Question

If $\alpha$ and $\beta (\alpha <\beta )$ are the roots of the equation $x^2+bx+c=0$, where $c<0<b$, then

• A. $\alpha <0<|\alpha |<\beta$
• B. $\alpha <\beta <0$
• C. $\alpha <0<\beta <|\alpha |$
• D. $0<\alpha <\beta$

Answer 1

The correct option is C

$\alpha <0<\beta <|\alpha |$

Given: $c<0<b$
Since, $\alpha +\beta =-b....\left(i\right)$
and $\alpha \beta =c....\left(ii\right)$
From Eq. $\left(ii\right),c<0\Rightarrow \alpha \beta <0$
$\Rightarrow$ Either $\alpha$ is -ve, $\beta$ is +ve or $\alpha$ is +ve, $\beta$ is -ve.
From Eq. $\left(i\right),b>0$ $\Rightarrow -b<0\Rightarrow -b<0\Rightarrow \alpha +\beta <0$ $\Rightarrow$ the sum is negative.
$\Rightarrow$ Modulus of negative quantity is $>$ modulus of positive quantity but $\alpha <\beta$ is given.
Therefore, it is a clear that $\alpha$ is negative and $\beta$ is positive and modulus of $\alpha$ is greater than modulus of $\beta$.
$\Rightarrow \alpha <0<\beta <|\alpha |$

$\text{NOTE:}$
This question is not on the theory of interval in which root lie, which appears looking at first sight. It is new type and first time asked in the paper. It is important for future. The actual type is interval in which parameter lie.