A : Displacement gradient is a dimensionless quantity.
R : Displacement is dimensionless quantity.

Options

A. If both Assertion & Reason are true and the reason is the correct explanation of the assertion, then mark (1).
B. If both Assertion & Reason are true but the reason is not the 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 and Reason are false statements, then mark (4).

Answer

To solve the question, we need to analyze the assertion (A) and the reason (R) provided.

**Assertion (A):** Displacement gradient is a dimensionless quantity.
**Reason (R):** Displacement is a dimensionless quantity.

### Step-by-step Solution:

1. **Understanding Displacement Gradient:**
- The term "displacement gradient" refers to the change in displacement with respect to a change in position (length). Mathematically, it can be expressed as:
\[ \text{Displacement Gradient} = \frac{\Delta \text{Displacement}}{\Delta x} \]
- Here, \(\Delta \text{Displacement}\) has dimensions of length (L), and \(\Delta x\) also has dimensions of length (L).

2. **Calculating the Dimensions:**
- When we calculate the displacement gradient, we divide displacement (L) by length (L):
\[ \text{Displacement Gradient} = \frac{L}{L} = 1 \]
- This means that the displacement gradient is indeed dimensionless.

3. **Evaluating the Assertion (A):**
- Since we have established that the displacement gradient is dimensionless, the assertion (A) is correct.

4. **Understanding Displacement:**
- Displacement itself is defined as the shortest distance from the initial to the final position of an object. It is a vector quantity with dimensions of length (L).

5. **Evaluating the Reason (R):**
- The reason states that displacement is a dimensionless quantity. However, we have established that displacement has dimensions of length (L).
- Therefore, the reason (R) is incorrect.

6. **Conclusion:**
- The assertion (A) is correct, while the reason (R) is incorrect. Thus, the correct answer is that the assertion is true, but the reason is false.

### Final Answer:
- Assertion (A) is correct.
- Reason (R) is incorrect.