Let $f (x) = {ax}^{2} + bx + c = 0$ . Then which of the following statements is (are) correct?

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$| b | > | a + c | \Rightarrow$ exactly one of the roots $\in$ (-1,1)

If a(a+b+c)<0<c(a+b+c), then one roots belongs to $(- \infty, 0)$ and other $\in (1, \infty)$

If c(a+b+c)<0<a(a+b+c), then one root $\in (- \infty, 0)$ and other $\in (0, 1)$

$| b | > | a + c | \Rightarrow$ both roots € (--, - 1) or both roots $\in (1, \infty)$

answer is A, B, C.

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Detailed Solution

$\begin{aligned} (a) | a + c | < | b | \Rightarrow (a + c)^{2} - b^{2} < 0 \\ \Rightarrow (a + c - b) (a + b + c) < 0 \end{aligned}$
of f(-1) and f(1) one ls positive and one is negative.

Exactly one root $\in (- 1, 1)$ (a) is correct
$\begin{array}{l} (b) a (a + b + c) < 0 \Rightarrow af (1) < 0 \\ c (a + b + c) > 0 \Rightarrow f (0) f (1) > 0 \Rightarrow (af (0)) (af (1)) > 0 \\ af (1) < 0 \Rightarrow af (0) < 0 \end{array}$

$One root \in (- \infty, 0) and other \in (1, \infty)$
(b) is correct
$\begin{aligned} (c) c (a + b + c) < 0 \Rightarrow f (0) f (1) < 0 \\ a (a + b + c) > 0 \Rightarrow af (1) > 0 \\ af (1) > 0 and af (0) < 0 \end{aligned}$