exRMwYZxq f™²vsk gƒj eBGqi AwZwiÚ Ask-2020 1

weMZ eQGii ˆR‰mwm cixÞvi cÉk² I mgvavb

AaÅvq-5 exRMwYZxq f™²vsk
cÉk
² 1 A = x2  5x + 6, B = x2  9, C = x2 + 4x + 3 A [Xv. ˆev. 17] 1
=
1
=
1(x  2) (x + 1)
B (x + 3) (x  3) (x + 3) (x  3) (x  2) (x + 1)
C
K. x2 + x ˆK jwNÓ¤ AvKvGi cÉKvk Ki| 2 (x  2) (x + 1)
=
1 1 (x + 3) (x  3) (x  2) (x + 1)
L. mij Ki: A + B 4 1 1 1(x  3) (x  2)
= =
1 1 1 C (x + 3) (x + 1) (x + 3) (x  3) (x  2) (x + 1)
M. A , B , C ˆK mvaviY niwewkÓ¡ f™²vsGk cÉKvk Ki| 4 (x  3) (x  2)
=
1 bs cÉGk²i mgvavb A (x + 3) (x  3) (x  2) (x + 1)

K ˆ`Iqv AvGQ, C = x2 + 4x + 3
  wbGYÆq mvaviY niwewkÓ¡ f™²vskmgƒn
C x2 + 4x + 3 x2 + 3x + x + 3 (x + 3) (x + 1) (x  2) (x + 1)
‰Lb, x2 + x = x(x + 1) = x(x + 1)
,
(x + 3) (x  3) (x  2) (x + 1) (x + 3) (x  3) (x  2) (x + 1)
,

x(x + 3) + 1(x + 3) (x  3) (x  2)
= (Ans.)
x(x + 1) (x + 3) (x  3) (x  2) (x + 1)
(x + 3) (x + 1) x + 3 cÉk
² 2 (i) a3  3a2  10a, a3 + 6a2 + 8a, a4  5a3  14a2
= = (Ans.)
x(x + 1) x
wZbwU exRMvwYwZK ivwk| B [Xv. ˆev. 16]
L ˆ`Iqv AvGQ, A = x2  5x + 6
 2
(ii) P = y  2, Q = y + 2y + 4 ‰es R = y + 8. 3

B = x2  9
1 1 1 1
K. (i) ‰i Z‡Zxq ivwkGK Drcv`GK weGkÏlY Ki| 2
 + = +
A B x2  5x + 6 x2  9 L. (i) ‰i wZbwU ivwki M.mv.à. wbYÆq Ki| 4
1 1 1 y2 6y
= +
x2  3x  2x + 6 x2  (3)2 M. mij Ki: P  Q + R . 4

=
1
+
1 2 bs cÉGk²i mgvavbA
x(x  3)  2(x  3) (x + 3) (x  3)
K ˆ`Iqv AvGQ,

1 1
= +
(x  3) (x  2) (x + 3) (x  3) (i) ‰i Z‡Zxq ivwk = a4  5a3  14a2
x+3+x2 2x + 1 = a2(a2  5a  14)
= =
(x + 3) (x  3) (x  2) (x + 3) (x  3) (x  2) = a2(a2  7a + 2a  14)
2x + 1 = a2{a(a  7) + 2 (a  7)}
= 2 (Ans.)
(x  9) (x  2)
= a2(a  7) (a + 2) (Ans.)
M ˆ`Iqv AvGQ, A = x2  5x + 6

L (i) ‰i ˆÞGò,

B = x2  9
1g ivwk = a3  3a2  10a = a(a2  3a  10)
C = x2 + 4x + 3
1,1,1 1 1 1 = a(a2  5a + 2a  10) = a{a(a  5) + 2(a  5)}
AZ‰e, A B C
f™²vskàGjv x2  5x + 6 , x2  9 , x2 + 4x + 3 = a(a  5) (a + 2)
‰LvGb, 1g f™²vsGki ni = x2  5x + 6 = x2  3x  2x + 6 2q ivwk = a3 + 6a2 + 8a = a(a2 + 6a + 8)
= (x  3) (x  2) = a(a2 + 4a + 2a + 8) = a{a(a + 4) + 2(a + 4)}
2q f™²vsGki ni = x2  9 = (x + 3) (x  3) = a(a + 2) (a + 4)

3q f™²vsGki ni = x2 + 4x + 3 = x2 + 3x + x + 3 ‰es ‘K’ ˆ^GK cvB, 3q ivwk = a2(a  7) (a + 2)

= (x + 3) (x + 1)  wbGYÆq M.mv.à = a(a + 2)
 niàGjvi j.mv.à = (x + 3) (x  3) (x  2) (x + 1) M ˆ`Iqv AvGQ, P = y  2, Q = y2 + 2y + 4 ‰es R = y3 + 8

1 1 1(x + 3) (x + 1) 1 y2 6y
‰Lb, A = (x  3) (x  2) = (x + 3) (x  3) (x  2) (x + 1) cÉ`î ivwk = P  Q + R
(x + 3) (x + 1) 1 y2 6y
= =  2 + 3
(x + 3) (x  3) (x  2) (x + 1) y  2 y + 2y + 4 y + 8
2 cvGéix ˆR‰mwm cixÞv mnvwqKv 2020  MwYZ

=
y2 + 2y + 4  (y  2)2
+ 3
6y niàGjvi j.mv.à = (x  2) (x  1) (x  3)
(y  2) (y2 + 2y + 4) y + 8
1 x3
y2 + 2y + 4  (y2  4y + 4) 6y  cÉ^g f™²vsk = (x  2) (x  1) = (x  2) (x  1) (x  3)
= + 3
(y  2) (y2 + 2y + 22) y +8
1 x1
y2 + 2y + 4  y2 + 4y  4 6y  w«¼Zxq f™²vsk = (x  2) (x  3) = (x  1) (x  2) (x  3)
= + 3
y3  23 y +8
1 x2
6y 6y  Z‡Zxq f™²vsk = (x  3) (x  1) = (x  1) (x  2) (x  3)
= 3 + 3
y 8 y +8
x3
6y4 + 48y + 6y4  48y  wbGYÆq f™²vskàGjv, (x  1) (x  2) (x  3) ,
=
(y3  8) (y3 + 8)
12y4 12y4 x1
= ,
3 2 2= 6 (Ans.) (x  1) (x  2) (x  3)
(y )  8 y  64
x2
cÉk
² 3 M = x2  3x + 2, N = x2  5x + 6 ‰es K = x2  4x + 3, (Ans.)
(x  1) (x  2) (x  3)
wZbwU exRMwYZxq ivwk| A [iv. ˆev. 17] 1 1 2x
M
cÉk
² 4 2x + 3y , 2x  3y , 4x2  9y2 wZbwU exRMvwYwZK f™²vsk|
K. x2
ˆK jwNÓ¤ AvKvGi cÉKvk Ki| 2
[w`.ˆev. 17]
1 1 1
L. mij Ki : + + .
M N K
4 K. 1g f™²vsk ˆ^GK 2q f™²vsk weGqvM Ki| 2
1 1 1 L. 1g I 2q f™²vsGki àYdjGK 3q f™²vsk «¼viv fvM Ki| 4
M. M , N , K ˆK mgniwewkÓ¡ f™²vsGk cÉKvk Ki| 4
M. f™²vsk wZbwUGK mvaviY niwewkÓ¡ f™²vsGk cÉKvk Ki| 4
3 bs cÉGk²i mgvavb A 4 bs cÉGk²i mgvavbA
K ˆ`Iqv AvGQ, M = x2  3x + 2
 1
K ˆ`Iqv AvGQ, cÉ^g f™²vsk = 2x + 3y

= x2  2x  x + 2
= x(x  2)  1(x  2) 1
w«¼Zxq f™²vsk = 2x  3y
= (x  2) (x  1)
M (x  2) (x  1)  weGqvMdj = 1g f™²vsk  2q f™²vsk
 = = x  1 (Ans.)
x2 (x  2) 1 1
= 
L ‘K’ nGZ cvB, M = (x  2) (x  1)
 2x + 3y 2x  3y

ˆ`Iqv AvGQ, N = x2  5x + 6 = x2  2x  3x + 6 2x  3y  2x  3y
=
(2x + 3y)(2x  3y)
= x(x  2)  3(x  2) = (x  2) (x  3)
 6y 6y
‰es K = x  4x + 3 = x2  3x  x + 3
2 =
(2x)2  (3y)2
= 2
4x  9y2
(Ans.)
= x(x  3)  1(x  3) = (x  3) (x  1) 2x
1 1 1 L ˆ`Iqv AvGQ, 3q f™²vsk = 4x2  9y2

‰Lb, + +
M N K
1 1
1 1 1  1g I 2q f™²vsGki àYdj = (2x + 3y)  (2x  3y)
= + +  
(x  2) (x  1) (x  2) (x  3) (x  3) (x  1)
1
x3+x1+x2 3x  6 =
= = (2x + 3y)(2x  3y)
(x  2) (x  1) (x  3) (x  2) (x  1) (x  3)
1 1
3(x  2) 3 = =
= = (Ans.) (2x)2  (3y)2 4x2  9y2
(x  2) (x  1) (x  3) (x  1) (x  3)
1
M ‘K’ I ‘L’ nGZ cvB, M = (x  2) (x  1)
 1g I 2q f™²vsGki àYdj 4x2  9y2
 fvMdj = 3q f™²vsk =
2x
N = (x  2) (x  3) 4x2  9y2
K = (x  3) (x  1)
1 (4x2  9y2) 1
1 1 = 2 2  = (Ans.)
 cÉ^g f™²vsk, =
M (x  2) (x  1)
(4x  9y ) 2x 2x
1 1 2x
1 1 M cÉ`î f™²vskàGjv 2x + 3y , 2x  3y , 4x2  9y2

w«¼Zxq f™²vsk, N = (x  2) (x  3)
1 1
‰LvGb, 1g f™²vsGki ni = 2x + 3y
Z‡Zxq f™²vsk, K = (x  3) (x  1) 2q f™²vsGki ni = 2x  3y
exRMwYZxq f™²vsk 3
3q f™²vsGki ni = 4x2  9y2 = (2x)2  (3y)2 M evgcÞ = Q  R  x  1

x2  9
= (2x + 3y)(2x  3y)
x2 + 2x  3 x2 + 12x + 35 x2  9
 niàGjvi j.mv.à. = (2x + 3y)(2x  3y) =
x2 + 6x  7
 2
x + 4x  5

x1
1 1(2x  3y)
AZ‰e, 1g f™²vsk = 2x + 3y = (2x + 3y)(2x  3y) =
(x + 3) x2 + 5x + 7x + 35 x2  32
 2  [‘L’ nGZ]
(x + 7) x + 5x  x  5 x1
2x  3y
= (x + 3) x(x + 5) + 7(x + 5) (x + 3) (x  3)
4x2  9y2 =  
(x + 7) x(x + 5)  1(x + 5) (x  1)
1 1(2x + 3y)
2q f™²vsk = 2x  3y = (2x  3y)(2x + 3y) (x + 3) (x + 5) (x + 7) (x  1) 1
=   = = WvbcÞ
(x + 7) (x + 5) (x  1) (x + 3) (x  3) x  3
2x + 3y
= x2  9 1
4x2  9y2  QR = (ˆ`LvGbv nGjv)
x1 x3
2x 2x  1 2x
3q f™²vsk = 4x2  9y2 = (4x2  9y2)  1 = 4x2  9y2
x3  y3 1 1
cÉk
² 6 A=
x4 + x2y2 + y4
,B=
1  x + x2
,C=
1 + x + x2
2x  3y 2x + 3y 2x
 wbGYÆq f™²vskàGjv: 4x2  9y2 , 4x2  9y2 I 4x2  9y2 (Ans.)
1
‰es D = 1 + x2 + x4 PviwU exRMwYZxq ivwk| [P. ˆev. 17]
2 2 2
x + 3x  4 x + 2x  3 , x + 12x + 35
cÉk
² 5 P = x2 + 7x + 12 , Q = 2 R= 2 .
x + 6x  7 x + 4x  5 K. A ˆK jwNÓ¤ AvKvGi cÉKvk Ki| 2
[Kz. ˆev. 17] L. cÉgvY Ki ˆh, B  C  2x  D = 0. 4
K. P ˆK jwNÓ¤ AvKvGi cÉKvk Ki| 2 1+x 2
M. mij Ki : D  (B + C) 4
L. P + Q ˆK mij Ki| 4
M. ˆ`LvI ˆh, Q  R  xx  19
2
=
1
. 4 6 bs cÉGk²i mgvavb A
x3
x3  y3 (x  y) (x2 + xy + y2)
5 bs cÉGk²i mgvavb A K A = x4 + x2y2 + y4 = (x2)2 + 2x2y2 + (y2)2  x2y2

K ˆ`Iqv AvGQ,
 (x  y) (x2 + xy + y2)
=
x2 + 3x  4 x2 + 4x  x  4 (x2 + y2)2  (xy)2
P= =
x2 + 7x + 12 x2 + 3x + 4x + 12 (x  y) (x2 + xy + y2)
=
x(x + 4)  1(x + 4) (x + xy + y2) (x2  xy + y2)
2

=
x(x + 3) + 4(x + 3) xy
=
(x + 4) (x  1) x  1 x2  xy + y2
= = (Ans.)
(x + 4) (x + 3) x + 3 1 1
L ˆ`Iqv AvGQ, B = 1  x + x2 ; C = 1 + x + x2

x2 + 3x  4 x2 + 2x  3
L
 P+Q= 2 +
x + 7x + 12 x2 + 6x  7
1 1 1
‰es D = 1 + x2 + x4 = 1 + 2x2 + x4  x2 = (1 + x2)2  x2
x  1 x2 + 3x  x  3
= +
x + 3 x2 + 7x  x  7
[‘K’ nGZ]
1
=
x  1 x(x + 3) 1 (x + 3) (1 + x + x2) (1  x + x2)
= +
x + 3 x(x + 7)  1 (x + 7)
evgcÞ = B  C  2x  D
x  1 (x + 3) (x  1)
= + 1 1 1
x + 3 (x + 7) (x  1)
=  2x 
1  x + x2 1 + x + x2 (1 + x + x2) (1  x + x2)
x1 x+3
= + 1 1 2x
x+3 x+7
=  2 
1  x + x2 1 + x + x (1 + x + x2) (1  x + x2)
(x  1) (x + 7) + (x + 3) (x + 3)
= 1 + x + x2  1 + x  x2  2x
(x + 3) (x + 7)
=
(1  x + x2) (1 + x + x2)
x2  x + 7x  7 + x2 + 3x + 3x + 9
=
x2 + 7x + 3x + 21 2x  2x 0
= =
2
2x + 12x + 2 (1  x + x2) (1 + x + x2) (1  x + x2) (1 + x + x2)
=
x2 + 10x + 21 = 0 = WvbcÞ
2(x2 + 6x + 1)
= 2
(x + 10x + 21)
(Ans.)  B  C  2x  D = 0 (cÉgvwYZ)
4 cvGéix ˆR‰mwm cixÞv mnvwqKv 2020  MwYZ
2
1+x 1 (x + 1)(x + 3)
M
 D
 (B + C) = 
(x – 2)(x – 3) (x + 1)(x + 3)
1 + x2 1 1 (x + 1)(x + 3)
=   =
1 1  x + x2 + 1 + x + x2 (x + 1)(x + 3)(x – 2)(x – 3)
2 2
(1 + x + x ) (1  x + x ) 1
2q f™²vsk, Q = (x + 3)(x – 3)
1 + x + x2 + 1  x + x2
= (1 + x2) (1 + x + x2) (1  x + x2) 
(1  x + x2) (1 + x + x2) 1 (x + 1)(x – 2)
= 
2 + 2x 2 (x + 3)(x – 3) (x + 1)(x – 2)
= (1 + x2) (1 + x + x2) (1  x + x2) 
(1  x + x2) (1 + x + x2) (x + 1)(x – 2)
=
2
(1  x + x ) (1 + x + x ) 2 (x + 1)(x + 3)(x – 2)(x – 3)
= (1 + x2)(1 + x + x2)(1 – x + x2) 
2(1 + x2) 1
‰es 3q f™²vsk, R = (x + 1)(x + 3)
(1  x + x2)2 (1 + x + x2)2
= (Ans.) 1 (x – 2)(x – 3)
2
= 
(x + 1)(x + 3) (x – 2)(x – 3)
1 1 1
cÉk
² 7 P = x2 – 5x + 6, Q = x2 – 9 ‰es R = x2 + 4x + 3 B =
(x – 2)(x – 3)
(x + 1)(x + 3)(x – 2)(x – 3)
[wm. ˆev. 17]
myZivs, P, Q I R f™²vsk wZbwUGK mgni wewkÓ¡ f™²vsGk
1 1 1
K. P + Q + R ‰i gvb wbYÆq Ki| 2 cÉKvk Kiv nGjv| (Ans.)
L. P + Q  R ˆK mij Ki| 4 x+1 x2 + x x2
cÉk
² 8 A = x  1, B = x2 + x  2 , C = x2 + 5x + 6 ‰es
M. P, Q ‰es R ˆK mvaviY niwewkÓ¡ f™²vsGk cÉKvk Ki| 4
x2  y2
7 bs cÉGk²i mgvavb D= PviwU exRMvwYwZK f™²vsk| A [wm. ˆev. 16]
x3  y3
1 1
K ˆ`Iqv AvGQ,
 P= 2
x – 5x + 6
;Q= 2
x –9 K. D ˆK jwNÓ¤ AvKvGi cÉKvk Ki| 2
1 x
‰es R = x2 + 4x + 3 L. cÉgvY Ki ˆh, A ÷ B  C = x + 3 . 4
1 1 1 1 1 1 M. A, B ‰es C ˆK mgniwewkÓ¡ f™²vsGk cÉKvk Ki| 4
 + + = + +
P Q R 1 1 1 8 bs cÉGk²i mgvavbA
x2 – 5x + 6 x2 – 9 x2 + 4x + 3
= x2 – 5x + 6 + x2 – 9 + x2 + 4x + 3
K ˆ`Iqv AvGQ,

= 3x2 – x(Ans.) x2  y2 (x + y) (x  y) x+y
D= = = 2 (Ans.)
x3  y3 (x  y) (x2 + xy + y2) x + xy + y2
L P+QR

x+1
1 1 1 L ˆ`Iqv AvGQ, A = x  1

= + 
x – 5x + 6 x2 – 9 x2 + 4x + 3
2

x2 + x x (x + 1)
1 1 1 Avevi, B = x2 + x  2 = x2 + 2x  x  2
= 2 + 
x – 2x – 3x + 6 x2 – 32 x2 + 3x + x + 3
x(x + 1)
1 1 1 =
= +  x(x + 2)  1(x + 2)
x(x – 2) – 3(x – 2) (x + 3)(x – 3) x(x + 3) + 1(x + 3)
1 1 (x + 3)(x + 1) x(x + 1)
= +  =
(x – 2)(x – 3) (x + 3)(x – 3) 1 (x + 2) (x  1)
x2 x2
=
1
+
x + 1 1 + (x – 2)(x + 1)
= ‰es C = x2 + 5x + 6 = x2 + 3x + 2x + 6
(x – 2)(x – 3) (x – 3) (x – 2)(x – 3)
1 + x2 – 2x + x – 2 x2 – x – 1 x2 x2
= = (Ans.) = =
(x – 2)(x – 3) (x – 2)(x – 3) x(x + 3) + 2(x + 3) (x + 3) (x + 2)
x+1 x(x + 1) x2
1
M ˆ`Iqv AvGQ, P = x2 – 5x + 6 = (x – 2)(x – 3)

1 ‰Lb, A  B  C = x  1  (x + 2) (x  1)  (x + 3) (x + 2)

1 1 (x + 1) (x + 2) (x  1) x2 x
Q= = =   =
2
x – 9 (x + 3)(x – 3) (x  1) x(x + 1) (x + 3) (x + 2) x + 3
1 1 x
‰es R = x2 + 4x + 3 = (x + 1)(x + 3)  ABC=
x+3
(cÉgvwYZ)

 f™²vskàGjvi nGii j. mv. à. = (x + 1)(x – 2)(x – 3)(x + 3) M ‘L’ ˆ^GK cvB,


1 x+1 x(x + 1) x2
 1g f™²vsk, P = (x – 2)(x – 3) A=
x1
,B=
(x + 2) (x  1)
,C=
(x + 3) (x + 2)
exRMwYZxq f™²vsk 5
 f™²vskàGjvi nimgƒGni j.mv.à. = (x  1) (x + 2) (x + 3) =
0
= 0 = WvbcÞ
(1 + x + x2) (1  x + x2)
x+1 (x + 1) (x + 2) (x + 3)
‰Lb, A = x  1 = (x  1) (x + 2) (x + 3)  3q ivwk + 2q ivwk  1g ivwk = 0 (ˆ`LvGbv nGjv)
x(x + 1) x(x + 1) (x + 3) M ‰LvGb, 2q ivwk  3q ivwk  4^Æ ivwk

B= =
(x + 2) (x  1) (x  1) (x + 2) (x + 3) 1 2x (x + 1)2  (x2 + x)
=  
x 2
x (x  1)2
1 + x + x2 1 + x2 + x4 x3 + 1
C= =
(x + 2) (x + 3) (x  1) (x + 2) (x + 3) 1 2x x2 + 2x + 1  x2  x
= 2  2 
1 + x + x (1 + x + x ) (1  x + x ) (x + 1) (x2  x + 1)
2
 wbGYÆq mgni wewkÓ¡ f™²vsk :
1 2x (x + 1)
(x + 1) (x + 2)(x + 3) x(x + 1)(x + 3) =  
,
(x  1) (x + 2) (x + 3) (x  1) (x + 2) (x + 3)
‰es 1 + x + x2 (1 + x + x2) (1  x + x2) (x + 1) (1  x + x2)
1 (1 + x + x2) (1  x + x2) (x + 1) (1  x + x2)
x2(x  1) =
(1 + x + x2)

2x

(x + 1)
(Ans.)
(x  1) (x + 2) (x + 3)
(1  x + x2)2
= (Ans.)
1 1 2x (x + 1)2  (x2 + x) 2x
cÉk²9 1  x + x2 , 1 + x + x2 , 1 + x2 + x4 ‰es
 x3 + 1
2p 1 1
PviwU exRMvwYwZK ivwk| A [h. ˆev. 17]
cÉk
² 10 x=
1 + p 2 + p4
,y=
1  p + p2
‰es z = 1 + p + p2

K. 1g I 2q ivwkGK mgni wewkÓ¡ f™²vsGk cÉKvk Ki| 2 wZbwU exRMwYZxq ivwk| [e. ˆev. 17]

L. ˆ`LvI ˆh, 3q ivwk + 2q ivwk  1g ivwk = 0| 4 K. y ‰es z ˆK mvaviY niwewkÓ¡ f™²vsGk cwiYZ Ki| 2
L. mij Ki : x  y + z 4
M. 2q ivwk  3q ivwk  4^Æ ivwk ‰i mijdj wbYÆq Ki| 4
M. mij Ki : (y  z)  x 4
9 bs cÉGk²i mgvavb A
1
10 bs cÉGk²i mgvavbA
K 1g f™²vsk = 1  x + x2
 1 1
K ˆ`Iqv AvGQ, y = 1  p + p2 ; z = 1 + p + p2

1
2q f™²vsk = 1 + x + x2
 y I z ‰i niàGjvi j.mv.à.
 niàGjvi j.mv.à. = (1 + x + x )(1  x + x ) = 1 + x + x
2 2 2 4 = (1 + p + p2)(1  p + p2) = {(1 + p2) + p} {(1 + p2)  p}
1 (1 + x + x2) = {(1 + p2)2  p2} = 1 + 2p2 + p4  p2 = 1 + p2 + p4
 1g f™²vsk = 1  x + x2 = (1 + x + x2)(1  x + x2) 1 (1 + p + p2) 1 + p + p2
 y= 2= 2 2 = 2
1  p + p (1 + p + p )(1  p + p ) 1 + p + p
4 (Ans.)
1 + x + x2
= (Ans.) (1  p + p2) 1  p + p2
1 + x2 + x4 1
 z = 1 + p + p2 = (1 + p + p2)(1  p + p2) = 1 + p2 + p4 (Ans.)
1 (1  x + x2)
 2q f™²vsk = 1 + x + x2 = (1 + x + x2)(1  x + x2) 2p 1 1
L x  y + z = 1 + p 2 + p4  1  p + p 2 + 1 + p + p 2

1  x + x2
= (Ans.) 2p 1 1
1 + x2 + x4 =  + 2
(1 + p + p2)(1  p + p2) 1  p + p2 1 + p + p
L evgcÞ = 3q ivwk + 2q ivwk  1g ivwk
 [‘K’ ˆ^GK cvB]
2x 1 1 2
2p  1  p  p + 1  p + p 2
2p  2p
= + 
1 + x2 + x4 1 + x + x2 1  x + x2 = =
(1 + p + p2)(1  p + p2) (1 + p + p2)(1  p + p2)
2x 1 1 0
= + 2  = =0
1 + 2x2 + x4  x2 1 + x + x 1  x + x2 (1 + p + p2)(1  p + p2)
2x 1 1  x  y + z = 0 (Ans.)
= + 2 
(1 + x2)2  x2 1 + x + x 1  x + x2
M (y  z)  x

2x 1 1
= + 2  1 1 2p
= 
1  p + p2  1 + p + p2  1 + p2 + p4 [ˆ`Iqv AvGQ]
(1 + x + x2) (1  x + x2) 1 + x + x 1  x + x2
2 2
2x + 1(1  x + x )  1(1 + x + x ) 2 2
= 1 + p + p  (1  p + p ) 2p
(1 + x + x2) (1  x + x2) = 2 2  2 4
 (1 + p + p ) (1  p + p )  1 + p + p
2x + 1  x + x2  1  x  x2 2 2
= 1 + p + p  1 + p  p  2p
(1 + x + x2) (1  x + x2) =  2 4  2 4
 1 + p + p  1+p +p
2x  2x 2p (1 + p2 + p4)
= = 4  = 1 (Ans.)
(1 + x + x2) (1  x + x2) 2
(1 + p + p ) 2p
6 cvGéix ˆR‰mwm cixÞv mnvwqKv 2020  MwYZ

DËi ms‡KZmn m„Rbkxj cÖkœ

Abykxjbx 5.1
a b cÉk
² 2 cÉ`î exRMvwYwZK ivwkàGjv chÆGeÞY Ki : A
cÉk
² 1 a2 – 9b2 , a2 + 6ab + 9b2 `yBwU exRMwYZxq f™²vsk| A
x y xy
K. f™²vsk `yBwUi niàGjvi j.mv.à. wbYÆq Ki| 2 x3 + y3 , x3  y3 , x4 + x2y2 + y4
L. f™²vsk `yBwUi ˆhvMdj wbYÆq Ki| 4
K. cÉ^g I w«¼Zxq ivwki AbycvZ wbYÆq Ki| 2
M. f™²vsk `yBwUi ˆhvMdGji mvG^ KZ ˆhvM KiGj ˆhvMdj
L. cÉ^g I w«¼Zxq ivwki ˆhvMdj nGZ Z‡Zxq ivwk weGqvM Ki| 4
1
a + 3b
nGe? 4 M. f™²vsk wZbwUGK mvaviY niwewkÓ¡ f™²vsGk cÉKvk Ki| 4
Dîi: K. (a + 3b) (a  3b)
2 3
x(x  y )3 4
x +y 4
Dîi: K. y(x3 + y3) ; L. x6  y6 ;
a2 + 4ab  3b2
L. (a + 3b)2 (a  3b)
x(x3  y3) y(x3 + y3) xy(x2  y2)
M. x6  y6 , x6  y6 , x6  y6
 2b(2a + 3b)
M. (a + 3b)2 (a  3b)

m†Rbkxj cÉGk²i mgvavb

cÉk²1 A = x2  5x + 6, B = x2  7x + 12 ‰es C = x2  9x + 
 M ‘K’ ‰es ‘L’ ˆ^GK cvB,
20 wZbwU exRMwYZxq ivwk| A = (x  3) (x  2)
K. 1g ivwkGK Drcv`GK weGkÏlY Ki| 2 B = (x  3) (x  4)
1 1 C = (x  4) (x  5)
L. B + C ˆK jwNÓ¤ AvKvGi cÉKvk Ki| 4 1 1 1
, ‰es ‰i nGii j.mv.à.
1 1 1 A B C
M. A , B ‰es C ˆK mvaviY niwewkÓ¡ f™²vsGk cÉKvk Ki| 4
= (x  2) (x  3) (x  4) (x  5)
1 bs cÉGk²i mgvavb 1 1
‰i ˆÞGò, (x  3) (x  2) ‰i je I niGK
K 1g ivwk = x2  5x + 6
 A
= x2  3x  2x + 6 (x  4) I (x  5) «¼viv àY KGi cvB,
= x(x  3)  2(x  3) (x  4) (x  5)
= (x  3) (x  2) (Ans.) (x  2) (x  3) (x  4) (x  5)
L B = x2  7x + 12
 1 1
‰i ˆÞGò, (x  3) (x  4) ‰i je I niGK
B
= x2  3x  4x + 12
= x(x  3)  4(x  3) (x  2) I (x  5) «¼viv àY KGi cvB,
= (x  3) (x  4) (x  2) (x  5)
C = x2  9x + 20 (x  2) (x  3) (x  4) (x  5)
= x2  4x  5x + 20 1 1
C
‰i ˆÞGò, (x  4) (x  5) ‰i je I niGK (x  2) I (x
= x(x  4)  5(x  4)
= (x  4) (x  5)  3) «¼viv àY KGi cvB,
1 1 1 1 (x  2) (x  3)
 + = +
B C (x  3) (x  4) (x  4) (x  5) (x  2) (x  3) (x  4) (x  5)
x5+x3
=  wbGYÆq f™²vskàGjv nGe
(x  3) (x  4) (x  5)
2x  8 (x  4) (x  5)
= ,
(x  3) (x  4) (x  5) (x  2) (x  3) (x  4) (x  5)

2(x  4) (x  2) (x  5)
= ,
(x  3) (x  4) (x  5) (x  2) (x  3) (x  4) (x  5)
1 1 2 (x  2) (x  3)
 + = (Ans.) (Ans.)
B C (x  3) (x  5) (x  2) (x  3) (x  4) (x  5)
exRMwYZxq f™²vsk 7

cÉk²2 A = 2x2  7x + 6, B = 2x2  11x + 12

 1
M ‰LvGb, A  B + C

1 2
‰es C = x2  6x + 8
1 1
A = 
K. B ˆK jwNÓ¤ AvKvGi cÉKvk Ki| 2 (2x  3) (x  2) (x  4) (2x  3)
2
1 1 1 + [‘K’ I ‘L’ ˆ^GK]
L. A , B I C ˆK mgni wewkÓ¡ f™²vsGk cÉKvk Ki| 4 (x  4) (x  2)
1(x  4)  1(x  2) + 2(2x  3)
1 1 2 =
M. mij Ki : A  B + C 4 (2x  3) (x  2) (x  4)
x  4  x + 2 + 4x  6
2 bs cÉGk²i mgvavb =
(2x  3) (x  2) (x  4)
K ˆ`Iqv AvGQ, A = 2x2  7x + 6, B = 2x2  11x + 12
 5x  x + 2  10
A 2x2  7x + 6 =
‰Lb, = 2 (2x  3) (x  2) (x  4)
B 2x  11x + 12
4x  8
2x2  3x  4x + 6 =
= 2 (2x  3) (x  2) (x  4)
2x  3x  8x + 12 4(x  2)
x(2x  3)  2(2x  3) =
= (2x  3) (x  2) (x  4)
x(2x  3)  4(2x  3) 4
(2x  3) (x  2) x  2 = (Ans.)
(2x  3) (x  4)
= = (Ans.)
(2x  3) (x  4) x  4
L ˆ`Iqv AvGQ, A = 2x2  7x + 6
 1 1
cÉk
² 3 S = 2 a4 + 2, P = 2 (2a2  10a + 12), Q = a2  11a  12
2
B = 2x  11x + 12
C = x2  6x + 8
‰es R = a2  9a + 20
1 1 1 1 K. S ˆK Drcv`GK weGkÏlY Ki| 2
AZ‰e, A , B , C f™²vsàGjv 2x2  7x + 6 , 1 1
L. +
Q 12
ˆK jwNÓ¤ AvKvGi cÉKvk Ki| 4
1 1
2x2  11x + 12
‰es x2  6x + 8 1 1 1
M. , ‰es ˆK mvaviY niwewkÓ¡ f™²vsGk cÉKvk Ki| 4
P Q R
‰LvGb, 1g f™²vsGki ni = 2x2  7x + 6
= (2x  3) (x  2) [‘K’ ˆ^GK] 3 bs cÉGk²i mgvavb
1 a4 + 4
2q f™²vsGki ni = 2x2  11x + 12 K ˆ`Iqv
 AvGQ, S = 2 a4 + 2 = 2
= (x  4) (2x  3) [‘K’ ˆ^GK] (a2)2 + (2)2 (a2 + 2)2  2.a2.2
= =
3q f™²vsGki ni = x  6x + 8
2 2 2
= x2  4x  2x + 8 (a2 + 2)2  4a2
=
= x(x  4)  2(x  4) 2
= (x  4) (x  2) (a2 + 2)2  (2a)2
=
2
 niàGjvi j.mv.à = (2x  3) (x  2) (x  4)
1 2
1 1 1  (x  4) = (a + 2a + 2) (a2  2a + 2) (Ans.)
‰Lb, A = (2x  3) (x  2) = (2x  3) (x  2) (x  4) 2
L ˆ`Iqv AvGQ, Q = a2  11a  12

(x  4)
= 1 1 1 1
(2x  3) (x  2) (x  4)  + = +
Q 12 a2  11a  12 12
1 1 1  (x  2)
= = 12 + a2  11a  12 a2  11a
B (x  4) (2x  3) (x  4) (2x  3) (x  2) = 2 = 2
12(a  11a  12) 12(a  11a  12)
(x  2)
= a(a  11)
(2x  3) (x  2) (x  4) =
12(a2  12a + a  12)
1 1 1  (2x  3)
= = a(a  11)
C (x  4) (x  2) (2x  3) (x  2) (x  4) =
12{a(a  12) + 1 (a  12)}
(2x  3)
= a(a  11)
(2x  3) (x  2) (x  4) = (Ans.)
12(a  12) (a + 1)
myZivs wbGYÆq mvaviY niwewkÓ¡ f™²vskmgƒn
M ˆ`Iqv AvGQ,

x4 x2
, , 1
(2x  3) (x  2) (x  4) (2x  3) (x  2) (x  4) P= (2a2  10a + 12)
2
2x  3 1
(2x  3) (x  2) (x  4) = . 2 (a2  5a + 6) = a2  5a + 6
2
8 cvGéix ˆR‰mwm cixÞv mnvwqKv 2020  MwYZ
Q = a2  11a  12 (a2 + b2) (a2  b2) (a2 + b2) (a + b) (a  b)
= =
R = a2  9a + 20 (a  b)2 (a  b) (a  b)

1 1 (a2 + b2) (a + b)
 = = (Ans.)
P a2  5a + 6 (a  b)


1
=
1 L ‰LvGb,

Q a2  11a  12
(a + b)2  4ab
1 1 2q f™²vsk = a3  b3
 = 2
R a  9a + 20
a+b
‰LvGb, 1g f™²vsGki ni = a2  5a + 6 3q f™²vsk = a2 + ab + b2
= a2  3a  2a + 6
2q f™²vsk I 3q f™²vsk ˆhvM KGi cvB,
= a(a  3)  2 (a  3)
(a + b)2  4ab a+b
= (a  3) (a  2) + 2
a3  b3 a + ab + b2
2q f™²vsGki ni = a2  11a  12 (a  b)2 a+b
2 = + 2 2
= a  12a + a  12 (a  b)(a2 + ab + b2) a + ab + b
= (a  12) + 1 (a  12) ab a+b
= +
= (a  12) (a + 1) a2 + ab + b2 a2 + ab + b2

3q f™²vsGki ni = a2  9a + 20 =
ab+a+b
2
a2 + ab + b2
= a  4a  5a + 20
2a
= (a  4) (a  5) = (Ans.)
a2 + ab + b2
 f™²vskàGjvi nGii j.mv.à.
M 1g f™²vsGki ni = a2 + b2  2ab = (a  b)2

= (a  3) (a  2) (a  12) (a + 1) (a  4) (a  5)
1 1 2q f™²vsGki ni = a3  b3 = (a  b) (a2 + ab + b2)
1g f™²vsk = a2  5a + 6 = (a  3) (a  2)
3q f™²vsGki ni = a2 + ab + b2
(a  12) (a + 1) (a  4) (a  5)
=
(a  3) (a  2) (a  12) (a + 1) (a  4) (a  5)
(Ans.)  niàGjvi j.mv.à. = (a  b)2 (a2 + ab + b2)
(a  b)2 (a2 + ab + b2)
2q f™²vsk = a2  11a  12
1 ‰Lb, (a  b)2
= a2 + ab + b2

1 (a4  b4)  (a2 + ab + b2)

=  1g f™²vsk = (a2 + b2  2ab) (a2 + ab + b2)
(a  12) (a + 1)
(a  3) (a  2) (a  4) (a  5) (a4  b4) (a2 + ab + b2)
= (Ans.) =
(a  3) (a  2) (a  12) (a + 1) (a  4) (a  5) (a  b)2 (a2 + ab + b2)
1 1 (a  b)2 (a2 + ab + b2)
3q f™²vsk = a2  9a + 20 = (a  4) (a  5) Avevi, (a  b)(a2 + ab + b2) = (a  b)

=
(a  3) (a  2) (a  12) (a + 1)
(Ans.)  2q f™²vsk
(a  3) (a  2) (a  12) (a + 1) (a  4) (a  5)
(a + b)2  4ab (a  b)2  (a  b)
= 3 3 =
a b (a  b) (a2 + ab + b2) (a  b)
a4  b4 (a + b)2  4ab a+b
cÉk
² 4 a2 + b2  2ab , a3  b3
, 2
a + ab + b2 (a  b)3
=
(a  b) (a2 + ab + b2)
2
exRMwYZxq f™²vsk|
(a  b)2 (a2 + ab + b2)
K. cÉ^g ivwkGK jwNÓ¤ AvKvGi cÉKvk Ki| 2 Avevi, (a2 + ab + b2)
= (a  b)2

L. w«¼Zxq I Z‡Zxq ivwki ˆhvMdj wbYÆq Ki| 4 a+b (a  b)2 (a + b)

 3q f™²vsk = a2 + ab + b2 = (a  b)2 (a2 + ab + b2)
M. ivwkàGjvGK mgniwewkÓ¡ f™²vsGk cwiYZ Ki| 4
4 bs cÉGk²i mgvavb (a4  b4) (a2 + ab + b2)
 mgni wewkÓ¡ wZbwU f™²vsk (a  b)2 (a2 + ab + b2)
K
 ˆ`Iqv AvGQ,
(a  b)3 (a  b)2 (a + b)
4
a b 4 22
(a )  (b ) 22 2 , (Ans.)
1g ivwk = a2 + b2  2ab = (a  b)2 (a  b) (a + ab + b ) (a  b)2 (a2 + ab + b2)
2 2
exRMwYZxq f™²vsk 9

Abykxjbx 5.2
DËi ms‡KZmn m„Rbkxj cÖkœ

cÉk
 ² 1 A = x + 2, B = x2 − 4, C = x2 − 2x + 4, D = x2 + 2x + x2 + 3x  4 x2  16 (x  4)2
4 ‰es E = x − 8 cuvPwU exRMwYZxq ivwk|
3 cÉ
 k² 2 P = 2
x  7x + 12
, Q = 2
x 9
‰es R =
(x  1)
A

K. A I C ‰i M.mv.à. wbYÆq Ki| 2 K. Q ˆ^GK ˆKvb mvswLÅK gvb weGqvM KiGj weGqvMdGji jGe
1 x + 2 6x
L. mij Ki : A − C + E 4 ˆKvGbv x ^vKGe bv? 2
A 1 1 C L. P  Q  R = KZ? 4
M. cÉgvY Ki, B  C  E  D = 1 4
M. P, Q I R ˆK mvaviY niwewkÓ¡ f™²vsGk cÉKvk Ki| 4
96x Dîi: K. L.
Dîi: K. 1 L. 6 x − 64
1; x + 3

m†Rbkxj cÉGk²i mgvavb

cÉk²1 P = x2 – 15x + 56, Q = x2 – x – 56, R = x2 – 12x + 32
 1 1
‰LvGb, P = (x  7) (x  8)
2x x x
‰es S = x2 + 5x + 4 – x2 – 1 + x2 + 3x – 4, PviwU exRMwYZxq ivwk| (x + 7) (x  4)
=
K. exRMwYZxq f™²vsk ejGZ Kx ˆevS? 2 (x  7) (x  8) (x + 7) (x  4)

L. S ‰i mijdj wbYÆq Ki| 4 1 1

=
Q (x  8) (x + 7)
1 1 1
M. P, Q ‰es R ˆK mgniwewkÓ¡ f™²vsGk cÉKvk Ki| 4 (x  7) (x  4)
=
(x  8) (x + 7) (x  7) (x  4)
1 bs cÉGk²i mgvavb
1 1
K exRMwYZxq f™²vsk : hw` m I n `ywU exRMwYZxq ivwk nq,
 =
R (x  4) (x  8)
m
ZGe n ‰KwU exRMwYZxq f™²vsk ˆhLvGb n  0| (x  7) (x + 7)
= (Ans.)
2 (x  4) (x  8) (x  7) (x + 7)
x+y x 5
ˆhgb : z  b , 2x + y BZÅvw`|
cÉk²2 A = x – 2, B = x2 + 2x + 4 ‰es C = x3 – 8 wZbwU

2x x x
L ˆ`Iqv AvGQ,
 S= 2  +
x + 5x + 4 x2  1 x2 + 3x  4 exRMwYZxq ivwk|
=
2x

x
+
x K. C ˆK Drcv`GK weGkÏlY Ki| 2
(x + 1) (x + 4) (x + 1) (x  1) (x + 4) (x  1)
L. B, C I x3 + 8 ‰i j.mv.à. wbYÆq Ki| 4
2x(x  1)  x(x + 4) + x(x + 1)
= 1 x+2 2–x 1
(x  1) (x + 1) (x + 4) M. cÉgvY Ki ˆh, A ÷ B × x3 – 8 = 4 – x2 4
2x2  2x  x2  4x + x2 + x
=
(x  1) (x + 1) (x + 4) 2 bs cÉGk²i mgvavb
2x2  5x K C = x3  8

=
(x  1) (x + 1) (x + 4)
= x3  23
2x2  5x
= 2 (Ans.) = (x  2) (x2 + 2x + 4) (Ans.)
(x  1) (x + 4)
L ‰LvGb, B = x2 + 2x + 4

M ‰Lb, P = x2  15x + 56
 C = (x  2) (x2 + 2x + 4) [‘K’ nGZ]
2
= x  7x  8x + 56 x3 + 8 = x3 + 23
= (x  7) (x  8) = (x + 2) (x2  2x + 4)
Q = x2  x  56
 B, C I x3 + 8 ‰i j.mv.à
= x2  8x + 7x  56
= x(x  8) + 7(x  8) = (x  2) (x + 2x + 4) (x + 2) (x2  2x + 4)
2

= (x  8) (x + 7) = (x3  23) (x3 + 23)

R = x2  12x + 32 = (x3  8) (x3 + 8)
= (x  4) (x  8) = x6  82
R, Q I R ‰i j.mv.à = (x  7) (x  8) (x + 7) (x  4) = x6  64 (Ans.)
10 cvGéix ˆR‰mwm cixÞv mnvwqKv 2020  MwYZ
1
M evgcÞ = A  B  x3  8

x+2 2x 4 bs cÉGk²i mgvavb
x xy
1 x2 + 2x + 4 { (x  2)} K
 y

y
=  
x2 x+2 (x  2) (x2 + 2x + 4)
xx+y y
1 = = = 1 (Ans.)
= 2 y y
x  22
L DóxcGKi ivwk wZbwUi àYdj

1 1
= 2 = x3 + y3 x3  y3 xy
2  x2 4  x2 =  
x2 + xy + y2 x2  xy + y2 x + y
= WvbcÞ (cÉgvwYZ)
(x + y) (x2  xy + y2) (x  y) (x2 + xy + y2) x  y
1 1 1 =  
cÉk
 ² 3 A = x2 – 11x + 28, B = x2 – 13x + 36, C = x2 – 15x + 44 x2 + xy + y2 x2  xy + y2 x+y

1 1 2 = (x  y)2 (ˆ`LvGbv nGjv)

‰es D = x2 – 1 + x4 – 1 + x8 – 1
M 1g ivwk  3q ivwk  2q ivwk

K. x + x y + y ˆK Drcv`GK weGkÏlY Ki|
4 2 2 4
2 x3 + y3 xy x3  y3
2 2  2
L. D ˆK mij Ki| 4 x + xy + y x + y x  xy + y2
M. A, B I C ˆK mgniwewkÓ¡ f™²vsGk cÉKvk Ki| 4 (x + y) (x2  xy + y2) x + y (x  y) (x2 + xy + y2)
=  
x2 + xy + y2 xy x2  xy + y2
3 bs cÉGk²i mgvavb
= (x + y)2 (Ans.)
K cÉ`î ivwk = x4 + x2y2 + y4

= (x2)2 + 2x2y2 + (y2)2  x2y2
x2 + 3x – 4 x2 + 2x – 3
= (x2 + y2)2  (xy)2 cÉk²5 P = x2 + 7x + 12, Q = x2 + 6x – 7 ‰es

= (x2 + xy + y2) (x2  xy + y2) (Ans.)
x2 + 12x + 35
1 1 2 R=
x2 + 4x – 5
wZbwU exRMvwYwZK ivwk|
L ‰LvGb,
 D= 2 + +
x  1 x4  1 x8  1
2x + y y
1 1 2 K.  x + y – 1  1 – x + y ˆK mij Ki| 2
= 2 + +
x  1 x4  1 (x4 + 1) (x4  1)
1 x4 + 1 + 2 L. (P + Q – R) = KZ? 4
= 2 + 4 2
x  1 (x + 1) (x4  1) x –9
M. cÉgvY Ki ˆh, Q  R  x2 – 4x + 3 = 1 4
1 x4 + 3
= 2 + 4
x  1 (x + 1) (x2  1) (x2 + 1) 5 bs cÉGk²i mgvavb
(x4 + 1) (x2 + 1) + x4 + 3 2x + y y
= 4
(x + 1) (x2 + 1) (x2  1) K  x + y – 1  1 – x + y

x6 + x4 + x2 + 1 + x4 + 3 2x + y – x – y x + y – y
= =
(x4 + 1) (x4  1)  x+y  x+y 
x + 2x4 + x2 + 4
6
x x
=
x8  1 = 
x+y x+y
M A, B I C ‰i niàGjv h^vKÌGg
 x x+y
= 
x2  11x + 28 = (x  7) (x  4) x+y x
x2  13x + 36 = (x  4) (x  9) = 1 (Ans.)
‰es x  15x + 44 = (x  4) (x  11)
2
L ˆ`Iqv AvGQ,

niàGjvi j.mv.à = (x  4) (x  7) (x  9) (x  11) x2 + 3x – 4 x2 + 4x – x – 4
1 (x  9) (x  11) P= 2 = 2
x + 7x + 12 x + 4x + 3x + 12
A= =
x2  11x + 28 (x  7) (x  4) (x  9) (x  11) x(x + 4) – 1(x + 4)
=
1 (x  7) (x  11) x(x + 4) + 3(x + 4)
B= 2 =
x  13x + 36 (x  4) (x  9) (x  7) (x  11) (x + 4)(x – 1) x – 1
= =
1 (x  7) (x  9) (x + 4)(x + 3) x + 3
C= 2 = (Ans.)
x  15x + 44 (x  4) (x  11) (x  7) (x  9) x2 + 2x – 3 x2 + 3x – x – 3
Q= =
3
x +y 3
x –y 3 3
x–y x2 + 6x – 7 x2 + 7x – x – 7
cÉk
² 4 , ,
x2 + xy + y2 x2 – xy + y2 x + y
wZbwU exRMwYZxq ivwk|
x(x + 3) – 1(x + 3)
x x–y =
K. ‰es y ‰i weGqvMdj wbYÆq Ki| 2 x(x + 7) – 1(x + 7)
y
(x + 3)(x – 1)
L. ˆ`LvI ˆh, ivwk wZbwUi àYdj (x – y) . 2
4 =
(x + 7)(x – 1)
M. cÉ^g ivwkGK Z‡Zxq ivwk «¼viv fvM KGi cÉvµ¦ fvMdjGK w«¼Zxq ivwk x+3
«¼viv àY Ki| 4 =
x+7
exRMwYZxq f™²vsk 11

x2 + 12x + 35 x2 + 7x + 5x + 35 L ‘K’ nGZ cvB,


‰es R = x2 + 4x – 5 = x2 + 5x – x – 5
a4  b4 (a2 + b2) (a + b)
x(x + 7) + 5(x + 7) 2 2=
= a  2ab + b (a  b)
x(x + 5) – 1(x + 5)
(x + 5)(x + 7) x+7 ivwk wZbwUi àYdj
= = (a2 + b2) (a + b) a  b a+b
(x + 5)(x – 1) x–1
=  3 
 cÉ`î ivwk (a  b) a + b3 a3 + b3
=P+Q–R (a2 + b2) (a + b) ab (a + b)
=  
x–1 x+3 x+7 (a  b) (a + b) (a2  ab + b2) (a + b) (a2  ab + b2)
= + –
x+3 x+7 x–1 a2 + b2
= 2 (Ans.)
(x – 1)(x + 7) + (x + 3)2 x + 7 (a  ab + b2)2
= –
(x + 3)(x + 7) x–1 a4  b4
2 2
x – x + 7x – 7 + x + 6x + 9 x + 7 M
 a  2ab + b2
2  (a3 + a2b + ab2 + b3)
= –
x2 + 3x + 7x + 21 x–1 (a2 + b2) (a + b)
2
2x + 12x + 2 x + 7 =  {a2 (a + b) + b2(a + b)} [‘K’ nGZ cvB,]
= 2 – (a  b)
x + 10x + 21 x – 1
(a2 + b2) (a + b)
(2x + 12x + 2)(x – 1) – (x + 7)(x2 + 10x + 21)
2
=  {(a + b) (a2 + b2)}
= (a  b)
(x2 + 10x + 21) (x – 1)
2x + 12x + 2x – 2x2 – 12x – 2 – x3 – 7x2
3 2 (a2 + b2) (a + b) 1
= 
– 10x2 – 70x – 21x – 147 (a  b) (a + b) (a2 + b2)
=
(x – 1)(x2 + 10x + 21) 1
=
x3 – 7x2 – 101x – 149 ab
= (Ans.)
(x – 1)(x + 3) (x + 7) a2
2
x –9 x2 – 32
‰Lb, cÉvµ¦ fvMdGji mvG^ a + b ˆhvM KiGj nq|
M
 ‰LvGb, x2 – 4x + 3 = x2 – 3x – x + 3
1 a2 a + b + a2 (a  b)
(x + 3)(x – 3) + =
ab a+b (a + b) (a  b)
=
x(x – 3) – 1(x – 3) a + b + a3  a2b a + a3 + b  a2b
(x + 3)(x – 3) x + 3 = = (Ans.)
= = (a + b) (a  b) a2  b2
(x – 3)(x – 1) x – 1 2 2
x+1 x +x x
‘L’ ˆ^GK cvB, cÉk
² 7 A = x  1 , B = x2 + x  2 , C = x2 + 5x + 6
x+3 x+7
Q=
x+7
‰es R = x – 1 x2  y2
‰es D = x3  y3 PviwU exRMvwYwZK f™²vsk|
x2 – 9
evgcÞ = Q  R  x2 – 4x + 3 K. D ˆK jwNÓ¤ AvKvGi cÉKvk Ki| 2
x + 3 x + 7 (x + 3) (x - 3) x
=  
x + 7 x – 1 (x – 3)(x - 1)
L. cÉgvY Ki ˆh, ABC=
x+3
| 4

=
(x + 3) (x + 7) (x + 3)
  M. A, B ‰es C ˆK mgniwewkÓ¡ f™²vsGk cÉKvk Ki| 4
(x + 7) (x – 1) (x – 1)
7 bs cÉGk²i mgvavb
(x + 3) (x – 1)
= 
(x – 1) (x + 3) K ˆ`Iqv AvGQ,

=1 x2  y2 (x + y) (x  y) x+y
D= = = 2 (Ans.)
= WvbcÞ x3  y3 (x  y) (x2 + xy + y2) x + xy + y2
x2 – 9 x+1
 Q  R  x2 – 4x + 3 = 1 (cÉgvwYZ) L
 ˆ`Iqv AvGQ, A = x  1
a4  b4 ab a+b x2 + x x (x + 1)
cÉk
² 6 a2  2ab + b2 , a3 + b3 , a3 + b3 Avevi, B = x2 + x  2 = x2 + 2x  x  2
K. 1g ivwkGK jwNÓ¤ AvKvGi cÉKvk Ki| 2 x(x + 1) x(x + 1)
= =
L. ivwk wZbwUi àYdj wbYÆq Ki| 4 x(x + 2)  1(x + 2) (x + 2) (x  1)
M. 1g ivwkGK a3 + a2b + ab2 + b3 «¼viv fvM KGi fvMdGji mvG^ x2 x2
‰es C = x2 + 5x + 6 = x2 + 3x + 2x + 6
a2
a+b
ˆhvM Ki| 4 x2 x2
= =
6 bs cÉGk²i mgvavb x(x + 3) + 2(x + 3) (x + 3) (x + 2)
a4  b4 (a2)2  (b2)2 x+1 x(x + 1) x2
K 1g ivwk
 = 2 2 =
‰Lb, A  B  C = x  1  (x + 2) (x  1)  (x + 3) (x + 2)
a  2ab + b (a  b)2
(a + b ) (a  b ) (a2 + b2) (a + b) (a  b)
2 2 2 2 (x + 1) (x + 2) (x  1) x2 x
= = =   =
(a  b)2 (a  b)2 (x  1) x(x + 1) (x + 3) (x + 2) x + 3
2 2
(a + b ) (a + b) x
=  ABC=
x+3
(cÉgvwYZ)
(a  b)
12 cvGéix ˆR‰mwm cixÞv mnvwqKv 2020  MwYZ

M ‘L’ ˆ^GK cvB,

 (1 + a2 + a4)
= (1 + a2) (1 + a2 + a4) 
x+1 x(x + 1) x2 2(1 + a2)
A= ,B= ,C= (1 + a2 + a4)2
x1 (x + 2) (x  1) (x + 3) (x + 2)
= (Ans.)
2
 f™²vskàGjvi nimgƒGni j.mv.à = (x  1) (x + 2) (x + 3)
2 3 2
x+1 (x + 1) (x + 2) (x + 3)
‰Lb, A = x  1 = (x  1) (x + 2) (x + 3) cÉk²9 x2  x  2 , x2 + x  6 ‰es x2 + 6x + 9 wZbwU
 f™²vsk|

x(x + 1) x(x + 1) (x + 3) K. 2q f™²vsGki nGii Drcv`K wbYÆq Ki| 2

B= =
(x + 2) (x  1) (x  1) (x + 2) (x + 3) L. f™²vskàGjvGK mvaviY niwewkÓ¡ f™²vsGk cÉKvk Ki| 4
x2 x2(x  1) 4x  8
C= =
(x + 2) (x + 3) (x  1) (x + 2) (x + 3) M. 1g `ywU f™²vsGki ˆhvMdj ˆ^GK (x  2) (x  1) (x + 3) weGqvM
 wbGYÆq mgni wewkÓ¡ f™²vsk : Ki| 4
(x + 1) (x + 2)(x + 3)
,
x(x + 1)(x + 3)
‰es 9 bs cÉGk²i mgvavb
(x  1) (x + 2) (x + 3) (x  1) (x + 2) (x + 3)
2
K 2q f™²vsGki ni = x2 + x  6

x (x  1)
(Ans.) = x2 + 3x  2x  6
(x  1) (x + 2) (x + 3)
9 9 = x(x + 3)  2(x + 3)
x y 1 1
cÉk
² 8 A = x3  y3, B = 1  a + a2, C = 1 + a + a2 ‰es = (x  2) (x + 3)
1 L
 1g f™²vsGki ni = x2  x  2
D= = x2  2x + x  2
1 + a2 + a4
K. A ˆK jwNÓ¤ AvKvGi cÉKvk Ki| 2 = x(x  2) + 1 (x  2)
= (x  2) (x + 1)
L. (B  C)  (2a  D) ‰i gvb wbYÆq Ki| 4
1 + a2 2q f™²vsGki ni = (x  2) (x + 3) [‘K’ nGZ]
M. mij Ki : D  (B + C) 4 3q f™²vsGki ni = x2 + 6x + 9
8 bs cÉGk²i mgvavb = x2 + 2.x.3 + 32
x9  y9 = (x + 3)2
K A = x3  y3
 = (x + 3) (x + 3)
(x3)3  (y3)3  f™²vsk wZbwUi nGii j.mv.à.
=
x3  y3 = (x  2) (x + 1) (x + 3) (x + 3)
(x3  y3) {(x3)2 + x3 . y3 + (y3)2} 2
=
(x3  y3)
1g f™²vsk = x2  x  2
(x3  y3) (x6 + x3y3 + y6) 2 (x + 3) (x + 3)
= = 
(x3  y3) (x  2) (x + 1) (x + 3) (x + 3)
= x + x y + y6 (Ans.)
6 3 3
2(x + 3)2
=
1 (x  2) (x + 1) (x + 3)2
L
 ˆ`Iqv AvGQ, B = 1  a + a2
3
1
2q f™²vsk = x2 + x 6
C=
1 + a + a2 3 (x + 1) (x + 3)
1 = 
‰es D = 1 + a2 + a4 (x  2) (x + 3) (x + 1) (x + 3)
3(x + 1) (x + 3)
1 1 2a =
 (B  C)  (2a  D) =  2 2 4
(x  2) (x + 1) (x + 3)2
1  a + a2 1 + a + a 1 + a + a 2
2
1 + a + a  (1  a + a ) 2
2a 3q f™²vsk = x2 + 6x + 9
=  2 4
(1  a + a2) (1+ a + a2) 1 + a + a 2 (x  2) (x + 1)
2
1+a+a 1+aa 2
2a = 
=  (x + 3) (x + 3) (x  2) (x + 1)
(1 + a2)2  a2 1 + a2 + a4
2(x  2) (x + 1)
2a 1 + a2 + a4 =
(x  2) (x + 1) (x + 3)2
= 2 4 = 1 (Ans.)
1+a +a 2a
2(x + 3)2
1+a 2
 wbGYÆq f™²vskàGjv = (x  2) (x + 1) (x + 3)2 ,
M
 D
 (B + C)
3(x + 1) (x + 3) , 2(x  2) (x + 1)
1 + a2 1 1
=  + 2 (x  2) (x + 1) (x + 3)2 (x  2) (x + 1) (x + 3)2
1 1  a + a2 1 + a + a 
1+a +a2 4 M 1g `ywU f™²vsGki ˆhvMdj

2
1+a+a +1a+a 
2 2 3
= (1 + a2) (1 + a2 + a4)   2 
+
2
(1  a + a ) (1 + a + a ) x2  x  2 x2 + x  6
2 + 2a2  2 3
= (1 + a2) (1 + a2 + a4)   = +
(x  2) (x + 1) (x  2) (x + 3)
1 + a2 + a4
exRMwYZxq f™²vsk 13

2x + 6 + 3x + 3 81  y4 (3 + y)2  12y 3+y

=
(x  2) (x + 1) (x + 3)
cÉk
² 11 P = 9  6y + y2, Q = 27  y3
,R=
(3 + y)2  3y

=
5x + 9 K. P ˆK jwNÓ¤ AvKvGi cÉKvk Ki| 2
(x  2) (x + 1) (x + 3) RQ
5x + 9 4x  8 L. mij Ki: R+Q
4
‰Lb, (x  2) (x + 1) (x + 3)  (x  2) (x  1) (x + 3)
M. P, Q ‰es R f™²vskGK mgni wewkÓ¡ f™²vsGk cÉKvk Ki| 4
5x2 + 9x  5x  9  (4x2  8x + 4x  8)
= 11 bs cÉGk²i mgvavb
(x  2) (x + 1) (x + 3) (x  1)
5x2 + 4x  9  4x2 + 4x + 8 K ˆ`Iqv AvGQ,

= 81  y4
(x  2) (x + 1) (x + 3) (x  1) P=
9  6y + y2
x2 + 8x  1
= (Ans.) 92  (y2)2
(x  2) (x + 1) (x + 3) (x  1) = 2
3  2.3y + y2
a3  b3 (a + b)2  3ab a + b (9 + y2) (9  y2)
cÉk
² 10 (a  b)2 + 3ab , a3  b3
, ‰es =
ab (3  y)2
a3 + b3 (9 + y2)(3 + y) (3  y)
a + a2b2 + b4
4 PviwU exRMwYZxq ivwk| =
(3  y)2
1 2m 1 1 2
(9 + y ) (3 + y)
K.  1 + m + 1  m2 m  m2 = KZ? 2 = (Ans)
  3y
L. cÉ^g ivwkGK Z‡Zxq ivwk «¼viv àY KGi cÉvµ¦ àYdGji mvG^ L ˆ`Iqv AvGQ,

KZ àY KiGj a3 + b3 àYdj nGe? 4 (3 + y)2  12y 9 + 6y + y2  12y 9  3y + y2
Q= 3 = 3 3 =
M. cÉ^g ivwk wZbwUi àYdjGK PZz^Æ ivwk «¼viv fvM Ki| 4 27  y 3 y (3  y) (9 + 3y + y2)
3+y 3+y 3+y
10 bs cÉGk²i mgvavb R= = = 2
(3 + y)2 3y 9 + 6y + y2  3y 9 + 3y + y
1 2m 1 1
K cÉ`î ivwk = 1 + m + 1  m2 m  m2
 3+y 9  3y + y2
  R+Q=
9 + 3y + y2
+
(3  y) (9 + 3y + y2)
1  m + 2m m  1
=  (3 + y)(3  y) + 9  3y + y2
(1  m2) m2 =
(3  y) (9 + 3y + y2)
1+m  (1  m)
=  9  y2 + 9  3y + y2
(1  m) (1 + m) m2 =
(3  y) (9 + 3y + y2)
1
= (Ans.) 18  3y
m2 =
(3 y) (9 + 3y + y2)
a  b3
3
a+b
L cÉkg² GZ,
 
(a  b)2 + 3ab a  b
 p = a3 + b3 3+y 9 – 3y + y2
RQ= 2–
9 + 3y + y (3 – y) (9 + 3y + y2)
[awi, p àY KiGZ nGe] (3 + y)(3 – y) – (9 – 3y + y2)
2 =
(a  b) + 3ab a  b (3 – y)(9 + 3y + y2)
ev, p = (a3 + b3)  a3  b3

a+b 9 – y2 – 9 + 3y – y2
=
ev, p = (a + b) (a2  ab + b2) (3 – y)(9 + 3y + y2)
a2 + ab + b2 (a  b) 3y – 2y2
=
 
(a  b) (a2 + ab + b2) (a + b) (3 – y)(9 + 3y + y2)
 p = a2  ab + b2 (Ans.) 3y  2y2
R Q (3 y)(9 + 3y + y2) 3y – 2y2
M ˆ`Iqv AvGQ,
  = = (Ans.)
R+Q 18  3y 18 – 3y
a3 − b3 (a + b)2 − 3ab 2
A = 2 ,B= , (3  y) (9 + 3y + y )
(a − b) + 3ab a3 − b3
(9 + y2) (3 + y)
C=
a+b
,D= 4
a3 + b3 M
 ‘K’ nGZ cvB, P = 3 y
a−b a + a2b2 + b4
a3 − b3 (a + b)2 − 3ab 9 – 3y + y2 3+y
 ABCD=  ‘L’ nGZ cvB, Q = (3 – y)(9 + 3y + y2) , R = 9 + 3y + y2
(a − b)2 + 3ab a3 − b3
a+b a3 + b3 P, Q I R ‰i niàGjvi j.mv.à. = (3  y) (9 + 3y + y2)
  4 (9 + y2) (3 + y) (9 + 3y + y2)
a − b a + a2b2 + b4
 P=
2
a − ab + b 2
a + b (a + ab + b ) (a2 − ab + b2)
2 2 (3  y) (9 + 3y + y2)
= 2 2  
a + ab + b a−b (a + b) (a2 − ab + b2) 9  3y + y2
Q=
[Drcv`GK weGkÏlY KGi] (3  y) (9 + 3y + y2)
a2 − ab + b2 (3 + y)(3 y) 9 y2
= (Ans.) R= 2 = (Ans.)
a−b (3 y)(9 + 3y + y ) (3  y)(9 + 3y + y2)
14 cvGéix ˆR‰mwm cixÞv mnvwqKv 2020  MwYZ

AaÅvGqi cÉÕ§wZ hvPvB: NGi eGm cixÞv

.gGWj cÉk-² 5.
m†Rbkxj eüwbeÆvPwb cÉk²cò
mgq: 30 wgwbU; gvb-30
x2 x 4 2 1 x+1 (x  1)2
1. 
x2  16 x + 4
= KZ? A K
x
L 0 M
x
N
x 22. x  1 ‰es x2 + x ‰i àYdj KZ?
2
2x 4x 1 – xy x2  1 x+1 x1 x+1
K 2
x  16
L 2
x  16
12. x–y
‰es x3 – y3 ‰i ˆhvMdj KZ? K L M N
x x x x1
2x(x  2) 4x – (x2 + y2) wbGPi ZG^Åi wfwîGZ (23-25) bs cÉGk²i Dîi `vI:
M 2 N 2 K
x  16 x  16 (x – y)(x2 + xy + y2) a3 + b 3 a3 – b 3
x2  6x + 5 x2 + y2 I
a2 – ab + b2 a2 + ab + b2
`yBwU exRMvwYZxq
2. ‰i jwNÓ¤ AvKvGi cÉKvwkZ i…c L 2
x – xy + y2
x2  1 f™²vsk|
wbGPi ˆKvbwU? x2 + y2 23. cÉ^g f™²vskwUi jwNÓ¤ i…c wbGPi ˆKvbwU?
M
x5 x+1 x1 x5 (x – y)(x2 + xy + y2) a+b
A K L M N x2 – y2 A K a+b L
x+1 x5 x+1 x1 N a2  b2 + ab
x y z (x – y)(x2 + xy + y2)
3. + + ‰i ˆhvMdj KZ nGe?
y z x x2  x  30 M
1
N
a+b
xz + xy + yz xyz
13. x2  36
‰i jwNÓ¤ i…c ˆKvbwU? a+b ab
A K L x+5 x+5 x5 x5
24. w«¼Zxq f™²vskwUi jwNÓ¤ i…c wbGPi ˆKvbwU?
xyz xz + xy + yz a2 – b2
2 2 2 A K L M N a–b a+b
x z + xy + yz xyz x6 x+6 x+6 x6 A K L M ab N 2 2
M N 2 mn2 , a3b
a+b ab a +b
xyz x z + xy2 + yz2
4. exRMwYZxq f™²vsGki ˆÞGò@ 14. n 3 a 2b 2
f™²vsk `yBwUi@ 25. f™²vsk«¼Gqi àYdj KZ?
a b a b a2 a A K a2 + b 2 L a2 – b 2
i.  =1 ii.  = 2 i. w«¼ZxqwUi jwNÓ¤ AvKvi M (a3 + b3) (a3  b2)
b a b a b b
1 1 1 m N (a + b)2
iii.  = ii. cÉ^gwUi jwNÓ¤ AvKvi (x + y)2  4xy x2  y2 x2  xy + y2
1 + x 1  x2 1  x n 26. x3 + y3

(x + y)2

x+y
wbGPi ˆKvbwU mwVK? iii. ˆhvMdj
an + bm
A K i I ii L i I iii bn ‰i mijK‡Z gvb ˆKvbwU?
M ii I iii N i, ii I iii wbGPi ˆKvbwU mwVK? A K
x+y
L
xy
M x  yN x + y
x2  5x  6 A K i I ii L i I iii xy x+y
5. f™²vskwUi jwNÓ¤ i…c ˆKvbwU? M ii I iii N i, ii I iii x x
x2  1 27. y  1 I 1  y ivwk `yBwUi@
(x  3) (x  2) x+6 wbGPi ZG^Åi wfwîGZ (15 I 16) bs cÉGk²i Dîi `vI:    
K L 1 1 4 32 i. ˆhvMdj 0 ii. fvMdj 1
(x + 1) (x  1) x+1 , , , PviwU
x6 x6 x – 2 x + 2 x2 + 4 x4 16 (x  y)2
M N iii. àYdj
x+1 x1 exRMvwYwZK ivwk| y2
wbGPi ZG^Åi AvGjvGK (6-8) bs cÉGk²i Dîi `vI: 15. 1g I 2q ivwki weGqvMdj wbGPi ˆKvbwU? wbGPi ˆKvbwU mwVK?
6(y2  6y + 5) 4 4 1 1 K i I ii L i I iii
8(y2  25)
‰KwU exRMvwYwZK f™²vsk| A K 2
x –4
L 2
x +4
M
x+2
N
x–2 M ii I iii N i, ii I iii
6. wbGPi ˆKvbwU f™²vskwUi nGii ‰KwU 16. ivwkgvjvi niàGjvi j.mv.à. ˆKvbwU? x z
Drcv`K nGe? A K x4 + 16 L x4 – 16 28. , f™²vsk `ywUi@
y y
A K y + 25 L 2(y + 25) M x2 + 16 N x2 – 16 y
M y5 N y1 m2 – n2 (m – n)2 i. cÉ^g f™²vskwUi àYvñK wecixZ f™²vsk x
17. 
m2 + n2 – 2mn (m + n)2 – 4mn
‰i
7. f™²vskwUi je I nGii M.mv.My. KZ? y
w«¼ZxqwUi àYvñK wecixZ f™²vsk z
A K y5 L 2(y  5) mijK‡Z gvb wbGPi ˆKvbwU? ii.
M 2(y + 5) N 2(y  1) m–n m+n x
8. f™²vskwUi jwNÓ¤ i…c ˆKvbwU? K m–n L m+ n M
m+n
N
m –n iii. fvMdj z
3(y  1) y1 x2 – 5x + 6
18. x2 – 9x + 20 f™²vskwUGK (x – 4)(x – 5) wbGPi ˆKvbwU mwVK?
A K L A K i I ii L ii I iii
4(y + 5) y+5
4(y + 5) 3(y + 1) «¼viv àY KiGj àYdj KZ nGe? M i I iii N i, ii I iii
M N 2
K x – 9x + 20 L x – 6x + 5 2
3(y + 1) 4(y + 5) x , x
x,y,z M x2 – 5x + 6 N x2 – 8x + 180 29. x+y xy
f™²vsk `yBwUi@
9. ˆK mvaviY niwewkÓ¡ f™²vsGk 2p2q3 6r2
p q r 19.  2 2 = KZ? i. nGii àYdj x2  y2
cÉKvk KiGj nGe@ 3r 4p q
x2 x+y
xqr , ypr , zpq xpq , yqr , zrp A K pq L qr M pr N pqr ii. àYdj 2 iii. fvMdj
A K L x+y 1 x  y2 xy
pqr pqr pqr pqr pqr pqr 20.  = KZ? wbGPi ˆKvbwU mwVK?
xrp , ypq , zqr xqr , ypq , zpr x2  y2 x  y
M N x+y x+y 1 A K i I ii L ii I iii
pqr pqr pqr pqr pqr pqr
x 1
A K
xy
L 2 2M
x y xy
N 1 M i I iii N i, ii I iii
10. 2+ 2 = KZ? 2 a2 a3
(x + 1) (x + 1)
21.
p – 2p + 1 p–1
ˆK x – 1 «¼viv fvM KiGj 30.  2
x3 x 9
‰i gvb KZ?
x x x2 – 2x + 1
K L fvMdj KZ nGe? a3 a
(x + 1) x+1 A K L
x 1 p–1 x–1 (x  3) (x2  9) x3
M N A K L a x+3
(x + 1)2 (x + 1) x–1 p–1 M N
1 1 2 (p – 1)3 p3 – 1 x+3 a
11. +  = KZ?
x x x M
(x – 1)3
N 3
x –1
1 L 2 K 3 M 4 K 5 N 6 M 7 L 8 K 9 K 10 N 11 L 12 M 13 L 14 N 15 K
Dîi

16 L 17 N 18 M 19 L 20 N 21 K 22 M 23 K 24 M 25 L 26 L 27 K 28 N 29 K 30 N
exRMwYZxq f™²vsk 15

m†Rbkxj iPbvgƒjK cÉk²cò

mgq: 2 N¥Ÿv 30 wgwbU; gvb-70
[ ˆh ˆKvGbv 7wU cÉGk²i Dîi `vI ]
1. M = x2  3x + 2, N = x2  5x + 6 ‰es K = x2  4x + 3, wZbwU K. 1g f™²vsk ˆ^GK 2q f™²vsk weGqvM Ki| 2
exRMwYZxq ivwk| L. 1g I 2q f™²vsGki àYdjGK 3q f™²vsk «¼viv fvM Ki| 4
M M. f™²vsk wZbwUGK mvaviY niwewkÓ¡ f™²vsGk cÉKvk Ki| 4
K. x  2 ˆK jwNÓ¤ AvKvGi cÉKvk Ki| 2 x3  y3 1 1
7. A = x4 + x2y2 + y4 , B = 1  x + x2 , C = 1 + x + x2 ‰es
1 1 1
L. mij Ki : M + N + K . 4 1
D=
1 + x2 + x4
PviwU exRMwYZxq ivwk|
1 1 1
M. , ,
M N K
ˆK mgniwewkÓ¡ f™²vsGk cÉKvk Ki| 4 K. A ˆK jwNÓ¤ AvKvGi cÉKvk Ki| 2
1 1 1 L. cÉgvY Ki ˆh, B  C  2x  D = 0. 4
2. x2  5x + 6 , x2  7x + 12 I x2  8x + 15 wZbwU f™²vsk| 1 + x2
M. mij Ki : D  (B + C) 4
K. cÉ^g f™²vsk `ywUi ˆhvMdj wbYÆq KGiv| 2
2p 1 1
L. ni wZbwUi M.mv.à. wbYÆq KGiv| 4 8. x = 1 + p2 + p4 , y = ‰es z = 1 + p + p2 wZbwU
1  p + p2
M. f™²vsk wZbwUGK mvaviY niwewkÓ¡ f™²vsGk cÉKvk KGiv| 4 exRMwYZxq ivwk|
1 1 2x
3. A = 1  x + x2 , B = 1 + x + x2 , C = 1 + x2 + x4 K. y ‰es z ˆK mvaviY niwewkÓ¡ f™²vsGk cwiYZ Ki| 2
L. mij Ki : x  y + z 4
x2
‰es D = 1  x6 PviwU f™²vsk| M. mij Ki : (y  z)  x 4
3a 2a a
K. C ‰i niGK Drcv`GK weGkÏlY Ki| 2 9. L = a2 + 3a  4, M = a2  1, N = a2 + 5a + 4,
L. cÉgvY Ki ˆh, A  B  C = 0 4 x3 + y3 + 3xy(x + y) (x  y)2 + 4xy
2  3 ... ... ... (a)
(x + y)  4xy x  y3  3xy(x  y)
M. A, B, C ˆK mgni wewkÓ¡ f™²vsGk cÉKvk Ki| 4
1 x y2
K. cÉ^g wZbwU ivwki nimgƒGni M.mv.à. wbYÆq Ki| 2
4. x  y , x2 + xy + y2 , x3  y3 wZbwU exRMvwYwZK ivwk| L. L + M + N ‰i gvb wbYÆq Ki| 4
K. f™²vsk wZbwUi nGii j.mv.à. wbYÆq Ki| 2 M. (a) ˆK mijxKiY Ki| 4
2 2 2
L. f™²vskàGjvi ˆhvMdj wbYÆq Ki| 4 10. P = x2 + 3x  4 , Q = x 2 16 ‰es R = (x  4)
x  7x + 12 x 9 (x  1)
M. cÉ`î f™²vskàGjvGK mgniwewkÓ¡ f™²vsGk cÉKvk Ki| 4 2
a b a a
5. P = a2  5a + 6, Q = a2  7a + 12 ‰es R = a2  9a + 20 wZbwU K. mij Ki: b + a + 1  b2 + b + 1. 2
exRMwYZxq ivwk| L. P  Q  R = KZ? 4
a a+b M. P, Q I R ˆK mvaviY ni wewkÓ¡ f™²vsGk cÉKvk Ki| 4
K. b  b ‰i weGqvMdj KZ? 2
3 3
a b 2ab a+b ab
1 1 11.  a  b  3ab, 1  a2 + b2 , a  b + a + b  wZbwU
L. ˆK jwNÓ¤ AvKvGi cÉKvk Ki|
+
Q R
4    
exRMwYZxq ivwk|
1 1 1
M. P , Q , R ˆK mvaviY niwewkÓ¡ f™²vsGk cÉKvk Ki| 4 K. cÉ^g ivwkGK jwNÓ¤i…Gc cÉKvk Ki| 2
1 1 2x L. w«¼Zxq ivwkGK cÉ^g ivwk «¼viv fvM Ki| 4
6. 2x + 3y , 2x  3y , 4x2  9y2 wZbwU exRMvwYwZK f™²vsk| M. ‘L’ ˆ^GK cÉvµ¦ fvMdGji mvG^ Z‡Zxq ivwki àYdj wbYÆq Ki|4
3 xy (1  x + x2)2 (1 + x + x2)2
1. K. x  1; L. (x  1) (x  3) 7. K. x2  xy + y2 ; M. 2
2 1 + p + p2 1  p + p2
2. K. (x  2) (x  4) ; L. (x  3) 8. K. 1 + p2 + p4; 1 + p2 + p4; L. 0; M. 1
3. K. (1 + x + x2) (1 – x + x2) 2a(3a + 5)
Dîigvjv

9. K. 1; L. (a + 4)(a + 1)(a  1); M. x2  y2

2(x2 + y2)
4. K. x3 – y3; L. 3 3
x –y b
10. K. a ; L. x + 3
2
5. K.  1; L.
(a  3) (a  5) 1 2
11. K. (a  b)2; L. a2 + b2 ; M. a2  b2
6y 1
6. K.  2
4x  9y2
; L. 2x;
16 cvGéix ˆR‰mwm cixÞv mnvwqKv 2020  MwYZ

2019
a2  b2 (a + b)2  4ab a+b
DËi ms‡KZmn m„Rbkxj cÖkœ cÉk
² 6 a2 + ab +b2
,
a3  b3
‰es a2 + ab + b2 wZbwU
f™²vsk| [BÕ·vnvwb cvewjK Õ•zj I KGjR, KzwgÍÏv]
Abykxjbx-5.1 K. 1g f™²vskwUGK jwNÓ¡ AvKvGi cÉKvk Ki| 2
x 1 x a2  b2 a+b (a + b)2  4ab
cÉk
² 1 xy, yz ‰es z wZbwU exRMwYZxq f™²vsk L. mij Ki: a2 + ab + b2 + a2 + ab + b2  4
a3  b3
K. f™²vsGki jwNÓ¤i…c wK? 2 (a  b) 2
M. cÉ^g wZbwU ivwki àYdjGK  a2 + ab + b2 «¼viv fvM Ki| 4
L. f™²vsk wZbwUGK mgni wewkÓ¡ f™²vsGk cÉKvk Ki| 4
2 2 2 2 2
M. f™²vsk wZbwUi ˆhvMdj wbYÆq Ki| 4 Dîi: K. 2 a  b 2 L. a2 – b + 2b2; M. 2 (a + b) 2 2
xz x x2y xy + z + 1 a + ab +b a + ab + b (a + ab + b )
Dîi: L. xyz, xyz ‰es xyz; M. yz x+y xy yz
² 7 (x  y)2 , x3 + y3 I x2  y2 wZbwU exRMwYZxq f™²vsk|
cÉk

2
1 1 2 + 2x 2x
cÉk
² 2 ,
1  x + x2 1 + x + x
2, , 2
x6  1 1 + x + x
4 PviwU K. f™²vsk wZbwUi niàGjvi j.mv.à wbYÆq Ki| 2
exRMvwYwZK ivwk| [cçMo miKvwi evwjKv DœP we`Åvjq] L. f™² v sk wZbwUGK mgniwewkÓ¡ f™² v sGk cÉ K vk Ki| 4
2 M. w«¼Zxq f™²vsGki ni ‰ y ‰i mnM 1 nGj f™²vskàGjvGK
3
K. 1g ivwk ˆ^GK 2q ivwk weGqvM Ki|
L. 1g, 2q I 4^Æ ivwkGK mgni wewkÓ¡ f™²vsGk cÉKvk Ki| 4 mgni wewkÓ¡ f™²vsGk cÉKvk Ki| 4
M. 1g ivwk ˆ^GK 2q I 3q ivwk weGqvM Ki| 4 Dîi: K. (x  y)2 (x3 + y3)
(x + y)2 (x2  xy + y2) (x  y)3 (y  z) (x  y) (x2  xy + y2)
2x
Dîi: K. 1 + x2 + x4 L. (x  y)2 (x3 + y3) ,(x  y)2 (x3 + y3) ‰es (x  y)2 (x3 + y3)
2 2 2
1 + x + x2 1 – x + x2 (x + y)(x + y) (x + xy + y ) (x + y) (x  y) (y  z) (x3  y3)
2x
L. 1 + x2 + x4, 1 + x2 + x4, 1 + x2 + x4 M. 2 2 3 3
(x  y ) (x  y )
, 2 2 3 3 , 2 2 3 3
(x  y ) (x  y ) (x  y ) (x  y )
2(x3  x2  x  1) x4  y4 (x + y)2  4xy x+y
M. x6  1 cÉk
² 8 x2 + y2  2xy, , 2
x + xy + y2
wZbwU
x3  y3
1 1 2x exRMvwYwZK ivwk| [bxjdvgvix miKvwi DœP we`Åvjq]
cÉk
² 3 A = 1  x + x2 , B = 1 + x + x2 , C = 1 + x2 + x4
K. 1g f™²vskGK jwNÓ¤ AvKvGi cÉKvk Ki| 2
x2 L. 1g I 2q f™²vsGki àYdGji mvG^ 3q f™²vsk fvM Ki| 4
‰es D = 1  x6 PviwU f™²vsk| [wfKvi‚bwbmv bƒb Õ•zj ‰´£ KGjR, XvKv]
M. f™²vskàGjvGK mvaviY ni wewkÓ¡ f™²vsGk cÉKvk Ki| 4
K. C ‰i niGK Drcv`GK weGkÏlY Ki| 2 (x2 + y2) (x + y) (x4  y4) (x2 + xy + y2)
L. cÉgvY Ki ˆh, A  B  C = 0 4 Dîi: K. (x  y)
; L. x2 + y2; M.
(x  y)2 (x2 + xy + y2)
,
M. A, B, C ˆK mgni wewkÓ¡ f™²vsGk cÉKvk Ki| 4 (x  y)3 2
(x  y) (x + y)
,
Dîi: K. (1 + x + x2) (1 – x + x2); (x  y)2 (x2 + xy + y2) (x  y)2 (x2 + xy + y2)
1 + x + x2 1 – x + x2 2x
M. 1 + x2 + x4, 1 + x2 + x4, 1 + x2 + x4 Abykxjbx-5.2
² 4 gGb Ki: A = x
cÉk
 2 2
 x  42, B = x + 11x + 30 cÉk
² 9 A=x + y  z2 + 2xy, B = y2 + z2  x2 + 2yz ‰es
2 2

2 2 2
[weqvg gGWj Õ•zj I KGjR, eàov] C = z + x  y + 2zx. [iscyi KÅvGWU KGjR]
x2  x  2 a2  7a + 12
K. jwNÓ¤ AvKvGi cÉKvk Ki: x2  1 . 2 K. a2  16 ivwkwU jwNÓ¤ AvKvGi cÉKvk Ki| 2
1 1 1 1 1
L. A ‰es B ˆK mvaviY ni wewkÓ¡ f™²vsGk cÉKvk Ki| 4 L. A , B I C ˆK mgniwewkÓ¡ f™²vsGk cÉKvk Ki| 4
A B x+6 xy CB 2
M. mij Ki: x2  36  x2  25  x2  12x + 35. 4 M. mij Ki : A  BC  C 4
x2 x+5 x7 a3 (y + z  x)
Dîi: K. x  1; L. (x2 + 11x + 30)(x  7) ‰es (x2 + 11x + 30)(x  7); Dîi: K. a + 4 M. (x + y + z) (x + y  z)
1
M. x  6 cÉk
² 10 A = x + 2, B = x2 − 4, C = x2 − 2x + 4, D = x2 + 2x
+ 4 ‰es E = x3 − 8
cuvPwU exRMwYZxq ivwk| [cvebv KÅvGWU KGjR]
4 2 2
² 5 x4 + x2 + 1 , x3  1 , x3 + 1 wZbwU exRMwYZxq f™²vsk|
cÉk
 K. A I C ‰i M.mv.à. wbYÆq Ki| 2
1 x + 2 6x
K. 1g f™²vsGki niGK Drcv`GK weGkÏlY Ki| 2 L. mij Ki : A − C + E 4
L. 1g I 2q f™²vsGki ˆhvMdj ˆ^GK 3q f™²vsk weGqvM Ki| 4 A 1 1 C
M. cÉgvY Ki, B  C  E  D = 1 4
M. f™²vsk wZbwUGK mgni wewkÓ¡ f™²vsGk cÉKvk Ki| 4
4x2 2(x3  1) 96x
Dîi: K. 1 L. x6 − 64
Dîi: K. (x2 + x + 1) (x2  x + 1); L. x6  1; M. x6  1
exRMwYZxq f™²vsk 17

1 1 1 1 1 2a
cÉk
² 11 A = x2  5x + 6, B = x2  7x + 12 , C = x2  9x + 20 
cÉk
² 16 1  a + a2
,
1 + a + a2
I 1 + a2 + a4
wZbwU
[eÐ-~ evWÆ Õ•zj ‰´£ KGjR, wmGjU]exRMvwYZxq ivwk|
K. C ‰i niGK Drcv`GK weGkÏlY Ki| 2 K. 3q ivwkwUi niGK Drcv`GK weGkÏlY Ki| 2
1 L. 1g ivwk ˆ^GK evwK `yBwU ivwk weGqvM KGi weGqvMdGji mvG^
L. cÉgvY Ki ˆh, A  B  C = x2  7x + 10 4
a  b + a + b ˆhvM Ki| 4
M. A, B ‰es C ˆK mgni wewkÓ¡ f™²vsGk cÉKvk Ki| 4  a b 
M. wZbwU ivwki ˆhvMdjGK ˆKvb ivwk «¼viv fvM KiGj fvMdj 2
Dîi: K. (x – 4) (x – 5)
cvIqv hvGe? 4
1 a3 + b3 a4 + a3 + a2  1
cÉk
² 12 a2  ab + b2, 2a
,
4a2
wZbwU
Dîi: K. (1 + a + a2) (1  a + a2) L.
a2 + 2ab  b2 1
M. 1  a + a2
ab
exRMwYZxq f™²vsk| [wf. ˆR miKvwi gvaÅwgK we`Åvjq, PzqvWvãv] x–y y–z
² 17 (y + z)(z + x) , (x + y)(z + x) `yBwU exRMwYZxq f™²vsk|
cÉk

K. Z‡Zxq f™²vsGki jeGK Drcv`GK weGkÏlY Ki| 2
L. 1g f™²vsGki ni ‰es 2q I Z‡Zxq f™²vsGki jGei j.mv.à. wbYÆq K. f™²vsk `ywUi nGii j.mv.à. ˆei Ki| 2
4 L. ˆ`LvI ˆh, ‰i
2 2 2 2 2 2
yz + xz + y z + x y + x z + xy + 2xyz
Ki|
1 3 3 4
a +b a +a b +b 2 2 4 Drcv`K f™² v sk `y w Ui nGii j.mv.à. ‰i mgvb| 4
M. a2 + ab + b2  2a  4a2 =? 4 M. f™²vsk `ywUi ˆhvMdGji mvG^ KZ ˆhvM KiGj ˆhvMdGji gvb
Dîi: K. (a + 1) (a + a − 1);
3 kƒbÅ nGe? 4
z–x
L. (a + b) (a2 − ab + b2) (a + 1) (a3 + a − 1); M. 2a(a + b)
1 Dîi: K. (y + z)(z + x) (x + y); M. (x + y)(y + z)
x2 – 4 x–7
cÉk
² 13 (i) a3  3a2  10a, a3 + 6a2 + 8a, a4  5a3  14a2 ² 18 x2 – 49 , x – 2 `yBwU exRMwYZxq f™²vsk|
cÉk

wZbwU exRMvwYwZK ivwk| K. f™²vsk `ywUGK àY Ki| 2
‰es R = y3 + 8.
(ii) P = y  2, Q = y2 + 2y + 4 L. ‘K’ àYdjwUi je I nGii mvG^ x àY KGi je ˆ^GK 3 weGqvM
K. (i) ‰i Z‡Zxq ivwkGK Drcv`GK weGkÏlY KGiv| 2 x2 – 6x + 5
‰es nGii mvG^ 12 ˆhvM KGi cÉvµ¦ f™²vskwUGK x2 – x – 20 «¼viv
L. (i) ‰i wZbwU ivwki M.mv.à. wbYÆq KGiv| 4
fvM Ki| 4
1 y2 6y
M. mij KGiv: P  Q + R . 4 x2 – 6x x+1
M. ˆ`LvI ˆh, x2 – 7x + 6 ˆ^GK x2 – 1 weGqvM KiGj cÉvµ¦ gvb
12y4
Dîi: K. a2(a  7) (a + 2); L. a(a + 2); M. y6  64 ‘L’ ˆ^GK cÉvµ¦ fvMdGji gvGbi mgvb| 4
x+2
1 1 2p Dîi: K. x + 7 L. 1
cÉk
² 14 1  p + p2 , 1 + p + p2 ‰es 1 + p2 + p4 wZbwU
1 1
exRMvwYZxq f™²vsk| cÉk
² 19 2x + 2 , 2x – 2 `yBwU exRMwYZxq f™²vsk|
K. 3q f™²vsGki niGK Drcv`GK weGkÏlY Ki| 2 K. f™²vsk `ywUGK ˆhvM Ki| 2
1 1 2p L. ‘K’ ˆ^GK cÉvµ¦ ˆhvMdj ‰i jeGK ni ‰es niGK je aGi
L. cÉgvY Ki ˆh, 1  p + p2  1 + p + p2  1 + p2 + p4 = 0 4 1
cÉvµ¦ f™²vsGki gvb 3 nGj x2 + x2 ‰i gvb wbYÆq Ki| 4
M. f™²vsk 3wUGK mgniwewkÓ¡ f™²vsGk cÉKvk Ki| 4 3 3 2
x +x x –x +x–1
Dîi: K. (1 + p + p2)(1  p + p2); L. 0; M. x2 + x ˆK x
«¼viv fvM KGi ‘L’ ˆ^GK cÉvµ¦ gvb
2p
M. 1 + p2 + p4 «¼viv fvMdj wbYÆq Ki| 4
x 1
Dîi: K. x2 – 1 L. 11 M. 3
cÉk
² 15 M = a  b, N = a + b ‰es R = x2  2x + 1 a4 + a2b2 + b4 a3 + b3
cÉk
² 20 a3 + b3
, 2
a  b2
`yBwU exRMwYZxq ivwk|
[KÝevRvi miKvwi DœP we`Åvjq, KÝevRvi]
K. MN-ˆK `yBwU eGMÆi A¯¦i…Gc cÉKvk Ki| 2 K. cÉ^g f™²vsGki je I nGii M.mv.à. ˆei Ki| 2
L. f™²vsk `yBwUGK ˆhvM Ki| 4
L. mij Ki:  a  a    b  b  +  N + M   N  M 4 a4 + a3b + a2b2
M N M N M N M N M. f™²vsk `yBwUi ˆhvMdGji mvG^ fvM KiGj
a3  b3
M. R = 0 nGj x5 + 15 -‰i gvb wnmve Ki| 4 a+b
x fvMdj a ‰i KZ àY? 4
3a2 + b2 3 2
Dîi: K. a2  b2; L. 2ab
; M. 2 2a
Dîi: K. a2  ab + b2 L. a2  b2 M. (a + b)2
2a