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.

The complex number z is purely imaginary , if

Z=-5+4i then Z^4 +9Z^3 +35Z^2 – Z + 4 =

If z is a unimodular number (!=+-i) then (z+i)/(z-i) is (A) purely real (B) purely imaginary (C) an imaginary number which is not purely imaginary (D) both purely real and purely imaginary

If bar(z)= -z ne 0 , show that z is necessarily a purely imaginary number

If z=|[-5, 3+4i,5-7i],[3-4i,6 ,8+7i],[5+7i,8-7i,9]|,t h e nz is (a) purely real (b)purely imaginary (c)an imaginary number that is not purely imaginary,a+i b , w h e r ea!=0,b!=0 (d)both purely real and purely 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

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: both alphaa n dbeta are purely real both alphaa n dbeta are purely imaginary alpha is purely real and beta is purely imaginary beta is purely real and alpha is purely imaginary

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

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