The question is one of membership, and the IN clause seems to be a very clear way to pose that question. After proper indexing, it yields the same query plan as the methods proposed, but in a more understandable query.

/* Students that have taken the SQL Server test */

SELECT s.*

FROM dbo.Students AS s

WHERE s.StID IN (SELECT StID FROM dbo.StudentExam WHERE ExamName = 'SQL Server')

/* Students that have not taken the SQL Server test */

SELECT s.*

FROM dbo.Students AS s

WHERE s.StID NOT IN (SELECT StID FROM dbo.StudentExam WHERE ExamName = 'SQL Server')