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 Product of roots \[{{A}^{3}}=-\frac{d}{a}\] \[\Rightarrow \] \[=A=-{{\left( \frac{d}{a} \right)}^{1/3}}\] Since, A is a root of the equation. \[\therefore \] \[a{{A}^{3}}+b{{A}^{2}}+cA+d=0\] \[\Rightarrow \] \[a\left( -\frac{d}{a} \right)+b{{\left( -\frac{d}{a} \right)}^{2/3}}+c{{\left( -\frac{d}{a} \right)}^{1/3}}+d=0\] \[\Rightarrow \] \[b{{\left( \frac{d}{a} \right)}^{2/3}}=c{{\left( \frac{d}{a} \right)}^{1/3}}\] \[\Rightarrow \] \[{{b}^{3}}\cdot \frac{{{d}^{2}}}{{{a}^{2}}}={{c}^{3}}\cdot \frac{d}{a}\] \[\Rightarrow \] \[{{b}^{3}}d={{c}^{3}}a\]
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