If x=a^{ \frac{1}{2}}+a^{- \frac{1}{2}}. y=a^{ \frac{1}{2}}-a^{- \frac{1}{2}} then value of (x^{4}-x^{2} y^{2}-1 )+ (y^{4}-x^{2} y^{2}+1 ) is

SSC CGL Tier II 25 Oct 2015 Paper 1

Question : 79 of 100

Marks: +1, -0

\ x=a^{ {1}/{2}}+a^{- {1}/{2}}. y=a^{ {1}/{2}}-a^{- {1}/{2}}

then value of

(x^{4}-x^{2} y^{2}-1 )+

(y^{4}-x^{2} y^{2}+1 )

Solution:

(1)

x=a^{ {1}/{2}}+a^{ {-1}/{2}}

x=a^{ {1}/{2}}-a^{ {-1}/{2}}

∴ x^{2}-y^{2}=4 a^{ {1}/{2}} . a^{ {-1}/{2}}=4

[∵(a+b)^{2}-(a-b)^{2}=4 a b ]

Again,

y^{2}-x^{2}=-4 . a^{ {1}/{2}}.a^{ {-1}/{2}}=-4

Expression

= (x^{4}-x^{2} y^{2}-1 )+ (y^{4}-x^{2} y^{2}+1 )

=x^{2} (x^{2}-y^{2} )-1+y^{2} (y^{2}-x^{2} )+1

=4 x^{2}-1-4 y^{2}+1

=4 (x^{2}-y^{2} )=4 × 4=16

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