$A$ cylindrical rod with one end in a steam chamber and the other end in ice results in the melting of $0.1 \, g$ of ice per second. If the rod is replaced by another with half the length and double the radius of the first,and if the thermal conductivity of the material of the second rod is $1/4$ that of the first,the rate at which ice melts in $g/s$ will be:

If the curves $x^2 - 6x + y^2 + 8 = 0$ and $x^2 - 8y + y^2 + 16 - k = 0$ $(k > 0)$ touch each other at a point,then the largest value of $k$ is

In the arrangement shown in the figure,the masses of blocks $B$ and $A$ are $2m$ and $m$ respectively. The surface between $B$ and the floor is smooth. Block $B$ is connected to block $C$ by means of a string-pulley system. If the whole system is released,find the minimum value of the mass of block $C$ so that block $A$ remains stationary with respect to $B$. The coefficient of friction between $A$ and $B$ is $\mu$.

$\sqrt{3} \operatorname{cosec} 20^{\circ} - \sec 20^{\circ}$ is equal to

If the mass defect of a nuclide is $3.32 \times 10^{-26} \ g$,its binding energy is (in $MeV$)?

An ideal gas of mass $m$ in a state $A$ goes to another state $B$ via three different processes as shown in the figure. If ${Q_1}, {Q_2}$ and ${Q_3}$ denote the heat absorbed by the gas along the three paths,then