11th Class Mathematics Sample Paper Mathematics Olympiad - Sample Paper-1

  • question_answer If\[\alpha \And \beta \]be the roots of the quadratic equation \[{{x}^{2}}+2x+3=0\]then the value of \[\frac{{{\alpha }^{2}}}{\beta }+\frac{{{\beta }^{2}}}{\alpha }\]is

    A) \[\frac{10}{3}\]                         

    B) \[\frac{3}{10}\]            

    C) \[\frac{2}{3}\]                          

    D) None of these

    Correct Answer: A

    Solution :

    [a] \[{{x}^{2}}+2x+3=0\] \[\therefore \alpha +\beta =-2\And \alpha .\beta =3\] Now \[\frac{{{\alpha }^{2}}}{\beta }+\frac{{{\beta }^{2}}}{\alpha }=\frac{{{\alpha }^{3}}+{{\beta }^{2}}}{\alpha .\beta }=\frac{{{(\alpha +\beta )}^{3}}-3\alpha \beta (\alpha +\beta )}{\alpha .\beta }\] \[=\frac{{{(-2)}^{3}}-3.3(-2)}{3}=\frac{-8+18}{3}=\frac{10}{3}\] Hence, option [a] is correct.

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