Show that the equation `a z^3+b z^2+ barb z+ bara =0` has a root `alpha` such that `|alpha|=1,a ,b ,z and alpha` belong to the set of complex numbers.

If alpha is a complex constant such that alphaz^2+z+baralpha=0 has areal root then

Show that the equation Z^4+2Z^3+3Z^2+4Z+5=0 has no root which is either purely real or purely imaginary.

If x alpha yz and y alpha zx , then show that (z!=0) is a constant.

Solve the equation z^(2)+|Z|=0 where z is a complex quantity.

If z is a complex number satisfying the equation z^6 +z^3 + 1 = 0 . If this equation has a root re^(itheta) with 90^@<0<180^@ then the value of theta is

Fill in the blanks: If x alpha y and x alpha z , then (y+z) alpha ______.

The common roots of the equations z^3+2z^2+2z+1=0 and z^1985+z^100+1=0 are

Suppose that z_(1), z_(2), z_(3) are three vertices of an equilateral triangle in the Argand plane. Let alpha=(1)/(2) (sqrt(3) +i) and beta be a non-zero complex number. The point alphaz_(1) +beta, alpha z_(2)+beta, az_(3)+beta will be

A root of unity is a complex number that is a solution to the equation, z^n=1 for some positive integer nNumber of roots of unity that are also the roots of the equation z^2+az+b=0 , for some integer a and b is

If the roots of equation x^3+a x^2+b=0a r ealpha_1,alpha_2 and alpha_3(a ,b!=0) , then find the equation whose roots are (alpha_1alpha_2+alpha_2alpha_3)/(alpha_1alpha_2alpha_3),(alpha_2alpha_3+alpha_3alpha_1)/(alpha_1alpha_2alpha_3),(alpha_1alpha_3+alpha_1alpha_2)/(alpha_1alpha_2alpha_3)