- **Option A:** In a purely capacitive circuit, the voltage (V) lags the current (I) by 90°. Therefore, **A-I** is correct.
- **Option B:** In a purely inductive circuit, the voltage (V) leads the current (I) by 90°. Therefore, **B-II** is correct.
- **Option C:** In an LCR series circuit at resonance, the inductive reactance (XL) equals the capacitive reactance (XC), resulting in a purely resistive behavior where the voltage and current are in phase. Therefore, **C-III** is correct.
- **Option D:** In a general LCR series circuit, the voltage and current are out of phase by an angle θ, which depends on the relative values of XL and XC. Therefore, **D-IV** is correct.
**Answer:** A-I, B-II, C-III, D-IV (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: