Take a look at the sentence again:
Defeated, Melody __________ her head on the desk. Even though she had studied
until 3 a.m., she could make no sense of the chemistry problems on the exam
before her.
Melody put her head on the desk. Thus, you need a form of lay. Since the sentence is past tense, you need the simple past form. Consult the table below or review the rules.
Infinitive | Definition | Simple Present | Simple Past | Past Participle | Present Participle |
to lay | to put something down | lay(s) | laid | laid | laying |
to lie | to rest or recline | lie(s) | lay | lain | lying |