If z is a complex number, then the number of common roots of the equations z 1985 + z 100 + 1 = 0 and z 3 + 2 z 2 + 2 z + 1 = 0 , is equal to [2024]

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Solution

Q.
If $z$ is a complex number, then the number of common roots of the equations $z^{1985} + z^{100} + 1 = 0$ and $z^{3} + 2 z^{2} + 2 z + 1 = 0,$ is equal to [2024]

1 2

2 3

3 1

4 0

Ans.
(1)

Given equation, $z^{1985} + z^{100} + 1 = 0$ ...(i)

and $z^{3} + 2 z^{2} + 2 z + 1 = 0$ ...(ii)

From (ii), we have $(z + 1) (z^{2} + z + 1) = 0$

$\Rightarrow z = - 1, z = ω, ω^{2}$

Clearly, $z = - 1$ does not satisfy equation (i).

$∴$ If $z = ω,$ then ${(ω)}^{1985} + {(ω)}^{100} + 1 = ω^{2} + ω + 1 = 0$

Also, $z = ω^{2}$ , then ${(ω^{2})}^{1985} + {(ω^{2})}^{100} + 1 = ω^{2} + ω + 1 = 0$

$∴$ Number of roots = 2

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