KVPY Sample Paper KVPY Stream-SX Model Paper-5

  • question_answer
    The number of continuous function \[f:[0,1]\to [0,1],\] such that \[f(x)<{{x}^{2}}\] for all x and \[\int_{0}^{1}{f(x)dx=\frac{1}{3}}\] is

    A) 0         

    B) 1

    C) 2                                 

    D) infinite

    Correct Answer: A

    Solution :

    [a] Given, \[f(x)<{{x}^{2}}f(x)\] is always positive for \[x\in [0,1]\] \[\int_{0}^{1}{f(x)dx<\int_{0}^{1}{{{x}^{2}}dx}}\] \[\frac{1}{3}<\left| \frac{{{x}^{3}}}{3} \right|_{0}^{1}\]\[\Rightarrow \]\[\frac{1}{3}<\frac{1}{3}\] It is not possible.            

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