Flight instructors are given the authority to develop and administer pre-solo exams with few restrictions imposed by the FAA. The instructor may choose which questions to ask, how many questions to ask, and whether to use multiple-choice or supply-type questions.
I have personally seen pre-solo exams scribbled on paper the size of an index card. Technically, this is not a problem because the regulations do not require a minimal number of questions on a pre-solo exam. Instructors are only required to test their student on the applicable regulations, airspace rules and procedures, and the flight characteristics and operational limitations for the make and model of aircraft to be soloed; however, when the pre-solo exam is too short, it will not be a useful teaching opportunity.
The Federal Aviation Regulations do not require the written exam to be closed-book. Many instructors and the AOPA Air Safety Foundation recommend that students look questions they are having trouble answering by referring to the applicable publications. This way, the student will become familiar with finding information on their own. The instructor should ensure that the student has access to a FAR / AIM book, an airport facility directory, and a sectional chart covering the flight training area.
Student pilots are required to complete the instructor's exam and correct their answers with the instructor before flying solo. Judging the responses to the student's answers, the instructor must decide if the student meets the aeronautical requirements to operate an aircraft in solo flight. If the student has not demonstrated an understanding of the test material, the instructor should provide additional ground instruction. If necessary, the student could be asked to review deficient knowledge areas at home and pass another exam given by the instructor, possibly even a brief oral quiz.
Pre-solo exams do not have to be kept by the student or instructor; however, when the instructor endorses the student's logbook for solo flight privileges, the instructor must record the student's name and keep the record for at least three years.
I always keep pre-solo exams that I administer for at least 90 days beyond the date of the student's solo endorsement. This corresponds to the expiration date of my endorsement. In most cases, the student earns his or her private pilot's license in this time frame and I will trash the test.
I have students that would like to keep their test for studying purposes. Since I like to keep exams that I administer, I created an exact copy of my pre-solo exam filled out with the appropriate answers to give the student in exchange for their completed copy. This keeps everyone happy.