AIIMS 2018 Physics Radioactive Decay MCQ Question

To determine the time required for a radioactive sample to decay by 75%, we can use the formula for exponential decay, $N(t) = N_0 e^{-\lambda t}$, where $N(t)$ is the remaining quantity, $N_0$ is the initial quantity, $\lambda$ is the decay constant (0.05/year), and $t$ is time. Setting $N(t) = 0.25 N_0$ (since 75% decay means 25% remains), we solve for $t$:

$$0.25 = e^{-0.05t} \implies \ln(0.25) = -0.05t \implies t = \frac{\ln(0.25)}{-0.05} \approx 27.7 \text{ years}.$$

Options B (57.7 years), C (60 years), and D (87 years) do not satisfy the decay equation for a 75% reduction, as they would imply a longer time than calculated for the given decay constant.