The radioactivity of a certain radioactive element drops to $1/64$ of its initial value in $30 \, s$. Its half-life is ......... $s$.

$A$ sample of a radioactive element contains $8 \times 10^{16}$ active nuclei. The half-life of the element is $15 \text{ days}$. The number of nuclei decayed after $60 \text{ days}$ is:

$A$ radioactive nucleus $A$ has a single decay mode with half-life $\tau_A$. Another radioactive nucleus $B$ has two decay modes $1$ and $2$. If decay mode $2$ were absent,the half-life of $B$ would have been $\tau_A / 2$. If decay mode $1$ were absent,the half-life of $B$ would have been $3 \tau_A$. If the actual half-life of $B$ is $\tau_B$,then the ratio $\tau_B / \tau_A$ is

$A$ radioactive nucleus $A$ with a half-life $T$ decays into a nucleus $B$. At $t = 0$,there is no nucleus $B$. At some time $t$,the ratio of the number of $B$ to that of $A$ is $0.3$. Then,$t$ is given by

The counting rate observed from a radioactive source at $t = 0$ was $1600 \ counts \ s^{-1}$,and at $t = 8 \ s$,it was $100 \ counts \ s^{-1}$. The counting rate observed as $counts \ s^{-1}$ at $t = 6 \ s$ will be

$A$ radioactive sample decays by two modes: $\alpha$-decay and $\beta$-decay. $66.6 \%$ of the time it decays by $\alpha$-decay and $33.3 \%$ of the time it decays by $\beta$-decay. If the half-life of the sample is $60 \text{ years}$,what will be the half-life of the sample if it decays only by $\alpha$-decay?