The volume of the parallelepiped whose coterminous edges are represented by the vectors $2i - 3j + 4k$,$i + 2j - 2k$,and $3i - j + k$ is ............ $cubic \ units$.

Let $\vec{a} = \hat{i} + 2\hat{j} + 4\hat{k}$,$\vec{b} = \hat{i} + \lambda\hat{j} + 4\hat{k}$,and $\vec{c} = 2\hat{i} + 4\hat{j} + (\lambda^2 - 1)\hat{k}$ be coplanar vectors. Then the non-zero vector $\vec{a} \times \vec{c}$ is:

If the origin and the points $(1, 2, 3)$,$(2, 3, 4)$,and $(x, y, z)$ are coplanar,then

If $d = \lambda (a \times b) + \mu (b \times c) + \nu (c \times a)$ and $[a, b, c] = \frac{1}{8}$,then $\lambda + \mu + \nu$ is equal to

If the four points $A(6,2,4)$,$B(1,3,5)$,$C(1,-2,3)$,and $D(6, k, 2)$ are coplanar,then $k=$

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 $\vec{a}+\vec{b}+\sqrt{3}\vec{c}=\overrightarrow{0}$	$(s)$ $\frac{\pi}{2}$
	$(t)$ $\pi$