Question

Find the complete set of values of a such that `(x^2-x)//(1-a x)` attains all real values.

Answer 1

Correct Answer - ` ain [1, infty)` ,
`y= (x^(2) - x)/ (1 - ax)`
or `x^(2) - x = y - axy`
or ` x^(2) + x (ay - 1) - y = 0`
Since x is real, so
`(ay - 1)^(2) + 4y ge 0`
`rArr a^(2) y^(2) + 2y (2-a) + 1 ge 0, AA y in ` R
`rArr a^(2) gt 0, 4(2 - a)^(2) - 4a^(2) le 0 rArr 4 - 4a le 0 rArr a in [ 1 , infty)` .