1. **Problem Statement:** Given resistors $R_2 = 50\,000\ \Omega$ and $R_3 = 3R_4 = 600\ \Omega$, find the value of $R_4$ and analyze the circuit behavior for the sinusoidal signal output from the operational amplifier. 2. **Find $R_4$:** Since $R_3 = 3R_4$, we have $$R_4 = \frac{R_3}{3} = \frac{600}{3} = 200\ \Omega.$$ 3. **Circuit Analysis:** The op-amp is configured with the inverting input connected through $R_3$ to point $a$, and the non-inverting input grounded. This suggests an inverting amplifier configuration. 4. **Gain Formula for Inverting Amplifier:** The voltage gain $A_v$ is given by $$A_v = -\frac{R_f}{R_{in}},$$ where $R_f$ is the feedback resistor and $R_{in}$ is the input resistor. 5. **Assigning Resistors:** Assuming $R_2$ is the feedback resistor $R_f = 50\,000\ \Omega$ and $R_4 = 200\ \Omega$ is the input resistor $R_{in}$, the gain is $$A_v = -\frac{50,000}{200} = -250.$$ 6. **Interpretation:** The output sinusoidal signal will be amplified by a factor of 250 and inverted in phase relative to the input. **Final answer:** $$R_4 = 200\ \Omega, \quad A_v = -250.$$