A radioactive substance is being produced at a constant rate of $10\, nuclei/s.$ The decay constant of the substance is $1/2\, sec^{-1}.$ After what time the number of radioactive nuclei will become $10$ $?$ Initially there are no nuclei present. Assume decay law holds for the sample.

Two radioactive substances $X$ and $Y$ originally have $N _{1}$ and $N _{2}$ nuclei respectively. Half life of $X$ is half of the half life of $Y$. After three half lives of $Y$, number of nuclei of both are equal. The ratio $\frac{ N _{1}}{ N _{2}}$ will be equal to

A sample initially contains only $U -238$ isotope of uranium. With time, some of the $U -238$ radioactively decays into $Pb -206$ while the rest of it remains undisintegrated.

When the age of the sample is $P \times 10^8$ years, the ratio of mass of $Pb -206$ to that of $U -238$ in the sample is found to be $7$ . The value of $P$ is. . . . . .

[Given : Half-life of $U-238$ is $4.5 \times 10^9$ years; $\log _e 2=0.693$ ]

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

A radioactive sample $\mathrm{S} 1$ having an activity $5 \mu \mathrm{Ci}$ has twice the number of nuclei as another sample $\mathrm{S} 2$ which has an activity of $10 \mu \mathrm{Ci}$. The half lives of $\mathrm{S} 1$ and $\mathrm{S} 2$ can be

Half-life is measured by