Number Theory T...

............are the Pentium binary program that can be embedded in a web page. 

Which of the following is a prime number?

If z satisfies $$\left| {z - 1} \right| < \left| {z + 3} \right|$$ then $$w = 2z + 3 - i$$ , ( where $$w = 2z + 3 - i$$ ) satisfies:

If $$z_1 \, z_2$$ be two distinct complex numbers and let z = (1 - t) $$z_1$$ + t$$z_2$$ for some real number t with 0 < t < 1. If arg $$(\omega)$$ denotes the principal argument of a non-zero complex number $$(\omega)$$, then

Simplify $$\left ( \dfrac{2i}{1 \, + \, i} \right )^2$$

Let $$z$$ be a complex number such that $$\left| z+\dfrac { 1 }{ z }  \right| =2$$. 

If $$\left| z \right| ={ r }_{ 1 }$$ and $$\left| \dfrac { 1 }{ z }  \right| =$$ $${r}_{2}$$ for $$\arg z=\dfrac { \pi  }{ 4 }$$ then 

$$\left| { r }_{ 1 }-{ r }_{ 2 } \right| =$$

Real part of  $$\dfrac{(1 + i)^2}{3 - i} =$$

If $$\dfrac{2z_1}{3z_2}$$ is a purely imaginary number,then $$\left|\dfrac{z_1-z_2}{z_1+z_2}\right|=$$

Find the real number $$x$$ if $$(x-2i)(1+i)$$ is purely imaginary.

The value of $$\dfrac{1}{i} + \dfrac{1}{{{i^2}}} + \dfrac{1}{{{i^3}}} + ... + \dfrac{1}{{i^{102}}}$$ is equal to