SATHEE: Theory of Equations 5 Question 9

Theory of Equations 5 Question 9

9. 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 interval(s) is/are a subset of $S$ ?

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

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

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

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Answer:

Correct Answer: 9. (a,d)

Solution:

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 $\frac{1}{α^{2}} < 5$

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

$∴ α \in - \infty, - \frac{1}{\sqrt{5}} \cup \frac{1}{\sqrt{5}}, \infty$

Also, $D > 0$

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

From Eqs. (i) and (ii), we get

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

Theory of Equations 5 Question 9

9. 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 interval(s) is/are a subset of $S$ ?

Answer:

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Theory of Equations 5 Question 9

9. 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 interval(s) is/are a subset of S ?

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

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9. 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 interval(s) is/are a subset of $S$ ?