JEE Main & Advanced Mathematics Complex Numbers and Quadratic Equations Question Bank Integral power of iota, Algebraic operations and Equality of complex numbers

  • question_answer \[\frac{3+2i\sin \theta }{1-2i\sin \theta }\] will be purely imaginary, if \[\theta =\] [IIT 1976; Pb. CET 2003]

    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]

    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  \[\sin \theta =\pm \frac{\sqrt{3}}{2}=\sin \left( \pm \frac{\pi }{3} \right)\] Þ  \[\theta =n\pi +{{(-1)}^{n}}\left( \pm \frac{\pi }{3} \right)=n\pi \pm \frac{\pi }{3}\]


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