Question

$\left(af\right(\mu )<0)$ is the necessary and sufficient condition for a particular real number $\mu$ to lie between the roots of a quadratic equation $f\left(x\right)=0$, where $f\left(x\right)=a{x}^2+bx+c$. Again if $f\left({\mu }_1\right)f\left({\mu }_2\right)<0$, then exactly one of the roots will lie between ${\mu }_1$ and ${\mu }_2$.

If $(a+b+c)c<0<a(a+b+c)$, then

• A. one root is less than $0$, the other is greater than $1$
• B. one root lies in $(-\infty ,0)$ and other is $(0,1)$
• C. both the roots lie in $(0,1)$
• D. one root lies in $(0,1)$ and other in $(1,\infty )$

Answer 1

The correct option is B one root lies in $(-\infty ,0)$ and other is $(0,1)$

Given,$f\left(x\right)=a{x}^2+bx+c$
Now,$(a+b+c)c<0$
$\Rightarrow f\left(1\right)f\left(0\right)<0$
$\Rightarrow$Exactly one root lies between $0$ and $1$.
$a(a+b+c)>0$
$\Rightarrow af\left(1\right)>0$
$\Rightarrow$Both the roots lie on the same side of $1$ i.e., both are less than $1$.
As exactly one root lies between $0$ and $1$.
Other root should be less than $0$.
$\Rightarrow$One root belongs to $(-\infty ,0)$ and other root belongs to $(0,1)$.
Hence, option 'B' is correct.