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

### Step-by-Step Solution:

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

2. **Determine the Period of the Combined Function:**
- The function \( f(x) \) will be periodic if the periods of both components are commensurable, meaning that their least common multiple (LCM) is finite.
- Therefore, we need to find the LCM of \( 2\pi \) and \( \frac{2\pi}{p} \).

3. **Calculate the LCM:**
- The LCM of \( 2\pi \) and \( \frac{2\pi}{p} \) can be expressed as:
\[ \text{LCM}(2\pi, \frac{2\pi}{p}) = 2\pi \cdot \text{LCM}(1, \frac{1}{p}) = 2\pi \cdot \frac{1}{\gcd(1, \frac{1}{p})} \]
- Since \( \gcd(1, \frac{1}{p}) = 1 \) for any non-zero \( p \), we have:
\[ \text{LCM}(2\pi, \frac{2\pi}{p}) = 2\pi \cdot p \]

4. **Condition for Periodicity:**
- For \( f(x) \) to be periodic, \( p \) must be such that \( \frac{2\pi}{p} \) is a rational number. This implies that \( p \) must be a rational number.

5. **Check the Options:**
- Option (a) Positive real number: This is not sufficient alone since \( p \) could also be negative.
- Option (b) Negative real number: This is not sufficient alone since \( p \) could also be positive.
- Option (c) Rational: This is necessary for the periodicity of \( f(x) \).
- Option (d) Prime: This is not a necessary condition for periodicity.

6. **Conclusion:**
- The only condition that ensures the periodicity of \( f(x) \) is that \( p \) must be a rational number.

### Final Answer:
Thus, the correct option is (c) Rational.