Question

The magnitude and amplitude of $ \frac{(1+i\sqrt{3})(2+2i)}{(\sqrt{3}-i)} $ are respectively

• A. $ 2,\frac{3\pi }{4} $
• B. $ 4,\frac{3\pi }{4} $
• C. $ 2\sqrt{2},\frac{\pi }{4} $
• D. $ 2\sqrt{2},\frac{\pi }{2} $
• E. $ 2\sqrt{2},\frac{3\pi }{4} $

Answer 1

Answer: (E)

$ \frac{(1+i\sqrt{3})(2+2i)}{(\sqrt{3}-i)}=\frac{2+2i+2\sqrt{3}i-2\sqrt{3}}{(\sqrt{3}-i)} $
$=\frac{\{(2-2\sqrt{3})+2i(1+\sqrt{3})\}}{(\sqrt{3}-i)}\times \frac{(\sqrt{3}+i)}{(\sqrt{3}+i)} $
$=\frac{2\sqrt{3}-6+2i-2\sqrt{3}i+2\sqrt{3}i+6i-2-2\sqrt{3}}{3+1} $
$=\frac{-8+8i}{4}=-2+2i $
$ \therefore $ Magnitude of $ \frac{(1+i\sqrt{3})(2+2i)}{(\sqrt{3}-i)} $
$=\sqrt{4+4}=2\sqrt{2} $ and amplitude of $ \frac{(1+i\sqrt{3})(2+2i)}{(\sqrt{3}-i)} $
$={{\tan }^{-1}}\left( \frac{2}{-2} \right)=\frac{3\pi }{4} $