The mean life of a radioactive material for alpha decay and beta decay are, respectively, $1620$ years and $520$ years. What is the half life of the sample (in years) ?

${ }_{92}^{238} U$ is known to undergo radioactive decay to form ${ }_{82}^{206} Pb$ by emitting alpha and beta particles. A rock initially contained $68 \times 10^{-6} g$ of ${ }_{92}^{238} U$. If the number of alpha particles that it would emit during its radioactive decay of ${ }_{92}^{238} U$ to ${ }_{82}^{206} Pb$ in three half-lives is $Z \times 10^{18}$, then what is the value of $Z$?

The half life of a radioactive isotope $'X'$ is $20$ years, It decays to another element $'Y'$ which is stable. The two elements $'X'$ and $'Y'$ were found to be in  the ratio $1:7$ in a simple of a given rock . The age of the rock is estimated to be............$years$

In a radioactive material, fraction of active material remaining after time $t$ is $\frac{9}{16}$ The fraction that was remaining after $\frac{t}{2}$ is 

If $'f^{\prime}$ denotes the ratio of the number of nuclei decayed $\left(N_{d}\right)$ to the number of nuclei at $t=0$ $\left({N}_{0}\right)$ then for a collection of radioactive nuclei, the rate of change of $'f'$ with respect to time is given as:

$[\lambda$ is the radioactive decay constant]

Half lives of two radioactive nuclei $A$ and $B$ are $10\, minutes$ and $20\, minutes$, respectively. If, initially a sample has equal number of nuclei, then after $60$ $minutes$ , the ratio of decayed numbers of nuclei $A$ and $B$ will be