The Fourth Amendment protects individuals from unreasonable searches and seizures. Therefore, the action that would be illegal under the Fourth Amendment is:
random searches of personal property
The other options involve different constitutional issues but do not pertain to the Fourth Amendment specifically. For example, "cruel and unusual punishment" relates to the Eighth Amendment, and "praying aloud in school" concerns the First Amendment. Book burning could be a First Amendment issue as well, relating to freedom of speech and expression.