If $\vec{a}=\alpha \hat{i}+\beta \hat{j}+3 \hat{k}$,$\vec{b}=\hat{j}+2 \hat{k}$,and $\vec{c}=3 \hat{i}+2 \hat{j}+\hat{k}$ are linearly dependent vectors and the magnitude of $\vec{a}$ is $\sqrt{14}$. If $\alpha$ and $\beta$ are integers,then $\alpha+\beta=$

Let the vectors $\overrightarrow{u}_1 = \hat{i} + \hat{j} + a\hat{k}$,$\overrightarrow{u}_2 = \hat{i} + b\hat{j} + \hat{k}$ and $\overrightarrow{u}_3 = c\hat{i} + \hat{j} + \hat{k}$ be coplanar. If the vectors $\overrightarrow{v}_1 = (a+b)\hat{i} + c\hat{j} + c\hat{k}$,$\overrightarrow{v}_2 = a\hat{i} + (b+c)\hat{j} + a\hat{k}$ and $\overrightarrow{v}_3 = b\hat{i} + b\hat{j} + (c+a)\hat{k}$ are also coplanar,then $6(a+b+c)$ is equal to $..............$.

If $\vec{r}$ is a vector perpendicular to both the vectors $2 \hat{i}+3 \hat{j}-4 \hat{k}$ and $3 \hat{i}-\hat{j}+\hat{k}$ and satisfies $\vec{r} \cdot(3 \hat{i}-3 \hat{j}+4 \hat{k})=5$, then $|\vec{r}|=$

If the given vectors $(-bc, b^2 + bc, c^2 + bc)$,$(a^2 + ac, -ac, c^2 + ac)$ and $(a^2 + ab, b^2 + ab, -ab)$ are coplanar,where none of $a, b$ and $c$ is zero,then:

Match the statements/expressions given in Column $I$ with the values given in Column $II$.

Column $I$	Column $II$
$(A)$ Root$(s)$ of the equation $2 \sin ^2 \theta + \sin ^2 2 \theta = 2$	$(p)$ $\frac{\pi}{6}$
$(B)$ Points of discontinuity of the function $f(x) = [\frac{6x}{\pi}] \cos [\frac{3x}{\pi}]$,where $[y]$ denotes the largest integer less than or equal to $y$	$(q)$ $\frac{\pi}{4}$
$(C)$ Volume of the parallelepiped with its edges represented by the vectors $\hat{i}+\hat{j}, \hat{i}+2\hat{j}$ and $\hat{i}+\hat{j}+\pi\hat{k}$	$(r)$ $\frac{\pi}{3}$
$(D)$ Angle between vectors $\vec{a}$ and $\vec{b}$ where $\vec{a}, \vec{b}$ and $\vec{c}$ are unit vectors satisfying $\vec{a}+\vec{b}+\sqrt{3}\vec{c}=\overrightarrow{0}$	$(s)$ $\frac{\pi}{2}$
	$(t)$ $\pi$

The value of $\lambda$ for which points $A(2, 2, 1)$,$B(1, 1, 1)$,$C(-\lambda, 2, 1)$,and $D(3, 0, -1)$ are coplanar is $\lambda = $ ............