Half-life of a substance is $10$ years. In what time, it becomes $\frac{1}{4}\,th$ part of the initial amount ……..$years$

Two radioactive materials $A$ and $B$ have decay constants $25 \lambda$ and $16 \lambda$ respectively. If initially they have the same number of nuclei, then the ratio of the number of nuclei of $B$ to that of $A$ will be "$e$" after a time $\frac{1}{a \lambda}$. The value of $a$ is $……$

Half-life is measured by

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; $\log _e 2=0.693$ ]

A certain radioactive material can undergo three different types of decay, each with a different decay constant $\lambda_1$, $\lambda_2$ and $\lambda_3$ . Then the effective decay constant is