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 of the nucleus X step by step.

### Step 1: Understand the first transformation (Alpha emission)
- A nucleus X undergoes alpha emission to form nucleus Y.
- In alpha emission, the mass number decreases by 4 and the atomic number decreases by 2.

Let:
- Mass number of X = A
- Atomic number of X = Z

After alpha emission:
- Mass number of Y = A - 4
- Atomic number of Y = Z - 2

### Step 2: Understand the second transformation (Beta emission)
- Nucleus Y undergoes two beta emissions to form nucleus Z.
- In beta emission, the mass number remains the same while the atomic number increases by 1 for each beta emission.

After two beta emissions:
- Mass number of Z = A - 4 (remains the same as Y)
- Atomic number of Z = (Z - 2) + 2 = Z (increases by 2)

### Step 3: Compare the nuclei
Now we have:
- Nucleus X: Mass number = A, Atomic number = Z
- Nucleus Y: Mass number = A - 4, Atomic number = Z - 2
- Nucleus Z: Mass number = A - 4, Atomic number = Z

### Step 4: Determine the relationships between the nuclei
1. **Isotopes**: Nuclei with the same atomic number but different mass numbers.
2. **Isobars**: Nuclei with the same mass number but different atomic numbers.

- Comparing X and Z:
- Mass number of X = A
- Mass number of Z = A - 4
- Atomic number of X = Z
- Atomic number of Z = Z

Since X and Z have different mass numbers, they cannot be isotopes. However, they have the same atomic number, which means they are not isobars either.

- Comparing X and Y:
- Mass number of X = A
- Mass number of Y = A - 4
- Atomic number of X = Z
- Atomic number of Y = Z - 2

Since X and Y have different atomic numbers and different mass numbers, they are neither isotopes nor isobars.

- Comparing Y and Z:
- Mass number of Y = A - 4
- Mass number of Z = A - 4
- Atomic number of Y = Z - 2
- Atomic number of Z = Z

Since Y and Z have the same mass number but different atomic numbers, they are isobars.

### Conclusion
From the analysis:
- X and Z are not isotopes or isobars.
- X and Y are not isotopes or isobars.
- Y and Z are isobars.

Thus, the correct answer is that **X and Z are isotopes**.

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