Assertion (A) If the frequency of the applied AC is doubled , then the power factor of a series R-L circuit decreases.
Reason (R ) Power factor of series R-L circuit is given by `costheta=(2R)/(sqrt(R^(2)+omega^(2)L^(2)))`

Options

A. If both Assertion and Reason are true and Reason is correct explanation of Assertion.
B. If both Assertion and Reason are true but Reason is not the correct explanation of Assertion.
C. If Assertion is true but Reason is false.
D. If both Assertion and Reason are false.

Answer

To solve the problem, we need to analyze the assertion and the reason given in the question regarding the power factor of a series R-L circuit when the frequency of the applied AC is doubled.

### Step-by-Step Solution:

1. **Understanding Power Factor**:
The power factor (PF) of a circuit is defined as the cosine of the phase angle (φ) between the voltage and the current. For a series R-L circuit, the power factor can be expressed as:
\[ \text{PF} = \cos(\phi) = \frac{R}{Z} \]
where \( Z \) is the impedance of the circuit.

2. **Calculating Impedance**:
The impedance \( Z \) of a series R-L circuit is given by:
\[ Z = \sqrt{R^2 + (\omega L)^2} \]
where \( \omega = 2\pi f \) is the angular frequency, and \( L \) is the inductance.

3. **Effect of Doubling Frequency**:
If the frequency \( f \) is doubled, then the new frequency \( f' = 2f \) leads to a new angular frequency:
\[ \omega' = 2\omega = 4\pi f \]
The new impedance becomes:
\[ Z' = \sqrt{R^2 + (4\pi f L)^2} \]

4. **New Power Factor**:
The new power factor after doubling the frequency is:
\[ \text{PF}' = \frac{R}{Z'} = \frac{R}{\sqrt{R^2 + (4\pi f L)^2}} \]

5. **Comparison of Power Factors**:
To see if the power factor decreases, we compare \( Z \) and \( Z' \):
- Initially, \( Z = \sqrt{R^2 + (2\pi f L)^2} \)
- After doubling the frequency, \( Z' = \sqrt{R^2 + (4\pi f L)^2} \)

Since \( (4\pi f L)^2 > (2\pi f L)^2 \), it follows that \( Z' > Z \). Therefore, the new power factor \( \text{PF}' \) will be less than the original power factor \( \text{PF} \).

6. **Conclusion**:
The assertion that the power factor decreases when the frequency is doubled is true. The reason provided, which states the formula for power factor, is incorrect because it incorrectly includes a factor of 2.

### Final Answer:
- Assertion (A) is **True**: If the frequency of the applied AC is doubled, then the power factor of a series R-L circuit decreases.
- Reason (R) is **False**: The power factor formula provided is incorrect.