A radio nuclide $A_1$ with decay constant $\lambda_1$  transforms into a radio nuclide $A_2$ with decay constant $\lambda_2$ . If at the initial moment the preparation contained only the radio nuclide $A_1$, then the time interval after which the activity of the radio nuclide $A_2$ reaches its maximum value is :-

A radioactive sample decays $\frac{7}{4}$ times its original quantity in $15$ minutes. The half-life of the sample is $……min$

${ }_{92}^{238} U$ 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 fossil bone has a ${}^{14}C:{}^{12}C$ ratio, which is $\left[ {\frac{1}{{16}}} \right]$ of that in a living animal bone. If the halflife of ${}^{14}C$ is $5730\, years$, then the age of the fossil bone is ……….$years$

$A$ and $B$ are two radioactive substances whose half lives are $1$ and $2$ years respectively. Initially $10\, g$ of $A$ and $1\,g$ of $B$ is taken. The time (approximate) after which they will have same quantity remaining is ……….. $years$