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JAMIA MILLIA ISLAMIA Jamia Millia Islamia Solved Paper-2006

  • question_answer
        If\[f(x)=\cos (\log x),\]then\[f(x)f(y)-\frac{1}{2}\left[ f\left( \frac{x}{y} \right)+f(xy) \right]\]has the value

    A)  \[-1\]                                   

    B)  \[\frac{1}{2}\]   

    C)  \[-2\]                                   

    D)  zero

    Correct Answer: D

    Solution :

                    Given\[f(x)=\cos (\log x)\] \[\therefore \]  \[f(x).f(y)-\frac{1}{2}\left[ f\left( \frac{x}{y} \right)+f(xy) \right]\] \[=\cos (\log x).\cos (\log y)\]                 \[-\frac{1}{2}\left[ \cos \log \left( \frac{x}{y} \right)+\cos \log (xy) \right]\] \[=\cos (\log x)\cos (\log y)-\frac{1}{2}2[\cos (\log x)\]                                                  \[\times \cos (\log y)]\] \[=\cos (\log x)\cos (\log y)\]                                 \[-\cos (\log x)\cos (\log y)\] \[=0\] \[\therefore \]\[f(x).f(y)-\frac{1}{2}\left[ f\left( \frac{x}{y} \right)+f(xy) \right]=0\]


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