Explanation

(b)
Since, $\alpha\ ,\ \beta$ are the roots of equation $x^{2} + bx + c = 0$.
Here, $D = b^{2} - 4c > 0\ because\ c < 0 < b$. So, roots are real and unequal.
Now, $\alpha + \beta = - b < 0$ and $\alpha\ \beta = c < 0$
$\therefore$ One root is positive and the other is negative, the negative root being numerically bigger. As $\alpha < \ \beta,\ \alpha$ is the negative root while $\beta$ is the positive root. So, $|\ \alpha| > \beta$ and $\alpha < 0 < \beta$.