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JEE Main & Advanced Mathematics Complex Numbers and Quadratic Equations Question Bank Condition for common roots, Quadratic expressions and Position of roots

  • question_answer
    If a, b  be the roots of the quadratic equation \[a{{x}^{2}}+bx+c=0\] and \[k\] be a real number, then the condition so that  \[\alpha <k<\beta \] is given by

    A) \[ac>0\]

    B) \[a{{k}^{2}}+bk+c=0\]

    C) \[ac<0\]

    D) \[{{a}^{2}}{{k}^{2}}+abk+ac<0\]

    Correct Answer: D

    Solution :

    Here \[a{{x}^{2}}+bx+c=a(x-\alpha )(x-\beta )\] Since \[\alpha ,\beta \] be the roots of\[a{{x}^{2}}+bx+c=0\]. Also \[\alpha <k<\beta ,\]so \[a(k-\alpha )(k-\beta )<0\] Also \[{{a}^{2}}{{k}^{2}}+abk+ac=a(a{{k}^{2}}+bk+c)\] \[={{a}^{2}}(k-\alpha )(k-\beta )<0\]Þ \[{{a}^{2}}{{k}^{2}}+abk+ac<0\]


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