A : rigid body can't be in a pure rolling on a rough inclined plane without giving any external force.
R : Since there is no torque providing force acting on the body in the above case, the body can't come in a rolling condition.

Options

A. If both Assertion & Reason are true and the reason is the correct explanation of the assertion,
B. If both Assertion & Reason are true but the reason is not the correct explanation of the assertion,
C. If Assertion is true statement but Reason is false,
D. If both Assertion and Reason are false statements,

Answer

To analyze the statements given in the question, we will break down the reasoning step by step.

### Step 1: Understanding the Statements
- **Statement A**: A rigid body cannot be in pure rolling on a rough inclined plane without giving any external force.
- **Statement R**: Since there is no torque providing force acting on the body in the above case, the body can't come in a rolling condition.

### Step 2: Analyzing Statement A
A rigid body, such as a disc or a sphere, can roll down a rough inclined plane without the need for any external force. The force of gravity acting on the body will cause it to roll. The component of gravitational force acting down the incline is given by \( mg \sin \theta \), where \( m \) is the mass of the body and \( \theta \) is the angle of the incline.

**Conclusion for Statement A**: This statement is **false** because a rigid body can indeed roll down a rough inclined plane without requiring any external force.

### Step 3: Analyzing Statement R
The reason provided states that there is no torque providing force acting on the body, and thus it cannot roll. However, the gravitational force acting on the body does create a torque about the point of contact with the inclined plane. The torque due to the gravitational force is given by:

\[ \tau = r \cdot F \]

Where \( r \) is the radius of the body and \( F \) is the component of the weight acting perpendicular to the radius, which is \( mg \sin \theta \).

**Conclusion for Statement R**: This statement is also **false** because the gravitational force does create a torque that allows the body to roll.

### Step 4: Final Conclusion
Both statements A and R are false. Therefore, the correct answer is that neither statement is true.