Number Theory T...

The number of complex numbers $$z$$ such that $$\left|z\right| = 1$$ and $$\left|z/\overline{z} + \overline{z}/z\right| = 1$$ is $$(arg(z) \in [0, 2\pi))$$ 

Consider a complex number $$z$$ which satisfies the equation $$\left |z-\left (\displaystyle\frac{4}{z}\right )\right |=2$$, then the absolute difference between the least and the greatest moduli of complex numbers is,

Find the value of $$x$$ such that $$\displaystyle \frac{(x + \alpha)^2 - (x + \beta)^2}{ \alpha + \beta} = \frac{sin  2 \theta}{sin^2  \theta}$$. when $$\alpha$$ and $$\beta $$ are the roots of $$t^2 - 2t + 2 = 0$$

If $$z_1, z_2$$ be two non zero complex numbers satisfying the equation $$\displaystyle \left | \frac{z_1 + z_2}{z_1 - z_2} \right | = 1$$ then $$\displaystyle \frac{z_1}{z_2} + \left ( \frac{z_1}{z_2} \right )$$ is

Find the range of real number $$\alpha$$ for which the equation $$z + \alpha |z - 1| + 2i = 0;  z= x + iy$$ has a solution. Find the solution.

Find the regions of the z-plane for which $$\displaystyle \left | \frac{z - a}{z + \overline a} \right | < 1, = 1$$ or $$> 1$$. when the real part of a is positive.

Find all complex numbers satisfying the equation $$2|z|^2 + z^2 - 5 + i \sqrt{3} = 0$$

Find the real values of the parameter $$a$$ for which at least one complex number $$z = x + iy$$ satisfies both the equality $$|z + \sqrt{2}| = a^2 - 3a + 2$$ and the inequality $$|z + i \sqrt{2}| < a^2$$.

If z be a complex number satisfying$$\displaystyle\ z^{4}+z^{3}+2z^{2}+z+1=0$$ then $$\displaystyle\ |z|$$ is 

$$\displaystyle { \left( \frac { \sqrt { 3 } +i }{ 2 }  \right)  }^{ 6 }+{ \left( \frac { i-\sqrt { 3 }  }{ 2 }  \right)  }^{ 6 }=$$