KVPY Sample Paper KVPY Stream-SX Model Paper-5

  • question_answer
    If \[f:R\to R\] is a differentiable function \[f(1)=4.\] Then, the value of \[\underset{x\to 1}{\mathop{\lim }}\,\int_{4}^{f(x)}{\frac{2t}{x-1}}dt\]

    A) \[8f'(1)\]

    B) \[4f'(1)\]

    C) \[2f'(1)\]            

    D) \[f'(1)\]

    Correct Answer: A

    Solution :

    We have, \[\underset{x\to 1}{\mathop{\lim }}\,\int_{4}^{f(x)}{\frac{2t}{x-1}dt}\]
    \[\underset{x\to 1}{\mathop{\lim }}\,\frac{\int_{4}^{f(x)}{2t}}{x-1}dt\]
    Apply Leibnitz's rule, we get \[\underset{x\to 1}{\mathop{\lim }}\,\frac{2f(x)\cdot f'(x)}{1}\]
    \[=2\cdot (4)\cdot f'(1)\]   \[[\because f(1)=4]\]

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