The graph in the figure shows how the count-rate $A$ of a radioactive source as measured by a Geiger counter varies with time $t$. The relationship between $A$ and $t$ is (Assume $\ln 12 = 2.6$):

$A$ piece of wood from the ruins of an ancient building was found to have a $^{14}C$ activity of $12$ disintegrations per minute per gram of its carbon content. The $^{14}C$ activity of the living wood is $16$ disintegrations per minute per gram. How long ago did the tree,from which the wooden sample came,die? Given half-life of $^{14}C$ is $5760$ years.

The decay constant of a radioactive sample is $\lambda$. The probability of decay per unit time is:

$A$ radioactive sample decays at a rate of $5000$ disintegrations/minute at a certain instant. After $5$ minutes, the rate becomes $1250$ disintegrations/minute. The decay constant is ....... (in $\ln 2$)

$A$ radioactive element having a half-life of $30 \ min$ is undergoing beta decay. The fraction of the radioactive element that remains undecayed after $90 \ min$ will be:

Consider a radioactive nuclide which follows the decay rate given by $A(t) = A_0 2^{-(t/t_0)}$, where $A(t)$ is the fraction of radioactive material remaining after time $t$ from the initial $A_0$ at zero time. Let $A_1$ be the fraction of original activity which remains after $120 \ h$. Likewise, $A_2$ is the fraction of original activity remaining after $200 \ h$. If $A_1/A_2 = 16$, then the half-life $(t_0)$ will be: (in $h$)