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JEE Main & Advanced AIEEE Solved Paper-2005

  • question_answer
    Let\[f:\text{ }R\to R\]be a differentiable function having\[f(2)=6,f'(2)=\left( \frac{1}{48} \right)\] Then,\[\underset{x\to 2}{\mathop{\lim }}\,\int_{6}^{f(x)}{\frac{4{{t}^{3}}}{x-2}}dt\] equals       AIEEE  Solved  Paper-2005

    A) 18                          

    B)        12                          

    C)        36                          

    D)        24

    Correct Answer: A

    Solution :

    \[\underset{x\to 2}{\mathop{\lim }}\,\int_{6}^{f(x)}{\frac{4{{t}^{3}}}{x-2}}dt\] \[=\underset{x\to 2}{\mathop{\lim }}\,\frac{\int_{6}^{f(x)}{4{{t}^{3}}}dt}{(x-2)}\] \[=\underset{x\to 2}{\mathop{\lim }}\,\frac{4{{\{f(x)\}}^{3}}}{1}f'(x)\] \[=4\{f{{(2)}^{3}}\}f'(2)\] \[=4\times {{(6)}^{3}}\times \frac{1}{48}\]              \[[\because f(2)=6f'(2)=\frac{1}{48}]\] \[=18\]


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