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)`

Options

A. (a) `(-1/2,-1/(sqrt(5)))`
B. (b) `(-1/(sqrt(5)),0)`
C. (c) `(0,1/(sqrt(5)))`
D. (d) `(1/(sqrt(5)),1/2)`

Answer

C

Explanation

`:' alpha x^(2)-x+alpha=0` has distinct real roots.
`:.Dgt0`
`implies1-4alpha^(2)gt0impliesalpha epsilon(-1/2,1/2)`…….i
Also `|x_(1)-x_(2)|lt1implies|x_(1)-x_(2)|^(2)lt1`
`impliesD/(a^(2))ltimplies(1-4alpha^(2))/(alpha^(2))lt1impliesalpha^(2)gt1/5`
`impliesalpha epsilon (-oo,-1/(sqrt(5)))uu(1/(sqrt(5)),oo)`........ii
From Eqs i and ii we get
`S=(-1/2,-1/(sqrt(5)))uu(1/(sqrt(5)),1/2)`