Match List-I with List-II and select the correct answer using the code given below the list.

	COLUMN-I		COLUMN-II
A)	If the length of any edge of a regular tetrahedron is 1 unit then the distance (in units) of any vertex from the opposite face is	P)	$\frac{1}{\sqrt{3}}$
B)	If the length of any edge of a regular tetrahedron is 1 unit and $\theta$ is the angle between any edge and a face not containing that edge then cos$\theta$  is equal to	Q)	$\frac{1}{2\sqrt{2}}$
C)	The edges of a parallelepiped are of unit length and are parallel to non-coplanar unit vectors  $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ such that  $\overrightarrow{a}.\overrightarrow{b}=\overrightarrow{b}.\overrightarrow{c}=\overrightarrow{c}.\overrightarrow{a}=\frac{1}{2}$. Then the volume of the parallelepiped (in cubic units) is	R)	$\frac{1}{\sqrt{2}}$
D)	If   $\theta$ is the acute angle between the lines x = 2t – 2, y = 3 – 4t, z = t – 4 and x = – 2 – t,  y = 3 + 2t, z = 3t – 4 then cos$\theta$ is equal to	S)	$\frac{1}{\sqrt{6}}$
		T)	$\frac{\sqrt{2}}{\sqrt{3}}$

Let ABCO be a tetrahedron where O is origin and P vs of A, B, C are  $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ respectively Let $\overrightarrow{m}$ be the p. v. of the foot of perpendicular from origin to face ABC.
 $$\begin{gathered} \overrightarrow{m}=\lambda (\overrightarrow{b}-\overrightarrow{a})\times (\overrightarrow{c}-\overrightarrow{a}) \\ \overrightarrow{m}=\lambda (\overrightarrow{a}\times \overrightarrow{b}+\overrightarrow{b}\times \overrightarrow{c}+\overrightarrow{c}\times \overrightarrow{a})\mathrm{.........................................}\left(1\right) \end{gathered}$$
The equation of plane ABC is  $(\overrightarrow{r}-\overrightarrow{a}).(\overrightarrow{b}-\overrightarrow{a})\times (\overrightarrow{c}-\overrightarrow{a})=0$
Which gives  $\overrightarrow{m}.(\overrightarrow{a}\times \overrightarrow{b}+\overrightarrow{b}\times \overrightarrow{c}+\overrightarrow{c}\times \overrightarrow{a})=\left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}\right]$
Taking dot product with  $(\overrightarrow{a}\times \overrightarrow{b}+\overrightarrow{b}\times \overrightarrow{c}+\overrightarrow{c}\times \overrightarrow{a})$ both sides of (1)
 $\left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}\right]=\lambda {|\overrightarrow{a}\times \overrightarrow{b}+\overrightarrow{b}\times \overrightarrow{c}+\overrightarrow{c}\times \overrightarrow{a}|}^2\Rightarrow \frac{1}{\sqrt{2}}=\lambda ,\frac{3}{4}$
 $\Rightarrow \lambda =\frac{2\sqrt{2}}{3}\therefore \left|\overrightarrow{m}\right|=\frac{2\sqrt{2}}{3}\times \frac{\sqrt{3}}{2}=\frac{\sqrt{2}}{\sqrt{3}}$
Let  $\alpha$  be the angle between edge OA and normal to face ABC.
$\cos \alpha =\frac{\overrightarrow{a}.(\overrightarrow{b}-\overrightarrow{a})\times (\overrightarrow{c}-\overrightarrow{a})}{\left|\overrightarrow{a}\right||(\overrightarrow{b}-\overrightarrow{a})\times (\overrightarrow{c}-\overrightarrow{a})|}=\frac{\left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}\right]}{|\overrightarrow{a}\times \overrightarrow{b}+\overrightarrow{b}\times \overrightarrow{c}+\overrightarrow{c}\times \overrightarrow{a}|}=\frac{\sqrt{2}}{\sqrt{3}}$ 
So, angle between edge OA and face ABC is $\frac{\pi }{2}-{\cos }^{-1}\frac{\sqrt{2}}{\sqrt{3}}$
i. e.  $\theta ={\sin }^{-1}\frac{\sqrt{2}}{\sqrt{3}} \sin \theta =\frac{\sqrt{2}}{\sqrt{3}}\cos \theta =\frac{1}{\sqrt{3}}$
(C)  ${\left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}\right]}^2=\left|\begin{array}{ccc}1& \frac{1}{2}& \frac{1}{2}\\ \frac{1}{2}& 1& \frac{1}{2}\\ \frac{1}{2}& \frac{1}{2}& 1\end{array}\right|=\frac{1}{2}$
Volume of the parallelepiped  $=\frac{1}{\sqrt{2}}$ cubic units
(D)   $\cos \theta =\frac{7}{\sqrt{21}\times \sqrt{14}}=\frac{1}{\sqrt{6}}$