Home
Class 12
MATHS
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)/(sqrt5))`

B

`(-(1)/(sqrt5),0)`

C

`(0,(1)/(sqrt5))`

D

`((1)/(sqrt5),(1)/(2))`

Text Solution

Verified by Experts

The correct Answer is:
A, D
Given,` x_(1)and x_(2)` are roots of `ax^(2)-x+alpha=0.`
`thereforex_(1)+x_(2)=(1)/(alpha)and x_(1)x_(2)=1`
`Aslo, |x_(1)-x_(2)|lt1`
`implies|x_(1)-x_(2)|^(2)lt1implies(x_(1)-x_(2))^(2)lt1`
`or (x_(1)+x_(2))^(2)-4x_(1)x_(2)lt1`
`implies(1)/(alpha^(2))-4 lt 1 or (1)/(alpha^(2))lt5`
`implies5alpha^(2)-1gt0or (sqrt5alpha-1)(sqrt5alpha+1)gt0`

`thereforealpha in (-ooo,-(1)/(sqrt5))uu((1)/(sqrt5),oo)" "...(i)`
Also, `implies1-4alpha^(2)gt0or alpha in(-(1)/(2),(1)/(2))" "...(ii)`
From Eqs. (i) and ?(ii), we get
`alpha in (-(1)/(2),(-1)/(sqrt(5)))uu((1)/(sqrt5),(1)/(2))`
Promotional Banner

Similar Questions

Explore conceptually related problems

Let S be the set of all non-zero real numbers such that the quadratic equation alphax^2-x+alpha=0 has two distinct real roots x_1a n dx_2 satisfying the inequality |x_1-x_2|<1. Which of the following intervals is (are) a subset (s) of S ? (1/2,1/(sqrt(5))) b. (1/(sqrt(5)),0) c. (0,1/(sqrt(5))) d. (1/(sqrt(5)),1/2)

The real roots of the quadratic equation x^(2)-x-1=0 are ___.

Complete set of values of x satisfying inequality ||x-1|-5| lt 2x -5 is

Find a quadratic equation whose product of roots x_1a n dx_2 is equal to 4 an satisfying the relation (x_1)/(x_1-1)+(x_2)/(x_2-1) =2.

The equation alpha x^(3)-2(alpha+1) x^(2)+4 alpha x=0 has real roots and alpha is any positive integer, then the sum of the roots of the equation is

If x satisfies the inequations 2x - 7 lt 11 , 3x + 4 lt -5 , then x lies in the interval

If both the roots of the quadratic equation x^(2) -mx+4=0 are real and distinct and they lie in the interval [1,5] then m lies in the interval

Let a ,\ b ,\ c be three non-zero real numbers such that the equation sqrt(3)\ acosx+2\ bsinx=c ,\ x in [-pi/2,pi/2] , has two distinct real roots alpha and beta with alpha+beta=pi/3 . Then, the value of b/a is _______.

Find all values of x that satisfies the inequality (2x-3)/((x-2)(x-4)) lt 0

Find all values of x that satisfies the inequality (2x-3)/((x-2)(x-4))lt0