Let $$ f(x)=\int \frac{d x}{x^{2 / 3}+2 x^{1 / 2}}, $$ be such that $f(0) = -26 + 24 \log_e(2)$. If $f(1) = a + b \log_e(3)$, where $a, b \in \mathbb{Z}$, then $a + b$ is equal to :

An ellipse has its center at $(1, -2)$, one focus at $(3, -2)$ and one vertex at $(5, -2)$. Then the length of its latus rectum is :

Given below are two statements :

Statement I :

$25^{13} + 20^{13} + 8^{13} + 3^{13}$ is divisible by 7.

Statement II :

The integral part of $(7 + 4\sqrt{3})^{25}$ is an odd number.

In the light of the above statements, choose the correct answer from the options given below :

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 :