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

  • question_answer
    If \[\alpha ,\beta \] be the roots of the equation \[2{{x}^{2}}-2({{m}^{2}}+1)x+{{m}^{4}}+{{m}^{2}}+1=0\],  then \[{{\alpha }^{2}}+{{\beta }^{2}}\]=

    A) 0

    B) 1

    C) m

    D) \[{{m}^{2}}\]

    Correct Answer: D

    Solution :

    \[\alpha +\beta =\frac{2({{m}^{2}}+1)}{2}={{m}^{2}}+1\] .....(i) and \[\alpha \beta =\frac{{{m}^{4}}+{{m}^{2}}+1}{2}\] .....(ii) Therefore \[{{\alpha }^{2}}+{{\beta }^{2}}={{(\alpha +\beta )}^{2}}-2\alpha \beta \] \[={{({{m}^{2}}+1)}^{2}}-2\frac{({{m}^{4}}+{{m}^{2}}+1)}{2}\] \[={{m}^{4}}+2{{m}^{2}}+1-{{m}^{4}}-{{m}^{2}}-1={{m}^{2}}\]


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