The energy spectrum of $\beta$-particles [number $N(E)$ as a function of $\beta$-energy $E$] emitted from a radioactive source is:

$A$ nucleus of lead $Pb_{82}^{214}$ emits two electrons followed by an $\alpha$-particle. The resulting nucleus will have

$A$ radioactive nuclide of element $X$ decays to form element $Y$. The graph representing the rate of production of $Y$ in a sample of $X$ is:

Outside a nucleus,which of the following statements is true?

Statement-$1$: $A$ nucleus having energy $E_{1}$ decays by $\beta^{-}$ emission to a daughter nucleus having energy $E_{2}$,but the $\beta^{-}$ rays are emitted with a continuous energy spectrum having an end-point energy $E_{1} - E_{2}$.
Statement-$2$: To conserve energy and momentum in $\beta$ decay,at least three particles must take part in the transformation.

Consider the decay of a free neutron at rest: $n \rightarrow p + e^-$. Show that the two-body decay of this type must necessarily give an electron of fixed energy and,therefore,cannot account for the observed continuous energy distribution in the $\beta$-decay of a neutron or a nucleus.