JEE Main & Advanced Sample Paper JEE Main - Mock Test - 43

  • question_answer
    If \[{{n}_{1}},{{n}_{2}},{{n}_{3}},....,{{n}_{100}}\] are positive real numbers such that \[{{n}_{1}}+{{n}_{2}}+{{n}_{3}}+.....+{{n}_{100}}=20\]and \[k={{n}_{1}}({{n}_{2}}+{{n}_{3}}+{{n}_{4}})\,({{n}_{5}}+{{n}_{6}}+....+{{n}_{9}})({{n}_{10}}+....+{{n}_{16}})\]\[...(...+{{n}_{100}}),\] then \[k\in \]

    A) \[(0,100]\]         

    B)        \[(0,128]\]

    C) \[[0,144]\]         

    D)        None of these

    Correct Answer: D

    Solution :

    [d] \[\frac{[{{n}_{1}}+({{n}_{2}}+{{n}_{3}}+{{n}_{4}})+({{n}_{5}}+{{n}_{6}}+....+{{n}_{9}})+...+(...+{{n}_{100}})]}{10}\] \[\ge \,\,\sqrt[10]{k}\,\,\,(A.M.\ge G.M.)\]             \[\Rightarrow \,\,\,\,\,\,\,\,2\ge \sqrt[10]{K}\]             \[\Rightarrow \,\,\,\,\,\,\,\,0<K\le 1024\]


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