The set of all $\alpha  \in R$, for which $w = \frac{{1 + \left( {1 – 8\alpha } \right)z}}{{1 – z}}$ is a purely imaginary number, for all $z \in C$ satisfying $\left| z \right| = 1$ and ${\mathop{\rm Re}\nolimits} \,z \ne 1$,  is

Let $S$ be the set of all complex numbers $z$ satisfying $\left|z^2+z+1\right|=1$. Then which of the following statements is/are $TRUE$?

$(A)$ $\left|z+\frac{1}{2}\right| \leq \frac{1}{2}$ for all $z \in S$  $(B)$ $|z| \leq 2$ for all $z \in S$

$(C)$ $\left|z+\frac{1}{2}\right| \geq \frac{1}{2}$ for all $z \in S$  $(D)$ The set $S$ has exactly four elements

If ${z_1}.{z_2}……..{z_n} = z,$ then $arg\,{z_1} + arg\,{z_2} + ….$+$arg\,{z_n}$ and $arg$$z$ differ by a

If a complex number $z$ statisfies the equation $x + \sqrt 2 \,\,\left| {z + 1} \right|\,+ \,i\, = \,0,$ then $\left| z \right|$ is equal to

Find the modulus and argument of the complex numbers:

$\frac{1}{1+i}$