An engine has an efficiency of $1/6$. When the temperature of the sink is reduced by $62^{\circ}C$,its efficiency is doubled. The temperature of the source is $.........^{\circ}C$.

If $\frac{1-x+6x^2}{x-x^3} = \frac{A}{x} + \frac{B}{1-x} + \frac{C}{1+x}$,then $A$ is equal to

Two cylinders $A$ and $B$ fitted with pistons contain equal number of moles of an ideal monoatomic gas at $400 ~K$. 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 $42 ~K$,the rise in temperature of the gas in $B$ is (in $~K$)

$A$ plane is parallel to two lines whose direction ratios are $1, 0, -1$ and $-1, 1, 0$ and it contains the point $(1, 1, 1)$. If it cuts the coordinate axes at $A, B, C$,then the volume of the tetrahedron $OABC$ (in cubic units) is

The coefficient of apparent expansion of mercury in a glass vessel is $153 \times 10^{-6}/^{\circ}C$ and in a steel vessel is $144 \times 10^{-6}/^{\circ}C$. If $\alpha$ for steel is $12 \times 10^{-6}/^{\circ}C$,then $\alpha$ for glass is:

If $f_n(x) = \log \log \log \ldots \log x$ (where $\log$ is repeated $n$ times),then $\int (x f_1(x) f_2(x) \ldots f_n(x))^{-1} dx$ is equal to