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2. mtpdhrp fTz;lu;
khupak;khs; fy;tpapay;
fy;Yup
nfhy;yk;ghisak;
<NuhL – 2.
ngau;: m.kupak;khs;
tFg;G : xd;gjhk; tFg;G
jiyg;G : ,aw;fzpjk;
gpupT : fzpjk;
2

4. 1.,aw;fzpj Nfhitfs;
2.gy;YWg;Gf; Nfhitfs;
3.kPjpj;Njw;wk;
4.fhuzpj;Njw;wk;
;
4

5. 5.,aw;fzpj Kw;nwhUikfs;
6. gy;YWg;Gf; Nfhitfis
fhuzpg;gLj;Jjy;
7.Nehpar; rkd;ghLfs;
5

6. cl;$Wfs;
ghlk;:
XUWg;Gf; Nfhit
xNu xU cWg;igf;nfhz;l gy;YWg;Gf; Nfhit XUWg;Gf;
Nfhit vdg;gLk;.
<UWg;Gf; Nfhit
,uz;L cWg;Gfis kl;LNk nfhz;l gy;YWg;Gf; Nfhit
<UWg;Gf; Nfhit vdg;gLk;.
6

7. %TWg;Gf; Nfhit
%d;W cWg;Gfis kl;LNk nfhz;l gy;YWg;Gf; Nfhit
%TWg;Gf; Nfhit vdg;gLk;.
kPjpj;Njw;wk;
P(x)-I (x-a) vd;w Nehpa gy;YWg;Gf; Nfhitahy;
tFj;jhy; fpilf;Fk; kPjp p(a) MFk;

8. 1.,aw;fzpj Nfhitfs;
$l;ly;>fopj;jy;>ngUf;fy;>tFj;jy; vd;w ehd;F
mbg;gil fzpjr; nray;fs;>mLf;Ff;Fwpfs; my;yJ
%yq;fisf; fz;nlLj;jy; Mfpatw;iw gad;gLj;jp
vz;fs; kw;Wk; khwpfspd;,izg;igf;nfhz;L
mikf;fg;gLk; Nfhitjhd; xU ,aw;fzpjf; Nfhit
v.fh:
7, x, 2x-3y+1
8

9. 2.gy;YWg;Gf; Nfhitfs;
tFj;jpfspy; khwpfs; ,y;yhjJk;>%yf;Fwpfspd;
cs;Ns khwpfs; ,y;yhjJk; kw;Wk; kpif KOf;fis
mLf;Ffshff; nfhz;l> khwpfisf; nfhz;L
mikAk; ,aw;fzpjf;Nfhit xU gy;Ywg;Gf; Nfhit
MFk;.
v.fh:
p(x)= 2x3 - x2 + 4x + 1
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10. 3.kPjpj; Njw;wk;
P(x) vd;gJ VNjDk; xU gy;YWg;Gf; Nfhit kw;Wk; a vd;gJ
VNjDk; xU nka;naz; vd;f.
P(x)-I (x-a) vd;w Nehpa gy;YWg;Gf; Nfhitahy; tFj;jhy;
fpilf;Fk; kPjp p(a) MFk;
v.fh:
4x3-5x2+6x-2 I (x-1) My; tFj;jhy; fpilf;Fk; kPjpiaf;
fhz;f.
10

11. jPh;T :
P(x) = 4x3-5x2+6x-2 vd;f. x-1 d; g+r;rpak; 1.
P(x) I x-1 My; tFj;jhy; fpilf;Fk; kPjp p(1) MFk;.
,g;NghJ>
P(1) = 4(1)3 - 5(1)2 + 6(1) – 2
=4–5+6–2= 3
/kPjp = 3.
11

12. 4.fhuzpj;Njw;wk;
P(x) vd;gJ VNjDk; xU gy;YWg;Gf; Nfhit kw;Wk;
a vd;gJ VNjDk; xU nka;naz; vd;f.
P(a)=0vdpy;> (x-a) vd;gJ p(x) -d; xU fhuzp MFk;.
Fwpg;
G:
P(x) –d; xU fhuzp (x-a) vdpy;>
P(a) = 0 MFk;.
2013-1-6 12

13. v.fh:
(x-5) vd;gJ p(x) = 2x3 - 5x2 - 28x + 5 vd;w gy;YWg;Gf;
Nfhitapd; xU fhuzpah vdj; jPu;khdp.
jPh;T :
fhuzpj;Njw;wj;jpd; gb p(5 ) = 0
vdpy; (x-5) vd;gJ p(x) -d;; xU fhuzpahFk;.
,g;NghJ>
P(5) = 2(5)3 – 5(5)2 – 28(5) + 15
= 2(125) – 5(25) – 140 + 15
= 250 – 125 – 140 + 15 = 0
/ (x-5) vd;gJ p(x) = 2x3 - 5x2 - 28x + 5 - d;
xU fhuzpMFk;.
13

14. 5.,aw;fzpj Kw;nwhUikfs;
xU rkd;ghL
mjpYs;skhwpfspd;vk;kjpg;gpw;f;Fk;
cz;ikahfNt ,Uf;Fkhdhy;> mr;rkd;ghL xU Kw;nwhUik
vdg;gLk.;
v.fh:1
(a+b)2 = a2 + 2ab + b2
(2a+3b)2 = (2a)2+2(2a)(3b)+(3b)2
= 4a2+12ab+9b2
v.fh:2
(a-b)2 = a2 - 2ab + b2
(3x-4y)2 = (3x)2-2(3x)(4y)+(4y)2
14
= 9x2-24xy+16y2

15. v.fh:3
(a+b) (a-b) = a2 - b2
(4x+5y)(4x-5y)=(4x)2-(5y)2
=16x2-25y2
v.fh:4
(x+a) (X+b) = x2 + (a+b)x + ab
(y+7)(y+5)=y2+(7+5)y+(7)(5)
=y2+12y+35
15

16. v.fh:5
(x + y + z)2 = x2 + y2 + z2 + 2xy + 2yz + 2zx
(2x+3y+5z)2 = (2x)2 + (3y)2 + (5z)2 + 2(2x)(3y) + 2(3y)(5z) +
2(5z)(2x)
= 4x2 + 9y2 + 25z2 + 12xy + 30yz + 20zx
,NjNghd;W>
(x - y + z)2 = x2 + y2 + z2 - 2xy - 2yz + 2zx
(x + y - z)2 = x2 + y2 + z2 + 2xy - 2yz - 2zx
(x - y - z)2 = x2 + y2 + z2 - 2xy + 2yz - 2zx 16

17. v.fh: 6
(x + a) (x+b) (x+c) = x3 + (a + b + c)x2 +( ab + bc + ca)x + abc
(x + 2) (x + 5) (x + 7) = x3 + (2 + 5 + 7)x2 +
[ (2)(5) + (5)(7 )+ (7)(2)]x
+ (2)(5)(7)
= x3 + 14x2 + (10 + 35 + 14) x + 70
= x3+14x2+59 x+70
17

18. v.fh:7
(x + y)3 = x3 + 3x2y + 3xy2 + y3
(x + y)3 = x3 + y3 + 3xy(x + y)
(3a + 4b)3 = (3a)3 + 3(3a)2(4b) + 3(3a)(4b)2 + 4(b)3
= 27a3 + 108a2b +144ab2 + 64b3
,NjNghd;W>
(x - y)3 = x3 - 3x2y+3xy2 - y3
(x - y)3 = x3- y3 - 3xy(x-y)
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19. v.fh: 8
x3 + y3 = (x + y)3 - 3xy (x + y)
x+y=4 kw;Wk;> xy=5 vdpy;> , x3+y3 If;fhz;f.
jPu;T :
x3 + y3 = (4)3-3(5)(4)
=64-60
=4
,NjNghd;W>
x3 - y3 = (x - y)3 + 3xy (x - y)
19

20. 6. gy;YWg;Gf; Nfhitfis
fhuzpg;gLj;Jjy; :
a (b + c) = ab + ac vd;gij khw;WKiwapy;
ab + ac = a(b + c) vd vOjyhk;. ,k;Kiwapy;
ab + ac vd;gij a(b + c) vd vOJtJ
fhuzpgLj;Jjy; vdg;gLk;. ab kw;Wk; ac
vd;w ,uz;L cWg;Gfspy; a vd;gJ xU
nghJf; fhuzp. 20

21. v.fh:
fhuzpg;gLj;Jf :
4x2 + 12xy +9y2
jPu;T :
4x2 + 12xy + 9y2 = (2x2) + 2(2x)(3y) + (3y)2
/ [(a +b)2 = a2 +2ab +b2 ]
= (2x+3y)2
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22. 7.Nehpar; rkd;ghLfs;
a , b Mfpad khwpypfshfTk;
kw;Wk;
a # 0 vdTk; nfhz;l ax + b = 0 vd;wike;j xU
khwpapy; cs;s Neupar; rkd;ghLfis
;
epidT$u;e;J > ,U khwpfspy; xU Nrhb
Neupar; rd;ghLfisf; fhzy;.
,U khwpfspy; xU Nrhb Neupar;
rkd;ghLfs; a1x + b1y = 0
a2 x + b 2 y = 0
22

23. v.fh :
gpd;tUk; xU Nrhb rkd;ghLfis gpujpapLk;
Kiwapy; jPu;f;f.
2x + 5y = 2 kw;Wk; x + 2y = 3
jPu;T :
2x + 5y = 2 ____ (1)
x + 2y = 3 ____ (2)
rkd;ghL (2) ypUe;J x = 3 - 2y ____ (3)
rkd;ghL (1) y; gpujpapl > 2(3 - 2y) + 5y = 2
6 - 4y + 5y = 2 23

24. -4y + 5y = 2 – 6
/ y=-4
y=-4 vd (3) y; gpujpapl>
X = 3 – 2 ( - 4) = 3 + 8 = 11 vdf; fpilf;fpwJ.
/ jPu;T x = 11 kw;Wk; y = -4
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