A radio isotope $X$ with a half-life $1.4 \times 10^{9}\; years$ decays of $Y$ which is stable. A sample of the rock from a cave was found to contain $X$ and $Y$ in the ratio $1: 7$. The age of the rock is ........ $ \times 10^9\; years$

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]

A radioactive sample disintegrates via two independent decay processes having half lives $T _{1 / 2}^{(1)}$ and $T _{1 / 2}^{(2)}$ respectively. The effective half- life $T _{1 / 2}$ of the nuclei is

The activity of a radioactive sample is measured as $N_0$ counts per minute at $t = 0$ and $N_0/e$ counts per minute at $t = 5\, minutes$. The time (in $minutes$) at which the activity reduces to half its value is

Starting with a sample of pure ${}^{66}Cu,\frac{7}{8}$ of it decays into $Zn$ in $15\, minutes$. The corresponding half life is..........$minutes$

The half life of a radioactive substance is $20$ minutes. The approximate time interval $(t_2 - t_1)$ between the time $t_2$ when $\frac{2}{3}$ of it had decayed and time $t_1$ when $\frac{1}{3}$ of it had decayed is ..........$min$