A Carnot engine, whose efficiency is $$40\% $$, takes in heat from a source maintained at a temperature of $$500$$ $$K.$$ It is desired to have an engine of efficiency $$60\% .$$ Then, the intake temperature for the same exhaust (sink) temperature must be :

A thermally insulated vessel contains an ideal gas of molecular mass $$M$$ and ratio of specific heats $$\gamma .$$ It is moving with speed $$v$$ and it's suddenly brought to rest. Assuming no heat is lost to the surroundings, Its temperature increases by:

Three perfect gases at absolute temperatures $${T_1},\,{T_2}$$ and $${T_3}$$ are mixed. The masses of molecules are $${m_1},{m_2}$$ and $${m_3}$$ and the number of molecules are $${n_1},$$ $${n_2}$$ and $${n_3}$$ respectively. Assuming no loss of energy, the final temperature of the mixture is:

A Carnot engine operating between temperatures $${{T_1}}$$ and $${{T_2}}$$ has efficiency $${1 \over 6}$$. When $${T_2}$$ is lowered by $$62$$ $$K$$ its efficiency increases to $${1 \over 3}$$. Then $${T_1}$$ and $${T_2}$$ are, respectively: