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JEE Main & Advanced Sample Paper JEE Main - Mock Test - 32

  • question_answer
    If \[\alpha ,\beta \] are the roots of \[a{{x}^{2}}+bx+c=0\] and \[\alpha +\beta ,{{\alpha }^{2}}+{{\beta }^{2}},{{\alpha }^{3}}+{{\beta }^{3}}\] are in G.P., where \[\Delta ={{b}^{2}}-4ac,\] then -

    A) \[\Delta \ne 0\]            

    B)        \[b\Delta =0\]

    C) \[cb\ne 0\]            

    D)        \[c\Delta =0\]

    Correct Answer: D

    Solution :

    [d] \[\frac{-b}{a},\frac{{{b}^{2}}-2ac}{{{a}^{2}}},\frac{-{{b}^{3}}+3abc}{{{a}^{3}}}\] are in G.P. \[{{\left( {{b}^{2}}-2ac \right)}^{2}}=\left( -{{b}^{3}}+3abc \right)\left( -b \right)\] \[{{b}^{4}}+4{{a}^{2}}{{\text{c}}^{2}}-4a{{b}^{2}}c={{b}^{4}}-3a{{b}^{2}}c\] \[a{{b}^{2}}c-4{{a}^{2}}{{c}^{2}}=0\] a.c. \[({{b}^{2}}-4ac)=0\] \[\Rightarrow c.\Delta =0\]


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