Statement-1: if moment of inertia of a rigid body is equal about two axis, then both the axis must be parallel.
Statement-2 from parallel axis theorem `I=I_(cm)+md^(2)`, where all terms have usual meaning.

Options

A. Statement-1 is true, Statement-2: is true, Statement-2 is a correct explanation for Statement-1.
B. Statement-1 is true, Statement-2: is true, Statement-2 is NOT a correct explanation for Statement-1.
C. Statement-1 is true but statement-2 is false
D. Both statement 1 false and statement 2 are true.

Explanation

To analyze the given statements, we need to evaluate each one separately.

### Step 1: Understanding Statement-1
**Statement-1:** "If the moment of inertia of a rigid body is equal about two axes, then both the axes must be parallel."

To understand this statement, we need to recall the definition of moment of inertia and how it varies with the choice of axes. The moment of inertia \( I \) of a rigid body depends on the axis about which it is calculated. If we have two different axes, the moment of inertia can be different unless certain conditions are met.

### Step 2: Applying the Parallel Axis Theorem
The Parallel Axis Theorem states that:
\[ I = I_{cm} + md^2 \]
where:
- \( I \) is the moment of inertia about the new axis,
- \( I_{cm} \) is the moment of inertia about the centroidal axis,
- \( m \) is the mass of the body,
- \( d \) is the distance between the two axes.

If the two axes are parallel, then the moment of inertia about the second axis can be calculated using this theorem.

### Step 3: Analyzing the Implications
If the moment of inertia about two axes is equal, say \( I_1 = I_2 \), we can express this using the parallel axis theorem:
\[ I_1 = I_{cm} + md_1^2 \]
\[ I_2 = I_{cm} + md_2^2 \]
Setting these equal gives:
\[ I_{cm} + md_1^2 = I_{cm} + md_2^2 \]
This simplifies to:
\[ md_1^2 = md_2^2 \]
Since \( m \) is the same for both axes, we can divide by \( m \):
\[ d_1^2 = d_2^2 \]
This implies that \( d_1 = d_2 \) or \( d_1 = -d_2 \). Thus, the axes must be parallel (or coincident) for the moment of inertia to be equal.

### Conclusion for Statement-1
Therefore, Statement-1 is **false** because the axes must be parallel for the moment of inertia to be equal, but it does not imply that they cannot be coincident.

### Step 4: Understanding Statement-2
**Statement-2:** "From the parallel axis theorem \( I = I_{cm} + md^2 \), where all terms have usual meaning."

This statement is a direct application of the parallel axis theorem and is indeed true. The formula correctly describes how to calculate the moment of inertia about an axis that is parallel to the centroidal axis.

### Conclusion for Statement-2
Thus, Statement-2 is **true**.

### Final Answer
- Statement 1 is false.
- Statement 2 is true.

The correct answer is option 4: Statement 1 is false, Statement 2 is true.

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