Question

'$af\left(k\right)<0$' is the necessary and sufficient condition for a particular real number $k$ to lie between the roots of a quadratic equation $f\left(x\right)=0$, where $f\left(x\right)=a{x}^2+bx+c$. If $f\left(k_1\right)f\left(k_2\right)<0$, then exactly one of the roots will lie between $k_1$ and $k_2$.

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

• A. one root of $f\left(x\right)=0$ is positive, the other root 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 $(-1,1)$

$|b|>|a+c|$
$\Rightarrow b^2>(a+c{)}^2$
$\Rightarrow (a+c{)}^2-b^2<0$
$\Rightarrow (a+c-b)(a+c+b)<0$
$\Rightarrow f(-1)f\left(1\right)<0$
So, there is exactly one root in $(-1,1)$