Match of following columns
`{:(,"Column-I",,"Column-II"),("(A)",underset(underset("field is increasing")("particle in gravitational"))("Kinetic enery of a"),"(p)",underset(underset("be positive")("gravitational force should"))("Work done by")),("(B)",underset(underset("field is increasing")("particle in gravitational"))("Potential energy of a"),"(q)",underset("force should be non-zero")("Work done by external")),("(C)",underset(underset("field is increasing")("particle in gravitational"))("Mechanical energy of a"),"(r)",underset(underset("be negative")("gravitational force should"))("Work done by")),(,,"(s)","Cannot say anything"):}`

Answer

To match the items in Column-I with those in Column-II, we need to analyze the relationships between kinetic energy, potential energy, mechanical energy, and the work done by gravitational and external forces in a gravitational field.

### Step-by-Step Solution:

1. **Understanding Kinetic Energy in a Gravitational Field:**
- When a particle moves in a gravitational field, its kinetic energy can increase if it is falling or being accelerated by the field.
- As kinetic energy increases, potential energy decreases because the total mechanical energy (kinetic + potential) remains constant in a conservative field.
- Therefore, for (A) "Kinetic energy of a particle in gravitational field is increasing," we can conclude that the work done by the gravitational force should be **positive**.
- **Match:** (A) → (p)

2. **Understanding Potential Energy in a Gravitational Field:**
- Potential energy in a gravitational field is given by the formula \( U = -\frac{GMm}{r} \).
- As a particle moves deeper into the gravitational field (closer to the mass), its potential energy decreases (becomes more negative).
- Therefore, for (B) "Potential energy of a particle in gravitational field is increasing," the work done by an external force should be **non-zero** to counteract the gravitational force.
- **Match:** (B) → (q)

3. **Understanding Mechanical Energy in a Gravitational Field:**
- Mechanical energy is the sum of kinetic and potential energy. If kinetic energy increases, potential energy decreases, but the total mechanical energy remains constant in the absence of non-conservative forces.
- Therefore, for (C) "Mechanical energy of a particle in gravitational field is increasing," we cannot definitively say anything without additional information about external forces acting on the system.
- **Match:** (C) → (s)

4. **Understanding Work Done by Gravitational Force:**
- Work done by gravitational force is typically considered negative when the force opposes the motion (for example, lifting an object).
- Therefore, for the statement regarding work done by gravitational force being negative, we can conclude that it is related to the potential energy decreasing.
- **Match:** (B) → (r)

### Final Matches:
- (A) → (p)
- (B) → (q)
- (C) → (s)