Sequence And Series Question 600
Question: If the roots of the cubic equation $ ax^{3}+bx^{2}+cx+d=0 $ are in G.P., then
Options:
A) $ c^{3}a=b^{3}d $
B) $ ca^{3}=bd^{3} $
C) $ a^{3}b=c^{3}d $
D) $ ab^{3}=cd^{3} $
Show Answer
Answer:
Correct Answer: A
Solution:
Let $ \frac{A}{R},\ A,\ AR $ be the roots of the equation $ ax^{3}+bx^{2}+cx+d=0 $ then $ A^{3}= $ Product of the roots $ =-\frac{d}{a} $
$ \Rightarrow $ $ A=-{{( \frac{d}{a} )}^{1/3}} $ Since $ A $ is a root of the equation.
$ \therefore aA^{3}+bA^{2}+cA+d=0 $
$ \Rightarrow $ $ a( -\frac{d}{a} )+b{{( -\frac{d}{a} )}^{2/3}}+c{{( -\frac{d}{a} )}^{1/3}}+d=0 $
Þ $ b{{( \frac{d}{a} )}^{2/3}}=c{{( \frac{d}{a} )}^{1/3}} $
Þ $ b^{3}\frac{d^{2}}{a^{2}}=c^{3}\frac{d}{a} $
Þ $ b^{3}d=c^{3}a $ .
BETA
