For a radioactive material,the half-life is $10$ minutes. If initially there are $600$ nuclei,the time taken (in minutes) for the disintegration of $450$ nuclei is:

$A$ 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 the 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; $\log_e 2 = 0.693$]

If there is $0.1 \ mg$ of radioactive $Th^{234}$, how much of it will remain undecayed after $120 \ \text{days}$, given its half-life is $24 \ \text{days}$? (Answer in $\mu g$)

$A$ sample of radioactive material has mass $m$,decay constant $\lambda$,and molar mass $M$. If the Avogadro constant is $N_A$,what is the initial activity of the sample?

$A$ 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 how many seconds (in $, s$)?

The half-life of a radioactive substance is $ 20 \text{ minutes} $. The time taken between $ 50\% $ decay and $ 87.5\% $ decay of the substance will be