Choose the most appropriate option to fill in the blank in the given…

The correct answer is Option (3) → unless

Conjunction of condition:

• Unless means "if not" or "except when"; it introduces the only condition under which the main statement does not hold.
• This sentence: the main clause gives a rule (never argue with heat and clamour), and the "unless" clause gives the one exception (when you think or know that you are right).
• Deciding clue: the rule opens with "Never", so the blank must bring in the one condition that lifts it.
• Complete sentence: Never maintain an argument with heat and clamour unless you think or know yourself to be in the right.
  Example: Do not sign the form unless you have read every line.

Why the Other Options are Incorrect?

Option 1: while — Incorrect. While shows that two things happen at the same time, or a contrast; the sentence needs a condition, not a time link.
Example: She hummed a tune while she worked.

Option 2: before — Incorrect. Before shows that one action comes earlier in time than another; "never argue before you think you are right" gives no exception to the rule.
Example: Wash your hands before you eat.

Option 4: because — Incorrect. Because gives a reason; the sentence would then forbid arguing for the very reason that you are right, which makes no sense.
Example: He stayed at home because he was unwell.

Contrast: "unless" sets a condition (if not); "while" links two things happening at the same time, "before" shows an earlier time, and "because" gives a reason.