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J & K CET Engineering J and K - CET Engineering Solved Paper-2012

  • question_answer
    Let \[f:(0,\,\,\infty )\to R\] and \[F(x)=\int_{0}^{x}{f(t)\,\,dt.}\]. If \[F({{x}^{2}})={{x}^{2}}(1+x),\]then \[f(1)\] equals to

    A)  \[\frac{5}{2}\]              

    B)  \[5\]

    C)  \[\frac{2}{5}\]                 

    D)  \[2\]

    Correct Answer: A

    Solution :

    Given,  \[F(x)=\int_{0}^{x}{f(t)\,\,dt}\] \[\therefore \] \[F{{(x)}^{2}}=\int_{0}^{{{x}^{2}}}{f(t)\,\,dt}\] \[\Rightarrow \] \[{{x}^{2}}(1+x)=\int_{0}^{{{x}^{2}}}{f(t)\,dt}\] On differentiating w. r. t.  x on both sides by Leibnitz rule, \[2x+3{{x}^{2}}=f{{(x)}^{2}}.2x\] Put \[x=1,\] \[2+3=f(1).2\] \[\Rightarrow \] \[f(1)=\frac{5}{2}\]


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