11th Class Mathematics Complex Numbers and Quadratic Equations Question Bank Critical Thinking

  • question_answer If a, b, g  are roots of equation \[{{x}^{3}}+a{{x}^{2}}+bx+c=0\], then \[{{\alpha }^{-1}}+{{\beta }^{-1}}+{{\gamma }^{-1}}=\] [EAMCET 2002]

    A) a/c

    B) - b/c

    C) b/a

    D) c/a

    Correct Answer: B

    Solution :

    a, b, g are the roots of the equation      \[{{x}^{3}}+a{{x}^{2}}+bx+c=0\] so \[\alpha +\beta +\gamma =-a\], \[\alpha \beta +\beta \gamma +\gamma \alpha =b\] and \[\alpha \beta \gamma =-\,c\] Now\[{{\alpha }^{-1}}+{{\beta }^{-1}}+{{\gamma }^{-1}}\] \[=\frac{1}{\alpha }+\frac{1}{\beta }+\frac{1}{\gamma }\] \[=\frac{\alpha \beta +\beta \gamma +\gamma \alpha }{\alpha \beta \gamma }\]                                  \[=-b/c\].

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