The correct matching is as follows:
- A. Lyman series corresponds to the UV region (II) as it involves transitions to the \( n=1 \) energy level.
- B. Balmer series corresponds to the Visible region (IV) as it involves transitions to the \( n=2 \) energy level.
- C. Paschen series corresponds to the Infrared region (III) as it involves transitions to the \( n=3 \) energy level.
- D. Pfund series corresponds to the Infrared region (I) as it involves transitions to the \( n=5 \) energy level.
The Correct Answer is: \( A - II, B - IV, C - III, D - I \)
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