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: \( \boxed{\text{B}} \)


The center of a disk of radius $ r $ and mass $ m $ is attached to a spring of spring constant $ k $, inside a ring of radius $ R>r $ as shown in the figure. The other end of the spring is attached on the periphery of the ring. Both the ring and the disk are in the same vertical plane. The disk can only roll along the inside periphery of the ring, without slipping. The spring can only be stretched or compressed along the periphery of the ring, following Hooke’s law. In equilibrium, the disk is at the bottom of the ring. Assuming small displacement of the disc, the time period of oscillation of center of mass of the disk is written as $ T = \frac{2\pi}{\omega} $. The correct expression for $ \omega $ is ( $ g $ is the acceleration due to gravity): 
Let $ a_0, a_1, ..., a_{23} $ be real numbers such that $$ \left(1 + \frac{2}{5}x \right)^{23} = \sum_{i=0}^{23} a_i x^i $$ for every real number $ x $. Let $ a_r $ be the largest among the numbers $ a_j $ for $ 0 \leq j \leq 23 $. Then the value of $ r $ is ________.