This question tests your understanding of modal verbs and their appropriate usage in context. Modal verbs express necessity, possibility, permission, or ability. Let's analyze the sentence:

"Each worker works carefully, for there _____ not be any mistakes."

The key phrase is "works carefully, for...". The word "for" here acts as a conjunction meaning "because". The workers are being careful because a certain condition must be met. The sentence implies a strong necessity or obligation to avoid mistakes in the book publishing process. This is a requirement, not just a possibility.

Now, let's evaluate the options:

• might: Expresses possibility or permission. ("It might rain later.") This is too weak for the context of a requirement.
• must: Expresses strong necessity or obligation. ("You must finish your homework.") This fits perfectly, as it states the necessity of having no mistakes.
• has to: Also expresses external obligation or necessity. While grammatically possible, it is less formal and slightly more awkward in this specific sentence structure compared to "must". "Must" is the more direct and conventional modal verb for this context.
• could: Expresses ability or possibility. ("I could help you.") This does not convey the required sense of obligation.

Therefore, the most appropriate word to convey the necessary obligation is must. The complete sentence reads: "Each worker works carefully, for there must not be any mistakes."

Related Topic: Modal Verbs

Modal verbs are auxiliary verbs that modify the main verb to express mood—such as ability, possibility, permission, obligation, or advice. The core modal verbs in English are: can, could, may, might, will, would, shall, should, must. They are followed by the base form of a verb and do not change form (e.g., no -s for third person singular).

Key Formulae/Theory

There is no mathematical formula, but a conceptual rule for choosing modal verbs:

Context → Modal Verb

You must analyze the context of the sentence to determine what is being expressed:

• Obligation/Necessity: Use must or have to.
• Possibility: Use might, may, or could.
• Ability: Use can or could.
• Advice: Use should.