$\frac{d}{dx}[(1 + x^2)\tan^{-1}x] = $

If $\frac{dy}{dx} = \frac{xy + y}{xy + x}$,then the solution of the differential equation is

Identify the mixture that shows positive deviation from Raoult's law.

If ${N_a} = \{ an : n \in N \}$,then ${N_3} \cap {N_4} = $

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 rigid boxes containing different ideal gases are placed on a table. Box $A$ contains one mole of nitrogen at temperature $T_0$,while box $B$ contains one mole of helium at temperature $(7/3) T_0$. The boxes are then put into thermal contact with each other,and heat flows between them until the gases reach a common final temperature (ignore the heat capacity of the boxes). The final temperature of the gases $T_f$ in terms of $T_0$ is: