The total decay rate R of a sample is related to the decay rate $$\text R_0$$​ at t = 0 and the disintegration constant or decay constant λ as:

Correct option is A

​$$\text{The rate of change of radioactive nuclei is proportional to the number of undecayed nuclei:} \\\frac{dR}{dt} = -\lambda R \\[10pt]\text{Where:} \\R \text{ = decay rate at time } t \\\lambda \text{ = decay constant} \\R_0 \text{ = initial decay rate at } t = 0$$

$$\int \frac{1}{R} \, dR = -\lambda \int dt \\[5pt]\ln R = -\lambda t + C \\[10pt]\text{Where } C \text{ is the integration constant.}$$

$$\text{At } t = 0, \, R = R_0 \\[5pt]\ln R_0 = C \\[10pt]\ln R = -\lambda t + \ln R_0 \\[5pt]\ln \left( \frac{R}{R_0} \right) = -\lambda t \\[5pt]R = R_0 e^{-\lambda t}$$​​​​