Let `z=9+b i ,and b` 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 dot

Let z be a complex number such that the imaginary part of z is non zero and a=z^2+z+1 is real then a cannot take the value

If n in N >1 , then the sum of real part of roots of z^n=(z+1)^n is equal to

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.

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

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.

Let a ,b ,a n dc be distinct nonzero real numbers such that (1-a^3)/a=(1-b^3)/b=(1-c^3)/c dot The value of (a^3+b^3+c^3) is _____________.

Let a ,b and c be any three nonzero complex number. If |z|=1 and' z ' satisfies the equation a z^2+b z+c=0, prove that a .bar a = c .bar c and |a||b|= sqrt(a c( bar b )^2)

z_1a n dz_2 are two distinct points in an Argand plane. If a|z_1|=b|z_2|(w h e r ea ,b in R), then the point (a z_1//b z_2)+(b z_2//a z_1) is a point on the line segment [-2, 2] of the real axis line segment [-2, 2] of the imaginary axis unit circle |z|=1 the line with a rgz=tan^(-1)2

Show that the equation of a circle passing through the origin and having intercepts a and b on real and imaginary axes, respectively, on the argand plane is given by z bar z =a(R e z)+b(I mz)dot