(1) Centre of gravity (C.G.) of a body is the point at which the weight of the body acts,
(2) Centre of mass coincides with the centre of gravity if the earth is assumed to have infinitely large radius,
(3) To evaluate the gravitational field intensity due to any body at an external point, the entire mass of the body can be considered to be concentrated at its C.G..,
(4) The radius of gyration of any body rotating about an axis is the length of the perpendicular dropped from the C.G. of the body to the axis. which one of the following pairs of statements is correct ?

Options

A. (1) and (4)
B. (1) and (2)
C. (1) and (3)
D. (3) and (4)

Explanation

To determine which pairs of statements are correct regarding the center of gravity (C.G.) and center of mass, we will analyze each statement one by one.

### Step 1: Analyze Statement 1
**Statement 1:** Centre of gravity (C.G.) of a body is the point at which the weight of the body acts.

- **Evaluation:** This statement is true. The center of gravity is defined as the point where the total weight of the body can be considered to act.

### Step 2: Analyze Statement 2
**Statement 2:** Centre of mass coincides with the centre of gravity if the earth is assumed to have infinitely large radius.

- **Evaluation:** This statement is incorrect. The center of mass and center of gravity coincide only under conditions of uniform gravitational field and uniform mass distribution. The assumption of the Earth having an infinitely large radius does not guarantee this condition.

### Step 3: Analyze Statement 3
**Statement 3:** To evaluate the gravitational field intensity due to any body at an external point, the entire mass of the body can be considered to be concentrated at its C.G.

- **Evaluation:** This statement is true. For calculating the gravitational field intensity at an external point, it is valid to treat the entire mass of the body as concentrated at its center of gravity.

### Step 4: Analyze Statement 4
**Statement 4:** The radius of gyration of any body rotating about an axis is the length of the perpendicular dropped from the C.G. of the body to the axis.

- **Evaluation:** This statement is incorrect. The radius of gyration is defined as the distance from the axis of rotation at which if the entire mass of the body were concentrated, the moment of inertia about that axis would remain the same. It is not merely the perpendicular distance from the C.G. to the axis.

### Conclusion
Based on the evaluations:
- Statement 1 is true.
- Statement 2 is false.
- Statement 3 is true.
- Statement 4 is false.

Thus, the correct pair of statements is **Statement 1 and Statement 3**.

### Final Answer
The correct option is **C: Statement 1 and Statement 3**.

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