A) \[{{\alpha }^{2}}\beta ,{{\beta }^{2}}\alpha \]
B) \[\alpha ,{{\beta }^{2}}\]
C) \[{{\alpha }^{2}}\beta ,\beta \alpha \]
D) \[{{\alpha }^{3}}\beta ,{{\beta }^{3}}\alpha \]
Correct Answer: A
Solution :
Dividing the equation \[{{a}^{3}}{{x}^{2}}+abcx+{{c}^{3}}=0\]by\[{{c}^{2}}\], we get \[a{{\left( \frac{ax}{c} \right)}^{2}}+b\left( \frac{ax}{c} \right)+c=0\] \[\Rightarrow \] \[\frac{ax}{c}=\alpha ,\,\,\beta \]are roots. \[\Rightarrow \] \[x=\frac{c}{a}\alpha \]and\[x=\frac{c}{a}\beta \] \[\Rightarrow \] \[x={{\alpha }^{2}}\beta \]and\[x=\alpha {{\beta }^{2}}\] [\[\because \,\,\alpha \beta =\frac{c}{a}\]as\[\alpha \],\[\beta \]are the roots of \[a{{x}^{2}}+bx+c=0\]] are roots of above equation.
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