JEE Main & Advanced Sample Paper JEE Main Sample Paper-47

  • question_answer
    The real roots of the equation, \[{{\cos }^{7}}x+{{\sin }^{4}}x=1\] in the interval \[(-\pi ,\,\,\pi )\]

    A) \[\frac{-\pi }{2},\,0\]              

    B)  \[\frac{-\pi }{2},\,0\,\frac{\pi }{2}\]

    C)  \[\frac{\pi }{2},\,0\]               

    D)  \[0,\frac{\pi }{2},\,\frac{\pi }{2}\]

    Correct Answer: B

    Solution :

     \[\because \]    \[{{\cos }^{7}}x\le {{\cos }^{2}}x\] and      \[{{\sin }^{4}}x\le \,{{\sin }^{2}}x\] On adding Eqs. (i) and (ii), we get \[\Rightarrow \]            \[{{\cos }^{7}}x+{{\sin }^{4}}x\le 1\] But given, \[{{\cos }^{7}}x+{{\sin }^{4}}x=1\] Equality holds only if \[{{\cos }^{7}}x={{\cos }^{2}}x\] and      \[{{\sin }^{4}}x={{\sin }^{2}}x\] Both are satisfies by \[x=\pm \,\frac{\pi }{2},\,0\]

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