If $|z| = 2$,then the points representing the complex numbers $-1 + 5z$ will lie on a

Let $C$ be the set of all complex numbers. Let $S_{1}=\{z \in C:|z-2| \leq 1\}$ and $S_{2}=\{z \in C: z(1+i)+\overline{z}(1-i) \geq 4\}$. Then,the maximum value of $\left|z-\frac{5}{2}\right|^{2}$ for $z \in S_{1} \cap S_{2}$ is equal to:

Let $A, B, C$ be three sets of complex numbers as defined below:
$A = \{z : \operatorname{Im}(z) \geq 1\}$
$B = \{z : |z - 2 - i| = 3\}$
$C = \{z : \operatorname{Re}((1 - i)z) = \sqrt{2}\}$
$1.$ The number of elements in the set $A \cap B \cap C$ is:
$(A) 0, (B) 1, (C) 2, (D) \infty$
$2.$ Let $z$ be any point in $A \cap B \cap C$. Then,$|z + 1 - i|^2 + |z - 5 - i|^2$ lies between:
$(A) 25 \text{ and } 29, (B) 30 \text{ and } 34, (C) 35 \text{ and } 39, (D) 40 \text{ and } 44$
$3.$ Let $z$ be any point in $A \cap B \cap C$ and let $w$ be any point satisfying $|w - 2 - i| < 3$. Then,$|z| - |w| + 3$ lies between:
$(A) -6 \text{ and } 3, (B) -3 \text{ and } 6, (C) -6 \text{ and } 6, (D) -3 \text{ and } 9$

The complex number $z = x + iy$ which satisfies the equation $\left| \frac{z - 5i}{z + 5i} \right| = 1$ lies on

For any integer $k$,let $w_k = \cos \left( \frac{k\pi}{11} \right) + i \sin \left( \frac{k\pi}{11} \right)$,where $i = \sqrt{-1}$. The value of the expression $\frac{\sum_{k=1}^8 |w_{2k+1} - w_{2k}|}{\sum_{k=1}^4 |w_{3k-1} - w_{3k-2}|}$ is

If $z$ is a complex number such that $\frac{z - 1}{z + 1}$ is purely imaginary,then