A) \[1-i\sqrt{3}\]
B) \[-1+i\sqrt{3}\]
C) \[i\sqrt{3}\]
D) \[-i\sqrt{3}\]
Correct Answer: C
Solution :
Given equation is \[4+5{{\left( -\frac{1}{2}+i\frac{\sqrt{3}}{2} \right)}^{334}}+3{{\left( -\frac{1}{2}+i\frac{\sqrt{3}}{2} \right)}^{365}}\] \[=4+5{{\left( \cos \frac{2\pi }{3}+i\sin \frac{2\pi }{3} \right)}^{334}}\]\[+3{{\left( \cos \frac{2\pi }{3}+i\sin \frac{2\pi }{3} \right)}^{365}}\] \[=4+5\left[ \cos \frac{668}{3}\pi +i\sin \frac{668}{3}\pi \right]\]\[3\left[ \cos \frac{730}{3}\pi +i\sin \frac{730}{3}\pi \right]\] \[=4+5\left[ \cos \left( 222\pi +\frac{2\pi }{3} \right)+i\sin \left( 222\pi +\frac{2\pi }{3} \right) \right]\]\[+3\left[ \cos \left( 243\pi +\frac{\pi }{3} \right)+i\sin \left( 243\pi +\frac{\pi }{3} \right) \right]\] \[=4+5\left( \cos \frac{2\pi }{3}+i\sin \frac{2\pi }{3} \right)+3\left( -\cos \frac{\pi }{3}-i\sin \frac{\pi }{3} \right)\] \[=4+5\left( -\frac{1}{2}+i\frac{\sqrt{3}}{2} \right)+3\left( -\frac{1}{2}-i\frac{\sqrt{3}}{2} \right)\] \[=4-4+2i\frac{\sqrt{3}}{2}=i\sqrt{3}\].
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