Complex Numbers And Quadratic Equations question 160
Question: If $ i=\sqrt{-1}, $ then $ 4+5{{( -\frac{1}{2}+\frac{i\sqrt{3}}{2} )}^{334}} $ $ +3{{( -\frac{1}{2}+\frac{i\sqrt{3}}{2} )}^{365}} $ is equal to [IIT 1999]
Options:
A) $ 1-i\sqrt{3} $
B) $ -1+i\sqrt{3} $
C) $ i\sqrt{3} $
D) $ -i\sqrt{3} $
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Answer:
Correct Answer: C
Solution:
Given equation is $ 4+5{{( -\frac{1}{2}+i\frac{\sqrt{3}}{2} )}^{334}}+3{{( -\frac{1}{2}+i\frac{\sqrt{3}}{2} )}^{365}} $ $ =4+5{{( \cos \frac{2\pi }{3}+i\sin \frac{2\pi }{3} )}^{334}} $ $ +3{{( \cos \frac{2\pi }{3}+i\sin \frac{2\pi }{3} )}^{365}} $ $ =4+5[ \cos \frac{668}{3}\pi +i\sin \frac{668}{3}\pi ] $ $ 3[ \cos \frac{730}{3}\pi +i\sin \frac{730}{3}\pi ] $ $ =4+5[ \cos ( 222\pi +\frac{2\pi }{3} )+i\sin ( 222\pi +\frac{2\pi }{3} ) ] $ $ +3[ \cos ( 243\pi +\frac{\pi }{3} )+i\sin ( 243\pi +\frac{\pi }{3} ) ] $ $ =4+5( \cos \frac{2\pi }{3}+i\sin \frac{2\pi }{3} )+3( -\cos \frac{\pi }{3}-i\sin \frac{\pi }{3} ) $ $ =4+5( -\frac{1}{2}+i\frac{\sqrt{3}}{2} )+3( -\frac{1}{2}-i\frac{\sqrt{3}}{2} ) $ $ =4-4+2i\frac{\sqrt{3}}{2}=i\sqrt{3} $ .
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