If (x+y)^3-\ (x-y)^3-6y\ (x^2-y^2)=\ k y...

If `(x+y)^3-\ (x-y)^3-6y\ (x^2-y^2)=\ k y^3,` then `k=` (a)1 (b) 2 (c) 4 (d) 8

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Correct option is D
`(x+y)^3-\ (x-y)^3-6y\ (x^2-y^2)=\ k y^3`
`=[(x+y)^3-(x-y)^3]-6y\(x^2-y^2)=ky^2`
We know, `a^3-b^3=(a-b)\(a^2+ab+b^2)`
`=2y(x^2+2xy+y^2+x^2-y^2-x^2-2xy+y^2)-6y(x^2-y^2)=ky^3`
`=2y(3x^2+y^2)-6y(x^2-y^2)+ky^3`
`=6x^2y+2y^3-6x^2y+6y^3=ky^3`
`8y^3=ky^3`
Therefore,
`k=8`.

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