$A$ radioactive element has a rate of disintegration of $9000$ disintegrations per minute at a particular instant. After $2$ minutes, it becomes $3000$ disintegrations per minute. The decay constant per minute is:

The graph between the instantaneous concentration $(N)$ of a radioactive element and time $(t)$ is:

$A$ radioactive substance decays to $\left(\frac{1}{16}\right)^{th}$ of its initial activity in $80\, days$. The half-life of the radioactive substance expressed in days is ... .

$A$ nucleus has a half-life of $30 \; min$. At $3 \; PM$, its decay rate was measured as $120000 \; cps$. What is the decay rate in $cps$ at $5 \; PM$?

The normal activity of living carbon-containing matter is found to be about $15$ decays per minute for every gram of carbon. This activity arises from the small proportion of radioactive $_{6}^{14}C$ present with the stable carbon isotope $_{6}^{12}C$. When the organism is dead,its interaction with the atmosphere (which maintains the above equilibrium activity) ceases and its activity begins to drop. From the known half-life ($5730$ years) of $_{6}^{14}C$,and the measured activity,the age of the specimen can be approximately estimated. This is the principle of $_{6}^{14}C$ dating used in archaeology. Suppose a specimen from Mohenjodaro gives an activity of $9$ decays per minute per gram of carbon. Estimate the approximate age (in years) of the Indus-Valley civilisation.

What mass in $\mu g$ of $Th^{227}$ is required to produce an activity of $1 \ mCi$ if its half-life is $1.9 \ \text{years}$?