At a given instant,there are $25\%$ undecayed radioactive nuclei in a sample. After $10 \, s$,the number of undecayed nuclei reduces to $6.25\%$. The mean life of the nuclei is...........$ s$.

For a certain radioactive process,the graph between $\ln R$ and $t \, (\text{sec})$ is obtained as shown in the figure. Then the value of the half-life for the unknown radioactive material is approximately $.... \, \text{sec}$.

The activity of a radioactive sample is measured as $N_0$ counts per minute at time $t=0$,and $\frac{N_0}{e}$ counts per minute at time $t=3$ minutes. The activity reduces to half its value in time (in minutes) is:

Which of the following statements are true regarding radioactivity?
$(I)$ All radioactive elements decay exponentially with time.
$(II)$ Half-life time of a radioactive element is the time required for one-half of the radioactive atoms to disintegrate.
$(III)$ The age of the Earth can be determined with the help of radioactive dating.
$(IV)$ Half-life time of a radioactive element is $50\%$ of its average life period.
Select the correct answer using the codes given below:

If $75 \%$ of a radioactive sample disintegrates in $16 \text{ days}$, the half-life of the radioactive sample is (in $\text{ days}$)

The variation of decay rate with the number of active nuclei is correctly shown in which graph?