Q 12-13-133JEE MainJEE Main 2018 (16 Apr, Shift 1)Medium
At some instant, a radioactive sample $S_1$ having an activity $5\ \mu\text{Ci}$ has twice the number of nuclei as another sample $S_2$ which has an activity of $10\ \mu\text{Ci}$. The half lives of $S_1$ and $S_2$ are:
Answer: (A) $4 : 1$
Activity $A = \lambda N$, so $\lambda = \dfrac AN$:
$$\frac{\lambda_1}{\lambda_2} = \frac{A_1}{A_2}\cdot\frac{N_2}{N_1} = \frac{5}{10}\cdot\frac12 = \frac14$$
Half-life $T = \dfrac{\ln2}{\lambda}$, so
$$T_1 : T_2 = 4 : 1$$
Solution by Sreeraj P, M.Sc Physics