8th mtaths samacheer complete tamil medium part 2

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.

www.errsm.org

1 www.errsm.org


2/41

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.

www.errsm.org

2 www.errsm.org


3/41

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.

www.errsm.org

3 www.errsm.org


4/41

ir

cW cW bfG

2.2.3 gYW nfhit mogil fUJf :

XUW nfhit,


5/41

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

www.errsm.org

5 www.errsm.org


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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?

www.errsm.org

6 www.errsm.org


7/41

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)

www.errsm.org

7 www.errsm.org


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9/41

afj

2.3 afj nfhitf bgUf2.3.1 U XUW nfhitf bgUf

cjhuz

Tl RUfgl

totnk bgUf MF.

www.errsm.org

9 www.errsm.org


10/41

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


12/41

[ ]ab baa =

(8ab, 15ab Mait Xd cWf. vdnt mtiw TLnwh)

F : nkfl vLJfho, U


13/41

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

www.errsm.org

13 www.errsm.org


14/41

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

www.errsm.org

14 www.errsm.org


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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

www.errsm.org

15 www.errsm.org


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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

www.errsm.org

16 www.errsm.org


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19/41

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.

www.errsm.org

19 www.errsm.org


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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.

www.errsm.org

20 www.errsm.org


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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)

www.errsm.org

21 www.errsm.org


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(a + b = 10, ab = 20Matiw ul)

vLJfhL 2.15

g 2.3

www.errsm.org

22 www.errsm.org


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24/41

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.

www.errsm.org

24 www.errsm.org


25/41

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 .

www.errsm.org

25 www.errsm.org


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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)


(cWfis khaikj)


(Kj U cWfisKj bjhFj bghJcWig btna vLj)

www.errsm.org

26 www.errsm.org


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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)

www.errsm.org

27 www.errsm.org


28/41

2.5.5 ( ) ( ) ( ) x a x b x a b x ab2+ + = + + + vw KbwhUikia gagL

fhugLj

vLJfhL 2.19

www.errsm.org

28 www.errsm.org


29/41

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

www.errsm.org

29 www.errsm.org


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g 2.4

www.errsm.org

30 www.errsm.org


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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

www.errsm.org

31 www.errsm.org


32/41

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

www.errsm.org

32 www.errsm.org


33/41

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

www.errsm.org

33 www.errsm.org


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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

www.errsm.org

34 www.errsm.org


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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

www.errsm.org

35 www.errsm.org


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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

www.errsm.org

36 www.errsm.org


37/41

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.

www.errsm.org

37 www.errsm.org


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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

www.errsm.org

38 www.errsm.org


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afjwww.errsm.org

39 www.errsm.org


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KbwhUikf

( a b2 2= a

( a b2 2 2= a

( )a 2

( ) ab= x +x

www.errsm.org

40 www.errsm.org


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8th mtaths samacheer complete tamil medium part 2

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