A radioactive element $X$ converts into another stable element $Y$. Half life of $X$ is $2 \mathrm{hrs}$. Initially only $X$ is present. After time $t$, the ratio of atoms of $X$ and $Y$ is found to be $1: 4$, then $t$ in hours is

A parent nucleus $X$ undergoes $\alpha -$ decay with a half-life of $75000$ years. The daughter nucleus $Y$ undergoes $\beta -$ decay with a half-life of $9$ months. In a particular sample, it is found that the rate of emission of $\beta -$ particles is nearly constant (over several months) at ${10}^7 {\mathrm{h}}^{-1}.$ What will be the number of $\alpha -$ particles emitted in an hour?

Mass spectrometric analysis of potassium and argon atoms in a moon rock sample shows that the ratio of the number of (stable) ${}^{40}\mathrm{A}\mathrm{r}$ atoms present to the number of (radioactive) ${}^{40}\mathrm{K}$ atoms is $10.3$. Assume that all the argon atoms were produced by the decay of potassium atoms, with a half-life of $1.25\times {10}^9 \mathrm{yr}$. How old is the rock? [Given, $\ln \left(11.3\right)=2.425$]

The decay constant of the parent nuclide in Uranium series is $\lambda$. Then, the decay constant of the stable end product of the series will be,