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JEE Main & Advanced Mathematics Complex Numbers and Quadratic Equations Question Bank Relation between roots and coefficients

  • 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  [IIT Screening 2005]

    A) \[\Delta \ne 0\]

    B) \[b\Delta =0\]

    C) \[cb\ne 0\]

    D) \[c\Delta =0\]

    Correct Answer: D

    Solution :

    \[{{({{\alpha }^{2}}+{{\beta }^{2}})}^{2}}=(\alpha +\beta )\,({{\alpha }^{3}}+{{\beta }^{3}})\] \[{{\left( \frac{{{b}^{2}}-2ac}{{{a}^{2}}} \right)}^{2}}=\left( \frac{-b}{a} \right)\,\left( \frac{-{{b}^{2}}+3abc}{{{a}^{3}}} \right)\] Þ  \[4{{a}^{2}}{{c}^{2}}=ac{{b}^{2}}\]Þ \[ac\,({{b}^{2}}-4ac)=0\] As \[a\ne 0\Rightarrow c\Delta =0\]


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