In modern conditions, the relevance of the monitoring of the quality of the preparation for EIA does not raise doubts about the mathematics of senior school pupils. Particularly, very important is the timely diagnosis of existing gaps in the preparation of theoretical material and during solving test tasks of various forms during all the term of studying in grade 11. In this article, we present one of the possible models for organizing a system for monitoring the preparation for the external assessment in mathematics of Ukrainian graduates. We will also develop methodological guidelines for the creation and implementation of diagnostic and training tests during preparation for the independent assessment of the quality of knowledge in mathematics. Special attention is paid to preparation for solving problems of open form with full explanation. In particular, we present here complete solutions, evaluation schemes and methodological comments for such tasks. The given scheme of monitoring organization and the mentioned above methodological recommendations should help to ensure the proper quality of training of graduate pupils to the EIA in mathematics.