$A$ given mass of gas expands reversibly from the state $A$ to the state $B$ by three paths $1, 2$ and $3$ as shown in the figure. If $w_1, w_2$ and $w_3$ respectively be the work done by the gas along three paths then

$\sinh^{-1}(2^{3/2})$ is equal to :

$A$ man measures the period of a simple pendulum inside a stationary lift and finds it to be $T \ s$. If the lift accelerates upwards with an acceleration $g/4$,then the period of the pendulum will be

If $\int \frac{dx}{x^2+2x+2} = f(x) + c$, then $f(x)$ is equal to :

Which of the following are considered petroleum plants?

At $298 \ K$,the molar conductivities at infinite dilution $(\Lambda_m^{\circ})$ of $NH_4Cl$,$KOH$,and $KCl$ are $152.8$,$272.6$,and $149.8 \ S \ cm^2 \ mol^{-1}$ respectively. The $\Lambda_m^{\circ}$ of $NH_4OH$ in $S \ cm^2 \ mol^{-1}$ and $\%$ dissociation of $0.01 \ M \ NH_4OH$ with $\Lambda_m = 25.1 \ S \ cm^2 \ mol^{-1}$ at the same temperature are: