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NDA 2020 Maths · Mathematics · Complex Numbers and Quadratic Equations · hard

The principal argument of (-1+i)^3 is

Single correct option

The principal argument of (−1+i)3(-1+i)^3 is
  1. A. 9π4\dfrac{9\pi}{4}
  2. B. 3π4\dfrac{3\pi}{4}
  3. C. π4\dfrac\pi4
  4. D. −π4-\dfrac{\pi}{4}
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Answer: (C) π4\dfrac\pi4

−1+i=2 ei3π/4-1+i=\sqrt2\,e^{i3\pi/4}, so the cube has argument 9π/49\pi/4. Reducing into (−π,π](-\pi,\pi] gives 9π/4−2π=π/49\pi/4-2\pi=\pi/4. Indeed (−1+i)3=2+2i(-1+i)^3=2+2i, which lies in the first quadrant. The value 9π/49\pi/4 is not a principal value.
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Original question written in the NDA 2020 Maths pattern and checked twice for the correct answer. Not an official question paper. How we write and check questions · Corrections policy · Report a mistake