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 $3 x+2 y+z=6$ is $3 x-8 y+7 z=4$. If $\theta$ is the acute angle between the line $L$ and the $y$-axis, then $415 \cos ^{2} \theta$ 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 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 __________.

Let $\mathrm{P}_{1}$ be the plane $3 x-y-7 z=11$ and be the plane passing through the points , and . If the foot of the perpendicular drawn from the point on the line of intersection of the planes and 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 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 _________.

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 $\alpha x+\beta y+\gamma z=1$ 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 _____________.

Two reactions are given below:

$\begin{aligned} & 2 \mathrm{Fe}_{(\mathrm{s})}+\frac{3}{2} \mathrm{O}_{2(\mathrm{~g})} \rightarrow \mathrm{Fe}_2 \mathrm{O}_{3(\mathrm{~s})}, \Delta \mathrm{H}^{\circ}=-822 \mathrm{~kJ} / \mathrm{mol} \\ & \mathrm{C}_{(\mathrm{s})}+\frac{1}{2} \mathrm{O}_{2(\mathrm{~g})} \rightarrow \mathrm{CO}_{(\mathrm{g})}, \Delta \mathrm{H}^{\circ}=-110 \mathrm{~kJ} / \mathrm{mol} \end{aligned}$