Statement I The number of solution of the equation `|sin x|=|x|` is only one.
Statement II `|sin x| ge 0 AA x in R `.

Options

A. Statement I is true , Statement II is true , Statement II is a correct explanation for Statement I.
B. Statement I is true , Statement II is true , Statement II is not a correct explanation for Statement II.
C. Statement I is true , Statement II is false
D. Statement I is false , Statement II true .

Answer

To solve the problem, we need to analyze the given statements and the equation \(|\sin x| = |x|\).

### Step 1: Analyze the equation \(|\sin x| = |x|\)

The equation states that the absolute value of \(\sin x\) is equal to the absolute value of \(x\).

### Step 2: Understand the properties of \(|\sin x|\)

The function \(|\sin x|\) oscillates between 0 and 1 for all \(x\). Therefore, \(|\sin x|\) is always non-negative and reaches its maximum value of 1.

### Step 3: Analyze the function \(|x|\)

The function \(|x|\) is a linear function that increases without bound as \(x\) moves away from 0 in either direction.

### Step 4: Graph the functions

To visualize the solutions, we can graph \(|\sin x|\) and \(|x|\):
- The graph of \(|\sin x|\) will oscillate between 0 and 1.
- The graph of \(|x|\) is a straight line that intersects the y-axis at the origin and increases linearly.

### Step 5: Identify points of intersection

The only point where \(|\sin x|\) can equal \(|x|\) is at \(x = 0\) because:
- For \(|x| > 1\), \(|\sin x|\) cannot equal \(|x|\) since \(|\sin x|\) is always less than or equal to 1.
- For \(x\) values between -1 and 1, \(|\sin x|\) oscillates and can equal \(|x|\) only at \(x = 0\).

### Conclusion for Statement I

Thus, the number of solutions to the equation \(|\sin x| = |x|\) is indeed only one, which is \(x = 0\). Therefore, Statement I is true.

### Step 6: Analyze Statement II

Statement II claims that \(|\sin x| \geq 0\) for all \(x \in \mathbb{R}\). This is indeed true because the absolute value function is always non-negative.

### Conclusion for Statement II

Thus, Statement II is also true.

### Final Evaluation

- Statement I is true.
- Statement II is true.
- However, Statement II does not provide a correct explanation for Statement I because the fact that \(|\sin x|\) is non-negative does not relate to the number of solutions of the equation \(|\sin x| = |x|\).

### Final Answer

Statement I is true, Statement II is true, but Statement II is not a correct explanation for Statement I.