Complex Numbers And Quadratic Equations question 221
Question: The value of $ {{[ \frac{1-\cos \frac{\pi }{10}+i\sin \frac{\pi }{10}}{1-\cos \frac{\pi }{10}-i\sin \frac{\pi }{10}} ]}^{10}}= $ [Karnataka CET 2001]
Options:
A) 0
B) - 1
C) 1
D) 2
Show Answer
Answer:
Correct Answer: B
Solution:
Let $ \cos \frac{\pi }{10}-i\sin \frac{\pi }{10}=z $ and $ \cos \frac{\pi }{10}+i\sin \frac{\pi }{10}=\frac{1}{z} $ Therefore, $ {{( \frac{1-z}{1-\frac{1}{z}} )}^{10}} $ $ ={{{ \frac{-(z-1)z}{(z-1)} }}^{10}}={{(-z)}^{10}} $ $ =z^{10}={{( \cos \frac{\pi }{10}-i\sin \frac{\pi }{10} )}^{10}} $ $ =\cos \pi -i\sin \pi =-1 $ .
BETA
