The natural length of a metallic root at `0^(@)C` is `l_(1)` at `theta^(@)C` is `l_(2)`. The given lengths of the rod at `theta^(@)` is `l_(3)`, then:

Options

A. The thermal stress in the rod is non zero if `l_(3)gtl_(2)`
B. The strain in the rod is zero if `l_(2)!=l_(3)`
C. The thermal stress in the rod is non-zero if `l_(3)ltl_(2)`
D. The thermal stress and strain will be zero if `l_(2)=l_(3)`

Explanation

To solve the problem regarding the lengths of a metallic rod at different temperatures, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**:
- We have a metallic rod with a natural length \( l_1 \) at \( 0^\circ C \).
- The length of the rod at a temperature \( \theta^\circ C \) is \( l_2 \).
- The given length of the rod at temperature \( \theta^\circ C \) is \( l_3 \).

2. **Identify the Relationship**:
- When the temperature of the rod changes, thermal stress affects its length. If there is no thermal stress, the length of the rod at temperature \( \theta \) should equal \( l_3 \) (the given length).
- Therefore, in the absence of thermal stress, we can state that:
\[ l_2 = l_3 \]

3. **Determine Conditions for Thermal Stress**:
- If \( l_3 > l_2 \), this indicates that the rod has expanded due to thermal stress. Thus, thermal stress is positive.
- Conversely, if \( l_3 < l_2 \), it indicates that the rod has contracted, implying that thermal stress is negative.

4. **Conclusion on Options**:
- If \( l_3 = l_2 \), there is no thermal stress, and thus the deformation does not occur.
- If \( l_3 \neq l_2 \), then thermal stress is present, and it can be either positive or negative depending on whether \( l_3 \) is greater or less than \( l_2 \).

5. **Final Options**:
- From the analysis, we can conclude:
- Option 4 is correct: \( l_2 = l_3 \) indicates no thermal stress.
- Options 1 and 3 are also correct as they describe the conditions under which thermal stress is non-zero.

### Summary of Correct Options:
- Correct options are: A, C, and D.