Number Theory T...

If $$(w - \overline{w}z)/(1-z)$$ is purely real where $$w = \alpha + i\beta, \beta \neq 0$$ and $$z \neq 1$$, then set of the values of  $$z$$ is 

Find the greatest and the least value of $$\left|z_1 + z_2\right|$$ if $$z_1 = 24 + 7i$$ and $$\left|z_2\right| = 6.$$

If $$(\sqrt{8} + i)^{50} = 3^{49}(a + ib)$$, then find the value of $$a^2 + b^2$$. 

Complex number $$z$$ satisfy the equation $$\left|z - (4/z)\right| = 2$$ The difference between the least and the greatest modulus of complex number is 

For all complex numbers $$z_1, z_2$$ satisfying $$\left|z_1\right| = 12$$ and $$\left|z_2 - 3 - 4i\right| = 5$$, then minimum value of $$\left|z_1 - z_2\right|$$ is  

If $$\displaystyle \displaystyle\ |z_{1}-1|<1, |z_{2}-2|<2, |z_{3}-3|<3$$ then $$\displaystyle\ |z_{1}+z_{2}+z_{3}|$$

Let $$z$$ be a complex number of constant modulus such that $$\displaystyle\ z^{2}$$ is purely imaginary then the number of possible values of z is 

If z is a complex number, then find the minimum value of $$\left|z\right| + \left|z - 1\right| + \left|2z - 3\right|.$$

Find the minimum value of $$|z-1|$$ if $$\left|\left|z - 3\right| - \left|z + 1\right|\right| = 2$$.

If $$\left|z_1 - 1\right| \le 1, \left|z_2 - 2\right| \le 2, \left|z_3 - 3\right| \le 3$$, then find the greatest value of $$\left|z_1 +  z_2 + z_3\right|$$.