If `a,b,c` are unequal then what is the condition that the value of the following determinant is zero `Delta =|(a,a^2,a^3+1),(b,b^2,b^3+1),(c,c^2,c^3+1)|`

If [[a,a^(2),a^(3)-1b,b^(2),b^(3)-1c,c^(2),c^(3)-1]]=0

If a, b, c are positive and unequal, show that value of the determinant Delta=|(a,b,c),(b,c,a),(c,a,b)| is negative

Without expanding evaluate the determinant "Delta"=|(1, 1, 1),(a, b, c),( a^2,b^2,c^2)| .

If a,b,c are positive and unequal,show that value of the determinant Delta=det[[b,b,cb,c,ac,a,b]]

If Delta is the value of the determinant |(a_(1), b_(1),c_(1)),(a_(2),b_(2),c_(2)),(a_(3),b_(3),c_(3))| then what is the value of the following determinant? |(pa_(1), b_(1), qc_(1)),(pa_(2), b_(2), qc_(2)),(pa_(3), b_(3),qc_(3))| (p ne 0 or 1, q ne 0 or 1)

If a^(2) + b^(2) + c^(3) + ab + bc + ca le 0 for all, a, b, c in R , then 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))| , is equal to

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

Let a_i^2+b_i^2+c_i^2=1 (for i=1,2,3) and a_ia_j+b_ib_j+c_ic_j=0 (i != j;i,j=1,2,3). Then the absolute value of determinant |(a_1,a_2,a_3),(b_1,b_2,b_3),(c_1,c_2,c_3)| (A) 1/2 (B) 0 (C) 1 (D) 2