Embibe Experts Solutions for Chapter: Atoms and Nuclei, Exercise 1: JEE Advanced Paper 1 - 2015

A heavy nucleus $N$, at rest, undergoes fission $N\to P+Q,$ where $P$ and $Q$ are two lighter nuclei. Let $\delta =M_N-M_P-M_Q$, where $M_P, M_Q$ and $M_N$ are the masses of $P,Q$ and $N$, respectively. $E_P$ and $E_Q$ are the kinetic energies of $P$ and $Q$, respectively. The speeds of $P$ and $Q$ are $v_P$ and $v_Q$, respectively. If $c$ is the speed of light, which of the following statement(s) is (are) correct?

In an $X$-ray tube, electrons emitted from a filament (cathode) carrying current $I$ hit a target (anode) at a distance $d$ from the cathode. The target is kept at a potential $V$ higher than the cathode resulting in emission of continuous and characteristic X-rays. If the filament current $I$ is decreased to $\frac{I}{2},$ the potential difference $V$ is increased to $2V,$ and the separation distance $d$ is reduced to $\frac{d}{2},$ then

A free hydrogen atom after absorbing a photon of wavelength ${\mathrm{\lambda }}_{\mathrm{a}}$ gets excited from the state $\mathrm{n}=1$ to the state $\mathrm{n}=4.$ Immediately after that the electron jumps to $\mathrm{n}=\mathrm{m}$ state by emitting a photon of wavelength ${\mathrm{\lambda }}_{\mathrm{e}}.$ Let the change in momentum of atom due to the absorption and the emission are $\mathrm{\Delta }{\mathrm{p}}_{\mathrm{a}}$ and $\mathrm{\Delta }{\mathrm{p}}_{\mathrm{e}},$ respectively. If $\frac{{\mathrm{\lambda }}_{\mathrm{a}}}{{\lambda }_e}=\frac{1}{5},$ which of the option(s) is/are correct?
[Use $\mathrm{h}\mathrm{c}=1242\mathrm{e}\mathrm{V}\mathrm{n}\mathrm{m},1\mathrm{n}\mathrm{m}={10}^{-9}\mathrm{m},$ $\mathrm{h}$ and $\mathrm{c}$ are Planck's constant and speed of light, respectively]

Suppose a ${}_{88}{}^{226}Ra$ nucleus at rest and in ground state undergoes $\mathrm{\alpha }$ -decay to a ${}_{86}{}^{222}Rn$ nucleus in its excited state. The kinetic energy of the emitted $\mathrm{\alpha }$ particle is found to be $4.44\mathrm{ }\mathrm{M}\mathrm{e}\mathrm{V}. {}_{86}{}^{222}Rn$ nucleus then goes to its ground state by $\mathrm{\gamma }$ -decay. The energy of the emitted $\mathrm{\gamma }$ -photon is _______ $\mathrm{k}\mathrm{e}\mathrm{V},$
[Given: atomic mass of ${\mathrm{ }}_{88}^{226}\mathrm{R}\mathrm{a}=226.005\mathrm{u},$ atomic mass of ${\mathrm{ }}_{86}^{222}\mathrm{R}\mathrm{n}=222.000\mathrm{u},$ atomic mass of $\mathrm{\alpha }$ particle $=4.000\mathrm{u},1\mathrm{u}=931\mathrm{M}\mathrm{e}\mathrm{V}/{\mathrm{c}}^2, \mathrm{c}$ is speed of the light]

A heavy nucleus $Q$ of half-life $20$ minutes undergoes alpha-decay with probability of $60\%$  and beta-decay with probability of $40\%$. Initially, the number of $Q$ nuclei is $1000$. The number of alpha-decays of $Q$ in the first one hour is:

Which of the following statement(s) is (are) correct about the spectrum of hydrogen atom?

A particle of mass $m$ moves in circular orbits with potential energy $U\left(r\right)=Fr$, where $F$ is a positive constant and $r$ is its distance from the origin. Its energies are calculated using the Bohr model. If the radius of the particle's orbit is denoted by $R$ and its speed and energy are denoted by $v$ and $E$ respectively, then for the $n^{\mathrm{th}}$ orbit (here $h$ is the Planck's constant)

In a radioactive sample, ${\mathrm{ }}_{19}^{40}\mathrm{K}$ nuclei either decay into stable ${\mathrm{ }}_{20}^{40}\mathrm{C}\mathrm{a}$ nuclei with decay constant $4.5\times {10}^{-10}$ per year or into stable ${\mathrm{ }}_{18}^{40}\mathrm{A}\mathrm{r}$ nuclei with decay constant $0.5\times {10}^{-10}$ per year. Given that in this sample all the stable ${\mathrm{ }}_{20}^{40}\mathrm{C}\mathrm{a}$ and ${\mathrm{ }}_{18}^{40}\mathrm{A}\mathrm{r}$ nuclei are produced by the ${\mathrm{ }}_{19}^{40}\mathrm{K}$ nuclei only. In time $\mathrm{t}\times {10}^9$ years, if the ratio of the sum of stable ${\mathrm{ }}_{20}^{40}\mathrm{C}\mathrm{a}$ and ${\mathrm{ }}_{18}^{40}\mathrm{A}\mathrm{r}$ nuclei to the radioactive ${\mathrm{ }}_{19}^{40}\mathrm{K}$ nuclei is $99,$ the value of t will be : [Given $\mathrm{l}\mathrm{n}\mathrm{ }10=2.3$ ]