Explanation

(a) From radioactive decay law.
$\ \ \ \ \ \ \ \ \ \ - \frac{dN}{dt}\ \propto N\ or - \frac{dN}{dt} = \lambda N$
Thus, $R = - \frac{dN}{dt}$
Or $\ \ \ \ \ R = \ \lambda N\ or\ R = \ \lambda N_{0}e^{- \lambda t}\ \ \ \ \ \ \ldots(i)$
Where $R_{0}\ = \lambda N_{0}\ $is the activity of the radioactive materialat time $t = 0.$
At time $t_{1},\ \ \ \ \ \ R_{1} = R_{0}e^{- \lambda t}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ldots(ii)$
At time $t_{2},\ \ \ \ \ \ R_{2} = R_{0}e^{- \lambda t}\ \ \ \ \ \ \ \ \ \ \ \ \ \ldots(iii)$
Dividing Eq. (ii) by (iii), we have
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{R_{1}}{R_{2}} = \frac{e^{- \lambda t_{1}}}{e^{- \lambda t_{2}}} = e^{- \lambda(t_{1} - t_{2})}$
or $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ R_{1} = R_{2}e^{- \lambda(t_{1} - t_{2})}$