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J & K CET Engineering J and K - CET Engineering Solved Paper-2006

  • question_answer
    If \[\alpha ,\beta ,\gamma \] are the roots of \[{{x}^{3}}+bx+c=0,\]then \[{{\alpha }^{2}}\beta +\alpha {{\beta }^{2}}+{{\beta }^{2}}\gamma +{{\gamma }^{2}}\alpha +\gamma {{\alpha }^{2}}\]  is equal to

    A)  \[c\]             

    B)  \[-c\]

    C)  \[-3c\]           

    D)  \[3c\]

    Correct Answer: D

    Solution :

    Let \[\alpha ,\beta ,\gamma \] are the roots of the equation \[{{x}^{3}}+bx+c=0\] Then, \[\Sigma \alpha =0,\,\,\Sigma \alpha \beta =b\] and \[\alpha \beta \gamma =-c\] ?.(i) Now,  \[\Sigma \alpha \,\,\Sigma \alpha \beta =(\alpha +\beta +\gamma ).(\alpha \beta +\beta \gamma +\gamma \beta )\] \[=\Sigma {{\alpha }^{2}}\beta +3\alpha \beta \gamma \] \[\Rightarrow \]  \[\Sigma {{\alpha }^{2}}\beta =\Sigma \alpha \Sigma \alpha B-3\alpha \beta \gamma =0.\Sigma \alpha \beta -3(-c)\] [from Eq. (i)] \[=3c\]


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