An element $A$ decays into element $C$ by a two-step process:
$A \to B + {\;_2}He^4$
$B \to C + 2e^-$
Then:

$A$ radioactive nucleus is initially at rest. It decays by emitting an electron and a neutrino at right angles to each other. The momentum of the electron is $3.2 \times 10^{-23} \ kg \cdot m/s$ and the momentum of the neutrino is $6.4 \times 10^{-23} \ kg \cdot m/s$. What is the direction of the recoil of the nucleus with respect to the direction of the electron?

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.

An element $X$ decays into element $Z$ by a two-step process:
$X \rightarrow Y + 4e$
$Y \rightarrow Z + 2e^{-}$
Then:

For the radioactive nuclei that undergo either $\alpha$ or $\beta$ decay, which one of the following cannot occur?

Which of the following rays are not electromagnetic waves?