Following are the four molecules ``P``, ``Q``, ``R`` and ``S``. Which one among the four molecules will react with H-Br(aq) at the fastest rate? Molecules:
\[ P: \text{Cyclic compound with two O groups attached to the ring.} \] \[ Q: \text{Cyclic compound with one O group and one CH\(_3\) group attached to the ring.} \] \[ R: \text{Cyclic compound with one O group attached to the ring and one CH\(_3\) group attached to the ring.} \] \[ S: \text{Cyclic compound with one CH\(_3\) group attached to the ring.} \]
Given:
- The reaction follows an electrophilic addition mechanism. - During the rate-determining step, a carbocation intermediate is formed.
The addition of \( H^+ \) and \( Br^− \) to the alkene follows the electrophilic addition mechanism, where the alkene reacts with an electrophile (such as \( H^+ \)) to form a carbocation intermediate.
The stability of the carbocation intermediate plays a crucial role in determining the course of the reaction. Among the compounds P, Q, R, and S, compound \( Q \) forms the most stable carbocation intermediate since it is resonance-stabilized.
Compound \( Q \) will form the most stable carbocation intermediate due to its resonance stabilization, making it the most stable product in this electrophilic addition reaction.
Let A be a 3 × 3 matrix such that \(\text{det}(A) = 5\). If \(\text{det}(3 \, \text{adj}(2A)) = 2^{\alpha \cdot 3^{\beta} \cdot 5^{\gamma}}\), then \( (\alpha + \beta + \gamma) \) is equal to: