Let

$A = \{ z \in \mathbb{C} : |z - 2| \leq 4 \}$ and

$B = \{ z \in \mathbb{C} : |z - 2| + |z + 2| = 5 \}$.

Then the max $\{|z_1 - z_2| : z_1 \in A \text{ and } z_2 \in B \}$ is :

Let A be the focus of the parabola $y^2 = 8x$. Let the line $y = mx + c$ intersect the parabola at two distinct points B and C. If the centroid of the triangle ABC is $\left( \frac{7}{3}, \frac{4}{3} \right)$, then $(BC)^2$ is equal to :

Let $f(x) = \lim\limits_{\theta \to 0} \left( \frac{\cos \pi x - x^\left( \frac{2}{\theta} \right) \sin(x-1)}{1 + x^\left( \frac{2}{\theta} \right) (x-1)} \right),\ x \in \mathbb{R}$. Consider the following two statements :

(I) $f(x)$ is discontinuous at $x=1$.

(II) $f(x)$ is continuous at $x = -1$.

Then,

$ \frac{6}{3^{26}} + \frac{10 \cdot 1}{3^{25}} + \frac{10 \cdot 2}{3^{24}} + \frac{10 \cdot 2^2}{3^{23}} + \ldots + \frac{10 \cdot 2^{24}}{3} $ is equal to :