The integration of generative artificial intelligence (GenAI) into web-based individual formative e-assessments in higher education is a nascent field that warrants further exploration. This study investigated the use of GenAI within an 8-week undergraduate-level research methods course at a university in the United States of America, aiming to understand how students leverage GenAI tools during individual formative e-assessments questions. The research revealed that a significant majority of students initially preferred traditional study resources over GenAI. However, a gradual shift towards more balanced use of both resources was observed, particularly in formative e-assessments involving statistical analysis and calculation questions. In their interactions with GenAI, students primarily used it for multiple-choice and true/false questions, often by directly copying and pasting the question prompt into the GenAI interface. Students were able to discern and accept accurate responses generated by GenAI and reject those that were incorrect or contradicted their existing knowledge. Students' reported primary motivations for turning to GenAI were to seek answers to assessment items as well as to corroborate the accuracy of their own responses. This study contributes to the growing body of literature empirically investigating actual usage behaviours with GenAI tools and the motivation behind these behaviours. We discuss the implications and limitations of these findings.