Match the compounds in List-I with the appropriate observations in List-II and choose the correct option.
P → 2; Q → 1; R → 5; S → 3
(P): The compound contains an $\alpha$-amino acid backbone → reacts with ninhydrin to give a purple color and phenol group gives violet color with FeCl\(_3\).
\[ \Rightarrow \boxed{P \rightarrow 2} \] (Q): It is an aryl hydrazone derivative. Upon hydrolysis, it gives an amine and a carbonyl. These give positive ninhydrin test but no phthalein dye.
\[ \Rightarrow \boxed{Q \rightarrow 5} \] (R): Aniline salt \( \text{Ph-NH}_3^+ \text{Cl}^- \) reacts with diazonium salts to form azo dyes (yellow).
\[ \Rightarrow \boxed{R \rightarrow 1} \] (S): Phenylhydrazine (NHNH\(_2\)) reacts with glucose to give hydrazones.
\[ \Rightarrow \boxed{S \rightarrow 3} \]
Final Answer: \( \boxed{\text{B}} \)
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 ________.