If z = 3 2 + i 2 i = - 1 , then 1 + i z + z 5 + i z 8 9 is equal to:

z = 3 2 + i 2 ∴ i z = - 1 2 + 3 2 i = ω ∴ 1 + i z + z 5 + i z 8 9 = 1 + ω + ω 5 i 5 + i × ω 8 i 8 9 = 1 + ω - i ω 2 + i ω 2 9 = - ω 2 9 = - ω 18 = - 1

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Mathematics - Complex Number Question with Solution | TestHub

MathematicsComplex NumberCube Root of UnityMedium2 minPYQ_2019

MathematicsMediumsingle choice

If $z = \frac{\sqrt{3}}{2} + \frac{i}{2} (i = \sqrt{- 1}),$ then ${(1 + i z + z^{5} + i z^{8})}^{9}$ is equal to:

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Answer:

Solution:

$z = \frac{\sqrt{3}}{2} + \frac{i}{2} ∴ i z = - \frac{1}{2} + \frac{\sqrt{3}}{2} i = ω$
$∴ {(1 + i z + z^{5} + i z^{8})}^{9} = {(1 + ω + \frac{ω^{5}}{i^{5}} + i \times \frac{ω^{8}}{i^{8}})}^{9}$
$= {(1 + ω - i ω^{2} + i ω^{2})}^{9}$
$= {(- ω^{2})}^{9} = - ω^{18}$
$= - 1$

Stream:JEESubject:MathematicsTopic:Complex NumberSubtopic:Cube Root of Unity

⏱ 2mℹ️ Source: PYQ_2019

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