Complex Numbers And Quadratic Equations question 34
Question: If $ 2a+3b+6c=0 $ then at least one root of the equation $ ax^{2}+bx+c=0 $ lies in the interval [Kurukshetra CEE 2002; AIEEE 2002, 04]
Options:
A) (0, 1)
B) (1, 2)
C) (2, 3)
D) (3, 4)
Show Answer
Answer:
Correct Answer: A
Solution:
$ f(x)=ax^{2}+bx+c $ Let $ F(x)=\int{f(x)dx=\frac{a}{3}x^{3}+\frac{b}{2}x^{2}+cx} $ Clearly $ F(0)=0 $ and $ F(1)=\frac{a}{3}+\frac{b}{2}+c $ $ =\frac{2a+3b+6c}{6}=0 $
Þ $ F(0)=F(1)=0 $ There exist at least one point c in between 0 and 1 such that $ {F}’(x)=0 $ or $ ax^{2}+bx+c=0 $ for some $ x\in (0,1) $ .
BETA
