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

During the disintegration of a radioactive nucleus of mass number $208$ at rest, two alpha particles each with kinetic energy $E$ are emitted. The total kinetic energy of the emitted alpha particles and the daughter nucleus after the disintegration is

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

Match List $I$ of the nuclear processes with List $II$ containing parent nucleus and one of the end products of each process and then select the correct answer using the codes given below the lists:

List $I$	List $II$
$P$. Alpha decay	$1$. ${ }_{8}^{15} O \rightarrow{ }_{7}^{15} N + \dots$
$Q$. $\beta^{+}$ decay	$2$. ${ }_{92}^{238} U \rightarrow{ }_{90}^{234} Th + \dots$
$R$. Fission	$3$. ${ }_{83}^{185} Bi \rightarrow{ }_{82}^{184} Pb + \dots$
$S$. Proton emission	$4$. ${ }_{94}^{239} Pu \rightarrow{ }_{57}^{140} La + \dots$

Codes: $P \quad Q \quad R \quad S$

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:

The graph between the number of decayed atoms $N'$ of a radioactive element and time $t$ is: