A radioactive sample, at a certain instant of time, has its disintegration rate $5000$ disintegrations per minute. After $5$ minutes, the rate is $1250$ disintegrations per minute. Then, the decay constant (in per minute) is,

The correct option is A $0.4\ln 2$
Initial activity, $A_0=5000$ dis. per min
After $5$ min. $A=1250$ dis. per min.

Now, $A=A_0{e}^{-\lambda t}$ where, $\lambda =$ decay constant per min.
$\Rightarrow 1250=5000e^{-5\lambda }$
$\Rightarrow \frac{1}{4}=e^{-5\lambda }$
$\Rightarrow 2^{-2}=e^{-5\lambda }$
$\Rightarrow 2\ln 2=5\lambda$

$\therefore \lambda =\frac{2\ln 2}{5}=0.4\ln 2$

Hence, $\left(A\right)$ is the correct answer.