

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.




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 ________.






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