Let Z = r e i θ ( r 0 and π θ ≤ 3 π ) is a root of the equation Z 8 - Z 7 + Z 6 - Z 5 + Z 4 - Z 3 + Z 2 - Z + 1 = 0 . If the sum of all values of θ is k π , then k is equal to

Question

Let Z = r e i θ ( r > 0 and π θ ≤ 3 π ) is a root of the equation Z 8 - Z 7 + Z 6 - Z 5 + Z 4 - Z 3 + Z 2 - Z + 1 = 0 . If the sum of all values of θ is k π , then k is equal to,

MARKS App · Accepted Answer

Given equation is - Z 9 - 1 - Z - 1 = 0 ⇒ Z 9 = - 1 w h e r e Z ≠ - 1 ⇒ Z = e i 2 k π 9 , where k= 5 , 6 , 7 , 8 , 10 , 11 , 12 , 13 ⇒ θ = 10 π 9 , 12 π 9 , 14 π 9 , 16 π 9 , 20 π 9 , 22 π 9 , 24 π 9 , 26 π 9 ⇒ the sum of all values of θ is equal to 16 π ⇒ k = 16

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