Let `z=9+b i ,w h e r eb` is nonzero real and `i^2=-1.` If the imaginary part of `z^2a n dz^3` are equal, then `b/3` is ______.

If a ,b ,c are nonzero real numbers and a z^2+b z+c+i=0 has purely imaginary roots, then prove that a=b^2c

If a ,b ,c are nonzero real numbers and a z^2+b z+c+i=0 has purely imaginary roots, then prove that a=b^2c dot

Prove that z=i^i ,w h e r ei=sqrt(-1) , is purely real.

Let z=9+ai, where i=sqrt(-1) and a be non-zero real. If Im(z^(2))=Im(z^(3)) , sum of the digits of a^(2) is

Find the imaginary part of the complex number if z=( 2-i)(5+i)

Let z_(1),z_(2) be two distinct complex numbers with non-zero real and imaginary parts such that "arg"(z_(1)+z_(2))=pi//2 , then "arg"(z_(1)+bar(z)_(1))-"arg"(z_(2)+bar(z)_(2)) is equal to

Let z be a complex number such that the imaginary part of z is nonzero and a = z^2 + z + 1 is real. Then a cannot take the value

Let z be a complex number such that the imaginary part of z is nonzero and a = z^2 + z + 1 is real. Then a cannot take the value

Prove that the locus of midpoint of line segment intercepted between real and imaginary axes by the line a z + a z+b=0,w h e r eb is a real parameterand a is a fixed complex number with nondzero real and imaginary parts, is a z+ a z =0.

Consider the equation 10 z^2-3i z-k=0,w h e r ez is a following complex variable and i^2=-1. Which of the following statements ils true? (a)For real complex numbers k , both roots are purely imaginary. (b)For all complex numbers k , neither both roots is real. (c)For all purely imaginary numbers k , both roots are real and irrational. (d)For real negative numbers k , both roots are purely imaginary.