Three-fourths of the active nuclei decay in a radioactive sample in $3/4 \, s$. The half-life of the sample is:

$A$ radioactive element emits $200$ particles per second. After $3$ hours,$25$ particles per second are emitted. The half-life period of the element will be .......... $minutes$.

$A$ radioactive isotope $X$ has a half-life of $20$ years and decays into a stable element $Y$. In a given rock sample, the ratio of the amounts of $X$ and $Y$ is found to be $1:7$. The estimated age of the rock is ......... years.

If ${N_0}$ is the original mass of the substance of half-life period ${T_{1/2}} = 5 \text{ years}$,then the amount of substance left after $15 \text{ years}$ is

$A$ certain radioactive substance reduces to $25\%$ of its initial value in $16\ days$. Its half-life is .......... $days$.

Two radioactive materials $X_1$ and $X_2$ have decay constants $5\lambda$ and $\lambda$ respectively. If initially they have the same number of nuclei,then the ratio of the number of nuclei of $X_1$ to that of $X_2$ will be $\frac{1}{e}$ after a time: