Phthalimide undergoes the following transformations:
In the presence of KOH, the acidic hydrogen of phthalimide is removed, forming the phthalimide anion.
The phthalimide anion then reacts with benzyl chloride via an S\(_N2\) reaction, leading to the formation of product 'P'.
In product 'P':
The benzene ring contributes \(3 \, \pi\)-bonds.
Each carbonyl group (C=O) provides \(2 \, \pi\)-bonds, giving a total of \(4 \, \pi\)-bonds for both groups.
The benzyl group attached to nitrogen contributes \(1\) additional \(\pi\)-bond.
Thus, the total number of \(\pi\)-bonds in product 'P' is:
\[3 + 2 + 2 + 1 = 8\]
Let \( y = f(x) \) be the solution of the differential equation
\[ \frac{dy}{dx} + 3y \tan^2 x + 3y = \sec^2 x \]
such that \( f(0) = \frac{e^3}{3} + 1 \), then \( f\left( \frac{\pi}{4} \right) \) is equal to:
Find the IUPAC name of the compound.
If \( \lim_{x \to 0} \left( \frac{\tan x}{x} \right)^{\frac{1}{x^2}} = p \), then \( 96 \ln p \) is: 32