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

  • A

    $1.5$

  • B

    $1$

  • C

    $0.5$

  • D

    $2$

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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 $......$

  • [JEE MAIN 2022]

A radioactive sample consists of two distinct species having equal number of atoms $N_0$ initially. The mean-life of one species is $\tau $ and of the other is $5\tau $. The decay products in both cases is stable. The total number of radioactive nuclei at $t = 5\tau $ is

The graph between number of decayed atoms $N'$ of a radioactive element and time $t$ is