$A$ balloon is filled with $500 \ m^3$ of helium gas at $27^{\circ}C$ and $1 \ atm$ pressure. What will be the volume of helium at $3^{\circ}C$ and $0.5 \ atm$ pressure in $m^3$?

The area (in square units) of the triangle formed by the lines $x=0$,$y=0$ and $3x+4y=12$ is:

The solution of the differential equation $\frac{dy}{dx} + \frac{3x^2}{1+x^3}y = \frac{\sin^2 x}{1+x^3}$ is

One mole of an ideal gas $(\gamma = 1.4)$ is adiabatically compressed so that its temperature rises from $27\,^oC$ to $35\,^oC$. The change in the internal energy of the gas is .... $J$. (Given $R = 8.3\,J/mol\cdot K$)

Consider the following reaction: $N_{2(g)} + 3H_{2(g)} \longrightarrow 2NH_{3(g)}$. The rate of this reaction in terms of $N_2$ at $T \ K$ is $-\frac{d[N_2]}{dt} = 0.02 \ mol \ L^{-1} \ s^{-1}$. What is the value of $-\frac{d[H_2]}{dt}$ (in units of $mol \ L^{-1} \ s^{-1}$) at the same temperature?

If the displacement $(x)$ and velocity $(v)$ of a particle executing simple harmonic motion are related through the expression $4 v^2 = 25 - x^2$,then the time period is