Two radioactive nuclei $A$ and $B$ both convert into a stable nucleus $C$. At time $t = 0$ nuclei of $A$ are $4N_0$ and that of $B$ are $N_0$. Half life of $A$ is $1\, min$ and that of $B$ is $2\, min$. initially number of nuclei of $C$ are zero. At what time rate of disintegrations of $A$ and $B$ are equal .......... $min$

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 half-life period of a radio-active element $X$ is same as the mean life time of another radio-active element $Y$ Initially they have the same number of atoms. Then

The half life period of radium is $1600$ years. The fraction of a sample of radium that would remain after $6400$ years is

$3.8$ days is the half-life period of a sample. After how many days, the sample will become $\frac{{1}}{{8}} \, th$ of the original substance

Two radioactive nuclei $P$ and $Q,$ in a given sample decay into a stable nucleus $R.$ At time $t = 0,$ number of $P$ species are $4\,\, N_0$ and that of $Q$ are $N_0$. Half-life of $P$ (for conversion to $R$) is $1$ minute where as that of $Q$ is $2$ minutes. Initially there are no nuclei of $R$ present in the sample. When number of nuclei of $P$ and $Q$ are equal, the number of nuclei of $R$ present in the sample would be