Let . What is the smallest positive integer for which is purely imaginary with a positive imaginary part?

Answer and explanation

Correct answer: D. 9

  1. Step 1: Express in polar/modulus-argument form: modulus , and argument .
  2. Step 2: Apply de Moivre's theorem: . For to be purely imaginary, the real part must vanish, requiring , which occurs when .
  3. Step 3: Multiply by : . For the smallest positive integer ( to give non-zero imaginary component ), (since gives , but checking orientation gives ).
Answer
9.

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