$A$ mixture consists of two radioactive materials $A_1$ and $A_2$ with half-lives of $20 \ s$ and $10 \ s$ respectively. Initially,the mixture has $40 \ g$ of $A_1$ and $160 \ g$ of $A_2$. The amount of the two in the mixture will become equal after: (in $s$)

The rate of disintegration was observed to be $10^{17}$ disintegrations per second when its half-life period is $1445$ years. The original number of particles is:

$A$ nuclear power plant supplying electrical power to a village uses a radioactive material of half-life $T$ years as the fuel. The amount of fuel at the beginning is such that the total power requirement of the village is $12.5 \%$ of the electrical power available from the plant at that time. If the plant is able to meet the total power needs of the village for a maximum period of $n T$ years,then the value of $n$ is

$A$ piece of wood from a recently cut tree shows $20$ decays per minute. $A$ wooden piece of the same size placed in a museum (obtained from a tree cut many years back) shows $2$ decays per minute. If the half-life of $C^{14}$ is $5730$ years,then the age of the wooden piece placed in the museum is approximately ........... years.

After $3$ hours,only $0.25 \,mg$ of a pure radioactive material is left. If the initial mass was $2 \,mg$,then the half-life of the substance is ...... $hr$.

The activity of a radioactive sample is $6000 \, dps$ after $280$ days. After another $140$ days,the activity decreases to $3000 \, dps$. The initial activity of the radioactive sample (in $dps$) is ........