The least value of n for which the number of integral terms in the Binomial expansion of $(\sqrt[3]{7}+\sqrt[12]{11})^n$ is 183, is :

The value of $\lim \limits_{n \rightarrow \infty}\left(\sum\limits_{k=1}^n \frac{k^3+6 k^2+11 k+5}{(k+3)!}\right)$ is :

Let y = y(x) be the solution of the differential equation :

$\cos x\left(\log _e(\cos x)\right)^2 d y+\left(\sin x-3 y \sin x \log _e(\cos x)\right) d x=0$, x ∈ (0, $\frac{\pi}{2}$ ). If $ y(\frac{\pi}{4}) $ = $-\frac{1}{\log_{e}2}$, then $ y(\frac{\pi}{6}) $ is equal to :

Let $ |z_1 − 8−2i| \leq 1 $ and $ |z_2−2+6i| \leq 2 $, $ z_1, z_2 \in \mathbb{C} $. Then the minimum value of $ |z_1 − z_2| $ is :