For an LCR circuit, the power transferred from the driving source to the driven oscillator is `P = I^(2) Z cos phi`.

Options

A. Hence, the power factor `cosphi gt- 0, P gt-0`
B. The driving force can give no energy to the oscillator (P=0) in some cases.
C. the driving force cannot syphon out`(Plt0)` the energy out of oscillator
D. The driving force can take away energy out of the oscillator.

Explanation

To solve the question regarding the power transferred from the driving source to the driven oscillator in an LCR circuit, we need to analyze the given equation and the implications of the power factor.

### Step-by-Step Solution:

1. **Understanding the Equation**:
The power transferred in an LCR circuit is given by:
\[ P = I^2 Z \cos \phi \]
where \( P \) is the power, \( I \) is the current, \( Z \) is the impedance, and \( \cos \phi \) is the power factor.

2. **Power Factor Analysis**:
The power factor \( \cos \phi \) is defined as the ratio of the resistance \( R \) to the impedance \( Z \):
\[ \cos \phi = \frac{R}{Z} \]
Since both \( R \) (resistance) and \( Z \) (impedance) are positive quantities, it follows that:
\[ \cos \phi \geq 0 \]
This indicates that the power factor cannot be negative.

3. **Implications of Power Factor**:
If \( \cos \phi \) is greater than or equal to zero, it implies that the average power \( P \) is also greater than or equal to zero:
\[ P \geq 0 \]
Therefore, the power transferred from the driving source to the driven oscillator is non-negative.

4. **Special Case of Zero Power**:
In some cases, such as in a wattless circuit where the phase difference between current and voltage is 90 degrees, the average power can be zero:
\[ P = 0 \quad \text{(when } \phi = 90^\circ\text{)} \]
This means that while energy is still being transferred, the average power consumed is zero.

5. **Energy Transfer**:
The driving force in the circuit does not siphon out energy from the oscillator; rather, it provides energy to the oscillator. Power can never be negative, so the driving force cannot take away energy.

6. **Conclusion**:
Based on the analysis:
- The power factor \( \cos \phi \) is always greater than or equal to zero.
- The average power can be zero in specific cases (wattless circuit).
- The driving force provides energy to the oscillator and cannot take energy away.

### Final Answer:
The correct options based on the analysis are:
- Option 1: Correct (Power factor \( \cos \phi \geq 0 \))
- Option 2: Correct (Driving force can give no energy to the oscillator in some cases)
- Option 3: Correct (Driving force cannot siphon out energy)
- Option 4: Incorrect (Driving force cannot take away energy)