




To identify the product (P) in the given reaction, we need to analyze the reaction conditions and the reactants involved.
This option depicts the cyclic structure with a bromine atom attached to the α-carbon, consistent with the reaction mechanism.
To solve this problem, we need to identify what product (P) is formed when the given reaction takes place. The reaction involves a carboxylic acid treated with \(\text{Br}_2/\text{Red P}\) followed by hydrolysis.
This type of reaction is known as the Hell-Volhard-Zelinsky (HVZ) reaction, which is typically used for the alpha-bromination of carboxylic acids.
Thus, the final product is an alpha-bromo carboxylic acid where the bromine atom is present at the alpha position. The correct product (P) matches the option shown in the image above.
In the given reaction sequence, the structure of Y would be:

Let \( C_{t-1} = 28, C_t = 56 \) and \( C_{t+1} = 70 \). Let \( A(4 \cos t, 4 \sin t), B(2 \sin t, -2 \cos t) \text{ and } C(3r - n_1, r^2 - n - 1) \) be the vertices of a triangle ABC, where \( t \) is a parameter. If \( (3x - 1)^2 + (3y)^2 = \alpha \) is the locus of the centroid of triangle ABC, then \( \alpha \) equals: