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NDA 2016 Maths · Mathematics · Complex Numbers and Quadratic Equations · hard

The product of all the sixth roots of unity is

Single correct option

The product of all the sixth roots of unity is
  1. A. 11
  2. B. 00
  3. C. 66
  4. D. −1-1
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Answer: (D) −1-1

The sixth roots of unity are the roots of z6−1=0z^6-1=0. The product of the roots of a monic polynomial of degree nn is (−1)n(-1)^n times the constant term, i.e. (−1)6(−1)=−1(-1)^6(-1)=-1. Equivalently the roots pair as conjugates giving 1⋅(−1)⋅∣ω∣2∣ω2∣2=−11\cdot(-1)\cdot|\omega|^2|\omega^2|^2=-1. The sum of the roots is 00, which is a common mix-up.
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Original question written in the NDA 2016 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