$A$ freshly prepared radioactive sample of half-life $1$ hour emits radiations that are $128$ times as intense as the permissible safe limit. The minimum time after which this sample can be safely used is ......... $hours$.

$A$ radioactive substance of half-life $138.6 \text{ days}$ is placed in a box. After $n$ days, only $20\%$ of the substance is present. Then the value of $n$ is $[\ln(5) = 1.61]$.

$A$ neutron beam has a kinetic energy of $0.0837 \, eV$. Its half-life is $693 \, s$ and its mass is $1.675 \times 10^{-27} \, kg$. What fraction of the neutrons will remain undecayed after traveling a distance of $40 \, m$?

$A$ 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 sample has an activity $A$ in air. If the sample is kept inside water,then its activity $A^{\prime}$

Radioactive material $A$ has decay constant $8 \lambda$ and material $B$ has decay constant $\lambda$. Initially,they have the same number of nuclei. After what time will the ratio of the number of nuclei of material $B$ to that of $A$ be $\frac{1}{e}$?