Given below are two statements:
Statement I :When the frequency of an a.c source in LCR circuit increases, the current in the circuit first increases, attains a maximum value and then decreases.
Statement II: In a series LCR circuit , the value of power factor at resonance is one.
In the light of given statements , choose the most appropriate answer from the option given below.

Options

A. Both statement `I` and statement `II` are False.
B. statement `I` is incorrect but statement `II` is true.
C. Both statement `i`i and statement `II` are true
D. Statement `I` is correct but statement `II` is false

Answer

To solve the problem, we need to analyze both statements regarding the behavior of an LCR circuit when subjected to an alternating current (AC) source.

### Step-by-Step Solution:

**Step 1: Analyze Statement I**
- Statement I claims that when the frequency of an AC source in an LCR circuit increases, the current first increases, reaches a maximum value, and then decreases.
- In an LCR circuit, the impedance (Z) is given by the formula:
\[ Z = \sqrt{R^2 + (X_L - X_C)^2} \]
where \(X_L = \omega L\) (inductive reactance) and \(X_C = \frac{1}{\omega C}\) (capacitive reactance).
- As frequency increases, \(X_L\) increases and \(X_C\) decreases. Initially, as frequency increases, the impedance decreases, leading to an increase in current.
- At a certain frequency (resonance frequency, \(\omega_r\)), the impedance reaches a minimum (equal to R), and the current reaches its maximum.
- Beyond this frequency, the impedance starts to increase again, leading to a decrease in current.
- Therefore, Statement I is **True**.

**Step 2: Analyze Statement II**
- Statement II states that in a series LCR circuit, the power factor at resonance is one.
- The power factor (PF) is defined as:
\[ PF = \cos(\phi) = \frac{R}{Z} \]
At resonance, the impedance \(Z\) is minimized and equals \(R\) (i.e., \(Z = R\)).
- Substituting this into the power factor formula gives:
\[ PF = \frac{R}{R} = 1 \]
- Therefore, Statement II is also **True**.

**Conclusion:**
Both statements are true. Thus, the correct answer is that both Statement I and Statement II are true.

### Final Answer:
Both Statement I and Statement II are true.

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