Let `f (x)= cos x+ sin px` be periodic, then p must be :

Options

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

Answer

To determine the value of \( p \) such that the function \( f(x) = \cos x + \sin(px) \) is periodic, we need to analyze the periods of the individual components of the function.

### Step-by-Step Solution:

1. **Identify the Period of \( \cos x \)**:
The period of \( \cos x \) is \( 2\pi \). This means that \( \cos(x + 2\pi) = \cos x \) for all \( x \).

**Hint**: Recall that the cosine function repeats its values every \( 2\pi \).

2. **Identify the Period of \( \sin(px) \)**:
The period of \( \sin(px) \) is given by \( \frac{2\pi}{p} \). This means that \( \sin(px + 2\pi) = \sin(px) \).

**Hint**: The period of a sine function can be found using the formula \( \frac{2\pi}{\text{coefficient of } x} \).

3. **Set Up the 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 \( 2\pi \) and \( \frac{2\pi}{p} \).

**Hint**: The function is periodic if both components share a common period.

4. **Calculate the LCM**:
The LCM of \( 2\pi \) and \( \frac{2\pi}{p} \) can be calculated as follows:
\[ \text{lcm}(2\pi, \frac{2\pi}{p}) = 2\pi \cdot \text{lcm}(1, \frac{1}{p}) = 2\pi \cdot \frac{p}{\gcd(1, p)} = 2\pi p \]
Here, we note that \( \gcd(1, p) = 1 \) for any \( p \).

**Hint**: The LCM can be thought of as the smallest value that both periods can divide into without a remainder.

5. **Determine Conditions on \( p \)**:
For \( f(x) \) to be periodic, \( p \) must be a rational number. Specifically, \( p \) should be a positive rational number so that \( \frac{2\pi}{p} \) is also a finite period.

**Hint**: Consider the implications of \( p \) being a rational number and how it affects the periodicity of \( \sin(px) \).

6. **Conclusion**:
Since \( p \) must be a rational number, and given the options, we can conclude that \( p \) should ideally be a prime number to satisfy the condition of being a rational number.

**Final Answer**: The value of \( p \) must be a prime number.