A telegraph line of length 100 km has a capacity of 0.01 $\mu$F/km and it carries an alternating current at 0.5 kilo cycle per second. If minimum impedance is required, then the value of the inductance that needs to be introduced in series is _____________ mH. (if $\pi$ = $\sqrt{10}$)

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 the shortest distance between the line joining the points (1, 2, 3) and (2, 3, 4), and the line ${{x - 1} \over 2} = {{y + 1} \over { - 1}} = {{z - 2} \over 0}$ is , then 28 is equal to ____________.

Let the equation of the plane P containing the line $x+10=\frac{8-y}{2}=z$ be and the distance of the plane from the point (1, 27, 7) be . 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 ___________.

$2.4 \mathrm{~g}$ 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 :