The half-lives of a radioactive substance are $T$ and $2T$ years for $\alpha$-emission and $\beta$-emission,respectively. The total decay constant for the simultaneous decay of the $\alpha$ and $\beta$ radioactive substance 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:

$A$ radioactive substance has a half-life of $60 \text{ minutes}$. During $3 \text{ hours}$,the amount of substance decayed would be: (in $\%$)

The activity $R$ of an unknown radioactive nuclide is measured at hourly intervals. The results found are tabulated as follows:

$t (h)$	$0$	$1$	$2$	$3$	$4$
$R (MBq)$	$100$	$35.36$	$12.51$	$4.42$	$1.56$

$(i)$ Plot the graph of $R$ versus $t$ and calculate the half-life from the graph.
$(ii)$ Plot the graph of $\ln \left( \frac{R}{R_0} \right)$ versus $t$ and obtain the value of the half-life from the graph.

The decay constant of a radioactive element is $1.5 \times 10^{-9} \text{ s}^{-1}$. Its mean life in seconds will be:

Which sample,$A$ or $B$ shown in the figure,has a shorter mean-life?