These questions consists of two statements each printed as Assertion and Reason. While answering these question you are required to choose any one of the following five reponses
(a) If both Assertion and Reason arecorrect and Reason is the correct explanation of Asserrtion.
(b) If both Assertion and Reason are correct but Reason is not the correct explanation of Assertion.
(c ) If Assertion is true but Reason is false.
(d) If Assertion is false but Reason is true. Assertion A body is moving along X-axis. Force `F=-2x^(2)` is acting on it and work done by this force in moving the body from x=-2 to x=-+2 is zero.
Reason From x=-2 to x=0, work done is negative and from x=0 to x=+2, work done is negative.

Explanation

To solve the problem, we need to analyze both the Assertion and the Reason provided in the question.

### Step 1: Analyze the Assertion
The assertion states that a body is moving along the X-axis under the influence of a force given by \( F = -2x^2 \). It claims that the work done by this force when moving the body from \( x = -2 \) to \( x = +2 \) is zero.

**Calculation of Work Done:**
The work done by a force can be calculated using the formula:
\[ W = \int_{x_1}^{x_2} F \, dx \]
In this case, we need to calculate the work done from \( x = -2 \) to \( x = +2 \):
\[ W = \int_{-2}^{2} -2x^2 \, dx \]

Calculating the integral:
\[ W = -2 \int_{-2}^{2} x^2 \, dx \]
The integral of \( x^2 \) is:
\[ \int x^2 \, dx = \frac{x^3}{3} \]
Thus,
\[ W = -2 \left[ \frac{x^3}{3} \right]_{-2}^{2} = -2 \left( \frac{(2)^3}{3} - \frac{(-2)^3}{3} \right) \]
Calculating the values:
\[ = -2 \left( \frac{8}{3} - \frac{-8}{3} \right) = -2 \left( \frac{8 + 8}{3} \right) = -2 \left( \frac{16}{3} \right) = -\frac{32}{3} \]
Since the work done is negative, the assertion that the work done is zero is **false**.

### Step 2: Analyze the Reason
The reason states that from \( x = -2 \) to \( x = 0 \), the work done is negative, and from \( x = 0 \) to \( x = +2 \), the work done is also negative.

**Verification of Reason:**
1. From \( x = -2 \) to \( x = 0 \):
\[ W_1 = \int_{-2}^{0} -2x^2 \, dx = -2 \left[ \frac{x^3}{3} \right]_{-2}^{0} = -2 \left( 0 - \frac{-8}{3} \right) = -2 \left( \frac{8}{3} \right) = -\frac{16}{3} \]
This is negative.

2. From \( x = 0 \) to \( x = +2 \):
\[ W_2 = \int_{0}^{2} -2x^2 \, dx = -2 \left[ \frac{x^3}{3} \right]_{0}^{2} = -2 \left( \frac{8}{3} - 0 \right) = -\frac{16}{3} \]
This is also negative.

Thus, the reason provided is **true**.

### Conclusion
- The assertion is false because the work done is not zero; it is negative.
- The reason is true as it correctly states that the work done in both intervals is negative.

### Final Answer
The correct option is (d) If Assertion is false but Reason is true.

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