Set of all real value of a such that `f(x)=((2a-1)+x^2+2(a+1)x+(2a-1))/(x^2-2x+40)` is always negative is `-oo,0` b. `0,oo` c. `-oo,1//2` d. none

Options

A. `(-oo, 0)`
B. `(0, oo)`
C. `(-oo, 1//2)`
D. None

Answer

1

Explanation

`f(x) = ((2a - 1)x^(2) + 2(a + 1)x + (2a - 1))/(x^(2) - 2x + 40)`
Since `x^(2) - 2x + 40 gt 0` for all real x.,
`(2a - 1)x^(2) + 2(a + 1)x + (2a - 1) lt 0 AA x in R`
`therefore 2a - 1 lt 0` or `a lt 1/2" "`(1)
and `D lt 0`
`rArr 4(a + 1)^(2) - 4(2a - 1)^(2) lt 0`
`rArr a(a - 2) gt 0`
`rArr a lt 0` or `a gt 2" "`(2)
From (1) and (2), `a lt 0`