KVPY Sample Paper KVPY Stream-SX Model Paper-5

  • question_answer
    If \[{{a}^{2}}+{{b}^{2}}=7\] and \[{{a}^{3}}+{{b}^{3}}=10,\] then the maximum value of \[|a+b|\] is

    A) 4                                 

    B) 5      

    C) 6                                 

    D) 3

    Correct Answer: B

    Solution :

    We have, \[{{a}^{2}}+{{b}^{2}}=7\] and \[{{a}^{3}}+{{b}^{3}}=10\]
    \[\Rightarrow \]\[{{a}^{3}}+{{b}^{3}}=(a+b)({{a}^{2}}+{{b}^{2}}-ab)\]\[\Rightarrow \]\[10=(a+b)(7-ab)\]
    \[\Rightarrow \]\[10=(a+b)\left( 7-\frac{{{(a+b)}^{2}}-7}{2} \right)\] \[[\because {{a}^{2}}+{{b}^{2}}={{(a+b)}^{2}}-2ab]\]
    \[\Rightarrow \]\[20=(a+b)(21-{{(a+b)}^{2}})\]\[\Rightarrow \]\[{{(a+b)}^{3}}-21(a+b)+20=0\]
    Let \[a+b=x\]
    \[\therefore \]\[{{x}^{3}}-21x+20=0\]
    \[\therefore \]\[x=1,4,-\,5\]
    \[\left| a+b \right|=1,4,5\]
    \[\therefore \]Maximum value of \[\left| a+b \right|=5\]

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