Browse Questions

# For which of the sets of a, is the statement, “The difference between the roots of the equation $x^2+ax+1 =$ is less than $\sqrt 5”$ true?

$\begin {array} {1 1} (A)\;(-3,\infty) & \quad (B)\;(3,\infty) \\ (C)\;(-\infty, -3) & \quad (D)\;(-3, 3) \end {array}$

Ans : (D)
Let $\alpha,\: \beta$ be the roots of the eqn. $x^2+ax+1=0$, then
$\alpha+\beta = -a\: and \: \alpha\beta = 1$
Now $| \alpha – \beta | = \sqrt{[( \alpha + \beta)2 - 4 \alpha\beta]}$
$\Rightarrow | \alpha – \beta | = \sqrt{a^2} – 4$
According to the question,
$\sqrt{a^2} – 4 < \sqrt5$
$\Rightarrow a ^2 – 4 < 5$
$\Rightarrow a^2 – 9 < 0$
$\Rightarrow a \in (-3, 3)$