A: A radioactive substance has half life of 4 hour. Therefore, if two nuclei of the substance are present initially , after 1 hour only one will remain undissociated.
R: When a nucleus makes a transition from excited state to ground state, it emits a `beta`-particle.

Options

A. if both assertion & reason are true and the reason is correct explanation of the assertion then mark (1)
B. if both assertion & reason are true and the reason is not correct explanation of the assertion then mark (2)
C. if assertion is true statement but reason is false then mark (3)
D. if both assertion & reason are false then mark (4)

Answer

To analyze the statements A and R, we need to evaluate each one based on the principles of radioactivity and nuclear transitions.

### Step-by-step Solution:

1. **Understanding Half-Life**:
- The half-life of a radioactive substance is the time required for half of the radioactive nuclei in a sample to decay.
- Given that the half-life of the substance is 4 hours, we can calculate how many nuclei remain after a certain period.

2. **Calculating Remaining Nuclei After 1 Hour**:
- If we start with 2 nuclei, we need to determine how many will remain after 1 hour.
- Since the half-life is 4 hours, after 1 hour (which is 1/4 of the half-life), we can use the formula for radioactive decay:
\[ N(t) = N_0 \left(\frac{1}{2}\right)^{\frac{t}{T_{1/2}}} \]
- Here, \( N_0 = 2 \) (initial nuclei), \( t = 1 \) hour, and \( T_{1/2} = 4 \) hours.
- Plugging in the values:
\[ N(1) = 2 \left(\frac{1}{2}\right)^{\frac{1}{4}} = 2 \times \left(\frac{1}{\sqrt{2}}\right) \approx 1.414 \]
- This means after 1 hour, approximately 1.414 nuclei remain. Since we can't have a fraction of a nucleus in practical terms, we can say that at least 1 nucleus will remain undissociated.

3. **Evaluating Statement A**:
- Statement A claims that after 1 hour, only one nucleus will remain undissociated. This is approximately correct, as we calculated about 1.414 nuclei remaining. Thus, we can conclude that Statement A is **True**.

4. **Understanding Beta Decay**:
- Statement R discusses the emission of beta particles when a nucleus transitions from an excited state to a ground state.
- Beta decay involves the transformation of a neutron into a proton (or vice versa), resulting in the emission of a beta particle (an electron or positron).
- However, the transition from an excited state to a ground state typically involves the emission of gamma rays (photons), not beta particles.

5. **Evaluating Statement R**:
- Since Statement R incorrectly states that beta particles are emitted during a transition from an excited state to a ground state, we conclude that Statement R is **False**.

### Final Conclusion:
- Statement A is **True**.
- Statement R is **False**.