Complex Numbers And Quadratic Equations question 232
Question: If $ \frac{1}{x}+x=2\cos \theta , $ then $ x^{n}+\frac{1}{x^{n}} $ is equal to [UPSEAT 2001]
Options:
A) $ 2\cos n\theta $
B) $ 2\sin n\theta $
C) $ \cos n\theta $
D) $ \sin n\theta $
Show Answer
Answer:
Correct Answer: A
Solution:
$ x+\frac{1}{x}=2\cos \theta $
Þ $ x^{2}-2x\cos \theta +1=0 $
Þ $ x=\cos \theta \pm i\sin \theta $
Þ $ x^{n}=\cos n\theta \pm i\sin n\theta $
Þ $ \frac{1}{x}=\frac{1}{\cos \theta \pm i\sin \theta } $
Þ $ \frac{1}{x}=\cos \theta \mp i\sin \theta $
Þ $ \frac{1}{x^{n}}=\cos n\theta \mp i\sin n\theta $ Thus, $ x^{n}+\frac{1}{x^{n}}=2\cos n\theta $ .
BETA
