Asseration : A body is moved from `x=2` to `x=1` under a force `F =4x, the work done by this force is negative.
Reason : Force and displacement are in opposite directions .

Options

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

Answer

To solve the question, we need to analyze both the assertion and the reason provided.

### Step 1: Understand the Assertion
The assertion states that a body is moved from \( x = 2 \) to \( x = 1 \) under a force \( F = 4x \), and the work done by this force is negative.

### Step 2: Calculate the Displacement
The displacement \( s \) can be calculated as:
\[ s = x_f - x_i = 1 - 2 = -1 \]
This indicates that the displacement is negative, meaning the body is moving in the negative direction.

### Step 3: Analyze the Force
The force acting on the body is given by:
\[ F = 4x \]
At \( x = 2 \):
\[ F = 4 \times 2 = 8 \quad (\text{positive force}) \]
At \( x = 1 \):
\[ F = 4 \times 1 = 4 \quad (\text{positive force}) \]
Thus, the force is positive throughout the movement.

### Step 4: Calculate the Work Done
The work done \( W \) by the force when moving from \( x = 2 \) to \( x = 1 \) can be calculated using the integral of the force:
\[ W = \int_{x_i}^{x_f} F \, dx = \int_{2}^{1} 4x \, dx \]
Calculating the integral:
\[ W = 4 \int_{2}^{1} x \, dx = 4 \left[ \frac{x^2}{2} \right]_{2}^{1} = 4 \left( \frac{1^2}{2} - \frac{2^2}{2} \right) = 4 \left( \frac{1}{2} - 2 \right) = 4 \left( \frac{1 - 4}{2} \right) = 4 \left( -\frac{3}{2} \right) = -6 \]
Thus, the work done \( W = -6 \), which is indeed negative.

### Step 5: Analyze the Reason
The reason states that the force and displacement are in opposite directions. Since the force is positive and the displacement is negative, they are indeed in opposite directions. This confirms that the work done is negative.

### Conclusion
Both the assertion and the reason are true, and the reason correctly explains the assertion.

### Final Answer
- Assertion: True
- Reason: True
- The reason is a correct explanation of the assertion.

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