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KVPY Sample Paper KVPY Stream-SX Model Paper-13

  • question_answer
    If \[a,b,c\] and d are four positive real numbers such that \[abcd=1,\] then the minimum value of \[(1+a)(1+b)(1+c)(1+d)\] is

    A) 4

    B) 1      

    C) 16

    D) 18

    Correct Answer: C

    Solution :

    Given, \[abcd=1\]
    \[\frac{1+a}{2}\le \sqrt{a}\]
    \[AM\ge GM\]
    \[1+a\ge 2\sqrt{a}\]
    Similarly, \[1+b\ge 2\sqrt{b}\]
    \[1+c\ge 2\sqrt{c}\]
    \[\Rightarrow \]\[1+d\ge 2\sqrt{d}\]
    \[\therefore \]\[(1+a)(1+b)(1+c)(1+d)\ge 16\sqrt{abcd}\]
    \[(1+a)(1+b)(1+c)(1+d)\ge 16\]


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