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)|`

Without expanding, show that the value of each of the determinants is zero: |[1,a, a^2-bc],[1,b,b^2-ac],[1,c,c^2-ab]|

Without expanding, show that the value of each of the determinants is zero: |1/a a^2 bc 1/b b^2 ac 1/c c^2 ab|

which of the following is // are true for Delta= |{:(a^(2),,1,,a+c),(0,,b^(2)+1,,b+c),(0,,b+c,,c^(2)+1):}| ?

If a,b,c are distinct real numbers and |{:(a,a^(2),a^(3)-1),(b,b^(2),b^(3)-1),(c,c^(2),c^(3)-1):}|=0 then

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=|a b c b c a c a b| is negative.

Without expanding the determinant, prove that |(a,a^2,bc),(b,b^2,ca),(c,c^2,ab)|=|(1,a^2,a^3),(1,b^2,b^3),(1,c^2,c^3)|

The determinant Delta=|(a^2(a+b),a b,a c),(a b,b^2(a+k),b c),(a c,b c,c^2(1+k))| is divisible by

Without expanding, show that the value of each of the following determinants is zero: |1//a , a^2b c1//bb^2a c1//cc^2a b| (ii) |a+b2a+b3a+b2a+b3a+b4a+b4a+b5a+b6a+b| (iii) |1a a^2 1bb^2 1cc^2\ \ \ -b c-a c-a b|

Evaluate the following |(1, 1, 1),(a^2,b^2,c^2),(a^3,b^3,c^3)|