Step 1: Analyzing the given reaction.
The given reaction involves the reduction of a molecule containing a carbonyl group (C=O) and an amide group (-NH\(_2\)) using lithium aluminum hydride (LiAlH\(_4\)), followed by hydrolysis with H\(_3\)O\(^+\).
1. Step 1: Reduction with LiAlH\(_4\)
Lithium aluminum hydride (LiAlH\(_4\)) is a strong reducing agent. When it reacts with a carbonyl compound (such as a ketone or aldehyde), it reduces the carbonyl group to a primary or secondary alcohol. \item In this case, the carbonyl group (C=O) of the given compound will be reduced to a hydroxyl group (-OH), resulting in an intermediate amide being reduced to a primary amine group (-NH\(_2\)). The structure of the intermediate product will be: \[ \text{H}_2\text{NC} - \text{CH}_2 - \text{CH}_2\text{OH} \] where the ketone group is reduced to an alcohol group.
2. Step 2: Hydrolysis with H\(_3\)O\(^+\)
After reduction, the product is treated with an acidic solution (H\(_3\)O\(^+\)), which will hydrolyze the intermediate and result in a final product where the amide group has been converted to an amine group (-NH\(_2\)) attached to a hydroxyl group (-OH). This confirms the product as: \[ \text{H}_2\text{N} - \text{CH}_2\text{OH} \] This is the final product, and it is a primary amine with a hydroxyl group attached to the adjacent carbon. Thus, the product 'X' is \( \text{H}_2\text{N} - \text{CH}_2\text{OH} \).
Let one focus of the hyperbola $ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 $ be at $ (\sqrt{10}, 0) $, and the corresponding directrix be $ x = \frac{\sqrt{10}}{2} $. If $ e $ and $ l $ are the eccentricity and the latus rectum respectively, then $ 9(e^2 + l) $ is equal to:
The largest $ n \in \mathbb{N} $ such that $ 3^n $ divides 50! is: