Challenging PROBLEMS for JEE.

The article focuses on a series of advanced problems and solutions related to complex numbers and quadratic equations, primarily aimed at students preparing for competitive exams like the Joint Entrance Examination (JEE). It covers topics such as roots of quadratic equations, properties of complex numbers in the Argand plane, polynomial equations with real coefficients, and inequalities involving real variables. Each problem is followed by a detailed solution that applies algebraic identities, geometric interpretations, and inequalities to determine values such as roots, minimum or maximum expressions, and the nature of solutions. The content emphasizes rigorous mathematical reasoning without introducing extraneous information beyond the stated problems and their solutions. [Extracted from the article]