Evaluate the following: ` |[a,b,c],[a^2, b^2, c^2],[a^3, b^3, c^3]| `

Show that |[a,b,c],[a^2,b^2,c^2],[a^3,b^3,c^3]|=abc(a-b)(b-c)(c-a)

Prove that |[a,b,c] , [a^2,b^2,c^2] , [a^3,b^3,c^3]|= abc(a-b)(b-c)(c-a)

Find the value of: |[1 a^2 a^3],[1 b^2 b^3],[1 c^2 c^3]|

If a^2+b^2+c^2+ab+bc+ca<=0 AA a, b, c in R then find the value of the determinant |[(a+b+2)^2, a^2+b^2, 1] , [1, (b+c+2)^2, b^2+c^2] , [c^2+a^2, 1, (c+a+2)^2]| : (A) abc(a^2 + b^2 +c^2) (B) 0 (C) a^3+b^3+c^3 + 3abc (D) 65

Factorize the following 3,a+b+c,a^(3)+b^(3)+c^(3)a+b+c,a^(2)+b^(2)+c^(2),a^(4)+b^(4)+c^(4)a^(2)+b^(2)+c^(2),a^(3)+b^(3)+c^(3),a^(5)+b^(5)+c^(5)]|

If A=|(1,1,1),(a,b,c),(a^3,b^3,c^3)|, B=|(1,1,1),(a^2,b^2,c^2),(a^3,b^3,c^3)|, C=|(a,b,c),(a^2,b^2,c^2),(a^3,b^3,c^3)| , then which relation is correct :

If a!=b!=c such that |[a^3-1,b^3-1,c^3-1] , [a,b,c] , [a^2,b^2,c^2]|=0 then

Match the following. (A) 1-c, 2-b, 3-a (B) 1-b, 2-c, 3-a (C) 1-a, 2-b, 3-c (D) 1-a, 2-c, 3-b

Prove that |{:(" "3,a+b+c,a^2+b^2+c^2),(a+b+c,a^2+b^2+c^2,a^3+b^3+c^3),(a^2+b^2+c^2,a^3+b^3+c^3,a^4+b^4+c^4):}|=|{:(1,1,1),(a,b,c),(a^2,b^2,c^2):}|^2