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Complex Numbers — Maths STD 11 Science — Question

Maharashtra BoardEnglish MediumSTD 11 ScienceMathsComplex Numbers2 Marks
MCQ
If $z=\left(\frac{\sqrt{3}}{2}+\frac{i}{2}\right)^5+\left(\frac{\sqrt{3}}{2}-\frac{i}{2}\right)^5$, then
  • A
    $\operatorname{Re}( z )=0$
  • ✓
    $\operatorname{Im}( z )=0$
  • C
    $\operatorname{Re}(z)>0, \operatorname{Im}(z)>0$
  • D
    $\operatorname{Re}(z)>0, \operatorname{Im}(z)<0$
✓

Answer

Correct option: B.
$\operatorname{Im}( z )=0$
(B)
Given that $z=\left(\frac{\sqrt{3}}{2}+i \frac{1}{2}\right)^5+\left(\frac{\sqrt{3}}{2}-i \frac{1}{2}\right)^5$
$=\left[\cos \left(\frac{\pi}{6}\right)+ i \sin \left(\frac{\pi}{6}\right)\right]^5+\left[\cos \left(\frac{\pi}{6}\right)- i \sin \left(\frac{\pi}{6}\right)\right]^5$
$=\cos \frac{5 \pi}{6}+i \sin \frac{5 \pi}{6}+\cos \frac{5 \pi}{6}-i \sin \frac{5 \pi}{6}$
Hence, $\operatorname{Im}( z )=0$

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