Complex Numbers...

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

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.

If the equation $${ x }^{ 2 }+nx+n=0,n\epsilon I$$, has integral roots then $${ n }^{ 2 }-4n$$ can assume

If, for quadratic equation $$ax^2+bx +c=0, b^2-4ac \nleq 0$$ and a, b, c are all positive, then 

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

If $$i^2= -1$$, then $$1+ i^2+ i^4 +i^6+i^8 +.............to ( 2n +1)$$ terms is equal to

$$i \, \log \left(\dfrac{x - i}{x + i}\right)$$ is equal to

The probability of choosing randomly a number c from the set $$\{1, 2, 3, ..........9\} $$ such that the quadratic equation $$x^2+ 4x +c=0$$ has real roots is: