The radioactive materials $x_1$ and $x_2$ have decay constants $10 \lambda$ 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 $1/e$ after a time $t$ equal to:

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

The half-life of a radioactive substance is $ 20 \text{ minutes} $. The time taken between $ 50\% $ decay and $ 87.5\% $ decay of the substance will be

$A$ radioactive nucleus can decay by two different processes. The half-lives of the first and second decay processes are $5 \times 10^3$ years and $10^5$ years respectively. Then,the effective half-life of the nucleus is

The normal activity of living carbon-containing matter is found to be about $15$ decays per minute for every gram of carbon. This activity arises from the small proportion of radioactive $_{6}^{14}C$ present with the stable carbon isotope $_{6}^{12}C$. When the organism is dead,its interaction with the atmosphere (which maintains the above equilibrium activity) ceases and its activity begins to drop. From the known half-life ($5730$ years) of $_{6}^{14}C$,and the measured activity,the age of the specimen can be approximately estimated. This is the principle of $_{6}^{14}C$ dating used in archaeology. Suppose a specimen from Mohenjodaro gives an activity of $9$ decays per minute per gram of carbon. Estimate the approximate age (in years) of the Indus-Valley civilisation.

$A$ radioactive element $X$ converts into another stable element $Y$. The half-life of $X$ is $2 \text{ hours}$. Initially,only $X$ is present. After a time $t$,if the ratio of atoms of $X$ to $Y$ is $1:4$,then the value of $t$ is