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8/6/2019 8th Mtaths samacheer complete tamil medium PART 2
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2afj
2.1 mKf
Algebra
al-jabr
m-th
Kitab al-jabr wa l-muqabala
#-fj
lnah~ghl
a fj jij
m-th
.. : 780-850)Kitab al-jabr-wa-l-muqubalavw jfijvGa mufj nkij. J a afjijnuftoaiyizJcUthfglJ. J fjtsfbgUjhfij
VgLaJ.
lnah~ghl(VwjhH .K. 3M
whL )
mby[h
oah thj
nuf fj
nkij. t
a fj
jij vW
miHfgLwh.
xn + y = z
vD rkghL
lnah~ghil
rkghL
vwiHf
gLwJ.
n > 2 DbghGJ
x, y, z-f if
mLF kfF
f VJiy.
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2.2 afj nfhitf - TlY fjY
cjhuz
2.2.1 afj nfhit kf
x- k nfhitx + 5 k
1 1 + 5 =
2 2 + 5 = 7
3 3 + 5 = 8
4 + 5 = 9
1 1 + 5 = 4
2 2 + 5 = 3
3 3 + 5 = 2
1 1 + =2
11
mltiz, x-
btntW kfSnfg
fhit k
hWtij kh 5-
kh w iy
gij ftf.
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afj
2.2.2 cWf, fhuf kW bfGf.
2x+ 3
cWf nrJ nfhitfshwd.
3ab 5a
X cW
vrh fhuna mj bfG vdgL.
(m) xU nfhit cs cWf vifia ma
m cs + (Tl) myJ (fj) Ffis
flJ v fzlyh.
(M) kh cW vgJ vj khlD nruhj jj
cWghF.
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ir
cW cW bfG
2.2.3 gYW nfhit mogil fUJf :
XUW nfhit,
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afj
cjhuz :
13 x x y 44 2 3 vw gYW nfhit goia fhQkhW X Ma
anghJ U khztf fLsthW ilajhf.
ah rahf brjh vd cdh fla aYkh?
cdJ il bfsj v, jtW vnf csJ?
Kj cW :
ulh cW :
(x, y Ma khf
mLFf TLj)
_wh cW :
bfsj brj jtW vd?
ulh cWghd x2 3- go ulhfnth myJ _ whfnth
UF vd bfsj idjh. FHgnk mtid jtwhd KoF
GJ brwJ. Mdh rahd t nkny sfgLsJ.
gYW nfhit l tot
bfsj
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cjhuz :
id bfhf: mL Ff VJ fh fgliy v mij 1vnw Jbfhs ntL.
cjhuz : 9 x1
xj cWf, khWgl cWf
xj khfis xj mLFfis bfhl cWf xj myJ
Xd cWf vdgLwd.
xj myJ Xd cWfis eh nHsthW goalyh :
2.2.4 afj nfhitf TlY fjY - ghit
xj myJ Xd cWfis kLnk Tlnth myJ ffnth
Ko.
. y, Mait V Xd
Wf my?
. 5x2 , y2- Mait V Xd
cWf my?
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afj
vLJfhL 2.1
khwhf
vLJfhL 2.2
u Kiw Tl2
2
vLJfhL 2.3
(myJ)
iuKiw Tl
(3 (2 ( ) x x yx
= (3 )x y
= ( ( )+
2 y .Xd cWfis xwhf nrj _ygYW nfhit Tl iwKiwbragLwJ
(8 , 5 xy xy- Ma uL Xd cWf)
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afj
2.3 afj nfhitf bgUf2.3.1 U XUW nfhitf bgUf
cjhuz
Tl RUfgl
totnk bgUf MF.
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F: 1. UW nfhitf bgU f gy X XUW nfhit MF .
2. gUf gy bfG = Kj XUW nfhit bfG #
ulh XUW nfhit bfG3. a am n n= w mLFF cWf bgUf gyid
fhz cjwJ.
4. , b bgUf gy a b, ab, a.b, (b , (a) b, (a) (b), ( b vbwyhF fgLwJ.
(myJ)
(myJ)
2.3.2 X XUW nfhitia X
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afj
vLJfhL 2.5
(gL ggo)
F:(i) XUW nfhitia X
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[ ]ab baa =
(8ab, 15ab Mait Xd cWf. vdnt mtiw TLnwh)
F : nkfl vLJfho, U
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afj
g 2.2
Kjyh XUW nfhit "ulh XUW nfhit
2x 3y 4x 5xy x2y 6x2y2
2x x2
3y
4x2
5xy 5x y2
x2y
6x2y 18x2y3
2.4 KbwhUikf
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KbwhUik: kh V F vyh kfS xU rk gh il iw br
v mrkghL xU KbwhUik vdgL.
2.4.1 afj KbwhUikf
Kjyh KbwhUik
[a ab = ba]
a b2
+^ h = 2a ab b2 2+ +
a b2
+^ h toa sf
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afj
2^ hb = a2 2a
ulh KbwhUik
a b2
-^ h = 2a ab b2 2- +
a b2
-^ h toa sf
2^ ha b = a2 2a
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_wh KbwhUik
[a
ab = ba]
a b a b+ -^ ^h h = a b2 2-
a b a b+ -^ ^h h toa sf
a hba = a 2
bghJ KbwhUik
x a x b+ +^ ^h h = x a b x ab2 + + +^ h
x a x b+ +^ ^h h toa sf
x a hbx = x x ab2 +b+a
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afj
vLJfhL 2.12
KbwhUikiaagLj :
( + b) (a b) = a2 b2
F, a = 5a, b = 3b
KbwhUikia
agLj :( + b) (a b) = a2 b2
F, = 50, b = 2
KbwhUikiaagLj :
( + b) (a b) = a2 b2
F, a = 997, b = 3
KbwhUikiaagLj :
(x + a) (x + b) =x2 + (a + b )x + b
F,x = m, a = 3, b = 5.
KbwhUikiaagLj :
(x + a) (x + b) =x2 + (a + b )x + b
F,x =p, a = 2, = 3.
KbwhUikia
agLj :(x + a) (x + b) =x2 + (a + b )x + b
F,x = 2x, = 3y, b = 4y.
KbwhUikiagagLj :
(x + a) (x + b) =x2 + (a + b )x + ab
F,x = 50, = 5, = 6.
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2.4.3 nkY y gaDs KbwhUikfis jUj
1 2a6 @ = a 2
4
1 2a a-6 @ = ab
ab2- = a 2
b4-^a = 2ha b
KbwhUikiagagLj
(x + ) (x + b) =x2 + ( + b)x + ab
F,x = 100, a = 5, = 3.
KbwhUikia
gagLj(x + ) (x + b) =x2 + ( + b)x + ab
F,x = 500, = 1, b = 5.
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afj
jUfgl KbwhUikf
ab2+a = a 2
ab4+a = 2^ hb
vLJfhL 2.13
(a + b = 7, a b = 4Matiw ul)
(a + b = 7, a b = 4Matiw ul)
vLJfhL 2.14
(a + b = 10, ab = 20Matiw ul)
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(a + b = 10, ab = 20Matiw ul)
vLJfhL 2.15
g 2.3
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2.5 fhugLj
ij eh tU vf
bgUfgydhf vGjyh.
20 = 1 20
20 = 2 10
20 = 4 5
1. xiwl bga vF 1 I jid ju ntW fhuf
iy v mj viz gfh v vnwh. cjhuz : 2, 3, 5, 7, ...2. xiwl bga vF uoF nkgl fhuf U
m btiz gF v v nwh . cjhuz : 4, , 8, 10, ...
3. xU viz mj fhuf bgUfgydhf vGJnghJ eh
1 I xU fhuahf bghJthf vGJtJ iy. Vbd 1 v gJ
vjbthU vF xU fhuahf mikwJ.
4. vjbthU a vQ gF vzhfnth myJ gfh vzhfnth
UF.
5. 1 vw v gF vQ my, gfh vQ my.
2.5.1 fhugLj vwh vd?
fhugLj : vjbthU gYW nfhitia mj fhuf
bgUfgydhf vGJtij fhugLj vnwh.
0 nj nfhigfis eh flthW
tntW tf tirgLjyh.
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afj
a fjfhit
Kjyhfhu
ulhhu
ekh tiw nkY fhugLj Kokh?
jyhfhu
ulhfhu
6x3 2x 3x2 M. 2x = 2 x M. 3x2 = 3 x x
3 b + 3ab (3ab ( + b)M.
3 b = 3 b
iy. ( + b) InkY fhugLj
KoahJ.
2x +x (2x 3) (x + 2)iy. (2x 3) InkY fhugLj
KoahJ.
iy. (x + 2) InkY fhugLj
KoahJ.
F : nkY fhufshf gF f Koahj fhufis gfh fhuf
vnwh. nkfl vLJfho ( 2), Mait gfh
fhuf MF.
2.5.2 bghJ fhuia btna vLJ fhugLj
vLJfhL 2.16
(2 vgJ U cWfF bghJthf csJ)
F : (i) F, 2, ( + 3) Mad (2 6)x + fhufshF.
(ii) 2, ( ) a fhufis nkY fhufshf gFf KoahJ.
vdnt 2, ( ) Mait gfh fhufshF .
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(bghJ cWghd 4x-I btna vLj)
(bghJ cWghd 3x-I btna vLj)
(bghJ cWghd ab-I btna vLj)
(bghJ cWghdx-I btna vLj)
(bghJ cWghd 7lm-I btna vLj)
2.5.3 cWfis bjhFJ fhugLj
vLJfhL 2.17
(cWfis khaikj)
(bghJ cWfis btna vLj)
(bghJ cWfis btna vLj)
(cWfis khaikj)
(bghJ cWfis btna vLj)
(Kj U cWfisKj bjhFj bghJcWig btna vLj)
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afj
2.5.4. KbwhUikfis gagL fhugLj
id Tf:
(i) ( a b2 2= a
(ii) ( b2 2 2a
(iii) ( ) ( 2=
vLJfhL 2.18
nfhit fhuf
a2 + 2ab + b (a + b), (a + b)
2ab + b2 (a b), (a b
a2
b ( + b , ( b)
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2.5.5 ( ) ( ) ( ) x a x b x a b x ab2+ + = + + + vw KbwhUikia gagL
fhugLj
vLJfhL 2.19
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afj
vLJfhL 2.20
b ab + b j
1
1 6 6 5
2 3 5
2 3 1
2 3 1
vLJfhL 2.21
8 fhf
fhufLj
1, 8 9
2, 4 6
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g 2.4
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afj
2.6 afj nfhitf tFj
2.6.1 X XUW nfhitia knwh XUW nfhitah tFj
(myJ)
(a
aa
n
m
m n
=- I gagLj)
2.6.2 xU gYW nfhitia X XUW nfhitah tFj
vLJfhL 2.22
khW Kiw :
x
x7x = 7x 1 5x1 1
= 7x 1 5x = 7x 5 (1)
[a a = 1]
= 7x 5
khW Kiw :x
2 vw bghJ cWig flJ eh thW
RUfyh
x
x x2
2
6 2x =
x
x
32
2 hx 2- x
=3
1 ^ hx 2x
= x3 3
24
- +x
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vLJfhL 2.23
g 2.5
khW Kiw :
x8x 6+x xbthU cWUJ
2x I bjLjh ekF,
x8x 62+x = x2x
2
52 3x
2+x- xh
= 2x jx 2 32
- +x vd ilF.
x
x
2
8x 6+x =)
x2
2x
= x42
32
+x
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afj
2.7 xUgo rkghLfis j
xU = Fah izfgLs U nfhitfis bfhl xU TWrkghL vdgL. ijna, xW myJ mjF nkgl khfis bfhl
rkjiks TW vW Twyh.
mLF myJ goia xwhf bfhl xW myJ mjF nkgl
khfsh Md rkghil nea myJ xUgo rkghL vnwh.
rkghLfis JbfhSj
f myJ _yf
x = 4 vgJ 2x 3 = 5 vwrkghil iw brwJ.vdnt,x = 4 vgJ rkgho MF.
lJg f = 2 (4) 3= 8 3 = 5 = yJ gf
x = 5 vgJ rkghoF my. Vbd x = 5 v,
lJ gf = 2 (5) 3= 10 3 = 7 tyJ gf
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rkghLfis f cj f
rkghoid FnghJ eh fl fS xiwnah myJ
uilnah myJ mid ij Tl ga gL j nelyh .
1. xU rkgho U gfK xnu viz Todhnyh myJ
f jhnyh rk gh o k khwhJ.2. akw xU vzh rkgho UgfK bgUdhnyh myJ
tFjhnyh mj k khwhJ.
3. lkhW Kiw : xU rk gh il F nghJ, khfis bfhl
cWfis vyh xU gfkhf, khfisbayh kWgfkhf
bfhL bry ntL . y cWfis xU g f U J kWg f F
lkhWj brtj _y jid brayh. rkgho vjbthU
cWig mj Fia kh xU gfUJ kWgfF vLJ
bryyh. thW cWfis khW KiwF l khW Kiw vW
bga.
2.7.1 xU kh mikj xUgo rkghL
U kh mik j xUgo rk gh o F xnu xU
kLnk cL
vLJfhL 2.24
lkhW Kiw:
(- 13 I tygfF khWj)
(Ugfij 5M tFj)
rghj :
lJ gf
= 5 11 13
= 55 13
= 42
= tyJ gf
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afj
vLJfhL 2.25
(UgfK 9I fj)
(Ugfij 5M tFj)
rghj :
lJ gf = 5 (3) + 9 = 24 = tyJ gf
vLJfhL 2.26
(- 5 I UwK nrj)
(Ugfij 3M tFj)
rghj : lJ gf = 2x + 5 = 2 (6) + 5 = 17,
tyJ gf = 23 x = 23 6 = 17.
vLJfhL 2.27
(lJgfF bghk vLj)
(FWF bgUfgo)
khWKiw :
5 24=
5 9 (9I tyJgfF khl)
5 15 kW y5
15= . vdnt 3y = .
khWKiw :
x2x
2 2x 3 5x (lkhW Kiw)
3 1x 8
x3
18= (Ugfij 3M tFj)
x 6=
rghj :
lJgf = m m2
9 = (29 4
= 18 + 4 = 22 = yJ gf
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vLJfhL 2.28
(lJgfF bghk vLj)
vLJfhL 2.29
vLJfhL 2.30
ghj :
lJ gf
=x x
2 5-
=)3
23 (
532
95- = -
=9
6 5
=9
1 = yJ gf
rghj :
15 + 17 = 32
rghj :
lJgf
= x3
1
5
1# # #
=3
1
5
1450# # #
= 15 = tyJgf
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afj
vLJfhL 2.31
vLJfhL 2.32
U taJ jij taJ
jbghGJ : 24 jbghGJ : 8
12 MLfF
24 12 = 12
48 12 = 36
36 = 3
(mU taJ)= 3 (12) = 36
rghj :
lJ gf
=2
1 5
4
5
3
2#
-+ =
-+
=12
8
12
7- +=
-
= tyJ gf.
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vLJfhL 2.33
mlfiy - el = gid iy
g 2.6
rghj :
el = 175000 20%
= 175000100
20#
= `35,000
.. = m.. - el
= 175000 35000
= `140000
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afjwww.errsm.org
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KbwhUikf
( a b2 2= a
( a b2 2 2= a
( )a 2
( ) ab= x +x
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