KVPY Sample Paper KVPY Stream-SX Model Paper-25

  • question_answer
    The greatest of the numbers  \[1,{{2}^{1/2}},{{3}^{1/3}},{{4}^{1/4}},{{5}^{1/5}},{{6}^{1/6}}\operatorname{and}{{7}^{1/7}}\]is

    A) \[{{2}^{1/2}}\]

    B) \[{{3}^{1/3}}\]

    C) \[{{7}^{1/4}}\]

    D) all but 1 are equal

    Correct Answer: B

    Solution :

    If \[f(x)={{x}^{1/x}}\],  then\[f'\left( x \right)=\frac{1}{{{x}^{2}}}\left[ {{x}^{1/x}}\left( 1-\ln x \right) \right]\]
    f is decreasing if \[x>e\]and f is increasing if \[x>e.\]
    As \[e<3<4<5<6<7\]
    \[\therefore \]\[Max\{{{3}^{1/3}},{{4}^{1/4}},{{5}^{1/5}},{{6}^{1/6}},{{7}^{1/7}}\}={{3}^{1/3}}\]
    Also \[1<2<e\]   \[\therefore \operatorname{Max}\{1,{{2}^{1/2}}\}={{2}^{1/2}}\] But \[{{2}^{1/2}}={{4}^{1/4}}<{{3}^{1/3}}\]
    \[\therefore \]     The greatest number is \[{{3}^{1/3}}\]

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