A nucleus `._Z^AX ` emits an `alpha`-particel. The resultant nucleus emits a `beta^(+)` particle. The respective atomic and mass numbers of the final nucleus will be

Options

A. `Z-3, A-4`
B. `Z-1,A-4`
C. `Z -2 , A-4`
D. `Z, A-2`

Explanation

To solve the question step by step, we will analyze the emissions of the alpha and beta particles from the nucleus \( _Z^A X \).

### Step 1: Emission of Alpha Particle
1. **Initial Nucleus**: The nucleus is represented as \( _Z^A X \), where \( A \) is the mass number and \( Z \) is the atomic number.
2. **Alpha Particle Emission**: When an alpha particle (which consists of 2 protons and 2 neutrons) is emitted, the mass number of the nucleus decreases by 4 and the atomic number decreases by 2.
- **New Mass Number**: \( A - 4 \)
- **New Atomic Number**: \( Z - 2 \)
3. **Resultant Nucleus**: After the emission of the alpha particle, the new nucleus can be denoted as \( _{Z-2}^{A-4} Y \).

### Step 2: Emission of Beta Plus Particle
1. **Beta Plus Particle Emission**: A beta plus particle (or positron) emission involves the conversion of a proton into a neutron, which results in the atomic number decreasing by 1 while the mass number remains unchanged.
- **Mass Number**: Remains \( A - 4 \)
- **Atomic Number**: Decreases from \( Z - 2 \) to \( Z - 3 \)
2. **Final Nucleus**: After the emission of the beta plus particle, the final nucleus can be denoted as \( _{Z-3}^{A-4} Z \).

### Conclusion
- The final atomic number of the resultant nucleus is \( Z - 3 \).
- The final mass number of the resultant nucleus is \( A - 4 \).

Thus, the respective atomic and mass numbers of the final nucleus are:
- **Atomic Number**: \( Z - 3 \)
- **Mass Number**: \( A - 4 \)

### Final Answer
- Atomic Number: \( Z - 3 \)
- Mass Number: \( A - 4 \)

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