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4-1.Complex numbers
hard
જો $A = \left\{ {0 \in \left( { - \frac{\pi }{2},\pi } \right):\frac{{3 + 2i{\mkern 1mu} \sin {\mkern 1mu} \theta }}{{1 - 2i{\mkern 1mu} \sin {\mkern 1mu} \theta }}} \right.$ શુધ્ધ કાલ્પનિક સંખ્યા છે.$\}$. તો $A$ ના ઘટકો નો સરવાળો મેળવો.
A
$\frac{{5\pi }}{6}$
B
$\pi$
C
$\frac{{3\pi }}{4}$
D
$\frac{{2\pi }}{3}$
(JEE MAIN-2019)
Solution
$z=\frac{3+2 i \sin \theta}{1-2 i \sin \theta} $ $\times \frac{1+2 i \sin \theta}{1+2 i \sin \theta}$
$z=\frac{\left(3-4 \sin ^{2} \theta\right)+8 i \sin \theta}{1+4 \sin ^{2} \theta}$
For purely imaginary real part should be zero. i.e. $3-4 \sin ^{2} \theta=0$
ie. $\sin \theta=\pm \frac{\sqrt{3}}{2}$
$\theta=-\frac{\pi}{3}, \frac{\pi}{3}, \frac{2 \pi}{3},$ Sum of all values is $-\frac{\pi}{3}+\frac{\pi}{3}+\frac{2 \pi}{3}=\frac{2 \pi}{3}$
Standard 11
Mathematics
