This question tests your understanding of passive voice and infinitive phrases in English grammar. The sentence structure requires a phrase that correctly modifies the noun "work" and indicates a necessity or obligation that exists in the present.

The main clause is: "There is a lot of work." We need to complete it with a phrase that explains what must happen to this work.

Let's analyze the options:

Option 1: "done today"
This is a past participle phrase. While it describes the work, it implies the work is already finished ("work that is done today"), which contradicts the first sentence "I have to do a lot of work today." This is incorrect.

Option 2: "be done today"
This is a bare infinitive ("be"). Infinitives used as noun modifiers almost always require the particle "to". A bare infinitive cannot correctly follow the noun "work" in this context. This is grammatically incorrect.

Option 3: "to be done today"
This is a passive infinitive phrase ("to" + "be" + past participle "done"). The passive voice is used here because the work is not doing the action; the action is being done to the work. The infinitive "to be done" functions as an adjective modifying the noun "work," indicating necessary future action. This perfectly matches the meaning of the first sentence, which expresses obligation.

Option 4: "being done today"
This is a present continuous passive participle phrase. It describes an action that is currently in progress ("work that is being done right now"). The first sentence, however, talks about work that still needs to be started and completed ("I have to do"), not work that is already underway. This creates a slight contradiction in meaning.

Conclusion: The most appropriate word is "to be done today". The full correct sentence is: I have to do a lot of work today. There is a lot of work to be done today.

Related Concepts & Formulae

1. The Passive Infinitive:
Structure: $\mathrm{to} \mathrm{be} + \mathrm{past} \mathrm{participle}$
Use: Used after nouns to indicate that something needs to or can be done.
Example: "This is a letter to be sent immediately."

2. Participle Phrases vs. Infinitive Phrases:
Participle Phrases (ending in -ing or -ed) describe a noun by showing an action that is happening simultaneously or is completed.
Infinitive Phrases (beginning with "to") often describe a noun by showing purpose, intention, or necessity.