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NDA 2023
Exam
NDA
Year
2023
Subject
Mathematics
Chapter
Complex Number

NDA 2023 · Mathematics · Complex Number

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

Consider equation-I: z3+2z2+2z+1=0z^3+2 z^2+2 z+1=0 and equation -II: z1985+z100+1=0z^{1985}+z^{100}+1=0 What are the roots of equation – I?

Options

  1. A1,ω,ω21, \omega, \omega^2
  2. B−1,ω,ω2-1, \omega, \omega^2
  3. C1,−ω,ω21,-\omega, \omega^2
  4. D−1,−ω,−ω2-1,-\omega,-\omega^2

Correct answer

B. −1,ω,ω2-1, \omega, \omega^2

Step-by-step solution

Given the equation−Iisequation-I isz3+2z2+2z+1=0⇒z3+z2+z2+z+z+1=0⇒z2(z+1)+z(z+1)+(z+1)=0⇒(z+1)(z2+z+1)=0⇒z=−1,ω,ω2\begin{aligned} & z^3+2 z^2+2 z+1=0 \Rightarrow z^3+z^2+z^2+z+z+1=0 \\ \Rightarrow & z^2(z+1)+z(z+1)+(z+1)=0 \\ \Rightarrow & (z+1)\left(z^2+z+1\right)=0 \\ \Rightarrow & z=-1, \omega, \omega^2 \end{aligned}$$
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