This question tests your understanding of participles and how they function in sentences to show condition or circumstance.

The sentence is: "Mr. Walker, _____ the opportunity, could become a first-class mathematician."

The blank needs a word that describes the condition under which Mr. Walker could become a great mathematician. The phrase is providing extra information about the subject, "Mr. Walker".

Let's analyze the options:

• to be give: This is grammatically incorrect. "To be" should be followed by a past participle ("given"), not a base verb ("give").
• given: This is the past participle of the verb "give". A past participle can be used to start a phrase that describes a condition. It implies a passive meaning: if Mr. Walker is given the opportunity.
• giving: This is the present participle. It implies an active meaning: if Mr. Walker is giving the opportunity. This doesn't make logical sense because Mr. Walker would be receiving the opportunity, not giving it away.
• to give: This is the infinitive form. It suggests purpose or intention (e.g., "He needs a platform to give a speech"). It does not fit here to describe a condition for Mr. Walker.

The correct choice is given. The phrase "given the opportunity" is a common participle phrase meaning "if he were provided with the opportunity".

Final Answer: The most appropriate word to fill in the blank is given.
The complete sentence is: "Mr. Walker, given the opportunity, could become a first-class mathematician."

Related Topic: Participles

A participle is a verb form that can function as an adjective to modify a noun or pronoun. There are two types:

• Present Participle: Ends in -ing (e.g., giving, running). It describes an action that is ongoing.
• Past Participle: Usually ends in -ed, -d, -t, -en, or -n (e.g., given, walked, burnt). It often describes a completed action or a passive state.

In this question, the past participle "given" is used to create an adjective phrase that modifies "Mr. Walker" and sets up a conditional statement.

Related Formulae

While there isn't a mathematical formula for grammar, you can use this logical structure:

Subject + [Participle Phrase] + Main Verb
The participle phrase describes the condition of the subject for the main action to be true.