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Let S be the set of all non-zero numbers...

Let S be the set of all non-zero numbers `alpha`such that the quadratic equation `alphax^2-x+alpha=0`has two distinct real roots `x_1, and x_2` satisfying the inequality `|x_1-x_2|lt1` which of the following intervals is(are) a subset of S?

A

`(-(1)/(2), - (1)/(sqrt(5)))`

B

`(-(1)/(sqrt(5)),0)`

C

`( 0, (1)/(sqrt(5)))`

D

`((1)/(sqrt(5)), (1)/(2))`

Text Solution

Verified by Experts

The correct Answer is:
1,4
`|x_(1) - x_(2)| lt 1`
` therefore (x_(1) + x_(2)) ^(2) - 4 x_(1) x_(2) lt 1`

`rArr (1)/(alpha^(2)) - 4 lt 1 `
`rArr 5 - (1) /(alpha^(2)) gt 0 `
` rArr (5alpha ^(2) - 1)/(alpha ^(2)) gt 0 `
`rArr alpha in (-infty, - (1)/(sqrt(5)) )cup ((1)/(sqrt(5)) , infty)` (1)
Also ` D gt 0 `
` therefore 1 - 4 alpha^(2) gt 0 ` (2)
From (1) and (2)
` alpha in (- (1)/(2) - (1)/(sqrt(5)) )cup ((1)/(sqrt(5)),(1)/(sqrt(5)) )` .
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