In a diode circuit, for the diode to be forward-biased, the P-end should be connected to a higher potential than the N-end. In option (4), the P-end of the diode is connected to the negative terminal of the power supply, while the N-end is connected to the positive terminal. This configuration causes the diode to be reverse-biased, as it is in opposition to the direction required for forward bias.
Thus the correct answer is option 4.
Let $ P_n = \alpha^n + \beta^n $, $ n \in \mathbb{N} $. If $ P_{10} = 123,\ P_9 = 76,\ P_8 = 47 $ and $ P_1 = 1 $, then the quadratic equation having roots $ \alpha $ and $ \frac{1}{\beta} $ is: