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

Prove that the equation Z^3+i Z-1=0 has no real roots.

Show that the equation z^(3) + 2 bar(z) = 0 has five solutions.

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

Given z is a complex number with modulus 1. Then the equation [(1+i a)/(1-i a)]^4=z has (a) all roots real and distinct (b)two real and two imaginary (c) three roots two imaginary (d)one root real and three imaginary

The quadratic equation p(x)=0 with real coefficients has purely imaginary roots. Then the equation p(p(x))=0 has A. only purely imaginary roots B. all real roots C. two real and purely imaginary roots D. neither real nor purely imaginary roots

If a is a complex number such that |a|=1 , then find thevalue of a, so that equation az^2 + z +1=0 has one purely imaginary root.

Consider two complex numbers alphaa n dbeta as alpha=[(a+b i)//(a-b i)]^2+[(a-b i)//(a+b i)]^2 , where a ,b , in R and beta=(z-1)//(z+1), w here |z|=1, then find the correct statement: (a)both alphaa n dbeta are purely real (b)both alphaa n dbeta are purely imaginary (c) alpha is purely real and beta is purely imaginary (d) beta is purely real and alpha is purely imaginary

If |z/| barz |- barz |=1+|z|, then prove that z is a purely imaginary number.