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4-1.Complex numbers
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સંકર સંખ્યાનો માનાંક અને કોણાંક શોધો. $z=-\sqrt{3}+i$
Option A
Option B
Option C
Option D
Solution
$z=-\sqrt{3}+i$
Let $r \cos \theta=-\sqrt{3}$ and $r \sin \theta=1$
On squaring and adding, we obtain
$r^{2} \cos ^{2} \theta+r^{2} \sin ^{2} \theta=(-\sqrt{3})^{2}+1^{2}$
$\Rightarrow r^{2}=3+1=4 \quad\left[\cos ^{2} \theta+\sin ^{2} \theta=1\right]$
$\Rightarrow r=\sqrt{4}=2 \quad[\text { Conventionally }, r>0]$
$\therefore$ Modulus $=2$
$\therefore 2 \cos \theta=-\sqrt{3}$ and $2 \sin \theta=1$
$\Rightarrow \cos \theta=\frac{-\sqrt{3}}{2}$ and $\sin \theta=\frac{1}{2}$
$\therefore \theta=\pi-\frac{\pi}{6}=\frac{5 \pi}{6}$ [As $\theta$ lies in the $II$ quadrant]
Thus, the modulus an argument of the complex number $-\sqrt{3}+i$ are $2$ and $\frac{5 \pi}{6}$ respectively.
Standard 11
Mathematics
