The largest value of $a$, for which the perpendicular distance of the plane containing the lines $\vec{r}=(\hat{i}+\hat{j})+\lambda(\hat{i}+a \hat{j}-\hat{k})$ and $\vec{r}=(\hat{i}+\hat{j})+\mu(-\hat{i}+\hat{j}-a \hat{k})$ from the point $(2,1,4)$ is $\sqrt{3}$, is _________.

If $\lambda_{1} < \lambda_{2}$ are two values of such that the angle between the planes and is , then the square of the length of perpendicular from the point to the plane is ______________.

The plane passing through the line $L: l x-y+3(1-l) z=1, x+2 y-z=2$ and perpendicular to the plane is . If is the acute angle between the line and the -axis, then is equal to _____________.

The line of shortest distance between the lines $\frac{x-2}{0}=\frac{y-1}{1}=\frac{z}{1}$ and makes an angle of with the plane , . If the image of the point in the plane is , then is equal to _________________.

Let the lines

${L_1}:\overrightarrow r = \lambda \left( {\widehat i + 2\widehat j + 3\widehat k} \right),\,\lambda \in R$

If $\overrightarrow a = 2\widehat i + \widehat j + 3\widehat k$, and are coplanar vectors and , , then is equal to ___________.

Let $\theta$ be the angle between the planes $P_{1}: \vec{r} \cdot(\hat{i}+\hat{j}+2 \hat{k})=9$ and . Let be the line that meets at the point and makes an angle with the normal of . If is the angle between and , then is equal to ____________.

One mole of an ideal gas at $350 \mathrm{~K}$ is in a $2.0 \mathrm{~L}$ vessel of thermally conducting walls, which are in contact with the surroundings. It undergoes isothermal reversible expansion from 2.0 L to $3.0 \mathrm{~L}$ against a constant pressure of . The change in entropy of the surroundings ( is ___________ (Nearest integer)