The mean lives of a radioactive sample are $30$ years and $60$ years for $\alpha$-emission and $\beta$-emission respectively. If the sample decays by both $\alpha$-emission and $\beta$-emission simultaneously,the time after which only one-fourth of the sample remains is:

The curve between the activity $A$ of a radioactive sample and the number of active atoms $N$ is

The activity of an element becomes $\frac{1}{64}$ of its original value in $60 \ s$. Then the half-life period is ............ $s$.

Two radioactive materials $R_1$ and $R_2$ have decay constants $6 \lambda$ and $\lambda$, respectively. The half-life of $R_2$ is $1.4 \times 10^{17} \,s$. Initially, they contain the same number of nuclei. The time at which the ratio of the remaining nuclei of $R_2$ to that of $R_1$ will be $e$ is (Let $\ln 2 = 0.7$):

$A$ freshly-prepared radioactive source of half-life $2 \, h$ emits radiation of intensity which is $64$ times the permissible safe level. The minimum time after which it would be possible to work safely with this source is ....... $h$.

$A$ radioactive material of half-life $2.5$ hours emits radiation that is $32$ times the safe maximum level. The time (in hours) after which the material can be handled safely is