The half-lives of two radioactive materials $A$ and $B$ are $T$ and $2T$ respectively. If the ratio of the initial masses of the materials $A$ and $B$ is $8:1$, then the time after which the ratio of the masses of the materials $A$ and $B$ becomes $4:1$ is

Two radioactive materials $M_1$ and $M_2$ have decay constants $9 \lambda$ and $\lambda$ respectively. Initially,they have the same number of nuclei. The ratio of the number of nuclei of $M_1$ to that of $M_2$ will be $\frac{1}{e}$ after a time interval of:

$A$ radioactive sample with a half-life of $1$ month has the label: "Activity $= 2 \, \mu Ci$ on $1-8-1991$." What will be its activity two months earlier in $\mu Ci$?

Following statements related to radioactivity are given below:
$(A)$ Radioactivity is a random and spontaneous process and is dependent on physical and chemical conditions.
$(B)$ The number of undecayed nuclei in the radioactive sample decays exponentially with time.
$(C)$ Slope of the graph of $\log_{e}$ (no. of undecayed nuclei) $Vs.$ time represents the negative reciprocal of mean life time $(-\frac{1}{\tau})$.
$(D)$ Product of decay constant $(\lambda)$ and half-life time $(T_{1/2})$ is constant,equal to $\ln(2)$.
Choose the most appropriate answer from the options given below.

Two radioactive materials $A_1$ and $A_2$ have decay constants of $10 \lambda_0$ and $\lambda_0$. If initially they have the same number of nuclei,the ratio of the number of their undecayed nuclei will be $(1/e)$ after a time $t$. Find $t$.

$A$ 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: (in $s$)