Amplitude of vibration is `A = (F_(0))/(p-q+r)` . Resonance will occur when

Options

A. `p=0,q=r`
B. `p=q=r`
C. `p= -r, q=0`
D. Both (1) & (3)

Answer

To determine when resonance occurs in the given scenario, we need to analyze the amplitude of vibration given by the formula:

\[ A = \frac{F_0}{P - Q + R} \]

Resonance occurs when the amplitude \( A \) becomes infinitely large, which mathematically implies that the denominator must approach zero:

\[ P - Q + R = 0 \]

### Step-by-Step Solution:

1. **Set the Denominator to Zero**:
To find the conditions for resonance, we set the denominator of the amplitude equation to zero:
\[ P - Q + R = 0 \]

2. **Rearranging the Equation**:
Rearranging this equation gives us:
\[ P = Q - R \]

3. **Analyzing the Options**:
We will evaluate each option to see if it satisfies the condition \( P - Q + R = 0 \).

- **Option A**: \( P = 0, Q = R \)
\[ 0 - Q + Q = 0 \quad \text{(True)} \]
Resonance occurs.

- **Option B**: \( P = Q, Q = R \)
\[ P - P + P = P \quad \text{(Not zero)} \]
Resonance does not occur.

- **Option C**: \( P = -R, Q = 0 \)
\[ -R - 0 + R = 0 \quad \text{(True)} \]
Resonance occurs.

- **Option D**: Both first and second options are correct.
Since only options A and C are correct, option D is incorrect.

4. **Conclusion**:
The conditions for resonance occur in options A and C. Therefore, the correct answer is:
\[ \text{Resonance will occur when: Option A and Option C are correct.} \]

### Final Answer:
Resonance will occur when \( P = 0, Q = R \) (Option A) or \( P = -R, Q = 0 \) (Option C).

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