https://file.aiquickdraw.com/mathbot/file/1726493791066k8dplkkx.jpg

Given the circuit in the image, we need to find the value of \( R \) given that the gain of the operational amplifier (op-amp) is -12.

### Step-by-step Analysis

1. **Circuit Configuration**: 
   - The circuit is an inverting op-amp configuration with multiple resistors.
   - Resistors: \( 10 \, k\Omega \) resistors are connected as feedback resistors along with the unknown resistor \( R \) that branches at point B.
   
2. **Inverting Op-amp Gain Formula**:
   For an inverting op-amp, the voltage gain (\( A_v \)) is given by:
   \[
   A_v = -\frac{R_f}{R_{in}}
   \]
   Where:
   - \( R_f \) is the total feedback resistance.
   - \( R_{in} \) is the input resistor (here, \( 10 \, k\Omega \)).

3. **Total Feedback Resistance**:
   - In the circuit, the resistors form a combination in the feedback loop.
   - The resistors at the point B (two \( 10 \, k\Omega \) resistors in series with \( R \) in parallel):
     \[
     R_f = 10 \, k\Omega + \left( \frac{10 \, k\Omega \times R}{10 \, k\Omega + R} \right)
     \]
   Here, the two \( 10 \, k\Omega \) resistors are in series, and their combination is in parallel with \( R \).

4. **Given Gain**:
   - The gain is provided as -12.
   - Plugging into the gain formula:
     \[
     -12 = -\frac{R_f}{10 \, k\Omega}
     \]
   Simplifying:
   \[
   R_f = 12 \times 10 \, k\Omega = 120 \, k\Omega
   \]

5. **Finding \( R \)**:
   - Substitute \( R_f = 120 \, k\Omega \) into the expression for total feedback resistance:
     \[
     120 \, k\Omega = 10 \, k\Omega + \left( \frac{10 \, k\Omega \times R}{10 \, k\Omega + R} \right)
     \]
   - Subtract \( 10 \, k\Omega \) from both sides:
     \[
     110 \, k\Omega = \frac{10 \, k\Omega \times R}{10 \, k\Omega + R}
     \]
   - Cross-multiply to solve for \( R \):
     \[
     110 \, k\Omega \times (10 \, k\Omega + R) = 10 \, k\Omega \times R
     \]
     \[
     1,100,000 \, k\Omega^2 + 110 \, k\Omega \times R = 10 \, k\Omega \times R
     \]
   - Isolate \( R \):
     \[
     1,100,000 \, k\Omega^2 = 10 \, k\Omega \times R - 110 \, k\Omega \times R
     \]
     \[
     1,100,000 \, k\Omega^2 = -100 \, k\Omega \times R
     \]
   - Divide by \( -100 \, k\Omega \) to get \( R \):
     \[
     R = -\frac{1,100,000 \, k\Omega^2}{100 \, k\Omega} = 11 \, k\Omega
     \]

### Final Answer
The value of \( R \) is \( 11 \, k\Omega \).

Would you like further explanation or have any questions?

### Related Questions
1. How would the gain change if \( R \) were doubled?
2. What is the effect of changing the input resistor on the overall gain?
3. How would you design a non-inverting amplifier with a gain of 12?
4. What happens to the op-amp gain if one of the resistors in the feedback loop is open-circuited?
5. How does the ideal op-amp assumption affect the accuracy of real-world circuit performance?

**Tip:** For accurate gain calculations in op-amp circuits, always check the configuration (inverting or non-inverting) and the arrangement of resistors in the feedback loop.

In the fig below, if gain is -12, find the value of R. Assume op-amp is ideal.

This problem involves calculating the value of a resistor (R) in an ideal inverting op-amp circuit with a given gain of -12. Learn how to apply inverting amplifier gain formulas and solve for the unknown resistor using series and parallel resistor combination formulas. Suitable for undergraduate electrical engineering students.

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