Kumar Mittal Solutions for Exercise 2: For DIFFERENT COMPETITIVE EXAMINATIONS

Two radioactive materials $A$ and $B$ have decay constants $10 \lambda$ and $\lambda$ respectively. If initially, they have the same number of nuclei, then the ratio of the number of nuclei of $A$ to that of $B$ will be $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$e$}\right.$ after a time:

Two radioactive substances $A$ and $B$ have decay constants $5\lambda$ and $\lambda$ respectively. At $t=0$, a sample has the same number of the two nuclei. The time taken for the ratio of the number of nuclei to become ${\left(\frac{1}{e}\right)}^2$ will be:

Half-lives of two radioactive nuclei $A$ and $B$ are $10$ minutes and $20$ minutes, respectively. If initially a sample has equal number of nuclei, then after $60$ minutes, the ratio of decayed numbers of nuclei $A$ and $B$ will be:

At a given instant, say $t=0$, two radioactive substances $A$ and $B$ have equal activities. The ratio $\frac{R_B}{R_A}$ of their activities after time $t$ itself decays with time $t$ as $e^{-3t}$. If decay constant of $A$ is $1$, the half-life of $B$ is:

Using a nuclear counter the count rate of emitted particles from a radioactive source is measured. At $t=0$ it was $1600$ counts per second and $t =8$ seconds it was $100$ counts per second. The count rate observed, as counts per second, at $t=6$ seconds is close to:

A sample of radioactive material $A$ that has an activity of $10 \mathrm{mCi}\left(1 \mathrm{Ci}=3.7\times {10}^{10}\text{ decays/s}\right)$, has twice the number of nuclei as another sample of a different radioactive material $B$ which has an activity of $20 \mathrm{mCi}$. The correct choices for half-lives of $A$ and $B$ would then be respectively:

In a radioactive decay chain, the initial nucleus is ${}_{90}{}^{232}\mathrm{Th}$. At the end there are $6 \alpha$-particles and $4 \beta$-particles which are emitted. If the end nucleus. If ${}_{\mathrm{Z}}{}^{\mathrm{A}}\mathrm{X}$, $A$ and $Z$ are given by:

In a radioactive decay chain, ${}_{90}{}^{232}\mathrm{Th}$ nucleus decays to ${}_{82}{}^{212}\mathrm{Pb}$ nucleus. Let $N_{\alpha }$ and $N_{\beta }$ be the number of $\alpha$ and $\beta$-particles, respectively, emitted in the decay process. Which of the following statement is (are) true?