Let the line $L: \frac{x-1}{2}=\frac{y+1}{-1}=\frac{z-3}{1}$ intersect the plane at the point . Let the point be the foot of perpendicular from the point on the line . If is the area of triangle , then is equal to __________.

If $\mathrm{d}_1$ is the shortest distance between the lines $x+1=2 y=-12 z, x=y+2=6 z-6$ and is the shortest distance between the lines , then the value of is :

Let a line with direction ratios $a,-4 a,-7$ be perpendicular to the lines with direction ratios $3,-1,2 b$ and . If the point of intersection of the line and the plane is , 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 d be the distance between the foot of perpendiculars of the points P(1, 2, $-$1) and Q(2, $-$1, 3) on the plane $-$x + y + z = 1. Then d² is equal to ___________.

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 ______________.

Let $\alpha x+\beta y+\gamma z=1$ be the equation of a plane passing through the point and perpendicular to the line joining the points and . Then the value of is equal to _____________.

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 _____________.

Let the lines

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

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)

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 ____________.

The heat of solution of anhydrous $\mathrm{CuSO}_4$ and $\mathrm{CuSO}_4 \cdot 5 \mathrm{H}_2 \mathrm{O}$ are and respectively.