$A$ stationary nucleus (mass number = $A$) emits an $\alpha$-particle with velocity $v$. Find the velocity of the daughter nucleus.

Consider the following statements:
$A.$ Atoms of each element emit a characteristic spectrum.
$B.$ According to Bohr's postulate, an electron in a hydrogen atom revolves in a certain stationary orbit.
$C.$ The density of nuclear matter depends on the size of the nucleus.
$D.$ A free neutron is stable, but a free proton decay is possible.
$E.$ Radioactivity is an indication of the instability of nuclei.
Choose the correct answer from the options given below:

Both the nucleus and the atom of some element are in their respective first excited states. They get de-excited by emitting photons of wavelengths $\lambda_N$ and $\lambda_A$ respectively. The ratio $\frac{\lambda_N}{\lambda_A}$ is closest to

Which of the alternatives gives the correct match of Column-$I$ with Column-$II$?

Column-$I$	Column-$II$
$a.$ Binding energy per nucleon for ${ }^{56} Fe$	$(i)$ $5.5 \,MeV$
$b.$ Energy of $\alpha$-particle in Geiger-Marsden experiment	$(ii)$ $200 \,MeV$
$c.$ Energy of photon of visible light	$(iii)$ $8.75 \,MeV$
$d.$ Energy released in fission of a uranium nucleus	$(iv)$ $2 \,eV$

From the statements given below:
$(A)$ The angular momentum of an electron in $n^{\text{th}}$ orbit is an integral multiple of $\hbar$.
$(B)$ Nuclear forces do not obey inverse square law.
$(C)$ Nuclear forces are spin dependent.
$(D)$ Nuclear forces are central and charge independent.
$(E)$ Stability of nucleus is inversely proportional to the value of packing fraction.
Choose the correct answer from the options given below:

$A$ radioactive nucleus emits an $\alpha$-particle to become a stable nucleus. If the velocity of the $\alpha$-particle is $\upsilon$ and the mass number of the original radioactive nucleus is $A$,what will be the velocity of the daughter nucleus?