JEE Main & Advanced JEE Main Paper Phase-I (Held on 07-1-2020 Evening)

  • question_answer
    If \[\frac{3+i\sin \theta }{4-i\cos \theta },\,\theta \in [0,\,2\pi ]\], is a real number, then an argument of \[\sin \,\theta +\,i\cos \theta \] [JEE MAIN Held on 07-01-2020 Evening]

    A) \[\pi -{{\tan }^{-1}}\left( \frac{3}{4} \right)\]

    B) \[\pi -{{\tan }^{-1}}\left( \frac{4}{3} \right)\]

    C) \[-{{\tan }^{-1}}\left( \frac{3}{4} \right)\]  

    D) \[{{\tan }^{-1}}\left( \frac{4}{3} \right)\]

    Correct Answer: B

    Solution :

    [b] \[\because \,\,\,Z=\frac{3+i\sin \,\theta }{4-i\cos \theta }\times \frac{4+i\cos \,\theta }{4+i\cos \,\theta }\]
    \[=\frac{(12\,-sin\theta \,\cos \theta )+i(4\,sin\theta \,+3\cos \theta )}{16+{{\cos }^{2}}\theta }\]
    \[\because \]  Z is purely real
    \[\therefore \,\,4\sin \,\theta +3\,\cos \,\theta =0\]
    \[\tan \theta =-\frac{3}{4}\]
    If\[\theta \,\in \left( \frac{\pi }{2},\pi  \right)\], then
    arg \[(sin\,\theta +i\cos \theta )\,=\pi \,{{\tan }^{-1}}\left( \frac{4}{3} \right)\]
    if\[\theta \in \left( \frac{3\pi }{2},2\pi  \right)\],  then
    arg  \[(sin\,\theta +i\cos \theta )=-\,ta{{n}^{-1}}\,\frac{4}{3}\]

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