Step 1. Understanding Vinyl Halide: Vinyl halides are compounds in which a halogen atom is directly attached to an sp2 hybridized aliphatic carbon atom, which forms part of a double bond.
Step 2. Examine Each Structure: Option (1): Contains a halogen attached to an sp2 hybridized carbon in a double-bonded system, classifying it as a vinyl halide. Option (2): Represents an allyl halide, with the halogen attached to an sp3 hybridized carbon adjacent to a double bond. Options (3) and (4): Both structures are aryl halides, with the halogen attached to an aromatic ring.
Step 3. Conclusion: Option (1) is the correct example of a vinyl halide.
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: