Rational Expression

MATHEMATICS Sample test papers with hints Class-X (CBSE) Rational Expression Formula used:(a + b)2 = a2 +b2 + 2ab (a - b)2 = a2 +b2 - 2ab a3 + b3 = (a + b) (a2 – ab + b2) a3 - b3 = (a - b) (a2 + ab + b2) (a + b)3 = a3 + 3a2b + 3ab2 + b3 (a - b)3 = a3 - 3a2b + 3ab2 - b3 (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca a3 + b3 + c3 - 3abc = (a + b + c) (a2 + b2 + c2 –ab –bc -ca)

Q.A Express the following in lowest terms (x

2 x 4 − 162 + 9 ( 2 x − 6)

) 1. x +3 ] 2

[ans.

2.

x −1 y

x(x 2 - 3x + 2y) x 2 y - 2xy

[Ans.

] ( x 2 + 3x + 2)( x 2 + 5 x + 6) x 2( x 2 + 4 x + 3)

3.

[Ans.

x+2 x

]

8 x 3 − 125

4. 4x + 10x + 25 2x-5]

[Ans.

2

5.

3 x 2 − 24 2 x 2 − 15 x + 22

3( x 2 + 2 x + 4)` 2 x − 11

6.

[Ans.

]

x 3 − 3x + 2 2x 3 − 4x 2 + 6x − 4

( x + 2)( x − 1) 2( x − 2)() x + 3

8.

]

5 x 2 − 15 x 3 1− 9x 2

5x 2 3x + 1

7.

[Ans.

[Ans.

]

x 3 + 216 x − 6 x 3 + 36 x 2

[Ans.

4

x+6 x2

]

9.

x 2 + 2x + 4 x 4 + 4 x 2 + 16

1 x 2 − 2x + 4

10.

[Ans.

]

x 3 + 4 x 2 + 3x x 4 − 10 x 2 + 9

[Ans.

x ( x − 1)( x − 3)

] Q.B Simplify 1.

1 1 2 4 − − − 1− x 1 + x x 2 +1 x 4 +1

[Ans. x

8

2.

1 1 2 4 + + + 1− z 1+ z z +1 z4 +1

[Ans. z

8

3.

1 1 2x − + 1+ x + x 2 1− x + x 2 1+ x 2 + x 4

[Ans. 0]

8 −1

]

8 −1

]

4.

1 1 1 ax bx cx + + + + + x + a x + b x + c x 3 + ax 2 x 3 + bx 2 x 3 + cx 2

5.

x3 x2 1 1 − − + x −1 x + 1 x −1 x +1

6.

7.

3

[Ans. x ] [Ans.x2+2]

y x z + + ( x − y )( x − z ) ( y − z )( y − x) ( z − x)( z − y )

[Ans.0]

2 3 + x2 − x − 6 x2 + x − 2 4 x 2 − 4x + 3

5 x − 11 4x + 8

[Ans.

]

8.

 x y  y z  z x   −  −  −   y x  z y  x z   1 1  1 1  1 1  2 − 2  2 − 2  2 − 2  y  y z  z x  x

9.

  1 1   1 − 3 y −  ÷  y −   y3 − y3  y   y 

[Ans.-x2y2z2] [Ans.

10.

x+3 x+2 x +1 + + x 2 − 3x + 2 x 2 − 4 x + 3 x 2 − 5 x + 6

11.

 x4 − y4 x 2 + xy   x 5 − y 3 x 3   x y  ÷  2   ÷ −  2 xy   x 3 + y 3   y x   x − 2 xy + y

12.

   y2 + x2 y2 − x2  x+ y y−x − 2 − 3  2 ÷ 3  3 y + xy + x 2 y − xy + x 2 y + x3    y +x

13.

1 2x − 1 2x − 2 2x −1 4x −1

1 2 x( 2 x − 1)

14.

16.

[Ans

x 2 y( x 2 + y 2 ) x3 + y3

x( x 3 + y 3 ) x4 + y3 + x 2 y2

]

]

] ]

]

a a +b b

)

a + b ( a − b)

+

2 b a+ b

x 3 + y 3 + z 3 − 3xyz x 2 + y 2 + z 2 − xy − yz − zx

Q.C

[Ans.

3 x 2 − 14 ( x − 1)( x − 2)( x − 3)

2

[Ans.

 x3 + y3   ( x + y ) 2 − 3xy   xy    ×   ( x − y ) 2 + 3xy  ÷  x 3 − y 3   x2 − y2     

15. (

[Ans.

 1  y − y   

−

ab a−b

  

[Ans. xy] [Ans.- (

2( a b − b a )

)

a + b (a − b)

]

[Ans + y +z]



x3 + y3 x3 − y3 x2 − y2 ,Q = and , R = xy ( x − y ) 2 + 3xy ( x + y ) 2 − 3xy

If lowest form. xy]

[Ans-

1



If A = 4 x + x , find A + 16 x 4 + 9 x 2 + 1 x(4 x 2 + 1)





 





If m = ]

1 A

[Ans.

] x +1 x −1

and n =

x −1 x +1

4x x 2 −1

If

If x =

26 54

,y=

28 54

x +1

7x x 2 + x − 12

to get

[Ans.

4 x+4

?

[Ans.

3 x−3

]

find the value of A and B. then find the value of

What should be added to

If P =

[Ans. x

to get x −1 ?

What should be subtracted from 5 x − 13 A B = + x 2 − 5x + 6 x − 2 x − 3

x 4 + 14 x 2 + 1 4 − 2x 2 +1

find the value of m2 + n2 –mn

What should be added to x −1 x +1 ]

− 2 x 3 + 10 x − 4 3x − 4



express [ P × Q ÷ R ] as a rational expression in

2 x 3 − 5x + 1 3x − 4

x3 + y3  y2   ÷  x + x− y  x − y 

to get additive inverses of

− 5x + 3 3x − 4

. [Ans.

]

x +1 x −1

and Q =

x −1 x +1

, then find

P−Q P+Q

[Ans.

2x x 2 +1

]