Maharashtra BoardEnglish MediumSTD 11 ScienceMathsComplex Numbers2 Marks
MCQ
If $|z|=4$ and $\arg z=\frac{5 \pi}{6}$, then $z=$
A
$2 \sqrt{3}-2 i$
B
$2 \sqrt{3}+2 i$
✓
$-2 \sqrt{3}+2 i$
D
$-\sqrt{3}+i$
✓
Answer
Correct option: C.
$-2 \sqrt{3}+2 i$
(C) Let $z =x+ i y$, then $| z |= r =\sqrt{x^2+y^2}=4$ and $\theta=\frac{5 \pi}{6}=150^{\circ}$ $\therefore \quad x= r \cos \theta=4 \cos 150^{\circ}=-2 \sqrt{3}$ and $y= r \sin \theta=4 \sin 150^{\circ}=\frac{4}{2}=2$ $\therefore \quad z =x+ i y=-2 \sqrt{3}+2 i$ Trick:Since $\arg z=\frac{5 \pi}{2}=150^{\circ}$, here the complex number must lie in second quadrant, so (A) and (B) are rejected. Also $|z|=4$ which satisfies (C) only.
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