At a given instant,say $t = 0$,two radioactive substances $A$ and $B$ have equal activities. The ratio $\frac{R_B}{R_A}$ of their activities after time $t$ decays with time $t$ as $e^{-3t}$. If the half-life of $A$ is $\ln 2$,the half-life of $B$ is:

$A$ small quantity of solution containing $Na^{24}$ radionuclide of activity $1 \, \mu Ci$ is injected into the blood of a person. $A$ sample of the blood of volume $1 \, cm^3$ taken after $5 \, hours$ shows an activity of $296$ disintegrations per minute. What will be the total volume of the blood in the body of the person? Assume that the radioactive solution mixes uniformly in the blood of the person: ............ $L$ (Take $1 \, Ci = 3.7 \times 10^{10}$ disintegrations per second and $e^{-\lambda t} = 0.7927$; where $\lambda$ is the disintegration constant).

The half-life of a radioactive nucleus is $50$ days. The time interval $(t_2 - t_1)$ between the time $t_2$ when $\frac{2}{3}$ of it had decayed and the time $t_1$ when $\frac{1}{3}$ of it had decayed is (in days):

For a substance,the average life for $\alpha$-emission is $1620 \ years$ and for $\beta$-emission is $405 \ years$. After how much time will $\frac{1}{4}$ of the material remain due to simultaneous emission?

State the relation between average life and decay constant.

The decay constants of a radioactive substance for $\alpha$ and $\beta$ emission are $\lambda_{\alpha}$ and $\lambda_{\beta}$ respectively. If the substance emits $\alpha$ and $\beta$ simultaneously,then the average half-life of the material will be: