The correct option is C X and Y are isotopes.
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 Y is formed from X as follows:
AZX α→ A−4Z−2X1
A−4Z−2X1 β→ A−4Z−2+1X2 = A−4Z−1X2
A−4Z−1X2 β→ A−4Z−1+1X∗3 = A−4ZX∗3
A−4ZX∗3 γ→ A−4ZX3 = Y
Therefore, the atomic number of Y= atomic number of X=Z, but their mass numbers are different (A - 4≠A). So, X and Y are isotopes.