When a system is taken from state $i$ to state $f$ along the path $iaf,$ it is found that $Q = 50 \, cal$ and $W = 20 \, cal$. Along the path $ibf$,$Q = 36 \, cal$. Work done along the path $ibf$ will be ........... $cal$.

$A$ reversible engine converts one-sixth of the heat input into work. When the temperature of the sink is reduced by $62^{\circ} C$,the efficiency of the engine is doubled. The temperatures of the source and sink are

If a line $l$ passes through $(k, 2k), (3k, 3k)$ and $(3, 1)$ where $k \neq 0$,then the distance from the origin to the line $l$ is

Laughing gas is prepared by heating

$A$ vertical spring with force constant $K$ is fixed on a table. $A$ ball of mass $m$ at a height $h$ above the free upper end of the spring falls vertically on the spring so that the spring is compressed by a distance $d$. The net work done in the process is

$A$ pendulum is oscillating with frequency $n$ on the surface of the earth. It is taken to a depth $d = \frac{R}{2}$ below the surface of the earth. What is the new frequency of oscillation at this depth? [$R$ is the radius of the earth]