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Complex Numbers — Maths STD 11 Science — Question

Maharashtra BoardEnglish MediumSTD 11 ScienceMathsComplex Numbers2 Marks
MCQ
If $z=\cos \frac{\pi}{6}+i \sin \frac{\pi}{6}$, then
  • A
    $|z|=1, \arg z=\frac{\pi}{4}$
  • ✓
    $|z|=1, \arg z=\frac{\pi}{6}$
  • C
    $|z|=\frac{\sqrt{3}}{2}, \arg z=\frac{5 \pi}{24}$
  • D
    $|z|=\frac{\sqrt{3}}{2}, \arg z=\tan ^{-1} \frac{1}{\sqrt{2}}$
✓

Answer

Correct option: B.
$|z|=1, \arg z=\frac{\pi}{6}$
(B)
$z=\cos \frac{\pi}{6}+ i \sin \frac{\pi}{6}=\frac{\sqrt{3}}{2}+\frac{ i }{2}$
$\therefore \quad|z|=\sqrt{\frac{3}{4}+\frac{1}{4}}=1$
and $\arg ( z )=\tan ^{-1}\left(\frac{y}{x}\right)=\tan ^{-1}\left(\frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}}\right)$
$=\tan ^{-1}\left(\frac{1}{\sqrt{3}}\right)$
$\Rightarrow \arg ( z )=\frac{\pi}{6}$

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