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 both the assertion (A) and the reason (R) provided.

### Step 1: Analyze the Assertion (A)
The assertion states that "displacement gradient is a dimensionless quantity."

- The displacement gradient is defined as the change in displacement per unit length. Mathematically, it can be expressed as:

\[ \text{Displacement Gradient} = \frac{\Delta \text{Displacement}}{\Delta \text{Length}} \]

- Displacement has the dimensions of length (L), and so does length in the denominator. Therefore, we can express this as:

\[ \text{Displacement Gradient} = \frac{L}{L} = 1 \]

- Since the units cancel out, we conclude that the displacement gradient is indeed a dimensionless quantity.

### Step 2: Analyze the Reason (R)
The reason states that "displacement is a dimensionless quantity."

- Displacement measures the distance between two points and is defined as the shortest path from the initial position to the final position. It has dimensions of length (L).

- Therefore, the statement that displacement is a dimensionless quantity is incorrect.

### Conclusion
- The assertion (A) is true: "displacement gradient is a dimensionless quantity."
- The reason (R) is false: "displacement is a dimensionless quantity."

Thus, the correct option is that the assertion is true and the reason is false.

### Final Answer
Assertion (A) is true, and Reason (R) is false.

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