Let α and β be the roots of the equation x 2 + 2 x + 2 = 0 , then α 15 + β 15 is equal to

Let α and β be the roots of the equation x2+2x+2=0, then α15+β15 is equal to
  1. -512
  2. 128
  3. 512
  4. -256

Solution

Let the two roots of  x2+2x+2=0 be α and β

⇒α,β=-2±4-82=-1±i

⇒α,β=2-12±i12

=2cos3π4±isin3π4

⇒α15+β15=2cos3π4+isin3π415+2cos 3π4-isin3π415

       ⇒α15+β15=2152cos45π4+isin45π4+cos45π4-isin45π4

⇒α15+β15=2.2152.cos45π4=2172.cos5π4=-2172.12

⇒α15+β15=-256

Asked in: JEE Main 2019 (09 Jan Shift 1)

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