Let f (x) = cos `alpha` x+sin x be periodic. Then `alpha` must be

Options

A. rational
B. Prime
C. positive real number
D. none

Answer

To determine the value of \( \alpha \) for which the function \( f(x) = \cos(\alpha x) + \sin(x) \) is periodic, we will analyze the periods of the individual components of the function.

### Step-by-Step Solution:

1. **Identify the Periods of the Functions:**
- The period of the sine function \( \sin(x) \) is \( 2\pi \).
- The period of the cosine function \( \cos(\alpha x) \) is given by \( \frac{2\pi}{\alpha} \).

2. **Determine the Period of the Combined Function:**
- For the function \( f(x) = \cos(\alpha x) + \sin(x) \) to be periodic, the periods of both components must be compatible. This means we need to find the least common multiple (LCM) of the two periods:
\[ \text{Period of } f(x) = \text{LCM}\left(2\pi, \frac{2\pi}{\alpha}\right) \]

3. **Calculate the LCM:**
- The LCM of \( 2\pi \) and \( \frac{2\pi}{\alpha} \) can be calculated as follows:
\[ \text{LCM}(2\pi, \frac{2\pi}{\alpha}) = 2\pi \cdot \frac{1}{\text{GCD}(1, \frac{1}{\alpha})} = 2\pi \cdot \alpha \]
- This means that for \( f(x) \) to be periodic, the LCM must also be a finite period.

4. **Condition for Periodicity:**
- For \( f(x) \) to be periodic, \( \alpha \) must be a rational number. This is because if \( \alpha \) is rational, then \( \frac{2\pi}{\alpha} \) will also yield a finite period.

5. **Conclusion:**
- Therefore, \( \alpha \) must be a positive rational number. This ensures that both components of the function share a common period.

### Final Answer:
Thus, the value of \( \alpha \) must be a positive rational number.

---