A) \[16i\]
B) 16
C) 1
D) 0
Correct Answer: D
Solution :
\[{{(1+i)}^{6}}+{{(1-i)}^{6}}\] \[={{(\sqrt{2})}^{6}}{{\left( \frac{1}{\sqrt{2}}+\frac{i}{\sqrt{2}} \right)}^{6}}+{{(\sqrt{2})}^{6}}{{\left( \frac{1}{\sqrt{2}}-\frac{i}{\sqrt{2}} \right)}^{6}}\] \[=8{{\left( \cos \frac{\pi }{4}+i\sin \frac{\pi }{4} \right)}^{6}}+8{{\left( \cos \frac{\pi }{4}-i\sin \frac{\pi }{4} \right)}^{6}}\] \[=8\left( \cos \frac{6\pi }{4}+i\sin \frac{6\pi }{4} \right)+8\left( \cos \frac{6\pi }{4}-i\sin \frac{6\pi }{4} \right)\] \[=8\left[ \cos \frac{3\pi }{2}+i\sin \frac{3\pi }{2}+\cos \frac{3\pi }{2}-i\sin \frac{3\pi }{2} \right]\] \[=8\left[ 2\cos \frac{3\pi }{2} \right]\] \[=8\times 0=0\]
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