A nucleus X undergoes following transformation `X overset(alpha)toY`
`Y underset(2beta)toZ` , then

Options

A. X and Y are isotopes
B. X and Z are isobars
C. X and Y are isobars
D. X and Z are isotopes

Answer

To solve the problem, we need to analyze the transformations that the nucleus X undergoes and determine the relationships between the nuclei X, Y, and Z.

### Step-by-Step Solution:

1. **Understanding the Transformation**:
- The nucleus X undergoes two transformations:
1. **Alpha Decay**: \( X \xrightarrow{\alpha} Y \)
2. **Beta Decay**: \( Y \xrightarrow{2\beta} Z \)

2. **Effects of Alpha Decay**:
- An alpha particle consists of 2 protons and 2 neutrons, which means it has a mass number of 4 and an atomic number of 2.
- When nucleus X emits an alpha particle, its atomic mass decreases by 4 and its atomic number decreases by 2.
- If we denote the atomic number of X as \( Z \) and its mass number as \( A \):
- After alpha decay, the atomic number of Y becomes \( Z - 2 \) and the mass number becomes \( A - 4 \).

3. **Effects of Beta Decay**:
- A beta particle is essentially an electron, and when a beta particle is emitted, a neutron in the nucleus is converted into a proton.
- This means that the atomic number increases by 1, while the mass number remains unchanged.
- Since Y undergoes 2 beta emissions:
- The atomic number of Z becomes \( (Z - 2) + 2 = Z \) (it increases by 2).
- The mass number remains \( A - 4 \).

4. **Analyzing Relationships**:
- Now we have:
- For nucleus Y: Atomic number = \( Z - 2 \), Mass number = \( A - 4 \)
- For nucleus Z: Atomic number = \( Z \), Mass number = \( A - 4 \)
- Since the mass number of Y and Z is the same (\( A - 4 \)), they are **isobars**.
- Since the atomic number of Z is \( Z \) and the atomic number of X is \( Z \), X and Z have the same atomic number, which means they are **isotopes**.

5. **Conclusion**:
- The relationships are:
- X and Y are **not** isobars (different mass numbers).
- X and Z are **isotopes** (same atomic number).
- X and Y are **not** isotopes (different atomic numbers).
- X and Y are **isobars** (same mass number).
- Therefore, the correct answer is that X and Z are isotopes.

### Final Answer:
X and Z are isotopes.