A particle of unit mass is moving along the x-axis under the influence of a force and its total energy is conserved. Four possible forms of the potential energy of the particle are given in column-I (a and U0 are constants). Match the potential energies in column-I to the corresponding statement(s) in column-II.
`{:((A),U_1(x)= (U_0)/(2)[1-(x/a)^(2)]^(2),(P),"the force acting on the particle is zero at x = a."),((B),U_2(x)= (U_0)/(2)(x/a)^(2),(Q),"the force acting on the particle is zero at x = 0."),((C),U_2(x)= (U_0)/(2)(x/a)^(2)exp[-(x/a)],(R),"the force acting on the particle is zero at x = 0."),((D),U_4(x)= (U_0)/(2)[x/a - 1/3 (x/a)^3],(S),"The particle experiences an attractive force towards x = 0 in the region | x | < a"),(,,(T),"The particle with total energy" (U_0)/4 " can oscillate about the point"x = -a.):}`

Explanation

To solve the problem, we need to analyze the potential energy functions given in Column-I and determine the corresponding statements in Column-II based on the behavior of the forces derived from these potential energy functions.

### Step-by-Step Solution:

1. **Identify the Potential Energy Functions**:
- \( U_1(x) = \frac{U_0}{2} \left[ 1 - \left( \frac{x}{a} \right)^2 \right]^2 \)
- \( U_2(x) = \frac{U_0}{2} \left( \frac{x}{a} \right)^2 \)
- \( U_3(x) = \frac{U_0}{2} \left( \frac{x}{a} \right)^2 e^{-\frac{x}{a}} \)
- \( U_4(x) = \frac{U_0}{2} \left[ \frac{x}{a} - \frac{1}{3} \left( \frac{x}{a} \right)^3 \right] \)

2. **Calculate the Force from Potential Energy**:
The force \( F \) is given by \( F = -\frac{dU}{dx} \).

3. **Analyze Each Potential Energy Function**:
- **For \( U_1(x) \)**:
- Calculate \( F_1 = -\frac{dU_1}{dx} \).
- Set \( F_1 = 0 \) to find where the force is zero.
- The roots will be \( x = -a, 0, a \). Thus, the force is zero at \( x = a \) (matches statement P).

- **For \( U_2(x) \)**:
- Calculate \( F_2 = -\frac{dU_2}{dx} \).
- Set \( F_2 = 0 \) to find where the force is zero.
- The force is zero at \( x = 0 \) (matches statement Q).

- **For \( U_3(x) \)**:
- Calculate \( F_3 = -\frac{dU_3}{dx} \).
- Set \( F_3 = 0 \) to find where the force is zero.
- The force is zero at \( x = 0 \) (matches statement R).

- **For \( U_4(x) \)**:
- Calculate \( F_4 = -\frac{dU_4}{dx} \).
- Set \( F_4 = 0 \) to find where the force is zero.
- The force is zero at \( x = -a, a \) (matches statements P and R).
- Check the behavior of the force: it is attractive towards \( x = 0 \) in the region \( |x| < a \) (matches statement S).

4. **Determine Oscillation Condition**:
- For \( U_1 \) and \( U_4 \), check the second derivative to confirm oscillation about \( x = -a \) (matches statement T).

### Final Matching:
- \( A \) matches with \( P \)
- \( B \) matches with \( Q \)
- \( C \) matches with \( R \)
- \( D \) matches with \( S \)
- The oscillation condition for \( U_4 \) matches with \( T \)

### Summary of Matches:
- \( A \) - \( P \)
- \( B \) - \( Q \)
- \( C \) - \( R \)
- \( D \) - \( S \)
- \( D \) also relates to \( T \) for oscillation.