A) \[{{c}^{3}}a={{b}^{3}}d\]
B) \[c{{a}^{3}}=b{{d}^{3}}\]
C) \[{{a}^{3}}b={{c}^{3}}d\]
D) \[a{{b}^{3}}=c{{d}^{3}}\]
Correct Answer: A
Solution :
Let \[\frac{A}{R},\ A,\ AR\] be the roots of the equation \[a{{x}^{3}}+b{{x}^{2}}+cx+d=0\] then \[{{A}^{3}}=\]Product of the roots \[=-\frac{d}{a}\]\[\Rightarrow \]\[A=-{{\left( \frac{d}{a} \right)}^{1/3}}\] Since \[A\] is a root of the equation. \[\therefore a{{A}^{3}}+b{{A}^{2}}+cA+d=0\] \[\Rightarrow \]\[a\left( -\frac{d}{a} \right)+b{{\left( -\frac{d}{a} \right)}^{2/3}}+c{{\left( -\frac{d}{a} \right)}^{1/3}}+d=0\] Þ \[b{{\left( \frac{d}{a} \right)}^{2/3}}=c{{\left( \frac{d}{a} \right)}^{1/3}}\] Þ \[{{b}^{3}}\frac{{{d}^{2}}}{{{a}^{2}}}={{c}^{3}}\frac{d}{a}\]Þ \[{{b}^{3}}d={{c}^{3}}a\].
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