One mole of an ideal diatomic gas undergoes a transition from $A$ to $B$ along a path $AB$ as shown in the figure. The change in internal energy of the gas during the transition is:

An air column in a tube $32 ~cm$ long,closed at one end,is in resonance with a tuning fork. The air column in another tube,open at both ends,of length $66 ~cm$ is in resonance with another tuning fork. When these two tuning forks are sounded together,they produce $8$ beats per second. Then the frequencies of the two tuning forks are (Consider fundamental frequencies only):

Two capacitors $C_1$ and $C_2$ in a circuit are joined as shown in the figure. The potential of point $A$ is $V_1$ and that of $B$ is $V_2$. The potential of point $D$ will be

The displacement-time equation of a particle executing $SHM$ is $x = A \sin \left( \omega t + \frac{\pi}{6} \right)$. The time taken by the particle to go directly from $x = -\frac{A}{2}$ to $x = +\frac{A}{2}$ is

$A$ plane passing through the points $(0, -1, 0)$ and $(0, 0, 1)$ and making an angle $\frac{\pi}{4}$ with the plane $y - z + 5 = 0$,also passes through the point:

$100 \ g$ of water is heated from $30^\circ C$ to $50^\circ C$. Ignoring the slight expansion of the water,the change in its internal energy is ...... $kJ$ (specific heat of water is $4184 \ J/kg/K$).