A) \[-{{a}^{3}}\]
B) \[{{a}^{3}}-3b\]
C) \[{{a}^{3}}\]
D) \[{{a}^{2}}-3b\]
Correct Answer: C
Solution :
Since, \[\alpha ,\]\[\beta ,\]\[\gamma \]are the roots of given equation therefore, \[\alpha +\beta +\gamma =-a\] \[\alpha \beta +\beta \gamma +\gamma \alpha =0\] and \[\alpha \beta \gamma =-b\] Now, \[\left| \begin{matrix} \alpha & \beta & \gamma \\ \beta & \gamma & \alpha \\ \gamma & \alpha & \beta \\ \end{matrix} \right|=-(\alpha +\beta +\gamma )\] \[({{\alpha }^{2}}+{{\beta }^{2}}+{{\gamma }^{2}}-\alpha \beta -\beta \gamma -\gamma \alpha )\] \[=(\alpha +\beta +\gamma )\,[{{(\alpha +\beta +\gamma )}^{2}}\] \[-3(\alpha \beta +\beta \gamma +\gamma \alpha )]\] \[=-\,(-a)\,({{a}^{2}}-0)={{a}^{3}}\]
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