Without expanding at any stage, prove that the value of the following determinant is zero. `|(1,omega,omega^(2)),(omega,omega^(2),1),(omega^(2),1,omega)|` , where `omega` is cube root of unity

Using properties, prove that |[1,omega,omega^2],[omega,omega^2,1],[omega^2,1,omega]|=0 where omega is a complex cube root of unity.

The value of |(1,omega,2omega^(2)),(2,2omega^(2),4omega^(3)),(3,3omega^(3),6omega^(4))| is equal to (where omega is imaginary cube root of unity

The value of the expression 1.(2- omega)(2- omega^(2)) + 2.(3-omega) (3-omega^(2)) + …+ (n-1)(n- omega)(n- omega^(2)) , where omega is an imaginary cube root of unity is

The value of (2 - omega) (2- omega ^(2)) (2-omega^(10)) (2- omega ^(11)) where omega is the complex cube root of unity, is : a)49 b)50 c)48 d)47

If omega is a cube root of unity, then find (3+5omega+3omega^(2))^(2)+ (3 +3omega+5omega^(2))^(2)

If omega is a complex cube root of unity, show that [[1 , omega, omega^2], [ omega, omega^2, 1],[ omega^2, 1, omega]] [[1],[ omega],[ omega^2]]=[[0],[ 0],[ 0]]

IF 1, omega , omega^2 are the cube roots of unity and if [{:(1+omega,2 omega),(-2 omega,-b):}]+[{:(a,-omega),(3 omega,2):}]=[{:(0, omega),(omega,1):}] then a^2+b^2 is equal to

If x= a+ b, y = a omega + b omega^(2), z= a omega^(2) + b omega , then the value of x^(3) + y^(3) + z^(3) is equal to (where omega is imaginary cube root of unity)

If omega is a complex cube root of unity, then value of Delta=|(a_(1)+b_(1)omega,a_(1)omega^(2)+b_(1),c_(1)+b_(1)omega),(a_(2)+b_(2)omega,a_(2)omega^(2)+b_(2),c_(2)+b_(2)omega),(a_(3)+b_(3)omega,a_(3)omega^(2)+b_(3),c_(3)+b_(3)omega)| is a)0 b)-1 c)2 d)None of these