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

A plot of the number of neutrons $(N)$ against the number of protons ( $P$ )of stable nuclei exhibits upward deviation from linearity for atomic number, $Z>20$. For an unstable nucleus having $N / P$ ratio less than $1$ , the possible mode($s$) of decay is(are)

($A$) $\beta^{-}$-decay ( $\beta$ emission)

($B$) orbital or $K$-electron sasture

($C$) neutron emission

($D$) $\beta^{+}$-decay (positron emission)

$\gamma$-decay occurs when

Given the masses of various atomic particles $m _{ p }=1.0072 u , m _{ n }=1.0087 u$ $m _{ e }=0.000548 u , m _{ v }=0, m _{ d }=2.0141 u$ where $p \equiv$ proton, $n \equiv$ neutron, $e \equiv$ electron, $\overline{ v } \equiv$ antineutrino and $d \equiv$ deuteron. Which of the following process is allowed by momentum and energy conservation $?$

Suppose a ${ }_{88}^{226} Ra$ nucleus at rest and in ground state undergoes $\alpha$-decay to a ${ }_{56}^{22} Rn$ nucleus in its excited state. The kinetic energy of the emitted $\alpha$ particle is found to be $4.44 MeV$. ${ }_{86}^{22} Rn$ nucleus then goes to its ground state by $\gamma$-decay. The energy of the emitted $\gamma$-photon is. . . . . . . .$keV$,

[Given: atomic mass of ${ }_{ gs }^{226} Ra =226.005 u$, atomic mass of ${ }_{56}^{22} Rn =222.000 u$, atomic mass of $\alpha$ particle $=4.000 u , 1 u =931 MeV / c ^2, c$ is speed of the light $]$

Pauli suggested the emission of nutrino during $\beta^{+}$decay to explain