$A$ radioactive element has a rate of disintegration $10,000$ disintegrations per minute at a particular instant. After four minutes it becomes $2500$ disintegrations per minute. The decay constant per minute is (in $log _e 2$)

$A$ radioactive sample decays at a rate of $5000$ disintegrations/minute at a certain instant. After $5$ minutes, the rate becomes $1250$ disintegrations/minute. The decay constant is ....... (in $\ln 2$)

The half-life of a radioactive isotope $X$ is $20$ years. It decays to another element $Y$,which is stable. The two elements $X$ and $Y$ were found to be in the ratio $1:7$ in a sample of a given rock. The age of the rock is estimated to be:

$A$ radioactive substance has a half-life of $3.8 \, days$. If the initial mass is $10.38 \, g$, how much mass (in grams) will remain after $19 \, days$?

Two radioactive materials $X_1$ and $X_2$ contain the same number of nuclei. If $6\lambda \, s^{-1}$ and $4\lambda \, s^{-1}$ are the decay constants of $X_1$ and $X_2$ respectively,the ratio of the number of undecayed nuclei of $X_1$ to that of $X_2$ will be $\left( \frac{1}{e} \right)$ after a time:

$A$ source contains two phosphorus radionuclides $_{15}^{32} P \left(T_{1/2} = 14.3 \ d\right)$ and $_{15}^{33} P \left(T_{1/2} = 25.3 \ d\right)$. Initially,$10\%$ of the decays come from $_{15}^{33} P$. How long must one wait until $90\%$ of the decays come from $_{15}^{33} P$?