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sinx+cosx=y^2-y+a has no value of x for ...

`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)`

A

`(0, sqrt(3))`

B

`(-sqrt(3), 0)`

C

`(-oo, -sqrt(3))`

D

`(sqrt(3), oo)`

Text Solution

Verified by Experts

The correct Answer is:
D
`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))`
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