JEE Advanced Model Paper 8 with solutions for online practice

We have (

x

x+1

)2 + (

x

x−1

)^2 = a (a - 1)
⇒ (

x

x+1

+

x

x−1

)2 - 2 (

x

x+1

)(

x

x−1

) = a (a - 1) ⇒ (

2x2

x2−1

)2 -

2x2

x2−1

= a (a - 1)
⇒ z2 - z - a (a - 1) = 0 , where z =

2x2

x2−1

⇒ z = a or 1 - a
When , z = a ,

2x2

x2−1

= a ⇒ 2x2 = ax2 - a ⇒ x = ± √

a

a−2

When , z = 1 - a ,

2x2

x2−1

= 1 - a ⇒ 2x2 = (1 - a) x2 - 1 + a ⇒ x = ± √

a−1

a+1

∴ x = ± √

a

a−2

, ± √

a−1

a+1

If a < - 1 ⇒ All the roots are real
If 1 < a < 2 ⇒ x = ± √

a

2−a

i , ± √

a−1

a+1

⇒ Only two roots are real
If a > 2 ⇒ All roots are real