Question

An element $\text{X}$ disintegrates by $\alpha$-emission and the new element further disintegrates twice by $\beta$-emission and once by $\gamma$-emission. If $\text{Y}$ is the final element formed, then:

Answer 1

In an alpha emission, the atomic number of the parent nucleus decreases by $2$, while the mass number decreases by $4$. And in a beta emission, the atomic number of the parent nucleus increases by 1, while the mass number remains the same. There is no change in the atomic and mass numbers, in a gamma emission.
According to the given information, the element $\text{Y}$ is formed from $\text{X}$ as follows:

$AZX\stackrel{\alpha }{\to }A-4Z-2X1$

$A-4Z-2X1\stackrel{\beta }{\to }A-4Z-2+1X2=A-4Z-1X2$

$A-4Z-1X2\stackrel{\beta }{\to }{A-4Z-1+1X\ast }_3^{\ast }={A-4ZX\ast }_3^{\ast }$

${A-4ZX\ast }_3^{\ast }\stackrel{\gamma }{\to }A-4ZX3=Y$

Therefore, the atomic number of $\text{Y}=$ atomic number of $\text{X}=\text{Z}$, but their mass numbers are different ($\text{A - 4}\ne \text{A}$). So, $\text{X}$ and $\text{Y}$ are isotopes.