In the radioactive decay of `._(Z)X^(A)`, which of the following could be considered as incorrect statement?

Options

A. `alpha`-decay involves the decrease of both A and Z by 2
B. `beta`-dacay involves the increase of Z by one, A remaining constant
C. K-electron capture results in the decrease of Z by one with no change in A and emission of `gamma`- rays
D. `gamma`- ray emission is followed by the emission of `alpha` or `beta` - particles

Answer

To solve the question regarding the incorrect statement in the radioactive decay of `._(Z)X^(A)`, we will analyze each option provided in the context of radioactive decay processes: alpha decay, beta decay, electron capture, and gamma emission.

### Step-by-Step Solution:

1. **Understanding Radioactive Decay:**
- Radioactive decay involves the transformation of an unstable nucleus into a more stable one, which can involve the emission of particles (alpha, beta, gamma) and changes in atomic mass (A) and atomic number (Z).

2. **Analyzing Each Option:**
- **Option 1:** Alpha decay involves the decrease of both A and Z by 2.
- In alpha decay, an alpha particle (which is a helium nucleus, `^4_2He`) is emitted. This results in:
- A decreases by 4 (since the mass number of the alpha particle is 4).
- Z decreases by 2 (since the atomic number of the alpha particle is 2).
- Therefore, the statement that both A and Z decrease by 2 is **incorrect**.

- **Option 2:** Beta decay involves the increase of Z by 1.
- In beta decay, a neutron is converted into a proton, and an electron (beta particle) is emitted. This results in:
- A remains unchanged (since the total number of nucleons remains the same).
- Z increases by 1 (since a neutron is converted into a proton).
- This statement is **correct**.

- **Option 3:** Electron capture involves the decrease of Z by 1.
- In electron capture, an electron is captured by a proton, converting it into a neutron. This results in:
- A remains unchanged.
- Z decreases by 1.
- This statement is **correct**.

- **Option 4:** Gamma ray emission is followed by alpha and beta particles.
- Gamma emission usually occurs after alpha or beta decay to release excess energy from the nucleus. It does not change A or Z.
- This statement is **correct**.

3. **Conclusion:**
- The only incorrect statement is from **Option 1**, where it incorrectly states that both A and Z decrease by 2 in alpha decay.

### Final Answer:
The incorrect statement is **Option 1**: "Alpha decay involves the decrease of both A and Z by 2."

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