In the uranium radioactive series,the initial nucleus is $_{92}U^{238}$ and the final nucleus is $_{82}Pb^{206}$. When the uranium nucleus decays to lead,the number of $\alpha$-particles emitted is $x$ and the number of $\beta$-particles emitted is $y$. Find $x$ and $y$.

Consider a radioactive nucleus $A$ which decays to a stable nucleus $C$ through the following sequence: $A \to B \to C$. Here $B$ is an intermediate nucleus which is also radioactive. Considering that there are $N_0$ atoms of $A$ initially,plot the graph showing the variation of the number of atoms of $A$ and $B$ versus time.

$A$ nucleus $_Z^AX$ emits an $\alpha$-particle. The resultant nucleus then emits a ${\beta ^ + }$ particle. What will be the respective atomic number and mass number of the final nucleus?

$\gamma$-decay occurs when

Assertion : The ionising power of $\beta$-particle is less compared to $\alpha$-particles but their penetrating power is more.
Reason : The mass of $\beta$-particle is less than the mass of $\alpha$-particles.

Explain $\beta$-decay and how a radioactive nucleus emits $\beta$-particles,even if there are no $\beta$-particles in the nucleus. Why does the mass number of the radioactive nuclide not change during $\beta$-emission?