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12th Class Mathematics Definite Integrals Question Bank Critical Thinking

  • question_answer
    If \[\int_{0}^{{{t}^{2}}}{xf(x)dx=}\frac{2}{5}{{t}^{5}},\,\,t>0,\]then\[f\left( \frac{4}{25} \right)=\] [IIT Screening 2004]

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

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

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

    D) None of these

    Correct Answer: A

    Solution :

    • \[\int_{0}^{{{t}^{2}}}{xf(x)dx=\frac{2}{5}{{t}^{5}},\,t>0}\]           
    • Differentiate both sides w.r.t. t, we get \[{{t}^{2}}f({{t}^{2}})2t=2{{t}^{4}}\]           
    • Þ \[f({{t}^{2}})=t\]           
    • Put \[t=\frac{2}{5},\] we get \[f\,\left( \frac{4}{25} \right)=\frac{2}{5}\].


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