Consider the following reactions at $1000\,^oC$:
$(I)$ $Zn_{(s)} + \frac{1}{2}O_{2(g)} \rightarrow ZnO_{(s)}; \Delta G^o = -360\,kJ\,mol^{-1}$
$(II)$ $C_{(s)} + \frac{1}{2}O_{2(g)} \rightarrow CO_{(g)}; \Delta G^o = -460\,kJ\,mol^{-1}$
Choose the correct statement at $1000\,^oC$.

$A$ solid cylinder of density $\rho_0$, cross-section area $A$ and length $l$ floats in a liquid of density $\rho$ $(> \rho_0)$ with its axis vertical, as shown. If it is slightly displaced downward and released, the time period of oscillation will be:

$A$ radioactive material decays by simultaneous emission of two particles with respective half-lives $1620 \, years$ and $810 \, years$. The time (in years) after which one-fourth of the material remains is:

Two particles of mass $m$ each are tied at the ends of a light string of length $2a$. The whole system is kept on a frictionless horizontal surface with the string held tight so that each mass is at a distance $a$ from the centre $P$ (as shown in the figure). Now,the mid-point of the string is pulled vertically upwards with a small but constant force $F$. As a result,the particles move towards each other on the surface. The magnitude of acceleration,when the separation between them becomes $2x$,is

The most important feature of all living systems is to

$\int_0^3 \frac{3x+1}{x^2+9} dx$ is equal to :