The electron configuration for potassium (\( \text{K} \), atomic number 19) is:
\(\text{K} : 1s^2, 2s^2, 2p^6, 3s^2, 3p^6, 4s^1.\)
The outermost orbital is the \( 4s \) orbital. The quantum numbers for the outermost electron are:
\( n = 4, \, l = 0, \, m = 0, \, s = +\frac{1}{2}. \)
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