Which one of the following is a root of equation-II?

Consider equation-I: z^{3}+2z^{2}+2z+1=0 and equation-II: z^{1985}+z^{100}+1=0.

  1. A. -1
  2. B. -\omega
  3. C. -\omega^{2}
  4. D. \omega ✓

Correct Answer: D. \omega

Explanation

Checking z = \omega in Equation-II: \omega^{1985} + \omega^{100} + 1 = 0. Since \omega^3 = 1, we divide the exponents by 3. 1985 = 3 \times 661 + 2, so \omega^{1985} = \omega^2. 100 = 3 \times 33 + 1, so \omega^{100} = \omega. The expression becomes \omega^2 + \omega + 1 = 0, which is a known identity. Thus, \omega is a root.

Related questions on Algebra

Practice more NDA Mathematics questions