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For a complex number Z , if all the roots of the equation Z 3 + a Z 2 + b Z + c = 0 are unimodular, then

Options

  1. Aa > 3 and c = 1
  2. Ba ≤ 3 and c = 3
  3. Ca > 3 and c = 1 3
  4. Da ≤ 3 & c = 1

Correct answer

D. a ≤ 3 & c = 1

Step-by-step solution

Given, Z 1 = Z 2 = Z 3 = 1 Now, Z 1 + Z 2 + Z 3 ≤ Z 1 + Z 2 + Z 3 ⇒ - a ≤ 1 + 1 + 1 ⇒ a ≤ 3 Also, Z 1 Z 2 Z 3 = Z 1 × Z 2 × Z 3 ⇒ - c = 1 × 1 × 1 ⇒ c = 1

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