If `i z^4+1=0,` then prove that `z` can take the value `cospi//8+isinpi//8.`

If |z-i R e(z)|=|z-I m(z)| , then prove that z , lies on the bisectors of the quadrants.

If |z-1|+|z+3|<=8 , then the range of values of |z-4| is

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

If z is a complex number such that z+|z|=8+ 12i , then the value of |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

Prove that z=i^i, where i=sqrt-1, is purely real.

If the roots of (z-1)^n=i(z+1)^n are plotted in an Argand plane, then prove that they are collinear.

If |z|lt=4 , then find the maximum value of |i z+3-4i|dot

If z_1 + z_2 + z_3 + z_4 = 0 where b_1 in R such that the sum of no two values being zero and b_1 z_1+b_2z_2 +b_3z_3 + b_4z_4 = 0 where z_1,z_2,z_3,z_4 are arbitrary complex numbers such that no three of them are collinear, prove that the four complex numbers would be concyclic if |b_1b_2||z_1-z_2|^2=|b_3b_4||z_3-z_4|^2 .

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