Let S be the set of all non-zero real numbers such that the quadratic equation x 2 - x + = 0 has two distinct real roots x 1 and x 2 satisfying the inequality | x 1 - x 2 | 1 . Which of the following intervals is(are) a subset(s) of S ?

Question

Let $S$ be the set of all non-zero real numbers $α$ such that the quadratic equation $α x^{2} - x + α = 0$ has two distinct real roots $x_{1}$ and $x_{2}$ satisfying the inequality $| x_{1} - x_{2} |$ $< 1$ . Which of the following intervals is(are) $a$ subset(s) of $S$ ? [2015]

Smart Achievers · Accepted Answer

(1, 4)

$Given, x_{1} and x_{2} are roots of α x^{2} - x + α = 0$

$∴$ $x_{1} + x_{2} = \frac{1}{α} and x_{1} x_{2} = 1$

$Also,$ $| x_{1} - x_{2} |$ $< 1$

$\Rightarrow | x_{1} - x_{2} |^{2} < 1 \Rightarrow {(x_{1} - x_{2})}^{2} < 1$

or ${(x_{1} + x_{2})}^{2} - 4 x_{1} x_{2} < 1$

$\Rightarrow \frac{1}{α^{2}} - 4 < 1$ or $\frac{1}{α^{2}} < 5$

or $5 α^{2} - 1 > 0$ or $(\sqrt{5} α - 1) (\sqrt{5} α + 1) > 0$

$∴$ $α \in (- \infty, - \frac{1}{\sqrt{5}}) \cup (\frac{1}{\sqrt{5}}, \infty) . . . (i)$

$Also, D > 0$

$\Rightarrow 1 - 4 α^{2} > 0$ or $α \in (- \frac{1}{2}, \frac{1}{2}) . . . (i i)$

$From (i) and (ii), we get$

$α \in (- \frac{1}{2}, - \frac{1}{\sqrt{5}}) \cup (\frac{1}{\sqrt{5}}, \frac{1}{2})$

Solution

Q.
Let $S$ be the set of all non-zero real numbers $α$ such that the quadratic equation $α x^{2} - x + α = 0$ has two distinct real roots $x_{1}$ and $x_{2}$ satisfying the inequality $| x_{1} - x_{2} |$ $< 1$ . Which of the following intervals is(are) $a$ subset(s) of $S$ [2015]

1 $(- \frac{1}{2}, - \frac{1}{\sqrt{5}})$

2 $(- \frac{1}{\sqrt{5}}, 0)$

3 $(0, \frac{1}{\sqrt{5}})$

4 $(\frac{1}{\sqrt{5}}, \frac{1}{2})$

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Solution

Q. Let S be the set of all non-zero real numbers α such that the quadratic equation αx2-x+α=0 has two distinct real roots x1 and x2 satisfying the inequality |x1-x2|<1. Which of the following intervals is(are) a subset(s) of S [2015]

1 (-12,-15)

2 (-15,0)

3 (0,15)