The modulus and argument of 3+2 i are - Vidyadip

MCQ

The modulus and argument of $\sqrt{3}+\sqrt{2} i$ are

✓
$\sqrt{5}, \tan ^{-1}\left(\sqrt{\frac{2}{3}}\right)$
B
$\sqrt{5}, \tan ^{-1}\left(\sqrt{\frac{3}{2}}\right)$
C
$\sqrt{7}, \tan ^{-1}\left(\sqrt{\frac{3}{2}}\right)$
D
$\sqrt{7}, \tan ^{-1}\left(\sqrt{\frac{2}{3}}\right)$

✓

Answer

Correct option: A.

$\sqrt{5}, \tan ^{-1}\left(\sqrt{\frac{2}{3}}\right)$

(A)
$|z|=\sqrt{a^2+b^2}=\sqrt{3+2}=\sqrt{5}$
Let $\theta$ be the argument of z .
$\therefore \quad \tan \theta=\left|\frac{ b }{ a }\right|=\left|\frac{\sqrt{2}}{\sqrt{3}}\right|=\sqrt{\frac{2}{3}}$
$\Rightarrow \theta=\tan ^{-1}\left(\sqrt{\frac{2}{3}}\right)$

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