The half-life of a radioactive sample is $5 \,s$. If the initial mass of the sample is $60 \,g$, then the time required to reduce the sample to $7.5 \,g$ is (in $\,s$)

After $3$ hours,only $0.25 \,mg$ of a pure radioactive material is left. If the initial mass was $2 \,mg$,then the half-life of the substance is ...... $hr$.

The initial activity of a radioactive element is $1600$. After $t = 8 \, s$,the activity becomes $100$. What is the activity at $t = 6 \, s$?

$A$ radioactive nuclide is produced at a constant rate of $n$ per second (e.g.,by bombarding a target with neutrons). If the number of nuclei at $t = 0$ is $N_0$,the number of nuclei $N$ at time $t$ is given by (where $\lambda$ is the decay constant):

If $f$ denotes the ratio of the number of nuclei decayed $(N_{d})$ to the number of nuclei at $t=0$ $(N_{0})$,then for a collection of radioactive nuclei,the rate of change of $f$ with respect to time is given as: [$\lambda$ is the radioactive decay constant]

Two radioactive materials $X_1$ and $X_2$ contain the same number of nuclei. If $6\lambda \, s^{-1}$ and $4\lambda \, s^{-1}$ are the decay constants of $X_1$ and $X_2$ respectively,the ratio of the number of undecayed nuclei of $X_1$ to that of $X_2$ will be $\left( \frac{1}{e} \right)$ after a time: