If $|\ln x| = px$ has exactly three distinct solutions, then find $[p]$ (where $[.]$ denote greater integer function).
Step-by-Step Solution
Key Concept: Tangency conditions and intersection counts are determined by analyzing when the linear approximation transitions from missing to touching to crossing the logarithmic curve.
The line $y = px$ is tangent to $y = |\ln x|$ when it touches the curve. For the tangent line to have exactly three intersections with $y = |\ln x|$, we require the slope $p$ to be in the range $\left(0, \frac{1}{e}\right)$. At the critical value $p = \frac{1}{e}$, the line touches $y = \ln x$ at $x = e$. Therefore $[p] = 0$.
Correct Answer: 0