Select the correct indirect form of the given sentence. “If you don’t…

The correct answer is Option (3) → He warned her calmly that he would shoot her if she didn’t keep quiet.

A threat with "if" in indirect speech:

Change	Direct	Indirect	Why
Reporting verb	said to her in a calm voice	warned her calmly	"I shall shoot you" is a threat, so "said to" becomes "warned". "In a calm voice" becomes the single word "calmly".
Joining word	" " (quotation marks)	that	"That" joins the warning. The main part comes first and the if-part follows it.
Tense change	shall shoot, don't keep	would shoot, didn't keep	"Said" is past. So "shall" becomes "would" (would + base form (V1)). The simple present "don't keep" becomes the simple past "didn't keep". The simple present is formed with the base form (V1), or V1 + s / es with he, she, it. The simple past is formed with the past form of the verb (V2).
Pronoun change	I, you, you	he, her, she	He is the speaker, so "I" becomes "he". She is the listener, so "you" becomes "her" as object and "she" as subject.
Full sentence	"If you don't keep quiet, I shall shoot you," he said to her in a calm voice.	He warned her calmly that he would shoot her if she didn't keep quiet.

Threatened can also report such words. In the same way, will → would and can → could. Example: The coach said to us, "I will drop you if you come late." → The coach warned us that he would drop us if we came late.

Why the Other Options are Incorrect?

Option 1: He warns her calmly that he will shoot her if she didn’t keep quiet. — Incorrect. "Warns" puts the reporting verb in the present. The original "said" is past.

Option 2: He warn her calmly that he will be shooting her if she didn’t keep quiet. — Incorrect. "He warn" is the bare verb with no tense. The reporting verb must be the past "warned".

Option 4: He warned her calmly that he would shoot her if she do not remain silent. — Incorrect. "Do not" is left in the present. After "warned" it becomes "did not".