Statement-1 `:` The work done in bringing a body down from the top to the base along a frictionless incline plane is the same as the work done in brining it down from the vertical side.
STATEMENT-2 `:` Work is defined as `intvec(F) . dvec(r )`

Options

A. Statement-1 is True , Statement-2 is True, Statement-2 is a correct explanation for Statement-1.
B. Statement-1 is True , Statement-2 is True, Statement-2 is NOT a correct explanation for Statement-1.
C. Statement-1 is True, Statement-2 is False
D. Statement-1 is False, Statement-2 is True.

Answer

To solve the question, we need to analyze both statements and derive the necessary conclusions step by step.

### Step-by-Step Solution:

1. **Understanding Statement 1**:
- Statement 1 claims that the work done in bringing a body down from the top to the base along a frictionless inclined plane is the same as the work done in bringing it down from the vertical side.
- We need to consider the forces acting on the body and the path taken.

2. **Defining Work Done**:
- Work done (W) is defined as the integral of the force (F) dot the displacement (dr), i.e., \( W = \int \vec{F} \cdot d\vec{r} \).

3. **Analyzing the Inclined Plane**:
- For a frictionless inclined plane, the only force doing work is the component of gravitational force acting along the incline.
- The gravitational force acting on the body is \( \vec{F} = m\vec{g} \), where \( m \) is the mass of the body and \( g \) is the acceleration due to gravity.

4. **Calculating Work Done on the Incline**:
- When the body moves down the incline, we can express the work done by gravity as:
\[ W = \int_{0}^{h} mg \cos(\alpha) \, dh \]
- Here, \( \alpha \) is the angle between the gravitational force and the direction of displacement.

5. **Calculating Work Done Vertically**:
- When the body is brought down vertically, the work done by gravity is:
\[ W = mgh \]
- This is because the entire weight of the body acts in the direction of the displacement.

6. **Comparing Work Done**:
- In both cases, whether the body is moved down the incline or vertically, the work done by gravity results in the same expression:
\[ W = mgh \]
- Thus, Statement 1 is true.

7. **Understanding Statement 2**:
- Statement 2 states that work is defined as \( \int \vec{F} \cdot d\vec{r} \).
- This is a correct definition of work in physics.

8. **Conclusion**:
- Both statements are true, and Statement 2 correctly explains Statement 1. Therefore, the correct option is that both statements are true and Statement 2 is the correct explanation of Statement 1.

### Final Answer:
Both Statement 1 and Statement 2 are true, and Statement 2 is the correct explanation of Statement 1.

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