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4-1.Complex numbers
medium
$\frac{{3 + 2i\sin \theta }}{{1 - 2i\sin \theta }}$ એ શુધ્ધ કાલ્પનિક સંખ્યા હોય તો , જો $\theta = $ [કે જ્યાં $n$ એ ધન પૂર્ણાક છે]
A
$2n\pi \pm \frac{\pi }{3}$
B
$n\pi + \frac{\pi }{3}$
C
$n\pi \pm \frac{\pi }{3}$
D
એકપણ નહીં.
(IIT-1976)
Solution
(c) $\frac{{3 + 2i\sin \theta }}{{1 – 2i\sin \theta }}$ will be purely imaginary, if $\theta = $will be purely imaginary, if the real part vanishes, i.e., $\frac{{3 – 4{{\sin }^2}\theta }}{{1 + 4{{\sin }^2}\theta }} = 0$
==> $3 – 4{\sin ^2}\theta = 0$ (only if $\theta $ be real)
==> $\sin \theta = \pm \frac{{\sqrt 3 }}{2} = \sin \left( { \pm \frac{\pi }{3}} \right)$
==> $\theta = n\pi + {( – 1)^n}\left( { \pm \frac{\pi }{3}} \right) = n\pi \pm \frac{\pi }{3}$
Standard 11
Mathematics
