Consider the following statements
I. sin |x| + cos |x| is always positive
II. `sin(x^2) + cos (x^2)` is always positive
Which of the above statement is/are correct?

Options

A. A) Only I
B. B) Only II
C. C) Both I and II
D. D) Neither I nor II

Answer

To determine the correctness of the given statements, we will analyze each statement step by step.

### Step 1: Analyze Statement I
**Statement I:** \( \sin |x| + \cos |x| \) is always positive.

1. The functions \( \sin |x| \) and \( \cos |x| \) are periodic functions with a period of \( 2\pi \).
2. The range of \( \sin |x| \) is between -1 and 1, and the range of \( \cos |x| \) is also between -1 and 1.
3. We need to find if there are values of \( x \) such that \( \sin |x| + \cos |x| \leq 0 \).
4. Consider \( x = \frac{3\pi}{2} \):
- \( |x| = \frac{3\pi}{2} \)
- \( \sin\left(\frac{3\pi}{2}\right) = -1 \)
- \( \cos\left(\frac{3\pi}{2}\right) = 0 \)
- Thus, \( \sin |x| + \cos |x| = -1 + 0 = -1 \), which is less than 0.
5. Therefore, Statement I is **not always positive**.

### Step 2: Analyze Statement II
**Statement II:** \( \sin(x^2) + \cos(x^2) \) is always positive.

1. Similar to the first statement, \( \sin(x^2) \) and \( \cos(x^2) \) are also periodic functions.
2. The range of \( \sin(x^2) \) and \( \cos(x^2) \) is also between -1 and 1.
3. We need to check if there are values of \( x \) such that \( \sin(x^2) + \cos(x^2) \leq 0 \).
4. Consider \( x = \sqrt{\frac{3\pi}{2}} \):
- \( x^2 = \frac{3\pi}{2} \)
- \( \sin\left(\frac{3\pi}{2}\right) = -1 \)
- \( \cos\left(\frac{3\pi}{2}\right) = 0 \)
- Thus, \( \sin(x^2) + \cos(x^2) = -1 + 0 = -1 \), which is less than 0.
5. Therefore, Statement II is **not always positive**.

### Conclusion
Both statements I and II are not always positive. Thus, the correct answer is that neither statement is correct.

### Final Answer
**Neither Statement I nor Statement II is correct.**

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