Select the most appropriate option to fill in the blank in the follow…

The correct answer is Option (2) → could

The sentence is a reported (indirect) question. The manager's direct words would be a question, "Can you complete the report by the end of the week?"

Reporting a yes/no question:

Change	Direct	Indirect	Why
Reporting verb	said to	asked	“Said to” becomes asked, because a question is reported.
Joining word	–	if	A yes/no question is joined with if (or whether).
Pronoun change	you	they	“You” becomes they, the people the manager spoke to.
Tense change	can	could	After the past verb “asked”, can becomes could (will → would, shall → should, may → might).
Word order	Can you complete ...?	they could complete ...	Statement order, “they could complete”. The question mark goes.
Full sentence	The manager said to them, “Can you complete the report by the end of the week?”	The manager asked if they could complete the report by the end of the week.

"By the end of the week" stays as it is. The blank takes could, which here asks about their ability to finish the work in time.

Example: She asked me if I could lend her my notes.

Why the Other Options are Incorrect?

Option 1: will — Incorrect. Will is not backshifted. After the past verb "asked" it should become would. "Asked if they will" mixes past and present time.
Example: I think it will rain this evening.

Option 3: shall — Incorrect. Shall is a present-time modal used mainly with I and we for offers and suggestions ("Shall we go?"). After "asked" it backshifts to should, so it does not fit a past reported question.
Example: Shall I close the window?

Option 4: would have — Incorrect. Would have is a perfect modal that talks about an imagined past result.

• It must be followed by a past participle (would have completed). "Asked if they would have complete the report" is ungrammatical.
• The question is about a task still ahead, not a past possibility.
  Example: She would have come if you had invited her.

Contrast: after the past verb "asked", can becomes could, just as will becomes would. Neither stays unchanged.

Note: In the original NTA paper, Option 4 read "should". NTA accepted this option as a correct answer as well. To leave only one correct answer and avoid confusion for students, it has been changed here to "would have".

Advice for the exam: In the real exam, mark the answer that is grammatically or factually correct. If NTA’s provisional answer key marks a different option, challenge it during the answer-key challenge window and give the reason in one or two lines.