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NDA 2025 · PYQ · Complex Numbers · medium

What is ((√3 + i)/(√3 - i))³ equal to?

Answer

The correct answer is A: -1. Rationalize: (√3 + i)/(√3 - i) = (√3 + i)²/((√3)² + 1) = (3 + 2√3 i - 1)/4 = (2 + 2√3 i)/4 = (1 + √3 i)/2. This equals cos60° + i sin60° = e^(iπ/3).

  1. A.-1✓ Correct
  2. B.0
  3. C.1
  4. D.3

Explanation

Rationalize: (√3 + i)/(√3 - i) = (√3 + i)²/((√3)² + 1) = (3 + 2√3 i - 1)/4 = (2 + 2√3 i)/4 = (1 + √3 i)/2. This equals cos60° + i sin60° = e^(iπ/3). Cubing: e^(iπ) = -1.
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