1.

The equation `asinx+Cos2x=2a-7` possesses a solution if

Answer» Given equation is,
`asinx + cos2x = 2a-7`
`=>asinx+1-2sin^2x =2a - 7`
`=>2sin^2x-asinx+2a-8 = 0`
`=> sinx = (-(-a)+-sqrtD)/(2(2))`
Here, `D = (-a)^2-4(2)(2a-8)`
`=> D = a^2-16a+64 = (a-8)^2`
`:. sinx = (a+-(a-8))/4`
As `sin x` lies between `-1` and `1`.
`:. sin x = (a+(a-8))/4`
`=>sinx = (a-4)/2`
`:. -1 le (a-4)/2 le 1`.
`=>-2 le a-4 le 2`
` => 2 le a le 6`
So, for `2 le a le 6`, given eqaution will possess a solution.


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