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JEE Main 2024 · Mathematics · Complex Numbers · medium

The principal argument of z=((1+i)^8/((1-i)^6 is

Single correct option

The principal argument of z=(1+i)8(1−i)6z=\dfrac{(1+i)^8}{(1-i)^6} is
  1. A. π2\dfrac{\pi}{2}
  2. B. 3π4\dfrac{3\pi}{4}
  3. C. π\pi
  4. D. −π2-\dfrac{\pi}{2}
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Answer: (D) −π2-\dfrac{\pi}{2}

(1+i)2=2i(1+i)^2=2i, so (1+i)8=(2i)4=16(1+i)^8=(2i)^4=16. Also (1−i)2=−2i(1-i)^2=-2i, so (1−i)6=(−2i)3=8i(1-i)^6=(-2i)^3=8i. Then z=168i=2i=−2iz=\dfrac{16}{8i}=\dfrac{2}{i}=-2i, which lies on the negative imaginary axis, so arg⁡z=−π/2\arg z=-\pi/2. Using π/2\pi/2 comes from forgetting the sign when dividing by ii.
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