A radio active element has rate of disintegration 8000 disintegrations per minute at a particular instant. After four minutes it becomes 2000 disintegrations per minute. The decay constant per minute is

The rate of radioactive disintegration follows exponential decay: $R = R_0 e^{-\lambda t}$.

Given $R_0 = 8000$ and $R = 2000$ after $t = 4$ minutes, we solve for $\lambda$:

$2000 = 8000 e^{-4\lambda}$

Simplifying gives $\frac{1}{4} = e^{-4\lambda}$.

Taking natural logarithms: $\ln\left(\frac{1}{4}\right) = -4\lambda$

Expanding: $-\ln 4 = -4\lambda$

Thus $\lambda = \frac{\ln 4}{4} = \frac{\ln (2^2)}{4} = \frac{1}{2} \ln 2$

This matches option C.