Complex Numbers

If a > 0 and z = $${{{{\left( {1 + i} \right)}^2}} \over {a - i}}$$, has magnitude $$\sqrt {{2 \over 5}}$$, then $$\overline z$$ is equal to :

If the equation, x² + bx + 45 = 0 (b $$\in$$ R) has conjugate complex roots and they satisfy |z +1| = 2$$\sqrt {10}$$ , then :

If $$\alpha$$, $$\beta$$ $$\in$$ R are such that 1 $$-$$ 2i (here i² = $$-$$1) is a root of z² + $$\alpha$$z + $$\beta$$ = 0, then ($$\alpha$$ $$-$$ $$\beta$$) is equal to :

Let n denote the number of solutions of the equation z² + 3$$\overline z$$ = 0, where z is a complex number. Then the value of $$\sum\limits_{k = 0}^\infty {{1 \over {{n^k}}}}$$ is equal to :

The area of the polygon, whose vertices are the non-real roots of the equation $$\overline z = i{z^2}$$ is :

The real part of the complex number $${{{{(1 + 2i)}^8}\,.\,{{(1 - 2i)}^2}} \over {(3 + 2i)\,.\,\overline {(4 - 6i)} }}$$ is equal to :

If $$z=2+3 i$$, then $$z^{5}+(\bar{z})^{5}$$ is equal to :

If the set $\left\{\operatorname{Re}\left(\frac{z-\bar{z}+z \bar{z}}{2-3 z+5 \bar{z}}\right): z \in \mathbb{C}, \operatorname{Re}(z)=3\right\}$ is equal to the interval $(\alpha, \beta]$, then $24(\beta-\alpha)$ is equal to :

Let $$S=\left\{z \in \mathbb{C}: \bar{z}=i\left(z^{2}+\operatorname{Re}(\bar{z})\right)\right\}$$. Then \sum_\limits{z \in \mathrm{S}}|z|^{2} is equal to :

For $$a \in \mathbb{C}$$, let $$\mathrm{A}=\{z \in \mathbb{C}: \operatorname{Re}(a+\bar{z}) > \operatorname{Im}(\bar{a}+z)\}$$ and $$\mathrm{B}=\{z \in \mathbb{C}: \operatorname{Re}(a+\bar{z})<\operatorname{Im}(\bar{a}+z)\}$$. Then among the two statements : (S1): If $$\operatorname{Re}(a), \operatorname{Im}(a) > 0$$, then the set A contains all the real numbers (S2) : If $$\operatorname{Re}(a), \operatorname{Im}(a) < 0$$, then the set B contains all the real numbers,