Explanation

(a) From Rutherford-Soddy law
$N = N_{0}\left( \frac{1}{2} \right)^{n}$
$\ \ \ \ \ \ \ \ \ \ \ n = \frac{t}{T}$
$\therefore\ 10^{6}\ = \ 1.414\ \times \ 10^{6}\left( \frac{1}{2} \right)^{t/T}$
$\Rightarrow \ \ \ \ \ \frac{1000}{1414} = \left( \frac{1}{2} \right)^{t/T}$
$\Rightarrow \ \left( \frac{1}{2} \right)^{2} = \left( \frac{10}{12} \right)^{2}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (Approximately)$
$\Rightarrow \ \ \ \ \ \ \ n = 2$
$\Rightarrow \ \ \ \ \ \ \ \ n = \frac{t}{T}$ =2
$\Rightarrow \ \ \ \ \ \ \ T = \ \frac{10}{2}\ = \ 5\ \min$