kj pg’yolqf & (&W WWW du[F s^a’9kdoAo]k;8hv’
9kj pzjvo8n,j vud !W gfnvoM]t (&W WWW du[D 4k,;jk ]k7k]qf9ad7aooUc,jogmqkj .fL

!WD 7totveo;pdko3I’IPo,afmtpq,csj’|bj’8hv’dko1kpxB,.shoadIPolusj hv’
.shzh6]t * sq;grnvj .sh1n, c]h;lq’j 7no.omhkpxuD 9eo;ooadIPoljushv’,ufa’j oUG #WF #@F
@_F #* 7qoD 4k,;jkxB,muj1kp.sh,uma’\qf9adsq;L

!!D 3I’]t7vocsj’|bj’F ,u[jvooaj’lk,xtgrffaj’oUG [jvooaj’]k7k &WWW du[ ,u @WW
[vj oD [vj ooa’j ]k7k (WWW du[,u !WW [vj oS [vj ooaj’]k7k _WWW du[,u *W [jvoD
4hk,u7qog0Qk-q,]t7vog8a,[jvooa’j 9twfhIa[g’yoma’\qf9addu[L

!@D zhkcrlv’w,hF w,hmu ! ,u]k7k @@&WWWW du[D w,hmu @ ,u]k7k @@WWWWWD 4k,;jk
crma’ @ w,h];,dao,uma’\qf9adc,af L I6h;jk]k7kc,af]t &WWWW du[D

!#D \;ddaodtmq[ (@ |;j p ,u]k7k];,ma’\qf &(*WWWW du[D 9t8hv’0kp\;d
|j;p]t gmqjk.f grnvj 1kdwfhdew]ma’\qf #*WWWW du[L

!$D .shltg\uzqo 3× (4 + x) = 12 + (3× x)
dD 9q’j d;f7no;kj ltg\uzqofa’j djk;4nd8hv’le]a[ x = 7 S x = 3S x = 11.
0D 9qj’-Uc9’;kj ltg\uzqooU4nd8hv’le]a[m5dM7kj 0v’ x D

!&D .shltg\uzqo 4 × (5 + y) = 30 + (2× y) D 9qj’d;f7no ;kj ltg\uzqooU 4nd8hv’ s^n[+
le]a[ y = 5 c]t y = 2D

!*D .shltg\uzqo (g +10)×5 = (g ×5) + 50 D 9qj’d;f7no ;kj ltg\uzqooU 4nd8hv’ s^n[+
le]a[ g = 5 c]t g = 18.

!(D .shltg\uzqo (t + 20) × 3 = (t × 2) + 80 D 9q’j d;f7no ;jkltg\uzqooU 4nd8hv’ s^n[+
le]a[ t = 20 c]t t = 10 D

78

!)D I6[fhko0hk’oUgxaog7nvj ’.-h.o 45 000 200 000  
7q;gInvo9eo;o|b’j g-yj’,uG 90 000
8a’j s;kp8ej @ |j;pF 8aj’s;kp
l6’ ! |j;p c]t rkg0Qk ! rkD 35000 35000
4hk,ug’yo $WW WWW du[ 9tlk,kf
-Ng7nvj ’.-h.ogInvo7q;ma’\qfgs^qjkoU
wfh[=L

!_D 9qj’8v[7e4k, 3fpoe.-h.[[yolqj’g7nvj ’]5,j oUD

]af;ylkstdyf3I’ry,lbdlk

[D lultgdf 4t|qo ,t3slqf g]dmu DDD}Irl

,D 9aomt[6]uF ot7vos^;’;P’9ao .[lq’j g7njv’

Fax: 219527, Box: 7409 g4y’G DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD

3mG W@! @!@_)# mj1u 6Gj DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDສະຫງສ.ວວນ.ລິສຂະສິດ

]}f ]kpdkort]yf8trao}lyo7hk 9e ]k7k 9eo;o \kpgsf
o;o g’yodu[

! rklk]k; xD! @ !W &WW @! WWW

@ 7toyflkf xD@ # _ &WW DDDDDDDDDDDD

# 7toyflkf xD& & * &WW #@ &WW

$ 75olq,[aflbdlk{3]dvhv,

8q;gIqk xD # ! & WWW DDDDDDDDDDDD

& rklk]k; ,D# ! & &WW DDDDDDDDDDDD

* -u;tlkf ,D& @ _ WWW !) WWW

( g7,ulkf ,D$ @ ) &WW !( WWW

];,ma’\qf !@( &WW

9eo;o0Pogxao8q;|a’ln DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD

DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD

muj DDDF ;aomu DDD ot7vos^;’;P’9aoF ;aomu DDD

z6hIa[g7nvj ’ z6hveo;pdko ]af;ylkstdyf3I’ry,lbdlk

79

dD 9eo;omulj tcf’.o4aomulk,\kpg4y’spa’L
0D 9eo;omulj tcf’.oc4;l5fmhkp\kpg4y’spa’L
7D 9eo;omu4j nd]n[wx.o4aomushkc,jo9eo;o.fcfjL

@WD 9qj’la’gdf.[[yo]5j,oU c]h;7yfw]j c]t 8n,j .lj[jvo9Eg,afD

Ihko7hkg]d &

]6d7hkG mhk; lul5rao [yo]k7k
4t|qo G xk’7e ;aomu @W}!W}@WW_
!@}& g,nv’9aomt[6]u

]kpdkoluo7hk 9eo;o ]k7k 9eo;og’yo}du[

3lh’0kpk; @ && WWW du[ DDDDDDDDDDDDDDDDDDDD du[

3lh’0klAo ! !) WWW du[ DDDDDDDDDDDDDDDDDDDD du[ສະຫງສ.ວວນ.ລິສຂະສິດ

glNoc0opk; # $! WWW du[ DDDDDDDDDDDDDDDDDDDD du[

glNv1nf3x3] @ #@ WWW du[ DDDDDDDDDDDDDDDDDDDD du[

];,ma’\qfG DDDDDDDDDDDDDDDDDDDD du[

80

[qfmu !! 7;k,leobddjP;da[g,af c]t glAo-nj

!D dyf9tde ສະຫງສ.ວວນ.ລິສຂະສິດ

dyf9tdemu !
.shoadIPola’gdfg,afcdoj zadF cdoj \kdw,hmuj,u0t|kfohvpMF 7yfg4y’vao.f.o
;y-kg]0k7toyf L
DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD
.o-u;yfxt9e;ao .shpqd8q;1kj ’vnoj Mlyj’mujgIaf.shgIqk,u7;k,leobddPj ;da[g,afD
DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD
.o;y-kg]0k7toyf groyj ltcf’g,af1jk’.f L
DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD
9q’j la’gdfI6[8+wxoU c]h;,u,t3ork[spa’cfj.o;y-kg]0k7toyf L

I6[oadIPo , 1 ltcf’,t3ork[0v’g,af I6[8klt,ksko16vj yogfp ! .o ( ,tsaflt9ao0v’3]d

81 

B D 
A  C 
ສະຫງສ.ວວນ.ລິສຂະສິດ
E

dyf9tdemu @
dD 9q’j la’gdfg[yj’lt4kotdko]5,j oU c]h;leobdg4y’vao.f.o;y-kg]0k7toyfD

{ oE16jrk-totX g;]koE16.j omkj [+8u’?F rkdlj;o8afdao]ts;jk’|hk0hk’0v’
rk-tot c]t |hkoED
{ g,njvoadIPolv’7qovvd,kfb’g-nvd.shg7a’j D

0D 9q’j la’gdfg[yj’I6[8klt,ksko1j6vyogfp0hk’gmy’ rkdlj;o|hkoE c]t 0v[lt
8afdaoF gIqk;kfrk[g4y’spa’.o;y-kg]0k7toyfL

dyf9tde # A 
!D 9qj’c8h,g,af A c]h;0ufglAo-nj ! glAoF @ glAoF

# glAoF DD.  zkj o AD gIqklk,kf0ufwfh9adglAoL  
DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD
DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD

82 

@D .shglAo-nj d  c]t d’D r;d,ao8afdao16j9adg,afL X.sh[vdd=]tou8jk’M?
DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD

#D .shc8h,glAo-njzkj o g,af A c]t BD gIqklk,kfc8h,wfhma’\qf9adglAoL
DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD

B

A

 

@D .97;k,G

.o;y-kg]0k7toyf gryoj ltcf’ g,af fh;pdko9E .ljczjog9hpF dtfkoF D D
3fp.-h8q;vadlvocmo g-joa A, B, C, D, E. . .

\kpgsfG g,af[+,u0t|kfD ສະຫງສ.ວວນ.ລິສຂະສິດ
glAo-nDj
glAo-nj c,oj d5,j 0v’[aofkg,afmuj]Po8yfdao8k,c4;-nj[+lYol5fD

 

-

gryoj ,adcmoglAo-nfj h;p8q;|a’ln g-ajo d, A , .. , a, b, c… 
4hk;kj glAo-nzj jkog,af A  c]t B gIqkg;Qk;jkglAo-nj AB c]t gIqk0Po   (AB)

B
Ax

mvj o-nj AB                     XI6[ !?
\kpgsfG glAo-n[j +,u7;k,0t|kfD

83 

mvj o-njD
mjvo-nj AB  c,oj rkdlj;o|bj’0v’glAo-nj (AB) xtdv[fh;pm5dg,afmuj16j]ts;jk’ A
c]t B   g-y’j gvUo;kj lQo0v’,aoF laopt]adfh;p [AB] XI6[ ! 0hk’gmy’?D

B

A

 mjvo-n j  AB 

g7y’j glAo-nDj

4hk;jk g,af A c[’j glAo-nj xy  gxaolv’rkdl;j oD c8j]trkdlj;ogvUo;jk g7y’j glAo-nj

g-y’j gIqk.shlaopt]ad [Ax) c]t [Ay).  g,af A gvUo;kj g,afg7Qk XI6[]5,j oU?D 

 

  xA ສະຫງສ.ວວນ.ລິສຂະສິດy

  

g7y’j glAo-nj [Ax) g7yj’glAo-nj [Ay)

w]ptskj ’c8j A sk B c,jo];’pk;0v’mjvo-nj [AB] g-yj’gIqklaopt]adfh;p AB.  

laopt]ad vjko I6[g]0k7toyf

  A B

(AB)               glAo-nj AB                                                   

   [AB]               mvj o-nj AB                  A B

    [Ax)              g7yj’glAo-nj Ax  A x 

    AB               ];’pk;  AB                  9eo;o 

 

              8q;1jk’G

gIqk;k’ # g,af A, B  c]t C 16gj my’glAo-nj  xy  3fp;kj AB = 5 sq;|j;pF

BC = 3 sq;|;j p c]t AC = 8 sq;|;j pD

               x                1 sq;|;j pF                                   y  
A C
     B
 

84 

s^adg7QkG

{ zjko|bj’g,af gIqklk,kflhk’wfhglAo-n1j kj ’[+lYol5fD
{ zjkolv’g,af8jk’dao gIqklk,kflhk’glAo-nj8jk’daowfhrP’glAofP;gmqkj oAoD
{ lv’glAo-nj8kj ’dao8afdaowfhrP’g,affP;gmqjkoAoD

#D [qfg/ydsaf t x  v 
A F 
!D 9q’j [vdglAo-n1j 6jdE0;koUfh;plv’ z 
;ymug-aoj  xy s^n  (AC) D B E 

@D 9q’j [vdm5dg7yj’glAo-nj c]t mjvo u C 
-n1j 6j.oI6[dE0;kX9qj’[vdg7yj’glAo D
-nmj ujdq’dao0hk,dao?D

#D gIqklk,kf0ufglAo-njwfh9adglAozkj o ສະຫງສ.ວວນ.ລິສຂະສິດy 
g,af16Ij 6[dE0;k,n 8k,c8]j td=]tou

]5j,oUG                    A B
        dD zkj og,af A
C
0D zjkog,af A, B c]t C                   D
        7D g,af A c]t B  

        ’D zjkog,af A, C  c]t D 
 
  $D 9q’j ;k’g,af  A, B  c]t C [+16glAo-nfj P;dao.lgj 9hp c]h;-vd,afg7y’j dk’0v’

[AB], [BC] c]t [AC]  fh;pdkora[g9hpD

&D 16gj my’cdo9eo;o]5j,oU 9qj’[vd s^n \kp8n,j G

dD g,afg7y’j dk’0v’ BF c]t 0v’ BE 
0D g,aflQo0v’mvj o-njmu,j u];’pk;gmqjk $ 4hk;kj g,aflQomulv’c,jo G.
7D g,afmu,j uw]ptsjk’9kd B gmqjk 3AC. 

  −1 0 1

  A BC DE F G

85 

*D 16gj my’cdo9eo;o]5j,oU 9q’j [vd s^n \kp8n,j G

dD g,afg7y’j dk’0v’ AE.  

0D g,afg7yj’dk’0v’ CF. 

7D g,aflQo0v’mjvo-ngj -yj’];’pk;gmqkj # 4hk;kj g,aflQomulv’c,jo F.

’D g,afmu,j uw]ptskj ’9kd E gmqkj 1 BD. 
2

     −1 0 1
 

 

  A BC DE F G

(. 9q’j la’gdfI6[]5,j oU c]h;8nj,g7njv’\kp ∈ s^n ∉.l[j jvo9Eg,af.sh4ndD 

  −1 0 1 H
 
A BC DE F G ສະຫງສ.ວວນ.ລິສຂະສິດ
   M

 
 

G….[AE]; F…(AB); E….(BC); D….(EG); B….[EG);

F….[CF); H….[DG); M….[AE]; C….[EG); M….[EG)

)D 9qj’;k’g,af L, A, O, P, D, R 3fp.shwfh # g,af]Po-ndj ao X.sh[vdc8j & d=]tou
8jk’dao?D

_D 9q’j c8h,I6[9t85]afmuj,u0hk’gmqjk 10 cm. \kpg,afg7yj’dk’0v’c8j]t0hk’D
0uf8+g,afg7yj’dk’0v’I6[9t85]afmujwfh.\jD ln[8+0ufg,afg7yj’dk’0v’I6[9t85]af.\j
c]t 0uf8+g,afg7y’j dk’ grnjv.shwfhI6[9t85]af.\vj ud9qo d;jk9twfhI6[9t85]af mu,j u
0hk’gmqkj 1 cm. c]h;mklu.sh’k,D

!WD dD zjko # g,af[+]Po-nj gIqklhk’wfh9adglAo-nj L
0D zjko $ g,af[+]Po-nj gIqklhk’wfh9adglAo-nj L
7D zkj o & g,af[+]Po-nj gIqklhk’wfh9adglAo-nj L
’D zkj o ) g,af[+]Po-nj gIqklhk’wfh9adglAo-nj L
9D zjko n g,af[+]Po-nj gIqklhk’wfh9adglAo-nj L

86 

[qfmu !@ glAo-nj0t|ko c]t glAo-n8j A’lkd

!D dyf9tde

dyf9tdemu !D 9q’j la’gdfg[y’j I6[rk[]5,j oU c]t ly’j 16vj hv,8q;gIqk c]h;lqomtok
dPj ;da[[aofks^adm5’F glqk0q;F lkpw2F 0v[mk’F glAo0k;c[’j g]o16jdk’4t|qoF
0v[dt fkoF 0v[38tD c8j]t1jk’gIqk,u7;k,leobd ;kj r;d,ao 8A’lkddao s^n
0t|kodaoD

ສະຫງສ.ວວນ.ລິສຂະສິດ

0q;mjk]kf c0;’;P’9ao Xglqk0q;F s^ad8k,0v[0q;0t\kodaoFDD?

4t\qowxxt86w- ot7vos^;’;P’9ao 4t\qowdlvorq,;ysko9kdxt86w-wx3rog7a’

87

dyf9tdemu @D 9q’j lqomtokdPj ;da[g,afIj;,dao0v’lv’glAo-nj.oI6[]5j,oUG

d′ d′
d

dd d′ 

I6[ ! I6[ @  I6[ # 

                                                                            

  16jI6[ ! ,u 16jI6[ @ [+gsaog,afIj;,dao 1j6I6[ # ,ug,afI;j ,dao[+L
|bj’g,afI;j ,dao c8j4hk;kj ln[8+0uf8+wx
9tgxaoco;.f L

dyf9tdemu #D 9q’j lqomtokdPj ;da[I6[]5j,oU ;jk,uglAo-nj.f0t|kodaoD

ສະຫງສ.ວວນ.ລິສຂະສິດ
I6[ ! I6[ @ I6[ #

dyf9tdemu $D 9qj’la’gdfI6[]5j,oU;jklv’glAo-jn d  c]t d’ 0t|kodaos^n[+ c]h; d;f
lv[fh;pdko;afcmdD

  d d’
88

dyf9tdemu &D
.shmjvo-nj AB c]t g,af OD 9q’j 0uf  AO c]h;0uf OA′ 8+ AO 3fpj;k OA′ = AO. 
g-aoj fP;dao.sh0uf BO c]h;0uf OB′ 3fp;kj OB′ = OB. 9q’j d;fg[y’j ;kj [AB]   c]t   
  [AB′]  gmqjkdao c]t 0t|kodaoD

dyf9tdemu *D
9qj’0ufglAo-nzj kj og,af A c]t B .ljg9hpD c]h;ra[g9hp3fp.shg,af A ,kg8a’g,af BD
rng9hpvvdc]h;0ufglAo-n8j k,Ivpra[ c]h;0ufIvpra[wfhglAo-nDj lv’glAo-nj8afdao
16gj ,af H X.sh\kpg,af H?D .-h[aomafcmd,5, c]h;xP[mP[,5,ma’lujD 9q’j .sh0=hla’gdfD

dyf9tdemu (D 16jI6[0hk’]5j,oU .sh\kpglAo-nj 8k,.93fp.sh-ngj -ajo a, b, c,… c]h;[vd
;kj glAo.f0t|koda[glAo.f s^nglAo.f8A’lkdda[glAo.f 8k,c8j]t76j3fp.sh.-hg7nvj ’
\kp ⊥ s^n // D

dyf9tdemu )D 9q’j la’gdfI6[]5j,oU c]h;8nj,.l[j jvo9Eg,af3fp.-hg7njv’\kp ⊥ s^n // D
ສະຫງສ.ວວນ.ລິສຂະສິດ
            
   
   
  d1 
d1   
d 2   
               d 2      d 3    
  d3 
 
      d1 D D  d 2  
                          d 2 . . .d 3  
        d1 D D  d 2       d1D D D d 3
                           d 2 . . .d 3  
        d1D D D d 3  
 

89

ສະຫງສ.ວວນ.ລິສຂະສິດdyf9tde _D 9q’j la’gdfg[y’j I6[ c]h;8nj,
.lj[jvo9Eg,affh;p7e;kj < 8A’lkd< s^n
<0t|ko<
{ lv’glAo-nj DDDDDDDDDDDDDDDDDD da[glAo-njfP;dao
8hv’ DDDDDDDDDDDDDDDDDD daoD
{ g,njvlv’glAo-nj0t|kodaoF m5dglAo-nmj ujD
DDDDDDDDDDDDDDDDDD da[glAomu|j b’j 8hv’ DDDDDDDDDDDDDDDDDD
da[glAomulv’D X,us^kp7e8v[?

dyf9tde !WD 9q’j c8h,I6[lujc9lkd READ 3fp;jk RE = 7 cm c]t EA =4 cm.  
  glAogoaj’9v,0v’r;d,ao8afdao16jg,af O F c]h;0ufglAo-nzj kj o D 8A’lkdda[ (RO),  
  0ufglAo-njzkj o O 8A’lkdda[ (AE) c]t 0ufglAo-nzj kj o A 8A’lkdda[ OD. 

dyf9tde !!D 9qj’c8h,glAo-nj d zjkog,af A F g,af M c]t N 16oj vd dD .sh.-h[aomaflkd
grjnvlhk’glAo-nzj kj o M c]h;0ufglAo|bj’vudzkj o N 8A’lkdda[ dD

•M  

                                                                                                 d

•A
                                                                                         

• N 

 

gIqklk,kfg;Qkco;.fdjP;da[ # glAo-njmuwj fh,kL

\kpgsfG 4hk;jklv’glAo-n j d1 c]t d 2 8A’lkdda[glAo-njfP;dao d 3 c]h;r;d,ao

8hv’0t|kodaoD      
                      
                                 
 

                                                                      d    1                    d  2                 d 3  

90

@D .97;k,

{ g,nvj lv’glAo-nuj,ug,afI;j ,daorP’g,affP; gryoj g;Qk;kj r;d,ao8afdaoD
{ g,nvj lv’glAo-nj d c]t d’ g4y’c,oj ;kj 9t0uf8+wx1jk’[+lYol5f d=[+,ug,afIj;,daoD
gryjog;Qk;kj r;d,ao0t|kodaoF gIqk0Po d // d'D  

xt3spdlaopt]ad vkj o I6[g]0k7toyf d
d’ 
d // d'                      d 0t|ko d’   

lv’glAo-nj d c]t d’ 8A’lkddao g,nvj r;d,ao8afdaowfh,5,gmqkj daoma’lujD
gIqk0Po d  ⊥  d’ 

xt3spdlaopt]ad vkj o I6[g]0k7toyf
d 8A’lkdda[ d'
d ⊥ d' d
ສະຫງສ.ວວນ.ລິສຂະສິດ
  
                                                                                                               

                                                                            d’

 

75o]adltot

                     d1 ⊥  d 3   

4hk;jk                      c]h;  d1 // d2    

           d 2 ⊥  d 3          

4hk;jklv’glAo-nj d1 c]t d 2 0t|kodaoc]h; m5dglAo-nj d 3 mu8j A’lkdda[glAo-n.j f|bj’

.olv’glAomuj.shdvj ooAo 8hv’8A’lkdda[glAomulv’D

                        d 3             d1            d 2                 d1//d2      

4hk;jk                    c]h; d1 ⊥ d2           

                                      d1 ⊥ d2     

91

#D [qfg/ydsaf a bc
d
!D 16Ij 6[0hk’0;hkoUF 9qj’[vd;jkG
{ @ glAo-n0j t|kodaoS e
{ @ glAo-nj8A’lkddaoS
{ @ glAo-n8j afdaoc8[j +8A’lkddaoD

@D 9q’j .-hg7nvj ’|kp // c]t ⊥   ln[8+0Po a // b ,  b  ⊥ c,  … 

                                    b ab d e
                      a                          c            d  c

  f
g
 

                                                    e

 

 

  I6[ @

I6[ !

ສະຫງສ.ວວນ.ລິສຂະສິດ
#D 9qj’c8h,I6[7no.\j.lgj 9hpmu,j u8kdk3iD c]h;;k’g,af N grnjv.sh (MN) 0t|koda[
glAo-nj dD

d
M

;k’g,af C grnjv.sh (MC) 8A’lkdda[ dD X[+.sh.-h;q’;Po?D

$D 9q’j .-h;q’;Pogrnjvlhk’g,afg7y’j dk’0v’mvj o-nj AB D
&D 9q’j ;k’g,af A 16oj vdglAo-nj d c]h;.-h[aomaflkdc8h,glAo-nj d’ c]t d” zkj o A

0t|ko c]t 8A’lkdda[ d 8k,]efa[D
*D dD 9qj’;k’g,af M 16jovdglAo-nj d. 
            0D 9qj’0ufglAo-nj d’ zkj o M c]t 8A’lkdda[ d. 
           7D 9q’j \kp N 16oj vd d c]t d’. 
            ’D 9qj’0ufglAo-nj d” zjko N 8A’lkdda[ d’. 
                 9D gIqkg;Qkwfhco;.fdjP;da[ d c]t d”? 

92

ສະຫງສ.ວວນ.ລິສຂະສິດ)D dD 9q’j ;k’g,af A, B c]t C [+]Po-njS 
            0D 9q’j 0ufglAo-nj (AB) c]t glAo-n8j A’lkdda[ (BC) 16gj ,af M zkj og,af  P; 
            7D 9qj’c8h,I6[lk,c9 ABC  c]h;0uf];’l6’ h zjko A. 
                 ’D gIqkg;Qkwfhco;.f8+glAo];’l6’ c]t glAo-nj (MP) ?

_D 16jg9hpmuj,u8kdk3iG
dD 9q’j c8h,I6[lk,c9 ABC lkd16 j  B, AB = 3 8kdk3i c]t BC = 2 8kdk3iD

E 
D
C

AB

0D 9kdI6[lk,c9 ABC .shc8h,I6[fa’j 0hk’gmy’oU9qoIvfg,af M.

93

[qfmu !# dkoc8h,glAo-nj0t|ko{glAo-n8j A’lkd

!D dyf9tde

dyf9tdemu ! Xlhk’lv’glAo-nj0t|kodaofh;p[aomaflkd?

9q’j xt8y[af8k,0Ao8vo8+wxoUgrnjvlhk’glAo-nzj kj og,af A c]t 0t\koda[glAo-n j d D
ສະຫງສ.ວວນ.ລິສຂະສິດ
{ ;k’g,af  A   16jovdglAo-nj d fa’j I6[mu !D 
{ gvqk[aomaflkd,kco[da[glAo-nj d, 0ufglAo-nj d′ c]h;g]njvo[aomaflkd8k,myf0v’

]6dlvo.oI6[ @ 9qod;jk]jP,0v’[aomaflkd-bj’g[Nv’8Qo8yfda[glAo-n j d ,kzjko

g,af A.
{ 0ufglAo-njzjkog,af A F gIqkwfhglAo-nj d′′  fa’j I6[ # D  

       I6[ @        I6[ # d’’ 
 A
I6[ ! A  
  d’  A
d                   
             

d  

  d’ 
d

gIqklk,kfg;Qkwfhco;.fdPj ;da[glAo-njmujwfh.\j mP[da[glAo-nj d L gxaospa’L

94 

dyf9tde mu @ Xoe.-h;q’;Po c]t [aomaf.odkolhk’glAo0t|ko?D
dD 9q’j oe.-h;q’;Po c]t [aomafgrjnvc8h,I6[lk,c9mPj ’ ABC g-y’j ;kj BA=AC. 
0D lv’0hk’g-y’j |hk0v’I6[lujc90hk’0t|ko,u];’pk; DDDDDDDDDDDDDDDDDDDD c]t DDDDDDDDDDDDDDDDDDD
7D 9qj’c8h,I6[lujc90hk’0t|ko ABCD ]5,j oU7no c]h;8n,j .l[j jvo9Eg,affh;pg7nvj ’

\kp < // s^n = <

(AB) DDDDDDDDD (DC)                  A                                        B 
 (BC) DDDDDDDDD (AD)   
AB DDDDDDDDD  DC    
AD DDDDDDDDD BC   
 
 
  D                                         C 
 

dyf9tdemu #

9q’j lhk’I6[lujc90hk’0t\ko ABCD  g,jnvI6h # 9v, A, B, C c]t 0hk’ AB= 4,5 cm;
ສະຫງສ.ວວນ.ລິສຂະສິດ
BC = 6,8 cmF 3fpxt8y[af8k,0Ao8vo8+wxoUG

{ 0ufmvj o-nj [AB] c]t mjvo-nj [BC] 8k,g’nvj ow0muj.shF c]h;0ufmjvodq’j mu,j u.9dk’ A  

c]t ]aflt\ugmqjk BC. 

 
 
 

 D

 

  C

 

B 

{ 0ufmjvodqj’muj,u.9dk’ C c]t ]aflt\ugmqjk AB.

 

  D
 

 
 

  BC
 

 

95 

   { lv’mjvodqj’8afdao16gj ,af D.  l5fmhkp.sh0umjvo-nj [ AD] c]t [CD] 

A  D

BC

I6[lucj 9 ABCD  gxaoI6[-toyf.fL .sh-Uc9’

dyf9tde $

9qj’oe.-h;q’;Po c]t [aomafgrnjvlhk’glAo-nj d’ zjkog,af A c]t 0t|koda[glAo-nj d

muj.shdjvo 3fpxt8y[af8k,0Ao8vo]5,j oUG

{ ;k’g,af A  [+16gj lAo-nj d A

d ສະຫງສ.ວວນ.ລິສຂະສິດ

{ ;k’g,af B c]t C 8jk’dao16gj lAo-nj d.
{ 0ufmjvodq’j muj,u.9dk’ C c]t ]aflt\u ABD
{ 0ufmjvodqj’muj,u.9dk’ A c]t ]aflt\u BCD
{ \kpg,af D -bj’gxaog,af8afdao0v’lv’mjvodqj’mujwfhoAoD
{ 0ufglAo-nj AD.

D

A

C

B
d

gIqklk,kfg;Qkwfhco;.fdjP;da[glAo-nj (AD) c]t d? gxaospa’ X.sh-Uc9’?L

{ l5fmhkp AD = ………...D

96