If `a , b , c` are real numbers, prove that `|a b c b c a c a b|=-(a+b+c)(c+b w+c w^2)(a+b w^2+c w),` where `w` is a complex cube root of unity.

If a , b , c are real numbers, prove that |[a,b,c],[b,c,a],[c,a,b]|=-(a+b+c)(c+b w+c w^2)(a+b w^2+c w), where w is a complex cube root of unity.

If a , b , c are real numbers, prove that \begin{vmatrix} a & b & c \\ b & c & a \\ c & a & b \\ \end{vmatrix}= -(a+b+c)(c+b w+c w^2)(a+ bw^2+cw), where \ \ w is a complex cube root of unity.

If f(x)=a+b x+c x^2AAa ,b ,c in R and a ,b ,c are distinct, then value of |a b c b c a c a b| is (where omega&omega^2 are complex cube roots of unity) f(1)f(omega)f(omega^2) (b) -f(omega)f(omega^2) -f(1)f(omega)f(omega^2) (d) f(omega)f(omega^2)

If f(x)=a+b x+c x^2AAa ,b ,c in R and a ,b ,c are distinct, then value of |a b c b c a c a b| is (where omega&omega^2 are complex cube roots of unity) f(1)f(omega)f(omega^2) (b) -f(omega)f(omega^2) -f(1)f(omega)f(omega^2) (d) f(omega)f(omega^2)

Prove that the lines ax+by + c = 0, bx+ cy + a = 0 and cx+ay+b=0 are concurrent if a+b+c = 0 or a+b omega + c omega^(2) + c omega = 0 where omega is a complex cube root of unity .

18.Prove that det[[b,a,cb,a,cc,b,a]]=(a+b+c)(a+b omega+c omega^(2))(a+b omega^(2)+c omega): where ,omega is a nonreal cube root of unity.

If a,b,c are real then ([a,b,cb,c,ac,a,b])=-(a+b+c)(a+b omega+c omega^(2))(a+b omega^(2)+c omega)

Prove that a^3 + b^3 + c^3 – 3abc = (a + b + c) (a + bomega + comega^2) (a + bomega^2 + "c"omega) , where omega is an imaginary cube root of unity.

Using properties of determinant, If f(x)= a + bx + cx^(2) , prove that |(a,b,c),(b,c,a),(c,a,b)|= -f(1)f(omega)f(omega^(2)), omega is a cube root of unity