Half-lives of two radioactive substances $A$ and $B$ are $20 \text{ minutes}$ and $40 \text{ minutes}$ respectively. Initially,the samples of $A$ and $B$ have an equal number of nuclei. After $80 \text{ minutes}$,the ratio of the remaining number of $A$ and $B$ nuclei is:

$A$ radioactive substance contains $10000$ nuclei and its half-life period is $20$ days. The number of nuclei present at the end of $10$ days is

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.

The half-life of ${}_{92}^{238}U$ against $\alpha$-decay is $13.86 \times 10^{16} \,s$. The activity of a $1 \,g$ sample of ${}_{92}^{238}U$ is:

The activity of a sample of radioactive material is $A_1$ at time $t_1$ and $A_2$ at time $t_2$ $(t_2 > t_1)$. If its mean life is $T$,then which of the following is true?

The decay constant of a radioactive sample is $\lambda$. The probability of decay per unit time is: