In the given reaction sequence, the structure of Y would be:

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 ________.
Aldehydes, Ketones, and Carboxylic Acids are carbonyl compounds that contain a carbon-oxygen double bond. These organic compounds are very important in the field of organic chemistry and also have many industrial applications.
Aldehydes are organic compounds that have the functional group -CHO.
Preparation of Aldehydes
Acid chlorides are reduced to aldehydes with hydrogen in the presence of palladium catalyst spread on barium sulfate.
Ketones are organic compounds that have the functional group C=O and the structure R-(C=O)-R’.
Preparation of Ketones
Acid chlorides on reaction with dialkyl cadmium produce ketones. Dialkyl cadmium themselves are prepared from Grignard reagents.
Carboxylic acids are organic compounds that contain a (C=O)OH group attached to an R group (where R refers to the remaining part of the molecule).
Preparation of Carboxylic Acids
Primary alcohols are readily oxidized to carboxylic acids with common oxidizing agents such as potassium permanganate in neutral acidic or alkaline media or by potassium dichromate and chromium trioxide in acidic media.