If `|(a^2,b^2,c^2),((a+b)^2 ,(b+1)^2,(c+1)^2),((a-1)^2 ,(b-1)^2,(c-1)^2)| =k(a-b)(b-c)(c-a)` then the value of k is a. 4 b. -2 c.-4 d. 2

Answer

`k=-4`

Explanation

`" Let " |{:(a^(2),,b^(2),,c^(2)),((a+1)^(2),,(b+1)^(2),,(c+1)^(2)),((a-1)^(2),,(b-a)^(2),,(c-1)^(2)):}|`
`= 4 |{:(a^(2),,b^(2),,c^(2)),(a,,b,,c),((a-1)^(2),,(b-1)^(2),,(c-1)^(2)):}|`
[Applying `R_(2) to R_(2)-R_(3),` then taking 4 common from `R_(2)]`
`=4 |{:(a^(2),,b^(2),,c^(2)),(a,,b,,c),(1,,1,,1):}| [R_(3) to R_(3)-R_(1)+2R_(2)]`
`=-4 |{:(1,,1,,1),(a,,b,,c),(a^(2),,b^(2),,c^(2)):}|`
`=- 4(a-b)(b-c)(c-a)`
Hence `k=-4.`