$A$ radioactive source has a half-life of $6 \,h$. $A$ freshly prepared sample of the same exhibits radioactivity $32$ times the permissible safe value. The minimum time after which it would be possible to work safely with the source is (in $\,h$)

What is the relationship between the decay constant $\lambda$ and the half-life $\tau_{1/2}$?

Two radioactive substances $X$ and $Y$ originally have $N_{1}$ and $N_{2}$ nuclei respectively. The half-life of $X$ is half of the half-life of $Y$. After three half-lives of $Y$,the number of nuclei of both are equal. The ratio $\frac{N_{1}}{N_{2}}$ will be equal to:

What is the half-life (in years) of a radioactive material if its activity drops to $1/16$th of its initial value in $30$ years (in $.5$)?

If there is $0.1 \ mg$ of radioactive $Th^{234}$, how much of it will remain undecayed after $120 \ \text{days}$, given its half-life is $24 \ \text{days}$? (Answer in $\mu g$)

Three-fourths of the active nuclei decay in a radioactive sample in $3/4 \, s$. The half-life of the sample is: