The reaction sequence provided involves the following steps:
The starting compound is a carboxylic acid (CH₃COOH). The reaction with red phosphorus and bromine results in the formation of an acyl bromide intermediate. This intermediate undergoes hydrolysis to form a product, denoted as \( P \). The next part of the reaction sequence involves treating \( P \) with concentrated \( \text{NH}_4\text{OH} \) solution followed by acidification to yield the product \( Q \), which is likely a compound such as an amide or a derivative of the starting acid.
This statement is incorrect. \( P \) (which is likely an acyl derivative) cannot be reduced to a primary alcohol using sodium borohydride (NaBH₄). NaBH₄ is effective in reducing aldehydes and ketones but not carboxylic acids or their derivatives.
This statement is correct. Treating acyl derivatives with ammonium hydroxide followed by acidification typically yields the corresponding amide or a related compound. This reaction is common in organic synthesis.
This statement is correct. The reaction of an amide (or a similar compound) with sodium nitrite (NaNO₂) in acidic conditions (HCl) results in the formation of nitrogen gas (N₂) due to the diazotization process.
This statement is correct. \( P \), being an acyl derivative, is more acidic than propanoic acid (CH₃CH₂COOH) because the acyl group (RCO-) enhances the ability to donate a proton (H⁺) compared to a simple carboxyl group.
The correct options are: B, C, and D.
Let $ S $ denote the locus of the point of intersection of the pair of lines $$ 4x - 3y = 12\alpha,\quad 4\alpha x + 3\alpha y = 12, $$ where $ \alpha $ varies over the set of non-zero real numbers. Let $ T $ be the tangent to $ S $ passing through the points $ (p, 0) $ and $ (0, q) $, $ q > 0 $, and parallel to the line $ 4x - \frac{3}{\sqrt{2}} y = 0 $.
Then the value of $ pq $ is
Let $ y(x) $ be the solution of the differential equation $$ x^2 \frac{dy}{dx} + xy = x^2 + y^2, \quad x > \frac{1}{e}, $$ satisfying $ y(1) = 0 $. Then the value of $ 2 \cdot \frac{(y(e))^2}{y(e^2)} $ is ________.
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