`sinx+cosx=y^2-y+a` has no value of `x` for any value of `y` if a belongs to `(0,""sqrt(3))` (b) `(-""sqrt(3),0)` `(-oo,-""sqrt(3))` (d) `(""sqrt(3),oo)`

Options

A. `(0, sqrt(3))`
B. `(-sqrt(3), 0)`
C. `(-oo, -sqrt(3))`
D. `(sqrt(3), oo)`

Answer

D

Explanation

`y^(2)-y+a=(y-1/2)^(2)+a-1/4`
Since `-sqrt(2) le sin x + cos x le sqrt(2)`, the given equation will have no real value x for any if `a - 1.4 gt sqrt(2)`.
i.e., `a in (sqrt(2)+1/4, oo)`
`rArr a in (sqrt(3), oo) ("as "sqrt(2) +1/4 lt sqrt(3))`