$A$ certain nuclide has a half-life period of $30 \ minutes$. If a sample containing $600$ atoms is allowed to decay for $90 \ minutes$,how many atoms will remain?

$8 \, g$ of the radioactive isotope,cesium$-137$ were collected on February $1$ and kept in a sealed tube. On July $1$,it was found that only $0.25 \, g$ of it remained. So the half-life period of the isotope is ....... $days$.

The half-life of a radioactive element is $100 \ yrs$. The time in which it disintegrates to $50 \%$ of its initial mass will be ........ $years$.

$t_{1/2}$ of $^{232}Th$ is $1.39 \times 10^{10} \ \text{years}$. Calculate the number of $\alpha$-particles emitted by $1.0 \ \text{g}$ of $^{232}Th$ per second.

If the quantity of a radioactive element is doubled,then its rate of disintegration per unit time will be

Assuming $1 \, \mu g$ of trace radioactive element $X$ with a half-life of $30 \ years$ is absorbed by a growing tree. The amount of $X$ remaining in the tree after $100 \ years$ is $n \times 10^{-1} \, \mu g$. Find the value of $n$. $[Given : \ln 10 = 2.303 ; \log 2 = 0.30]$