For a triangle $A B C$, let $\vec{p}=\overrightarrow{B C}, \vec{q}=\overrightarrow{C A}$ and $\vec{r}=\overrightarrow{B A}$. If $|\vec{p}|=2 \sqrt{3},|\vec{q}|=2$ and $\cos \theta=\frac{1}{\sqrt{3}}$, where $\theta$ is the angle between $\vec{p}$ and $\vec{q}$, then $|\vec{p} \times(\vec{q}-3 \vec{r})|^2+3|\vec{r}|^2$ is equal to :

The largest $n \in \mathbb{N}$, for which $7^n$ divides $101!$, is :

Let the line $L_1$ be parallel to the vector $-3\hat{i} + 2\hat{j} + 4\hat{k}$ and pass through the point $(2, 6, 7)$, and the line $L_2$ be parallel to the vector $2\hat{i} + \hat{j} + 3\hat{k}$ and pass through the point $(4, 3, 5)$. If the line $L_3$ is parallel to the vector $-3\hat{i} + 5\hat{j} + 16\hat{k}$ and intersects the lines $L_1$ and $L_2$ at the points $C$ and $D$, respectively, then $\left|\overrightarrow{CD}\right|^2$ is equal to:

Let $[\cdot]$ denote the greatest integer function and $f(x) = \lim\limits_{n \to \infty} \frac{1}{n^{3}} \sum\limits_{k=1}^n \left[ \frac{k^2}{3^x} \right]$. Then $12 \sum\limits_{j=1}^{\infty} f(i)$ is equal to ________.