1.

Let a, b, c be non-zero real numbers such that `a+b+c=0," let "q=a^(2)+b^(2)+c^(2)" and " r=a^(4)+b^(4)+c^(4)`. Then-A. `q^(2)lt2r` alwaysB. `q^(2)=2r` alwaysC. `q^(2)gt2r` alwaysD. `q^(2)-2r` can take both positive and negative values

Answer» Correct Answer - B
`a+b+c=0`
`=a^(2)+b^(2)+c^(2),r=a^(4)+b^(4)+c^(4)`
`q^(2)-2r=(a^(2)+b^(2)+c^(2))^(2)-2(a^(4)+b^(4)+c^(4))`
`=2a^(2)b^(2)+2b^(2)c^(2)+2a^(2)c^(2)-a^(4)-b^(4)-c^(4)`
`=2a^(2)c^(2)+2b^(2)c^(2)-(a^(2)-b^(2))^(2)-c^(4)`
`=2c^(2)(a^(2)+b^(2))-c^(2)(a-b)^(2)-c^(4)`
`=c^(2)[2a^(2)+2b^(2)-(a-b)^(2)-c^(2)]`
`=c^(2)[2ab+a^(2)+b^(2)+c^(2)]`
`=c^(2)[(a+b)^(2)-c^(2)]`
`=0`


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