If the degrees of freedom of a gas molecule is 6, then the total internal energy of the gas molecule at a temperature of \(47^{\circ} \mathrm{C~}(\mathrm{in~} \mathrm{eV})\) is
\(\left(\right.\) Boltzmann constant \(\left.=1.38 \times 10^{-23} \mathrm{~J} \mathrm{~K}^{-1}\right)\)

Two cylinders \(A\) and \(B\) fitted with pistons, contain equal number of moles of an ideal monoatomic gas at \(400 \mathrm{~K}\). The piston of \(A\) is free to move while that of \(B\) is held fixed. Same amount of heat energy is given to the gas in each cylinder. If the rise in temperature of the gas in \(A\) is \(42 \mathrm{~K}\), the rise in temperature of the gas in \(B\) is \((\gamma=5 / 3)\)

An ideal gas is kept in a cylinder of volume \(3 \mathrm{~m}^3\) at a pressure of \(3 \times 10^5 \mathrm{~Pa}\). The energy of the gas is

An infinitely long straight conductor is bent into the shape as shown below. It carries a current of I ampere and the radius of the circular loop is R metre. Then, the magnitude of magnetic induction at the centre of the circular loop is

A charge \(\mathrm{Q}\) is to be divided between two objects. The values of the charges on the objects so that the electrostatic force between them will be maximum is

A box of mass \(2 \mathrm{~kg}\) is placed on a inclined plane that makes \(30^{\circ}\) with the horizontal. The coefficient of friction between the box and inclined plane is 0.2 . A force \(F\) is applied on the box perpendicular to incline to prevent the box from sliding down. The minimum value of \(F\) is
(acceleration due to gravity \(=10 \mathrm{~ms}^{-2}\) )

An element X of a half-life of \(1.4 \times 10^9\) years decays to form another stable element Y. A sample is taken from a rock that contains both X and Y in the ratio \(1: 7\). If at the time of formation of the rock, \(Y\) was not present in the sample, then the age of the rock in years is

The coefficient of \(x^2\) in the expansion of \((1-3 x)^{\frac{1}{3}}(1+2 x)^{-\frac{1}{2}}\) is

Identify the compound (B) in given reaction.
\(\mathrm{CH}_3 \mathrm{Cl} \xrightarrow{\mathrm{KCN}}(A) \xrightarrow{\mathrm{H}^{+} / \mathrm{H}_2 \mathrm{O}}(B)\)

Match the following.

\(\mathrm{K}_{\mathrm{c}}\) for the following reaction is 99.0
\(\mathrm{A}_2(\mathrm{~g}) \stackrel{\mathrm{T}(\mathrm{~K})}{\rightleftharpoons} \mathrm{B}_2(\mathrm{~g})\)
In a one litre flask, 2 moles of \(\mathrm{A}_2\) was heated to \(\mathrm{T}(\mathrm{K})\) and the above equilibrium is reached. The concentrations at equilibrium of \(\mathrm{A}_2\) and \(\mathrm{B}_2\) are \(\mathrm{C}_1\left(\mathrm{~A}_2\right)\) and \(\mathrm{C}_2\left(\mathrm{~B}_2\right)\) respectively. Now, one mole of \(\mathrm{A}_2\) was added to flask and heated to \(\mathrm{T}(\mathrm{K})\) to establish the equilibrium again. The concentrations of \(\mathrm{A}_2\) and \(\mathrm{B}_2\) are \(\mathrm{C}_3\left(\mathrm{~A}_2\right)\) and \(\mathrm{C}_4\left(\mathrm{~B}_2\right)\) respectively. What is the value of \(\mathrm{C}_3\left(\mathrm{~A}_2\right)\) in \(\mathrm{mol}^{-1}\) ?

The particular solution of the different equation
\[
\frac{d x}{d y}=\frac{\sin y(1+y \cot y)}{x \log \left(x^2 e\right)}, y(1)=0
\]

\(\cos h^{-1} 2=\)