The Joint Entrance Examination (JEE), the gateway to the IITs—India’s prestigious institutes of learning—is considered as one of the toughest examinations conducted in India. Inevitably, there is competition to succeed in it. While such a competition is not bad in itself, the undesirable consequence is that in the mad rush to solve problems, the educative value of a problem is often lost. The thought process of students is thus skewed into solving problems (mostly by analogy) without thoroughly understanding the basic concepts lying behind the problems and without a corresponding development of the thinking ability. Consequently, they are often unable to offer logical negation of a statement, even though they are intellectually capable of it. This book is an attempt to lay before students both the concepts as well as the process of solving problems at the JEE. The problem have been collected mostly from JEE papers ranging over a period of two decades. The thrust is not so much on solving the problems as in the assimilation of the theory behind them and learning a few related new concepts. The solutions then come as natural corollaries. The book also contains some helpful tips aimed at sharpening the thinking ability and increasing the mathematical maturity of the student. New problems and an occasional new solution to an existing problem were added in the second edition. In this edition, two appendices, one on matrices and the other on solid coordinate geometry, have been added in view of the changes in JEE syllabus. Further, all the errors in the earlier two editions, listed as errata in the author’s blog, have been corrected, and occasionally, a new solution to an existing problem has been included.