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Complex Numbers — Maths STD 11 Science — Question

Maharashtra BoardEnglish MediumSTD 11 ScienceMathsComplex Numbers2 Marks
MCQ
$(-\sqrt{3}+i)^{53}$, where $i^2=-1$, is equal to
  • A
    $2^{53}(\sqrt{3}+2 i)$
  • B
    $2^{52}(\sqrt{3}- i )$
  • ✓
    $2^{53}\left(\frac{\sqrt{3}}{2}+\frac{1}{2} i \right)$
  • D
    $2^{53}(\sqrt{3}-i)$
✓

Answer

Correct option: C.
$2^{53}\left(\frac{\sqrt{3}}{2}+\frac{1}{2} i \right)$
(C)
$(-\sqrt{3}+ i )^{53}=2^{53}\left(\frac{-\sqrt{3}}{2}+\frac{ i }{2}\right)^{53}$
$=2^{53}\left(\cos 150^{\circ}+ i \sin 150^{\circ}\right)^{53}$
$=2^{53}\left[\cos \left(150^{\circ} \times 53\right)+i \sin \left(150^{\circ} \times 53\right)\right]$
$=2^{53}\left[\cos \left(22 \pi+30^{\circ}\right)+i \sin \left(22 \pi+30^{\circ}\right)\right]$
$=2^{53}\left[\cos 30^{\circ}+i \sin 30^{\circ}\right]$
$=2^{53}\left[\frac{\sqrt{3}}{2}+ i \frac{1}{2}\right]$

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