The half-lives of two radioactive materials $A$ and $B$ are $T$ and $2T$ respectively. If the ratio of the initial masses of the materials $A$ and $B$ is $8:1$, then the time after which the ratio of the masses of the materials $A$ and $B$ becomes $4:1$ 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.

$10 \, g$ of radioactive material of half-life $15 \, \text{years}$ is kept in store for $20 \, \text{years}$. The disintegrated material is ............ $g$.

The rate of disintegration was observed to be $10^{17}$ disintegrations per second when its half-life period is $1445$ years. The original number of particles is:

If the measurement errors in all the independent quantities are known,then it is possible to determine the error in any dependent quantity. This is done by the use of series expansion and truncating the expansion at the first power of the error. For example,consider the relation $z = x / y$. If the errors in $x, y$ and $z$ are $\Delta x, \Delta y$ and $\Delta z$,respectively,then $z \pm \Delta z = \frac{x \pm \Delta x}{y \pm \Delta y} = \frac{x}{y} (1 \pm \frac{\Delta x}{x}) (1 \pm \frac{\Delta y}{y})^{-1}$. The series expansion for $(1 \pm \frac{\Delta y}{y})^{-1}$,to first power in $\Delta y / y$,is $1 \mp (\Delta y / y)$. The relative errors in independent variables are always added. So the error in $z$ will be $\Delta z = z (\frac{\Delta x}{x} + \frac{\Delta y}{y})$. The above derivation makes the assumption that $\Delta x / x \ll 1, \Delta y / y \ll 1$. Therefore,the higher powers of these quantities are neglected.
$(1)$ Consider the ratio $r = \frac{(1 - a)}{(1 + a)}$ to be determined by measuring a dimensionless quantity $a$. If the error in the measurement of $a$ is $\Delta a$ $(\Delta a / a \ll 1)$,then what is the error $\Delta r$?
$(2)$ In an experiment,the initial number of radioactive nuclei is $3000$. It is found that $1000 \pm 40$ nuclei decayed in the first $1.0 \ s$. For $|x| < 1$,$\ln(1 + x) = x$ up to first power in $x$. The error $\Delta \lambda$,in the determination of the decay constant $\lambda$,in $s^{-1}$,is:

${ }_{92}^{238} U$ is known to undergo radioactive decay to form ${ }_{82}^{206} Pb$ by emitting alpha and beta particles. $A$ rock initially contained $68 \times 10^{-6} \text{ g}$ 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 $Z$?