The radioactive nucleus $_7N^{13}$ decays to $_6C^{13}$ through the emission of

In a radioactive decay chain reaction,${ }_{90}^{230} Th$ nucleus decays into ${ }_{84}^{214} Po$ nucleus. The ratio of the number of $\alpha$ to number of $\beta^{-}$ particles emitted in this process is. . . . .

$A$ nucleus $_Z{X^A}$ emits $9 \alpha$-particles and $5 \beta$-particles. The ratio of total protons to neutrons in the final nucleus is:

$_{90}^{232}Th$,an isotope of thorium,decays in ten stages,emitting six $\alpha$-particles and four $\beta$-particles in total. The end product of the decay is:

For the $\beta^{+}$ (positron) emission from a nucleus,there is another competing process known as electron capture (an electron from an inner orbit,say,the $K$-shell,is captured by the nucleus and a neutrino is emitted).
$_{z}^{A} X + e^{-} \rightarrow _{z-1}^{A} Y + \nu$
Show that if $\beta^{+}$ emission is energetically allowed,electron capture is necessarily allowed,but not vice-versa.

Radioactivity is