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CEE Kerala Engineering CEE Kerala Engineering Solved Paper-2004

  • question_answer
    If the tangent to the graph function\[y=f(x)\]makes angles\[\frac{\pi }{4}\]and\[\frac{\pi }{3}\]with the\[x-\]axis is at the point\[x=2\]and\[x=4\]respectively, the value of \[\int_{2}^{4}{f(x)}f\,(x)dx:\]

    A)  \[f(4)f(2)\]                        

    B)  \[f(4)\]

    C)  \[f(2)\]

    D)         \[0\]

    E)  \[1\]

    Correct Answer: E

    Solution :

    It is given that \[f(2)=\tan \frac{\pi }{4}=1,\]\[f(4)=\tan \frac{\pi }{3}=\sqrt{3}\] \[\therefore \]\[\int_{2}^{4}{f(x)f(x)}\,dx=\left[ \frac{1}{2}{{[f(x)]}^{2}} \right]_{2}^{4}\]                                 \[=\frac{1}{2}{{[f(4)]}^{2}}-\frac{1}{2}{{[f(2)]}^{2}}\]                                 \[=\frac{3}{2}-\frac{1}{2}=1\]


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