Given below are two statements :
Statement I : Hyperconjugation is not a permanent effect.
Statement II : In general, greater the number of alkyl groups attached to a positively charged C-atom, greater is the hyperconjugation interaction and stabilization of the cation.
In the light of the above statements, choose the correct answer from the options given below
Statement I : Hyperconjugation is not a permanent effect. Hyperconjugation is a permanent effect that involves the delocalization of \( \sigma \)-electrons of a C-H bond of an alkyl group directly attached to an unsaturated system or to an atom with an unshared p-orbital.
This delocalization occurs even in the absence of an external reagent or condition.
Therefore, Statement I is false. Statement II : In general, greater the number of alkyl groups attached to a positively charged C-atom, greater is the hyperconjugation interaction and stabilization of the cation.
A carbocation is stabilized by hyperconjugation due to the donation of \( \sigma \)-electrons from the adjacent C-H bonds. Alkyl groups attached to the positively charged carbon atom have C-H bonds that can participate in hyperconjugation. The more alkyl groups attached, the greater the number of \( \alpha \)-hydrogen atoms available for hyperconjugation.
This leads to greater delocalization of the positive charge and hence greater stability of the carbocation.
Therefore, Statement II is true. In conclusion, Statement I is false, but Statement II is true. This corresponds to option (3).
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: