Two radioactive nuclei $A$ and $B$ both convert into a stable nucleus $C$. At time $t = 0$,the number of nuclei of $A$ is $4N_0$ and that of $B$ is $N_0$. The half-life of $A$ is $1 \, min$ and that of $B$ is $2 \, min$. Initially,the number of nuclei of $C$ is zero. At what time are the rates of disintegration of $A$ and $B$ equal?

$A$ radioactive element initially has $4 \times 10^{16}$ active nuclei. If its half-life is $10 \ days$,find the number of nuclei that have decayed in $30 \ days$.

If $10\%$ of a radioactive material decays in $5\, days$,then the amount of the original material left after $20\, days$ is approximately .......... $\%$

$A$ 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).

For a radioactive element,the mean life is $\tau$. At time $t = 0$,the number of nuclei decaying per unit time is $n$. The number of nuclei that have decayed between time $0$ and $t$ is:

$A$ radioactive sample disintegrates via two independent decay processes having half-lives $T_{1/2}^{(1)}$ and $T_{1/2}^{(2)}$ respectively. The effective half-life $T_{1/2}$ of the nuclei is