KVPY Sample Paper KVPY Stream-SX Model Paper-5

  • question_answer
    Let \[f(x):\frac{\tan \,[{{e}^{2}}]{{x}^{3}}-\tan \,[-\,{{e}^{2}}]{{x}^{3}}}{{{\sin }^{3}}x},x\ne 0\] ([.] represents greatest integer function). The value of \[f(0)\] for which \[f(x)\] is continuous, is

    A) 15                    

    B) 12       

    C) \[-\,12\]                         

    D) 14

    Correct Answer: A

    Solution :

    We have, \[f(x)=\frac{\tan [{{e}^{2}}]{{x}^{3}}-\tan [-{{e}^{2}}]{{x}^{3}}}{{{\sin }^{3}}x}\]
    \[7<{{e}^{2}}<8,\] so \[[{{e}^{2}}]=7\] and \[[-\,{{e}^{2}}]=-\,8\]
    So \[f(0)=\underset{x\to 0}{\mathop{\lim }}\,\frac{\tan 7{{x}^{3}}-\tan (-\,8){{x}^{3}}}{{{\sin }^{3}}x}\]
    \[\underset{x\to 0}{\mathop{\lim }}\,\left[ \frac{\tan 7{{x}^{3}}}{{{\sin }^{3}}x}+\frac{\tan 8{{x}^{3}}}{{{\sin }^{3}}x} \right]\]\[=7+8=15\]

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