The half-life of radium is about $1600$ years. Of $100 \, g$ of radium existing now, $25 \, g$ will remain unchanged after .......... $years$.

Number of nuclei of a radioactive substance at time $t = 0$ are $1000$ and $900$ at time $t = 2 \, s$. Then number of nuclei at time $t = 4 \, s$ will be

$A$ radioactive substance has decay constants $\lambda_{\alpha}$ and $\lambda_{\beta}$ for $\alpha$ and $\beta$ emission,respectively. If the substance emits both $\alpha$ and $\beta$ particles,find the effective half-life of the substance.

At time $t = 0$,$N_1$ nuclei of decay constant $\lambda_1$ and $N_2$ nuclei of decay constant $\lambda_2$ are mixed. The decay rate of the mixture is:

$A$ radioactive sample at any instant has its disintegration rate $5000$ disintegrations per minute. After $5$ minutes,the rate is $1250$ disintegrations per minute. Then,the decay constant (per minute) is (in $, \ln 2$)

The half-life of a radioactive element is $10 \, days$. The time required for the mass of the sample to reduce to $\frac{1}{10}$ of its initial mass is approximately ........ days.