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4-1.Complex numbers
normal
If complex numbers $z_1$, $z_2$ are such that $\left| {{z_1}} \right| = \sqrt 2 ,\left| {{z_2}} \right| = \sqrt 3$ and $\left| {{z_1} + {z_2}} \right| = \sqrt {5 - 2\sqrt 3 }$, then the value of $|Arg z_1 -Arg z_2|$ is
A
$\frac{{2\pi }}{3}$
B
$\frac{\pi }{3}$
C
$\frac{\pi }{4}$
D
$\frac{{3\pi }}{4}$
Solution
$\cos \theta=-\frac{(5-2 \sqrt{3})+2+3}{2 \sqrt{6}}$
$=\frac{2 \sqrt{3}}{2 \sqrt{2} \sqrt{3}}=\frac{1}{\sqrt{2}}$
$\theta=\frac{\pi}{4} \Rightarrow \arg \left(\frac{z_{1}}{z_{2}}\right)=\frac{3 \pi}{4}$
Standard 11
Mathematics
