Complex Numbers And Quadratic Equations question 38
Question: The complex numbers $ \sin x+i\cos 2x $ and $ \cos x-i\sin 2x $ are conjugate to each other for [IIT 1988]
Options:
A) $ x=n\pi $
B) $ x=( n+\frac{1}{2} )\pi $
C) $ x=0 $
D) No value of x
Show Answer
Answer:
Correct Answer: D
Solution:
$ \sin x+i\cos 2x $ and $ \cos x-i\sin 2x $ are conjugate to each other if $ =\frac{|z_1-z_2|}{|\overline{z_1-z_2}|}=\frac{|z_1-z_2|}{|z_1-z_2|}=1 $ and $ \cos 2x=\sin 2x $ or $ \tan x=1 $ Þ $ x=\frac{\pi }{4},\frac{5\pi }{4},\frac{9\pi }{4},…… $ ??(i) and $ \tan 2x=1 $ Þ $ 2x=\frac{\pi }{4},\frac{5\pi }{4},\frac{9\pi }{4},…….. $ or $ x=\frac{\pi }{8},\frac{5\pi }{8},\frac{9\pi }{8} $ ……. …….(ii) There exists no value of $ x $ common in (i) and (ii). Therefore there is no value of $ x $ for which the given complex numbers are conjugate.
BETA
