The given reaction is a Wolff-Kishner reduction, which is used to reduce carbonyl groups (aldehydes and ketones) to alkanes.
The first step involves the formation of a hydrazone derivative: \[ \text{CH}_3 - \text{CO} - \text{CH}_2 - \text{CH}_3 \xrightarrow{\text{N}_2\text{H}_4} \text{CH}_3 - \text{C}(\text{NHNH}_2) - \text{CH}_2 - \text{CH}_3. \]
In the presence of ethylene glycol and KOH, the hydrazone undergoes decomposition to form: \[ \text{CH}_3 - \text{CH}_2 - \text{CH}_2 - \text{CH}_3. \]
Thus, the product 'A' is butane.
For $ \alpha, \beta, \gamma \in \mathbb{R} $, if $$ \lim_{x \to 0} \frac{x^2 \sin \alpha x + (\gamma - 1)e^{x^2} - 3}{\sin 2x - \beta x} = 3, $$ then $ \beta + \gamma - \alpha $ is equal to:
The largest $ n \in \mathbb{N} $ such that $ 3^n $ divides 50! is: