A radioactive nucleus can decay by two different processes. Half-life for the first process is $3.0\, hours$ while it is $4.5\, hours$ for the second process. The effective half- life of the nucleus will be $.........\,hours.$

From a newly formed radioactive substance (Half life $2$ hours), the intensity of radiation is $64$ times the permissible safe level. The minimum time after which work can be done safely from this source is ..........$hours$

An archaeologist analyses the wood in a prehistoric structure and finds that $C^{14}$ (Half life $= 5700\, years$) to $C^{12}$ is only one-fourth of that found in the cells of buried plants. The age of the wood is about ..........$years$

Some nuclei of a radioactive material are undergoing radioactive decay. The time gap between the instances when a quarter of the nuclei have decayed and when half of the nuclei have decayed is given as:

(where $\lambda$ is the decay constant)

A radioactive sample with a half life of $1$ month has the label : “Activity$=2\, micro\,\,curies$ on $1-8-1991$.'' What will be its activity two months earlier ............ $micro\,\, curies$.

The activity of a radioactive sample is measured as $N_0$ counts per minute at $t = 0$ and $N_0/e$ counts per minute at $t = 5$ minutes. The time (in minutes) at which the activity reduces to half its value is