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 $|b|>|a+c|$, then

• A. one root of $f\left(x\right)=0$ is positive, the other is negative
• B. exactly one of the roots of $f\left(x\right)=0$ lies in $(-1,1)$
• C. $1$ lies between the roots of $f\left(x\right)=0$
• D. both the roots of $f\left(x\right)=0$ are less than $1$

Answer 1

The correct option is B exactly one of the roots of $f\left(x\right)=0$ lies in $(-1,1)$

Given that $|b|>|a+c|$
Therefore,

$b^2-(a+c{)}^2>0$

$a+c+b>0$ & $a-b+c<0$

Now, $f\left(1\right)f(-1)=(a+b+c)(a-b+c)<0$
Therefore, exactly one roots of $f\left(x\right)=0$ lies in $(-1,1)$.
Ans: B