$A$ certain radioactive material can undergo three different types of decay,each with a different decay constant $\lambda_1$,$\lambda_2$,and $\lambda_3$. Then the effective decay constant is:

$A$ nucleus with $Z = 92$ emits $\alpha, \alpha, \beta^-, \beta^-, \alpha, \alpha, \alpha, \alpha, \beta^-, \beta^-, \alpha, \beta^+, \beta^+, \alpha$ particles in sequence. What is the $Z$ of the resulting nucleus?

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

Explain the conservation of linear momentum for the radioactive decay of a radium nucleus.

The $\beta$-decay process,discovered around $1900$,is basically the decay of a neutron $(n)$. In the laboratory,a proton $(p)$ and an electron $(e^-)$ are observed as the decay products of the neutron. Therefore,considering the decay of a neutron as a two-body decay process,it was predicted theoretically that the kinetic energy of the electron should be a constant. But experimentally,it was observed that the electron kinetic energy has a continuous spectrum. Considering a three-body decay process,i.e.,$n \rightarrow p + e^- + \bar{\nu}_e$,around $1930$,Pauli explained the observed electron energy spectrum. Assuming the anti-neutrino $(\bar{\nu}_e)$ to be massless and possessing negligible energy,and the neutron to be at rest,momentum and energy conservation principles are applied. From this calculation,the maximum kinetic energy of the electron is $0.8 \times 10^6 \ eV$. The kinetic energy carried by the proton is only the recoil energy.
$1.$ What is the maximum energy of the anti-neutrino?
$(A)$ Zero
$(B)$ Much less than $0.8 \times 10^6 \ eV$
$(C)$ Nearly $0.8 \times 10^6 \ eV$
$(D)$ Much larger than $0.8 \times 10^6 \ eV$
$2.$ If the anti-neutrino had a mass of $3 \ eV/c^2$ (where $c$ is the speed of light) instead of zero mass,what should be the range of the kinetic energy,$K$,of the electron?
$(A)$ $0 \leq K \leq 0.8 \times 10^6 \ eV$
$(B)$ $3.0 \ eV \leq K \leq 0.8 \times 10^6 \ eV$
$(C)$ $3.0 \ eV \leq K < 0.8 \times 10^6 \ eV$
$(D)$ $0 \leq K < 0.8 \times 10^6 \ eV$
Give the answer for question $1$ and $2$.