Let `f (x)= cos x+ sin px` be periodic, then p must be :
(a).Positive real number (b). Negative real number (c).Rational (d).Prime

Options

A. Positive real number
B. Negative real number
C. Rational
D. Prime

Answer

To determine the values of \( p \) for which the function \( f(x) = \cos x + \sin(px) \) is periodic, we need to analyze the periodicity of both components of the function.

### Step-by-Step Solution:

1. **Identify the Period of Each Component**:
- The function \( \cos x \) has a period of \( 2\pi \).
- The function \( \sin(px) \) has a period of \( \frac{2\pi}{p} \).

2. **Condition for Periodicity**:
- For \( f(x) \) to be periodic, the periods of \( \cos x \) and \( \sin(px) \) must have a common period. This means we need to find the least common multiple (LCM) of the two periods.
- The LCM of \( 2\pi \) and \( \frac{2\pi}{p} \) must also be a period of \( f(x) \).

3. **Finding the LCM**:
- The LCM can be expressed as:
\[ \text{lcm}(2\pi, \frac{2\pi}{p}) = 2\pi \cdot \text{lcm}(1, \frac{1}{p}) = 2\pi \cdot \frac{p}{\text{gcd}(p, 1)} \]
- Since \( \text{gcd}(p, 1) = 1 \), we have:
\[ \text{lcm}(2\pi, \frac{2\pi}{p}) = 2\pi \cdot p \]

4. **Condition for \( p \)**:
- For the function \( f(x) \) to be periodic, \( p \) must be a rational number. This is because the period \( \frac{2\pi}{p} \) must also be a finite value, which occurs when \( p \) is rational.
- If \( p \) is irrational, \( \frac{2\pi}{p} \) would not yield a rational period, thus making \( f(x) \) non-periodic.

5. **Conclusion**:
- Therefore, \( p \) must be a rational number for \( f(x) \) to be periodic.

### Final Answer:
The correct option is (c) Rational.