Let a line with direction ratios $a,-4 a,-7$ be perpendicular to the lines with direction ratios $3,-1,2 b$ and $b, a,-2$. If the point of intersection of the line $\frac{x+1}{a^{2}+b^{2}}=\frac{y-2}{a^{2}-b^{2}}=\frac{z}{1}$ and the plane $x-y+z=0$ is $(\alpha, \beta, \gamma)$, then $\alpha+\beta+\gamma$ 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 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 the lines

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

If the shortest between the lines ${{x + \sqrt 6 } \over 2} = {{y - \sqrt 6 } \over 3} = {{z - \sqrt 6 } \over 4}$ and is 6, then the square of sum of all possible values of is :

If the equation of the plane passing through the point $(1,1,2)$ and perpendicular to the line $x-3 y+ 2 z-1=0=4 x-y+z$ is , then is equal to ___________.

$\lim_{x \to 0} \frac{48}{x^4} \int_{0}^{x} \frac{t^3}{t^6 + 1} \, dt$ 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 _________________.

$2.4 \mathrm{~g}$ coal is burnt in a bomb calorimeter in excess of oxygen at $298 \mathrm{~K}$ and $1 \mathrm{~atm}$ pressure. The temperature of the calorimeter rises from to . The enthalpy change during the combustion of coal is . The value of is ___________. (Nearest Integer)

17.0 g of NH₃ completely vapourises at $-$33.42$^\circ$C and 1 bar pressure and the enthalpy change in the process is 23.4 kJ mol^$-$1. The enthalpy change for the vapourisation of 85 g of NH under the same conditions is _________ kJ.

If three moles of an ideal gas at $300 \mathrm{~K}$ expand isothermally from $30 \mathrm{~dm}^3$ to against a constant opposing pressure of , then the amount of heat transferred is _______ J.

At $25^{\circ} \mathrm{C}$, the enthalpy of the following processes are given :