`|[a,a^(2),b+c],[b,b^(2),c+a],[c,c^(2),a+b]|` is equal to

Suppose a,b,c are distinct real number and Delta=|(a,a^(2),b+c),(b,b^(2),c+a),(c,c^(2),a+b)|=0 Then a+b+c equals

Prove the identities: |[a^2,a^2-(b-c)^2,b c], [b^2,b^2-(c-a)^2,c a],[ c^2,c^2-(a-b)^2,a b]|=(a-b)(b-c)(c-a)(a+b+c)(a^2+b^2+c^2)

[[a,a^(2),(b+c)b,b^(2),(a+c)c,c^(2),(a+b)]]=(b-c)(c-a)(a-b)(a+b+c)

If a,b,c in R^(+), then the minimum value of a(b^(2)+c^(2))+b(c^(2)+a^(2))+c(a^(2)+b^(2)) is equal to (a)abc (b)2abc (c)3abc (d)6abc

Prove that |[(b+c)^2, a^2, bc],[(c+a)^2, b^2, ca],[(a+b)^2, c^2, ab]|=(a-b)(b-c)(c-a)(a+b+c)(a^2+b^2+c^2)

Value of |(1,a,a^2),(1,b,b^2),(1,c,c^2)| is (A) (a-b)(b-c)(c-a) (B) (a^2-b^2)(b^2-c^2)(c^2-a^2) (C) (a-b+c)(b-c+a)(c+a-b) (D) none of these