Complex Numbers And Quadratic Equations question 231
Question: If n is a positive integer, then $ {{(1+i)}^{n}}+{{(1-i)}^{n}} $ is equal to [Orissa JEE 2003]
Options:
A) $ {{(\sqrt{2})}^{n-2}}\cos ( \frac{n\pi }{4} ) $
B) $ {{(\sqrt{2})}^{n-2}}\sin ( \frac{n\pi }{4} ) $
C) $ {{(\sqrt{2})}^{n+2}}\cos ( \frac{n\pi }{4} ) $
D) $ {{(\sqrt{2})}^{n+2}}\sin ( \frac{n\pi }{4} ) $
Show Answer
Answer:
Correct Answer: C
Solution:
$ {{(1+i)}^{n}}+{{(1-i)}^{n}} $ $ ={{(2)}^{n/2}}{ \cos \frac{n\pi }{4}+i\sin \frac{n\pi }{4}+\cos \frac{n\pi }{4}-i\sin \frac{n\pi }{4} } $ $ ={2^{\frac{n}{2}}}.2\cos \frac{n\pi }{4}={2^{\frac{n}{2}+1}}\cos \frac{n\pi }{4}={{(\sqrt{2})}^{n+2}}\cos \frac{n\pi }{4} $ .
BETA
