$A$ freshly prepared radioactive sample of half-life $1$ hour emits radiations that are $128$ times as intense as the permissible safe limit. The minimum time after which this sample can be safely used is ......... $hours$.

An alloy is composed of two radioactive materials $A$ and $B$ having equal weight. The half-lives of $A$ and $B$ are $10 \ yrs$ and $20 \ yrs$ respectively. After time $t$, the alloy was found to consist of $(1/e) \ kg$ of $A$ and $1 \ kg$ of $B$. If the atomic weights of $A$ and $B$ are the same, then the value of $t$ is (Assume $\ln 2 = 0.7$):

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:

The half-life of the isotope $^{11}Na^{24}$ is $15 \text{ hr}$. How much time does it take for $(7/8)$ of a sample of this isotope to decay (in $\text{ hour}$)?

For a substance,the fraction of its initial quantity $(N_0)$ which will disintegrate in its average lifetime is about $(e = 2.71)$.

$A$ solution containing active cobalt ${}_{27}^{60}Co$ having activity of $0.8\,\mu Ci$ and decay constant $\lambda$ is injected into an animal's body. If $1\,cm^3$ of blood is drawn from the animal's body after $10\,hrs$ of injection,the activity found is $300\,decays$ per minute. What is the total volume of blood in the animal's body in litres? (Given: $1\,Ci = 3.7 \times 10^{10}$ decays per second and at $t = 10\,hrs$,$e^{-\lambda t} = 0.84$)