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JEE Main & Advanced Sample Paper JEE Main - Mock Test - 33

  • question_answer
    If a, b, c, d and p are distinct real numbers such that \[({{a}^{2}}+{{b}^{2}}+{{c}^{2}}){{p}^{2}}-2p(ab+bc+cd)+({{b}^{2}}+{{c}^{2}}+{{d}^{2}})\le 0\] then a, b, c, d are in-

    A) A.P.                 

    B)        GP.

    C) H.P.                 

    D)        None of these

    Correct Answer: B

    Solution :

    Here the given condition \[({{a}^{2}}+{{b}^{2}}+{{c}^{2}}){{p}^{2}}-2p(ab-bc-cd)+{{b}^{2}}+{{c}^{2}}+{{d}^{2}}\le 0\]\[\Rightarrow \,\,\,{{(ap-b)}^{2}}+{{(bp-c)}^{2}}+{{(cp-d)}^{2}}\le 0\] Since the squares cannot be negative \[\Rightarrow \,\,\,{{(ap-\,b)}^{2}}+{{(bp\,-c)}^{2}}+{{(cp-d)}^{2}}\le \,\,0\] \[\Rightarrow \,\,\,\frac{1}{p}=\frac{a}{b}=\frac{b}{c}=\frac{c}{d}\] \[\therefore \] a, b, c, d are in GP.


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