Complex Numbers And Quadratic Equations question 417
Question: $ \frac{3+2i\sin \theta }{1-2i\sin \theta } $ will be purely imaginary, if $ \theta = $ [IIT 1976; Pb. CET 2003]
Options:
A) $ 2n\pi \pm \frac{\pi }{3} $
B) $ n\pi +\frac{\pi }{3} $
C) $ n\pi \pm \frac{\pi }{3} $
D) None of these [Where $ n $ is an integer]
Show Answer
Answer:
Correct Answer: C
Solution:
$ \frac{3+2i\sin \theta }{1-2i\sin \theta } $ will be purely imaginary, if the real part vanishes, i.e., $ \frac{3-4{{\sin }^{2}}\theta }{1+4{{\sin }^{2}}\theta }=0 $
Þ $ 3-4{{\sin }^{2}}\theta =0 $ (only if $ \theta $ be real)
Þ $ \sin \theta =\pm \frac{\sqrt{3}}{2}=\sin ( \pm \frac{\pi }{3} ) $
Þ $ \theta =n\pi +{{(-1)}^{n}}( \pm \frac{\pi }{3} )=n\pi \pm \frac{\pi }{3} $
BETA
