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CET Karnataka Engineering CET - Karnataka Engineering Solved Paper-2006

  • question_answer
    If \[f(x)\] is defined [-2,2] by \[f(x)=4{{x}^{3}}-3x+1\] and\[g(x)=\frac{f(-x)-f(x)}{{{x}^{2}}+3}\],then \[\int_{-2}^{2}{g\,(x)\,dx}\] is equal to :

    A)  64                                         

    B)  -48

    C)  0                                            

    D)  24

    Correct Answer: C

    Solution :

    Given that \[f(x)=4{{x}^{2}}-3x+1,\]\[g(x)=\frac{f(-x)-f(x)}{{{x}^{2}}+3}\] \[\therefore \]\[g(x)=\frac{(4{{x}^{2}}+3x+1)-(4{{x}^{2}}-3x+1)}{{{x}^{2}}+3}\]                 \[=\frac{6x}{{{x}^{2}}+3}\] Now,  \[g(-x)=-\frac{6x}{{{x}^{2}}+3}\]                 \[-g(x)\] Which is an odd function \[\therefore \]\[\int_{-2}^{2}{g(x)\,\,dx=0}\]


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