$A$ nuclear reaction given by $_Z{X^A} \to {_{Z+1}}{Y^A} + _{-1}{e^0} + \bar{\nu}$ represents:

$A$ free neutron decays spontaneously into:

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.

Give a brief explanation about radioactivity.

$A$ radioactive nucleus (initial mass number $A$ and atomic number $Z$) emits $3$ $\alpha$-particles and $2$ positrons. The ratio of the number of neutrons to that of protons in the final nucleus will be:

$A$ nucleus $_n^mX$ emits one $\alpha$ particle and two $\beta^-$ particles. The resulting nucleus is