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Complex Numbers — Maths STD 11 Science — Question

Maharashtra BoardEnglish MediumSTD 11 ScienceMathsComplex Numbers2 Marks
MCQ
If $z=\frac{\sqrt{3}+i}{2}$, then $z^{69}$ is equal to
  • ✓
    $- i$
  • B
    i
  • C
    1
  • D
    $- i \sqrt{3}$
✓

Answer

Correct option: A.
$- i$
(A)
$z^{69}=\left(\frac{\sqrt{3}+i}{2}\right)^{69}=\left[\frac{1}{i}\left(\frac{-1+\sqrt{3} i}{2}\right)\right]^{69}$
$=\left(\frac{\omega}{i}\right)^{69}$
$=\frac{\omega^{69}}{\left(i^4\right)^{17} i}=\frac{1}{i}=-i$

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