A force F=kx (where k is a positive constant) is acting on a particle Work done:
`{:(,"Column-1",," Column-2"),("(A)","in displacing the body from x=2 to x=4",,"(P) Negative"),("(B)","In displacing the body from x=-4 to x=-2",,"(Q) Positive"),("(C)","In displacing the body from x=-2 to x=+2",,"(R) Zero"):}`

Explanation

To solve the problem, we need to calculate the work done by the force \( F = kx \) for three different displacements. The work done \( W \) by a force is given by the integral of the force over the displacement.

### Step-by-Step Solution:

1. **Understanding the Force**:
The force acting on the particle is given by \( F = kx \), where \( k \) is a positive constant.

2. **Work Done Formula**:
The work done \( W \) by a variable force is calculated using the integral:
\[ W = \int F \, dx \]
Substituting the expression for \( F \):
\[ W = \int kx \, dx \]

3. **Calculating Work Done from \( x = 2 \) to \( x = 4 \)**:
\[ W = \int_{2}^{4} kx \, dx = k \int_{2}^{4} x \, dx \]
The integral of \( x \) is:
\[ \int x \, dx = \frac{x^2}{2} \]
Therefore,
\[ W = k \left[ \frac{x^2}{2} \right]_{2}^{4} = k \left( \frac{4^2}{2} - \frac{2^2}{2} \right) = k \left( \frac{16}{2} - \frac{4}{2} \right) = k \left( 8 - 2 \right) = 6k \]
Since \( k \) is positive, \( W = 6k > 0 \). Thus, this corresponds to option (A) matching with (Q).

4. **Calculating Work Done from \( x = -4 \) to \( x = -2 \)**:
\[ W = \int_{-4}^{-2} kx \, dx = k \int_{-4}^{-2} x \, dx \]
Using the same integral:
\[ W = k \left[ \frac{x^2}{2} \right]_{-4}^{-2} = k \left( \frac{(-2)^2}{2} - \frac{(-4)^2}{2} \right) = k \left( \frac{4}{2} - \frac{16}{2} \right) = k \left( 2 - 8 \right) = -6k \]
Since \( k \) is positive, \( W = -6k < 0 \). Thus, this corresponds to option (B) matching with (P).

5. **Calculating Work Done from \( x = -2 \) to \( x = 2 \)**:
\[ W = \int_{-2}^{2} kx \, dx = k \int_{-2}^{2} x \, dx \]
The integral of \( x \) over symmetric limits around zero results in zero:
\[ W = k \left[ \frac{x^2}{2} \right]_{-2}^{2} = k \left( \frac{2^2}{2} - \frac{(-2)^2}{2} \right) = k \left( \frac{4}{2} - \frac{4}{2} \right) = k(2 - 2) = 0 \]
Thus, this corresponds to option (C) matching with (R).

### Summary of Results:
- (A) Work done from \( x=2 \) to \( x=4 \): **Positive** (matches with Q)
- (B) Work done from \( x=-4 \) to \( x=-2 \): **Negative** (matches with P)
- (C) Work done from \( x=-2 \) to \( x=2 \): **Zero** (matches with R)