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8. Sequences and Series
hard
If the roots of the cubic equation $a{x^3} + b{x^2} + cx + d = 0$ are in $G.P.$, then
A
${c^3}a = {b^3}d$
B
$c{a^3} = b{d^3}$
C
${a^3}b = {c^3}d$
D
$a{b^3} = c{d^3}$
Solution
(a) 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$
==> $b{\left( {\frac{d}{a}} \right)^{2/3}} = c{\left( {\frac{d}{a}} \right)^{1/3}}$
==> ${b^3}\frac{{{d^2}}}{{{a^2}}} = {c^3}\frac{d}{a}$
==> ${b^3}d = {c^3}a$.
Standard 11
Mathematics
