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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 \[2a+3b+6c=0\] then at least one root of the equation \[a{{x}^{2}}+bx+c=0\]lies in the interval [Kurukshetra CEE 2002; AIEEE 2002, 04]

    A) (0, 1)

    B) (1, 2)

    C) (2, 3)

    D) (3, 4)

    Correct Answer: A

    Solution :

    \[f(x)=a{{x}^{2}}+bx+c\] Let \[F(x)=\int{f(x)dx=\frac{a}{3}{{x}^{3}}+\frac{b}{2}{{x}^{2}}+cx}\] Clearly \[F(0)=0\] and \[F(1)=\frac{a}{3}+\frac{b}{2}+c\]                     \[=\frac{2a+3b+6c}{6}=0\] Þ \[F(0)=F(1)=0\] There exist at least one point c in between 0 and 1 such that \[{F}'(x)=0\] or \[a{{x}^{2}}+bx+c=0\]for some \[x\in (0,\,\,1)\].


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